A Conservative Finite Element ALE Scheme for Mass-Conserving Reaction-Diffusion Equations on Evolving Two-Dimensional Domains

Abstract

Mass-conservative reaction-diffusion systems have recently been proposed as a general framework to describe intracellular pattern formation. These systems have been used to model the conformational switching of proteins as they cycle from an inactive state in the cell cytoplasm, to an active state at the cell membrane. The active state then acts as input to downstream effectors. The paradigm of activation by recruitment to the membrane underpins a range of biological pathways - including G-protein signalling, growth control through Ras and PI 3-kinase, and cell polarity through Rac and Rho; all activate their targets by recruiting them from the cytoplasm to the membrane. Global mass conservation lies at the heart of these models reflecting the property that the total number of active and inactive forms, and targets, remains constant. Here we present a conservative arbitrary Lagrangian Eulerian (ALE) finite element method for the approximate solution of systems of bulk-surface reaction-diffusion equations on an evolving two-dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a moving mesh partial differential equation (MMPDE) approach. Global conservation of the fully discrete finite element solution is established independently of the ALE velocity field and the time step size. The developed method is applied to model problems with known analytical solutions; these experiments indicate that the method is second-order accurate and globally conservative. The method is further applied to a model of a single cell migrating in the presence of an external chemotactic signal.