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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:

• 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
• Repressor1

Parameters:

• Ki: Binding constant for IPTG to the repressor
• 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 an effect similar to a repression reaction. It modify the transcription rate law again.

Then, the equations are:

Increase of temperature:

repressor2 → null

IPTG Induction:

RNAp + promoter1 + repressor1 + IPTG → + promoter1 + repressor1 + mRNA1 + IPTG_n

Rate Law (transcription + repression + IPTG Induction):

Simulations

We add the previous equations to the basic bistable model of the previous section and we define 3 events in the simulation. These events are:

• Time=0, temp=0 IPTG=2e3. We start the simulation adding IPTG.
• Time=70000, temp=1 IPTG=0. We remove all the IPTG and simulate the increasing of temperature making the degradation sontant get higher following an exponential rate.
• Time=140000, temp=0 IPTG=2e3. We add IPTG and decrease the temperature.

The results showed by using Matlab's Symbiology are:

We can see how the behavior of the system is the expected one. The result shows the difference between one induction and the other. The change due to the temperature is harder than the IPTG induction. That is why the IPTG must ligand to repressor1 and this reaction might be slower than the other which attacks to a sensible property of the protein.

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