Team:Buenos Aires/Project/Model

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= Modeling a synthetic ecology =
= Modeling a synthetic ecology =
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To gain insight into the behavior of our crossfeeding design, to understand which  parameters of the system are important, and to study the feasibility of the project, we decided to implement a model of the system. We found this interesting per se, as the model we need is that of a microbial community, an ecology.  
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To gain insight into the behavior of our crossfeeding design, to understand which  parameters of the system are important, and to study the feasibility of the project, we decided to implement a model of the system. We found this interesting per se, as the model we need is that of a microbial community and ecology.  
The mathematical modeling of ecological system is an active field of research with a very rich and interesting history; from the works of Lotka and Volterra, who modeled predator-prey dynamics with ordinary differential equations, to Robert May’s chaotic logistic maps. In fact a lot of synthetic systems of interacting bacteria created in recent years were used to study these sort of models.
The mathematical modeling of ecological system is an active field of research with a very rich and interesting history; from the works of Lotka and Volterra, who modeled predator-prey dynamics with ordinary differential equations, to Robert May’s chaotic logistic maps. In fact a lot of synthetic systems of interacting bacteria created in recent years were used to study these sort of models.
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* Each strain only consumes the AA it doesn't produce
* Each strain only consumes the AA it doesn't produce
* The system reaches steady state.
* The system reaches steady state.
-
 
[[File:Bsas2012-model1.png]] (model)
[[File:Bsas2012-model1.png]] (model)
 +
In the equation for the temporal evolution of each population we take the growth rate as a Hill function of the amount of AA to which the strain is auxotrophic (it can't produce). The function captures the increase of the growth rate with the relevant AA concentration, until it reaches a plateau at the maximal growth rate, that corresponds to a doubling time of 90 minutes (doubling time = ln(2)/kmax). These functions are of the form
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In the equation for the temporal evolution of each population we take the growth rate as a Hill function of the amount of AA to which the strain is auxotrophic (it can't produce). The function captures the increase of the growth rate with the relevant AA concentration, until it reaches a plateau at the maximal growth rate of 90 minutes per cell division cycle (1/kmax).
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[[File:bsas2012-modeling-eq1.png|200px]](1)
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[[File:bsas2012-modeling-eq1.png]](1)
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where '''Kaa''' is the effective concentration of AA at which half maximal growth rate is obtained, and '''l''' is the Hill coefficient, that describes how "steep" the curve is.  
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K_AA is the effective concentration of AA at which half maximal growth rate is obtained.
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The term (1-(Nhis+Ntrp)/Cc) of the model accounts for other factors, not explicit in our model that can limit the concentration of cells in a given volume (carrying capacity), enabling the population to reach equilibrium. We included a term related to cell death (- death Ntrp), which is makes biologically sense (cell do die) and is critical for a correct behavior of the model.
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The second term accounts for other factors, not explicit in our model that can limit the concentration of cells in a given volume (carrying capacity) enabling the population to reach equilibrium. Given that our model relies on nutrient deficiency we can't rule out starvation so we added a term related to cell death.
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The terms in the equations for the evolution of each [AA] in the medium are the fluxes of AA entering and leaving the medium. The first term (Ptrp*Nhis-) is a measure of the AA produced and exported in the form of peptides by a cell, for example the rate of tryptophan secretion by the histidine dependent cell.  
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The terms in the equations for the evolution of each [AA] in the medium are the fluxes of AA entering and leaving the medium. Each rate '''p''' is a measure of the AA produced and exported in the form of a protein by a cell. In our simulations it is defined as a fraction of the calculated maximum of elongation events.
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The second term is the flux leaving the medium and entering the auxotrophic cells. Each cell has a more or less fixed amount of each amino acid, which we call '''d'''. If a cell is to replicate itself and cannot produce and amino acid, it will have to absorb it from the medium. Therefore the flux of AA entering the cell, is '''d''' divided by the doubling time τ (the time it takes to "construct" a new cell).
 +
One of the hypotheses built into our model is that τ will vary greatly with the concentration of nutrients available. We've used
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The second term is the flux leaving the medium and entering the auxotrophic cells. We have assumed that the total amount of AA absorbed, '''d''', during the cell cycle time τ  is the required to ''build'' the daughter. One of the hypotheses built into our model is that τ  will vary greatly with the concentration of nutrients available. We've used
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[[File:bsas2012-modeling-eq2.png|200px]](2)
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[[File:bsas2012-modeling-eq2.png]](2)
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The amount of AA entering the cells that produce it was considered negligible. This means that a cell that produces Trp doesn't import it from the medium in significant quantities. This is likely to hold when AA concentrations are limiting. 
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The amount of AA entering the cells that produce it was considered negligible.
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=== Steady State Solution ===
 +
The 4 equations in the model were equaled to zero and the non-linear algebraic system solved using [http://www.wolfram.com/mathematica/ Mathematica] to find their equilibrium values.
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===== Steady State Solution =====
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Here we change the notation a little bit to two generic strains '''a''' and '''b''' where population '''Na''' produces amino acid '''a''' and requires '''b''', and population '''Nb''' produces '''b''' and requires '''a'''. We are interested in the value of the cell populations in equilibrium, or more importantly the fraction of each population in steady state. Thus a change was made to more convenient variables: the sum of both population and a strain's fraction.
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The 4 equations in [] were equaled to zero and the non-linear algebraic system solved using MATEMATICA to find their equilibrium values.
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Nt = Na + Nb
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Here we change the notation a little to two generic strains '''a''' and '''b''' where population '''Na''' produces amino acid '''a''', etc. We are interested in the value of the cell populations in equilibrium, more importantly in the fraction of each population. Thus a change was made to more convenient variables: the sum of both population and a strain's molar fraction.
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Xa = Na / Nt
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{N_a  ;  N_b} → { N_t  ,  x_a }
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Only 3 fixed points were found for the system (See [[Team:Buenos_Aires/Project/ModelAdvance#Appendix | Appendix]]). There are some solutions for which it isn't true that the four variables reach a steady state (SS) or equilibrium. There are initial conditions for which the AA concentrations will not stop growing! So caution is required when assuming steady state.
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Only 3 fixed points were found for the system {N_t, X_a, AA_a, AA_b} (See Appendix)  That means that for some solutions it’s not true that the four variables reach a steady state (SS) or equilibrium. There are initial conditions for which some of the variables will not stop growing!
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*In the first solution we found the culture doesn't thrive, for example because one strain was missing or the initial density was to low.
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*The first solution is clear. The culture will not thrive under certain conditions… to be determined!
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*The second one has no biological relevance since it yields negative concentrations for the Amino Acids.
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*The second one has no physical relevance since it yields negative concentrations for the Amino Acids.
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*The third one is the significant one (see equations below). However for some parameters the concentrations of Nt and AA can be negative, therefore restricting the parameter space in which the model works.  
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*The third one is the significant one. However the concentrations of Nt and AA can be negative; we must work within the region where is positive.
 
