  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, the entropy
of protein distribution, along with an explicit aggregation potential and
derive the governing equations for the coupled system. 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,
when coupled with curvature generation, can alter the landscape of
membrane-protein interactions.
