% Systems Biology 7.81/8.591/9.531 Fall 2004 % Problem set 1 % % Problem 3: Positive feedback and multistability % file: ps1.m clear; close; % keep the following variables at their default values Dtot=1; % concentration of DNA A=1; % association constant gamma=1; % degradation rate % change the values of the following variables v= ; % basal transcription rate k= ; % maximal transcription rate n= ; % Hill coefficient % n=1 for Scheme A, n=2 for Scheme B % we later solve for roots, a procedure that requires % that n be an integer if( rem(n,1)~=0 ) error('n should be an integer'); end % these variables define the range of x values; % change them from the default to 'zoom in' on interesting values xmin=0; % lower limit xmax=2; % upper limit nx=1000; % number of points in mesh x=linspace(xmin,xmax,nx); % generation rate gen=v+k*Dtot*(x.^n)./(A^n+x.^n); % degradation rate deg=gamma*x; % plot curves plot(x,deg,x,gen); xlabel('concentration (x)'); ylabel('rate of change (dx/dt)'); title('Generation and Degradation of X'); legend('deg(x)','gen(x)',4); % now solve for the roots of the system if(n==1) c(1)=gamma; c(2)=-(v+k*Dtot)+gamma*A^n; c(3)=-v*A^n; else c(1)=gamma; c(2)=-(v+k*Dtot); c(n+1)=gamma*A^n; c(n+2)=-v*A^n; end xsol=roots(c); % count and plot roots nsol=length(xsol); nroots=0; for i=1:nsol if( (imag(xsol(i))==0)&(real(xsol(i))>=0) ) nroots=nroots+1; xbar=[xsol(i);xsol(i)]; ybar=[0;gamma*xsol(i)]; hl=line(xbar,ybar); set(hl,'Color',[1 0 0]); end end if( nroots==1 ) etext=['']; else etext=['s']; end nrtext=['The system has ' int2str(nroots) ' root' etext '.']; text(xmin+0.1*(xmax-xmin),0.9*xmax,nrtext)