http://2012.igem.org/wiki/index.php?title=Special:Contributions/D.olivera1320&feed=atom&limit=50&target=D.olivera1320&year=&month=2012.igem.org - User contributions [en]2024-03-29T01:08:02ZFrom 2012.igem.orgMediaWiki 1.16.0http://2012.igem.org/Team:Colombia/Modeling/ScriptingTeam:Colombia/Modeling/Scripting2012-10-27T03:28:50Z<p>D.olivera1320: /* OPTIMIZATION */</p>
<hr />
<div>{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Scripting ==<br />
<br />
<br />
== RALSTONIA DIFFERENTIAL EQUATION SOLUTION ==<br />
<br />
<br />
These scripts create the differential equations, find the steady state concetrations and then solve them by a 4th order Runge Kutta:<br />
----<br />
<br />
'''DIFFERANTIAL EQUATIONS DECLARATIONS'''<br />
<br />
%THIS CODE CREATE ALL THE DIFFERENTIAL EQUATIONS FOR RALSTONIA'S SYSTEM<br />
function y=ecuaDifR(t,v) <br />
%---------Parameters------%<br />
global alfS %Basal concentration of the sensor phcS<br />
global alfRA %Basal concnetration of the comple pchA-pchR <br />
global alfR %Basal concentration of LuxR<br />
global alfI %Basal concentration of CI <br />
global alfCI %Basal concentration of CI <br />
global alfHA %Basal concnetration of HipA7<br />
global alfHB %Basal concnetration of HipB <br />
global alfAS %Basal concnetration of Salycilic acid <br />
global gammaS %Degradation of the sensor pchS<br />
global gammaRA %Degradation of the complex pchR-pchA<br />
global gammaR %Degradation of LuxR<br />
global gammaI %Degradation of LuxI <br />
global gammaCI %Degradation of CI <br />
global gammaHA %Degradation of HipA7<br />
global gammaHB %Degradation of HipB <br />
global gammaAS %Dergradation of Salycilic acid <br />
global mOHS %Kinetic constant for the detection of 3-OH-PAME by the sensor pchS (phosphorylation)<br />
global mSFR %Kinetic constant for the phosphorylation of the complex pchR-pchA by the sensor<br />
global mA %Kinetic constant for the activation of the promoter by the pchA<br />
global mIR %Kinetic constant for the formation of the complex LuxILuxR<br />
global mI %Constant that represent the union of the complex LuxILuxR with the promoter<br />
global mHAHB %Kinetic constant for the inhibition of HipA7 <br />
global betaI %Max production of LuxI<br />
global betaCI %Max production of CI <br />
global betaHB %Max peoduction of HipB<br />
global betaHA %Max production of HipA7 <br />
global betaAS %Max production of Salicylic acid<br />
global kA %Constant k of the hill ecuation for the promoter promoted by pchA<br />
global kIR %Constant k of the hill equiation for the promorer prmoted by the complex luxIluxR<br />
global kCI %Constant k of the hill equation for the promoter promoted by CI <br />
global hA %Hill constant for the promoters promoted by pchA <br />
global hIR %Hill constant for the promoter promoted by the complex IR<br />
global hCI %Hill constant fot the promoter CI <br />
global eI %Export factor of LuxI<br />
global jI %Import factor of LuxI<br />
global deltaI %Difusion of LuxI outside the cell <br />
global eAS %Export of Salicylic acid <br />
global numcel %number of cells <br />
<br />
if (t<(10) || ((t)>20))<br />
<br />
OH=0;<br />
<br />
else<br />
<br />
OH=15;<br />
<br />
<br />
end <br />
<br />
%------ Variables%------<br />
<br />
S=v(1); %Cocentration of the sensor pchS the cell<br />
SF=v(2); %Concentration of phosphorylated sensor the cell<br />
RA=v(3); %Concentration of the comple pchR-pchA<br />
A=v(4); %Concentratio of the promoter avtivator pchA<br />
Ii=v(5); %Concentration of LuxI inside the cell<br />
Io=v(6); %Concentration of LuI outsied the cell<br />
IR=v(7); %Concentration of the complex LuxI-LuxR <br />
R=v(8);%Concentration of the protein CI <br />
CI=v(9);%Concentration of HipA7<br />
HB=v(10);%Concnetratio of HipB<br />
HA=v(11);%Concentration of salicylic acid<br />
AS=v(12); %Concentratio of quitin monomers<br />
<br />
%---Equations---%<br />
<br />
<br />
dS=alfS- gammaS*S - mOHS*OH*S; %Change of the sensor pchS<br />
dSF = mOHS *OH*S - mSFR *SF*RA ; %Change of phosphorylated sensor<br />
dRA=alfRA - gammaRA*RA - mSFR*SF*RA;%Change of the comple pchR-pchA<br />
dA= mSFR*SF*RA-mA*A; %Change of the activator pchA inside the cell<br />
dIi= alfI+ (betaI*(A^hA))/(kA^hA+(A^hA)) -gammaI*Ii +jI*Io- eI*Ii- mIR*Ii*R; %Change of LuxI inside <br> the cell<br />
dIo= numcel*(eI*Ii-jI*Io)-deltaI*Io; %Change of LuxI outside the cell<br />
dIR= mIR*Ii*R - mI*IR; %Change of the complex LuxI luxR<br />
dR= alfR-gammaR*R -mIR*Ii*R +(betaI*(A^hA))/(kA^hA+(A^hA)); %Change of LuxR<br />
dCI= alfCI -gammaCI*CI+ (betaCI*(CI^hCI))/(kCI^hCI+(CI^hCI)) +(betaCI*(IR^hIR))/(kIR^hIR+(IR^hIR));<br>%Change of CI<br />
dHB=alfHB-gammaHB*HB+(betaHB*(CI^hCI))/(kCI^hCI+(CI^hCI))+(betaHB*(IR^hIR))/(kIR^hIR+(IR^hIR))<br>-mHAHB*HA^2*HB^2; %Chanche of HipB <br />
dHA=alfHA-gammaHA*HA+ (betaHA*(CI^hCI))/(kCI^hCI+(CI^hCI))-mHAHB*HA^2*HB^2; %Change of HipA7<br />
dAS=alfAS-gammaAS*AS +(betaAS*(CI^hCI))/(kCI^hCI+(CI^hCI))-eAS*AS+(betaAS*(IR^hIR))/(kIR^hIR+(IR^hIR));<br>%Change of Salicylic acid <br />
y1(1)=dS;<br />
y1(2)=dSF;<br />
y1(3)=dRA;<br />
y1(4)=dA;<br />
y1(5)=dIi;<br />
y1(6)=dIo;<br />
y1(7)=dIR;<br />
y1(8)=dR;<br />
y1(9)=dCI;<br />
y1(10)=dHB;<br />
y1(11)=dHA;<br />
y1(12)=dAS; <br />
y=y1'; <br />
<br />
end<br />
<br />
<br />
<br />
----<br />
<br />
'''RUNGE KUTTA'''<br />
<br />
%FILE THAT SOLVES THE DIFFERENTIAL EQUATION AND GRAPHS THEM <br />
alfS=0.9; %Basal concentration of the sensor pchS<br />
alfRA=0.9; %Basal concnetration of the complez pchA-pchR<br />
alfR=0.6; %Basal concentration of LuxR<br />
alfI=0.4; %Basal concentration of LuxI<br />
alfCI=0.5; %Basal concentration of CI<br />
alfHA=1; %Basal concnetration of HipA7<br />
alfHB=0.4; %Basal concnetration of HipB<br />
alfAS=0.4; %Basal concnetration of Salycilic acid<br />
gammaS=1; %Degradation of Chitinase inside the cell<br />
gammaRA=1; %Degradation of chitoporin<br />
gammaR=1; %Degradation of LuxR<br />
gammaI=1; %Degradation of LuxI<br />
gammaCI=1; %Degradation of CI<br />
gammaHA=1; %Degradation of HipA7<br />
gammaHB=4; %Degradation of HipB<br />
gammaAS=1; %Dergradation of Salycilic acid <br />
mOHS=4; %Kinetic constant for the formation of the phoshorilation of the sensor<br />
mSFR=2.6; %Kinetic constant of the reaction of the phosphorilarion of the comple R<br />
mA=3.5; %Kinetic constant for the activation by A <br />
mIR=3; %Kinetic constant for the formation of the complex LuxILuxR<br />
mI=3; %Constant that represent the union of the complex LuxILuxR with the promoter<br />
mHAHB=12; %Kinetic constant for the inhibition of HipA7<br />
betaI=10; %Max production of LuxI<br />
betaCI=9.96; %Max production of CI<br />
betaHB=9.95; %Max production of HipB<br />
betaHA=10; %Max production of HipA7<br />
betaAS=11.2; %Max production of Salicylic acid<br />
kA=0.1; %Constant k of the hill ecuation for the promoter promoted by S<br />
kIR=0.39; %Constant k of the hill equiation for the promorer prmoted by the complex luxIluxR<br />
kCI=0.055; %Cosntant k of the hill equation for the promoter promoted by CI<br />
hA=1.2; %Hill constant for the promoters promoted by S<br />
hIR=3.4; %Hill constant for the promoter promoted by the complex IR<br />
hCI=2.3; %Hill constant fot the promoter CI<br />
eI=0.5; %Export factor of LuxI<br />
jI=0.8; %Import factor of LuxI<br />
deltaI=0.2;%Difusion of LuxI outside the cell <br />
eAS=0.8;<br />
numcel=1; %number of cells<br />
%----%<br />
<br />
h=100; %Tiempo maximo<br />
<br />
m=0.01; %Longitud de paso [s] <br />
<br />
t=0:m:h; %Vector tiempo<br />
<br />
xi=[1,1,1,1,1,1,1,1,1,1,1,1];<br />
<br />
y=fsolve(@CondInR,xi);<br />
<br />
conInd=y;<br />
<br />
l=(0:m:h)'; %Vector de tiempo<br />
<br />
x=zeros(length(l),length(conInd)); %Matriz de variables, en las columnas varia<br />
%la variable y en las filas varia el tiempo<br />
<br />
OH=zeros(1,length(l));<br />
<br />
x(1,:)=conInd;<br />
<br />
for k=1:length(l)-1<br />
<br />
xk=x(k,:); %Captura de la ultima posicion de la matirz, es decir, los<br />
%valores actuales de las variables<br />
<br />
k1=ecuaDifR(l(k),xk); %Primera pendiente del metodo de RK4<br />
k2=ecuaDifR(l(k)+m/2,xk+(m/2*k1)'); %Segunda pendiente del metodo de RK4<br />
k3=ecuaDifR(l(k)+m/2,xk+(m/2*k2)'); %Tercera pendiente del metodo de RK4<br />
k4=ecuaDifR(l(k)+m,xk+(m*k3)'); %Cuarta pendiente del metodo de RK4<br />
<br />
xk1=xk+m/6*(k1+2*k2+2*k3+k4)'; %Calculo de nuevos valores para las<br />
%variables<br />
<br />
%xk1=xk+m*ecuaDif(l(k),xk)'; %Method of Newton<br />
<br />
xk2=zeros(1,length(xk1));<br />
<br />
<br />
for p=1:length(xk1)<br />
<br />
if(xk1(p)<0.00000001)<br />
<br />
xk2(p)=0;<br />
else<br />
<br />
xk2(p)=xk1(p);<br />
end<br />
<br />
end<br />
<br />
<br />
x(k+1,:)=xk2; %Actualizacion del nuevo vector de variables en la matriz<br />
<br />
<br />
<br />
end<br />
<br />
for j=1:length(l)<br />
<br />
if (l(j)<(10) || l(j)>(20))<br />
<br />
OH(j)=0;<br />
<br />
else<br />
<br />
OH(j)=15;<br />
<br />
<br />
end<br />
<br />
<br />
end<br />
<br />
S=x(:,1);<br />
SF=x(:,2);<br />
RA=x(:,3);<br />
A=x(:,4);<br />
Ii=x(:,5);<br />
Io=x(:,6);<br />
IR=x(:,7);<br />
R=x(:,8);<br />
CI=x(:,9);<br />
HB=x(:,10);<br />
HA=x(:,11);<br />
AS=x(:,12); <br />
<br />
figure(1) <br />
plot(l,S,l,SF)<br />
legend('Sensor (pchS)',' Phosporilated pchS')<br />
xlabel('Time')<br />
ylabel('Concetratio (micromolar)')<br />
title('Response of Sensor pchS')<br />
figure(5)<br />
plot(l,RA,l,A)<br />
legend('Complex pchR-pchA','Activator pchA')<br />
xlabel('Time')<br />
ylabel('Concetration (micromolar)')<br />
title('Activator response')<br />
figure(2)<br />
plot (l,R,l,Ii,l,Io,l,IR)<br />
legend('LuxR','LuxI nside the cell','LuxI outside the cell','Complex(Lux-LuxR)')<br />
xlabel('Time')<br />
ylabel('Concetration (micromolar)')<br />
title('LuxI-LuxR system response')<br />
figure(3) <br />
plot (l,HA,l,HB) <br />
legend('Toxin HipA7','Antitoxin HipB')<br />
xlabel('Time')<br />
ylabel('Concetration (micromolar)')<br />
title('Toxin-Antitoxin module')<br />
figure(4) <br />
plot (l,OH,l,AS,l,CI) <br />
legend('3-OH-PAME','Salicylic acid','CI')<br />
xlabel('Time')<br />
ylabel('Concetration (micromolar)')<br />
title('CI and Salicylic Acid response')<br />
<br />
== RUST DIFFERENTIAL EQUATION SOLUTION ==<br />
<br />
'''Chitin impulse function''': This function returns the chitin concentration depending on the time imput <br />
<br />
function answer = functionChi( t )<br />
%<br />
answer=0;<br />
%<br />
if t>10 && t<20<br />
% <br />
answer=15;<br />
% <br />
end<br />
%<br />
<br />
end<br />
<br />
<br />
----<br />
<br />
'''Differential equation declaration:'''<br />
<br />
%THIS CODE CREATE ALL THE DIFFERENTIAL EQUATIONS FOR THE SYSTEM<br />
%<br />
function y=ecuaDif(t,v) <br />
%---------Parameters------%<br />
global alfA %Basal concentration of Chitinase inside the cell (micromolar)<br />
global alfP %Basal concnetration of chitoporin <br />
global alfC %Basal concentration of the CBP<br />
global alfR %Basal concentration of LuxR<br />
global alfI %Basal concentration of CI <br />
global alfCI %Basal concentration of CI <br />
global alfHA %Basal concnetration of HipA7<br />
global alfHB %Basal concnetration of HipB <br />
global alfAS %Basal concnetration of Salycilic acid <br />
global alfCS<br />
global gammaA %Degradation of Chitinase inside the cell<br />
global gammaP %Degradation of chitoporin <br />
