  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-2D
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.
