Hydrodynamics of a single filament moving in a supported spherical bilayer

Dynamic assembly of biopolymers and rod-like proteins on membranes is key to many cellular processes. The protein dynamics is determined, in part, by its membrane-mediated hydrodynamic resistance/mobility. Using a slender-body formulation, we compute the translational resistance of a single rod of length L moving in the outer layer of a bilayer membrane with 2D viscosity η_m, supported by a rigid inner sphere of radius R, and surrounded by a Newtonian fluid of viscosity η on the exterior. This geometry models membrane-coated beads that are frequently used in in-vitro studies. The outer- and inner-layer are coupled through a Brinkman-like friction term with a coefficient μ. We find that the dimensionless resistance in the directions parallel to the rod’s axis and perpendicular to it, γ_‖,⊥/(4πη_m), depends only on the ratio L/L^†, where L^† is the length scale over which momentum is transferred from the membrane to the exterior fluid and is determined as the minimum of the following three length scales: L^†=min(η_m/η, R, (η_m/μ)^½). Furthermore, we study the relaxation spectrum of transverse correlation of a semi-flexible filament on the geometry and find a non-monotonic transition regime between two dynamical regimes of long and short wave-vector around qR~1 (q is wave-vector). This novel behavior arises from flow confinement in spherical geometry and is absent in planar membranes.