Stochastic model of the λ-phage lysis/lysogeny decision network.

The same fundamental model can be implemented as a stochastic model by use of the Gillespie algorithm.

We use the same equations to define the fundamental rates of each reaction:

v₁ = ω_cl[cl_rna]     v₂ = Xcl,prot[Cl_prot]

v₃ = μ_cl(1 - |cro_prot|²/(k²_cro,1/2 + [cro_prot]²))    v₄ = X_cI,rna[cI_rna]

v₅ = ω_cro[cro_rna]        v₆ = X_cro,[cro_prot]

v₇ = μ_cro(1 - |cI_prot|²/(k²_cI,1/2 + [cI_prot]²))    v₈ = X_cro,rna[cro_rna]

where the concentrations correspond the the numbers of molecules present. The algorithm can be generally outlined as:

for each step

• find tine increment
• choose next reaction
• increment or decrement molecule numbers by 1, as appropriate for the chosen reaction

The time increment is chosen from an exponential distribution with coefficient R_tot^-1. where R_tot is the total of all reaction rates (R_tot = Σ_i=1⁸ v_i. The next reaction is chosen based on a probability proportional to v, (P(i) = v_i/R_tot.

1. Implement this stochastic model in MATLAB. using the same constants as for the deterministic model. Beginning with all concentrations equal to zero. run a simulation for 50.000 steps, and plotting all concentrations versus time. Discuss what you observe.

2. Stochastic simulations will never give identical behavior each time. so repeat the simulation from (I) at least 10 times. plotting all simuluations on the same plot of concentration versus time: also plot the results on a single plot of the cro_prot vs cI_prot, phase plane. Discuss in your observations in detail.

3. Repeat these simulations with starting concentrations of 20 molecules of either cro or cl RNA (do both, but independently), and discuss your results. Starting concentrations of both cro and cl protein should be set to zero.

4. Add the enhanced degradation rate chosen in the last part. of the of examination of the deterministic system. and begin a set of simulations (at least 10) from starting conditions corresponding to the lysogenic state. Describe the behavior you observe, do you think that the length of simulation is significant here? Repeat these simulations for a longer period of time (such as 100.000 steps) and discuss any differences.

5. Discuss how the results obtained from the stochastic model compare to those seen in the deterministic case What are the implications of these differences for the biological system? Are there specific challenges in understanding the results of the stochastic simulations? Be as detailed as passible in your response.