[[File:bsas2012-modeling-eqsol3.png]](3)
[[File:bsas2012-modeling-eqsol3.png]](3)
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=== Regulation ===
=== Regulation ===
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Note that the percentage of each population in the community is a function of √((da pb)/(db pa)).
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Note that the fraction of each strain in the community is a function of the AA secretion rates ('''pa''' and '''pb''') and the amount of AA required to "construct" a cell ('''da''' and '''db''').
-
 
+
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In particular
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[[File:bsas2012-modeling-eq4.png]] (4)
[[File:bsas2012-modeling-eq4.png]] (4)
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For positive values of ''d'' and ''p'' the range of the function is (0; 1), so far it’s consistent with what we expect.
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For positive values of '''d''' and '''p''' the range of the function is (0; 1), consistent with what we expect for a fraction.
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The percentages of the strains in the culture in equilibrium are regulated by the production and export of AA – represented in the model through pa and pb; independent of initial conditions!!
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The fraction of each strain in the culture in equilibrium are regulated by the production and export of each AA – represented in the model through '''pa''' and '''pb'''. These fractions are independent of initial conditions! This means that the system auto-regulates itself, as intended. 
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In fact we need only know the ratio between these two parameters to control it:   
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In fact we only need to control the ratio between these parameters to control the culture composition:  
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[[File:bsas2012-modeling-eq5.png]](5)
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[[File:bsas2012-modeling-eq5revised.png]](5)
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The next figure illustrates how the mole fraction x_a varies with the variable ε.
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[[File:bsas2012-modeling-fig1.png]]
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The next figure illustrates how the fraction Xa varies with the variable ε.
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  Figure 1. Mole fraction x_a as a fucntion of the ratio of protein export for values of D=0.1,1,10.
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[[File:bsas2012-modeling-fig1revised.png]]
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The ratio D is fixed for two A.A., this determines the range of percentages of each strain accessible for any selection of biobricks.  
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Figure 1. Fraction Xa as a fuction of the ratio of protein export for values of D=0.1,1,10 (in green, purple and  red).
 +
To programme the percentage we need a second set of biobricks that can sense an external stimulus and transduce this signal  to modify ε. The range of percentages we can control is determined by the range of ε accessible with this second device and the ratio D that is set for any two A.A.
=== Parameters ===
=== Parameters ===
-
The parameter estimation process is shown in xx section(linked)
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The parameter estimation process is detailed in [[Team:Buenos_Aires/Project/Model#Parameter_selection | parameter selection]]
{| class="wikitable"
{| class="wikitable"
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|-
|-
|style="color:white; background-color: purple;"|d trp
|style="color:white; background-color: purple;"|d trp
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|2.630e7
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|2.630e7  
|AA/cell
|AA/cell
|}
|}
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   Table 1.
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   Table 1. Model parameters used for the simulations
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== Model Analysis ==
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=== Parameter selection ===
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The standard procedure when dealing with a non-linear system is to find the fixed points(FP) and later analyze their stability; same as with a linear system (**). The goal is to determine all possible trajectories, i.e. for any initial conditions how do the variables evolve?
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How is this behavior governed by the model’s parameters?
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We estimatated values for all the parameter in the model, doing dedicated experiments or using values from the literature. This allowed to check the feasibility of the system.
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How do the trajectories change as the parameters vary? Does any new FP or close orbit appear?
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*The '''Kaa''' found in the equations are related to the concentration of AAs in the medium required to reach half the maximum rate of growth. Curves of OD600 vs [AA](t=0) were measured experimentally for each amino acid and the data fitted to a Hill equation (6) using MATLAB’s TOOLBOX: Curve Fitting Tool.
 +
[[File:Bsas2012-modeling-eqfit.png|200px]](6)
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There was a major difficulty in the analysis of the model because of the dimension of the problem - 4 variables. The analytical results yielded by the usual tools (nullclines, linearization) could not be simplified so that all the relevant properties of the system could be understood in terms of the parameters.  
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This data from [[Team:Buenos_Aires/Results/Strains#Growth dependence on the Trp and His concentrations | experimental results]] and the best fit obtained for each are shown below in Figure 2.
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We obtained the 4x4 Jacobian - evaluated in the FP - and later its eigenvalues. The expression found for them, a function of the relevant parameters, is too long and can’t be simplified in a way that reveals changes in stability, etc.  
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[[File:Bsas2012-modeling-fig_aux1.png]]
 +
[[File:Bsas2012-modeling-fig_aux2.png]]
 +
Figure 2. Single strain culture density after over night growth vs the initial concentration of AA in the medium (dilutions 1:X).
 +
The best fit to the data is also shown.
 +
The values taken from the fits are 
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Besides that, we couldn’t do a visual analysis of the nullclines to have a qualitative idea of what goes on.
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His-  :
 +
      K_dil =    0.2255 
 +
      n =          1.52
 +
  R-square: 0.997 
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A very common way to proceed in such a case is to reduce the model, ideally to a two-dimensional system. Sometimes one of the variables varies so slowly or that much faster than the ones we are interested in, that it can be considered constant when studying the dynamics of the rest. This is a standard procedure in the study of chemical reactions called Quasi Steady State Approximation.  
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Trp- :
 +
      K_dil =    0.0469
 +
      n =          1.895
 +
  R-square: 0.998
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# QSSA: Knowing that initially the variation of AA in the medium is much faster than the yeast‘s populations.
 
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[[File:bsas2012-modeling-eq15.png]](6)
 
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We supposed that it reaches SS before the other. Therefore taking  dAAj/dt=0  we can rewrite the AminoAcids as a function of (Na,Nb) in a reduced  2x2 model for the evolution of the populations.
 
-
 
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[[File:bsas2012-modeling-eq16.png]](model_aprox)
 
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Thus a phase portrait for the nullclines and trajectories was drawn.
 
