Reaction-Diffusion Models for Biological Pattern Formation

We consider the use of reaction-diffusion equations to model biological pattern formation and describe the derivation of the reaction-terms for several illustrative examples. After a brief discussion of the Turing instability in such systems we extend the model formulation to incorporate domain growth. Comparisons are drawn between solution behaviour on growing domains and recent results on self-replicating patterns on domains of fixed size.

[1]  James D. Murray,et al.  On a Model Mechanism for the Spatial Patterning of Teeth Primordia in the Alligator , 1996 .

[2]  B Bunow,et al.  Pattern formation by reaction-diffusion instabilities: application to morphogenesis in Drosophila. , 1980, Journal of theoretical biology.

[3]  John E. Pearson,et al.  Self-replicating spots in reaction-diffusion systems , 1997 .

[4]  Li S-R,et al.  On a Model Mechanism for the Spatial Patterning of Teeth Primordia in the Alligator , 2022 .

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

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

[7]  H. Swinney,et al.  Experimental observation of self-replicating spots in a reaction–diffusion system , 1994, Nature.

[8]  H. Engel.  Chemical Chaos , 1995 .

[9]  Christopher A. Gilligan,et al.  Spatial heterogeneity in three species, plant–parasite–hyperparasite, systems , 1998 .

[10]  J. E. Pearson.  Complex Patterns in a Simple System , 1993, Science.

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

[12]  P K Maini,et al.  Stripe formation in juvenile Pomacanthus explained by a generalized turing mechanism with chemotaxis. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Valery Petrov,et al.  Excitability, wave reflection, and wave splitting in a cubic autocatalysis reaction-diffusion system , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

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

[15]  E. Sel'kov,et al.  Self-oscillations in glycolysis. 1. A simple kinetic model. , 1968, European journal of biochemistry.

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

[17]  Andrew J. Wathen,et al.  A model for colour pattern formation in the butterfly wing of Papilio dardanus , 2000, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[18]  Stephen K. Scott,et al.  Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C , 1984 .

[19]  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.

[20]  P K Maini,et al.  Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation. , 1991, Bulletin of mathematical biology.

[21]  Daishin Ueyama,et al.  A skeleton structure of self-replicating dynamics , 1997 .

[22]  T. Lacalli.  DISSIPATIVE STRUCTURES AND MORPHOGENETIC PATTERN IN UNICELLULAR ALGAE , 1981 .

[23]  Lee,et al.  Lamellar structures and self-replicating spots in a reaction-diffusion system. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[24]  Erik Mosekilde,et al.  WAVE-SPLITTING IN THE BISTABLE GRAY-SCOTT MODEL , 1996 .

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

[26]  I. Epstein,et al.  Modeling of Turing Structures in the Chlorite—Iodide—Malonic Acid—Starch Reaction System , 1991, Science.

[27]  Hans Meinhardt,et al.  The Algorithmic Beauty of Sea Shells , 1998, The Virtual Laboratory.

[28]  G. Oster,et al.  Mechanical aspects of mesenchymal morphogenesis. , 1983, Journal of embryology and experimental morphology.

[29]  V. V. Osipov,et al.  Theory of self-replication and granulation of spike autosolitons , 1996 .

[30]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[31]  Stephen K. Scott,et al.  Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability , 1983 .

[32]  P K Maini,et al.  Mathematical Biology , 2006 .

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

[34]  G F Oster,et al.  Pattern formation models and developmental constraints. , 1989, The Journal of experimental zoology.

[35]  Reynolds,et al.  Dynamics of self-replicating patterns in reaction diffusion systems. , 1994, Physical review letters.

[36]  D Thomas,et al.  Artificial enzyme membranes. , 1976, Methods in enzymology.

[37]  P K Maini,et al.  Cellular mechanisms of pattern formation in the developing limb. , 1991, International review of cytology.

[38]  J. Boissonade,et al.  Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction , 1991 .

[39]  R. A. Barrio,et al.  Confined Turing patterns in growing systems , 1997 .

[40]  W. Burridge,et al.  “Excitability” , 1933 .