Saffman-Delbrück and beyond: A pointlike approach

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 η2 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 < η2/(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. arXiv:1912.01893v1 [cond-mat.soft]

How to best estimate the viscosity of lipid bilayers

“…For lipid-like dyes, there is often a low dependence of the translational diffusion coefficient on the length of the lipids [51][52][53][54]. According to different authors [55] this could be due to the fact that when the probe moves in a monolayer of a bilayer and if the inter-monolayer friction strongly couples the movement of the probe to the adjacent monolayer, then for all practical purposes, the probe can be treated as a cross-layer particle.…”

Subdiffusive-Brownian crossover in membrane proteins: a generalized Langevin equation-based approach

“…Yet, for time lags exceeding some characteristic time, the standard Brownian dynamics is recovered. This Brownian regime is well described by the celebrated Saffman and Delbrück model ( 24 , 25 ) and its extensions ( 26 , 27 , 28 , 29 , 30 ), and a logarithmic dependence of the diffusion coefficient as a function of the protein radius ( D ∝ log(1/ R )) is found. Nevertheless, in protein-crowded membranes, a deviation from the Saffman and Delbrück model law has been shown ( 31 ), and one finds D ∝ 1/ R .…”

“…Yet, for time lags exceeding some characteristic time, the standard Brownian dynamics is recovered. This Brownian regime is well described by the celebrated Saffman and Delbr€ uck model (24,25) and its extensions (26)(27)(28)(29)(30), and a logarithmic dependence of the diffusion coefficient as a function of the protein radius (D f log(1/R)) is found. Nevertheless, in protein-crowded membranes, a deviation from the Saffman and Delbr€ uck model law has been shown (31), and one finds D f 1/ R. The crossover from the subdiffusive to the standard Brownian dynamics can take place over a quite large time window, and the transition onset strongly depends on packing and crowding, ranging from a tenth to hundreds of nanoseconds for lipids in protein-free membranes or proteins in membranes at infinite protein dilution up to arbitrarily long timescales for crowded real systems (20,22,32).…”

Basic Residue Clusters in Intrinsically Disordered Regions of Peripheral Membrane Proteins: Modulating 2D Diffusion on Cell Membranes

“…For larger membrane inserted objects, the dependency approaches a Stokes-like form in which the diffusion coefficient is proportional to (1/R) [48,49]. Recently, an analytical function using a pointlike form was found to correctly predict the diffusion of proteins interacting with only one or both lípid leaflets, including the possible deformation from planarity of the lípid bilayer [50].…”