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Mass-conservative reaction-diffusion systems have recently been proposed as ageneral framework to describe intracellular pattern formation. These systemshave been used to model the conformational switching of proteins as they cyclefrom an inactive state in the cell cytoplasm, to an active state at the cellmembrane. The active state then acts as input to downstream effectors. Theparadigm of activation by recruitment to the membrane underpins a range ofbiological pathways - including G-protein signalling, growth control throughRas and PI 3-kinase, and cell polarity through Rac and Rho; all activate theirtargets by recruiting them from the cytoplasm to the membrane. Global massconservation lies at the heart of these models reflecting the property that thetotal number of active and inactive forms, and targets, remains constant. Herewe present a conservative arbitrary Lagrangian Eulerian (ALE) finite elementmethod for the approximate solution of systems of bulk-surfacereaction-diffusion equations on an evolving two-dimensional domain. Fundamentalto the success of the method is the robust generation of bulk and surfacemeshes. For this purpose, we use a moving mesh partial differential equation(MMPDE) approach. Global conservation of the fully discrete finite elementsolution is established independently of the ALE velocity field and the timestep size. The developed method is applied to model problems with knownanalytical solutions; these experiments indicate that the method issecond-order accurate and globally conservative. The method is further appliedto a model of a single cell migrating in the presence of an externalchemotactic signal.

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