Two protein species A and B diffuse independently in water at (so ) and react the…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Biophysics & Medical Physics · Physical foundations of the living cell

Forces, Machines and Signals, Ch Diffusion: friction, capture and the crowded cytoplasm

Problem

Two protein species A and B diffuse independently in water at T=298 KT=298\,\mathrm{K} (so kBT=4.11×10−21 Jk_BT = 4.11\times10^{-21}\,\mathrm{J}) and react the instant their surfaces touch. Protein A is a sphere of radius a1=3.0 nma_1=3.0\,\mathrm{nm} and protein B a sphere of radius a2=2.0 nma_2=2.0\,\mathrm{nm}; both have Stokes drag coefficients ζi=6πηai\zeta_i = 6\pi\eta a_i in water of viscosity η=1.0×10−3 Pa s\eta = 1.0\times10^{-3}\,\mathrm{Pa\,s} (quoted, for a sphere). ((a)) Each particle performs an independent three-dimensional random walk. Writing the relative position as ρ=r1−r2\boldsymbol\rho = \mathbf r_1 - \mathbf r_2, use the fact that independent displacements have additive variances to show that ρ\boldsymbol\rho itself diffuses with coefficient Drel=D1+D2D_{\rm rel} = D_1+D_2, and that the pair therefore meets no faster than a single particle of diffusion coefficient D1+D2D_1+D_2 reaching a fixed absorbing sphere of radius a1+a2a_1+a_2, giving k=4π(D1+D2)(a1+a2)k = 4\pi(D_1+D_2)(a_1+a_2). ((b)) Use the Einstein relation to find D1D_1 and D2D_2. ((c)) Evaluate kk in m3 s−1\mathrm{m^3\,s^{-1}} per pair and convert it to M−1 s−1\mathrm{M^{-1}\,s^{-1}}, showing the conversion factor.

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