Skip to main content
Log in

Discussion: Turing's theory of morphogenesis—Its influence on modelling biological pattern and form

  • Developmental Biology
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

The evolution of spatial pattern is a central issue in developmental biology. Turing's (Phil. Trans. R. Soc. Lond. B237, 37–72, 1952) chemical theory of morphogenesis is a seminal contribution. In this talk I give a personal and necessarily limited view of its impact on mathematical and developmental biology. I briefly describe some of the interesting mathematical aspects of Turing's reaction-diffusion mechanism and discuss some of the different models which Turing's vision inspired. The emphasis throughout is on the practical biological applications of the various theories.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Literature

  • Arcuri, P. and J. D. Murray. 1986. Pattern sensitivity to boundary and initial conditions in reaction-diffusion models.J. math. Biol. 24, 141–165.

    Article  MATH  MathSciNet  Google Scholar 

  • Bowen, J., J. R. Hinchliffe, T. J. Horder and A. M. F. Reeve. 1989. The fate map of the chick forelimb-bud and its bearing on the hypothesized developmental control method.Anat. Embryol. 179, 269–283.

    Article  Google Scholar 

  • Britton, N. F. 1986.Reaction-Diffusion Equations and their Applications to Biology. New York: Academic Press.

    MATH  Google Scholar 

  • Britton, N. F. 1989. A singular dispersion relation arising in a caricature of a model for morphogenesis.J. math. Biol. 26, 387–403.

    MathSciNet  Google Scholar 

  • Britton, N. F. 1981. The mechanism of feather pattern development 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.

    Google Scholar 

  • Dhouailly, D. 1975. Formation of cutaneous appendages in dermoepidermal recombination between reptiles, birds and mammals.Wilhelm Roux Arch. EntwMech. Org. 177, 323–340.

    Article  Google Scholar 

  • Edelman, G. M. 1988.Topobiology. An Introduction to Molecular Biology. New York: Basic Books.

    Google Scholar 

  • Field, R. J. and M. Burger (Eds). 1985.Oscillations and Travelling Waves in Chemical Systems. New York: Wiley.

    Google Scholar 

  • Fisher, R. A. 1937. The wave of advance of advantageous genes.Ann. Eugenics 7, 353–369.

    Google Scholar 

  • Goodwin, B. C., J. D. Murray and D. Baldwin. 1985. Calcium: the elusive morphogen inAcetabularia. In: Proceedings of the 6th International Symposium onAcetubalaria S. Bonotto, F. Cinelli and R. Billiau (Eds), Pisa 1984, pp. 101–108. Belgian Nuclear Center, C.E.N.-S.C.K. Mol, Belgium.

    Google Scholar 

  • Harris, A. K., P. Ward and D. Stopak. 1980. Silicon rubber substrata: a new wrinkle in the study of cell locomotion.Science 208, 177–179.

    Google Scholar 

  • Harris, A. K., D. Stopak and D. Wild. 1981. Fibroblast traction as a mechanism for collagen morphogenesis.Nature 290, 249–251.

    Article  Google Scholar 

  • Ide, H. and H. Aono. 1988. Retinoic acid promotes proliferation and chondrogenesis in the distal mesodermal cells of chick limb bud.Devl Biol. 130, 767–773.

    Article  Google Scholar 

  • Kareiva, P. M. (1983). Local movement in herbivorous insects: applying a passive diffusion model to mark-recapture field experiments.Oecologia (Berlin) 57, 322–327.

    Article  Google Scholar 

  • Keller, E. F. and L. A. Segel. 1971. Travelling bands of chemotactic bacteria: a theoretical analysis.J. theor. Biol. 30, 235–248.

    Article  Google Scholar 

  • Kolmogoroff, A., I. Petrovsky and N. Piscounoff. 1937. Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique.Moscow Univ. Bull. Math. 1, 1–25.

    MATH  Google Scholar 

  • Kulyk, W. M. and R. A. Kosher. 1987. Temporal and spatial analysis of hyaluronate activity during development of the embryonic chick limb bud.Devl Biol. 120, 535–541.

    Article  Google Scholar 

  • Levin, S. A. and L. A. Segel. 1985. Pattern generation in space and aspect.SIAM Rev. 27, 45–67.