   
   
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[[File:bsas2012-modeling-fignull.png]]
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The results were then converted to the units chosen for the simulations.
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Figure 2
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K = K_dil * [AA 1x] * Navog / (Molar mass AA)
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Although the reduced model presents the same fixed point, it’s a saddle point.
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where [AA 1x] = 0.02 mg/ml is the concentration of the AA (His or Trp) in the 1x medium and Navog is Avogadro's number. We get
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What does that mean? Pick any point for the plane; those are your initial conditions for the concentration of each population. Now to see how the community evolves with time follow the arrows. No matter how close to the FP we start, the community is overtaken by one of the two strains.  
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*K trp= .0469* 5.88e16 AA / ml =  2.76e15 molecules/ml
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* The stable FP are {Na-> Nt, Nb-> 0} and {Na-> 0, b->Nt}.  
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*K his= 0.2255* 7.8e16  AA / ml = 1.76e16 molecules /ml
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* Only by taking i.c. along the stable manifold –direction marked between the red arrows for clarification in Figure 2 – do we reach the FP we wanted as t-> inf.
 
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This is not a representation of auxotrophic community!
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The competition within a strain and with the other were taken as equal. The system's general ''carrying capacity'' considered in the model is the one often used here, in a lab that works with these yeast strains:
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Indeed a linearization of the set of ODEs around the FP yields instability regardless of the parameters. One of them is always positive.  
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**Cc=  3e7  cell/ml.
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{2death,death (1-√((pb pa)/(db da))/death)}
 
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There is no agreement between the predictions found here and the simulations; we won’t pursue this reduction any further.
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*The death rate was taken between 3 and 7 days.
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# We also noticed that the Hill coefficients are similar to one. So we explored the possibility of approximating them to one; is there a  relevant change or not?
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===Simulations===
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*The rate of “production and export” was given an upper bound value (P_MAX) of 1% of the maximum estimated number of protein elongation events (peptidil transferase reactions) for a yeast cell per hour.  That is, if all the biosynthetic capacity of the cell was used to create the AA rich exportation peptide, the export rate would equal P_MAX. Of course this is not possible because the cell has to do many other things, therefore we considered 1% of P_MAX as a reasonable upper bound for '''p'''.
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We’ve relied on numerical simulations to explore possible behaviors.
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To run the simulations all that is left is to choose values {p, ε, i.c.}.
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From [http://www.biomedcentral.com/1752-0509/2/87| von der Haar 2008]  we get an estimate for the total number of elongation events (peptidil transferase reactions):  6e6  1 / cell sec [ ]
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Since this is merely a framewok we have taken values p  from a log scale from -3 to 1.  
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* P_MAX = 2.16e8 1/ cell hour
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ε was ranged from 0.0001 to 4; which varies the fraction of each population from 10% to 90%.  
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These parameters will be regulated in the simulation by ε and '''p''', to alter the fraction between populations.
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Cells’ initial concentration: dilutions from 1/10000 to 1/100 of the capacity (Cc).
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*p trp = '''p'''* '''ε'''*2.16e8  AA / cell hour
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AAs’ initial concentration: null.
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*p his = '''p'''*2.16e8  AA / cell hour
 +
where '''p''' < 1 controls the fraction of the P_MAX value and &epsilon; the ratio between the export of each amino acid.
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=== Die or thrive?===
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'''d''' is the total number of amino acids in a yeast cell –same as the ones needed to create a daughter -per cell:
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Some basic properties were observed by simplifying the system; we wanted to check that our model was sound.
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[[File:bsas2012-modeling-eq6.png]](7)
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'''d''' = # A.A. per cell = (mass of protein per yeast cell ) * relative abundance of AA * Navog / AA's molar mass.
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where we can define [[File:bsas2012-modeling-eq7.png]]
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*d trp= 2.630e7 AA / cell
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[[File:bsas2012-modeling-fig3.5.png]]
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*d his= 6.348e8 AA / cell
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Figure 3.
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==Numerical Simulations==
 +
:Initially we relied on numerical simulations (NS) performed with [http://www.mathworks.com/products/matlab/| MATLAB] to explore possible behaviors of the model, ODEs rarely have solutions that can be expressed in a closed form.
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The condition for culture growth is  &lambda;  >  1.  This ratio &lambda; between two rates determines whether the system is able to produce and absorbed the nutrients needed for the cell cycle before its death. If it doesn’t hold the culture dies, ie the model predicts a negative amount of cells; which doesn’t make sense.
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:To run the simulations all that is left is to choose values {p, &epsilon;, initial conditions}.
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The other condition is related to extracellular AAs. Let’s define 
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[[File:bsas2012-modeling-eq8.png]](8)
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* Since this is merely a framewok we have taken values '''p''' from a log scale from -3 to 1.  
 +
* '''&epsilon;''' was ranged from 0.0001 to 4; which varies the fraction of each population from 10% to 90%.
 +
* Cells initial concentration (i.c.c.): dilutions from 1:10000 to 10x of the carrying capacity (Cc).
 +
* Amino acids (AA) initial concentration: null, unless stated otherwise.
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The concentration of extracellular AAs  is positive only if &delta; < 1. In fact as &delta; -> 1the concentration -> infinity because &delta; = 1 is a vertical asymptote.  This ratio &delta; between two rates is related to our proposed  regulation; if the amount of nutrients produced  exceeds the required one for a 90 minutes cycle, the medium is saturated. Then the regulation fails.  
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== Die or thrive?==
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:Some basic properties were observed by simplifying the system; we wanted to check that our model was sound. We considered a "symmetric interaction" in which both secretion rates and amino acid requirement were equivalent.  
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===Parameter Space and Solutions ===
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[[File:bsas2012-modeling-eq6.png]](7)
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Uniting the conditions for &lambda; and &delta; we see that '''p''' and &epsilon  must be bound for our solution to make sense:
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-
[[File:bsas2012-modeling-eq14.png]](9)
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where we can define
 +
[[File:bsas2012-modeling-eq7.png]] (8)
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[[File:bsas2012-modeling-fig3.png]]
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:This parameter &lambda; has an intuitive interpretation. The ratio '''d'''/'''p''' is the time it takes a cell to export enough AA for a cell of the other strain to build a daughter. On the other hand, 1/death is the average life span of a cell. Therefore &lambda; is the ratio between the "life span" of a cell and the time it takes to export sufficient AA to build a cell, thus &lambda; represents the amount of cells that can be constructed with the AA secreted by a cell in its lifetime. If this value is grated than one (&lambda; &gt;  1) the culture grows, otherwise it dies (This is completely analogous to the "force of infection" as defined in epidemiology).    
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  Figure 4. Regions in the parameter space that present different types of solutions. Only Region III both conditions are true.
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-
Now, let’s classify these regions numerically to see whether the ideas we just presented are supported. Taking random values from each region we found the following:
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:Keeping '''d''' and '''death''' constant, the total steady state number of cells in the culture depends on the secretion rate '''p''' as shown in the following figure 3. There is a threshold value p given by &lambda;=1; with lower values the culture dies.
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I . The culture doesn’t grow, it decays with varying velocities for any i.c.. This is consistent with solution 1, where NT =  0.
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[[File:bsas2012-modeling-fig3revised.png]]
 +
Figure 3. Total number of cells in the mix vs the strengh '''p''' of the production and export of AAs for a set of parameters Cc,d, death.
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II. The extracellular AA keep growing and the medium is saturated. The culture reaches equilibrium for every i.c. no matter how low.
 
