The resistivity and mobility functions for a model system of two equal-sized proteins in a lipid bilayer

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Abstract
We have calculated the mobility and resistivity functions for two equal-sized cylinders moving in a thin viscous sheet surrounded by a lower-viscosity fluid. These functions describe the hydrodynamic interactions between proteins embedded in a lipid bilayer surrounded by an aqueous solution. For a protein of radius a embedded in a biological membrane of thickness h and viscosity μ, a key parameter λ = μh/aμ′, which characterizes the viscosity ratio between the bilayer and surounding solution, is O(100). The method of solution for the hydrodynamic interactions differs depending on the separation distance, r, between the cylinders. When r = O(a), the particles are in the near-field regime, and the solution of the Stokes equations is divided between an inner and outer domain based on the asymptotically large value of λ. The inner solution neglects the flow in the lower-viscosity fluid and is solved numerically using a bipolar expansion. The outer solution is based on the fluid flows in all phases, but the cylinders are approximated by net forces. When r [Gt ] O(a), the particles are in the far-field regime, and we use the method of reflections to solve for the hydrodynamic interactions. The uniformly valid approximations, constructed from a combination of the near-field and far-field solutions, agree with the analytic solutions obtained within the lubrication and far-field regimes. Our results show that the hydrodynamic interactions between the cylinders are long range, perarting on a lengthscale of O(λ). The range of the hydrodynamic interactions is much longer than that of non-hydrodynamic interparticle forces, suggesting that hydrodynamic interactions will be significant determinants of the structure and dynamics of biological membranes.