Saffman-Delbr\"uck and beyond: a pointlike approach

TL;DR

This paper presents an analytical approximation of the Saffman-Delbrück law for membrane inclusion mobility, extending it to account for bilayer structure, intermonolayer friction, and spontaneous curvature effects using a pointlike force approach.

Contribution

It introduces a simple pointlike force method to generalize the SD law, incorporating bilayer properties, intermonolayer friction, and spontaneous curvature effects.

Findings

SD law approximated with small wavelength cutoff

Mobility influenced by intermonolayer friction when b is low

Total friction includes SD and membrane deformation contributions

Abstract

We show that a very good analytical approximation of Saffman-Delbr\"uck's (SD) law (mobility of a bio-membrane inclusion) can be obtained easily from the velocity field produced by a pointlike force in a 2D fluid embedded in a solvent, by using a small wavelength cutoff of the order of the particle's radius~$a$. With this method, we obtain analytical generalizations of the SD law that take into account the bilayer nature of the membrane and the intermonolayer friction $b$. We also derive, in a calculation that consistently couples the quasi-planar two-dimensional (2D) membrane flow with the 3D solvent flow, the correction to the SD law arising when the inclusion creates a local spontaneous curvature. For an inclusion spanning a flat bilayer, the SD law is found to hold simply upon replacing the 2D viscosity $\eta_2$ of the membrane by the sum of the monolayer viscosities, without influence of $b$ as long as $b$ is above a threshold in practice well below known experimental values. For an inclusion located in only one of the two monolayers (or adhering to one monolayer), the SD law is influenced by $b$ when $b<\eta_2/(4a^2)$. In this case, the mobility can be increased by up to a factor of two, as the opposite monolayer is not fully dragged by the inclusion. For an inclusion creating a local spontaneous curvature, we show that the total friction is the sum of the SD friction and that due to the pull-back created by the membrane deformation, a point that was assumed without demonstration in the literature.