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The growth rate tends to the theoretical 90 min and the AA dependence disappears from the equations for each population. Remarkably the mole fraction is still independent of initial concentration of cells.
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:Another condition is related to extra-cellular concentration of amino acids. Looking at the equations for the evolution of the AA in the model, we can identify the parameter &delta; as follows
 +
[[File:bsas2012-modeling-eq8.png]](9)
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III. The four variables reach SS. The mole fraction is solely dependent on &epsilon; according to the formula on eq [].  
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: As before, the ratio '''&radic;(da db)/&radic;(pa pb)''' is the average time required for a cell to export enough AA to construct another cell. '''1/kmax''' is the time it takes a cell to replicate in optimal growth conditions. &delta; can be interpret as the amount of cells that can be made with the material secreted a by a single cell before it divides (in optimal conditions). If &delta; &gt; 1 the amount of nutrients produced exceeds the consumption, the medium gets saturated with AA and the regulation fails. So the system is auto-regulated only if &delta; &lt; 1.
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The total population of the community (Nt) is the same as in Region II.
+
-
This is the region where the auxotrophy leads to regulation of the community.
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:Combining these two conditions (&lambda;  &gt;  1 and &delta; &lt; 1), we conclude that the production rate has to be  in a defined region for the system to work. To low and the culture dies out, to big and it gets out of control.
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However the culture doesn’t grow for all i.c. – will get back to this point shortly.
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== Parameter Space and Solutions ==
 +
:Uniting the conditions for &lambda; and &delta; we see that in general '''p''' and &epsilon;  must be bound for our solution to make sense (i.e. all concentrations &gt; 0):
-
We had postulated that the solution doesn’t hold in II and expected a dramatic change in the system’s behavior. However there is a smooth transition between II and III. For a fixed &epsilon; we can step from III into II by increasing ‘’p’’. Once we cross the threshold we notice that the total population of the community does not vary from one region to the next and though the AA continues to grow over time (and accumulate) it does so slowly (compare to what is expected for a vertical asymptote).  More so, the formula [] is a good approximation for the mole fraction in the immediate area; the difference between the two increases gradually.
+
[[File:bsas2012-modeling-eq14.png]](10)
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There’s no pointed change caused by the asymptote, in fact the two regions can’t be distinguished experimentally; so small is the difference between the mole fractions of a strain in each side of the frontier.
 
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An important test: under which initial conditions do the populations thrive or die has to be understood in terms of Figure 4, not merely the conditions associated with Figure 3.
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[[File:bsas2012-modeling-fig3.png]]
 +
Figure 4.  Regions in the parameter space that present different types of solutions. Only in Region III the system works as intended.
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We noticed that the culture’s survival in region III is also dependent on the Ks even though is not explicit in ay e formula so far. Rather for a fixed value of K there is a i.c. threshold below which the culture doesn’t prosper.
 
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This is reasonable considering that it introduces another time scale. Not only have the cells to produce and absorb the required amounts of AA before they die, their ability to grow is restricted by the concentration of AA in the medium. The cells must produce enough amino acids so that [AA] is of the order of  K in the same time frame to insure their prosperity. 
 
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Why is this important? Originally we took both Hill coefficients equal to one, is close enough, right?
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:Now, let's classify these regions numerically to see whether the ideas we just presented are supported. Taking random values from each region we found that all four variables are always positive, but each region is associated with a different kind of solution. The conditions shape the behavior and not the existence of a solution.
-
Well, the whole system didn’t seem viable; it took over 100 hours for the fastest cultures to grow. Now using the fitted coefficients; the time scale becomes reasonable again.
+
:Taking AA(t=0)=0:
 +
::I. The culture doesn't grow, it decays with varying velocities for any initial concentration of cells (i.c.c.). This is consistent with solution 1, where Nt= 0 as t &rarr; &infin;. This is the &lambda; &lt; 1 case.
-
[[File:bsas2012-modeling-fig5.png]]
 
-
  Figure 5. Evolution in time of two cultures that only differ in the i.c. for n_ hill =1 (left)and  n_hill >1 (right).  
+
::II. The extracellular AA concentration keeps growing and the medium gets saturated. The culture reaches equilibrium for every i.c.c. no matter how low, but there is no regulation of the composition of the culture. This is the &delta; &gt; 1 case.  
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When n_ hill =1 only the one with higher i.c. reaches a non-trivial steady state (ss). The other slowly dies. However for n_ hill =1.5 there is no clear difference between the cases with high and low i.c.. Note that it  reaches SS faster as well.
+
::The growth rate tends to the theoretical 90 min and the AA dependence disappears from the equations for each population. Remarkably the fraction of each strain is still independent of the i.c.c.
-
The survival of an auxotrophic community is dependent on both the apparent dissociation constants  (Ks) and the Hill coefficient as shown clearly on Figure 5. There are two effects here:
 
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Shorter time scale to SS.
+
::III. The four variables reach SS. The mole fraction is solely dependent on &epsilon; according to the formula on equation (5). The total population of the community (Nt) is the same as in Region II.
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Larger subspace of i.c. that lead to SS.
+
::'''This is the region where the auxotrophy leads to regulation of the community.''' However the culture doesn’t grow for all i.c.c. – will get back to this point shortly.
-
So when creating an auxotrophy, pick a nutrient with an appropriate Hill coefficient and K. Check the range and stability of your model estimating the values  of the rest of the parameters.
+
===Conclusions===
 +
::Some conclusions we've drawn from these numerical simulations:
-
Our case is similar to the one shown in figure 5 with both hill coefficients > 1. Region III is less sensitive to i.c.
+
# We had postulated that the solution doesn't hold in region II and expected a dramatic change in the system’s behavior. However there is a smooth transition between regions II and III. For a fixed &epsilon; we can go from region III to region II by increasing '''p'''. Once we cross the threshold we notice that the total population of cells (Nt) doeen't vary from one region to the next. Though the AA continues to grow over time (and accumulate) it does so slowly; more so than we'd expect for a vertical asymptote.
 +
# More so, the formula (5) is a good approximation for the fraction of each strain in the region II close to the limit with region III; the difference between the two increases gradually as '''p''' gets further away from this threshold value. The formula is not true in II; thus regulation fails.
 +
# There is no dramatic change caused by the asymptote, in fact the two regions can't be distinguished experimentally through a strain's fraction measurement in each side of the frontier; so small is the difference between the values across.
 +
# Taking into account that the threshold is based on estimations; perhaps is safer to choose points well inside Region III to ensure proper regulation. This has a cost, for a fixed percentage the production and export of AAs is now lower and the dynamic of the culture slower.
-
=== Relaxation time===
+
:The original purpose of this analysis was to undertand '''under which initial conditions do the populations thrive or die'''. This has to be analyzed in terms of Figure 4, not merely the conditions for &lambda;.
-
Not only are we interested in predicting the fraction of each strain in the culture, we want it within a time frame.
+
:We noticed that the culture's survival in region III also depends on the values of '''Kaa''', even though is not explicit in any formula so far. Remember that '''Kaa''' is the concentration of an AA in the medium required for half maximal growth rate. For a fixed value of '''Kaa''' there is a critical initial concentration of cells below which the culture doesn't prosper. This effect is important when the percentage desired is extreme (close 100%), where the '''p''' value required for the strain gets really low.  
   
