Dynamic interplay between plasma membrane and underlying cytoskeleton is essential for cell shape change and cell migration. We will investigate the physics of cell shape changes by developing a general model of cell motility that consists of differential equations, specifically those of reaction-diffusion type, posed on the surface cell boundary coupled to an evolution law for the cell membrane. We first develop a theoretical and computational model of polymerization of a single actin filament (F-actin) surrounded by a pool of G-actins and its interaction with plasma membrane, which is accompanied by membrane bending. Biophysical mechanisms leading to and controlling the formation and growth of a single actin filament and filament networks with in the cell will be studied by means of mesoscale modelling frame-work of Brownian dynamics. We will model the plasma membrane by Helfrich theory of membrane elasticity. We will use Finite

Difference and Finite Element approaches to solve the resulting differential equations which cannot be treated analytically. This general frame work of modelling will then be modified to describe specific structures such as Lamellipodia and Podosomes. The theoretical and /or computational result will be compared with and tuned according to the experimental observations.