Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T14:04:40.967Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alber, M., Hentschel, H.G.E., Kazmierczak, B., Newman, S.A.. Existence of solutions to a new model of biological pattern formation. J. Math. Anal. Appl., 308 (2005), No. 1, 175–194. CrossRef
Alber, M., Glimm, T., Hentschel, H.G.E., Kazmierczak, B., Zhang, Y.-T., Zhu, J., Newman, S.A.. The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb. Bulletin of Mathematical Biology, 70 (2008), No. 2, 460–483. CrossRef
Cheng, Y., Shu, C.-W.. A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Mathematics of Computation, 77 (2008), No. 262, 699–730. CrossRef
B. Cockburn, G. Karniadakis, C.-W. Shu. The development of discontinuous Galerkin methods, in Discontinuous Galerkin Methods: Theory, Computation and Applications, B. Cockburn, G. Karniadakis, and C.-W. Shu, Editors. Lecture Notes in Computational Science and Engineering, 11 (2000), Springer, 3–50.
Cockburn, B., Shu, C.-W.. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing, 16 (2001), No. 3, 173–261. CrossRef
Cockburn, B., Shu, C.-W.. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numererical Analysis, 35 (1998), No. 6, 2440–2463. CrossRef
Hentschel, H.G.E., Glimm, T., Glazier, J.A., Newman, S.A.. Dynamical mechanisms for skeletal pattern formation in the vertebrate limb. Proc. R. Soc. B, 271 (2004), No. 1549, 1713–1722. CrossRef
Hundsdorfer, W.. Trapezoidal and midpoint splittings for initial-boundary value problems. Mathematics of Computation, 67 (1998), No. 223, 1047–1062. CrossRef
P.K. Kundu. Fluid Mechanics. Academic Press, Inc, London, 1990.
Levy, D., Shu, C.-W., Yan, J.. Local discontinuous Galerkin methods for nonlinear dispersive equations. Journal of Computational Physics, 196 (2004), No. 2, 751–772. CrossRef
Madzvamuse, A., Wathen, A.J., Maini, P.K.. A moving grid finite element method applied to a model biological pattern generator. Journal of Computational Physics, 190 (2003), No. 2, 478–500. CrossRef
Madzvamuse, A., Maini, P.K., Wathen, A.J.. A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains. J. Sci. Comput., 24 (2005), No. 2, 247–262. CrossRef
Madzvamuse, A.. Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains. Journal of Computational Physics, 214 (2006), No. 1, 239–263. CrossRef
Nelson, C.E., Morgan, B.A., Burke, A.C., Laufer, E., DiMambro, E., Murtaugh, L.C., Gonzales, E., Tessarollo, L., Parada, L.F., Tabin, C.. Analysis of Hox gene expression in the chick limb bud. Development, 122 (1996), No. 5, 1449–1466.
S.A. Newman, G.B. Müller. Origination and innovation in the vertebrate limb skeleton: an epigenetic perspective. J. Exp. Zoolog. B Mol. Dev. Evol. 304 (2005), No. 6, 593–609.
Newman, S.A., Bhat, R.. Activator-inhibitor dynamics of vertebrate limb pattern formation. Birth Defects Res C Embryo Today, 81 (2007), No. 4, 305–319. CrossRef
Newman, S.A., Christley, S., Glimm, T., Hentschel, H.G.E., Kazmierczak, B., Zhang, Y.-T., Zhu, J., Alber, M.. Multiscale models for vertebrate limb development. Curr. Top. Dev. Biol., 81 (2008), 311–340. CrossRef
Ros, M.A., Lyons, G.E., Mackem, S., Fallon, J.F.. Recombinant limbs as a model to study homeobox gene regulation during limb development. Dev. Biol., 166 (1994), No. 1, 59–72. CrossRef
Strang, G.. On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 8 (1968), No. 3, 506–517. CrossRef
Summerbell, D.. A descriptive study of the rate of elongation and differentiation of the skeleton of the developing chick wing. J. Embryol. Exp. Morphol., 35 (1976), No. 2, 241–260.
Svingen, T., Tonissen, K.F.. Hox transcription factors and their elusive mammalian gene targets. Heredity, 97 (2006), No. 2, 88–96. CrossRef
Tickle, C.. Patterning systems - from one end of the limb to the other. Dev. Cell, 4 (2003), No. 4, 449–458. CrossRef
Xu, Y., Shu, C.-W.. Local discontinuous Galerkin methods for three classes of nonlinear wave equations. Journal of Computational Mathematics, 22 (2004), No. 2, 250–274.
Xu, Y., Shu, C.-W.. Local discontinuous Galerkin methods for nonlinear Schrodinger equations. Journal of Computational Physics, 205 (2005), No. 1, 72–97. CrossRef
Xu, Y., Shu, C.-W.. Local discontinuous Galerkin methods for two classes of two dimensional nonlinear wave equations. Physica D, 208 (2005), No. 1-2, 21–58. CrossRef
Xu, Y., Shu, C.-W.. Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. Computer Methods in Applied Mechanics and Engineering, 195 (2006), No. 25-28, 3430–3447. CrossRef
Yan, J., Shu, C.-W.. A local discontinuous Galerkin method for KdV type equations. SIAM Journal on Numerical Analysis, 40 (2002), No. 2, 769–791. CrossRef
Yan, J., Shu, C.-W.. Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. Journal of Scientific Computing, 17 (2002), No. 1-4, 27–47. CrossRef
Zhu, J., Zhang, Y.-T., Newman, S.A., Alber, M.. Application of discontinuous Galerkin methods for reaction-diffusion systems in developmental biology. Journal of Scientific Computing, 40 (2009), No. 1-3, 391–418. CrossRef
Zwilling, E.. Development of fragmented and of dissociated limb bud mesoderm. Dev. Biol., 9 (1964), No. 1, 20–37. CrossRef