global gammaC %Degradation concentration of the CBP<br />
global gammaR %Degradation of LuxR<br />
global gammaI %Degradation of LuxI <br />
global gammaCI %Degradation of CI <br />
global gammaHA %Degradation of HipA7<br />
global gammaHB %Degradation of HipB <br />
global gammaAS %Dergradation of Salycilic acid<br />
global gammaCS %Degradation of the complex CS <br />
global mCS %Kinetic constant for the formation of the complex CS <br />
global mCSQ %Kinetic constant of the reaction of the complex CS with the chitin <br />
global mAQQ %Kinetic constant for the reaction of the chitinase and th chitin<br />
global mIR %Kinetic constant for the formation of the complex LuxILuxR<br />
global mI %Constant that represent the union of the complex LuxILuxR with the promoter<br />
global mHAHB %Kinetic constant for the inhibition of HipA7 <br />
global betaP %Max production of the chitoporin <br />
global betaA %Max production of chitinase<br />
global betaI %Max production of LuxI<br />
global betaCI %Max production of CI <br />
global betaHB %Max peoduction of HipB<br />
global betaHA %Max production of HipA7 <br />
global betaAS %Max production of Salicylic acid <br />
global kS %Constant k of the hill ecuation for the promoter promoted by S<br />
global kIR %Constant k of the hill equiation for the promorer prmoted by the complex luxIluxR<br />
global kCI %Constant k of the hill equation for the promoter promoted by CI <br />
global hS %Hill constant for the promoters promoted by S <br />
global hIR %Hill constant for the promoter promoted by the complex IR<br />
global hCI %Hill constant fot the promoter CI <br />
global eA %Export factor of the chitinase <br />
global jQ %Import factor of the chitin monomers <br />
global deltaA %Difusion factor of the chinitanse outside the cell<br />
global eI %Export factor of LuxI<br />
global jI %Import factor of LuxI<br />
global deltaI %Difusion of LuxI outside the cell <br />
global Stotal %Total concentration of the sensor in the cell <br />
global eAS %Export of Salicylic acid <br />
global numcel %number of cells <br />
% <br />
% <br />
QQ=functionChi(t);<br />
%<br />
%------ Variables%------<br />
%<br />
% <br />
C=v(1); %Cocentration of chitinase outside the cell<br />
CS=v(2); %Concentration of chitinase inside the cell<br />
P=v(3); %Concentration of chitiporin<br />
Ai=v(4); %Concentratio of chitin binding protein (CBP)<br />
Ao=v(5); %Concentration the complex CBP-s<br />
Q=v(6); %Concentration of LuxR<br />
Ii=v(7); %Concentration of LuxI inside the cell<br />
Io=v(8); %Concentration of LuI outsied the cell<br />
IR=v(9); %Concentration of the complex LuxI-LuxR <br />
R=v(10);%Concentration of the protein CI <br />
CI=v(11);%Concentration of HipA7<br />
HB=v(12);%Concnetratio of HipB<br />
HA=v(13);%Concentration of salicylic acid<br />
AS=v(14); %Concentratio of quitin monomers<br />
%<br />
% <br />
% %---Equations---%<br />
<br />
$ <br />
% <br />
S=Stotal-CS;<br />
% <br />
% <br />
%<br />
dC=alfC- gammaC*C - mCS*C*S; %Change of CBP <br />
dCS=alfCS+ mCS*C*S- mCSQ*CS*Q-gammaCS*CS; %Change of the complex CS<br />
dP=alfP - gammaP*P + (betaP*(S^hS))/(kS^hS+(S^hS));%Change of chitoporin<br />
dAi=(alfA- gammaA*Ai+ (betaA*(S^hS))/(kS^hS+(S^hS)))- eA*Ai; %Change of chitinase inside the cell <br />
dAo= eA*Ai-deltaA*Ao- mAQQ*Ao*QQ; %Change of chitinase outside the cell<br />
dQ= 2*jQ*P*(mAQQ*QQ*Ao)-mCSQ*CS*Q; %Change of chitin monomer inside the cell<br />
dIi= alfI+ (betaI*(S^hS))/(kS^hS+(S^hS)) -gammaI*Ii +jI*Io- eI*Ii- mIR*Ii*R; %Change of LuxI inside <br>the cell<br />
dIo= numcel*(eI*Ii-jI*Io)-deltaI*Io; %Change of LuxI outside the cell<br />
dIR= mIR*Ii*R - mI*IR; %Change of the complex LuxI luxR<br />
dR= alfR-gammaR*R -mIR*Ii*R +(betaI*(S^hS))/(kS^hS+(S^hS)); %Change of LuxR<br />
dCI= alfCI -gammaCI*CI+ (betaCI*(CI^hCI))/(kCI^hCI+(CI^hCI)) +(betaCI*(IR^hIR))/(kIR^hIR+(IR^hIR));<br>%Change of CI<br />
dHB=alfHB-gammaHB*HB+(betaHB*(CI^hCI))/(kCI^hCI+(CI^hCI))+(betaHB*(IR^hIR))/(kIR^hIR+(IR^hIR))<br>-mHAHB*HA^2*HB^2; %Chanche of HipB <br />
dHA=alfHA-gammaHA*HA-mHAHB*HA^2*HB^2+(betaHA*(CI^hCI))/(kCI^hCI+(CI^hCI)); %Change of HipA7<br />
dAS=alfAS-gammaAS*AS +(betaCI*(CI^hCI))/(kCI^hCI+(CI^hCI))+(betaAS*(IR^hIR))/(kIR^hIR+(IR^hIR))-eAS*AS;<br>%Change of Salicylic acid<br />
%<br />
y1(1)=dC;<br />
y1(2)=dCS;<br />
y1(3)=dP;<br />
y1(4)=dAi;<br />
y1(5)=dAo;<br />
y1(6)=dQ;<br />
y1(7)=dIi;<br />
y1(8)=dIo;<br />
y1(9)=dIR;<br />
y1(10)=dR;<br />
y1(11)=dCI;<br />
y1(12)=dHB;<br />
y1(13)=dHA;<br />
y1(14)=dAS;<br />
%<br />
y=y1'; <br />
% <br />
% <br />
end<br />
<br />
<br />
----<br />
'''Diferential equation solution'''<br />
<br />
tic;<br />
%<br />
%File that solves the differential equations and graphs them<br />
%<br />
alfA = 0.9; %Basal concentration of Chitinase inside the cell (micromolar)<br />
alfP = 0.9; %Basal concnetration of chitoporin<br />
alfC = 0.9; %Basal concentration of the CBP<br />
alfR = 0.6; %Basal concentration of LuxR<br />
alfI = 0.4; %Basal concentration of LuxI<br />
alfCI = 0.5; %Basal concentration of CI<br />
alfHA = 1.4; %Basal concnetration of HipA7<br />
alfHB = 0.4; %Basal concnetration of HipB<br />
alfAS = 0.4; %Basal concnetration of Salycilic acid<br />
alfCS=1.4;<br />
gammaA=1; %Degradation of Chitinase inside the cell<br />
gammaP=1; %Degradation of chitoporin<br />
gammaC=1; %Degradation concentration of the CBP<br />
gammaR=1; %Degradation of LuxR<br />
gammaI=1; %Degradation of LuxI<br />
gammaCI=1; %Degradation of CI<br />
gammaHA=1; %Degradation of HipA7<br />
gammaHB=4; %Degradation of HipB<br />
gammaAS=0.8; %Dergradation of Salycilic acid<br />
gammaCS=1; %Degradation of the complex Cs <br />
mCS=13; %Kinetic constant for the formation of the complex CS<br />
mCSQ=12; %Kinetic constant of the reaction of the complex CS with the chitin<br />
mAQQ=0.2; %Kinetic constant for the reaction of the chitinase and th chitin<br />
mIR=3; %Kinetic constant for the formation of the complex LuxILuxR<br />
mI=3; %Constant that represent the union of the complex LuxILuxR with the promoter<br />
mHAHB=12; %Kinetic constant for the inhibition of HipA7 <br />
betaP=12; %Max production of the chitoporin<br />
betaA=10; %Max production of chitinase<br />
betaI=10; %Max production of LuxI<br />
betaCI=9.96; %Max production of CI<br />
betaHB=9.5; %Max production of HipB<br />
betaHA=11; %Max production of HipA7<br />
betaAS=11.2; %Max production of Salicylic acid <br />
kS=0.08; %Constant k of the hill ecuation for the promoter promoted by S<br />
kIR=0.39; %Constant k of the hill equiation for the promorer prmoted by the complex luxIluxR<br />
kCI=0.055; %Cosntant k of the hill equation for the promoter promoted by CI <br />
hS=1; %Hill constant for the promoters promoted by S<br />
hIR=3.4; %Hill constant for the promoter promoted by the complex IR<br />
hCI=2.3; %Hill constant fot the promoter CI<br />
eA=0.5; %Export factor of the chitinase<br />
jQ=0.1; %Import factor of the chitin monomers<br />
deltaA=0.2; %Difusion factor of the chinitanse outside the cell <br />
eI=0.5; %Export factor of LuxI<br />
jI=0.8; %Import factor of LuxI<br />
deltaI=0.2;%Difusion of LuxI outside the cell<br />
Stotal= 1.5; %Total concentration of the sensor in the cell<br />
eAS=0.8;<br />
numcel=1; %number of cells<br />
%<br />
h=100; %Tiempo maximo<br />
%<br />
m=0.01; %Longitud de paso [s]<br />
%<br />
t=0:m:h; %Vector tiempo<br />
%<br />
xi=[1,1,1,1,1,1,1,1,1,1,1,1,1,1];<br />
%<br />
y=fsolve(@CondIn,xi);<br />
%<br />
%<br />
conInd=y;<br />
%<br />
%<br />
%<br />
l=(0:m:h)'; %Vector de tiempo<br />
%<br />
x=zeros(length(l),length(conInd)); %Matriz de variables, en las columnas varia<br />
%la variable y en las filas varia la longitud<br />
%<br />
QQ=zeros(1,length(l));<br />
%<br />
x1=conInd;<br />
%<br />
for k=1:length(l)-1<br />
% <br />
xk=x(k,:); %Captura de la ultima posicion de la matirz, es decir, los<br />
%valores actuales de las variables<br />
<br />
k1=ecuaDif(l(k),xk); %Primera pendiente del metodo de RK4<br />
k2=ecuaDif(l(k)+m/2,xk+(m/2*k1)'); %Segunda pendiente del metodo de RK4<br />
k3=ecuaDif(l(k)+m/2,xk+(m/2*k2)'); %Tercera pendiente del metodo de RK4<br />
k4=ecuaDif(l(k)+m,xk+(m*k3)'); %Cuarta pendiente del metodo de RK4<br />
<br />
xk1=xk+m/6*(k1+2*k2+2*k3+k4)'; %Calculo de nuevos valores para las<br />
%variables<br />
<br />
%xk1=xk+m*ecuaDif(l(k),xk)'; %Method of Newton<br />
<br />
xk2=zeros(1,length(xk1));<br />
<br />
<br />
for p=1:length(xk1)<br />
<br />
if(xk1(p)<0.00000001)<br />
<br />
xk2(p)=0;<br />
else<br />
<br />
xk2(p)=xk1(p);<br />
end<br />
<br />
end<br />
<br />
<br />
x(k+1,:)=xk2; %Actualizacion del nuevo vector de variables en la matriz<br />
<br />
end<br />
% <br />
%<br />
%<br />
for j=1:length(l)<br />
% <br />
QQ(1,j)=functionChi(l(j));<br />
% <br />
end<br />
%<br />
%<br />
C=x(:,1);<br />
CS=x(:,2);<br />
P=x(:,3);<br />
Ai=x(:,4);<br />
Ao=x(:,5);<br />
Q=x(:,6);<br />
Ii=x(:,7);<br />
Io=x(:,8);<br />
IR=x(:,9);<br />
R=x(:,10);<br />
CI=x(:,11);<br />
HB=x(:,12);<br />
HA=x(:,13);<br />
AS=x(:,14); <br />
%<br />
%<br />
figure(1) <br />
plot(l,C,l,CS,l,P,l,Ai,l,Ao,l,Q,l,QQ)<br />
legend('CBP','Complex Sensor(CBP)','Chitoporin','Chitinase inside the cell','Chitinase outside the<br>cell','Chitin monomers','Chitin')<br />
xlabel('Time')<br />
ylabel('Concentration')<br />
title('Detection system substances')<br />
%<br />
figure(2)<br />
plot (l,R,l,Ii,l,Io,l,IR)<br />
legend('LuxR','LuxI Inside the cell','LuxI outside the cell','Complex LuxI-LuxR')<br />
xlabel('Time')<br />
ylabel('Concentration')<br />
title('LuxI-LuxR system substances') <br />
%<br />
figure(3) <br />
plot (l,HA,l,HB) <br />
legend('HipA7','HipB')<br />
xlabel('Time')<br />
ylabel('Concentration')<br />
title('Toxin-Antitoxin module substances')<br />
%<br />
figure(4) <br />
plot (l,QQ,l,AS,l ,CI) <br />
legend('QQ','Salicylic acid','CI')<br />
xlabel('Time')<br />
ylabel('Concentration')<br />
title('CI and Salicylic acid response')<br />
%<br />
<br />
== OPTIMIZATION ==<br />
<br />
The optimization was done using the software GAMS. Here we present the code:<br />
<br />
sets<br />
t /1*201/<br />
;<br />
*<br />
variable<br />
z Objective function<br />
;<br />
*<br />
positive variables<br />
alfc constituive production of CBP<br />
alfp constitutive production chitoporin<br />
alfa constitutive production chitinase<br />
alfi basal production luxI<br />
alfr basal production luxR<br />
alfci basal production CI<br />
alfha basal production HipA7<br />
alfhb basal production HipB<br />
alfas basal production Salicylic acid<br />
mcs Kinetic constant<br />
mcsq Kinetic constant<br />
maqq Kinetic constant<br />
mir Kinetic constant<br />
mi Kinetic constant<br />
mha Kinetic constant<br />
betap Maximal production of chitoporin<br />
betaa Maximal production of chitinase<br />
betai Maximal production of LuxI<br />
betahb Maximal production of HipB<br />
betaha Maximal production of HipA7<br />
betaas Maximal production of Salicylic acid<br />
ks Hill's constant for the activation by S<br />
kir Hill's constant for the activation by IR<br />
hs Hill's constant for the activation by S<br />
hir Hill's constant for the activation by IR<br />
ea Export factor of chitinase<br />
jq Import factor of chitine<br />
deltaa Diffusion factor of chitinase<br />
ei Export factor of LuxI<br />
ji Import factor of LuxI<br />
deltai Difussion factor of LuxI<br />
