Biological Theory

, Volume 4, Issue 1, pp 54–67 | Cite as

The Concept of Morphospaces in Evolutionary and Developmental Biology: Mathematics and Metaphors

  • Philipp Mitteroecker
  • Simon M. Huttegger


Formal spaces have become commonplace conceptual and computational tools in a large array of scientific disciplines, including both the natural and the social sciences. Morphological spaces (morphospaces) are spaces describing and relating organismal phenotypes. They play a central role in morphometrics, the statistical description of biological forms, but also underlie the notion of adaptive landscapes that drives many theoretical considerations in evolutionary biology. We briefly review the topological and geometrical properties of the most common morphospaces in the biological literature. In contemporary geometric morphometrics, the notion of a morphospace is based on the Euclidean tangent space to Kendall’s shape space, which is a Riemannian manifold. Many more classical morphospaces, such as Raup’s space of coiled shells, lack these metric properties, e.g., due to incommensurably scaled variables, so that these morphospaces typically are affine vector spaces. Other notions of a morphospace, like Thomas and Reif’s (1993) skeleton space, may not give rise to a quantitative measure of similarity at all. Such spaces can often be characterized in terms of topological or pretopological spaces.

The typical language of theoretical and evolutionary biology, comprising statements about the “distance” among phenotypes in an according space or about different “directions” of evolution, is not warranted for all types of morphospaces. Graphical visualizations of morphospaces or adaptive landscapes may tempt the reader to apply “Euclidean intuitions” to a morphospace, whatever its actual topology might be. We discuss the limits of metaphors such as the developmental hourglass and adaptive landscapes that ensue from the geometric properties of the underlying morphospace.


adaptive landscapes affine space developmental hourglass morphometrics phenetic space sequence space shape space skeleton space theoretical morphology topology 


