Someone compares the bending energy of two bilayer vesicles of bending modulus

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Macromolecules, membranes and cell architecture · Macromolecules, membranes and cell architecture

Soft Matter at the Thermal Scale, Ch Membranes, fluctuations and soft matter in biology

Problem

Someone compares the bending energy of two bilayer vesicles of bending modulus κb=20kBT\kappa_b = 20k_BT, one of radius R1=50 nmR_1 = 50\,\mathrm{nm} and one of radius R2=5 μmR_2 = 5\,\mu\mathrm{m}, and argues: "Each point on the sphere has mean curvature c=1/Rc = 1/R, so the bending energy per unit area is 12κb(2c)2=2κb/R2\tfrac12\kappa_b(2c)^2 = 2\kappa_b/R^2. Since a membrane bends along its rim, I multiply this energy density by the vesicle's circumference 2πR2\pi R to get the total energy, F=4πκb/RF = 4\pi\kappa_b/R. So F1/F2=R2/R1=100F_1/F_2 = R_2/R_1 = 100: the small vesicle costs a hundred times more bending energy than the large one." ((a)) Find the error in this reasoning. ((b)) Give the correct bending energy for each vesicle and compare F1F_1 and F2F_2.

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