A Finite Element Model Based on Discontinuous Galerkin Methods on Moving Grids for Vertebrate Limb Pattern Formation

Skeletal patterning in the vertebrate limb, i.e., the spatiotemporal regulation of cartilage differentiation (chondrogenesis) during embryogenesis and regeneration, is one of the best studied examples of a multicellular developmental process. Recently [Alber et al., The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb, Bulletin of Mathematical Biology, 2008, v70, pp. 460-483], a simplified two-equation reaction-diffusion system was developed to describe the interaction of two of the key morphogens: the activator and an activator-dependent inhibitor of precartilage condensation formation. A discontinuous Galerkin (DG) finite element method was applied to solve this nonlinear system on complex domains to study the effects of domain geometry on the pattern generated [Zhu et al., Application of Discontinuous Galerkin Methods for reaction-diffusion systems in developmental biology, Journal of Scientific Computing, 2009, v40, pp. 391-418]. In this paper, we extend these previous results and develop a DG finite element model in a moving and deforming domain for skeletal pattern formation in the vertebrate limb. Simulations reflect the actual dynamics of limb development and indicate the important role played by the geometry of the undifferentiated apical zone.