The membrane of the cell is highly dynamic and heterogeneous. Just below the cell membrane lies a meshwork of filaments called actin which together with associated motor proteins (myosin) and cross-linking proteins, gives rise to stress on the cell membrane. The active stress generated by the actomyosin cortex, coupled to the cell membrane allows the cell to deform, respond to the environment and play important roles in cell movement and cell division. In the first part of my talk, I will discuss the structural deformations of the cell membrane due to contractile and protruding forces generated by the acto-myosin cortex. We solve a non conservative time dependent Ginzberg-Landau equation for the deformation of the membrane keeping the energetic contribution of bending and stretching and an active force due to the actomyosin cortex. Using numerical and analytical calculations we demonstrate different dynamical phases depending on the activity. The uniform state gets destabilized to form pattern, localized pulsation and traveling waves, behaviors observed in living cells.
The second part of the talk focuses on the problem related to deformation of the extracellular matrix during cell migration. The problem we have dealt with, investigates the role of active force generation by myosin motors as they slide the actin filaments in the cell cortex and these filaments are linked to the extracellular matrix via clutch like proteins such as integrin. We model this system by writing dynamical equations in overdamped limit and perform linear stability analysis on them to identify different dynamical phases. The motor clutch system exhibits a stable, an unstable and stable limit cycle behavior if the active velocity of the motors are changed by changing the ATP concentration of the ambient fluid.