Bulletin of Mathematical Biology

, Volume 65, Issue 1, pp 57–65 | Cite as

From sphere to torus: A topological view of the metazoan body plan

  • Harald JockuschEmail author
  • Andreas Dress


From the viewpoint of mathematical topology, membrane systems in intact living cells can be described as closed and orientable surfaces, i.e., as surfaces with two sides and no boundary lines so that an ‘inside’ and an ‘outside’ can be distinguished. Usually, biomembranes represent topological spheres, often one embedded in another one. Toroidal membranes are occasionally observed, e.g., in specialized structures of plant cells like the prolamellar body. Here, we propose that rules analogous to those that govern the topology of biomembranes hold for the epithelial cell sheets that cover anatomically external as well as internal surfaces of multicellular animals. We suggest to study the emergence of morphological complexity during metazoan development using concepts from mathematical topology, and propose experimental analyses of those topological transitions that appear to be relevant in development and evolution.


Euler Characteristic Body Plan Orientable Surface Klein Bottle Topological Transition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Arendt, D., U. Technau and J. Wittbrodt (2001). Evolution of the bilaterian larval foregut. Nature 409, 81–85.CrossRefGoogle Scholar
  2. Chaplain, M., G. Singh and J. McLachlan (1999). On Growth and Form: Spatio-temporal Pattern Formation in Biology, J. Wiley & Son.Google Scholar
  3. Coxeter, H. S. M. (1989). Introduction to Geometry, New York: Wiley.Google Scholar
  4. Devlin, K. (1999). Mathematics: The New Golden Age, New York: Columbia University Press.Google Scholar
  5. Edelmann, G. M. (1988). Topobiology. An Introduction to Molecular Embryology, New York: Basic Books.Google Scholar
  6. Fiorini, P. (1992). Allgemeine und Vergleichende Embryologie der Tiere, Berlin: Springer.Google Scholar
  7. Gilbert, S. F. (1991). Developmental Biology, Sinauer Associates.Google Scholar
  8. Gunning, B. and M. Steer (1996). Plant Cell Biology, Sudbury, MA: Jones & Bartlett.Google Scholar
  9. Hobmayer, B., F. Rentzsch, K. Kuhn, C. M. Happel, C. C. von Laue, P. Snyder, U. Rothbacher and T. W. Holstein (2000). WNT signalling molecules act in axis formation in the diploblastic metazoan hydra. Nature 407, 186–189.CrossRefGoogle Scholar
  10. Lipowsky, R. (1991). The conformation of membranes. Nature 349, 475–481.CrossRefGoogle Scholar
  11. Meinhardt, H. (2002). The radial-symmetric hydra and the evolution of the bilateral body plan: an old body became a young brain. Bioessays 24, 185–191.CrossRefGoogle Scholar
  12. Odell, G., G. Oster, P. Alberch and B. Burnside (1981). The mechanical basis of morphogenesis. I. Epithelial folding and invagination. Dev. Biol. 84, 446–462.CrossRefGoogle Scholar
  13. Schnepf, E. (1984). The cytological viewpoint of functional compartmentation, in Compartments in Algal Cells and their Interaction, W. Wiessner, D. Robinson and R. Starr (Eds), Berlin: Springer.Google Scholar
  14. Stewart, I. (1995). Concepts of Modern Mathematics, New York: Dover Publications.Google Scholar
  15. Technau, U. and H. R. Bode (1999). HyBra1, a brachyury homologue, acts during head formation in hydra. Development 126, 999–1010.Google Scholar
  16. Westheide, W. and R. Rieger (1996). Spezielle Zoologie: Einzeller und Wirbellose Tiere, Stuttgart: G. Fischer.Google Scholar

Copyright information

© Society for Mathematical Biology 2003

Authors and Affiliations

  1. 1.Forschungsschwerpunkt Mathematisierung-Strukturbildungsprozesse, Developmental Biology and Molecular Pathology, W7Bielefeld UniversityBielefeldGermany
  2. 2.Forschungsschwerpunkt Mathematisierung-Strukturbildungsprozesse, Faculty for MathematicsBielefeld UniversityBielefeldGermany

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