Estimate the lateral diffusion coefficient of a receptor dimer of radius spanning a…

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Forces, Machines and Signals, Ch The fluid membrane: bending, flicker and shape

Problem

Estimate the lateral diffusion coefficient of a receptor dimer of radius a=5 nma = 5\,\mathrm{nm} spanning a bilayer with ηmh=4×10−10 Pa s m\eta_m h = 4\times10^{-10}\,\mathrm{Pa\,s\,m} (membrane surface viscosity times thickness), bathed on both sides in cytoplasm and extracellular fluid of viscosity ηw=1 mPa s\eta_w = 1\,\mathrm{mPa\,s}, at body temperature T=310 KT = 310\,\mathrm{K} (kBT=4.28 pN nmk_BT = 4.28\,\mathrm{pN\,nm}). Use the quoted Saffman-Delbruck result D=kBT4πηmh[ln⁡ ⁣(ηmhηwa)−γ],γ=0.577,D = \frac{k_BT}{4\pi\eta_m h}\left[\ln\!\left(\frac{\eta_m h}{\eta_w a}\right) - \gamma\right], \qquad \gamma = 0.577, valid for aa much smaller than the Saffman-Delbruck length ηmh/ηw\eta_m h/\eta_w. ((a)) List the assumptions the estimate relies on and check that aa is indeed small compared with the Saffman-Delbruck length. ((b)) Estimate DD for the dimer, to two significant figures. ((c)) Compare it with the lipid value D≈5 μm2 s−1D \approx 5\,\mathrm{\mu m^2\,s^{-1}} (radius 0.5 nm0.5\,\mathrm{nm}) and comment on how weakly DD depends on size, contrasting with what would happen in a three-dimensional Stokes-Einstein estimate.

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