Topology in Biology: Singularities and Surgery Transformations in Metazoan Development and Evolution


The review presents a topological description and interpretation (analysis) of some events in metazoan development and evolution through the use of well-known mathematical concepts and theorems (using topological approach). It is the topological language that can provide strict and adequate description of various phenomena in developmental and evolutionary transformations. Topological singularities inevitably arising and transforming during early development destroy the preexisting pattern of symmetry. The symmetry breaking of preexisting spatial pattern plays a critical role in biological morphogenesis in development and evolution. Some events of early development are interpreted in terms of symmetry breakdown and related to well-known mathematical theorems. A topological inevitability of some developmental events through the use of classical topological concepts is discussed. The topological approach makes it possible to consider the succession of spherical surgeries, which change the topological genus of an animal body surface. We model the biological shape as a set of smooth, closed, oriented surfaces—membrane or epithelial layers. Membrane and epithelial surfaces are boundary layers, interfaces between a living structure and its environment, ensuring metabolism. Toroid forms as well as fractal structures in metazoans can be considered as functionally optimized biological design and attractors in biological morphogenesis. The epithelial surface is an interface between the internal medium of an organism and the outside environmental medium; topological and fractal transformations during metazoan evolution and development increase this interface, ensuring better adaptation of organism to the environment. Fractal structures as well as toroid forms can be considered as a functionally optimized design in Metazoa. Topological methodology reveals a certain set of topological rules constraining and directing biological morphogenesis during evolution and development.

