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\sect{The zipper model for DNA molecule}

The DNA molecule forms a double stranded helix with hydrogen bonds
stabilizing the double helix. Under certain conditions the two
strands get separated resulting in a sharp "phase transition" (in
the thermodynamic limit). As a model for this unwinding, use the
"zipper model" consisting of ${N}$ parallel links which can be
opened from one end.  If the links ${1, 2, 3, ..., p}$ are all open
the energy to open to ${p+1}$ link is ${\varepsilon}$ and if the
earlier links are closed the energy to open the link is infinity.
The last link ${p=N}$ cannot be opened. Each open link can assume
${G}$ orientations corresponding to the rotational freedom about the
bond. Construct the canonical partition function. Find then the
average number of open links ${\langle p\rangle}$ as function of
${x=Gexp\left[-\varepsilon/T\right]}$. Plot ${\langle p\rangle}$ as
function of ${x}$ (assuming ${N}$ very large). What is the value of
${x}$ at the transition? Study ${\langle p\rangle}$ near the
transition: what is its slope as ${N --> \infty ?}$ Derive the
entropy ${S}$. What is it at the transition region and at the
transition? Do the same for the heat capacity. What is the order of
the transition?



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