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Simulations

These simulations are made in Simbiology.We use the same actions.

• 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 simulation shows that the toggle switch continues showing the same behavior. It should be mention that the levels of the Repressor proteins on the steady state of the system are not equals due to the different repressions’ strength.

But to confirm the robustness we use a stochastic solver.

To assert the effect pointed out we build the same simulation based on Euler method:

    Km_1=Km1*(((1+Rep2(t)/Kiv)^beta));
    Km_2=Km2*(((1+Rep1(t)/Kiu)^gamma));
    
    mRNA1(t)=abs(random('norm',mRNA1(t),0.3));
    mRNA2(t)=abs(random('norm',mRNA2(t),0.3));
    
    mRNA1(t+1)=mRNA1(t)+poissrnd(tau*((Lambda1*RNAp/(Km_1+RNAp))))
-tau*rand*delta1*mRNA1(t);
    mRNA2(t+1)=mRNA2(t)+poissrnd(tau*((Lambda2*RNAp/(Km_2+RNAp))))
-tau*rand*delta2*mRNA2(t);
   
    asRNA(t+1)=abs(asRNA(t)+(mRNA1(t+1)-mRNA1(t))
-das*rand*asRNA(t));
    protease(t+1)=abs(protease(t)+(mRNA2(t+1)-mRNA2(t))
-dtr*rand*protease(t));
   
%     asRNA reaction
    mRNA2(t+1)=mRNA2(t+1)-kasr*rand*asRNA(t+1);
    if(mRNA2(t+1)<=0)  
        mRNA2(t+1)=0;   
    end
    asRNA(t+1)=asRNA(t+1)-kasr*asRNA(t+1);
 
    trigger
    
    k1=Kc1*mRNA1(t+1,1);
    k2=Kc2*mRNA2(t+1,1);
 
    Rep1(t)=abs(random('norm',Rep1(t),0.3));
    Rep2(t)=abs(random('norm',Rep2(t),0.3));
    
    Rep1(t+1)=Rep1(t)+poissrnd(tau*(k1*Rib/(Kmu+Rib)))
-tau*rand*d1*Rep1(t);
    Rep2(t+1)=Rep2(t)+poissrnd(tau*(k2*Rib/(Kmv+Rib)))
-tau*rand*(d2+ktemp)*Rep2(t);
    
    % Proteolysis
    Rep1(t+1)=Rep1(t+1)-kprs*protease(t+1);
    if(Rep1(t+1)<=0)
        Rep1(t+1)=0;
    end
    
    Rep1(t+1)=(Rep1(t+1)/K)/((1+IPTG/Ki)^eta);

Evaluating each species at every step, we obtained:

If we analyze the behavior of the system in different simulations, we realize that the levels of the system are more similar. We can see how the system is able to set a higher level for each repressor’s expression.

As we did with the basic toggle switch we made an analysis of the system when we simulate weaker stochastic actions and no inductions on the system.

It shows how the system is still no stable; however the system doesn’t change its state by itself. It must be mention that in this situation, the simulations of this model are very similar between them; it always wins the repressor2, and it never changes the state.