   
-
Let’s call &Tau; the time it takes one of the populations to reach 90% of its SS value (relaxation time). We rely again on numerical simulations to obtain those set of parameters where &Tau; is less than 65 hours, for example.
+
:This is reasonable considering that another time scale is introduced. Not only have the cells to produce and absorb the required amounts of AA before they die, they also need to modify the AA concentration of the medium. The cells must produce enough amino acids so that the concentration of AA approximates '''Kaa'', before the original cells die out. If the initial cell density is to low, this won't happen.
-
Creating a mesh we explored the parameter space (p, &epsilon;) for different initial conditions to visualize when the model’s prediction and the numerical result of the simulations differ in an amount lower than the given error; before 65 hours. We won’t limit the mesh to Region III only.
 
-
We’ve taken 5% of the value as error; considering it an accurate estimation of an experimental measure’s precision.
+
== Lag Time ==
-
[[File:bsas2012-modeling-fig6.png]]
+
{|
 +
|Let's finally look at some numerical simulations to see how the system evolves in time. We present a sample simulation with parameters from Region III to the rigth.  There's a new characeristic that our previous analysis didn't show: the system reaches the desired steady state, however there will be significant lag phase.
-
Figure 6.
+
We found numerically that it's related to time it takes for the Amino Acids in the medium to reach a certain concentration; one similar to the parameters '''Kaa''' and '''Kbb''' respectively. To this effect we note that it's possible to reduce this ''lag time'' by
-
=== Oscillations===
+
*Using higher initial concentration for each population.
-
One interesting effect appears if the initial concentration of AA in the medium are of the order of the Ks.  Damp oscillations are observed!!
+
*Lowering parameters '''Kaa''' and '''Kbb'''.
-
   
+
-
[[File:bsas2012-modeling-fig7.png]]
+
-
  Figure 7.
+
Given the relationship between these parameters and the AA absortion rates, we deviced a way to facilitate absortion.  
 +
We decided to work with
-
== Variable Transformation ==
+
'''Troyan Peptides'''
 +
|
 +
[[File:timeevol.jpg|350px]]
 +
|-
 +
|This lag time was quantified using the slowest strain's third cell division as a lag time.
-
Let’s review what we’ve learned so far from our system.
+
The unit of the color scale is hours, it's clear than for lower initial concentrations and export rates
-
 
+
|
-
* Seems stable,
+
[[File:BsAs2012_11LagK.jpg.jpg|350px]]
-
 
+
|
-
* capable of oscillations,
+
|}
-
 
+
-
* there are two cases with a stable population and a defined mole fraction for each strain, one where the AAs are in SS and one where they aren’t.
+
-
 
+
-
* The set of i.c. for which the culture thrives is dependent on the Ks.
+
-
 
+
-
Yet all this information is not so trivial to find – especially the oscillations discovery was a fluke. There had to be a transformation where all these properties were more accessible or evident.
+
-
 
+
-
What we did so far was to write the results in more convenient variables.
+
-
 
+
-
Now we will try to find a transformation to a new set of variables, whose evolution is simpler to study and write a new set of ODEs for them.
+
-
 
+
-
For instance the AAs only appear as the argument of Hill functions, so why not just work with those rations. We defined:
+
-
 
+
-
[[File:bsas2012-modeling-eqt1.jpg]]
+
-
 
+
-
This is better because now &xi; is constant when:
+
-
 
+
-
# AAa reachs steady state.
+
-
 
+
-
# Na exports more that could Nb possibly adsorb and AAa accumulates in the medium. For AAi >> Ki the ratio &xi; ->  1.
+
-
 
+
-
* The AAs can be zero, so can the new variables. Rewriting the equation we see that each ratio is elevated to 2- (s+1)/s  > 0 for s>1; so the variables are not really in the denominator.
+
-
+
-
* &chi; stop growing (is constant) if and only if &mu; = &xi;, n can’t equal 1 since n=1 doesn’t satisfy the DE for n.
+
-
 
+
-
Here’s the great improvement over the old variables. The SS for  &chi;, &xi; and &mu; is independent of n; i.e. the conditions for which their DE equal zero. This means that the nullclines are independent as well. Now we have a reduction in the system's dimension. Same SS  as 4x4 but this 3x3 system can be analyzed visually,
+
-
 
+
-
* We can represent the vector field  ''eq''  in space and track a trajectory that starts at (&chi;0, &xi;0, &mu;0)
+
-
 
+
-
* We plot the nullclines, see where they meet. 
+
-
 
+
-
[[File:bsas2012-modeling-fig9.jpg]]
+
-
 
+
-
Figure 9. Nullclines
+
-
 
+
-
== Stability of the percentage in equilibrium ==
+
-
 
+
-
Let’s go the linearization of our 3x3 set of ODEs close to the FP. We said that n wasn’t relevant to the location of the fixed point,  but now we have to consider it while examining the eigenvalues.
+
-
 
+
-
Once again the expresion for the three eigenvalues &gamma; is very long. We replace the mayority of the parameters with the value we estimated, and n for its SS value - this is not always the case.
+
-
 
+
-
We are in condition to study the stability of the FP when we vary p_a and p_b. The results are shown vs variables '''p''' and &epsilon; in figure 10.
+
-
 
+
-
The FP is not stable if al least one of the eigenvalues is positive.
+
-
 
+
-
[[File:bsas2012-modeling-fig10.jpg]]
+
-
Figure 10.
+
-
 
+
-
Since the three eigenvalues are negative, it's a stable node for this set of parameters. There are however some points for which there's no value plotted. Taking ('''p''', &epsilon;) from that region we obtain complex eigenvalues.
+
-
 
