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

Write $u=x/L$ and $A=k_{B}T{L}_p$. The worm-like chain force-extension interpolation isA fit to an elastic rising branch estimates its contour length $L$ and, if not fixed independently, its persistence length $L_p$. Comparing successive fitted contour lengths gives the length released by a domain and helps distinguish whole-repeat unfolding from partial events. $L_p$ estimates the flexibility of the unfolded chain and the bending rigidity $A$; at a stated temperature, $A=k_{B}T{L}_p$.

For small extension, expansion gives $f\simeq [3k_{B}T/(2L_{p}L)]x$, so the effective entropic spring constant is $3k_{B}T/(2L_{p}L)$. Close to full extension, $f\sim k_{B}T/[4L_p(1-x/L{)}^2]$, giving strong strain stiffening. These regimes constrain the fit differently and correlate its parameters if the measured interval is short. The fit does not by itself identify a unique folding barrier or unfolding kinetic rate.

Persistence-length convention. In a three-dimensional isotropic worm-like chain the standard definition is $⟨\mathbf{t}(s)\cdot \mathbf{t}(0)⟩=e^{-s/L_p}$. Thus at $s=L_p$, $\arccos ⟨\cos \theta ⟩=\arccos (e^{-1})\simeq 68.4^{\circ }$. This is not literally the mean angle $⟨\theta ⟩$. For a simple counterexample, equally probable angles $0^{\circ }$ and $90^{\circ }$ have mean angle $45^{\circ }$ but $\arccos ⟨\cos \theta ⟩=60^{\circ }$. The source's angle description is shorthand for the tangent-correlation convention.