Hydrodynamics of a single filament moving in a fluid spherical membrane

TL;DR

This study models the hydrodynamics of a single filament moving in a spherical membrane, revealing how geometry and curvature influence resistance and flow asymmetries, with implications for cellular processes.

Contribution

It extends previous planar membrane models to spherical membranes, analyzing how curvature and size ratios affect filament resistance and flow behavior.

Findings

Flow confinement effects increase with filament length to radius ratio.

Perpendicular drag resistance diverges as filament length approaches membrane circumference.

Filament curvature induces flow asymmetry and torque, coupling translational and rotational dynamics.

Abstract

Dynamic organization of the cytoskeletal filaments and rod-like proteins in the cell membrane and other biological interfaces occurs in many cellular processes. Previous modeling studies have considered the dynamics of a single rod on fluid planar membranes. We extend these studies to the more physiologically relevant case of a single filament moving in a spherical membrane. Specifically, we use a slender-body formulation to compute the translational and rotational resistance of a single filament of length $L$ moving in a membrane of radius $R$ and 2D viscosity $\eta_m$, and surrounded on its interior and exterior with Newtonian fluids of viscosities $\eta^{-}$ and $\eta^{+}$. We first discuss the case where the filament's curvature is at its minimum $\kappa=1/R$. We show that the boundedness of spherical geometry gives rise to flow confinement effects that increase in strength with increasing the ratio of filament's length to membrane radius $L/R$. These confinement flows only result in a mild increase in filament's resistance along its axis, $\xi_{\parallel}$, and its rotational resistance, $\xi_{\Omega}$. As a result, our predictions of $\xi_\parallel$ and $\xi_{\Omega}$ can be quantitatively mapped to the results on a planar membrane. In contrast, we find that the drag in perpendicular direction, $\xi_\perp$, increases superlinearly with the filament's length, when $L/R >1$ and ultimately $\xi_\perp \to \infty$ as $L/R \to \pi$. Next, we consider the effect of the filament's curvature, $\kappa$, on its parallel motion, while fixing the membrane's radius. We show that the flow around the filament becomes increasingly more asymmetric with increasing its curvature. These flow asymmetries induce a net torque on the filament, coupling its parallel and rotational dynamics. This coupling becomes stronger with increasing $L/R$ and $\kappa$.