General Models of Pattern Formation: Some Uses, Problems and Successes

  • J. D. Murray
Part of the NATO ASI Series book series (NSSA, volume 259)


Understanding the evolution of spatial patterns and the mechanisms which create them are among the most crucial issues in developmental biology. Whether the process is the generation of cartilage patterns in the developing limb, the formation of feathers and scales, the myriad of colour patterns on butterfly wings, regeneration in Hydra, the segmentation in the egg of a developing Drosophila, the patterns on the coats of animals, the formation of the grex in Dictyostelium and so on, a key question is “How are these patterns formed and what is the mechanism (or mechanisms) that creates them?” Enormous progress has been made in understanding some of the basic principles that any mechanism must possess to be able to generate spatial patterns. In spite of this we still do not know, with any certainty, definitive details of a single pattern formation mechanism which is involved in development. Model mechanisms — morphogenetic models — for biological pattern generation can suggest to the embryologist possible scenarios as to how, and sometimes when, pattern is laid down and how the embryonic form might be created. Although genes control pattern formation, genetics says nothing about the actual mechanisms involved nor how the vast range of pattern and form that we see evolves from a homogeneous mass of dividing cells. However, with the enormous strides made in our understanding of a wide variety of developmental situations there is room for optimism that in the not too distant future we shall have isolated not only the biochemical and physical elements involved but also have discovered some of the actual mechanisms. There has, in the past few years, been an increasing recognition among experimentalists and theoreticians that the most dramatic progress in biology will come about through a genuine interdisciplinary approach in which biomedical scientists, mathematicians and physical scientists all play a role. The book by Murray (1989) discusses in detail many successful case studies of such an interdisciplinary approach.


Pattern Formation Developmental Constraint Reaction Diffusion Model Butterfly Wing Vertebrate Limb 
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  1. Berg, H., & Budrene, E. 1991. Complex patterns formed by motile cells in E. coli. Nature, 349, 630–633.PubMedCrossRefGoogle Scholar
  2. Cruywagen, G. C., & Murray, J. D. 1991. A new tissue interaction model for epidermal-dermal spatial patterns. In Proc. 1st European Conference on the Applications of Mathematics to Medicine & Biology, Demongeot, J., & Capasso, V. (eds), Heidelberg, 1991. Springer-Verlag. (In press).Google Scholar
  3. Davidson, D. 1983. The mechanism of feather pattern formation in the chick. I The time of determination of feather position. II Control of the sequence of pattern formation. J. Embryol. Exp. Morph., 74, 245–273.PubMedGoogle Scholar
  4. de Kepper, P., Castets, V., Dulos, E., & Boissonade, J. 1991. Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction. Physica, D 49, 161–169.Google Scholar
  5. Goodwin, B. C., Kauffman, S., & Murray, J. D. 1993. Is morphogenesis an intrinsically robust process? J. Theor. Biol. (In press).Google Scholar
  6. Hinchliffe, J. R., Hurle, J. M., & Summerbell, D. 1991. Developmental patterning of the vertebrate limb. New York: Plenum.Google Scholar
  7. Maini, P. K., Myerscough, M. R., Winters, K. H., & Murray, J. D. 1991. Bifurcating spatially heterogeneous solutions in a Chemotaxis model for biological pattern formation. Bull. Math. Biol., 53, 701–719.PubMedGoogle Scholar
  8. Meinhardt, H. 1983. Models of biological pattern formation. London: Academic Press.Google Scholar
  9. Murray, J. D. 1981. On pattern formation mechanisms for lepidopteran wing patterns and mammalian coat markings. Phil. Trans. Roy. Soc. (Lond.), B295, 473–496. (Also in the book of the Proceedings of the Royal Sociey Meeting on Theories of Biological Pattern Formation. March 1981.).CrossRefGoogle Scholar
  10. Murray, J. D. 1989. Mathematical biology. New York.: Springer-Verlag.Google Scholar
  11. Murray, J. D. 1991. Complex pattern formation and tissue interaction. In Proc. 1st European Conference on the Applications of Mathematics to Medicine & Biology, Demongeot, J., & Capasso, V. (eds), Heidelberg, 1991. Springer-Verlag.Google Scholar
  12. Murray, J. D., & Myerscough, M. R. 1991. Pigmentation pattern formation on snakes. J. Theor. Biol., 149, 339–360.PubMedCrossRefGoogle Scholar
  13. Murray, J. D., & Oster, G. F. 1984a. Cell traction models for generating pattern and form in morphogenesis. J. Math. Appl. in Medic. & Biol., 1, 51–75.CrossRefGoogle Scholar
  14. Murray, J. D., & Oster, G. F. 1984b. Generation of biological pattern and form. IMA J. Math. Biol., 19, 265–279.Google Scholar
  15. Murray, J. D., Oster, G. F., & Harris, A. K. 1983. A mechanical model for mesenchymal morphogenesis. J. Math. Biol., 17, 125–129.PubMedCrossRefGoogle Scholar
  16. Murray, J. D., Deeming, D. C., & Ferguson, M. W. J. 1990. Size dependent pigmentation pattern formation in embryos of Alligator mississippiensis: time of initiation of pattern generation mechanism. Proc. Roy. Soc. (Lond.), B239, 279–293.Google Scholar
  17. Nagorcka, B. N. 1986. The role of reaction-diffusion system in the initiation of skin organ primordia. I. the first wave of initiation. J. Theor. Biol., 121, 449–475.CrossRefGoogle Scholar
  18. Nagorcka, B. N., Manoranjan, V. S., & Murray, J. D. 1987. Complex spatial patterns from tissue interactions — an illustrative example. J. Theor. Biol., 128, 359–374.CrossRefGoogle Scholar
  19. Oster, G. F., & Murray, J. D. 1989. Pattern formation models and developmental constraints. J. Exp. Zool., 251, 186–202.PubMedCrossRefGoogle Scholar
  20. Oster, G. F., Murray, J. D., & Harris, A. K. 1983. Mechanical aspects of mesenchymal morphogenesis. J. Embryol. Exp. Morph., 78, 83–125.PubMedGoogle Scholar
  21. Oster, G. F., Shubin, N., Murray, J. D., & Alberch, P. 1988. Evolution and morphogenetic rules. The shape of the vertebrate limb in ontogeny and phylogeny. Evolution, 42, 862–884.CrossRefGoogle Scholar
  22. Shaw, L. J., & Murray, J. D. 1990. Analysis of a model for complex skin patterns. SIAM J. Appl. Math., 50, 628–648.CrossRefGoogle Scholar
  23. Tranquillo, R. T., & Murray, J. D. 1992. Continuum model of fibroblast-driven wound contraction: inflammation mediation. J. Theor. Biol., 158, 135–172.PubMedCrossRefGoogle Scholar
  24. Turing, A. M. 1952. The chemical basis for morphogenesis. Phil. Trans. Roy. Soc. Lond., B237, 37–72.Google Scholar
  25. Wolpert, L. 1981. Positional information and pattern formation. Phil. Trans. R. Soc. Lond., B295, 441–450.Google Scholar
  26. Wolpert, L., & Hornbruch, A. 1990. Double anterior chick limb buds and models for cartilage rudiment specification. Development, 109, 961–966.PubMedGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • J. D. Murray
    • 1
  1. 1.Applied Mathematics FS-20University of WashingtonSeattleUSA

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