# Assignment 2 : Due Oct. 12 at 11:30 am

Note: You may consult with other students when devising strategies to solve the homework, but the work you turn in must be your own.

1. For each of the following chemical reaction networks, find the reachability graph. If the graph is finite, draw the whole thing. If the graph is infinite, explain why it is infinite it, and draw the graph schematically. If there are any states that can be reached, but never left, point them out. Assume that there are initially two A molecules and zero of all others. The symbol O is intended to mean "nothing", so O → A means that A molecules can shown up from nowhere.

a) A ↔ B
b) A ↔ B, B → C
c) A ↔ 2B
d) O → A → B → O

2. 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. Which of the systems has the most intrinsic noise? Assume all the production rates are 1 and all of the degradation rates are 0.1 for RNA and 0.0 for gfp and rfp (they mostly just dilute).

a) O → RNA1 → O, O → RNA2 → O, RNA1 → GFP + RNA1, RNA2 → RFP + RNA2
b) O → RNA → O, RNA → GFP + RFP + RNA

3. Via simulation and similar plots, show what happens to the intrinsic noise in the systems in problem 2 when the production rates are doubled. Also, show what happens to the intrinsic noise when the degradation rate of RNA is halved.

4. In gro, implement the reaction O → gfp with rate 5v, where v is the volume of the cell. Grow the colony to 250 cells, and print out the number of gfp molecules in each cell to a file. In your favorite software, plot a histogram of the data (normalized so that it looks like a probability distribution) and fit the data to a Poission distribution with parameter alpha. What might account for any differences between the data and the ideal distribution? Repeat this exercise with a model that includes RNA production and degradation.