stotal Total concetration of the sensor in the cell<br />
eas Export factor os Salicylic acid <br />
*<br />
c(t) Concentration of CBP<br />
cs(t) Concentration of complex CBP-S<br />
p(t) Concentration of chitoporin<br />
ai(t) Concentration of chitinase<br />
ao(t) Concentration of chitinase outside the cell<br />
q(t) Concentration of chitine monomer<br />
ii(t) Concentration of LuxI inside the cell<br />
io(t) Concentration of LuxI outside the cell<br />
ir(t) Concentration of complex LuxI-LuxR<br />
r(t) Concentration of LuxR<br />
ci(t) Concentration of CI<br />
hb(t) Concentration of HB<br />
ha(t) Concentration of HA<br />
as(t) Concentration of Salicylic Acid<br />
qq(t) Concentration of Chitin<br />
s(t) <br />
;<br />
*<br />
alfc.lo=0.7;<br />
alfc.up=1.2;<br />
alfc.l=1;<br />
*<br />
alfp.lo=0.7;<br />
alfp.up=1.2;<br />
alfp.l=1;<br />
*<br />
alfa.lo=0.7;<br />
alfa.up=1.2;<br />
alfa.l=1;<br />
*<br />
alfi.lo=0.01;<br />
alfi.up=0.6;<br />
alfi.l=0.4;<br />
*<br />
alfr.lo=0.01;<br />
alfr.up=0.6;<br />
alfr.l=0.4;<br />
*<br />
alfci.lo=0.01;<br />
alfci.up=0.6;<br />
alfci.l=0.4;<br />
*<br />
alfha.lo=0.01;<br />
alfha.up=0.6;<br />
alfha.l=0.4;<br />
*<br />
alfhb.lo=0.01;<br />
alfhb.up=0.6;<br />
alfhb.l=0.4;<br />
*<br />
alfas.lo=0.01;<br />
alfas.up=0.6;<br />
alfas.l=0.4; <br />
*<br />
mcs.lo=50;<br />
mcs.up=1000;<br />
mcs.l=500;<br />
*<br />
mcsq.up=1000;<br />
mcsq.l=500;<br />
*<br />
maqq.up=1000;<br />
maqq.l=500;<br />
*<br />
mir.up=1000;<br />
mir.l=500;<br />
*<br />
mi.up=1000;<br />
mi.l=20; <br />
*<br />
mha.up=1000;<br />
mha.l=500; <br />
*<br />
betap.lo=5;<br />
betap.up=23;<br />
betap.l=14;<br />
*<br />
betaa.lo=5;<br />
betaa.up=23;<br />
betaa.l=14;<br />
*<br />
betai.lo=5;<br />
betai.up=23;<br />
betai.l=14;<br />
*<br />
betahb.lo=5;<br />
betahb.up=23;<br />
betahb.l=14;<br />
*<br />
betaha.lo=5;<br />
betaha.up=23;<br />
betaha.l=14; <br />
*<br />
betaas.lo=5;<br />
betaas.up=23;<br />
betaas.l=14;<br />
* <br />
ks.lo=0.01;<br />
ks.up=0.9;<br />
ks.l=0.05;<br />
*<br />
kir.lo=0.01;<br />
kir.up=0.9;<br />
kir.l=0.05;<br />
*<br />
hs.up=3;<br />
hs.l=1;<br />
*<br />
hir.up=5;<br />
hir.l=3; <br />
*<br />
ea.lo=0.01;<br />
ea.up=1;<br />
ea.l=0.05;<br />
*<br />
jq.lo=0.01;<br />
jq.up=1;<br />
jq.l=0.05;<br />
*<br />
deltaa.lo=0.01;<br />
deltaa.up=1;<br />
deltaa.l=0.05;<br />
*<br />
ei.lo=0.01;<br />
ei.up=1;<br />
ei.l=0.05;<br />
*<br />
ji.lo=0.01;<br />
ji.up=1;<br />
ji.l=0.05;<br />
*<br />
deltai.lo=0.01;<br />
deltai.up=1; <br />
deltai.l=0.05;<br />
*<br />
eas.lo=0.01;<br />
eas.up=1;<br />
eas.l=0.05;<br />
*<br />
stotal.lo=0.4;<br />
stotal.up=2.5;<br />
stotal.l=1;<br />
*<br />
c.l(t)=0.1616;<br />
cs.l(t)=0.738;<br />
p.l(t)=4.7233;<br />
ai.l(t)=3.1489;<br />
ao.l(t)=7.8722;<br />
q.l(t)=0;<br />
ii.l(t)=0.9662;<br />
io.l(t)=0.4831;<br />
ir.l(t)=3.6605;<br />
r.l(t)=1.2628;<br />
ci.l(t)=20.2;<br />
hb.l(t)=0.5722;<br />
ha.l(t)=2.5117;<br />
as.l(t)=0.16933;<br />
s.l(t)=1;<br />
*<br />
*<br />
scalar<br />
*<br />
gammaa /1/<br />
gammac /1/<br />
gammap /1/<br />
gammar /1/<br />
gammai /1/<br />
gammaci /1/<br />
gammaha /1/<br />
gammahb /4/<br />
gammaas /1/<br />
gammacs /1/ <br />
*<br />
betaci /9.96/<br />
kci /0.055/<br />
hci /2.3/<br />
*<br />
numcel /1/<br />
*<br />
dt /0.5/<br />
*<br />
*<br />
qq1 /0/<br />
qq2 /20/<br />
qq3 /60/<br />
qq4 /5/ <br />
;<br />
*<br />
equations<br />
*<br />
cbss Steady state equation of CBP<br />
css<br />
pss<br />
aiss<br />
aoss<br />
qss<br />
iiss<br />
ioss<br />
irss<br />
rss<br />
ciss<br />
hbss<br />
hass<br />
asss<br />
dc(t) Diferential equations<br />
dcs(t)<br />
dp(t)<br />
dai(t)<br />
dii(t)<br />
dio(t)<br />
dir(t)<br />
dr(t)<br />
dci(t)<br />
dhb(t)<br />
dha(t)<br />
das(t)<br />
dao1(t)<br />
dq1(t)<br />
dao2(t)<br />
dq2(t)<br />
dao3(t)<br />
dq3(t)<br />
sf(t) function of the change of S in the cell<br />
*<br />
res1 restricion 1<br />
res2 restriction 2<br />
res3 restriction 3<br />
*<br />
*<br />
fobj Objective function<br />
;<br />
*<br />
*Steady state equations<br />
*<br />
cbss.. alfc-gammac*c('1')-mcs*c('1')*s('1')=e=0;<br />
css.. mcs*c('1')*s('1')-mcsq*cs('1')*q('1')-gammacs*cs('1')=e=0;<br />
pss.. alfp-gammap*p('1')+ betap*(s('1')**hs)/(ks**hs+s('1')**hs)=e=0;<br />
aiss.. alfa- gammaa*ai('1')+ betaa* (s('1')**hs)/((ks**hs)+(s('1')**hs))-ea*ai('1')=e=0;<br />
aoss.. ea*ai('1')-deltaa*ao('1')=e=0;<br />
qss.. -mcsq*cs('1')*q('1')=e=0;<br />
iiss.. alfi + betai* (s('1')**hs)/((ks**hs)+ (s('1')**hs))-gammai*ii('1')+ji*io('1')-ei*ii('1')<br>-mir*ii('1')*r('1')=e=0;<br />
ioss.. numcel*(ei*ii('1')-ji*io('1'))-deltai*io('1')=e=0;<br />
irss.. mir*ii('1')*r('1')-mi*ir('1')=e=0;<br />
rss.. alfr-gammar*r('1')- mir*ii('1')*r('1')+ betai* (s('1')**hs)/((ks**hs)+(s('1')**hs))=e=0;<br />
ciss.. alfci-gammaci*ci('1')+ betaci* (ci('1')**hci)/((kci**hci)+(ci('1')**hci))+<br>betaci* (ir('1')**hir)/((kir**hir)+(ir('1')**hir))=e=0;<br />
hbss.. alfhb-gammahb*hb('1')+ betahb* (ci('1')**hci)/((kci**hci)+(ci('1')**hci))+ <br>betahb* (ir('1')**hir)/((kir**hir)+ (ir('1')**hir))-mha*power(ha('1'),2)*power(hb('1'),2)=e=0;<br />
hass.. alfha-gammaha*ha('1')+ betaha* (ci('1')**hci)/((kci**hci)+(ci('1')**hci))<br>-mha*power(ha('1'),2)*power(hb('1'),2)=e=0;<br />
asss.. alfas-gammaas*as('1')+ betaas* (ci('1')**hci)/((kci**hci)+(ci('1')**hci))- eas*as('1')=e=0;<br />
* <br />
*Differential equations<br />
*<br />
dc(t).. alfc-gammac*c(t)-mcs*c(t)*s(t)=e=(c(t+1)-c(t))/dt;<br />
dcs(t).. mcs*c(t)*s(t)-mcsq*cs(t)*q(t)-gammacs*cs('1')=e=(cs(t+1)-cs(t))/dt ;<br />
dp(t).. alfp-gammap*p(t)+ betap* (s(t)**hs)/((ks**hs)+(s(t)**hs))=e=(p(t+1)-p(t))/dt;<br />
dai(t).. alfa- gammaa*ai(t)+ betaa* (s(t)**hs)/((ks**hs)+(s(t)**hs))-ea*ai(t)=e=(ai(t+1)-ai(t))/dt ;<br />
dii(t).. alfi + betai*(s(t)**hs)/((ks**hs)+(s(t)**hs))-gammai*ii(t)+ji*io(t)-ei*ii(t)<br>-mir*ii(t)*r(t)=e=(ii(t+1)-ii(t))/dt;<br />
dio(t).. numcel*(ei*ii(t)-ji*io(t))-deltai*io(t)=e=(io(t+1)-io(t))/dt;<br />
dir(t).. mir*ii(t)*r(t)-mi*ir(t)=e=(ir(t+1)-ir(t))/dt;<br />
dr(t).. alfr-gammar*r(t)- mir*ii(t)*r(t)+ betai* (s(t)**hs)/((ks**hs)+(s(t)**hs))=e=(r(t+1)-r(t))/dt ;<br />
dci(t).. alfci-gammaci*ci(t)+ betaci* (ci(t)**hci)/((kci**hci)+(ci(t)**hci))<br>+ betaci* (ir(t)**hir)/((kir**hir)+ (ir(t)**hir))=e=(ci(t+1)-ci(t))/dt ;<br />
dhb(t).. alfhb-gammahb*hb(t)+ betahb* (ci(t)**hci)/((kci**hci)+(ci(t)**hci))<br>+ betahb* (ir(t)**hir)/((kir**hir)+(ir(t)**hir))-mha*power(ha(t),2)*power(hb(t),2)=e=(hb(t+1)-hb(t))/dt ;<br />
dha(t).. alfha-gammaha*ha(t)+ betaha* (ci(t)**hci)/((kci**hci)+(ci(t)**hci))<br>-mha*power(ha(t),2)*power(hb(t),2)=e=(ha(t+1)-ha(t))/dt;<br />
das(t).. alfas-gammaas*as(t)+ betaas* (ci(t)**hci)/((kci**hci)+(ci(t)**hci))<br>- eas*as(t)=e=(as(t+1)-as(t))/dt ;<br />
*<br />
*Differential equations depending on chitin impulse<br />
*<br />
dao1(t) $(ord(t)<51).. ea*ai(t)-deltaa*ao(t)-maqq*ao(t)*qq1=e=(ao(t+1)-ao(t))/dt;<br />
dao2(t)$(ord(t)>50 and ord(t)<151).. ea* ai(t) -deltaa*ao(t)-maqq*ao(t)*qq2=e=(ao(t+1)-ao(t))/dt;<br />
dao3(t) $(ord(t)>151 and ord(t)<201).. ea*ai(t)-deltaa*ao(t)-maqq*ao(t)*qq1=e=(ao(t+1)-ao(t))/dt;<br />
dq1(t)$(ord(t)<51).. 2*jq*p(t)*maqq*qq1*ao(t) -mcsq*cs('1')*q('1')=e=(q(t+1)-q(t))/dt;<br />
dq2(t)$(ord(t)>51 and ord(t)<151).. 2*jq*p(t)*maqq*qq2*ao(t) -mcsq*cs('1')*q('1')=e=(q(t+1)-q(t))/dt;<br />
dq3(t)$(ord(t)>151 and ord(t)<201).. 2*jq*p(t)*maqq*qq1*ao(t) -mcsq*cs('1')*q('1')=e=(q(t+1)-q(t))/dt;<br />
*S equation<br />
sf(t).. s(t)=e=stotal-cs(t);<br />
*Some restrictions<br />
res1.. ha('1')=g= hb('1');<br />
res2.. hb('131')=g=ha('131');<br />
res3.. ha('181')=g=hb('181');<br />
*objetive function<br />
fobj.. z=e=power((as('1')-as('41')),2)+power((1.6*as('1')-as('111')),2)<br>+ power((1.8*as('1')-as('121')),2)+power((1.1*as('1')-as('161')),2)+power((as('1')-as('181')),2) ;<br />
*<br />
option nlp=ipopt;<br />
model igem /all/;<br />
solve igem using nlp minimizing z;<br />
<br />
</div><br />
<br />
<br />
==STOCHASTIC MODEL ==</div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/StochasticTeam:Colombia/Modeling/Stochastic2012-10-26T19:00:38Z<p>D.olivera1320: /* Stochastic Model */</p>
<hr />
<div>{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== '''Stochastic Model''' ==<br />
<br />
<p align="justify"><br />
<br />
The previous sections showed how to know the mean behavior of the system for one cell, but this is just an average of the total proteins within the cell. All the biological systems are controlled by probability events. The cell is a huge space where there are a lot of small molecules. If we want a biological process to happen, two of these molecules have to find each other among millions in a huge pool. The differential equations do not take into account these uncontrollable events that can change the response dramatically. <br />
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If we look only at one cell, it may not behave like we want and the system may not respond. Even worse, the probability of dying exist and our cell may die. But dealing with one cell is not real and we always work with hundreds of cells. Within this population, some cells may not behave as expected but others will and the average of cells would be able to respond to the presence of the pest.<br />
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The stochastic algorithms are a way to model these probability events within a population. This simulation is made in order to confirm that the system dynamics are robust, consistent and show us if the response is still behaving like we want (taking probabilities into consideration). We use the Gillespie algorithm to develop our model. Here is a brief explanation of how it works: <br />
<br />
The complete method consists of eight steps.<br />
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::#Define the number of cells.<br />
::#Define the number of steps<br />
::#Define and name all the constants involved.<br />
::#Define creation and destruction events for each substance involved: The differential equations in this part have to be divided in two, the creation and the destruction expression. <br />
::#Apply [[Gilliespie algorithm:]] <br />
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:::• Calculate the sample space of the analyzed system: This is the sum of all the changes presented at specific time "t". <br />
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[[File:sampspa.png|center|200x100pxpx]]<br />
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:::•Calculate time distribution that depends on a random number between 0 and 1.<br />
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[[File:sampspa1.png|center|600x200pxpx]]<br />
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:::•Calculate which event occurs: The Gillespie algorithm does not consider simultaneous events, this is that in each time only one event occurs. In this case, the events are the creation or destruction of one protein. Each event has a probability of occurrence within the sample space between 0 and 1. To know which event occurs at the time "t+1", we take into account the random number use for the time distribution and look for the event that has this probability of occurrence . For example, if you see the sample space figure below, there are 5 possible events with their probability; if the random number is 0.4 then A is going to be destructed but if the random number is 0.55. then B will be created. <br />
