### From SOSwiki

# A5 : Due 10/29

1. In gro, simulate each of the following networks until there are 500 cells. For each system, make a plot like Figure 3A in the Elowitz paper and compute the intrinsic and extrinsic noise, as well as hoe it varies as the the parameters are changed.

- a) This example was suggested by Garrett in class, in which gfp and rfp are initially fused, and then separate. We assume that X does not fluoresce in either channel. What happens to the intrinsic and extrinsic noise as the rate at which the fusion protein is cleaved is varied?
- O → X
- X → gfp + rfp // this rate should be varied

- b) In this example the mRNAs are engineered to annihilate each other at some rate k.
- O → RNA
_{1}→ O - O → RNA
_{2}→ O - RNA
_{1}+ RNA_{2}→ O // this rate, which you should call k, should be varied. Include zero as a possibility. - RNA
_{1}→ GFP + RNA_{1} - RNA
_{2}→ RFP + RNA_{2}

- O → RNA

Assume RNA production rates are 1 times the volume, protein production rates are just 1 times the amount RNA (and not the volume), and all of the degradation rates are 0.1 for RNA and 0.05 for gfp and rfp (in addition to dilution).

2. Consider the reaction network

- A ↔ 2 B
- B → C

starting with two As and no Bs or Cs.

- a) Enumerate the states and draw the Markov process the results from this system (i.e. draw the reachability graph with the rates of each transition noted).
- b) Determine the rate matrix.
- c) Find the probabilities of being in each state as a function of time by solving the master equation. You can do this analytically and/or numerically using an ODE solver.
- d) Find the mean and variance for the number of Cs as a function of time and plot the mean and the one standard deviation window versus time Note: To simplify things, you may set the rates of the reactions 1.0.

3. Show that if a system of reactions over a finite number of species admits a mass vector that has no zeros, then the Markov process describing its stocahstic behavior has a finite number of states. Give an example of a non mass conserving system that still produces a finite number of states.

4. [EXTRA CREDIT] Consider the system

- O → X → O

with rates k1 and k2 for production and degradation, respectively. This example produces an infinite dimensional Markov process. Kolmogorov’s equation for this system has an infinite dimensional Q matrix. In this exercise, you will approximate this matrix wirth a series of finite matrices. Define Q(n) to be the rate matrix for the following approximation to the above system where we disallow the production reaction if there are n+1 or mor X molecules. Consider the case when k1 = 5 and k2 = 1 and there are initially no X molecules. Using p ̇ = eQ(n)tp(0), plot the mean number of X molecules (plus standard deviation windows) versus time for n = 5, 10, 20, 30, and 40. How could this idea be used to approximate the behavior any reaction network, with a finite number of states or not?

Turn in homework A5 here: https://catalyst.uw.edu/collectit/assignment/koishi/23996/97900