Hydrodynamic aggregation of membrane inclusions due to non-Newtonian surface rheology

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: in particular, the surface shear viscosity of phospholipids that comprise the membrane depends strongly on the surface pressure. We use the Lorentz reciprocal theorem to extract the effective long-ranged hydrodynamic interaction among membrane inclusions that arises due to such non-trivial 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. Finally, we employ numerical simulations to demonstrate the formation of hexatic crystals due to the effective hydrodynamic interactions within the membrane.