Use the differential equations on Slide 15 from Seminar 6&7 slides to build a model of the homeostasis with Hill function. You may use the model of Homeostasis from seminar as example.
Use the following parameters:
K = 1;
k0 = 2;
k2 = 1;
k = 0.05; (signal S growth rate)
n = 8
and the initial condition: R = 2; S = 0.
Assume that S grows with the rate k for the whole simulation. Run the simulation for t=40 minutes. Present a simulation of the system as a graph (R and S concentrations vs t).
Study the effect of the Hill coefficient (change Hill coefficient as: n = 2, 4, 6, 8, 10). Find the max&min values of the response R between the time points t=10 and t=30. Present the data as a table and as a plot (max-min) (Y) vs n (X).
Now let S=0.6 be constant. Plot the nullcline (dR/dt=0) and the line E(R) for n=8. Plot E on Y-axis and R on X-axis. (Set R=linspace(0,2) and you might need to use ./ and .^ for division and power. E can take values between 0 and 1.)