Team:Slovenia/ModelingPositiveFeedbackLoopSwitch

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Modeling - positive feedback loop switch

  1. Deterministic model
  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;
  • Kr 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;
  • kBFP is BFP production rate from construct 1 (i.e. production rate when construct 1 promoter is active);
  • kbBFP is basal BFP production rate from construct 1 (i.e. production rate when construct 1 promoter is inactive);
  • degBFP is BFP degradation rate;
  • kcit is mCitrine production rate from construct 2 (i.e. production rate when construct 2 promoter is active);
  • kbcit is basal mCitrine production rate from construct 2 (i.e. production rate when construct 2 promoter in inactive);
  • degcit is mCitrine degradation rate;
  • k2AKR is TAL-A:KRAB production rate from construct 2;
  • kb2AKR is basal TAL-A:KRAB production rate from construct 2;
  • k4AKR is TAL-A:KRAB production rate from construct 4;
  • kb4AKR is basal TAL-A:KRAB production rate from construct 4;
  • degAKR is TAL-A:KRAB degradation rate;
  • k1BKR is TAL-B:KRAB production rate from construct 1;
  • kb1BKR Is basal TAL-B:KRAB production rate from construct 1;
  • k3BKR is TAL-B:KRAB production rate from construct 3;
  • kb3BKR is basal TAL-B:KRAB production rate from construct 3;
  • degBKR is TAL-B:KRAB degradation rate;
  • kPIP is PIP:KRAB production rate;
  • degPIP is PIP:KRAB degradation rate;
  • kE is E:KRAB production rate;
  • degE is E:KRAB degradation rate.

See model derivation for details.

Simulation results


Next: Stochastic model of the positive feedback loop switch >>