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  1. Alberch P (1991) From genes to phenotype: Dynamical systems and evolvability. Genetica 85: 5–11CrossRefGoogle Scholar
  2. Amari S (1985) Differential-Geometrical Methods in Statistics. Berlin: Springer.CrossRefGoogle Scholar
  3. Arnold SJ, Pfrender ME, Jones A (2001) The adaptive landscape as a conceptual bridge between micro and macroevolution. Genetica 112–113: 9–32.CrossRefGoogle Scholar
  4. Bininda-Emonds ORP, Jeffery EJ, Richardson MK (2003) Inverting the hourglass: Quantitative evidence against the phylotypic stage in vertebrate development. Proceedings of the Royal Society London B 270: 341–346.CrossRefGoogle Scholar
  5. Blackith RE, Reyment RA (1971) Multivariate Morphometrics. London: Academic Press.Google Scholar
  6. Bookstein FL (1989) Principal warps: Thin plate splines and the decomposition of deformations. IEEE Transactions on Pattern Analysis and Machine Intelligence 11: 567–585.CrossRefGoogle Scholar
  7. Bookstein F (1991) Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge: Cambridge University Press.Google Scholar
  8. Bookstein F (1998) A hundred years of morphometrics. Acta Zoologica Academiae Scientarium Hungaricae 44: 7–59.Google Scholar
  9. Dryden IL, Mardia KV (1998) Statistical Shape Analysis. New York: Wiley.Google Scholar
  10. Duboule D (1994) Temporal colinearity and the phylotypic progression: A basis for the stability of a vertebrate Bauplan and the evolution of morphologies through heterochrony. Development (Suppl): 135–142.Google Scholar
  11. Fontana W, Schuster P (1998a) Continuity in evolution: On the nature of transitions. Science 280: 1451–1455.CrossRefGoogle Scholar
  12. Fontana W, Schuster P (1998b) Shaping space: The possible and the attainable in RNA genotype-phenotype mapping. Journal of Theoretical Biology 194: 491–515.CrossRefGoogle Scholar
  13. Foote M (1993) Contributions of individual taxa to overall morphological disparity. Paleobiology 19: 403–419.Google Scholar
  14. Galis F, Metz JAJ (2001) Testing the vulnerability of the phylotypic stage: On modularity and evolutionary conservation. Journal of Experimental Zoology 291: 195–204.CrossRefGoogle Scholar
  15. Galton F (1888) Co-relations and their measurement, chiefly from anthropometric data. Proceeding of the Royal Society 45: 135–145.CrossRefGoogle Scholar
  16. Galton F (1907) Classification of portraits. Nature 76: 617–618.CrossRefGoogle Scholar
  17. Gavrilets S (2004) Fitness Landscapes and the Origin of Species. Princeton, NJ: Princeton University Press.Google Scholar
  18. Gould SJ (1977) Ontogeny and Phylogeny. Cambridge, MA: Harvard University Press.Google Scholar
  19. Hofbauer J, Sigmund K (1998) Evolutionary Games and Population Dynamics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  20. Huttegger S, Mitteroecker P (in preparation) The geometry of phenotype spaces: Invariance and meaningfulness.Google Scholar
  21. Johnson N (2008) Sewall Wright and the development of shifting balance theory. Nature Education 1(1).Google Scholar
  22. Johnson RA, Wichern DW (1998) Applied Multivariate Statistical Analysis. Upper Saddle River, NJ: Prentice Hall.Google Scholar
  23. Kaplan J (2008) The end of the adaptive landscape metaphor? Biology and Philosophy 23: 625–638.CrossRefGoogle Scholar
  24. Kendall D (1981) The statistics of shape. In: Interpreting Multivariate Data (Barnett V, ed), 75–80. New York: Wiley.Google Scholar
  25. Kendall D (1984) Shape manifolds: Procrustean metrics and complex projective spaces. Bulletin of the London Mathematical Society 16: 81–121.CrossRefGoogle Scholar
  26. Klingenberg CP, McIntyre GS (1998) Geometric morphometrics of developmental instability: Analyzing patterns of fluctuating asymmetry with Procrustes methods. Evolution 52: 1363–1375.CrossRefGoogle Scholar
  27. Manfreda E, Mitteroecker P, Bookstein FL, Schaefer K (2006) Functional morphology of the first cervical vertebra in humans and non-human primates. Anatomical Record, Part B: The New Anatomist 289: 184–194.CrossRefGoogle Scholar
  28. Marcus LF (1990) Traditional morphometrics. In: Proceedings of the Michigan Morphometrics Workshop (Rohlf FJ, Bookstein FL, eds), 77–122. Ann Arbor, MI: University of Michigan Museums.Google Scholar
  29. Marcus LF, Hingst-Zaher E, Zaher H (2000) Application of landmark morphometrics to skulls representing the orders of living mammals. Hystrix, Italian Journal of Mammology 11: 27–47.Google Scholar
  30. Mardia KV, Bookstein F, Moreton I (2000) Statistical assessement of bilateral symmetry of shapes. Biometrika 87: 285–300.CrossRefGoogle Scholar
  31. Mardia KV, Kent JT, Bibby JM (1979) Multivariate Analysis. London: Academic Press.Google Scholar
  32. McGhee GR (1999) Theoretical Morphology: The Concept and Its Applications. New York: Columbia University Press.Google Scholar
  33. McGhee GR (2007) The Geometry of Evolution: Adaptive Landscapes and Theoretical Morphospaces. Cambridge: Cambridge University Press.Google Scholar
  34. Milne-Edwards H (1844) Considérations sur quelques principes relatifs à la classification naturelle des animaux. Annales des Sciences Naturelles (Zoologie) (Série 3) 1: 65–99.Google Scholar
  35. Mitteroecker P, Bookstein FL (2007) The conceptual and statistical relationship between modularity and morphological integration. Systematic Biology 56: 818–836.CrossRefGoogle Scholar
  36. Mitteroecker P, Bookstein FL (2008) The evolutionary role of modularity and integration in the hominoid cranium. Evolution 62: 943–958.CrossRefGoogle Scholar
  37. Mitteroecker P, Bookstein FL (2009) The ontogenetic trajectory of the phenotypic covariance matrix, with examples from craniofacial shape in rats and humans. Evolution 63: 727–737.CrossRefGoogle Scholar
