Discuss the below:
1.) Integrate set of diff. equations with given initial conditions and constants and plot concentration profiles of all 5 dependent variables
2.) Plot phase plot of s1 vs s2. What does the phase plot demonstrate?
3.) Perform a stability analysis examining the eigenvalues (matlab eig function) of Jacobian matrix around the steady state.% The following describes the Sel'kov model of a glycolysis. The law of mass
%action was applied to this reaction scheme to obtain the following set of
%ODE's that describe the dynamics of the reactions.
%Set of Simultaneous first-order nonlinear ODE's:
%Rate of glycolysis is oscillatory
ds1/dt = v1-k1*s1*x1+kneg1*x2
ds2/dt = k2*x2-k3*s2^(gamma)*e+kneg3*x1-v2s2
dx1/dt = -k1*s1*x1+(kneg1+k2)*x2+k3*s2^(gamma)*e-kneg3*x1
dx2/dt = k1*s1*x1-(kneg1+k2)*x2
de/dt = -dx1/dt - dx2/dt %used to describe rate of change of free enzyme
%obtained from the balance equation for total enzyme in the cell (e0),
%assuming total amount of enzyme remains constant (e+x1+x2=e0)
%where: square brackets used to denote concentration
%s1 = [S1] = [ATP]
%s2 = [S2] = [ADP]
%e = [E]
%x1 = [E*S2^(gamma)]
%x2 = [S1*E*S2^(gamma)]
%Initial conditions:
s1(0) = 1.0
s2(0) = 0.2
x1(0) = 0
x2(0) = 0
e0(0) = 1.4
%Constants: units of time (s) and concentrations (nM)
gamma = 2.0
v1 = 0.003
v2 = 2.5*v1
k1 = 0.1
kneg1 = 0.2
k2 = 0.1
k3 = 0.2
kneg3 = 0.2