Biochem 104b Problem Set 1
Problem 1:
Remember the simple model we used to explain why rubber is elastic and returns to its
original length after being stretched or condensed. Now use a simple model to work out
why solutions have a tendency to mix. Use this model as your starting point. The
property you can observe macroscopically is the fraction of black and white balls on
either side.
a)What happens when you pull out the barrier in the middle and why? How much higher
is the multiplicity of a fully-mixed arrangement (equal numbers of black and white on
both sides). How much higher is the multiplicity for achieving a fully mixed solution than
it is for the starting state shown above. Solve this problem both by drawing the possible
configurations on paper and by using the formula for multiplicity. This is a little trickier
than you may think. Remember that the macroscopically observable property is the
number of black and white balls left and right of the barrier. Hint, the multiplicity of a
system is calculated by multiplying the multiplicity of its subsystems (Wsubsystem A+ subsystem B
= Wsubsystem A ⋅Wsubsystem B)
b) Now repeat this problem for the following starting configuration
Which arrangement has the highest multiplicity? Try out a number of arrangements and
calculate their W. Is the maximal increase in W for mixing in this case greater or smaller
than the in the case shown under section a) of this question.
Problem 2:
Back to the tetrahedron game we talked about in class. What if you would go and put a
weight under the white face of every tetrahedron. Lets say you keep adjusting the size of
those weights until a random toss of all your tetrahedrons makes 50% of the pieces end
up with their white faces down.
Does this 50% 50% state have a higher or lower entropy than the original system, where
all sides were equally heavy? Try to reason it out in your head. Then use the formula for
entropy to calculate the entropies of both cases.
Problem 3:
Pentose sugars undergo puckering to relieve steric strain. This puckering is the main
source of conformational entropy for these molecules. Ribose and deoxyribose are both
pentoses and chemically differ only in their substituents on the C2 carbon. In Ribose this
carbon carries a hydroxyl group and a hydrogen, while in the deoxyribose the equivalent
carbon carries two hydrogen atoms. The steric collisions caused by the bulkier hydroxyl
substitutent on ribose restricts the number of puckering conformations that the ring can
adopt to 4 while the deoxyribose can adopt 8 different puckered conformations. Lets
think of an enzyme that can bind both ribose and deoxy-ribose and that this enzyme binds
both molecules via the face of the rings that is chemically identical and the binding
enthalpy is identical for the binding of both ribose and deoxyribose.
a) Calculate the difference in standard entropy of binding for the two sugars.
(Assume that, except for the loss of conformational entropy from sugar puckering,
all contributions to binding entropy are the same for the two sugars and can
therefore be neglected.)
b) By how much does this difference in binding entropy shift the binding
equilibrium of the ribose relative to deoxyribose. Lets assume that the binding
[sugar " enzyme]
= 1 . What is the Kequil. for
constant for deoxyribose is K equil. =
[enzyme][sugar]
binding ribose? Does the ribose bind more tightly and why?
!
Problem 4:
You have a population of NA molecules that is at equilibrium between a number of
different states (A-E) and you observe the following probability distribution.
A 36%
B 14%
C 28%
D 5%
E 17%
a) What is the entropy of that distribution?
b) What distribution would you have expected a priori? What does this deviation from
the expected distribution tell you (e.g. about D vs. A?)
c) What is the entropy change if you convert this population to a second population with
the following distribution:
A 25%
B 25%
C 25%
D 25%
E 0%
d) If the molecules in your system can only adopt states A-through-E what is the highest
entropy you can achieve.
Problem 5
Here are three distributions that describe the probability with which a molecule adopts
one of 6 different conformations. The relative probabilities of these 6 states are given by
the numbers on top of the bars.
A)
B)
C)
a) Without doing any calculation, which of the three distributions do you think has
the highest entropy and which has the lowest? Why?
Problem 7
You have a strand of DNA in a solution containing positively charged ions. Think about
what sort of forces and effects the ions will be exposed to. After equilibrating under the
influence of these forces, the ions will adopt a concentration distribution around the
DNA.
a) Which one of the three distributions in problem 6 looks like it could be used to
describe this distribution. (Provided you replace 1,2,3… with the distance from
the DNA and the probabilities with a concentration value). What are the two
effects that cause the distribution to adopt this shape.
b) How would a change in temperature change the shape of the ion distribution
around the DNA molecule? Draw a qualitative graph by hand in which you show
the distribution at three different temperatures (cold, medium, hot). Explain in
qualitative terms why the shape changes that way. Which effect dominates at low
temperature, and which effect dominates at high temperature?