The effect of hydrodynamic interactions on the tracer and gradient diffusion of integral membrane proteins in lipid bilayers

Abstract

Biological membranes can be considered two-dimensional fluids with suspended integral membrane proteins (IMPs). We have calculated the effect of hydrodynamic interactions on the various diffusion coefficients of IMPs in lipid bilayers. The IMPs are modelled as hard cylinders of radius a immersed in a thin sheet of viscosity μ and thickness h bounded by a fluid of low viscosity μ′. We have ensemble averaged the N-body Stokes equations to the pair level and have renormalized them following the methods of Batchelor (1972) and Hinch (1977). The lengthscale for the hydrodynamic interactions is λa = μh / μ′, Which is O (100a), and the slow decay of the interactions introduces new features in the renormalizations compared to the analogous analyses for three-dimensional suspensions of spheres.We have calculated the asymptotic limits for the short- and long-time tracer diffusivities, Ds and Dl, respectively, and for the gradient diffusivity, Dg, for ϕ [Lt ] 1 and λ [Gt ] 1, where ϕ is the IMP area fraction and λ = μh / (μ′a). The diffusivities are \begin{eqnarray*} D_s/D_0 &=& 1-2\phi[1-(1+\ln (2)-9/32)/(\ln(\lambda)-\gamma)], D_l/D_0 &=& D_s/D_0 - 0.07/(\ln(\lambda)-\gamma), D_g/D_0 &=& 1+\phi[-7+(6\ln(2)+7/16+0.37)/(\ln(\lambda)-\gamma)], \end{eqnarray*} where D0 is the diffusivity in the limit of zero area fraction, and γ = 0.577216 is Euler's constant. The results for Dl and Ds differ only slightly. The decrease in Dg/Do as ϕ increases contrasts with the result for spheres for which Dg/Do > 1.