Please see the attached file.
In general, the partition function of a system is given by:
Z = Sum over r of Exp(-beta E_r) (1)
where r denotes the possible quantum states of the system and we have to sum over all possible states. In this case we have a big DNA molecule that can move, rotate, have all sorts of complicated internal motions (vibrations) and it can also partially unzip. The state of the DNA molecule will thus be described by a large number of quantum numbers that tell you how much each of the vibrational, rotational, etc, modes are excited. One of these modes is the unzipping mode and you can introduce a (quantum) number, nz, that tells you by how much the molecule has unzipped. If we denote the other quantum numbers by n1, n2, n3, etc. then we can write the energy of some state specified by nz, n1, n2, formally as:
So, formally, the r in Eq. (1) is just a sequence of a large number of integers. Now, unless the quantum numbers are large, the energy will be of the form
E(nz, n1,n2,n3,...) = Ez(nz) + E1(n1) + E2(n2) + E3(n3) + .... (2)
I.e., you can consider each of the modes separately, find the ...
The problem is solved in detail from first principles.