Past exam of the biological physics course of the University of Oxford / 2016 / 4 / b / Solution

On each branch the worm-like chain force-extension interpolation relates force to the extension of the already released chain. Fitting a sufficiently long branch estimates its polymer contour length $L$ and, if the data constrain it independently, its persistence length $L_p$. In practice $L_p$ may be fixed across branches so that the changes in $L$ are better identified. Consecutive fitted polymer contour lengths reveal how much additional chain became available at an protein denaturation event; using a justified polymer contour length per residue can then estimate residue counts. A projected folded length and the pulling geometry must be included before assigning a structural boundary.

Writing $u=x/L$, expansion at small extension givesNear full extension, $f\sim k_{B}T/[4L_p(1-x/L{)}^2]$. Larger $L$ shifts a branch toward larger extension at fixed force. The effective flexural rigidity is $\kappa =k_{B}T{L}_p$, with units of force times length squared. This is entropic straightening of a worm-like chain; bond stretching and barrier rupture themselves are not included in the formula.

The persistence-length wording requires care. The three-dimensional model gives $⟨\mathbf{t}(s)\cdot \mathbf{t}(0)⟩=e^{-s/L_p}$. At $s=L_p$, $\arccos ⟨\cos \theta ⟩=\arccos (e^{-1})=68.4^{\circ }$. This is not a statement that $⟨\theta ⟩=68.4^{\circ }$. For example, angles zero and $90^{\circ }$ with equal probabilities have mean angle $45^{\circ }$, but $\arccos ⟨\cos \theta ⟩=60^{\circ }$. The distinction prevents a false literal interpretation of the printed definition.