  Membrane bending is an extensively studied problem from both modeling and
experimental perspectives because of the wide implications of curvature
generation in cell biology. Many of the curvature generating aspects in
membranes can be attributed to interactions between proteins and membranes.
These interactions include protein diffusion and formation of aggregates due to
protein-protein interactions in the plane of the membrane. Recently, we
developed a model that couples the in-plane flow of lipids and diffusion of
proteins with the out-of-plane bending of the membrane. Building on this work,
here, we focus on the role of explicit aggregation of proteins on the surface
of the membrane in the presence of membrane bending and diffusion. We develop a
comprehensive framework that includes lipid flow, membrane bending energy, the
entropy of protein distribution, and an explicit aggregation potential and
derive the governing equations. We compare this framework to the Cahn-Hillard
formalism to predict the regimes in which the proteins form patterns on the
membrane. We demonstrate the utility of this model using numerical simulations
to predict how aggregation and diffusion, coupled with curvature generation,
can alter the landscape of membrane-protein interactions.
