  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.
