A patch of a floppy lipid bilayer held at zero tension has bending modulus at

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

A patch of a floppy lipid bilayer held at zero tension has bending modulus κb=10kBT\kappa_{\rm b} = 10k_BT at T=298 KT = 298\,\mathrm{K}, for which kBT=4.11×10−21 Jk_BT = 4.11\times10^{-21}\,\mathrm{J}. Take the patch to be square with side L=2 μmL = 2\,\mu\mathrm{m} and periodic boundary conditions, so the allowed Fourier wavevectors are q=(2π/L)(nx,ny)q = (2\pi/L)(n_x, n_y) with nx,nyn_x, n_y integers, and the smallest is qmin⁡=2π/Lq_{\min} = 2\pi/L. The mean-square amplitude of a mode of wavevector qq at zero tension is ⟨∣hq∣2⟩=kBT/(Aκbq4)\langle|h_q|^2\rangle = k_BT/(A\kappa_{\rm b}q^4), with A=L2A = L^2 the patch area. ((a)) Starting from ⟨h2⟩=∑q⟨∣hq∣2⟩\langle h^2\rangle = \sum_q\langle|h_q|^2\rangle, replace the sum over modes by an integral over qq-space (mode density A/(2π)2A/(2\pi)^2) with lower cutoff qmin⁡q_{\min}, dropping the molecular-scale upper cutoff because the integral is dominated by its lower limit, and so derive ⟨h2⟩=kBTL2/(16π3κb)\langle h^2\rangle = k_BTL^2/(16\pi^3\kappa_{\rm b}). ((b)) Evaluate the root-mean-square height fluctuation ⟨h2⟩1/2\langle h^2\rangle^{1/2} for the given patch.

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