1. Differentiation
A. Basic Model
Aumptions:
1. Without control elements, A:B=A%:B% in the stable state ??
(1) No control element.
x – concentration of repressor 1(CI)
y – concentration of repressor 2(434)
a1,2 – effective rate of synthesis of x,y b, – cooperativity of repression from
promoter 1(PCI), promoter 2(P434
Figure 1. Nullclines
Parameters
a1=a2=100
b=g=2 (b, g>1)
- three steady states; one unstable, two stable – bistability
- in the case of b, g=1
there is one stable steady state
Figure 2. Bifurcation diagram
Bistability requires:
- a1, a2 do not differ greatly
- b,g are not drastically different
- how “drastic” difference of a1 vs. a2 is tolerated depends on the magnitude of b and g
———————————————————————————————————————————
B. Improved Model
Equation(2) and (3) represent the chemical processes of gene expression.Reactions (4) and (5) are pressing processes of the promoters by forming Pro1r1 (PCI) and Pro2r1(P434), and (6) and (7) reactivate the promoters’ expression ability with inducers Ind1 ( in our program, Ind2=0) and Ind1(pluX). The last two reactions, (8) and (9) are the decay processes of the gene expression products (A, B).n1 and n2 equal to 1. In our simulations, the stochastic of all reactions are equal to 1s-1, except the decay reaction, with a stochastic rate constant of 0.001 s-1.RNAP=50, Pro1=1, Pro2=1 and Ind1=1.(Without control element Ind2=0).
Figure 3. With control element , inducer 1 is present. That is removed about time 100s.
——————————————————————————————————————————
2. Geneguard
Introduction:
This system includes 4 proteins:
1. Toxin1 can sentence the bacteria to death in certain concentration.
2. Toxin2 is another kind of toxin which has the same effect as toxin 1.
3. Antidote1 is the antidote to toxin1.
4. Antidote2 is the antidote to toxin2.
In our system, toxin1 and antidote2’ genes are settled on the genome, antidote1 and toxin2 are on the plasmid, thus two antidote genes can prevent bacteria from death caused by two toxin genes. Like Siamese twins, whenever plasmids and genome are separated (wild type cells get the plasmid or artificial cells lost their plasmid), toxin genes on both side would kill them.
Parameters:
C1, C1’, C2, C2’: the concentration of toxin1, antidote1, toxin2, antidote2
C3, C4: the concentration of compound (toxin1-antidote1), compound (toxin2-antidote2)
T1, T2: the lethal dose of toxin1 and toxin2.
γ1, γ1’, γ2, γ2’: The degradation rate of toxin1, antidote1, toxin2, antidote2.
k1, k1’, k2, k2’: forward/backward rate constant of toxin1 and toxin2 inhibition.
g1, g1’, g2, g2’: generation rate of toxin1, antidote1, toxin2, antidote2.
n1, n2: the minimum/maximum numbers of plasmid in one cell (n will stands for current plasmids’ number in cell).
m1, m2: the ratio of g1 an g1’, the ratio of g2 and g2’.
Description:
This model contains three parts, one is the harmonious situation in our modified cells and the other two are toxin1/toxin2’s transcription and cells’ death. We only take consideration of stable situation.
Wild type cells with our plasmids:
Plasmids lost modified cells:
Harmonious world:
The last two functions have showed the relationship between m1 (m2) and C1 (C2). Thus proper C1 (C2) we want that makes cells grow normally can lead us to proper m1 (m2), provide directions about RBS/promoter plasmids selection.
"
