A Meshfree Technique for Numerical Simulation of Reaction-Diffusion Systems in Developmental Biology

In this work, element free Galerkin (EFG) method is posed for solving nonlinear, reaction-diffusion systems which are often employed in mathematical modeling in developmental biology. A predicator-corrector scheme is applied, to avoid directly solving of coupled nonlinear systems. The EFG method employs the moving least squares (MLS) approximation to construct shape functions. This method uses only a set of nodal points and a geometrical description of the body to discretize the governing equation. No mesh in the classical sense is needed. However a background mesh is used for integration purpose. Numerical solutions for two cases of interest, the Schnakenberg model and the Gierer-Meinhardt model, in various regions is presented to demonstrate the effects of various domain geometries on the resulting biological patterns.

[1]  I. Graham,et al.  Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth , 2001, Journal of mathematical biology.

[2]  Philip K. Maini,et al.  Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains , 2007, J. Comput. Phys..

[3]  P. Villon,et al.  GENERALIZING THE FEM: DIFFUSE APPROXIMATION AND DIFFUSE ELEMENTS , 1992 .

[4]  P. Maini,et al.  Reaction and diffusion on growing domains: Scenarios for robust pattern formation , 1999, Bulletin of mathematical biology.

[5]  Wing Kam Liu,et al.  Reproducing kernel particle methods for structural dynamics , 1995 .

[6]  Mehdi Dehghan,et al.  The finite volume spectral element method to solve Turing models in the biological pattern formation , 2011, Comput. Math. Appl..

[7]  M J Gómez-Benito,et al.  Computational simulation of fracture healing: influence of interfragmentary movement on the callus growth. , 2007, Journal of biomechanics.

[8]  Anotida Madzvamuse,et al.  Pigmentation pattern formation in butterflies: experiments and models. , 2003, Comptes rendus biologies.

[9]  Andrew J. Wathen,et al.  A moving grid finite element method applied to a model biological pattern generator , 2003 .

[10]  S. Atluri,et al.  A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach , 1998 .

[11]  Ted Belytschko,et al.  Numerical integration of the Galerkin weak form in meshfree methods , 1999 .

[12]  Satya N. Atluri,et al.  A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method , 1998 .

[13]  J. Murray.  A Pre-pattern formation mechanism for animal coat markings , 1981 .

[14]  S. Atluri,et al.  A meshless local boundary integral equation (LBIE) method for solving nonlinear problems , 1998 .

[15]  Ted Belytschko,et al.  An introduction to programming the meshless Element F reeGalerkin method , 1998 .

[16]  Aly-Khan Kassam.  Solving reaction-diffusion equations 10 times faster , 2003 .

[17]  J. Verwer,et al.  Numerical solution of time-dependent advection-diffusion-reaction equations , 2003 .

[18]  J. Mantilla,et al.  Turing pattern formation for reaction–convection–diffusion systems in fixed domains submitted to toroidal velocity fields , 2011 .

[19]  P K Maini,et al.  A two-dimensional numerical study of spatial pattern formation in interacting Turing systems , 1999, Bulletin of mathematical biology.

[20]  J. Monaghan.  Smoothed particle hydrodynamics , 2005 .

[21]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[22]  J. Schnakenberg,et al.  Simple chemical reaction systems with limit cycle behaviour. , 1979, Journal of theoretical biology.

[23]  H. Meinhardt,et al.  A theory of biological pattern formation , 1972, Kybernetik.

[24]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[25]  Oden,et al.  An h-p adaptive method using clouds , 1996 .

[26]  L. Libersky,et al.  Smoothed Particle Hydrodynamics: Some recent improvements and applications , 1996 .

[27]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[28]  Mark S. Gockenbach,et al.  Understanding and implementing the finite element method , 1987 .

[29]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[30]  Ted Belytschko,et al.  Overview and applications of the reproducing Kernel Particle methods , 1996 .

[31]  M. L. Martins,et al.  Reaction-diffusion model for the growth of avascular tumor. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Andrew J. Wathen,et al.  A Moving Grid Finite Element Method for the Simulation of Pattern Generation by Turing Models on Growing Domains , 2005, J. Sci. Comput..

[33]  E. Oñate,et al.  A FINITE POINT METHOD IN COMPUTATIONAL MECHANICS. APPLICATIONS TO CONVECTIVE TRANSPORT AND FLUID FLOW , 1996 .

[34]  S. Atluri,et al.  The meshless local Petrov-Galerkin (MLPG) method , 2002 .

[35]  Junjie Wei,et al.  Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system ✩ , 2009 .

[36]  Thilo Gross,et al.  Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations. , 2007, Journal of theoretical biology.

[37]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[38]  H. Meinhardt.  Models of biological pattern formation , 1982 .

[39]  A. Turing.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[40]  L. L. Yaw,et al.  Co-rotational meshfree formulation for large deformation inelastic analysis of two -dimensional structural systems , 2008 .

[41]  M. Dehghan,et al.  The Finite Point Method for Reaction-Diffusion Systems in Developmental Biology , 2011 .

[42]  S. Kondo,et al.  A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus , 1995, Nature.