Abstract
Topological patterns in the development and evolution of metazoa, from sponges to chordates, are considered by means of previously elaborated methodology, with the genus of the surface used as a topological invariant. By this means metazoan morphogenesis may be represented as topological modification(s) of the epithelial surfaces of an animal body. The animal body surface is an interface between an organism and its environment, and topological transformations of the body surface during metazoan development and evolution results in better distribution of flows to and from the external medium, regarded as the source of nutrients and oxygen and the sink of excreta, so ensuring greater metabolic intensity. In sponges and some Cnidaria, the increase of this genus up to high values and the shaping of topologically complicated fractal-like systems are evident. In most Bilateria, a stable topological pattern with a through digestive tube is formed, and the subsequent topological complications of other systems can also appear. The present paper provides a topological interpretation of some developmental events through the use of well-known mathematical concepts and theorems; the relationship between local and global orders in metazoan development, i.e., between local morphogenetic processes and integral developmental patterns, is established. Thus, this methodology reveals a “topological imperative”: A certain set of topological rules that constrains and directs biological morphogenesis.
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Allaerts, W., 1999. Local and global patterns during morphogenesis of the retinotectal topographical mapping in the vertebrate brain. Acta Biotheor. 47, 99–122.
Arnold, V.I., 1992. Ordinary Differential Equations. Springer Textbook, 3rd edition. Springer-Verlag, New York.
Atiyah, M., 2002. Mathematics in the 20th century. Bull. Lond. Math. Soc. 34, 1–15.
Blazis, D.E.J., 2002. Introduction. The limits to self-organization in biological systems. Biol. Bull. 202, 245–246.
Bourbaki, N., 1948. L`architecture de mathematiques. La mathematique ou les mathematiques? Les Grands Courants de la Pensée Mathematiques, Cahiers du Sud, Paris, pp. 35–47.
Camazine, S., Deneubourg, J.L., Franks, N.R., Sneyd, J., Theraulaz, G., Bonabeau, E., 2001. Self-Organization in Biological Systems. Princeton University Press, Princeton.
Child, C.M., 1941. Patterns and Problems of Development. University of Chicago Press, Chicago.
Chernyshev, A.V., Isaeva, V.V., 2002. Formation of chaotic patterns of the gastrovascular system in the ontogenesis of the medusa Aurelia aurita. Russian J. Mar. Biol. 28, 347–351.
Chin-Sang, I.D., Chisholm, A.D., 2000. Form of the worm: Genetics of epidermal morphogenesis in C. elegans. Trends Genet. 16, 544–551.
Collins, A.G., Valentine, J.W., 2001. Defining phyla: Evolutionary pathways to metazoan body plan. Evol. Dev. 3, 432–442.
Crick, F.H.C., 1976. Linking numbers and nucleosomes. Proc. Natl. Acad. Sci. USA 75, 2639–2643.
Damiani, G., 1994. Evolutionary meaning, functions and morphogenesis of branching structures in biology. In: Nonnenmacher, T.F., et al. (Eds.), Fractals in Biology and Medicine. Birkhäuser Verlag, Basel, pp. 104–115.
D'Arcy Thompson, W., 1917. On Growth and Form, 2nd edition (1942). Cambridge University Press, Cambridge.
Drasdo, D., Forgacs, G., 2000. Modeling the interplay of generic and genetic mechanisms in cleavage, blastulation, and gastrulation. Dev. Dyn. 219, 182–191.
Driesch, H., 1894. Analytische Theorie der Organischen Entwicklung. Verlag von Engelman, Leipzig.
Dubertret, B., Rivier, N., 1997. The renewal of the epidermis: A topological mechanism. Biophys. J. 73, 38–44.
Dubrovin, B.A., Novikov, S.P., Fomenko, A.T., 1990. Modern Geometry–Methods and Applications, Graduate Texts in Mathematics, vol. 124. Part III: Introduction to Homotopy Theory. Springer-Verlag, New York.