+
-
 
+
-
 
+
-
[[File:bsas2012-modeling-fig11.jpg]]
+
-
 
+
-
== Parameter selection ==
+
-
 
+
-
We've estimatated values for all the parameter in the model to check is viability.
+
-
 
+
-
*The Ks found in the equations are a measure of the concentration of A.A. in the medium required to reach half the maximum rate of growth. Curves OD vs [AA] were measured experimentally for each amino acid and the data fitted to a Hill equation using MATLAB’s TOOLBOX: Curve Fitting Tool.
+
-
 
+
-
 
+
-
Figure  aux his and  aux trp.
+
-
 
+
-
The values were taken from the fits are 
+
-
 
+
-
His-  :
+
-
      K_dil =    0.3279 
+
-
      ODmax =      62.53 
+
-
      n =          1.243
+
-
  R-square: 0.9376 
+
-
 
+
-
Trp-  :
+
-
      K_dil =    0.1770
+
-
      ODmax =      57.46 
+
-
      n =          1.636 
+
-
  R-square: 0.9905
+
-
 
+
-
+
-
 
+
-
The results were then converted to the units chosen for the simulations.
+
-
 
+
-
K = K_dil * [AA 1x] # Avog / (Molar mass AA), where [AA 1x] = 0.02 mg/ml
+
-
 
+
-
**K trp= 0.1770 * 5.88e16 AA / ml.
+
-
 
+
-
**K his= 0.3279 * 7.8e16  AA / ml.
+
-
 
+
-
 
+
-
*The ''capacity'' of the system taken in the model is the one often used in the lab that works with these yeast strains:
+
-
 
+
-
 
+
-
**Cc=  3e7  #cell/ml.
+
-
 
+
-
 
+
-
*The death rate was taken between 3 and 7 days.
+
-
 
+
-
 
+
-
*The rate of “production and export” was given a top value (P_MAX) 1%  of  the maximum estimated number of elongation events for a yeast cell per hour. Our model does not explicitly include time as a variable, that means that the production is switched ON all the time.
+
-
 
+
-
Elongation events (peptidil transferase reactions):  6e6  1 / cell sec [ref]
+
-
 
+
-
P_MAX = 2.16e8 1/ cell hour
+
-
 
+
-
These parameters will be regulated experimentally ('''&epsilon;''' , ''p'') to alter the fraction between populations.
+
-
 
+
-
**p trp = ''p''* '''&epsilon;'''*2.16e8  aa / cell hour
+
-
 
+
-
**p his = ''p''*2.16e8  aa / cell hour
+
-
 
+
-
where ''p'' < 1 controls the fraction of the top value and '''&epsilon;''' the ratio between  the export of each amino acid.
+
-
 
+
-
 
+
-
*'''d''' is the total number of amino acids in a yeast cell –same as the ones needed to create a daughter -per cell:
+
-
 
+
-
'''d''' = # A.A. per cell = (mass of protein per yeast cell [ref]) * relative abundance of AA * # Avog / AA's molar mass.
+
-
 
+
-
 
+
-
**d trp= 2.630e7 AA / cell 
+
-
 
+
-
**d his= 6.348e8 AA / cell
+
-
 
+
-
== Appendix ==
+
-
 
+
-
=== solutions model1 ===
+
-
Sol2
+
-
 
+
-
Sol3
+
-
 
+
-
=== Jacobian ===
+
-
 
+
-
 
+
-
= References =
+

Latest revision as of 03:13, 27 October 2012

Contents

Modeling a synthetic ecology

To gain insight into the behavior of our crossfeeding design, to understand which parameters of the system are important, and to study the feasibility of the project, we decided to implement a model of the system. We found this interesting per se, as the model we need is that of a microbial community and ecology.

The mathematical modeling of ecological system is an active field of research with a very rich and interesting history; from the works of Lotka and Volterra, who modeled predator-prey dynamics with ordinary differential equations, to Robert May’s chaotic logistic maps. In fact a lot of synthetic systems of interacting bacteria created in recent years were used to study these sort of models.

The crossfeeding model

In the crossfeeding design, each strain produces and releases to the medium an aminoacid the other strain(s) need to grow. The growth rate of each strain depends on the aminoacid (AA) concentration of those AA it can’t produce. In turn these concentrations depend on the abundance of the other strains. Therefore the growth of the strains is interdependent.

For simplicity and to be consistent with the experimental work, we decided to model two interacting strains, one that produces tryptophan (Trp) but requires histidine (His), and another that produces His but requires Trp.

We decided to use ordinary differential equations to model the system. This is a normal approach when studying population dynamics of this kind.

The model has four variables:

  • [Nhis-] : the concentration of histidine dependent (Trp producing) yeast cells, in cells per ml
  • [Ntrp-] : the total amount of tryptophan dependent (His producing) yeast cells, in cell per ml
  • [his]: the concentration of histidine in the medium, in molecules per ml
  • [trp]: the concentration of tryptophan in the medium, in molecules per ml.

To build the model we did the following assumptions:

  • The growth rate of each strain depends on the concentration in the medium of the AA they can’t produce. As the concentration of this AA in the medium increases, so does the growth rate of the strain, until it settles at the growth rate observed in optimal conditions (doubling time of 90 min for yeast).
  • There is a maximal density of cells the medium can support (carrying capacity)
  • Each cell has a fixed probability of dying per time interval
  • Each cell releases to the medium the AA it produces at a fixed rate
  • Each strain only consumes the AA it doesn't produce
  • The system reaches steady state.

Bsas2012-model1.png (model)

In the equation for the temporal evolution of each population we take the growth rate as a Hill function of the amount of AA to which the strain is auxotrophic (it can't produce). The function captures the increase of the growth rate with the relevant AA concentration, until it reaches a plateau at the maximal growth rate, that corresponds to a doubling time of 90 minutes (doubling time = ln(2)/kmax). These functions are of the form

Bsas2012-modeling-eq1.png(1)

where Kaa is the effective concentration of AA at which half maximal growth rate is obtained, and l is the Hill coefficient, that describes how "steep" the curve is.

The term (1-(Nhis+Ntrp)/Cc) of the model accounts for other factors, not explicit in our model that can limit the concentration of cells in a given volume (carrying capacity), enabling the population to reach equilibrium. We included a term related to cell death (- death Ntrp), which is makes biologically sense (cell do die) and is critical for a correct behavior of the model.