    Article  MATH  MathSciNet  Google Scholar 

  • Lubina, J. A. and S. A. Levin. 1988. The spread on an reinvading organism: range expansion of the California sea otter.Am. Nat. 131, 526–543.

    Article  Google Scholar 

  • Luther, R.-L. 1906. Raümliche Fortpflanzung Chemischer Reaktionen.Z. Elektrochem. Phys. Chem. 12(32), 506–600; English translation: R. Arnold, K. Showalter and J. J. Tyson. 1989. Propagation of chemical reactions in space.J. chem. Educ., in press.

    Google Scholar 

  • Maini, P. K. 1989. Spatial and temporal patterns in a cell-haptotaxis model.J. math. Biol., in press.

  • Martiel, J.-L. and A. Goldbeter. 1987. A model based on receptor desensitization for cyclic-AMP signalling inDictvostelium cells.Biophys. J. 52, 807–828.

    Article  Google Scholar 

  • Meinhardt, H. 1982.Models of Biological Pattern Formation. London: Academic Press.

    Google Scholar 

  • Murray, J. D. 1981a. On pattern formation mechanisms for lepidopteran wing patterns and mammalian coat markings.Phil. Trans. R. Soc. Lond. B295, 473–496.

    Google Scholar 

  • Murray, J. D. 1981b. A pre-pattern formation mechanism for animal coat markings.J. theor. Biol. 88, 161–199.

    Article  Google Scholar 

  • Murray, J. D. 1982. Parameter space for Turing instability in reaction diffusion mechanisms: a comparison of models.J. theor. Biol. 98, 143–163.

    Article  Google Scholar 

  • Murray, J. D. 1989a.Mathematical Biology. Heidelberg: Springer-Verlag.

    MATH  Google Scholar 

  • Murray, J. D. 1989b. Modelling the pattern generating mechanism in the formation of stripes on alligators. In: Proceedings of the IXth International Congress on Mathematical Physics, Swansea, pp. 208–213.

  • Murray, J. D., G. F. Oster and A. K. Harris. 1983. A mechanical model for mesenchymal morphogenesis.J. math. Biol. 17, 125–129.

    Article  MATH  Google Scholar 

  • Murray, J. D. and G. F. Oster. 1984a. Generation of biological pattern and form.IMA J. Math. appl. Med. Biol. 1, 51–75.

    MATH  MathSciNet  Google Scholar 

  • Murray, J. D. and G. F. Oster. 1984b. Cell traction models for generating pattern and form in morphogenesis.J. math. Biol. 19, 265–279.

    MATH  MathSciNet  Google Scholar 

  • Murray, J. D., E. A. Stanley and D. L. Brown. 1986. On the spatial spread of rabies among foxes.Proc. R. Soc. Lond. B229, 111–150.

    Article  Google Scholar 

  • Murray, J. D., P. K. Maini and R. T. Tranquillo. 1988. Mechanochemical models for generating biological pattern and form.Phys. Reports 171, 60–84.

    Article  MathSciNet  Google Scholar 

  • Murray, J. D., D. C. Deeming and M. W. J. Ferguson. 1989. Size dependent pigmentation pattern formation in embryos ofAlligator mississippiensis: time of initiation of pattern generation mechanism, to appear.

  • Murray, J. D. and P. K. Maini. 1989. Pattern formation mechanisms—a comparison of reaction diffusion and mechanical models. In:Cell to Cell signalling: From Experiments to Theoretical Models, A. Goldbeter (Ed.), pp. 159–170. Academic Press: London.

    Google Scholar 

  • Nagorcka, B. N. 1989. Wavelike isomorphic prepatterns in development.J. theor. Biol. 137, 127–162.

    MathSciNet  Google Scholar 

  • Oster, G. F., J. D. Murray and A. K. Harris. 1983. Mechanical aspects of mesenchymal morphogenesis.J. Embryol. exp. Morphol. 78, 83–125.

    Google Scholar 

  • Oster, G. F., J. D. Murray and P. K. Maini. 1985. A model for chondrogenic condensations in the developing limb: the role of extracellular matrix and cell tractions.J. Embryol. exp. Morphol. 89, 93–112.