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::6. Take the outputs from the simulation and convert them into regular interval vectors.<br />
::7. Obtain the Gillespie function mean values.<br />
::8. Plot the obtained functions.<br />
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</p> <br />
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</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-26T16:58:45Z<p>D.olivera1320: /* How did we do it? */</p>
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<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
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<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
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There are three possible ways to find this parameters: <br />
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::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
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</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
'''As said above finding the right set of parameters for a system is a headache for all the synthetic biologists. That is why we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we need to find as much information in literature as possible.''' <br />
<br />
<br />
<br> <br />
<br><br />
<br />
For our particular case, ee found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to develop the standard method. Here are the steps proposed with the results of our system as an example<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First you need to define our desired response, in our case of the Salicylic Acid. For this we stablished some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. The main objective is to find this area. Suppose you only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis''' <br />
<br />
In this step the importance of each parameters in the main outputs behavior (Our case: Salicylic Acid, the toxin HipA7 and the antitoxin HipB) is tested. The objective is to find how much each parameter affect the desired response.<br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the change of each parameter.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: LuxI Maximal production rate<br />
<br><br />
<br />
Range: 0-22 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:luxi.png|center|350x350pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The figure above shows how the HipB, HipA7 and Salicylic Acid concentration change during the presence of the pathogen is significantly influenced by the maximal production rate of LuxI protein. We make this procedure for all the parameters in both systems (Ralstonia and Fungus). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The sensitivity analysis was performed for the system with the detection module for fungus. The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|480x650pxpx]]<br />
<br />
Here we present some of the graphics found in the analysis. The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|650x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive to the rest:<br />
<br />
[[File:funnochangeone.png|center|650x380pxpx]] <br />
<br />
Finally, we found some parameters that affect the final response of all the main outputs of interest: <br />
<br />
[[File:funnochangetwo.png|center|650x380pxpx]] <br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensitivity test for Ralstonia. The next table identifies those parameters that make a relevant change in the model from those that do not. The main outputs tested were the HipB, HipA7 and the salicylic acid change in concentration due to the impulse. <br />
<br />
[[File:resultsabn3.png|center|480x650pxpx]]<br />
<br />
The following figures show the results obtained for almost all the relevant parameters. Like in the previous case we found some parameters that had no effect at all in the response and others than only had effect in one of the main outputs. <br />
<br />
[[File:ralchangeone.png|center|650x380pxpx]]<br />
<br />
We also found some interesting results, there were some parameters that had a significant influence in a small range, but when the rest of the range was evaluated the response was not sensitive to these parameters: <br />
<br />
[[File:ralchangetwo.png|center|650x380pxpx]]<br />
<br />
Finally we found parameters that had significant influence in the response of the three main outputs and in the complete range of the parameter. Here we present some of the most interesting behaviors of these kind of parameters: <br />
<br />
[[File:ralchange3.png|center|650x380pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to the final step.<br />
<br />
<br><br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-26T16:54:36Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
'''As said above finding the right set of parameters for a system is a headache for all the synthetic biologists. That is why we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we need to find as much information in literature as possible.''' <br />
<br />
<br />
<br> <br />
<br><br />
<br />
For our particular case, ee found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to develop the standard method. Here are the steps proposed with the results of our system as an example<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis''' <br />
<br />
In this step the importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) is tested. The objective is to find how much each parameter affect the desired response.<br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the change of each parameter.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: LuxI Maximal production rate<br />
<br><br />
<br />
Range: 0-22 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:luxi.png|center|350x350pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The figure above shows how the HipB, HipA7 and Salicylic Acid concentration change during the presence of the pathogen is significantly influenced by the maximal production rate of LuxI protein. We make this procedure for all the parameters in both systems (Ralstonia and Fungus). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The sensitivity analysis was performed for the system with the detection module for fungus. The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|480x650pxpx]]<br />
<br />
Here we present some of the graphics found in the analysis. The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|650x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive to the rest:<br />
<br />
[[File:funnochangeone.png|center|650x380pxpx]] <br />
<br />
Finally, we found some parameters that affect the final response of all the main outputs of interest: <br />
<br />
[[File:funnochangetwo.png|center|650x380pxpx]] <br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensitivity test for Ralstonia. The next table identifies those parameters that make a relevant change in the model from those that do not. The main outputs tested were the HipB, HipA7 and the salicylic acid change in concentration due to the impulse. <br />
<br />
[[File:resultsabn3.png|center|480x650pxpx]]<br />
<br />
The following figures show the results obtained for almost all the relevant parameters. Like in the previous case we found some parameters that had no effect at all in the response and others than only had effect in one of the main outputs. <br />
<br />
[[File:ralchangeone.png|center|650x380pxpx]]<br />
<br />
We also found some interesting results, there were some parameters that had a significant influence in a small range, but when the rest of the range was evaluated the response was not sensitive to these parameters: <br />
<br />
[[File:ralchangetwo.png|center|650x380pxpx]]<br />
<br />
Finally we found parameters that had significant influence in the response of the three main outputs and in the complete range of the parameter. Here we present some of the most interesting behaviors of these kind of parameters: <br />
<br />
[[File:ralchange3.png|center|650x380pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to the final step.<br />
<br />
<br><br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-26T16:51:40Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
'''As said above finding the right set of parameters for a system is a headache for all the synthetic biologists. That is why we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible.''' <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis''' <br />
<br />
In this step the importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) is tested. The objective is to find how much each parameter affect the desired response.<br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the change of each parameter.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: LuxI Maximal production rate<br />
<br><br />
<br />
Range: 0-22 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:luxi.png|center|350x350pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The figure above shows how the HipB, HipA7 and Salicylic Acid concentration change during the presence of the pathogen is significantly influenced by the maximal production rate of LuxI protein. We make this procedure for all the parameters in both systems (Ralstonia and Fungus). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The sensitivity analysis was performed for the system with the detection module for fungus. The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|480x650pxpx]]<br />
<br />
Here we present some of the graphics found in the analysis. The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|650x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive to the rest:<br />
<br />
[[File:funnochangeone.png|center|650x380pxpx]] <br />
<br />
Finally, we found some parameters that affect the final response of all the main outputs of interest: <br />
<br />
[[File:funnochangetwo.png|center|650x380pxpx]] <br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensitivity test for Ralstonia. The next table identifies those parameters that make a relevant change in the model from those that do not. The main outputs tested were the HipB, HipA7 and the salicylic acid change in concentration due to the impulse. <br />
<br />
[[File:resultsabn3.png|center|480x650pxpx]]<br />
<br />
The following figures show the results obtained for almost all the relevant parameters. Like in the previous case we found some parameters that had no effect at all in the response and others than only had effect in one of the main outputs. <br />
<br />
[[File:ralchangeone.png|center|650x380pxpx]]<br />
<br />
We also found some interesting results, there were some parameters that had a significant influence in a small range, but when the rest of the range was evaluated the response was not sensitive to these parameters: <br />
<br />
[[File:ralchangetwo.png|center|650x380pxpx]]<br />
<br />
Finally we found parameters that had significant influence in the response of the three main outputs and in the complete range of the parameter. Here we present some of the most interesting behaviors of these kind of parameters: <br />
<br />
[[File:ralchange3.png|center|650x380pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to the final step.<br />
<br />
<br><br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T15:11:37Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis''' <br />
