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 curvature elastic deformation of lipid bilayers. The proteins in the membrane are modeled 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.
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