A molecule undergoes a three-dimensional random walk: at each step

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Biophysics & Medical Physics · Physical foundations of the living cell

Life at Thermal Energy, Ch Random walks, diffusion and the crowded cell

Problem

A molecule undergoes a three-dimensional random walk: at each step, of duration Δt\Delta t, it moves independently by ±ℓ\pm\ell along xx, by ±ℓ\pm\ell along yy, and by ±ℓ\pm\ell along zz, with the sign of each chosen independently and with equal probability of ++ or −-. ((a)) Using the one-dimensional result ⟨x2⟩=Nℓ2\langle x^2\rangle = N\ell^2 derived in the text for NN independent steps, write down ⟨y2⟩\langle y^2\rangle and ⟨z2⟩\langle z^2\rangle for the same walk, and hence show that ⟨r2⟩=⟨x2⟩+⟨y2⟩+⟨z2⟩=6Dt\langle r^2\rangle = \langle x^2\rangle + \langle y^2\rangle + \langle z^2\rangle = 6Dt, where D=ℓ2/(2Δt)D = \ell^2/(2\Delta t) and t=NΔtt = N\Delta t. ((b)) A protein diffusing freely in the plane of a lipid bilayer effectively takes independent steps along only two directions (xx and yy), so its mean square displacement obeys ⟨r2⟩=4Dt\langle r^2\rangle = 4Dt. If the protein's diffusion coefficient in the membrane is D=5.0×10−13 m2 s−1D = 5.0\times10^{-13}\,\mathrm{m^2\,s^{-1}}, find the time for its mean square displacement in the plane to reach (2.0 μm)2(2.0\,\mathrm{\mu m})^2.

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