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

Published online by Cambridge University Press:  11 July 2009

J. Zhu ,
Y.-T. Zhang ,
S. A. Newman  and
M. S. Alber
J. Zhu
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618, USA
Y.-T. Zhang*
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618, USA
S. A. Newman
Affiliation:
Department of Cell Biology and Anatomy, Basic Science Building, New York Medical College, Valhalla, NY 10595, USA
M. S. Alber
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618, USA
*
yzhang10@nd.edu malber@nd.edu
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Abstract

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

Keywords

discontinuous Galerkin finite element methodsreaction-diffusionequationsoperator splittingtriangular meshesmoving domaincomplexgeometrylimb development
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 4 , Issue 4: Morphogenesis , 2009 , pp. 131 - 148
Copyright
© EDP Sciences, 2009

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