Eur. Phys. J. E (2019) 42: 156
https://doi.org/10.1140/epje/i2019-11922-8

Regular Article

Saffman-Delbrück and beyond: A pointlike approach

Laboratoire “Matière et Systèmes Complexes” (MSC), UMR 7057 CNRS, Université de Paris, 75205, Paris Cedex 13, France

^* e-mail: jean-baptiste.fournier@u-paris.fr

Received: 15 May 2019
Accepted: 29 November 2019
Published online: 17 December 2019

Abstract

We show that a very good analytical approximation of Saffman-Delbrück'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 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 < /(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.