Past exam of the biological physics course of the University of Oxford / 2022 / 5 / d / Solution

Identifying the fitted limiting extension with the polymer contour length givesThese are statistical links of the freely jointed chain, not necessarily chemical polymer repeat units. A measured maximum just before detachment need not equal the true contour length exactly, so the identification is itself an estimate.

The stated temperature request mixes two distinct quantities. At unchanged $N$ and $b$, the true freely jointed chain maximum is geometric:In contrast, the low-force relation $⟨z⟩=N{b}^{2}f/(3k_{B}T)$ predicts a ratio at the same small force, not a new maximum:Formally multiplying $500\mathrm{nm}$ by this ratio gives about $400\mathrm{nm}$, but $500\mathrm{nm}=Nb$ is outside the small-extension domain, so that is not a justified maximum-extension prediction. A concrete internally consistent counterexample uses the same chain at $f=0.10\mathrm{pN}$: the low-force extensions are about $2.59\mathrm{nm}$ and $2.07\mathrm{nm}$, while both contour lengths remain $500\mathrm{nm}$. Any temperature dependence of the segment length, solvent interactions, attachment strength or rupture force needs information absent from the model.