Past exam of the biological physics course of the University of Oxford / 2017 / 3 / c / Solution

After a domain unfolds, its backbone joins the same continuous chain as previously unfolded material. The new branch differs mainly by an added contour length, while the local backbone bending statistics remain similar. It is therefore consistent to fit successive branches with different $L_j$ and one common $L_p\simeq 0.8\mathrm{nm}$. Tension is transmitted through serial portions of the same chain; one should not add their tensile forces as though they were independent parallel springs.

The persistence length is short compared with the contour increment per repeat, so each released domain contributes many flexible bending segments. Stretching suppresses those thermal modes and raises free energy through loss of entropy, which is precisely the worm-like-chain mechanism. The common $L_p$ is a property of the unfolded backbone under the measurement conditions, not the folded domain's axial length or the number of domains already released. With $T=300\mathrm{K}$ only as an illustrative value, $k_{B}T{L}_p\simeq 3.3\mathrm{pN}\mathrm{n}{\mathrm{m}}^2$.

A common persistence-length fit is evidence for approximately repeat-independent chain flexibility over the observed branches; it is not proof that a continuum inextensible model describes every molecular scale. Residual folded structure, heterogeneous linkers, force-induced backbone stretching, and very short-chain effects can cause deviations. The formal force divergence at $x=L$ signals the model's inextensibility assumption, not a physical infinite tensile force.