There is evidence that a critical concentration of a trigger protein is needed for cell di...

There is evidence that a critical concentration of a trigger protein is needed for cell division. This 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.

Let’s choose a simple mechanism to consider quantita tively. The trigger protein U is being synthesized by a zero- order mechanism with rate coefficient kQ. It is being degraded by a first-order mechanism with rate coefficient k1

a. Write a differential equation consistent with the mechanism.

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b. The solution to the correct differential equation is

if [U] is equal to zero at zero time. Show that your equa tion in part (a) is consistent with this.

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c. If U is being synthesized at a constant rate with k0 = 1.00 nM s-1 and its half-life for degradation is 0.500 h, calculate the maximum concentration that U will reach. How long will it take to reach this concentration? Make a plot of [U] vs. time.

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d. If a concentration of U of 1.00 μM is needed to trigger DNA synthesis and cell replication, how long will it take to reach this concentration?

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e. If the rate of synthesis is cut in half (k0 = 0.50 nM s) how long will it take for U to reach a concentration of 1.00 μM?

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f. What is the smallest rate of U synthesis k0 that will allow (slow) cell replication? Assume k1 remains constant.