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Cellular membranes are elastic lipid bilayers that contain a variety of proteins, including ion channels, receptors and scaffolding proteins. These proteins are known to diffuse in the plane of the membrane and to influence the bending of the membrane. Experiments have shown that lipid flow in the plane of the membrane is closely coupled with the diffusion of proteins. Thus, there is a need for a comprehensive framework that accounts for the interplay between these processes. Here, we present a theory for the coupled in-plane viscous flow of lipids, diffusion of transmembrane proteins and elastic deformation of lipid bilayers. The proteins in the membrane are modelled such that they influence membrane bending by inducing a spontaneous curvature. We formulate the free energy of the membrane with a Helfrich-like curvature elastic energy density function modified to account for the chemical potential energy of proteins. We derive the conservation laws and equations of motion for this system. Finally, we present results from dimensional analysis and numerical simulations and demonstrate the effect of coupled transport processes in governing the dynamics of membrane bending and protein diffusion. 

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.