  38. Mitteroecker P, Gunz P (2009) Advances in geometric morphometrics. Evolutionary Biology 36: 235–247.CrossRefGoogle Scholar
  39. Mitteroecker P, Gunz P, Bernhard M, Schaefer K, Bookstein F (2004) Comparison of cranial ontogenetic trajectories among great apes and humans. Journal of Human Evolution 46: 679–697.CrossRefGoogle Scholar
  40. Niklas KJ, Kerchner V (1984) Mechanical and photosynthetic constraints on the evolution of plant shape. Paleobiology 10: 79–101.Google Scholar
  41. Pearson K, Morant GM (1934) The Wilkinson head of Oliver Cromwell and its relationship to busts, masks and painted portraits. Biometrika 26: 1–116.CrossRefGoogle Scholar
  42. Pigliucci M (2008) Sewall Wright’s adaptive landscapes: 1932 vs. 1988. Biology and Philosophy 23: 591–603.CrossRefGoogle Scholar
  43. Raff R (1996) The Shape of Life: Genes, Development, and the Evolution of Animal Form. Chicago: University of Chicago Press.Google Scholar
  44. Rasskin-Gutman D, Buscalioni AD (1996) Affine transformation as a model of virtual form change for generating morphospaces. In: Advances in Morphometrics (Marcus LF, Corti M, Loy A, Slice D, Naylor G, eds), 169–178. New York: Plenum Press.CrossRefGoogle Scholar
  45. Raup DM (1966) Geometric analysis of shell coiling: General problems. Journal of Paleontology 40: 1178–1190.Google Scholar
  46. Raup DM, Michelson A (1965) Theoretical morphology of the coiled shell. Science 147: 1294–1295.CrossRefGoogle Scholar
  47. Reyment RA (1991) Multidimensional Paleobiology. New York: Pergamon Press.Google Scholar
  48. Richardson MK, Hanken J, Gooneratne ML, Pieau C, Raynaud A, Selwood L, Wright GM (1997) There is no highly conserved embryonic stage in the vertebrates: Implications for current theories of evolution and development. Anatomy and Embryology 196: 91–106.CrossRefGoogle Scholar
  49. Rohlf FJ (1999) Shape statistics: Procrustes superimpositions and tangent spaces. Journal of Classification 16: 197–223.CrossRefGoogle Scholar
  50. Rohlf FJ, Marcus LF (1993) A revolution in morphometrics. Trends in Ecology and Evolution 8: 129–132.CrossRefGoogle Scholar
  51. Rohlf FJ, Slice DE (1990) Extensions of the Procrustes method for the optimal superimposition of landmarks. Systematic Zoology 39: 40–59.CrossRefGoogle Scholar
  52. Rohlf FJ, Sokal RR (1965) Coefficients of correlation and distance in numerical taxonomy. University of Kansas Science Bulletin 45: 3–27.Google Scholar
  53. Schindel DE (1990) Unoccupied morphospace and the coiled geometry of gastropods: Architectural constraints or geometric covariation? In: Causes of Evolution (Ross RA, Allmon WD, eds), 270–304. Chicago: University of Chicago Press.Google Scholar
  54. Simpson GG (1944) Tempo and Mode in Evolution. New York: Columbia University Press.Google Scholar
  55. Slice DE (2001) Landmark coordinates aligned by procrustes analysis do not lie in Kendall’s shape space. Systematic Biology 50: 141–149.CrossRefGoogle Scholar
  56. Slice DE (2005) Modern Morphometrics in Physical Anthropology. Dordrecht, the Netherlands: Kluwer.CrossRefGoogle Scholar
  57. Small C (1996) The Statistical Theory of Shape. New York: Springer.CrossRefGoogle Scholar
  58. Sneath P, Sokal R (1973) Numerical Taxonomy. San Francisco, CA: Freeman.Google Scholar
  59. Sokal RR (1961) Distance as a measure of taxonomic similarity. Systematic Zoology 10: 70–79.CrossRefGoogle Scholar
  60. Stadler PF (2002) Fitness landscapes. In: Biological Evolution and Statistical Physics (Lässig M, Valleriani A, eds), 187–207. Berlin: Springer.Google Scholar
  61. Stadler BMR, Stadler PF (2004) The topology of evolutionary biology. In: Modeling in Molecular Biology (Ciobanu G, Rozenberg G, eds), 267–286. Berlin: Springer.CrossRefGoogle Scholar
  62. Stadler BMR, Stadler PF, Shpak M, Wagner GP (2002) Recombination spaces, metrics, and pretopologies. Zeitschrift för Physikalische Chemie 216: 217–234.Google Scholar
  63. Stadler BMR, Stadler PF, Wagner G, Fontana W (2001) The topology of the possible: Formal spaces underlying patterns of evolutionary change. Journal of Theoretical Biology 213: 241–274.CrossRefGoogle Scholar
  64. Suppes P, Krantz DH, Luce RD, Tversky A (1989) Foundations of Measurement, Vol. II: Geometrical, Threshold, and Probabilistic Representations. New York: Academic Press.Google Scholar
  65. Thomas RDK, Reif W-E (1993) The skeleton space: A finite set of organic designs. Evolution 47: 341–360.CrossRefGoogle Scholar
  66. Thomas RDK, Shearman RM, Stewart GW (2000) Evolutionary exploitation of design options by the first animals with hard skeletons. Science 288: 1239–1242.CrossRefGoogle Scholar
  67. Thompson DAW (1917) On Growth and Form. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  68. von Baer KE (1828) Entwicklungsgeschichte der Thiere: Beobachtung und Reflexion. Königsberg, Germany: Bornträger.Google Scholar
  69. Wagner GP, Altenberg L (1996) Complex adaptations and the evolution of evolvability. Evolution 50: 967–976.CrossRefGoogle Scholar
  70. Wright S (1932) The roles of mutation, inbreeding, crossbreeding, and selection in evolution. Proceedings of the Sixth International Congress of Genetics 1: 356–366.Google Scholar
  71. Wright S (1988) Surfaces of selective value revisited. American Naturalist 131: 115–123.CrossRefGoogle Scholar

Copyright information

© Konrad Lorenz Institute for Evolution and Cognition Research 2009

Authors and Affiliations

  1. 1.Department of Theoretical BiologyUniversity of ViennaViennaAustria
  2. 2.Department of Logic and Philosophy of ScienceUniversity of California, IrvineIrvineUSA

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