Hydrodynamic Coupling of Particle Inclusions Embedded in Curved Lipid Bilayer Membranes

We develop theory and computational methods to investigate particle inclusions embedded within curved lipid bilayer membranes. We consider the case of spherical lipid vesicles where inclusion particles are coupled through (i) intramembrane hydrodynamics, (ii) traction stresses with the external and trapped solvent fluid, and (iii) intermonolayer slip between the two leaflets of the bilayer. We investigate relative to flat membranes how the membrane curvature and topology augment hydrodynamic responses. We show how both the translational and rotational mobility of protein inclusions are effected by the membrane curvature, ratio of intramembrane viscosity to solvent viscosity, and inter-monolayer slip. For general investigations of many-particle dynamics, we also discuss how our approaches can be used to treat the collective diffusion and hydrodynamic coupling within spherical bilayers.Comment: 32 pages, double-column format, 15 figure

Topological fluid mechanics of point vortex motions

Topological techniques are used to study the motions of systems of point vortices in the infinite plane, in singly-periodic arrays, and in doubly-periodic lattices. The reduction of each system using its symmetries is described in detail. Restricting to three vortices with zero net circulation, each reduced system is described by a one degree of freedom Hamiltonian. The phase portrait of this reduced system is subdivided into regimes using the separatrix motions, and a braid representing the topology of all vortex motions in each regime is computed. This braid also describes the isotopy class of the advection homeomorphism induced by the vortex motion. The Thurston-Nielsen theory is then used to analyse these isotopy classes, and in certain cases strong conclusions about the dynamics of the advection can be made

Why Use Sobolev Metrics on the Space of Curves

We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient numerical methods for higher order Sobolev type metrics is an extremely desirable goal