Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 355 / 2 / Solution

In the small-slope approximation the worm-like chain bending energy isThe clamped end obeys $h(0)=h_x(0)=0$, while a force-free and torque-free tip has the natural conditions $h_{xx}(L)=h_{xxx}(L)=0$. A static transverse tip force $f$ produces the cantilever compliance $h(L)/f=L^3/(3A)$. The equipartition theorem, or equivalently the static fluctuation--response relation, therefore gives the exact tip variance

For the dynamics, resistive-force theory gives the transverse drag per lengthat logarithmic accuracy and the stochastic beam equationLet ${\varphi }_n$ be clamped--free bending modes with $q_{n}L={\alpha }_n$, whereWriting $N_n={\int }_0^L{\varphi }_n^{2}dx$, independent thermally driven modes giveThe first mode contains almost all of the tip variance, so

For a microtubule, $A=k_{B}T{L}_p$. With $L=10\mu \mathrm{m}$ and $L_p=3\mathrm{mm}$,Taking water at room temperature and $a=0.5\mu \mathrm{m}$ gives ${\zeta }_{\perp }\simeq 3.6\times 10^{-3}\mathrm{Pa}\mathrm{s}$ andwith an order-one uncertainty from the logarithmic slender-body drag approximation.