Vortex flows and streamline topology in curved biological membranes

Abstract

When considering flows in biological membranes, they are usually treated as flat, though more often than not, they are curved surfaces, even extremely curved, as in the case of the endoplasmic reticulum. Here, we study the topological effects of curvature on flows in membranes. Focusing on a system of many point vortical defects, we are able to cast the viscous dynamics of the defects in terms of a geometric Hamiltonian. In contrast to the planar situation, the flows generate additional defects of positive index. For the simpler situation of two vortices, we analytically predict the location of these stagnation points. At the low curvature limit, the dynamics resemble that of vortices in an ideal fluid, but considerable deviations occur at high curvatures. The geometric formulation allows us to construct the spatio-temporal evolution of streamline topology of the flows resulting from hydrodynamic interactions between the vortices. The streamlines reveal novel dynamical bifurcations leading to spontaneous defect-pair creation and fusion. Further, we find that membrane curvature mediates defect binding and imparts a global rotation to the many-vortex system, with the individual vortices still interacting locally.

Hydrodynamics of a single filament moving in a fluid spherical membrane

“…where s ∈ [−L/2, L/2] denotes the filament's arclength, x m and x 0 m are points on the membrane surface, and G(x m , x 0 m (s)) is the Green's function of membrane-3D fluids coupled system in response to a point-force applied on the membrane at position x 0 m . Assuming flow incompressibility on the membrane surface and the adjacent 3D flows and negligible inertia, the associated momentum and continuity equation for the membrane and 3D fluid domains are [36,37]:…”

“…Here, θ ∈ [0, π] and φ ∈ (0, 2π] are the polar and azimuthal angles in spherical coordinate. The detailed derivation of the Green's function is outlined in [37] and [36]. The expressions for all components of the Green's function (G θθ , G θφ , G φθ , G φφ ) are provided in Appendix A for completeness.…”

“…We note that applying a net force on filament bound to the spherical membrane produces a net torque on the membrane, which leads to a net rotational flow of the entire system [36]. This effect is absent in a planar membrane [37]. The resistance and mobility functions are defined based on the relative velocity of the filament with respect to the ambient fluid.…”

“…In the special cases of spherical and cylindrical membranes the Gaussian curvature remains constant over the surface. This greatly simplifies the equations, allowing for analytical calculations of fundamental solutions to a point-force, a point-torque, and a torque-dipole [35][36][37]. These fundamental solutions can be used to describe the flow disturbances due to the presence of other bodies in the membrane, which is similar to the singular methods for 3D Stokes flow suspensions [38,39].…”

“…At larger curvatures R/ 0 1, the mobility is reduced with the logarithmic term now being defined as ln(R/a) in contrast to ln( 0 /a) on planar membranes. In a recent study, Samanta and Oppenheimer [37] used the same framework to study the effect of the membrane curvature on the hydrodynamic interactions (HIs) and the ensued nonlinear dynamics of rotors bound to spherical membranes. Bagaria and Samanta [40] investigated the curvature confinement effect on the aggregation of force dipoles on spherical membrane.…”