Vortex Flows and Streamline Topology in Curved Biological Membranes

TL;DR

This paper investigates how curvature in biological membranes influences vortex flow dynamics, revealing new topological effects, defect behaviors, and bifurcations driven by membrane geometry.

Contribution

It introduces a geometric Hamiltonian framework for vortex dynamics on curved membranes, highlighting curvature-induced defect creation, fusion, and global rotation effects.

Findings

Curvature causes additional positive defects in vortex flows.

Analytical prediction of stagnation points for two vortices.

Membrane curvature mediates defect binding and induces global rotation.

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.