Past exam of the biological physics course of the University of Oxford / 2015 / 4 / b / Solution

Fit each pre-unfolding branch with its own polymer contour length $L$ and a persistence length $L_p$, allowing for the measured extension zero. The worm-like chain force-extension interpolation has the limitsThus the low-extension slope and the high-extension curvature constrain the flexible segment's length and bending stiffness. The persistence length is related to the bending modulus by $\kappa =k_{B}T{L}_p$. Successive increases in fitted $L$ measure contour released by unfolding, after subtracting the original folded domain's projected length. With an independently known contour increment per residue, this also estimates how many residues became available. A common fitted $L_p$ across branches supports a common type of unfolded chain; a systematic change can signal altered interactions or a poor model.

For a thermal worm-like chain, $⟨\mathbf{t}(s)\cdot \mathbf{t}(0)⟩=e^{-s/L_p}$. Consequently $s=L_p$ gives $⟨\cos \theta ⟩=1/e$, and $\arccos (1/e)=68.4^{\circ }$. The latter is an angle associated with the mean cosine, not generally the mean angle itself. For example, angles $0$ and $90^{\circ }$ with equal probabilities have mean angle $45^{\circ }$, but the inverse cosine of their mean cosine is $60^{\circ }$. This distinction makes the source's informal angular description precise.

Without Figure B, no numerical fit parameters can be read. The requested physical interpretation of those parameters follows without its numerical data.