Modeling - mutual repressor switch

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

Deterministic model of the mutual repressor switch

The model

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

• 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:

• [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

Simulation results are shown as reporter (BFP, mCitrine) concentrations – 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 (signal 1 and signal 2), modeled as a unit step present for a certain duration of time. Each signal was used to induce a different stable state. The presence of signal 1 caused PIP:KRAB to dissociate from promoter 3 binding site and activated transcription. The presence of signal 2 caused E:KRAB to dissociate from promoter 4 binding site and activated transcription.

Initial species' concentrations were 0 for all simulations.

Detailed parameter values for each simulation can be found in MATLAB files here.

Simulation set 1 - switching between stable states

The purpose of this simulation was to show that conditions exist in deterministic setting where the switch can assume each of the two stable states. Functional cooperativity of each transcription factor (TF) was initialy taken as 2.5. Product production:degradation ratio was equal to 100 (i.e. maximal possible expression level was 100). Ideal circumstances with zero leaky expression were assumed. Figure 1 shows the result of this simulation, demonstrating bistability. The following state-switching scenario was used:

• at time = 0, signal 1 was introduced (signal 1 = 1) to induce the first stable state, depicted by a high level of BFP;
• at time = 100, signal 1 was removed from the system (signal 1 = 0); high-BFP state persisted even after that, meaning a stable state was achieved;
• at time = 400, signal 2 was introduced (signal 2 = 1) to induce the second, high-mCitrine stable state;
• at time = 500, signal 2 was removed from the system (signal 2 = 0); high-mCitrine state persisted, meaning a stable state was achieved;
• at time = 800, signal 1 was re-introduced (signal 1 = 1) to show that a transition to high-BFP state could follow a high-mCitrine state;
• at time = 900, signal 1 was removed (signal 1 = 0); high-BFP stable state persisted.