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

Correction of the printed limits: ordinary positive polymer contour length requires integration from $R_{\mathrm{in}}$ to $R_{\mathrm{out}}$, because $R_{\mathrm{out}}>R_{\mathrm{in}}$ and the ring count is positive. For example, taking $n(R)=1$, $R_{\mathrm{out}}=2$ and $R_{\mathrm{in}}=1$ makes the printed outward-to-inward integral negative. Thus its sign cannot be retained as a physical length.

For hexagonally packed strands choose axial ring spacing $s$ and radial layer spacing $\sqrt{3}s/2$. With $n=h/s$ and radial layer density $2/(\sqrt{3}s)$,The same continuum replacement in b givesEliminating the inner radius yieldsThe resisting packaging force is the marginal free energy per added polymer contour length, at fixed $h$, $s$ and $R_{\mathrm{out}}$:The force opposing insertion is positive; its contribution to outward ejection has the same magnitude. The domain is $0\le L<2\pi h{R}_{\mathrm{out}}^2/(\sqrt{3}{s}^2)$. The divergence as the inner hole closes reflects the singular curvature of ideal rings and marks breakdown of the continuum model. If spacing also relaxes, the full free energy must include interstrand interactions before optimization; simply varying $s$ in this bending-only formula is insufficient.