Consider a simple model for gene expression with reactions

Let α = 1/2, κ = 20 log(2)/120, δ = log(2)/120 and γ = log(2)/600, and answer the following questions:

(i) Use the stochastic simulation algorithm (SSA) to obtain realizations of the stochastic process of gene expression and numerically compare with the deterministic ODE solution. Explore how the realizations become close to or a part from the ODE solution when the volume is changed. Determine the stationary probability distribution for the protein (you can do this numerically).

(ii) Now consider the additional binding reaction of protein P with downstream DNA binding sites D:

Note that the system is no longer linear due to the presence of a bimolecular reaction. Use the SSA algorithm to obtain sample realizations and numerically compute the probability distribution of the protein. Compare it to what you obtained in part (i). Explore how this probability distribution and the one of C change as the rate constants a and d become larger with respect to γ,α,κ,δ . Do you think we can use a QSS approximation similar to what we have done for ODE models?

(iii) Determine the Langevin equation for the system in part (ii) and obtain sample realizations. Explore numerically how good this approximation is when the volume decreases/increases.