Question

1. Very simple toy model for base-pairing of double stranded DNA. (This problem is originally due to Kittel.) DNA is an important biological heteropolymer. A single DNA molecule base-pairs with a complementary DNA molecule to form a duplex double stranded structure. Consider two strands of complementary DNA. Each strand has N monomers. Each monomer is cross- linked by base-pairing to the corresponding monomer on the complementary DNA molecule. Suppose that the energy for breaking a cross link is є and suppose that a cross-link can only be broken if a neighboring cross link is broken Suppose that the DNA duplex is in contact with a heat bath at temperature T. Finally, suppose, for simplicity, that DNA duplex is tethered to a wall on one side so that the DNA duplex can only unwind from the opposite side. (a) Compute the partition function for the DNA (b) Compute the Helmholtz potential for the DNA. (c) Determine the probability that n cross links are broken. (d) Calculate the average energy of the system as a function of temperature. (e) Compute the specific heat of the system as a function of temperature. Use the result for geometric series to obtain simple results: -1 k 0

Answer 1

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