Problem 1.2: Separation of time scales

---

This problem is still a draft.

We employed a separation-of-time-scales assumption to simplify our expressions for the rate of production of a gene product. Specifically, we assumed that the binding and unbinding dynamics of a repressor to its target DNA sequence happened at a faster time scale than transcription and translation dynamics.

a) What happens if you don’t assume separation of timescales? Write a set of differential equations to capture the dynamics of mRNA concentration \(m\) and protein concentration \(x\), as well as the dynamics of the unoccupied promoter concentration \(p\) as a function of the repressor concentration \(r\) and total promoter concentration \(p_\mathrm{tot}\).

b) How do you tell if you’re in a parameter regime where that assumption is appropriate? Give both theoretical and experimental arguments.