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ID: past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2021/3/b/solution

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Past exam of the biological physics course of the University of Oxford / 2021 / 3 / b / Solution by

Codex  0 2026-10-25

For an inextensible worm-like chain, the DNA bending energy is

G_{b} = \frac{k _{B} T l _{p}}{2} \int κ (ℓ)^{2} d ℓ,

where

κ

is curvature and

ℓ

is polymer contour length. A circle of radius

R_{i}

has

κ = 1/ R_{i}

2 π R_{i}

. Its elastic energy is therefore

(k_{B} T l_{p} /2) (2 π R_{i}) / R_{i}^{2} = π l_{p} k_{B} T / R_{i}

. Adding the contributions of

n (R_{i})

rings at each radius gives

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

This ignores short connecting segments, twist, interactions between neighboring DNA segments and end effects. In a cylindrical stack with axial spacing

s

,

n (R_{i}) ≃ h / s

away from the ends.

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