Past exam of the biological physics course of the University of Oxford / 2019 / 5 / d / Solution

The weak-force spring constant isUsing the literal printed $N=186$, $b=7\mathrm{nm}$ at $25^{\circ }\mathrm{C}=298.15\mathrm{K}$ givesThis is a calculation with the quoted parameters, not a self-consistent fit to the plotted $26\mu \mathrm{m}$ contour. If instead $N=186$ and $L_c=26\mu \mathrm{m}$ are retained, then $b=L_c/N$ and $k=3.40\times 10^{-9}\mathrm{N}{\mathrm{m}}^{-1}$. Retaining $b=7\mathrm{nm}$ and $L_c=26\mu \mathrm{m}$ gives $k=6.79\times 10^{-8}\mathrm{N}{\mathrm{m}}^{-1}$. These distinct alternatives quantify the unresolved parameter inconsistency rather than silently choosing a correction.

The restoring force is entropic. Stretching selects a small fraction of the many disordered orientations, reducing conformational entropy. The Gaussian chain free-energy cost is $\Delta \mathcal{F}\simeq 3k_{B}T{R}^2/(2N{b}^2)$, whose gradient gives the Hooke-law coefficient above. In an ideal inextensible chain, individual bonds need not change their length to generate this force.