Lipid bilayers are formed from the self-assembly of two layers of amphiphilic lipid molecules. They are excellent model systems for studying the behavior of cellular membranes. The mechanics of lipid bilayers fascinates physicists and engineers because of their in-plane flow and out-of-plane bending. In cellular membranes, lipid bilayers contain a heterogeneous composition of integral and peripheral proteins that perform specific biophysical processes. These proteins are known to generate curvature on the membrane, and they also sense the curvature. Additionally, proteins undergo diffusion and aggregation in the membrane. Experiments have shown that the in-plane viscous flow of lipid influences the dynamics of protein distribution through advection. Therefore, the ability of the protein to deform the membrane, combined with the ability of the membrane curvature and flow to influence protein distribution, leads to a closed coupled problem. In the first part of the thesis, I present a comprehensive theory of the coupled problem of elastic bending of lipid bilayers, diffusion and aggregation of proteins, and in-plane viscous flow of lipids. The curvature generation of the proteins on the membranes is modeled with the help of a spontaneous curvature that emulates the asymmetry in the lipid leaflets. We formulate a Helfrich-like free energy of the membrane, which is modified to include the entropic contribution that leads to diffusion of the protein and the aggregation potential that mimics the forces of protein-protein interaction. The free energy is minimized to get the conservation equation of proteins and the equation of motion for the membrane shape. The mass conservation relation is further extended to account for the binding of proteins from the bulk volume. We perform a stability analysis to find the necessary condition to form an aggregate and compare our system with the Cahn-Hilliard-like formalism. We demonstrate the utility of the model by presenting numerical simulations in the limit of small deformation of the membrane and large deformation axisymmetric geometry. We rigorously investigate the effect of bending and curvature-induced feedback in protein distribution and related energy landscape.

In the second part of the thesis, we model the formation and the shape transition in the membrane tubes generated by aggregated domains BAR-proteins that induce anisotropic curvatures. BAR-proteins are 1-dimensional rod-like proteins that bend the membrane because of their intrinsic shape and binding orientation. We first formulate a continuum bending energy due to anisotropic spontaneous curvatures. Then we include the effect of orientation by the values of spontaneous curvatures. The resultant shape equations that minimize the free energy are solved numerically in an axisymmetric geometry. We observe that the membrane undergoes a snap-through transition of its shape from a tent to a tube, and the transition is observed for all parameters. We further analyze the nature of the transition and report a hysteresis-like behavior that is commonly observed across snap-through transitions in elastic structures.