# Physics 434, 2014: Homework 6

2. Download and work with the Matlab code that simulates this bursty birth-death process. Use it, or its appropriate modification, to estimate numerically the Fano factor (${\displaystyle \sigma ^{2}/\mu }$) of the mRNA distribution some reasonably long time after the start of the process. Use ${\displaystyle \alpha =30\,{\rm {s}}^{-1}}$, ${\displaystyle r=1\,{\rm {s}}^{-1}{\rm {molecule}}^{-1}}$, ${\displaystyle k_{on}=0.1\,{\rm {s}}^{-1}}$, ${\displaystyle k_{off}=0.1\,{\rm {s}}^{-1}}$. Recall that the Fano factor for a Poisson distribution is 1. Is your number, for this bursty process, larger or smaller than 1? Explain.
3. Consider a biochemical system for production of a protein that is somewhat different from what we explored in class: number of proteins ${\displaystyle n}$ increases by one with a rate ${\displaystyle \alpha }$ and it decreases with the rate ${\displaystyle rn^{2}}$. This happens when the protein is involved in a self-degradation, as is widely believed to be true for development (Eldar et al., 2003). In other words, two protein molecules must meet to destroy each other at the same time, and this will happen in proportion to the square of their concentration.
• Write down the master equation describing the system. Be careful that two molecules are removed at the same time, so that one transitions from the state ${\displaystyle n}$ to ${\displaystyle n-2}$.