Modelling and simulations of a viscous model for cell migration

This thesis presents a mathematical model for cell migration that couples a system of reaction-advection-diffusion equations describing the interations between F-actin and myosin II toa force balance equation describing the velocity vector of the actin-myosin network. Cell migration plays a crucial role in many biological processes. In eukaryotic cells, this migration is largely powered by a system of actin and myosin II. At the leading edge of the cell, cross-inked actin filaments polymerise by adding actin monomers to their ends while at the back of the cell, myosin II binds to a bundle of actin filaments. These processes create protrusive and contractile forces generated by the action of actin polymerisation and myosin II contraction. Based on the idea that cell migration is powered by the actin-myosin network of the cell, we formulate model equations for migrating cell which comprises reaction-advection-diffusion equations that are coupled to a force balance describing the velocity vector of the network. This is a viscous model with active stresses coming from the actin-myosin system. These equations describing the migrating cells are highly nonlinear partial differential equations with no closed form solutions and we therefore result to numerical methods in order to compute the approximate solution. F-actin and myosin II solution are the solution for the reaction-advection-diffusion equations while the speeds of the cell come from the solution of the force balance equation. We begin simulations on a unit disk at zero initial velocity with different data for the initial conditions of F-actin and myosin II concentrations. We also vary some parameters at a time while keeping all the other parameters constant: for example (i) total amount of actin ?tota and (ii) contraction coefficient for myosin II ?0m. Actin polymerisation causes protrusive stress at the cell periphery which results in expansion of the cell. We observe that the initial conditions play an important role in the spatiotemporal dynamics of F-actin as well as the evolution of the cell shape. Actin changes from the active state (F-actin) to inactive state (G-actin) and vice-versa through polymerisation and depolymerisation processes and hence the total amount of actin is conserved at any time. We note that in our model, myosin II only diffuses inside the cell and exerts contractile stress in the cell. Its total concentration in the entire cell is conserved.