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ID: past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2016/5/b/solution

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Past exam of the biological physics course of the University of Oxford / 2016 / 5 / b / Solution by

Codex  0 2026-10-25

Use the elastic-rod bending functional for an abstract inextensible worm-like chain,

G_{b} = \frac{κ}{2} \int C (s^{'})^{2} d s^{'}, κ = l_{p} k_{B} T .

A circular ring of radius

R

1/ R

2 π R

, hence energy

πκ / R

. With

n (R_{i})

rings of each radius, add their energies:

G_{b} = π l_{p} k_{B} T i \sum \frac{n ( R _{i} )}{R _{i}} .

For a cylindrical annulus of height

h

, choose neighboring rings separated vertically by

s

. Then

n = h / s

, while adjacent radial layers are separated by

3 s /2

. Replacing the radial sum by its continuum integral gives

L = \frac{2}{3 s} \int_{R_{in}}^{R_{out}} 2 π R \frac{h}{s} d R = \frac{2 πh}{3 s ^{2}} (R_{out}^{2} - R_{in}^{2}),

and

G_{b} = \frac{2 π l _{p} k _{B} T h}{3 s ^{2}} \int_{R_{in}}^{R_{out}} \frac{d R}{R} = \frac{2 π l _{p} k _{B} T h}{3 s ^{2}} ln \frac{R _{out}}{R _{in}} .

Eliminate the inner radius to obtain

G_{b} = \frac{π l _{p} k _{B} T h}{3 s ^{2}} ln [\frac{R _{out}^{2}}{R _{out}^{2} - 3 L s ^{2} / ( 2 πh )}] .

The dimensions are energy, and the expression is positive for

0 < L < 2 πh R_{out}^{2} / (3 s^{2})

. It tends to zero with

L

and diverges as the inner hole closes in this continuum model. The divergence signals the growing cost of forcing an inextensible chain to zero radius; it is not a physical prediction at arbitrarily small molecular scales. Interstrand, hydration and configurational contributions have been excluded, and disconnected ideal rings represent local curvature without modeling the connections between them.

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