# Molecular Population Dynamics as a Markov Process

## Stochastic simulation of transcriptional bursting

We are going to simulate the gene regulation Markov process that we described in class. The process includes 6 reactions.

r reaction propensity $\eta(R)$ $\eta(P)$
1 $\mbox{Gene(I)} \overset{k_b}{\longrightarrow} \mbox{Gene(A)}$ $k_b$ $0$ $0$
2 $\mbox{Gene(A)} \overset{k_u}{\longrightarrow} \mbox{Gene(I)}$ $k_u$ $0$ $0$
3 $\mbox{Gene(A)} \overset{k_1}{\longrightarrow} RNA$ $k_1$ $+1$ $0$
4 $RNA \overset{k_2}{\longrightarrow} \emptyset$ $k_2 R$ $-1$ $0$
5 $RNA \overset{k_3}{\longrightarrow} Protein$ $k_3 R$ $0$ $+1$
6 $Protein \overset{k_4}{\longrightarrow} \emptyset$ $k_4 P$ $0$ $-1$
• Implement the Gillespie algorithm to generate 5 time trajectories for (a) gene activity, (b) RNA population, and (c) protein populations.

• Plot the trajectories of those stochastic processes, and compare to their steady-state solutions.

• Design one set of rate values that would produce transcriptional bursting, and another set of values that would not. Reason your findings.

I have also added hints about how to sample from a exponential distributions easily, and about implementing the Gillespie algorithm.