    Google Scholar 

  • Oster, G. F. and J. D. Murray. 1989. Pattern formation models and developmental constraints.J. exp. Zool. [J. P. Trinkaus Anniversary Volume] 251, 186–202.

    Article  Google Scholar 

  • Oster, G. F., N. Shubin, J. D. Murray and P. Alberch. 1988. Evolution and morphogenetic rules. The shape of the vertebrate limb in ontogeny and phylogeny.Evolution, in press.

  • Perelson, A. S., P. K. Maini, J. D. Murray, J. M. Hyman and G. F. Oster. 1986. Nonlinear pattern selection in a mechanical model for morphogenesis.J. math. Biol. 24, 525–541.

    Article  MATH  MathSciNet  Google Scholar 

  • Rawles, M. 1963. Tissue interactions in scale and feather development as studied in dermal-epidermal recombinations.J. Embryol. exp. Morph. 11, 765–789.

    Google Scholar 

  • Sengel, P. 1976.Morphogenesis of Skin. Cambridge University Press.

  • Shaw, L. J. and J. D. Murray. 1990. Analysis of a model for complex skin patterns.SIAM J. appl. Math., in press.

  • Solursh, M. and R. S. Reiter. 1988. Inhibitory and stimulatory effects of limb ectoderm in vitro chondrogenesis.J. exp. Zool. 248, 147–154.

    Article  Google Scholar 

  • Swalla, B. J. and M. Solursh. 1984. Inhibition of limb morphogenesis by fibronection.Differentiation 26, 42–48.

    Article  Google Scholar 

  • Thaller, C. and G. Eichele. 1987. Identification and spatial distribution of retinoids in the developing chick limb bud.Nature 327, 625–628.

    Article  Google Scholar 

  • Tickle, C. and A. Crawley. 1988. The effects of local application of retinoids to different positions along the proximo-distal axis of embryonic chick wings.Roux Arch. devl. Biol. 197, 27–36.

    Article  Google Scholar 

  • Tomchik, K. J. and P. N. Devreotes. 1981. Adenosine 3′,5′-monophosphate waves inDictyostelium discoideium: a demonstration of isotope dilution-fluorography.Science 212, 443–446.

    Google Scholar 

  • Turing, A. M. 1952. The chemical basis of morphogenesis.Phil. Trans. R. Soc. Lond. B237, 37–72.

    Google Scholar 

  • Tyson, J. J., K. A. Alexander, V. S. Manoranjan and J. D. Murray. 1989. Spiral waves of cyclic AMP in a model of slime mold aggregation.Physica D34, 193–207.

    Article  MATH  MathSciNet  Google Scholar 

  • Tyson, J. J. and J. D. Murray. 1989. Cyclic-AMP waves during aggregation of Dictyostelium amoebae.Development,106, 421–426.

    Google Scholar 

  • Winfree, A. T. 1972. Spiral waves of chemical activity.Science 175, 634–636.

    Google Scholar 

  • Winfree, A. T. 1974. Rotating chemical reactions.Sci. Am. 230 (6), 82–95.

    Article  Google Scholar 

  • Winfree, A. T. 1980.The Geometry of Biological Time. Heidelberg: Springer.

    MATH  Google Scholar 

  • Winfree, A. T. and S. H. Strogatz. 1984. Organizing centres for three-dimensional chemical waves.Nature 311, 611–615.

    Article  Google Scholar 

  • Wolpert, L. 1969. Positional information and the spatial pattern of cellular differentiation.J. theor. Biol. 25, 1–47.

    Article  Google Scholar 

  • Wolpert, L. 1981. Positional information and pattern formation.Phil. Trans. R. Soc. Lond. B295, 441–450.

    Google Scholar 

  • Wolpert, L. and A. Hornbruch. 1981. Positional signalling along the anteroposterior axis of the chick wing. The effect of multiple polarizing region grafts.J. Embryol. exp. Morph. 63, 145–159.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Murray, J.D. Discussion: Turing's theory of morphogenesis—Its influence on modelling biological pattern and form. Bltn Mathcal Biology 52, 117–152 (1990). https://doi.org/10.1007/BF02459571

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02459571

Keywords

Navigation