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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.

Basic Toggle Switch - Stochastic Model

In other script (ecoli.m), we’ve tried to show a different integration. This time we add some stochastic elements to our general equation model. We supposed every species as random Gaussian variables while its changes are random Poisson variables. At each integration step we also decided to understand the degradation reactions as random processes. Finally:

As we can see, we got a different result.

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

It shows a bistable system. If one of the states is “on” the other is “off”.

At the following graphic, the system is able to change its states when we act on it, and each state has its own level.

If we simulate weaker stochastic effects, and no inductions on the system, we can see the following result:

The system is not stable without actions. Even, the system is able to change the state randomly. This picture shows a unstable behavior in case of lack of actions.

Basic Toggle Switch - Experimental Data

Let’s see the experimental results performed by Amalia.

The data represented are just a relation between the number of molecules and its fluorescence medition. So we can accept that the difference between RFP and GFP levels may be lower. The green channel seems to have an offset while the red channel seems to be weaker than the green one.

Accepting the previous assumptions, we have fit he experimental data (42 case) on a simulation.

o green channel datas

+ red channel datas>/p>

And a detail:

The experimental values are scaled