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

An inextensible freely jointed chain of $N$ links of Kuhn length $b$ has contour length $L_c=Nb$ and cannot be extended beyond it at any finite force. There is an explicit numerical contradiction in the printed parameters:The discrepancy is present in the original page, not introduced by digitization. To obtain $26\mu \mathrm{m}$ with $186$ links requires $b\simeq 140\mathrm{nm}$; to obtain it with $b=7\mathrm{nm}$ requires $N\simeq 3714$. No single inextensible FJC can satisfy all three stated numbers. The graph's much larger extension is an immediate counterexample to treating the literal $N,b$ as an accurate global fit.

Separately, accepting the graph's nominal $26\mu \mathrm{m}$ contour as an effective low-force model, its continuation to $32.7\mu \mathrm{m}$ is about $26\%$ beyond that contour. An inextensible ideal chain contains orientational entropy only; it excludes stretching of segments, energetic compliance of bonds, changes in helix structure or base pairing, and heterogeneous attachments. An extensible freely jointed chain or a semiflexible-chain model with appropriate energetic terms can describe some departures. Calibration or contour-length assignment can also be wrong. The graph alone does not identify which mechanism contributes, and its displayed moderate forces do not justify assigning a specific high-force DNA structural transition. Arithmetic inconsistency in $Nb$ is not explained away by these physical mechanisms.