The effect of curvature and topology on membrane hydrodynamics

Abstract

We study the mobility of extended objects (rods) on a spherical liquid-liquid interface to show how this quantity is modified in a striking manner by both the curvature and the topology of the interface. We present theoretical calculations and experimental measurements of the interfacial fluid velocity field around a moving rod bound to the crowded interface of a water-in-oil droplet. By using different droplet sizes, membrane viscosities, and rod lengths, we show that the viscosity mismatch between the interior and exterior fluids leads to a suppression of the fluid flow on small droplets that cannot be captured by the flat interface predictions.The dynamics of mobile inclusions in lipid membranes are fundamental to a variety of biological processes, including signal transduction [1] and the endocytosis of bacterial toxins [2]. Membrane inclusions, such as proteins [3] or lipid "rafts" [4], are in many cases significantly larger than the lipids making up the membrane, so their dynamics can be studied in terms of macroscopic objects moving in a continuum fluid environment. Additionally, elucidating the mobilities and hydrodynamic interactions of colloidal particles at a fluid-fluid interface has important technological ramifications for the design and formation of membranes and capsules composed of particles assembled on droplets [5].Low Reynolds number hydrodynamics in viscous membranes or interfaces differs substantially from the betterknown problem of hydrodynamics in three dimensions. Because of the coupling between the two-dimensional membrane and its surrounding viscous solvent, in-plane momentum in the membrane is lost to the surrounding fluid. The flows induced in the surrounding fluid generate nonlocal couplings between the membrane velocity and stress. The net result of these effects is to introduce an inherent length scale 0 -the Saffman-Delbrück (SD) length -which is set by the ratio of the (2D) membrane viscosity η m to the (3D) fluid viscosity η, 0 ∼ η m /η [6]. For cellular plasma membranes 0 1µm [7]. In contrast, low-Reynolds number hydrodynamics in 3D fluids is a scale-invariant theory. The existence of an inherent length scale in membrane and interfacial hydrodynamics has complex and rather subtle effects on a variety of problems, including protein diffusion in cell membranes [6,8], the flow of monolayers through channels [9], the dynamics of monolayer domains [10], microrheology of fluid-fluid interfaces [11], and the mobilities of rigid and flexible extended objects in membranes [12].In this letter we explore the effect of nontrivial interfacial geometry and topology on the hydrodynamics of viscous interfaces. We find two principal results with broad implications for particulate transport on curved or spherical interfaces. The first is that the compact topology of a spherical interface fundamentally alters the nature of the 2D interfacial velocity field. On a sphere, any vector field must include at least two vortices [13]. In contrast, there are no such singularities in the velocity field on...

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Membrane Viscosity Determined from Shear-Driven Flow in Giant Vesicles

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Mobility Measurements Probe Conformational Changes in Membrane Proteins due to Tension

“…Here, we instead seek a classical hydrodynamic explanation for the additional drag, and hence reduced diffusion, of curvature-inducing proteins. In order to take account of the geometry of the membrane, we employ a covariant formulation of low Reynolds number hydrodynamics in two dimensions [7][8][9][10]. In doing so, we neglect both membrane fluctuations and any chemical interactions occurring between the protein and the amphiphilic molecules that make up the membrane [11,12].…”