Modeling - positive feedback loop switch

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

Stochastic model of the positive feedback loop switch

Stochastic simulation, in agreement with the deterministic analysis, proved that the positive feedback loop switch was more robust than the mutual repressor switch. It exhibited bistability without cooperativity (i.e. cooperativity equal to 1) even for leaky expression that caused the mutual repressor switch to lose bistable behavior. Higher cooperativity further increased the robustness and leaky expression tolerance, allowing for higher expression levels to be reached.

The model

The basis for the stochastic simulation of the positive feedback loop switch was the following set of reactions that describe the dynamics of the switch:

Here:

• Pro1 is construct 1 promoter (i.e. promoter 1 - minimal);
• Pro2 is construct 2 promoter (i.e. promoter 2 - minimal);
• Pro3 is 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);

It is implicitly assumed that when both activator (TAL:VP16) and repressor (TAL:KRAB) bind to the promoter, the effect of the repressor will be more significant.

See the mutual repressor switch stochastic model for additional description.

Simulation results

The same assumptions apply as for the mutual repressor switch simulation. Detailed parameter values for each simulation can be found in the corresponding simulation files that can be found here.

Figure 1 shows the result of the first simulation, where zero leaky expression and no cooperativity were assumed. Bistability was exhibited, just like it was in the deterministic simulation. The state-switching scenario used was the same as for the mutual repressor switch:

• at time = 500, signal 2 was introduced, inducing stable state 2 (high mCitrine state);
• at time = 2500, signal 1 was introduced, inducing stable state 1 (high BFP state);
• at time = 4500, signal 2 was introduced, switching the system to stable state 2;
• at time = 6500, signal 1 was introduced, switching the system to stable state 1.

Figure 1. Positive feedback loop switch transitioning between stable states. No cooperativity and zero leaky expression were assumed.

While the introduction of leaky expression of 0.03 for each gene caused the mutual repressor switch to stop exhibiting bistable behavior, this was not the case for the positive feedback loop switch. Here, while the stable-state levels dropped moderately, bistability was observed in such case – shown in Figure 2 - even with no cooperativity (i.e. cooperativity equal to 1). This was in agreement with the deterministic model.

The simulation showed that the constitutive promoters leakage was more detrimental to bistability than the minimal promoters leakage. The leaky expression tolerance depended on both production and degradation rates (with production to degradation rate ratios too high or too low resulting in no bistable behavior). Figure 3 shows the positive feedback switch exhibiting bistability without cooperativity for minimal promoters leaky transcription of 10% and constitutive promoters leaky transcription of 5%. The mutual repressor switch did not exhibit bistability for these parameters.

When leaky expression was above a certain threshold (depending on other parameter values), cooperativity was required for bistability. Figure 4 shows a case where leaky expression of 0.08 did not result in bistability loss if cooperativity was equal to 2. Increasing leaky expression resulted in lower stable-state levels. Higher cooperativity improved this, allowing high (maximal) levels to be reached. Cooperativity being too high, like for the mutual repressor switch, caused the loss of bistability.

Figure 2. Positive feedback loop switch exhibited bistability when leaky expression was equal to 0.03 – the same value that caused the mutual repressor switch to lose bistability - despite no cooperativity.

Figure 3. Positive feedback loop switch exhibited bistability without cooperativity for minimal promoters leaky transcription equal to 10% and constitutive promoters leaky transcription equal to 5%. Protein production rates (i.e. rates for non-repressed constitutive promoters and activated minimal promoters) were 1.0 and protein degradation rates were 0.15. Increasing protein degradation rates from 0.1 to 0.15 improved the tolerance to leakage.

Figure 4. Bistability was observed for leaky expression of all genes equal to 0.08 as long as high-enough cooperativity was used. Here, cooperativity was equal to 2, with protein production rates equal to 1.0 and degradation rates equal to 0.1.

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