<br />
In this step the importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) is tested. The objective is to find how much each parameter affect the desired response.<br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the change of each parameter.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: LuxI Maximal production rate<br />
<br><br />
<br />
Range: 0-22 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:luxi.png|center|350x350pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The figure above shows how the HipB, HipA7 and Salicylic Acid concentration change during the presence of the pathogen is significantly influenced by the maximal production rate of LuxI protein. We make this procedure for all the parameters in both systems (Ralstonia and Fungus). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The sensitivity analysis was performed for the system with the detection module for fungus. The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|480x650pxpx]]<br />
<br />
Here we present some of the graphics found in the analysis. The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|650x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive to the rest:<br />
<br />
[[File:funnochangeone.png|center|650x380pxpx]] <br />
<br />
Finally, we found some parameters that affect the final response of all the main outputs of interest: <br />
<br />
[[File:funnochangetwo.png|center|650x380pxpx]] <br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensitivity test for Ralstonia. The next table identifies those parameters that make a relevant change in the model from those that do not. The main outputs tested were the HipB, HipA7 and the salicylic acid change in concentration due to the impulse. <br />
<br />
[[File:resultsabn3.png|center|480x650pxpx]]<br />
<br />
The following figures show the results obtained for almost all the relevant parameters. Like in the previous case we found some parameters that had no effect at all in the response and others than only had effect in one of the main outputs. <br />
<br />
[[File:ralchangeone.png|center|650x380pxpx]]<br />
<br />
We also found some interesting results, there were some parameters that had a significant influence in a small range, but when the rest of the range was evaluated the response was not sensitive to these parameters: <br />
<br />
[[File:ralchangetwo.png|center|650x380pxpx]]<br />
<br />
Finally we found parameters that had significant influence in the response of the three main outputs and in the complete range of the parameter. Here we present some of the most interesting behaviors of these kind of parameters: <br />
<br />
[[File:ralchange3.png|center|650x380pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to the final step.<br />
<br />
<br><br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/File:Ralchange3.pngFile:Ralchange3.png2012-10-15T15:00:45Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/File:Ralchangetwo.pngFile:Ralchangetwo.png2012-10-15T15:00:04Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/File:Ralchangeone.pngFile:Ralchangeone.png2012-10-15T14:58:34Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/File:Resultsabn3.pngFile:Resultsabn3.png2012-10-15T14:57:04Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T14:56:16Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: LuxI Maximal production rate<br />
<br><br />
<br />
Range: 0-22 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:luxi.png|center|350x350pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The figure above shows how the HipB, HipA7 and Salicylic Acid concentration change during the presence of the pathogen is significantly influenced by the maximal production rate of LuxI protein. We make this procedure for all the parameters in both systems (Ralstonia and Fungus). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|700x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
[[File:funnochangeone.png|center|700x380pxpx]] <br />
<br />
And there are some parameters that affect the final response of all the main outputs of interest: <br />
<br />
[[File:funnochangetwo.png|center|700x380pxpx]] <br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table identifies those parameters that make a relevant change in the model from those that do not. Th main outputs tested were the HipB, HipA7 and the salicylic acid change in concentration due to the impulse. <br />
<br />
[[File:resultsabn3.png|center|500x700pxpx]]<br />
<br />
The following figures show the results obtained for almost all the relevant parameters. Like in the previous case we found some parameters that had no effect at all in the response and others than only had effect in one of the main outputs. <br />
<br />
[[File:ralchangeone.png|center|700x380pxpx]]<br />
<br />
We also found some interesting results, there were some parameters that had a significant influence in a small range, but when the rest of the range was evaluated the response was not sensitive to these parameters: <br />
<br />
[[File:ralchangetwo.png|center|700x380pxpx]]<br />
<br />
Finally we found parameters that had significant influence in the response of the three main outputs and in all the range of the parameter. Here we present some of the most interesting behaviors of these kind of parameters: <br />
<br />
[[File:ralchange3.png|center|700x380pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to the final step.<br />
<br />
<br><br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T03:26:38Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: LuxI Maximal production rate<br />
<br><br />
<br />
Range: 0-22 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:luxi.png|center|350x350pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The figure above shows how the HipB, HipA7 and Salicylic Acid concentration change during the presence of the pathogen is significantly influenced by the maximal production rate of LuxI protein. We make this procedure for all the parameters in both systems (Ralstonia and Fungus). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|700x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
[[File:funnochangeone.png|center|700x380pxpx]] <br />
<br />
And there are some parameters that affect the final response of all the main outputs of interest: <br />
<br />
[[File:funnochangetwo.png|center|700x380pxpx]] <br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsabn3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to the final step.<br />
<br />
<br><br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T03:17:44Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: LuxI Maximal production rate<br />
<br><br />
<br />
Range: 0-22 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:luxi.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The figure above shows how the HipB, HipA7 and Salicylic Acid concentration change during the presence of the pathogen is significantly influenced by the maximal production rate of LuxI protein. We make this procedure for all the parameters in both systems (Ralstonia and Fungus). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|700x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
[[File:funnochangeone.png|center|700x380pxpx]] <br />
<br />
And there are some parameters that affect the final response of all the main outputs of interest: <br />
<br />
[[File:funnochangetwo.png|center|700x380pxpx]] <br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsabn3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to the final step.<br />
<br />
<br><br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T02:45:02Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: LuxI Maximal production rate<br />
<br><br />
<br />
Range: 0-22 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:luxi.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The figure above shows how the HipB, HipA7 and Salicylic Acid concentration change during the presence of the pathogen is significantly influenced by the maximal production rate of LuxI protein. We make this procedure for all the parameters in both systems (Ralstonia and Fungus). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|700x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
[[File:funnochangeone.png|center|700x380pxpx]] <br />
<br />
And there are some parameters that affect the final response of all the main outputs of interest: <br />
<br />
[[File:funnochangetwo.png|center|700x380pxpx]] <br />
<br />
<br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T02:41:58Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: LuxI Maximal production rate<br />
<br><br />
<br />
Range: 0-22 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:luxi.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The figure above shows how the HB, HA and SA concentration changes during the presence of the pathogen is significantly influenced by the maximal production rate of LuxI protein.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Fungus). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|700x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
[[File:funnochangeone.png|center|700x380pxpx]] <br />
<br />
And there are some parameters that affect the final response of all the main outputs of interest: <br />
<br />
[[File:funnochangetwo.png|center|700x380pxpx]] <br />
<br />
<br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/File:Luxi.pngFile:Luxi.png2012-10-15T02:39:45Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T02:37:51Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: LuxI Maximal production rate<br />
<br><br />
<br />
Range: 0-22 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:luxi.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|700x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
[[File:funnochangeone.png|center|700x380pxpx]] <br />
<br />
And there are some parameters that affect the final response of all the main outputs of interest: <br />
<br />
[[File:funnochangetwo.png|center|700x380pxpx]] <br />
<br />
<br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/File:Funnochangetwo.pngFile:Funnochangetwo.png2012-10-15T02:34:58Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/File:Funnochangeone.pngFile:Funnochangeone.png2012-10-15T02:34:38Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T02:31:01Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: CI degradation rate<br />
<br><br />
<br />
Range: 1-10 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|700x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
[[File:funnochangeone.png|center|700x380pxpx]] <br />
<br />
And there are some parameters that affect the final response of all the main outputs of interest: <br />
<br />
[[File:funnochangetwo.png|center|700x380pxpx]] <br />
<br />
<br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T02:25:25Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: CI degradation rate<br />
<br><br />
<br />
Range: 1-10 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|center|700x380pxpx]] <br />
<br />
<br><br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
<br />
<br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/File:Funnochange.pngFile:Funnochange.png2012-10-15T02:23:35Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T02:22:44Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: CI degradation rate<br />
