  Morphological change of bilayer membrane in vivo is not a spontaneous
procedure but modulated by various types of proteins in general. Most of these
modulations are associated with the localization of related proteins in the
crowded lipid environment in bilayer membrane. This work presents an
mathematical model for the localization of multiple species of diffusion
molecules on membrane surfaces. We start with the energetic description of the
distributions of molecules on curved membrane surface, by assembling the
bending energy of bilayer membrane and the entropic energy of diffusive
molecules. We introduce the spontaneous curvature of molecules in membrane, and
define the spontaneous curvature of bilayer membrane as a function of the
molecule concentrations on membrane surfaces. This connection gives rise to a
drift-diffusion equation to govern the gradient flows of the surface molecule
concentrations. We recast the energetic formulation and the related governing
equations in the Eulerian framework by using a phase field function that
defines the membrane morphology. Computational simulations with the proposed
mathematical model and related numerical techniques predict the molecular
localization on membrane surfaces at locations with preferred mean curvature.
