Modeling - mutual repressor switch

1. Deterministic model
2. Stochastic model
   • Summary
   • The model
   • Simulation results
3. C#Sim model

Stochastic model of the mutual repressor switch

Stochastic simulation revealed that the mutual repressor switch exhibited bistability only as long as there was no leaky expression present. Otherwise, cooperativity higher than 1 was required. Higher cooperativity improved leaky expression tolerance, but only to a certain threshold. Higher leaky expression resulted in lower stable-state levels.

The model

The basis for the stochastic simulation of the mutual repressor switch was the following set of reactions that describe the dynamics of the switch:

Here:

• Pro1 is construct 1 promoter (i.e. promoter 1 - constitutive);
• Pro2 is construct 2 promoter (i.e. promoter 2 - constitutive);
• Pro3 s construct 3 promoter (i.e. promoter 3 - constitutive);
• Pro4 is construct 4 promoter (i.e. promoter 4 - constitutive);
• Pro5 is construct 5 promoter (i.e. promoter 5 - constitutive);
• Ind1 Is inducer 1 (i.e. signal 1), used to induce stable state 1;
• Ind2 is inducer 2 (i.e. signal 2), used to induce stable state 2.

[multi] is a variable, equal to a degree of oligomerization and is used to model cooperativity. Normal gene expression means constitutive gene expression, occuring when no repressor is bound.

When inducer 1 is present, it binds PIP:KRAB and forms a complex, denoted as Ind1.PIP:KRAB. This complex is referred to as inactive PIP:KRAB, meaning PIP:KRAB that cannot bind to the promoter 3. When inducer 2 is present, it binds E:KRAB and forms a complex, denoted as Ind2.E:KRAB. This complex is referred to as inactive E:KRAB, meaning E:KRAB that cannot bind to the promoter 4.

Simulation results

Simulation results are shown as concentrations of reporter for different states (BFP, mCitrine) indicating one of the two states as a function of time. No specific units were used, hence no absolute interpretation of the results' values in terms of units is in place. Switching between states was achieved using two signals (inducers) that were introduced into the system at appropriate time. The signals were modeled as a high number of molecules that triggered state induction and were removed at an appropriate time.

Cooperativity was modeled as a degree of oligomerization, with repressor oligomers binding to a promoter.

Protein production:degradation rate ratio was set to 10. Promoter unbinding (e.g. bound transcription factor unbinding from the target binding site) reactions were assumed to be 100 times slower than promoter binding reactions, which is in the range of typical transcription factors. Initial concentrations of proteins were 0 for all simulations.

Detailed parameter values for each simulation can be found in the corresponding simulation files that can be found here.

The purpose of the first simulation was to show that the mutual repressor switch can exhibit bistability in a stochastic environment. No leaky expression was assumed, and no cooperativity (i.e. no TAL reprssor oligomerization). The following state-switching scenario was used in all stochastic simulations:

• at time = 500, signal 2 was introduced, inducing stable state 2 (high mCitrine state); the signal gradually degraded, but stable state was preserved even after degradation;
• at time = 2500, signal 1 was introduced to induce stable state 1 (high BFP state); the switch remained in a stable state after signal degradation;
• at time = 4500, signal 2 was introduced again, switching the system back to stable state 2;at time = 6500, signal 1 was introduced again, switching the system to stable state 1.
• at time = 6500, signal 1 was introduced again, switching the system to stable state 1.

As shown in Figure 1, bistability was exhibited for this scenario, despite no cooperativity. However, as soon as sufficient leaky expression (e.g. 0.03 compared to protein production rate of 1) was introduced for each gene, bistability was lost (Figure 2).

Figure 1. Mutual repressor switch exhibiting transitions between stable states. No cooperativity and zero leaky expression were assumed here.

Figure 2. Mutual repressor switch exhibited no bistability when no cooperativity was used together with a relatively low leaky expression of 0.03.

For the switch to exhibit bistability in the presence of leaky expression, higher cooperativity values were required. Figure 3 shows bistable behavior of the switch for the cooperativity of 3 in the presence of leaky expression. Higher cooperativity led to higher leaky expression tolerance, but only to a certain leaky expression threshold, depending on other parameter values, such as production and degradation rates. Higher leaky expression also lowered the reached stable-state levels. The threshold for maximal cooperativity allowing bistability depended, just like leaky expression threshold, on other parameter values, such as production and degradation rates. Figure 4 shows that for very high cooperativity of 10, no bistability was observed at leaky expression rate of 0.05.

Figure 3. Bistability was obtained for cooperativity equal to 3 and leaky expression equal to 0.05.

Figure 4. Bistability was lost for cooperativity equal to 10 and leaky expression equal to 0.05.

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