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Introduction

The next step in this point is to model the actuations that allow us to modify the state of the system, inducing a switch in the flip-flop.

These actuators are:

• Using a thermosensitive repressor protein. By increasing the temperature of the bacteria, the protein is rapidly degraded, thus helping the change.
• Adding IPTG. The IPTG molecules bind competitively to the other repressor. Therefore, the inhibition of the transcription by the repression is not permitted. The IPTG induction is achieved assuming a Hill kinetic for the ligand-binding.

Diagram

Then, the model is complemented with 3 new elements:

• r1: it represents the level of free repressor protein (may inhibit the transcription stage). repressor1 does not have the same sense anymore, but now means the entire level of repressor, free or not.
• IPTG: it represents the level of IPTG specie.
• Reaction from Repressor2: It represents the forced degradation of Repressor2 when we increase the temperature.

Equations

Before describing the equations of the modified model, the next list shows the new species, and parameters.

Species:

• IPTG
• r1 (repressor1)

Parameters:

• Ki: Binding constant for IPTG to the repressor
• K: Multimerization constant
• η: Maximum number of bound IPTG molecule per repressor
• ktemp: degradation constant due to the temperature effect

Also, some further assumptions are made:

• The thermosensitive protein reacts to a rise temperature disappearing faster.
• To model the IPTG induction, we use a simple expression for the ligand-binding. This expression is taken from the document ‘Design and Construction of synthetic gene regulatory network’ by Timothy S. Gardner. This formula assumes a new level to free repressor on the steady state, based on the whole repressor concentration.

Then, the equations are:

Increase of temperature:

repressor2 → null

IPTG Induction:

Repressor1 + n· IPTG → Repressor1_IPTG_n

Free repressor level:

Simulations

We add the previous equations to the basic bistable model of the previous section and we define 2 events in the simulation. These events are:

• Time=40000, temp=1. We increase the temperature, meaning that the thermosensitive proteins start to disappear.
• Time=80000, IPTG=2e5 We simulate the addition of IPTG to the system.

The results showed by using Matlab's Symbiology are:

Although we only act at the described moments, at the beginning we have favored repressor2 in the simulation.

You can download the full Simbiology model of the Toggle Switch here