Stability of n-dimensional patterns in a generalized Turing system: implications for biological pattern formation

The stability of Turing patterns in an n-dimensional cube (0 ,π ) n is studied, where n 2. It is shown by using a generalization of a classical result of Ermentrout concerning spots and stripes in two dimensions that under appropriate assumptions only sheet-like or nodule-like structures can be stable in an n-dimensional cube. Other patterns can also be stable in regions comprising products of lower-dimensional cubes and intervals of appropriate length. Stability results are applied to a new model of skeletal pattern formation in the vertebrate limb.

[1]  H L Frisch,et al.  Dynamics of skeletal pattern formation in developing chick limb. , 1979, Science.

[2]  S A Kauffman,et al.  Control of sequential compartment formation in Drosophila. , 1978, Science.

[3]  Qing Nie,et al.  Do morphogen gradients arise by diffusion? , 2002, Developmental cell.

[4]  B. Ermentrout.  Stripes or spots? Nonlinear effects in bifurcation of reaction—diffusion equations on the square , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[5]  S A Newman,et al.  On multiscale approaches to three-dimensional modelling of morphogenesis , 2005, Journal of The Royal Society Interface.

[6]  Jesús A. Izaguirre,et al.  COMPUCELL, a multi-model framework for simulation of morphogenesis , 2004, Bioinform..

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

[8]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[9]  S A Newman,et al.  Lineage and pattern in the developing vertebrate limb. , 1988, Trends in genetics : TIG.

[10]  G. Iooss,et al.  Elementary stability and bifurcation theory , 1980 .

[11]  Peter Grindrod,et al.  Patterns and Waves: The Theory and Applications of Reaction-Diffusion Equations , 1991 .

[12]  James A Glazier,et al.  Interplay between activator-inhibitor coupling and cell-matrix adhesion in a cellular automaton model for chondrogenic patterning. , 2004, Developmental biology.

[13]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[14]  Edmund J. Crampin,et al.  Reaction-Diffusion Models for Biological Pattern Formation , 2001 .

[15]  S A Newman,et al.  Morphogenetic differences between fore and hind limb precartilage mesenchyme: relation to mechanisms of skeletal pattern formation. , 1994, Developmental biology.

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

[17]  H. Swinney,et al.  Transition from a uniform state to hexagonal and striped Turing patterns , 1991, Nature.

[18]  T. K. Callahan,et al.  Symmetry-breaking bifurcations on cubic lattices , 1997 .

[19]  James A Glazier,et al.  Dynamical mechanisms for skeletal pattern formation in the vertebrate limb , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[20]  Dulos,et al.  Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. , 1990, Physical review letters.

[21]  David H. Sattinger,et al.  Group theoretic methods in bifurcation theory , 1979 .

[22]  D. Sattinger.  Topics in stability and bifurcation theory , 1973 .