Application of discontinuous Galerkin finite element methods for vertebrate limb pattern formation

Major outstanding questions regarding vertebrate limb development are how the numbers of skeletal elements along the proximodistal (P-D) and anteroposterior (A-P) axes are determined and how the shape of a growing limb affects skeletalelement formation. A mechanism based on local autoregulation of a molecular activator of cell aggregation coupled to a laterally acting inhibitor (a LALI system), is consistent with in vivo and in vitro experimental results and provides qualitative interpretations of several genetic anomalies affecting limb development.Nonlinear reaction-diffusion systems are often employed in mathematical modeling to study the activator-inhibitor subnetwork in developmental biology. These systems are usually highly stiff in both diffusion and reaction terms and are typically considered on multidimensional complex geometrical domains because of complex shapes of embryos. Using an empirically based mathematical representation of such reaction-diffusion mechanism and combining discontinuous Galerkin(DG) finite element methods with Strang type symmetrical operator splitting technique that permits simulation of LALI systems in domains of varying shape and size, we show that major aspects of the limb pattern, including those of aberrant and evolutionary transitional forms, emerge in a robust fashion from the inherent self-organizing properties of a core skeletal patterning mechanism in different geometric settings, without a requirement for positional information.