Rotational mobility in spherical membranes: The interplay between Saffman-Delbr\"uck length and inclusion size

TL;DR

This paper extends classical membrane mobility theory to spherical membranes, calculating the rotational mobility of finite-sized particles as a function of SD length and inclusion size, with applications to biological membrane dynamics.

Contribution

It introduces a semi-analytical method for computing rotational mobility of particles in spherical membranes, accounting for inclusion size and SD length, extending flat membrane models.

Findings

Mobility depends on SD length and inclusion size

Method recovers flat-space mobility as a special case

Applicable to biological membrane processes

Abstract

The mobility of particles in fluid membranes is a fundamental aspect of many biological processes. In a 1975 paper [1], Saffman and Delbr\"uck demonstrated how the presence of external Stokesian solvents is crucial in regularising the apparently singular flow within an infinite flat membrane. In the present paper, we extend this classical work and compute the rotational mobility of a rigid finite-sized particle located inside a spherical membrane embedded in Stokesian solvents. Treating the particle as a spherical cap, we solve for the flow semi-analytically as a function of the Saffman-Delbr\"uck (SD) length (ratio of membrane to solvent viscosity) and the solid angle formed by the particle. We study the dependence of the mobility and flow on inclusion size and SD length, recovering the flat-space mobility as a special case. Our results will be applicable to a range of biological problems including rotational Brownian motion, the dynamics of lipid rafts, and the motion of aquaporin channels in response to water flow. Our method will provide a novel way of measuring a membrane's viscosity from the rotational diffusion of large inclusions, for which the commonly used planar Saffman-Delbr\"uck theory does not apply.