Questions

1. Show by integration of Equation 2 in the lecture

Equation 2
Equation 2


that the time-dependent protein concentration cp(t) is given by:

Equation 5
Equation 5


where k is the protein production rate and μ its decay rate, and cp(0) is the protein concentration at time zero.

2. In the lecture we discussed a protein which repressed its own production by binding to its own gene in the promoter region of the DNA sequence. The differential equation for the dynamics of protein production was:

Equation 14
Equation 14


Let us now consider a protein which activates its own production: protein is only produced when a protein molecule is bound to the promoter region.

(a) write down the equivalent differential equation for the dynamics of protein production in this case.

(b) Solve for the steady state protein concentration cP(ss). What interesting feature do you notice about your solution?

Answers

1

Equation


2 (a)

Equation


(b)

Equation


or

Equation


An interesting feature is that in principle there can be two solutions: no protein or a finite concentration of protein. “Positive feedback loops” like this one can act as bistable switches in cellular gene regulation networks. However this particular version does not work as a switch if there is a low constitutive level of protein production (why not?). For a more robust switch, cooperative binding of several protein molecules to the promoter is needed.