The moving grid finite element method applied to cell movement and deformation

• A novel application of the moving grid finite element method to a visco-elastic model for cell movement and deformation.
• The cell moves and deforms on a two-dimensional substrate in complete agreement with experiments.
• The cell can move, translate, expand, contract and relocate, its position continuously changing.
• The generality of the numerical methodology allows us to solve similar complex problems in multi-dimensions.

In this article we present a novel application of the moving grid finite element method [1] for solving a cytomechanical model that describes actin dynamics in order to generate cell movement and deformation. The cytomechanical model describes 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) [2] and the filamentous form is arranged in a paired helix of two protofilaments [3]. By assuming slow deformations of the cell, we validate numerical results by comparing qualitatively with predictions from linear stability theory close to bifurcation points. Far from bifurcation points, the mathematical model and computational algorithm are able to describe and generate the complex cell deformations typically observed in experiments. Our numerical results illustrate cell expansion, cell contraction, cell translation and cell relocation as well as cell protrusions. A key model bifurcation parameter identified is the contractile tonicity formed by the association of actin filaments to the myosin II motor proteins. The robustness, generality and applicability of the numerical methodology allows us to tackle similar problems in developmental biology, biomedicine and plant biology where similar mechanisms are routinely used.