Dynamic organization of the cytoskeletal filaments and rodlike proteins in the cell membrane and other biological interfaces occurs in many cellular processes, including cell division, membrane transport, and morphogenesis. The filament dynamics are determined, in part, by their membrane-mediated hydrodynamic interactions. 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 result only in a mild increase in filament's resistance along its axis, ${\xi }_{\parallel }$, and its rotational resistance, ${\xi }_{\mathrm{\Omega }}$. As a result, our predictions of ${\xi }_{\parallel }$ and ${\xi }_{\mathrm{\Omega }}$ can be quantitatively mapped to the results on a planar membrane, when the momentum transfer length scale is modified from ${\ell }_0=({\eta }^{+}+{\eta }^{-})/{\eta }_m$ in planar membranes to ${\ell }^{★}={({\ell }_0^{-1}+R^{-1})}^{-1}$. In contrast, we find that the drag in the 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 $\kappa$.

• Received 2 March 2022
• Accepted 5 July 2022

DOI:https://doi.org/10.1103/PhysRevFluids.7.084004