Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 355 / 3

For a nearly planar membrane, the quadratic Helfrich energy of a Fourier mode isThe equipartition theorem and $A-A_{\mathrm{proj}}=\int |\nabla h{|}^2{d}^{2}x/2$ giveUsing the stated molecular and system-size cutoffs, $q_{\max }=\pi /a$ and $q_{\min }=\pi /\sqrt{A}$, gives

Subtracting the two fluctuation areas givesWhen both tensions are much smaller than ${\pi }^2{k}_c/a^2$, the molecular-scale factors cancel to leading order, leavingIncreasing tension suppresses thermal membrane undulations, releasing their hidden excess area into the tether.

Set ${\sigma }_0=0$. Equating the cylindrical tether area $2\pi rL$ to $A\Delta \alpha$ givesorBecause a cylinder has $2H=1/r$, the energy becomesThe two stationarity equations implyEliminating $r$ gives the implicit entropic membrane-tether force--extension relationAs tether length grows, fluctuation area is depleted, tension rises, the tether narrows, and the required force increases.

For a membrane connected to a reservoir at fixed tension, its energy per length is $\pi k_c/r+2\pi \sigma r-F$. Minimization instead givesThe reservoir supplies area without changing $\sigma$, so the force is independent of extension; the finite vesicle has an entropic, strain-stiffening response.