Team:Slovenia/ModelingPositiveFeedbackLoopSwitch
From 2012.igem.org
Modeling - positive feedback loop switch
- Deterministic model
- Stochastic model
- 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:- f_{1}, f_{2}, f_{3} and f_{4} 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;
- k_{1}, k_{2}, k_{3} and k_{4} are association constants;
- n_{1}, n_{2}, n_{3} and n_{4} 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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