1. (a) Estimate the (thermally-averaged) dipole-dipole interaction energy between 2 water molecules  separated by ~ 3 A.

         Compare the dipole-dipole energy with that of a hydrogen bond.
 
 

2.   Show that the thermally averaged interaction energy between a charge and a dipole, when U << k_BT, is given by

3. (a) Derive an expression for the interaction energy between a charge q and a molecule with polarizaibility , separated by a distance r.

    (b) What is the form of the distance dependence with and without thermal averaging?
 
 

4. Self-complementary sequences of RNA or single-stranded DNA can form hairpins that are in equilibrium with a open, random coil conformation, with equilibrium constant K dependent on temperature.

The figure below plots the fraction of molecules that are in the hairpin conformation as a function of temperature for a DNA strand with 7 base-pairs in the stem and 8 bases in the loop.

(a) Calculate the free energy difference at 40 C, 50 C, and 60 C.

(b) From the temperature-dependence of estimate

       (i) the gain in energy (from base-pairs) when a hairpin is formed

       (ii) the ratio 

where W_hairpin (W_random) are the number of different configurations accessible to the molecule in the hairpin (random) states.
 

5. (a) Consider the statistical mechanics of a two state system, where the (free) energy of state 1 is  relative to that of state 0.  Find the probability that the system is in state 0 or state 1 in thermodynamical equilibrium.

Suppose that these two states refer to non-occupation (0), or occupation (1) of a particular site on a DNA molecule, by a protein which binds to a particular sequence.   Consider the free energy to be made up of an energy associated with the interactions which hold the DNA to the protein which is 0 when the protein is off and -E₀ when the protein is bound, and an entropy for the protein which is 0 when it is bound, and equal to  S₀ + k_B log c where S₀ is a constant, and where c is the molarity  (concentration in moles per litre) of protein.

(b) Find a formula for the protein being bound as a function of molarity.

(c) Explain in a few words why it is reasonable for the entropy of the unbound protein to depend as log c (think about the number of states!)

(d)  Suppose the DNA site is occupied 95% of the time at a protein concentration of 1x10^-6 M (micromolar).   Plot the binding probability as a function of molarity, on a logarithmic scale from 1x10^-12 M (picomolar) to 1x10^-3 M (millimolar)

(e)  At what concentration will the protein bind the DNA 50% of the time?  This special concentration is sometimes called the dissociation constant K_d of the protein.