Collective hydrodynamics of driven particles in viscous membranes: effect of non-Newtonian surface rheology and particle shape

Biological membranes are self-assembled complex fluid interfaces that host proteins, molecular motors, and other macromolecules essential for cellular function. These membranes have a distinct in-plane fluid response with a surface viscosity that has been well characterized. The resulting quasi-two-dimensional fluid dynamical problem describes the motion of embedded proteins or particles. However, the viscous response of biological membranes is often non-Newtonian and the inclusions are rarely simple discs. We use the Lorentz reciprocal theorem to extract the effective long-ranged hydrodynamic interaction among membrane inclusions that arises due to a particular class of surface-pressure dependent rheology. We show that the corrective force that emerges ties back to the interplay between membrane flow and non-constant viscosity, which suggests a mechanism for biologically favorable protein aggregation within membranes. We quantify and describe the mechanism for such a large-scale concentration instability using a mean-field model, which we verify with numerical simulations. Finally, we extend the mean-field model to describe the role of particle shape. The anisotropic mobility and orientability of rod-like particles lead to co-ordinated dynamics that depend on the ratio of membrane viscosity to that of the surrounding fluid. New mechanisms for separation and aggregation emerge from this analysis, suggesting creative strategies to tune assembly on viscous membranes.