Question

8. DNA Zipper Model: a very simple model of the unwinding of double-stranded DNA (a process which is required before translation or copying) treats the DNA molecule as a "zipper" with N sequential links. Each link has a state where it is closed (with a nucleotide bonded to the complementary strand), with energy 0, and a state where it is open (unbonded with the complementary strand), with positive energy $\mathcal{E}$. However, in this model we require that the DNA can only "unzip" from the left end, meaning that the kth link can only be open if links 1, ..., k \textendash{} 1 are already open. (a) Find the partition function Z(T, N) for this system, at temperature T. (b) Approximate the average number of open links as a function of T, for $k_B T < \mathcal{E}$, retaining the leading-order temperature dependence. (c) Approximate the average number of open links for $k_B T >> \mathcal{E}$, retaining the dominant temperature dependence.

          8. DNA Zipper Model: a very simple model of the unwinding of double-stranded DNA (a process which
is required before translation or copying) treats the DNA molecule as a "zipper" with N sequential links.
Each link has a state where it is closed (with a nucleotide bonded to the complementary strand), with
energy 0, and a state where it is open (unbonded with the complementary strand), with positive energy
$\mathcal{E}$. However, in this model we require that the DNA can only "unzip" from the left end, meaning that
the kth link can only be open if links 1, ..., k \textendash{} 1 are already open.
(a) Find the partition function Z(T, N) for this system, at temperature T.
(b) Approximate the average number of open links as a function of T, for $k_B T < \mathcal{E}$, retaining the
leading-order temperature dependence.
(c) Approximate the average number of open links for $k_B T >> \mathcal{E}$, retaining the dominant temperature
dependence.

8. DNA Zipper Model: a very simple model of the unwinding of double-stranded DNA (a process which
is required before translation or copying) treats the DNA molecule as a "zipper" with N sequential links.
Each link has a state where it is closed (with a nucleotide bonded to the complementary strand), with
energy 0, and a state where it is open (unbonded with the complementary strand), with positive energy
ℰ. However, in this model we require that the DNA can only "unzip" from the left end, meaning that
the kth link can only be open if links 1, ..., k – 1 are already open.
(a) Find the partition function Z(T, N) for this system, at temperature T.
(b) Approximate the average number of open links as a function of T, for kB T < ℰ, retaining the
leading-order temperature dependence.
(c) Approximate the average number of open links for kB T >> ℰ, retaining the dominant temperature
dependence.

Added by Mary G.

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