An Eulerian/XFEM formulation for the large deformation of cortical cell membrane

Most animal cells are surrounded by a thin layer of actin meshwork below their membrane, commonly known as the actin cortex (or cortical membrane). An increasing number of studies have highlighted the role of this structure in many cell functions including contraction and locomotion, but modelling has been limited by the fact that the membrane thickness (about 1 μm) is usually much smaller than the typical size of a cell (10-100 μm). To overcome theoretical and numerical issues resulting from this observation, we introduce in this paper a continuum formulation, based on surface elasticity, that views the cortex as an infinitely thin membrane that can resists tangential deformation. To accurately model the large deformations of cells, we introduced equilibrium equations and constitutive relations within the Eulerian viewpoint such that all quantities (stress, rate of deformation) lie in the current configuration. A solution procedure is then introduced based on a coupled extended finite element approach that enables a continuum solution to the boundary value problem in which discontinuities in both strain and displacement (due to cortical elasticity) are easily handled. We validate the approach by studying the effect of cortical elasticity on the deformation of a cell adhering on a stiff substrate and undergoing internal contraction. Results show very good prediction of the proposed method when compared with experimental observations and analytical solutions for simple cases. In particular, the model can be used to study how cell properties such as stiffness and contraction of both cytoskeleton and cortical membrane lead to variations in cell's surface curvature. These numerical results show that the proposed method can be used to gain critical insights into how the cortical membrane affects cell deformation and how it may be used as a means to determine a cell's mechanical properties by measuring curvatures of its membrane.