  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 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.
