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Other Models

Now, we show a different vision of the model presented in the previous sections. All of them are implemented in Matlab scripts or using the Simulink tool.

Basic Toggle switch general Equation Model

Here we have integrated the equations using the Euler approximation method. Even if the details may change, the results obtained show a similar behavior to the previous graphics.

We are using this new method to show a different integration method than the used before. Here it is easier to show how the repression acts as a modifier of the Michaelis constant of transcription process.

The algorithm used evaluates each species and its change in every step of the algorithm. So the algorithm is:

for t=1:(tsim-1)
    
    Km_1=Km1*(1+Rep2(t)/Kiv)^beta;
    Km_2=Km2*(1+r1(t)/Kiu)^gamma;

    mRNA1(t+1)=
     abs(mRNA1(t)+tau*(Lambda1*RNAp/(Km_1+RNAp)-delta1*mRNA1(t)));
    mRNA2(t+1)=
     abs(mRNA2(t)+tau*(Lambda2*RNAp/(Km_2+RNAp)-delta2*mRNA2(t)));

    trigger
    
    k1=Kc1*mRNA1(t+1,1);
    k2=Kc2*mRNA2(t+1,1);

    Rep1(t+1)=abs(Rep1(t)+tau*(k1*Rib/(Kmu+Rib)-d1*Rep1(t)));
    Rep2(t+1)=abs(Rep2(t)+tau*(k2*Rib/(Kmv+Rib)-(d2+ktemp)*Rep2(t)));

    r1(t+1)=(Rep1(t+1)/K)/((1+IPTG/Ki)^eta);

end

The ‘trigger’ is just the moments where we simulate the effect of adding IPTG or increasing temperature. The parameters used are the same as in the last Simbiology simulation.

If we run this program, we obtain the following result:

Basic Toggle Switch General Equation Model with Simulink

Here we present another implementation on Simulink, using Matlab again.

In the model we have included the overall process of the transcription and translation as a first-order kinetic model. The only exception to that vision is that the repression is treated as a modifier of the Michaelis constant of the transcription.

The diagram of the process is the following:

These results show no relevant information with respect to the models already presented. But the way to implement the system is closer to the vision of systems theory. Under this environment, we show all processes as first order systems.

Basic Toggle Switch Simplified Equations Model

Now we show an overall vision of the general behavior of the system. It is based on the simplification made by Timothy S. Gardner. It only includes the changes on the repressors concentration:

Repressor1:

Repressor2:

And including the effects of IPTG or temperature:

Repressor1:

Repressor2:

These simple equations offer an external view of the system (a black-box model), and not in the internal processes in the bacteria analyzing the input-output relations.

The parameters α1 or α2 just show the force of each repressor of the proteins’ expression, while β and γ model the cooperativity of the mutual inhibition. If we simulate this scheme:

Although the behavior is not the same than the one showed before, due to the fact that there exists a delay at the moment of the switch, it is a good approximation for a qualitative analysis.

We have also analyzed an stochastic integration method:

In the following image, we show the concentration of one repressor against the other.

At first, the repressor2 wins the battle, but this is just casual, and in a different simulation may occur this: