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Unzipping of a DNA molecule as a function of temperature

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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:

E(nz, n1,n2,n3,...)

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 ...

Solution Summary

The problem is solved in detail from first principles. Unzipping of a DNA molecule as a function of temperature is examined.

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Statistical Physics

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Note only the energies associated with the unzipping have to be considered in the partition function; any energies from the bending, stretching, etc. of the molecule can be ignored.

A toy model for DNA molecules. A DNA molecule is made of two complementary strands, which are hold together by hydrogen bonds. A very simple model of a DNA molecule is that of a zipper which has N links; each link has a state in which it is closed with energy 0 and a state in which it is open with energy e. We require that the zipper only unzip from one side( say from the left) and the link can only open if all the links to the left of it (1, 2, ..., n-1) are already open.

a. find the partition function
b. find the average number of open links <n> and show that for low temperature kT<<e, <n> is independent of N.

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