<br><br />
<br />
Range: 1-10 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochange.png|left|700x380pxpx]] <br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
<br />
<br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T02:16:41Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: CI degradation rate<br />
<br><br />
<br />
Range: 1-10 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochang.png|left|350x350pxpx]] [[File:funnochang2.png|right|350x350pxpx]]<br />
<br />
<br><br />
<br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
<br />
<br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T02:16:01Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: CI degradation rate<br />
<br><br />
<br />
Range: 1-10 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochang.png|left|400x400pxpx]] [[File:funnochang2.png|right|400x400pxpx]]<br />
<br />
<br><br />
<br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
<br />
<br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/File:Funnochang2.pngFile:Funnochang2.png2012-10-15T02:14:38Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/File:Funnochang.pngFile:Funnochang.png2012-10-15T02:14:07Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T02:13:22Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: CI degradation rate<br />
<br><br />
<br />
Range: 1-10 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
[[File:funnochang.png|left|500x500pxpx]] [[File:funnochang2.png|right|500x500pxpx]]<br />
<br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
<br />
<br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/File:Resultsab.pngFile:Resultsab.png2012-10-15T02:10:31Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T02:07:43Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: CI degradation rate<br />
<br><br />
<br />
Range: 1-10 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsab.png|center|500x700pxpx]]<br />
<br />
The pictures below show two parameters that do not have significant influence in the desired response:<br />
<br />
<br />
We also found some parameters that have a significant influence in one of the main outputs but were not sensitive the rest:<br />
<br />
<br />
<br />
<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T01:54:14Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: CI degradation rate<br />
<br><br />
<br />
Range: 1-10 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
<br><br />
<br><br />
<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T01:53:50Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: CI degradation rate<br />
<br><br />
<br />
Range: 1-10 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
<br><br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T01:53:12Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: CI degradation rate<br />
<br><br />
<br />
Range: 1-10 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Fungus:'''<br />
<br />
The following table shows the different parameters involved in the system with the fungal detection system and how they affect the final behavior <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T01:49:21Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) Determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code. iv) See how the difference between the steady state's concentrations and the concentrations during the impulse of the pathogen changed with the iteration of the parameters.<br />
<br />
<br><br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
<br><br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: CI degradation rate<br />
<br><br />
<br />
Range: 1-10 <br />
<br />
<br><br />
<br />
Step size: 0.5<br />
<br />
[[File:.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T01:45:17Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following stages: i.) Establish the ranges of the parameters (see above), ii) determine appropriate division for the ranges, iii) Iterate each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: Chitinase production<br />
<br><br />
<br />
Range: 0.1-0.9 μM<br />
<br />
<br><br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T01:42:44Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following staged: Establishing the ranges of the parameters (see above), then determine appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Finally, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|500x300pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/File:Param5.pngFile:Param5.png2012-10-15T01:40:25Z<p>D.olivera1320: </p>
<hr />
<div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T01:40:07Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following staged: Establishing the ranges of the parameters (see above), then determine appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Finally, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, so the probability of going out of the working area is smaller. <br />
<br />
[[File:param5.png|center|400x200pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T01:38:46Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following staged: Establishing the ranges of the parameters (see above), then determine appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Finally, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal of these standard procedure is to set the changeable parameters in such a way that the cross sectional area of the area of parameters that has desired response is maximized. <br />
<br />
Suppose you only have two parameters, one that can be changed and one that does not. If you set the first parameter in a black dot (see figure below), a small change of the uncontrollable parameter would take the system out of its working area. The optimum solution is to set the first parameter in the red line, with this the probability of going out of the working area is smaller. <br />
<br />
[[File:param4.png|center|400x200pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-15T01:31:01Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following staged: Establishing the ranges of the parameters (see above), then determine appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Finally, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
</p><br />
<br />
<p align="center"><br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
</p><br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
There are some parameters that can me modified experimentally like the maximal rate production but most of them depend on molecular details, are unchangeable and can only be known with experiments. The final goal is to <br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-14T21:35:47Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how each parameter affected the model. <br />
<br />
This test considered the following staged: Establishing the ranges of the parameters (see above), then determine appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Finally, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
<p align="center"><br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
<p align="justify"><br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-14T20:57:58Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that make the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how the parameters affect the model. <br />
This test considered the following: establishing the ranges of the parameters (see above), we determined appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Then, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
----<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-14T20:57:17Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that makes the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how the parameters affect the model. <br />
This test considered the following: establishing the ranges of the parameters (see above), we determined appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Then, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
----<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
<br />
:'''Step 4: Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the second step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensitive to the parameter, we take longer steps than the case which the response is more sensitive. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-14T20:55:11Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that makes the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area to begin the screening. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variables. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (when there is no signal). This algorithm was programmed in a specialized software. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
:'''Step 3: Sensitivity anlysis'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how the parameters affect the model. <br />
This test considered the following: establishing the ranges of the parameters (see above), we determined appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Then, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
----<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
<br />
:'''3.Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the last step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensible to the parameter, we take longer steps than the case which the response is largely sensible. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-14T20:51:17Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that makes the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. <br />
<br />
The goal in this step is to find a point within a specific area and to begin the screening.Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
Now we want a point within these area. It seems easy to identify in two dimensions, but this process can take a lot of time (weeks or even months) when it is a space of more than 10 parameters. So, one way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variable. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (chitin is zero). This algorithm was programmed in the software Gamside. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
----<br />
<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how the parameters affect the model. <br />
This test considered the following: establishing the ranges of the parameters (see above), we determined appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Then, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
----<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
<br />
:'''3.Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the last step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensible to the parameter, we take longer steps than the case which the response is largely sensible. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-14T20:49:45Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters: A point within an area'''<br />
<br />
Within the parameter space, there are many sets of parameters that makes the system behaves just like expected; and these can be represented as an area. Our main objective is to find this area. The goal in this step is to find a point within a specific area and to begin the screening.<br />
<br />
Suppose we only have two parameters. The plane represents the parameter space and the green area is where the system works. <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
