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Simulations

These simulations are made in Simbiology.

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:

 trigger
 
 Km_1=Km1*(1+Rep2(t)/Kiv)^beta;
 Km_2=Km2*(1+r1(t)/Kiu)^gamma;
 
 mRNA1(t+1)= abs(random('norm',
   (mRNA1(t)+tau*(Lambda1*RNAp/(Km_1+RNAp)-delta1*mRNA1(t))),sig));

 asRNA(t+1)= abs(random('norm',
   (asRNA(t)+tau*(Lambda1*RNAp/(Km_1+RNAp)-delta1*asRNA(t))),sig));

 mRNA2(t+1)= abs(random('norm',
   (mRNA2(t)+tau*(Lambda2*RNAp/(Km_2+RNAp)-delta2*mRNA2(t))),sig));

protease(t+1)= abs(random('norm',
(protease(t)+tau*(Lambda2*RNAp/(Km_2+RNAp)-delta2*protease(t))),sig));
 
 % asRNA reaction
 mRNA2(t+1)=mRNA2(t+1)-kasr*asRNA(t+1);
 asRNA(t+1)=(1-kasr)*asRNA(t+1);
    
 % Proteolysis
 Rep1(t+1)=Rep1(t+1)-kprs*protease(t+1);
 
 k1=Kc1*mRNA1(t+1,1);
 k2=Kc2*mRNA2(t+1,1);
 
 Rep1(t+1)=Rep1(t)+tau*(k1*Rib/(Kmu+Rib)-d1*Rep1(t));
 Rep2(t+1)=abs(random('norm',
     Rep2(t)+tau*(k2*Rib/(Kmv+Rib)-(d2+ktemp)*Rep2(t)),sig));
 
 r1(t+1)=abs(random('norm',(Rep1(t+1)/K)/((1+IPTG/Ki)^eta),sig));

Evaluating each species at every step, we obtained:

If we analyze the behavior of the system in different simulations, we realize that the robustness makes the system slower to respond the changes. We need to apply stronger actions to make the system answer. But these acts are not confirmed in lab.