# Team:Slovenia/ModelingPositiveFeedbackLoopSwitch

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The model

The model

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

+

-

+ -

+ -

+ where: 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;
• +
• f1, f2, f3 and f4 are probabilities of constructs’ promoters 1, 2, 3 and 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;
• +
• [TAL-A:KRAB], [TAL-B:KRAB], [TAL-A:VP16], [TAL-B:VP16], [PIP:KRAB] and [E:KRAB] are protein concentrations at a given time;
• -
• k1, k2, k3 and k4 are association constants;
• +
• k1, k2, k3, k4, k5 and k6 are association constants;
• -
• n1, n2, n3 and n4 are exponents representing the degree of functional cooperativity;
• +
• n1, n2, n3, n4, n5 and n6 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);
• +
• Kr is the amount of repressor required for 50% repression of constitutive promoter;
• +
• Ka is the amount of activator required for 50% activation of minimal promoter.
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-

+

+

+

Line 467: Line 468: where: 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);
• +
• [BFP], [mCitrine], [TAL-A:KRAB], [TAL-B:KRAB], [TAL-A:VP16], [TAL-B:VP16], [PIP:KRAB] and [HetAct] are protein concentrations; -
• 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);
• +
• kBFP is BFP production rate from construct 1; -
• 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;
• +
• kbBFP is leaky BFP production rate from construct 1; -
• 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;
• +
• degBFP is BFP degradation rate;
• -
• 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;
• +
• kcit is mCitrine production rate from construct 2; -
• 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;
• +
• kbcit is leaky mCitrine production rate from construct 2; -
• degE is E:KRAB degradation rate.
• + + + +
• degcit is mCitrine degradation rate; +
• + + +
• k2AKR is TAL-A:KRAB production rate from construct 2; +
• + + +
• kb2AKR is leaky TAL-A:KRAB production rate from construct 2; +
• + + +
• k4AKR is TAL-A:KRAB production rate from construct 4; +
• + + +
• kb4AKR is leaky 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 leaky TAL-B:KRAB production rate from construct 1; +
• + + +
• k3BKR is TAL-B:KRAB production rate from construct 3; +
• + + +
• kb3BKR is leaky TAL-B:KRAB production rate from construct 3; +
• + + +
• degBKR is TAL-B:KRAB degradation rate; +
• + + +
• k1AVP is TAL-A:VP16 production rate from construct 1; +
• + + +
• kb1AVP is leaky TAL-A:VP16 production rate from construct 1; +
• + + +
• k3AVP is TAL-A:VP16 production rate from construct 3; +
• + + +
• kb3AVP is leaky TAL-A:VP16 production rate from construct 3; +
• + + +
• degAVP is TAL-A:VP16 degradation rate; +
• + + +
• k2BVP is TAL-B:VP16 production rate from construct 2;
• + + + +
• kb2BVP is leaky TAL-B:VP16 production rate from construct 2; +
• + +
• k4BVP is TAL-B:VP16 production rate from construct 4; +
• + + +
• kb4BVP is leaky TAL-B:VP16 production rate from construct 4; +
• + + +
• degBVP is TAL-B:VP16 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; +
• +

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Simulation results

Simulation results

+

+ Similar simulation methods and assumptions as for the mutual repressor switch were used. Again, switching between states was achieved using two signals, each of which was used to induce a different stable state. The presence of signal 1 caused PIP:KRAB to dissociate from promoter 3 binding site. The presence of signal 2 caused E:KRAB to dissociate from promoter 4 binding site. +

+ +

+ The ratio between the amount of activator required for 50% activation in case of a minimal promoter to the amount of repressor required for 50% repression in case of a constitutive promoter was set to 3. +

+ +

+ 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 deterministic conditions exist where the switch would exhibit stable states. In order to compare this switch with the mutual repressor switch, the same parameter values were used: functional cooperativity of each transcription factor (TF) was equal to 2.5; product production-degradation ratio was equal to 100 (maximal possible expression level); zero leaky expression was assumed. Figure 1 shows the result of this simulation, indicating that stable states are possible. The state-switching scenario used was the same as the one in simulation 1 for mutual repressor switch: +

+
• 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 ce be done from a high-mCitrine state;
• +
• at time = 900, signal 1 was removed (signal 1 = 0); high-BFP stable state persisted.
• + +
+

+ + +

+ + +

+ + + + + +
+ + + +

+ + +

+ Comparing the result to the mutual repressor switch under similar conditions, it is obvious that here, each steady state is reached sooner. For the mutual repression switch, the transition to a recently-induced state – i.e. rise in its expression level – only began after the level of the previous stable state was very low (close to zero). For the positive feedback loop switch, the rise coincided in time with the introduction of the signal that causes the transition. Hence, stable state was reached more quickly. +

+ +

+ The mutual repressor switch exhibited no bistability for low cooperativity values, such as 1.15 (when leaky expression was present) or very low stable-state level bistability (when no or extremely little leaky expression was present). The positive feedback loop switch, however, exhibited bistability with high stable-state levels even at this cooperativity (1.15), as shown in Figure 2. This indicated that the positive feedback loop switch was much less dependent on cooperativity and could exhibit high expression-level bistability even at low cooperativity values. Bistability was exhibited with no noticeable change in reached levels even when PIP:KRAB and E:KRAB production rate was only half of TAL production rates. +

+

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Line 511: Line 667: Line 519: Line 675: +

Simulation set 2 - cooperativity and leaky expression

+

+ As with the mutual repressor switch, higher cooperativity values were found to increase the robustness of the switch. However, the robustness at a given cooperativity was much higher for the positive feedback loop switch. Figure 3 shows that even for a low cooperativity of 1.15, the switch was exhibiting bistability with high-expression levels in the presence of leaky expression of 0.1. Increasing cooperativity to a still relatively low value of 1.85 dramatically increased switch’s capability to retain bistable behavior. In such case, as shown in Figure 4, even a leaky expression of 1.2 for each TAL did not prevent bistability from occuring, although the expression levels were reduced slightly. +

