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

At fixed $N$ and $b$, the entropic spring constant is proportional to absolute temperature, because the free-energy penalty for a given loss of conformational entropy is $-T\Delta S$. ThereforeFor the literal printed chain this is $\boxed{1.287\times 10^{-6}\mathrm{N}{\mathrm{m}}^{-1}}$. The same factor multiplies either alternative contour-consistent spring constant from part (d), provided its Kuhn length is held fixed. Real DNA's persistence length, base pairing and solvent interactions can themselves vary with temperature and salt; then $b(T)$ contributes through $k\propto T/b(T{)}^2$ at fixed segment count, or $k\propto T/[L_{c}b(T)]$ if the physical contour is instead held fixed. The simple linear temperature result belongs to the specified ideal-chain model.