Hints on HW1

Dear class-

Here are some hints on HW1 for the problems.

Problem 1:
In this problem, you are tackling a problem of combinatorics. The problem essentially asks you how many spots there must be in a DNA sequence that are variable (i.e., one of the four possible bases) in order to have 10^10 unique sequences.

Problem 2:
To compute how far proteins are from each other, you want to assume each protein sits in a little cube of volume. The distance between proteins is then V^(1/3). Therefore, to solve this problem, you have to first compute the volume per protein.

Problem 3:
For part b, first sketch out what the Coulombic interaction looks like. Then, sketch what the total LJ + Coulombic interaction would look like for small q, medium q, and large q. At some point, as you increase q, the minimum in the net potential should vanish. What are the mathematical criteria for the vanishing of a minimum? In other words, what two conditions involving the derivatives of a function must apply when a minimum becomes and inflection point? This provides two equations... what are the two unknowns?

Problem 6:
(b) Imagine that the protein is fixed at the origin. What conformations must the ligand give up when it binds?

Problem 7:
(a) To get from heat capacities to entropies and enthalpies, you will have to integrate from a reference temperature. What are the definitions of CP?

(b) To eliminate DeltaS(Tf) from your final expressions, you will need to let T = Tf and DeltaG= the value at the folding temperature, then solve your expression for DeltaS.

(d) You will want to use either the slope or intercept of some line to estimate DeltaH(Tf)

Problem 8
You are going to first want to write all of your equations in terms of K1, K2, K3, K4 where K1=exp(-DeltaG1/RT) and so on and so forth. Next, express the concentrations of the various bound hemoglobin species in terms of equilibrium constants, [O2], and [H]. What constraint governs the total concentration of hemoglobin? Write down a mass balance on concentrations of hemoglobin. Let [H]0 be equal to the total amount. Then, f0 = [H]/[H]0, f1 = [HO2]/[H]0 and so on and so forth.

Problem 9
This problem amounts to finding the probability of one molecular state relative to another one, using Boltzmann populations, P ~ exp(-U/kB T). Be sure to normalize the probabilities such that they sum to one.

That's all the hints for now. Good luck,

MSS




Problem