Dubrovin, B.A., Novikov, S.P., Fomenko, A.T., 1992. Modern Geometry–Methods and Applications, Graduate Texts in Mathematics, vol. 93, 2nd edition. Part I: The Geometry of Surfaces, Transformation Groups, and Fields. Springer-Verlag, New York.
Dubrovin, B.A., Fomenko, A.T., Novikov, S.P., 1995. Modern Geometry–Methods and Applications, Graduate Texts in Mathematics, vol. 104. Part II: The Geometry and Topology of Manifolds. Springer-Verlag, New York.
Duvdevani-Bar, S., Segel, L., 1994. On topological simulations in developmental biology. J. Theor. Biol. 166, 33–50.
Edelman, G.M., 1988. Topobiology. An Introduction to Molecular Embryology. Basic Books, New York.
Ferrier, D.E.K., Holland, P.W.H., 2001. Sipunculan ParaHox genes. Evol. Dev. 3, 263–270.
Gilbert, S.F., 1991. Developmental Biology. Sinauer Assoc. Inc. Publ., Sunderland.
Goldberger, A.L., Rigney, D.R., West, B.J., 1990. Chaos and fractals in human physiology. Sci. Am. 262, 43–49.
Goldenfeld, N., Kadanoff, L.P., 1999. Simple lessons from complexity. Science 284, 87–89.
Gong, Y.M., Si, Y.L., 2002. Classification and evolution of metazoan traces at a topological level. Lethaia 35, 263–274.
Gurwitsch, A.G., 1922. Über den Begriff des embryonalen Feldes. W. Roux' Archiv für Entwicklungsmechanik der Organismen 52, 383–415.
Hiiragi, T., Solter, D., 2004. First cleavage plane of the mouse egg is not predetermined but defined by the topology of the two apposing pronuclei. Nature 430, 360–364.
Ingber, D.E., 2005. Mechanical control of tissue growth: Function follows form. Proc. Natl. Acad. Sci. USA 102, 11571–11572.
Isaeva, V.V., 2005. Personal page. http://chaos.dvo.ru/isaeva-index.htm.
Jockusch, H., Dress, A., 2003. From sphere to torus: A topological view of the metazoan body plan. Bull. Math. Biol. 65, 57–65.
Jockusch, H., Dress, A., 2004. Letter to the editor. Bull. Math. Biol. 66, 1455.
Knoll, A.H., Carroll, S.B., 1999. Early animal evolution: Emerging view from comparative biology and geology. Science 284, 2129–2136.
Listing, I.B., 1847. Vorstudien zur Topologie. Göttingen Studien, Universität Göttingen, Göttingen, pp. 811–875.
Mandelbrot, B.B., 1983. The Fractal Geometry of Nature. Freeman, New York.
Maresin, V.M., Presnov, E.V., 1985. Topological approach to embryogenesis. J. Theor. Biol. 114, 387–398.
Meinhardt, H., 1982. Models of Biological Pattern Formation. Academic Press, London.
Milnor, J., 1963. Morse Theory. Princeton University Press, Princeton.
Murray, J.D., 2003. Mathematical Biology, 3rd edition. Springer-Verlag, Berlin.
Needham, J., 1936. Order and Life. Cambridge University Press, Cambridge.
Nelson, C.M., Jean, R.P., Tan, J.L., Liu, W.F., Sniadecki, N.J., Spector, A.A., Chen, C.S., 2005. Emergent patterns of growth controlled by multicellular form and mechanics. Proc. Natl. Acad. Sci. USA 102, 11594–11599.
Nuccitelli, R., 1984. The involvement of transcellular ion currents and electric fields in pattern formation. In: Malacinski, G.M., Bryant, S.V. (Eds.), Pattern Formation. A Primer in Developmental Biology. MacMillan, London, pp. 23–46.
Nüsslein-Volhard, C., 1991. Determination of the embryonic axes of Drosophila. Development (Suppl. 1), 1–10.
