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

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.