The terms in the equations for the evolution of each [AA] in the medium are the fluxes of AA entering and leaving the medium. The first term (Ptrp*Nhis-) is a measure of the AA produced and exported in the form of peptides by a cell, for example the rate of tryptophan secretion by the histidine dependent cell.

The second term is the flux leaving the medium and entering the auxotrophic cells. Each cell has a more or less fixed amount of each amino acid, which we call d. If a cell is to replicate itself and cannot produce and amino acid, it will have to absorb it from the medium. Therefore the flux of AA entering the cell, is d divided by the doubling time τ (the time it takes to "construct" a new cell). One of the hypotheses built into our model is that τ will vary greatly with the concentration of nutrients available. We've used

Bsas2012-modeling-eq2.png(2)

The amount of AA entering the cells that produce it was considered negligible. This means that a cell that produces Trp doesn't import it from the medium in significant quantities. This is likely to hold when AA concentrations are limiting.

Steady State Solution

The 4 equations in the model were equaled to zero and the non-linear algebraic system solved using [http://www.wolfram.com/mathematica/ Mathematica] to find their equilibrium values.

Here we change the notation a little bit to two generic strains a and b where population Na produces amino acid a and requires b, and population Nb produces b and requires a. We are interested in the value of the cell populations in equilibrium, or more importantly the fraction of each population in steady state. Thus a change was made to more convenient variables: the sum of both population and a strain's fraction.

Nt = Na + Nb

Xa = Na / Nt

Only 3 fixed points were found for the system (See Appendix). There are some solutions for which it isn't true that the four variables reach a steady state (SS) or equilibrium. There are initial conditions for which the AA concentrations will not stop growing! So caution is required when assuming steady state.

  • In the first solution we found the culture doesn't thrive, for example because one strain was missing or the initial density was to low.
  • The second one has no biological relevance since it yields negative concentrations for the Amino Acids.
  • The third one is the significant one (see equations below). However for some parameters the concentrations of Nt and AA can be negative, therefore restricting the parameter space in which the model works.


Bsas2012-modeling-eqsol3.png(3)

Regulation

Note that the fraction of each strain in the community is a function of the AA secretion rates (pa and pb) and the amount of AA required to "construct" a cell (da and db).

Bsas2012-modeling-eq4.png (4)

For positive values of d and p the range of the function is (0; 1), consistent with what we expect for a fraction.

The fraction of each strain in the culture in equilibrium are regulated by the production and export of each AA – represented in the model through pa and pb. These fractions are independent of initial conditions! This means that the system auto-regulates itself, as intended.

In fact we only need to control the ratio between these parameters to control the culture composition:

Bsas2012-modeling-eq5revised.png(5)

The next figure illustrates how the fraction Xa varies with the variable ε.

Bsas2012-modeling-fig1revised.png

Figure 1. Fraction Xa as a fuction of the ratio of protein export for values of D=0.1,1,10 (in green, purple and   red).

To programme the percentage we need a second set of biobricks that can sense an external stimulus and transduce this signal to modify ε. The range of percentages we can control is determined by the range of ε accessible with this second device and the ratio D that is set for any two A.A.

Parameters

The parameter estimation process is detailed in parameter selection

Parameters selected
Value Units
kmax 0.4261 1/hr
K his 2.588e16 AA/ml
K trp 1.041e16 AA/ml
n his 1.243
n trp 1.636
Cc 3.0 10^7 cell/ml
death 3 to 7 days
p his p 2.16e8 AA/ cell hr
p trp p ε 2.16e8 AA/ cell hr
d his 6.348e8 AA/cell
d trp 2.630e7 AA/cell
 Table 1. Model parameters used for the simulations

Parameter selection

We estimatated values for all the parameter in the model, doing dedicated experiments or using values from the literature. This allowed to check the feasibility of the system.

  • The Kaa found in the equations are related to the concentration of AAs in the medium required to reach half the maximum rate of growth. Curves of OD600 vs [AA](t=0) were measured experimentally for each amino acid and the data fitted to a Hill equation (6) using MATLAB’s TOOLBOX: Curve Fitting Tool.

Bsas2012-modeling-eqfit.png(6)

This data from experimental results and the best fit obtained for each are shown below in Figure 2.

Bsas2012-modeling-fig aux1.png Bsas2012-modeling-fig aux2.png

Figure 2. Single strain culture density after over night growth vs the initial concentration of AA in the medium (dilutions 1:X).
The best fit to the data is also shown.

The values taken from the fits are

His-  :

      K_dil =     0.2255  
      n =          1.52
 R-square: 0.997  

Trp-  :

      K_dil =     0.0469
      n =          1.895 
 R-square: 0.998


The results were then converted to the units chosen for the simulations.

K = K_dil * [AA 1x] * Navog / (Molar mass AA)

where [AA 1x] = 0.02 mg/ml is the concentration of the AA (His or Trp) in the 1x medium and Navog is Avogadro's number. We get

  • K trp= .0469* 5.88e16 AA / ml = 2.76e15 molecules/ml
  • K his= 0.2255* 7.8e16 AA / ml = 1.76e16 molecules /ml


The competition within a strain and with the other were taken as equal. The system's general carrying capacity considered in the model is the one often used here, in a lab that works with these yeast strains:

    • Cc= 3e7 cell/ml.


  • The death rate was taken between 3 and 7 days.


  • The rate of “production and export” was given an upper bound value (P_MAX) of 1% of the maximum estimated number of protein elongation events (peptidil transferase reactions) for a yeast cell per hour. That is, if all the biosynthetic capacity of the cell was used to create the AA rich exportation peptide, the export rate would equal P_MAX. Of course this is not possible because the cell has to do many other things, therefore we considered 1% of P_MAX as a reasonable upper bound for p.

From [http://www.biomedcentral.com/1752-0509/2/87| von der Haar 2008] we get an estimate for the total number of elongation events (peptidil transferase reactions): 6e6 1 / cell sec [ ]

  • P_MAX = 2.16e8 1/ cell hour

These parameters will be regulated in the simulation by ε and p, to alter the fraction between populations.

  • p trp = p* ε*2.16e8 AA / cell hour
  • p his = p*2.16e8 AA / cell hour

where p < 1 controls the fraction of the P_MAX value and ε the ratio between the export of each amino acid.


d is the total number of amino acids in a yeast cell –same as the ones needed to create a daughter -per cell:

d = # A.A. per cell = (mass of protein per yeast cell ) * relative abundance of AA * Navog / AA's molar mass.

