Mathematical Modeling of Bistable Toggle Switch

Nns=5*10^6; %Number of non-specific sites
Kb=1.3806*10^-23; %(JK-1) %Boltzman's constant
T=298; %(K) %Temperature in Kelvin
P=3000; %Number of RNA polymerase (2000-4000)

Kspd=100*10^-9; %dissociation constant for specific binding of polymerase on DNA
Knspd=10000; %dissociation constant for non-specific binding of polymerase on DNA
delEpd=Kb*T*log(Kspd/Knspd); %Binding energy between polymerase and DNA

Ksrd=(3)*10^-9; %TALE %dissociation constant for specific binding of repressor on DNA (TALE: 1/150nM~, paper
example: 0.02)
Knsrd=10000; %dissociation constant for non-specific binding of repressor on DNA
delErd=Kb*T*log(Ksrd/Knsrd); %Binding energy between repressor and DNA

R=logspace(-6, 5,100); %Number of repressors
% R=linspace(0,100);
Freg1=1./(1+(R./Nns).*exp(-delErd/(Kb*T))); %Regulation factor 1x (<1 for repression)
Freg3=(1./(1+(R./Nns).*exp(-delErd/(Kb*T)))).^3; %Regulation factor 3x (<1 for repression)
Freg5=(1./(1+(R./Nns).*exp(-delErd/(Kb*T)))).^5; %Regulation factor 5x (<1 for repression)

p_bound_base=1./(1+(Nns/P)*exp(delEpd/(Kb*T))); %basal level
p_bound_1=1./(1+(Nns./(P.*Freg1)).*exp(delEpd/(Kb*T))); %level with repressor bound (1x)
p_bound_3=(1./(1+(Nns./(P.*Freg3)).*exp(delEpd/(Kb*T)))); %level with repressor bound (3x)
p_bound_5=(1./(1+(Nns./(P.*Freg5)).*exp(delEpd/(Kb*T)))); %level with repressor bound (5x)

Fold_Change1=p_bound_1./p_bound_base; %Fold Change 1x
Fold_Change3=p_bound_3./p_bound_base; %Fold Change 3x
Fold_Change5=p_bound_5./p_bound_base; %Fold Change 5x

fit1=1./(0.0111.*(R).^1+1);
fit3=1./(4.7215e9.*(R).^2.9929+1);
fit5=1./(6.6001e20.*(R).^4.8635+1);


figure1=figure(1);
% subplot(2,1,1),semilogx(R,Fold_Change1,'k.', R,fit1,'k-', R,Fold_Change3,'b.', R,fit3,'b-',
R,Fold_Change5,'r.',R,fit5,'r-')
semilogx(R,Fold_Change1,'k.', R,fit1,'k-', R,Fold_Change3,'b.', R,fit3,'b-', R,Fold_Change5,'r.',R,fit5,'r-')
axis([10^-6 10^5 0 1])
title(char('Thermodynamic Model and Apparent Hill Coefficient',' Fold Change vs Repressor Level (Lin - Log)'))
xlabel('R (number of repressor molecules)')
ylabel('Fold-Change of P_b_o_u_n_d')
%ylabel(char(' Fold-Change of P_b_o_u_n_d','(probability of Polymerase binding to promoter)'))
legend('1x Binding Site Model','1x Binding Site Hill Function Fit', '3x Binding Site Model', '3x Binding Site Hill
Function Fit','5x Binding Site Model','5x Binding Site Hill Function Fit','Location','NorthEastOutside')



% subplot(2,1,2),loglog(R,Fold_Change1,'k--', R,Fold_Change3,'b--', R,Fold_Change5,'r--')
figure(2)
loglog(R,Fold_Change1,'k--', R,Fold_Change3,'b--', R,Fold_Change5,'r--')
axis([10^-6 10^5 10^-3 10^0])
title(char(' Thermodynamic Model and Sensitivity','Fold Change vs Repressor Level (Log - Log)'))
legend('1x Binding Site', '3x Binding Site', '5x Binding Site','Location','NorthEastOutside')
xlabel('R (number of repressor molecules)')
ylabel('Fold-Change of P_b_o_u_n_d')
% ylabel(char(' Fold-Change of P_b_o_u_n_d','(probability of Polymerase binding to promoter)'))