Due 11:59 PM, Thursday, November 14.
Submit as pdf to csm2013fall-instructors@googlegroups.com.
1. (10 pts) Code an implementation of Gillespie's Stochastic Simulation Algorithm (the one presented in class) in Matlab or Python. Apply your code to the simulation of the two step model of transcription and translation presented in class (see p. 4 of lecture notes). Set w_-x = w_-y = 1/sec. Taking w_+x = w_+y = 10 / sec, plot the steady state distribution of x and y and show that you obtain the correct mean and variance for both (Note that var(y) = mean(y)(1+ w_+y/(w_-x + w_-y).) Compute the Fano factor with the model with these parameters and also when you set w_+x=100/ sec, w_-x=1 / sec. Explain your results.
2. (10 pts) Use your Gillespie simulation routine to determine the effect of negative autoregulation on the two-step model from problem 1. Assume that protein y modulates the rate of expression of x through a Hill function with n=2 and K=1/50 (1/molecule). How does this regulation affect the Fano factor for y at steady state? Compute the Fano factor for at least two other values of n and explain your results.
3. (10 pts) Schlogl model revisited. Consider the reaction system A + 2X <-> 3X k1, k2 ; B <-> X k3, k4. Use the following values of the parameters: k1=3e-7 1/molecule^2 1/s, k2=1e-4 1/molecule^2 1/s , k3=1e-3 1/s, k4=3.5 1/s, Atot=1e5 molecule, Btot=2e5 molecule
(a) Assuming the the concentrations of A and B are fixed, derive the propensities of the four reactions as a function of X assuming the rate constants are given on a per reactant set basis. Hint: Don't forget the symmetry considerations!
(b) Derive the matrix form of the CME, dX/dt = A*X. That is, determine the elements of the matrix A.
(c) Plot the steady state solution of the system assuming a finite starting amount of X.
(d) Compute the steady state probability distribution using the Gillespie algorithm and compare the result to what you obtained in (c). Hint: They should agree to within statistical error.