Abstract
We present a numerical study of the nonlinear mechanical model for morphogenesis proposed by Oster et al. (1983) with the aim of establishing the pattern forming capability of the model. We present a technique for mode selection based on linear analysis and show that, in many cases, it is a reliable predictor for nonlinear mode selection. In order to determine the set of model parameters that can generate a particular pattern we develop a technique based on nonlinear least square fitting to a dispersion relation. As an application we present a scenario for sequential pattern formation of dermal aggregations in chick embryos which leads to the hexagonal array of cell aggregations observed in feather germ formation in vivo.
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Davidson, D.: The mechanism of feather pattern development in the chick I. The time of determination of feather position. J. Embryol. Exp. Morph. 74, 245–259 (1983a)
Davidson, D.: The mechanism of feather pattern development in the chick II. Control of the sequence of pattern formation. J. Embryol. Exp. Morph. 74, 261–273 (1983b)
Dhouailly, D.: Early events in retinoic acid-induced ptilopody in the chick embryo. Roux's Arch. Dev. Biol. 192, 21–73 (1983)
Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik 12, 30–39 (1972)
Harris, A. K., Ward, P., Stopak, D.: Silicone rubber substrata: a new wrinkle in the study of cell locomotion. Science 208, 177–179 (1980)
Hyman, J. M.: MOL1D: A general purpose subroutine package for the numerical solution of partial differential equations. Manual, Los Alamos Nat. Lab., LA-7595-M (1979)
Hyman, J. M.: Numerical methods for nonlinear differential equations. In: Bishop, A. R., Campbell, D. K., Nicolaenko, B. (eds.). Nonlinear problems: present and future, pp. 91–107. Amsterdam: North Holland 1980
Maden, M.: Vitamin A and pattern formation in the regenerating limb. Nature 295, 672–675 (1982)
Maini, P. K., Murray, J. D., Oster, G. F.: A mechanical model for biological pattern formation: a nonlinear bifurcation analysis. In: Sleeman, B. D., Jarvis, R. J. (eds.). Ordinary and partial differential equations. Proceedings, Dundee 1984 (Lect. Notes Math., vol. 1151, pp. 252–269), Berlin Heidelberg New York: Springer (1984)
Meinhardt, H.: Models of biological pattern formation. London: Academic Press 1983
Moré, J. J.: The Levenberg-Marquardt algorithm: implementation and theory. In: Watson, G. A. (ed.). Numerical Analysis. Proceedings, Dundee 1977 (Lect. Notes Math., vol. 630, pp. 105–116), Berlin Heidelberg New York: Springer (1978)
Murray, J. D.: Non-linear-differential-equation models in biology. Oxford: Clarendon Press 1977
Murray, J. D.: A pre-pattern formation mechanism for animal coat markings. J. Theor. Biol. 88, 161–199 (1981)
Murray, J. D.: Parameter space for Turing instability in reaction-diffusion mechanisms: a comparison of models. J. Theor. Biol. 98, 143–163 (1982)
Murray, J. D., Oster, G. F.: Generation of biological pattern and form. IMA J. Math. Appl. Med. Biol. 1, 51–75 (1984a)
Murray, J. D., Oster, G. F.: Cell traction models for generating pattern and form in morphogenesis. J. Math. Biol. 19, 265–279 (1984b)
Oster, G. F., Murray, J. D., Harris, A. K.: Mechanical aspects of mesenchymal morphogenesis. J. Embryol. Exp. Morph. 78, 83–125 (1983)
Oster, G. F., Murray, J. D., Maini, P. K.: A model for chondrogenic condensations in the developing limb: the role of the extracellular matrix and cell tractions. J. Embryol. Exp. Morph. 89, 93–112 (1985)
Purcell, E.: Life at low Reynolds number. Am. J. Phys. 45, 1–11 (1977)
Sengel, P.: Morphogenesis of skin. Develop. Cell Biol. Ser. 3. Cambridge: Cambridge Univ. Press 1976
Shubin, N., Alberch, P.: A morphogenetic approach to the origin and basic organisation of the tetrapod limb. In: Hecht, M., Wallace, B., Steere, W. (eds.). 20. New York: Plenum Press 1986
Solursh, M.: Cell-cell interactions and chondrogenesis. In: Hall, B. K. (ed.). Cartilage 2: development, differentiation and growth. New York: Academic Press 1983
Turing, M. A.: The chemical basis of morphogenesis. Philos. Trans. Roy. Soc. (Lond.) B237, 37–73 (1952)
Wolpert, L.: Positional information and pattern formation. Philos. Trans. Roy. Soc. (Lond.) B295, 441–450 (1981)
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Perelson, A.S., Maini, P.K., Murray, J.D. et al. Nonlinear pattern selection in a mechanical model for morphogenesis. J. Math. Biology 24, 525–541 (1986). https://doi.org/10.1007/BF00275682
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DOI: https://doi.org/10.1007/BF00275682