HW10

There is evidence that a critical concentration of a trigger protein is needed for cell division. The unstable protein is continually being synthesized and degraded. The rate of protein synthesis controls how long it takes for the trigger protein to build up to the concentration necessary to start DNA synthesis and eventually to cause cell division. A simple mechanism to explore is one in which the trigger protein, P, is being synthesized by a zero-order mechanism with rate constant k₀. It is being degraded by a first-order mechanism with rate constant k₁.

(a) Write a differential equation consistent with the proposed mechanism.

(b) Solve the differential equation by simple integration. (Look up the relevant integral or substitute y = a+bx into dx/(a+bx) to get something familiar.)

(c) Normally, P is being synthesized at a constant rate with k₀ = 1.00 nM/s (nM = nanomolar = 10^-9 M) and its half-life for degradation is 0.500 hr. If a concentration of P of 1.00 micromolar (uM = 10^-6 M) is needed to trigger DNA synthesis and cell replication, how long will it take to reach this concentration?