iGEM 2012

Modeling - positive feedback loop switch

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

Deterministic model of the positive feedback loop switch

Research suggests that bistability is in theory possible without cooperativity (see Cooperativity). Deterministic analysis showed that the positive feedback switch - due to positive feedback loops and competitive binding of activators and repressors – was exhibiting bistability even at very low functional cooperativity values close to 1. The switch proved much more robust than the mutual repressor switch, exhibiting bistable behavior with high stable-state expression levels even when leaky TAL expression was high. As with the mutual repressor switch, increase in cooperativity further improved robustness. However, even at a relatively low cooperativity, the switch was robust enough to suggest that an experimental realization may be possible. Another advantage of the positive feedback loop switch over the mutual repressor switch, according to our deterministic model, was faster transition from one stable state to another after induction. Our experimental results showed bistable behavior of the switch, as predicted by the model.

The model

We can describe the relations for the mutual repressor switch by the following equations. Fractional occupancies of promoters are:

where:

• f₁, f₂, f₃ and f₄ are probabilities of promoters 1 (construct 1), 2 (construct 2), 3 (construct 3) and 4 (construct 4), respectively, being in an active state, resulting in gene expression;
• [TAL-A:KRAB], [TAL-B:KRAB], [PIP:KRAB] and [E:KRAB] are protein concentrations at a given time;
• k₁, k₂, k₃ and k₄ are association constants;
• n₁, n₂, n₃ and n₄ are exponents representing the degree of functional cooperativity;
• K_r is the amount of repressor required for 50% repression of constitutive promoter (equal to 1 in our simulations);

ODEs representing protein production are described by a set of equations:

where:

• [BFP], [mCitrine], [TAL-A:KRAB], [TAL-B:KRAB], [PIP:KRAB] and [E:KRAB] are protein concentrations;
• k_BFP is BFP production rate from construct 1 (i.e. production rate when construct 1 promoter is active);
• kb_BFP is basal BFP production rate from construct 1 (i.e. production rate when construct 1 promoter is inactive);
• deg_BFP is BFP degradation rate;
• k_cit is mCitrine production rate from construct 2 (i.e. production rate when construct 2 promoter is active);
• kb_cit is basal mCitrine production rate from construct 2 (i.e. production rate when construct 2 promoter in inactive);
• deg_cit is mCitrine degradation rate;
• k_2AKR is TAL-A:KRAB production rate from construct 2;
• kb_2AKR is basal TAL-A:KRAB production rate from construct 2;
• k_4AKR is TAL-A:KRAB production rate from construct 4;
• kb_4AKR is basal TAL-A:KRAB production rate from construct 4;
• deg_AKR is TAL-A:KRAB degradation rate;
• k_1BKR is TAL-B:KRAB production rate from construct 1;
• kb_1BKR Is basal TAL-B:KRAB production rate from construct 1;
• k_3BKR is TAL-B:KRAB production rate from construct 3;
• kb_3BKR is basal TAL-B:KRAB production rate from construct 3;
• deg_BKR is TAL-B:KRAB degradation rate;
• k_PIP is PIP:KRAB production rate;
• deg_PIP is PIP:KRAB degradation rate;
• k_E is E:KRAB production rate;
• deg_E is E:KRAB degradation rate.

See model derivation for details.

Simulation results

Figure 2. Degraded bistability of the mutual repressor toggle switch for cooperativity of 1.15. In comparison to Figure 1 - the same state-switching scenario applies - bistability was exhibited, but very low difference between the states was achieved when cooperativity was set to 1.15 (other parameters were identical to parameters of Figure 1 ) and no leaky transcription was present. Increasing the the production rate of E:KRAB and PIP:KRAB slightly increased the levels, but even a 100-fold increase in E:KRAB and PIP:KRAB production rate compared to other protein production rates did not produce significantly higher stable-state levels. Introducing residual TAL expression of as little as 1% resulted in the loss of bistability for the cooperativity of 1.15.

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