+

This results suggested that positive feedback loop switch was far more robust and leaky expression-tolerant than the mutual repressor switch. +

+ +

+ + +

+ + + + + +
+ + + +

+ +
+ +

+ + +

+ + + + + +
+ + + +

# 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

Fractional occupancies of promoters are:

where:
• f1, f2, f3 and f4 are probabilities of constructs’ promoters 1, 2, 3 and 4, respectively, being in an active state, resulting in gene expression;
• [TAL-A:KRAB], [TAL-B:KRAB], [TAL-A:VP16], [TAL-B:VP16], [PIP:KRAB] and [E:KRAB] are protein concentrations at a given time;
• k1, k2, k3, k4, k5 and k6 are association constants;
• n1, n2, n3, n4, n5 and n6 are exponents representing the degree of functional cooperativity;
• Kr is the amount of repressor required for 50% repression of constitutive promoter;
• Ka is the amount of activator required for 50% activation of minimal promoter.

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

where:
• [BFP], [mCitrine], [TAL-A:KRAB], [TAL-B:KRAB], [TAL-A:VP16], [TAL-B:VP16], [PIP:KRAB] and [HetAct] are protein concentrations;
• kBFP is BFP production rate from construct 1;
• kbBFP is leaky BFP production rate from construct 1;
• degBFP is BFP degradation rate;
• kcit is mCitrine production rate from construct 2;
• kbcit is leaky mCitrine production rate from construct 2;
• degcit is mCitrine degradation rate;
• k2AKR is TAL-A:KRAB production rate from construct 2;
• kb2AKR is leaky TAL-A:KRAB production rate from construct 2;
• k4AKR is TAL-A:KRAB production rate from construct 4;
• kb4AKR is leaky 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 leaky TAL-B:KRAB production rate from construct 1;
• k3BKR is TAL-B:KRAB production rate from construct 3;
• kb3BKR is leaky TAL-B:KRAB production rate from construct 3;
• degBKR is TAL-B:KRAB degradation rate;
• k1AVP is TAL-A:VP16 production rate from construct 1;
• kb1AVP is leaky TAL-A:VP16 production rate from construct 1;
• k3AVP is TAL-A:VP16 production rate from construct 3;
• kb3AVP is leaky TAL-A:VP16 production rate from construct 3;
• degAVP is TAL-A:VP16 degradation rate;
• k2BVP is TAL-B:VP16 production rate from construct 2;
• kb2BVP is leaky TAL-B:VP16 production rate from construct 2;
• k4BVP is TAL-B:VP16 production rate from construct 4;
• kb4BVP is leaky TAL-B:VP16 production rate from construct 4;
• degBVP is TAL-B:VP16 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

Similar simulation methods and assumptions as for the mutual repressor switch were used. Again, switching between states was achieved using two signals, each of which was used to induce a different stable state. The presence of signal 1 caused PIP:KRAB to dissociate from promoter 3 binding site. The presence of signal 2 caused E:KRAB to dissociate from promoter 4 binding site.

The ratio between the amount of activator required for 50% activation in case of a minimal promoter to the amount of repressor required for 50% repression in case of a constitutive promoter was set to 3.

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 deterministic conditions exist where the switch would exhibit stable states. In order to compare this switch with the mutual repressor switch, the same parameter values were used: functional cooperativity of each transcription factor (TF) was equal to 2.5; product production-degradation ratio was equal to 100 (maximal possible expression level); zero leaky expression was assumed. Figure 1 shows the result of this simulation, indicating that stable states are possible. The state-switching scenario used was the same as the one in simulation 1 for mutual repressor switch:

• 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 ce be done from a high-mCitrine state;
• at time = 900, signal 1 was removed (signal 1 = 0); high-BFP stable state persisted.

Comparing the result to the mutual repressor switch under similar conditions, it is obvious that here, each steady state is reached sooner. For the mutual repression switch, the transition to a recently-induced state – i.e. rise in its expression level – only began after the level of the previous stable state was very low (close to zero). For the positive feedback loop switch, the rise coincided in time with the introduction of the signal that causes the transition. Hence, stable state was reached more quickly.

The mutual repressor switch exhibited no bistability for low cooperativity values, such as 1.15 (when leaky expression was present) or very low stable-state level bistability (when no or extremely little leaky expression was present). The positive feedback loop switch, however, exhibited bistability with high stable-state levels even at this cooperativity (1.15), as shown in Figure 2. This indicated that the positive feedback loop switch was much less dependent on cooperativity and could exhibit high expression-level bistability even at low cooperativity values. Bistability was exhibited with no noticeable change in reached levels even when PIP:KRAB and E:KRAB production rate was only half of TAL production rates.

### Simulation set 2 - cooperativity and leaky expression

As with the mutual repressor switch, higher cooperativity values were found to increase the robustness of the switch. However, the robustness at a given cooperativity was much higher for the positive feedback loop switch. Figure 3 shows that even for a low cooperativity of 1.15, the switch was exhibiting bistability with high-expression levels in the presence of leaky expression of 0.1. Increasing cooperativity to a still relatively low value of 1.85 dramatically increased switch’s capability to retain bistable behavior. In such case, as shown in Figure 4, even a leaky expression of 1.2 for each TAL did not prevent bistability from occuring, although the expression levels were reduced slightly.

This results suggested that positive feedback loop switch was far more robust and leaky expression-tolerant than the mutual repressor switch.

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