Use Matlab for all questions, include graphs and comment what you did and address all questions. Upload your m-files with your submission. In this portion of the homework you will reproduce the minimalistic cell cycle model of Goldbeter, published in PNAS in 1992, a publication posted under the Biosim Resources folder.

1) Use code from Berkeley Madonna listed below as a starting point to develop your own Matlab implementation. Use ode45 (RK4 solver) or your own RK4 solver from the previous homework to solve the set of ODEs, and graph the solutions for C, M and X over time. Of note, your solution might slightly divert from the one published in the PNAS paper.

2) Perform a sensitivity analysis to determine the parameter most sensitive to perturbation. We use a definition of the Sensitive Objective Function (SOF) suggested by Aldridge (see posted Nature paper on Physicochemical Modeling): SOF = (% change in outcome) / (% change of rate). Perturb each parameter/constant you use in the model (excluding the initial concentrations of C, M and X) by 5%, and determine the amount of the cyclin C at the end (last timepoint) of the simulation, to determine the SOF. Reset the parameter back to its original value and proceed to the next one. It is relevant to consistently increase or decrease by 5%. Produce a table in which you rank all parameters for their SOF sensitivity. What is the most sensitive parameter? Remark: Perform each perturbation manually, or program a loop to change parameters automatically.

3) Plot C dependent on X. This is called a phase plot revealing a limit cycle (see also Fig. 4 in the paper). Produce a second plot to demonstrate how a more dramatic perturbation of the initial conditions of C or X causes the system to not get back to cycle. This is a demonstration for the limit of robustness of the system.

Here is the Berkeley Madonna code:

DT=1e-4

STOPTIME=100

d/dt (C) = Synthesis - Degradation

INIT C = 0.01

Synthesis = 0.025

Degradation = vd*X*(C/(Kd+C)) - kdd*C

d/dt (M) = Phosphatase1 - Kinase1

INIT M = 0.01

Phosphatase1 = VM1*(C/(Kc+C))*((1-M)/(K1+(1-M)))

Kinase1 = V2*(M/(K2+M))

d/dt (X) = Phosphatase2 - Kinase2

INIT X = 0.01

Phosphatase2 = M*VM3*((1-X)/(K3+(1-X)))

Kinase2 = V4*(X/(K4+X))

K1 = 0.005

K2 = 0.005

K3 = 0.005

K4 = 0.005

Kc = 0.5

Kd = 0.02

kdd = 0.01

V2 = 1.5

V4 = 0.5

vd = 0.25

VM1 = 3

VM3 = 1

(Remark: DT=h)