Petersen, P., 1999. Aspects of global Riemannian geometry. Bull. Am. Math. Soc. 36, 297–344.
Peterson, K.J., Eernisse, D.J., 2001. Animal phylogeny and the ancestry of bilaterians: Inference from morphology and 18S rDNA gene sequences. Evol. Dev. 3, 170–205.
Presnov, E.V., 1982. Classification of biological shapes. In: Zotin, A.I., Presnov, E.V. (Eds.), Mathematical Developmental Biology. Nauka Publishers, Moscow, pp. 126–135 (in Russian).
Presnov, E.V., Isaeva, V.V., 1990. Positional information as symmetry of morphogenetic fields. Forma 5, 59–61.
Presnov, E.V., Isaeva, V.V., 1991. Local and global aspects of biological morphogenesis. Speculat. Sci. Tech. 14, 68–75.
Presnov, E., Isaeva, V., 1996. Topological classification: Onto- and phylogenesis. Memorie della Società Italiana di Scienze Naturali e del Museo Civico di Storia Naturale di Milano 27, 89–94.
Presnov, E.V., Isaeva, V.V., 2004. Topological models in developmental and evolutionary biology. In: Jockusch, H., Dress, A. (Eds.), Presentation for “Workshop on Topology and Morphogenesis of Biological Systems” 6th and 7th December 2004, Bielefeld University. http://chaos.dvo.ru/isaeva-index.htm.
Presnov, E.V., Malyghin, S.N., Isaeva, V.V., 1988. Topological and thermodynamic structures of morphogenesis. In: Lamprecht, I., Zotin, A.I. (Eds.), Thermodynamics and Pattern Formation in Biology. Walter de Gruyter, Berlin, pp. 337–370.
Pyshnov, M.B., 1980. Topological solution for cell proliferation in intestinal crypt. 1. Elastic growth without cell loss. J. Theor. Biol. 87, 189–200.
Rieger, R., Ladurner, P., 2001. Searching for the stem species of the Bilateria. Belgian J. Zool. 131(Suppl. 1), 27–34.
Sará, M., 1999. New perspectives on the role of constraints in evolution. Rivista Biologica/ Biological Forum 92, 29–52.
Schatten, G., Donovan, P., 2004. Embryology plane talk. Nature 430, 301–302.
Siegel, J.S., 2004. Chemical topology and interlocking molecules. Science 304, 1256–1257.
Spemann, H., 1938. Embryonic Development and Induction. Yale University Press, New Haven.
Stadler, B.M.R., Stadler, P.F., Wagner, G.P., Fontana, W., 2001. The topology of possible: Formal spaces underlying patterns of evolutionary change. J. Theor. Biol. 213, 241–274.
Thom, R., 1969. Topological models in biology. Topology 8, 313–335.
Thom, R., 1996. Qualitative and quantitative in evolutionary theory with some thoughts on Aristotelian biology. Memorie della Società Italiana di Scienze Naturali e del Museo Civico di Storia Naturale di Milano 27, 115–117.
Thomas, R.D.K., Reif, W.E., 1993. The skeleton space: A finite set of organic design. Evolution 47, 341–359.
Waddington, C.H., 1940. Organisers and Genes. Cambridge University Press, Cambridge.
Wasserman, S.A., Cozzarelli, N.R., 1986. Biochemical topology: Application to DNA recombination and replication. Science 232, 951–960.
West, G.B., Brown, J.H., Enquist, B.J., 1999. The fourth dimension of life: Fractal geometry and allometric scaling of organisms. Science 284, 1677–1679.
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Isaeva, V., Presnov, E. & Chernyshev, A. Topological Patterns in Metazoan Evolution and Development. Bull. Math. Biol. 68, 2053–2067 (2006). https://doi.org/10.1007/s11538-006-9063-2
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DOI: https://doi.org/10.1007/s11538-006-9063-2