week 10:

Molecular Population Dynamics as a Markov Process

Preliminars

Present all your reasoning, derivations, plots and code as part of the homework. Imagine that you are writing a short paper that anyone in class should to be able to understand. If you are stuck in some point, please describe the issue and how far you got. A jupyter notebook if you are working in Python is not required, but recommended.

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.