It seems easy to identify in two dimensions, but optimization can takes a lot of time (weeks or even months) when it is a space of more than 10 parameters. <br />
<br />
One way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variable. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (chitin is zero). This algorithm was programmed in the software Gamside. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
----<br />
<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how the parameters affect the model. <br />
This test considered the following: establishing the ranges of the parameters (see above), we determined appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Then, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
----<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
<br />
:'''3.Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the last step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensible to the parameter, we take longer steps than the case which the response is largely sensible. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-14T20:46:07Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablish some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters'''<br />
<br />
Within the parameter space, there are many sets of parameters that makes the system behaves just like expected; and these can be represented as an area. The figure below shows an example of the acceptable area for two parameters . <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
It seems easy to identify in two dimensions, but optimization can takes a lot of time (weeks or even months) when it is a space of more than 10 parameters. <br />
<br />
The goal in this step is to find a point within a specific area and to begin the screening. One way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variable. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (chitin is zero). This algorithm was programmed in the software Gamside. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how the parameters affect the model. <br />
This test considered the following: establishing the ranges of the parameters (see above), we determined appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Then, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
----<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
<br />
:'''3.Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the last step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensible to the parameter, we take longer steps than the case which the response is largely sensible. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-14T20:41:39Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we define our desired response of the Salicylic Acid. We stablishe some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse. We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters'''<br />
<br />
<br />
<br />
Within the ranges of the parameters, there is a number combined with the other parameters that makes the system behaves just like expected. Then, there are many possible combinations where this condition can happened. The combination can be represented as an area where the system works. The figure below shows an example of the acceptable area for two parameters . <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
It seems easy to identify in two dimensions, but optimization can takes a lot of time (weeks or even months) when it is a space of more than 10 parameters. <br />
<br />
The goal in this step is to find a point within a specific area and to begin the screening. One way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variable. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (chitin is zero). This algorithm was programmed in the software Gamside. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how the parameters affect the model. <br />
This test considered the following: establishing the ranges of the parameters (see above), we determined appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Then, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
----<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
<br />
:'''3.Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the last step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensible to the parameter, we take longer steps than the case which the response is largely sensible. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-14T20:39:54Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function: Define your desired behavior'''<br />
<br />
First we defined our desired response of the Salicylic Acid. We stablished some ranges where it was suppose to be, depending of the chitin or 3-OH-PAME impulse.<br />
<br><br />
We only want our system to respond if the signal is long and intense. Otherwise we don't want our cells to be "awake". The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]] <br />
<br />
<br />
:'''Step 2. Optimization of parameters'''<br />
<br />
<br />
<br />
Within the ranges of the parameters, there is a number combined with the other parameters that makes the system behaves just like expected. Then, there are many possible combinations where this condition can happened. The combination can be represented as an area where the system works. The figure below shows an example of the acceptable area for two parameters . <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
It seems easy to identify in two dimensions, but optimization can takes a lot of time (weeks or even months) when it is a space of more than 10 parameters. <br />
<br />
The goal in this step is to find a point within a specific area and to begin the screening. One way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variable. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (chitin is zero). This algorithm was programmed in the software Gamside. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how the parameters affect the model. <br />
This test considered the following: establishing the ranges of the parameters (see above), we determined appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Then, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
----<br />
<br />
<br />
<br />
<br />
<br />
----<br />
<br />
<br />
:'''3.Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the last step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensible to the parameter, we take longer steps than the case which the response is largely sensible. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-14T19:20:52Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
<br />
----<br />
<br />
Once the ranges of the parameters were set, we proceed to do the screening. We tried a lot of techniques and failed, finally we found an acceptable one. Here we present its steps:<br />
<br />
<br />
:'''Step 1: Objective function '''<br />
<br />
:'''Step 2. Optimization of parameters'''<br />
<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how the parameters affect the model. <br />
This test considered the following: establishing the ranges of the parameters (see above), we determined appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Then, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
----<br />
<br />
<br />
<br />
<br />
Within the ranges of the parameters, there is a number combined with the other parameters that makes the system behaves just like expected. Then, there are many possible combinations where this condition can happened. The combination can be represented as an area where the system works. The figure below shows an example of the acceptable area for two parameters . <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
It seems easy to identify in two dimensions, but optimization can takes a lot of time (weeks or even months) when it is a space of more than 10 parameters. <br />
<br />
The goal in this step is to find a point within a specific area and to begin the screening. One way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variable. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (chitin is zero). This algorithm was programmed in the software Gamside. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
<br />
----<br />
<br />
<br />
:'''3.Screening'''<br />
<br />
::''If we have a set of parameters, why do we do a screening?''<br />
<br />
In the last step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
<br />
To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
<br />
How much do we move depends on the sensitivity analysis. If the response is not very sensible to the parameter, we take longer steps than the case which the response is largely sensible. <br />
<br />
[[File:param3.png|center|400x200pxpx]]<br />
<br />
Now it is needed to know if we are in the acceptable area or not. To solve this problem, we defined some ranges where Salicylic Acid depends of the chitin or 3-OH-PAME impulse. The figure below shows this: <br />
<br />
[[File:param4.png|center|480x450pxpx]]<br />
<br />
-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
<br />
</p> <br />
<br />
</div></div>D.olivera1320http://2012.igem.org/Team:Colombia/Modeling/ParamterersTeam:Colombia/Modeling/Paramterers2012-10-14T19:16:24Z<p>D.olivera1320: /* How did we do it? */</p>
<hr />
<div>== Team Colombia @ 2012 iGEM ==<br />
<br />
{{https://2012.igem.org/User:Tabima}}<br />
<br />
<div class="right_box"><br />
<br />
== Parameters of the equations ==<br />
<br />
When we want to model a biological process, it is necessary to write the differential equation that model the system which requires a number of constants. If the real value for the constants are unknown, the system can not have any biological sense. These entries are called parameters and they are crucial elements in the mathematical model made in this project.<br />
<br />
There are three possible ways to find this parameters: <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::i) Literature. There are a lot of studies trying to find biological parameters, such as basal rate of protein production, kinetic constants, Monod's constants, etc. Moreover, there are so many biological systems and only a few of them have been characterized. Thus, it is difficult to find the parameters that are needed. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
::ii) Experimental way. If an experiment is made using the biological system of interest, it is possible to find the parameters for the equations that models the whole system. For this project, it was necessary to model the biological system first. Thus, experiments couldn't be performed to find the constant for the differential equations. <br />
<br />
<html><br />
<br><br />
</br><br />
</html><br />
<br />
<br />
::iii) Screening of parameters. Sometimes we don't have the exact number that we need, but we have a rank where it could be or the parameter for a similar biological system, then we can perfom a screening of parameter, where we try to find the value that perfectly fits the reponse of our circuit.<br />
<br />
== How did we do it? ==<br />
<br />
<br />
<p align="justify"><br />
<br />
<br />
To find the Pest-busters parameters we developed a standard procedure that can be use any synthetic biology mathematical model. It has four main steps. But before starting to work in it we needed to find as much information in literature as possible. <br />
