Hydrodynamics in curved membranes: The effect of geometry on particulate mobility

Abstract

We determine the particulate transport properties of fluid membranes with nontrivial geometries that are surrounded by viscous Newtonian solvents. Previously, this problem in membrane hydrodynamics was discussed for the case of flat membranes by Saffman and Delbrück [P. G. Saffman and M. Delbrück, Proc. Natl. Acad. Sci. U.S.A. 72, 3111 (1975)]. We review and develop the formalism necessary to consider the hydrodynamics of membranes with arbitrary curvature and show that the effect of local geometry is twofold. First, local Gaussian curvature introduces in-plane viscous stresses even for situations in which the velocity field is coordinate-independent. Secondly, even in the absence of Gaussian curvature, the geometry of the membrane modifies the momentum transport between the bulk fluids and the membrane. We illustrate these effects by examining in detail the mobilities of particles bound to spherical and cylindrical membranes. These two examples provide experimentally testable predictions for particulate mobilities and membrane velocity fields on giant unilamellar vesicles and membrane tethers. Finally, we use the examples of spherical and cylindrical membranes to demonstrate how the global geometry and topology of the membrane influences the membrane velocities and the particle mobilities.

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“…23,35) Thus, the motion of an object in a fluid membrane is considered to cause significant flow in the surrounding 3D fluids if it is much larger than = , but negligible flow if it is much smaller than = . 24) From Eq. (4.21), the dimensionless diffusion coefficient for the smallest wave-number is found to be …”

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“…The low-Reynolds-number hydrodynamics for incompressible fluid flow, with velocity u α , in a curved membrane with viscosity η, is described by the following equations [4,10]:…”

“…Additionally, K is the local Gaussian curvature, p is the pressure, and we have also introduced a Stokeslet term [3,4,12,13],…”

“…In this work we use covariant, classical, low-Reynolds number, hydrodynamic theory in order to rigorously elucidate the intrinsic curvature dependence of the diffusion coefficient of an embedded inclusion, in a generally (albeit weakly) curved fluid membrane, or surface. This is carried out using a Stokeslet-type approach, which has been successfully applied in previous work to calculate the diffusion constant for a small particle embedded in non-planar membranes [3,4].…”

Processivity and collectivity of biomolecular motors extracting membrane nanotubes

“…The coefficient ζ is nontrivial, because it involves the following: (i) the size λ of the protein domain moving in the membrane and (ii) the local radius of curvature r of the membrane relative to the Saffman-Delbrück length η ≡ η m /η w (the scale below which η m dominates the viscous dissipation [24]). According to recent experiments [25] and theoretical analyses [24,26], the axial drag coefficient in a PHYSICAL REVIEW E 86, 010901(R) (2012) cylindrical membrane of radius r η is given by [24] …”