3. There is evidence that a critical concentration of a trigger protein is needed for cell division (see Molecular Biology of the Cell by Alberts et al., cited in chapter 2). 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 quantitatively. The trigger protein U is being synthesized by a zero-order mechanism with rate constant kn. It is being degraded by a first-order mechanism with rate constant in. (a) Write a differential equation consistent with the mechanism. (b) The solution to the correct differential equation is [U] = Eu e'iv) in if {U] is equal to zero at zero time. Show that your equation in part $3) is consistent with this. (c) If U is being synthesized at a constant rate with kg = \"1.00 nlvl s' and its half-life for degradation is 0.500 n, calculate the maximum concentration that U will reach. How long will it take to reach this concentration? Make a plot of [U] vs. time. (d) If a concentration of U of 1.99 pM is needed to trigger DNA synthesis and cell replication, how long will it talte to reach this concentration? (e) If the rate of synthesis is cut in half [Kg = {1.500 nl'v'l s"), how long will it take for U to reach a concentration of 1.00 uM? (f) What is the smallest rate of U synthesis ito that will allow (slow) cell replication? Assume k1 remains constant