<br />
<br> <br />
<br><br />
<br />
We found all the CI promoter box parameters, we normalized all the degradation terms with the cell division time and for the other we found acceptable ranges. We used a lot of resources and methods to approximate the limits for this ranges. Here we show an explanation:<br />
<br />
[[File:tabla2.png|center|300x250pxpx]]<br />
<br />
'''''a.Basal levels of the proteins:'''''<br />
<br />
Determining a rank which this value could be, we described mathematically a very simple process. Suppose there is a usual differential equation:<br />
<br />
[[File:alfbsal1.png|center|120x120px]]<br />
<br />
Where αo is the basal level, γ is the protein parameter for degradation and P is the protein concentration. Thus, we can assume that in steady state conditions we have that dP/dt= 0:<br />
<br />
[[File:alfbasal2.png|center|80x60pxpx]]<br />
<br />
But γ is approximately 1/T, then:<br />
<br />
[[File:alfbasal3.png|center|80x60pxpx]]<br />
<br />
Establishing the rank, we assumed T as φ and we looked for a normal rank of concentration of a protein in a cell [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=104520&ver=10&trm=protein]. Another important aspect is that there were constitutive proteins and inducible proteins. Thus, we divided the basal levels in two class: i) one for basal proteins and ii) one for constitutive proteins. Finally, following ranges were defined: <br />
<br />
Basal proteins range (LuxR, LuxI, HipB and Salicylic Acid):<br />
<br />
[[File:range1.png|center|156x104pxpx]]<br />
<br />
Constitutive proteins range (HipA7,Sensor, Chitinase, Chitoporin and CBP):<br />
<br />
[[File:z.png|center|153x105pxpx]]<br />
<br />
By the other hand, the basal production of HipA has to be constitutive and inducible because the cell has to have three stages. The first stage is always sleep (constitutive) due to HipA effect. Once an impulse of chitin or 3-OH-PAME is presented, it is going to awake the cell due to concentration decrease of HipA . This is the second stage of the cell. The third stage is when cell is slept again which it is achieve through a positive control (inducible). The basal concentration used in the equations was the constitutive one because it is going to have more effect in the response. <br />
<br />
'''b.Protein degradation:'''<br />
<br />
The chosen unit of time was the bacteria life time because it represents when the proteins are diluted and decrease in the cell. We can establish easily the protein degradation using values related to life time of E.Coli. We defined the protein degradation for almost all values as follows: <br />
<br />
[[File:degrada.png|center|100x80pxpx]]<br />
<br />
Again, we found a special case. Degradation of HB is greater, then we assigned it a value of 4/φ. <br />
<br />
'''c. Reaction constant:'''<br />
<br />
This value can vary depending of the described process. Fortunately, we found in literature how to approximate this value [http://www.biokin.com/dynafit/scripting/html/node23.html#SECTION00431000000000000000]. However, we took the approximation and turned it into units described above. <br />
<br />
[[File:reactioncte.png|center|186x126pxpx]]<br />
<br />
'''d. Hill coefficients k:'''<br />
<br />
We looked for some reference in different data sources and we found the coefficient of CI ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). Then, we decided to approximate a rank taking this value as a reference.<br />
<br />
[[File:zzzzzzz.png|center|145x95pxpx]]<br />
<br />
'''e. Hill coefficients n:'''<br />
<br />
We based on the value "n" found for CI( [http://www.sciencemag.org/content/307/5717/1962.short NItzal et al.2005]). In this case, there was an analysis that took in account the number of molecules that interfere with gene activation.<br />
<br />
[[File:zzzzRRzzz.png|center|260x172pxpx]]<br />
<br />
'''f. Maximum cell production (β):'''<br />
<br />
Considering the concentration of Rubisco, which is about 20 times the average concentration and rate of production of a protein, [http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=107431&ver=1&trm=rubisco], we assign the limits for this range as follows.<br />
<br />
[[File:RRMoizzz.png|center|136x90pxpx]]<br />
<br />
'''CI PARAMETERS'''<br />
<br />
All the parameters for CI activation system were found in the literature. ([http://www.sciencemag.org/content/307/5717/1962.short Nitzal et al.2005]). We only made a change of units. <br />
<br />
[[File:CI.png|center|150x150pxpx]]<br />
<br />
Once the ranks of the parameters were set, we proceed to do the screening. We tried a lot of techniques but we failed. Here we present the steps for the successful one:<br />
<br />
----<br />
:'''Step 1.Sensitivity Analysis'''<br />
The importance of each parameters in the main outputs behavior (Salicylic Acid, the toxin HipA7 and the antitoxin HipB) was tested. Thus, we did a sensitivity analysis in order to understand how the parameters affect the model. <br />
This test considered the following: establishing the ranges of the parameters (see above), we determined appropriate division for the ranges. Next, we obtained enough data for analysis (according to step size). Then, we iterated each parameter while leaving the others fixed in the MATLAB code.<br />
<br />
'''Note: the exact value for each parameter is not important. It is important their relevance and how they change the response.'''<br />
<br />
Here is an example:<br />
<br />
Parameter: Chitinase production<br />
<br />
Range: 0.1-0.9 μM<br />
<br />
Step size: 0.01<br />
<br />
[[File:sa.png|center|400x400pxpx]]<br />
<br />
The diagram above shows that the chitinase production does not have an important effect in salicylic acid, toxin, or antitoxin behavior.<br />
<br />
We make this procedure for all the parameters in both systems (Ralstonia and Rust). The results are shown below:<br />
<br />
'''Rust:'''<br />
The following table shows the different parameters involved in the coffee rust detection system and how they affect the final behavior. <br />
<br />
[[File:resultsa.png|center|500x700pxpx]]<br />
<br />
The following graphs represent more relevant examples. It shows that rust detection is not directly sensible to some parameters. The blue line is the Salicylic Acid, the green is HipA7 and the red is HipB:<br />
<br />
[[File:resutsa1.png|center|500x700pxpx]]<br />
<br />
'''Ralstonia'''<br />
<br />
Below, it is exposed the results obtained from sensibility test for Ralstonia. The next table shows the results obtained. From the results, it is possible extract two important aspects between parameters. The first aspect is identify those parameters that make a relevant change in the model from those that do not. From parameters that are relevant, identify those that are directly or inversely proportional to the salicylic acid response (denoted as “+” and “-“ in the table) is the main objective. This criterion is the second aspect. <br />
<br />
[[File:resultsa3.png|center|500x700pxpx]]<br />
<br />
The following figure shows the results obtained for almost all the relevant parameters. It sketched the response of the salicylic acid in function of the value for a determined parameter.<br />
<br />
[[File:resultsa4.png|center|500x700pxpx]]<br />
<br />
<br />
Using this results it is possible to know the parameters that affect the response in a major way. Although it is still unknown the exact value of the parameter, it is possible predict how the system is supposed to response. Then, we could proceed to step 2. <br />
<br />
----<br />
<br />
<br />
:'''Step 2. Optimization of parameters'''<br />
<br />
Within the ranges of the parameters, there is a number combined with the other parameters that makes the system behaves just like expected. Then, there are many possible combinations where this condition can happened. The combination can be represented as an area where the system works. The figure below shows an example of the acceptable area for two parameters . <br />
<br />
[[File:param.png|center|400x200pxpx]]<br />
<br />
It seems easy to identify in two dimensions, but optimization can takes a lot of time (weeks or even months) when it is a space of more than 10 parameters. <br />
<br />
The goal in this step is to find a point within a specific area and to begin the screening. One way to achieve this process is making an optimization. <br />
<br />
[[File:param2.png|center|400x200pxpx]]<br />
<br />
A process of optimization takes a function and found maximum or minimum values changing some variable of it. The changing variables are named optimization variables and function is called objective function. If we have some limitations, it is possible to add restrictions to the system, and the function will be optimized without breaking this limitations.<br />
<br />
In this case, the objective function is a minimum difference of squares between the points of the expected behavior and the response of the equations with a set of optimization variable. When the distance between these points is minimum, the parameters are found. <br />
<br />
This optimization process was done discretizing the differential equations and putting them as restrictions. The same procedure was made for the stable state concentration where the differential equation must be equal to zero (chitin is zero). This algorithm was programmed in the software Gamside. <br />
<br />
Unfortunately the firsts results were not satisfactory and a new code it is being developed. <br />
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:'''3.Screening'''<br />
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::''If we have a set of parameters, why do we do a screening?''<br />
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In the last step we found a set of parameters that give the expected response of the system. But this is only a point. What does happen with the rest of points of the area? Does the optimization method “know” if these parameters are real or have biological sense? As long as we specified one single response for the parameters, we don’t know if there are other possible responses of valid solutions. Do they exist?<br />
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To answer these questions we performed a screening of the parameters. Beginning at the resulting point of the optimization, we started to move in all directions in a fixed range and then we examined when the acceptable area ends. <br />
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How much do we move depends on the sensitivity analysis. If the response is not very sensible to the parameter, we take longer steps than the case which the response is largely sensible. <br />
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[[File:param3.png|center|400x200pxpx]]<br />
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Now it is needed to know if we are in the acceptable area or not. To solve this problem, we defined some ranges where Salicylic Acid depends of the chitin or 3-OH-PAME impulse. The figure below shows this: <br />
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[[File:param4.png|center|480x450pxpx]]<br />
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-Note: Due to long computational time, the code is being parallelized <br> and we expect to run the tests shortly. <br />
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</div></div>D.olivera1320