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Isaeva, V. , Kasyanov, N. and Presnov, E. (2014) Topology in Biology: Singularities and Surgery Transformations in Metazoan Development and Evolution. Applied Mathematics, 5, 2664-2674. doi: 10.4236/am.2014.517255.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Bouligand, Y. (1996) Morphological Singulatities and Macroevolution. Memorie della Societa Italiana di Scienze Naturali e de Museo Civico di Storia Naturale di Milano, 27, 89-94.
[2] Thom, R. (1969) Topological Models in Biology. Topology, 8, 313-335.
[3] Thom, R. (1996) Qualitative and Quantitative in Evolutionary Theory with Some Thoughts on Aristotelian Biology. Memorie della Societa Italiana di Scienze Naturali e de Museo Civico di Storia Naturale di Milano, 27, 115-117.
[4] Thom, R. (1997) Structural Stability and Morphogenesis: An Outline of a General Theory of Models. Addison-Wesley Publishing Co., New York.
[5] Wasserman, S.A. and Cozzarelli, N.R. (1986) Biochemical Topology: Applications to DNA Recombination and Replication. Science, 232, 951-960.
[6] Monastyrsky, M.I. (2007) Topology in Molecular Biology. Springer-Verlag, Berlin.
[7] Maresin, V.M. and Presnov, E.V. (1985) Topological Approach to Embryogenesis. Journal of Theoretical Biology, 114, 387-398.
[8] Presnov, E.V., Malyghin, S.N. and Isaeva, V.V. (1988) Topological and Thermodynamic Structure of Morphogenesis. In: Lamprecht, I. and Zotin, A.I., Eds., Thermodynamics and Pattern Formation in Biology, Walter de Gruyter, Berlin, 337-370.
[9] Presnov, E., Isaeva, V. and Kasyanov N. (2010) Topological Determination of Early Morphogenesis in Metazoa. Theory in Bioscience, 129, 259-270.
[10] Presnov, E., Isaeva, V. and Kasyanov, N. (2014) Topological Invariance in Biological Development. Axiomathes, 24, 117-135.
[11] Presnov, E.V. and Isaeva, V.V. (1991) Local and Global Aspects of Biological Morphogenesis. Speculations in Science and Technology, 14, 68-75.
[12] Presnov, E.V. and Isaeva, V.V. (1996) Topological Classification: Onto- and Phylogenesis. Memorie della Societa Italiana di Scienze Naturali e de Museo Civico di Storia Naturale di Milano, 27, 89-94.
[13] Isaeva, V., Presnov, E. and Chernyshev, A. (2006) Topological Patterns in Metazoan Evolution and Development. Bulletin of Mathematical Biology, 68, 2053-2067.
[14] Isaeva, V.V., Kasyanov, N.V. and Presnov, E.V. (2008) Analysis Situs of Spatial-Temporal Architecture in Biological Morphogenesis. In: Kelly, J.T., Ed., Progress in Mathematical Biology Research, Nova Science Publishers, New York, 141-189.
[15] Isaeva, V.V., Kasyanov, N.V. and Presnov, E.V. (2012) Topological Singularities and Symmetry Breaking in Development. Biosystems, 109, 280-298.
[16] Jockush, H. and Dress, A. (2003) From Sphere to Torus: A Topological View of the Metazoan Body Plan. Bulletin of Mathematical Biology, 65, 57-65.
[17] Greenblum, S., Turnbaugh, P.J. and Borenstein, E. (2012) Metagenomic Systems Biology of the Human Gut Microbiome Reveals Topological Shifts Associated with Obesity and Inflammatory Bowel Disease. Proceedings of the National Academy of Sciences of the United States of America, 109, 594-599.
[18] Dabaghian, Y., Memoli, F., Frank, L. and Carlsson, G. (2012) A Topological Paradigm for Hippocampal Spatial Map Formation Using Persistent Homology. PLoS Computational Biology, 8.
[19] Niizato, T., Murakami, H. and Gunji, Y.P. (2014) Emergence of the Scale-Invariant Proportion in a Flock from the Metric-Topological Interaction. Biosystems, 119, 62-68.
[20] Li, R. and Bowerman, B. (2010) Symmetry Breaking in Biology. Cold Spring Harbor Perspectives in Biology, 2, a003475.
[21] Nüsslein-Volhard, C. (1991) Determination of the Embryonic Axes of Drosophila. Development. Supplement, 1, 1-10.
[22] Mullins, R.D. (2010) Cytoskeletal Mechanisms for Breaking Cell Symmetry. Cold Spring Harbor Perspectives in Biology, 2, a003392.
[23] Nuccitelli, R. (1984) The Involvement of Transcellular Ion Currents and Electric Fields in Pattern Formation. In: Malacinski, G.M. and Bryant, S.V., Eds., Pattern Formation: A Primer in Developmental Biology, MacMillan, London, 23-46.
[24] Vacquer, V.D. (1981) Dynamic Changes of the Egg Cortex. Developmental Biology, 84, 1-26.
[25] Kirschner, M.W. and Gerhart, J.C. (2005) The Plausibility of Life. Yale University Press, Yale, New Haven.
[26] Gilbert, S.F. (2000) Developmental Biology. Sinauer Associates, Inc., Sunderland.
[27] Kline, D. and Nuccitelli, R. (1985) The Wave of Activation Current in the Xenopus Egg. Developmental Biology, 111, 471-487.
[28] Rivier, N., Miri, M.F. and Oguey, C. (2005) Plasticity and Topological Defects in Cellular Structures: Extra Matter, Folds and Crab Moulting. Colloids and Surfaces A, 263, 39-45.
[29] Vogel, G. (2000) Tracking the Movements that Shape an Embryo. Science, 288, 86-87.
[30] Kolega, J. (1986) The Cellular Basis of Epithelial Morphogenesis. In: Browder, W., Ed., Developmental Biology, Vol. 2, Plenum Press, New York, 103-143.
[31] Zallen, J.A. (2007) Planar Polarity and Tissue Morphogenesis. Cell, 129, 1051-1063.
[32] Nelson, C.M. (2009) Geometric Control of Tissue Morphogenesis. Biochimica et Biophysica Acta-Molecular and Cell Research, 1793, 903-910.
[33] Vladar, E.K., Antic, D. and Axelrod, J.D. (2009) Planar Cell Polarity Signaling: The Developing Cell’s Compass. Cold Spring Harbor Perspectives in Biology, 1, a002964.
[34] Halanych, K.M. (2004) The New View of Animal Phylogeny. Annual Review of Ecology, Evolution and Systematics, 35, 229-256.
[35] Tashiro, Y. (1983) Subcellular Compartments and Protein Topogenesis. Cell Structure and Function, 8, 91-107.
[36] Whatley, J.M. and Whatley, E.R. (1984) Evolutionary Aspects of the Eukaryotic Cell and Its Organelles. In: Lindskens, H.F. and Heslop-Harrrison, J., Eds., Cellular Interactions, Springer Verlag, Berlin, 18-58.
[37] Shubin, N.H. (1998) Vertebrate Palaeontology: Evolutionary Cut and Paste. Nature, 394, 12-13.
[38] Raff, R.A. and Sly, B.J. (2000) Modularity and Dissociation in the Evolution of Gene Expression Territories in Development. Evolution and Development, 2, 102-113.
[39] Peter, I.S. and Davidson, E.H. (2011) Evolution of Gene Regulatory Networks Controlling Body Plan Development. Cell, 144, 970-985.
[40] Minelli, A. (2003) The Development of Animal Form: Ontogeny, Morphology, and Evolution. Cambridge University Press, Cambridge.
[41] Deutsch, J.S. and Mouchel-Vielh, E. (2003) Hox Genes and the Crustacean Body Plan. BioEssays, 25, 878-887.
[42] Sará, M. (1999) New Perspectives on the Role of Constraints in Evolution. Rivista Biologica/Biological Forum, 92, 29-52.
[43] Thomas, R.D.K. and Reif, W.E. (1993) The Skeleton Space: A Finite Set of Organic Designs. Evolution, 47, 341-360.

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