Mathematical modelling and numerical simulations of actin dynamics in the eukaryotic cell.

The aim of this article is to study cell deformation and cell movement by considering both the mechanical and biochemical properties of the cortical network of actin filaments and its concentration. Actin is a polymer that can exist either in filamentous form (F-actin) or in monometric form (G-actin) (Chen et al. in Trends Biochem Sci 25:19-23, 2000) and the filamentous form is arranged in a paired helix of two protofilaments (Ananthakrishnan et al. in Recent Res Devel Biophys 5:39-69, 2006). By assuming that cell deformations are a result of the cortical actin dynamics in the cell cytoskeleton, we consider a continuum mathematical model that couples the mechanics of the network of actin filaments with its bio-chemical dynamics. Numerical treatment of the model is carried out using the moving grid finite element method (Madzvamuse et al. in J Comput Phys 190:478-500, 2003). Furthermore, by assuming slow deformations of the cell, we use linear stability theory to validate the numerical simulation results close to bifurcation points. Far from bifurcation points, we show that the mathematical model is able to describe the complex cell deformations typically observed in experimental results. Our numerical results illustrate cell expansion, cell contraction, cell translation and cell relocation as well as cell protrusions. In all these results, the contractile tonicity formed by the association of actin filaments to the myosin II motor proteins is identified as a key bifurcation parameter.