  • d trp= 2.630e7 AA / cell
  • d his= 6.348e8 AA / cell

Numerical Simulations

Initially we relied on numerical simulations (NS) performed with [http://www.mathworks.com/products/matlab/| MATLAB] to explore possible behaviors of the model, ODEs rarely have solutions that can be expressed in a closed form.
To run the simulations all that is left is to choose values {p, ε, initial conditions}.
  • Since this is merely a framewok we have taken values p from a log scale from -3 to 1.
  • ε was ranged from 0.0001 to 4; which varies the fraction of each population from 10% to 90%.
  • Cells initial concentration (i.c.c.): dilutions from 1:10000 to 10x of the carrying capacity (Cc).
  • Amino acids (AA) initial concentration: null, unless stated otherwise.

Die or thrive?

Some basic properties were observed by simplifying the system; we wanted to check that our model was sound. We considered a "symmetric interaction" in which both secretion rates and amino acid requirement were equivalent.

Bsas2012-modeling-eq6.png(7)

where we can define

Bsas2012-modeling-eq7.png (8)

This parameter λ has an intuitive interpretation. The ratio d/p is the time it takes a cell to export enough AA for a cell of the other strain to build a daughter. On the other hand, 1/death is the average life span of a cell. Therefore λ is the ratio between the "life span" of a cell and the time it takes to export sufficient AA to build a cell, thus λ represents the amount of cells that can be constructed with the AA secreted by a cell in its lifetime. If this value is grated than one (λ > 1) the culture grows, otherwise it dies (This is completely analogous to the "force of infection" as defined in epidemiology).
Keeping d and death constant, the total steady state number of cells in the culture depends on the secretion rate p as shown in the following figure 3. There is a threshold value p given by λ=1; with lower values the culture dies.

Bsas2012-modeling-fig3revised.png

Figure 3. Total number of cells in the mix vs the strengh p of the production and export of AAs for a set of parameters Cc,d, death. 


Another condition is related to extra-cellular concentration of amino acids. Looking at the equations for the evolution of the AA in the model, we can identify the parameter δ as follows

Bsas2012-modeling-eq8.png(9)

As before, the ratio √(da db)/√(pa pb) is the average time required for a cell to export enough AA to construct another cell. 1/kmax is the time it takes a cell to replicate in optimal growth conditions. δ can be interpret as the amount of cells that can be made with the material secreted a by a single cell before it divides (in optimal conditions). If δ > 1 the amount of nutrients produced exceeds the consumption, the medium gets saturated with AA and the regulation fails. So the system is auto-regulated only if δ < 1.
Combining these two conditions (λ > 1 and δ < 1), we conclude that the production rate has to be in a defined region for the system to work. To low and the culture dies out, to big and it gets out of control.

Parameter Space and Solutions

Uniting the conditions for λ and δ we see that in general p and ε must be bound for our solution to make sense (i.e. all concentrations > 0):

Bsas2012-modeling-eq14.png(10)


Bsas2012-modeling-fig3.png

Figure 4.  Regions in the parameter space that present different types of solutions. Only in Region III the system works as intended.


Now, let's classify these regions numerically to see whether the ideas we just presented are supported. Taking random values from each region we found that all four variables are always positive, but each region is associated with a different kind of solution. The conditions shape the behavior and not the existence of a solution.
Taking AA(t=0)=0:
I. The culture doesn't grow, it decays with varying velocities for any initial concentration of cells (i.c.c.). This is consistent with solution 1, where Nt= 0 as t → ∞. This is the λ < 1 case.


II. The extracellular AA concentration keeps growing and the medium gets saturated. The culture reaches equilibrium for every i.c.c. no matter how low, but there is no regulation of the composition of the culture. This is the δ > 1 case.
The growth rate tends to the theoretical 90 min and the AA dependence disappears from the equations for each population. Remarkably the fraction of each strain is still independent of the i.c.c.


III. The four variables reach SS. The mole fraction is solely dependent on ε according to the formula on equation (5). The total population of the community (Nt) is the same as in Region II.
This is the region where the auxotrophy leads to regulation of the community. However the culture doesn’t grow for all i.c.c. – will get back to this point shortly.

Conclusions

Some conclusions we've drawn from these numerical simulations:
  1. We had postulated that the solution doesn't hold in region II and expected a dramatic change in the system’s behavior. However there is a smooth transition between regions II and III. For a fixed ε we can go from region III to region II by increasing p. Once we cross the threshold we notice that the total population of cells (Nt) doeen't vary from one region to the next. Though the AA continues to grow over time (and accumulate) it does so slowly; more so than we'd expect for a vertical asymptote.
  2. More so, the formula (5) is a good approximation for the fraction of each strain in the region II close to the limit with region III; the difference between the two increases gradually as p gets further away from this threshold value. The formula is not true in II; thus regulation fails.
  3. There is no dramatic change caused by the asymptote, in fact the two regions can't be distinguished experimentally through a strain's fraction measurement in each side of the frontier; so small is the difference between the values across.
  4. Taking into account that the threshold is based on estimations; perhaps is safer to choose points well inside Region III to ensure proper regulation. This has a cost, for a fixed percentage the production and export of AAs is now lower and the dynamic of the culture slower.
The original purpose of this analysis was to undertand under which initial conditions do the populations thrive or die. This has to be analyzed in terms of Figure 4, not merely the conditions for λ.
We noticed that the culture's survival in region III also depends on the values of Kaa, even though is not explicit in any formula so far. Remember that Kaa is the concentration of an AA in the medium required for half maximal growth rate. For a fixed value of Kaa there is a critical initial concentration of cells below which the culture doesn't prosper. This effect is important when the percentage desired is extreme (close 100%), where the p value required for the strain gets really low.
This is reasonable considering that another time scale is introduced. Not only have the cells to produce and absorb the required amounts of AA before they die, they also need to modify the AA concentration of the medium. The cells must produce enough amino acids so that the concentration of AA approximates 'Kaa, before the original cells die out. If the initial cell density is to low, this won't happen.


Lag Time

Let's finally look at some numerical simulations to see how the system evolves in time. We present a sample simulation with parameters from Region III to the rigth. There's a new characeristic that our previous analysis didn't show: the system reaches the desired steady state, however there will be significant lag phase.

We found numerically that it's related to time it takes for the Amino Acids in the medium to reach a certain concentration; one similar to the parameters Kaa and Kbb respectively. To this effect we note that it's possible to reduce this lag time by

  • Using higher initial concentration for each population.
  • Lowering parameters Kaa and Kbb.

Given the relationship between these parameters and the AA absortion rates, we deviced a way to facilitate absortion. We decided to work with

Troyan Peptides

Timeevol.jpg

This lag time was quantified using the slowest strain's third cell division as a lag time.

The unit of the color scale is hours, it's clear than for lower initial concentrations and export rates

BsAs2012 11LagK.jpg.jpg