Some examples of applications
Bilateral symmetry of bilaterians was the first type of symmetry to be studied using the tools of geometric morphometrics (especially with Drosophila and the mouse as models). These studies are based on the methodological extension of fluctuating asymmetry analyses (Leamy 1984; Palmer and Strobeck 1986), originally based on the measurement of right–left differences of simple traits (linear measurements), to the study of shape as a highly multidimensional phenotypic trait (Klingenberg and McIntyre 1998; Mardia et al. 2000). Leamy et al. (2015), for example, using the fluctuating asymmetry in the size and shape of mouse mandibles to explore the genetic architecture of developmental instability. Quantitative trait locus (QTL) analysis underlines the epistatic genetic basis of fluctuating asymmetry, and suggests that the genes involved in the developmental stability of the mandible are the same as those controlling its shape and size.
The generalization of this morphometric framework to any type of symmetry has extended its scope to a wide variety of taxa and, in particular, to flowering plants (Savriama and Klingenberg 2011; Savriama 2018). Corolla symmetry is indeed involved in multiple aspects of the evolution of flowering plants, and morphometrics allows the statistical testing of adaptive hypotheses. For example, in Erysimum mediohispanicum, Gómez and Perfectti (2010) have shown the impact of the shape of the corolla (and its deviation from the expected symmetry) on the selective value of the plant: pollinators (bees, bombyliids) significantly prefer flowers with bilaterally symmetric corollas (zygomorphism).
The decomposition of the variation into symmetric and asymmetric components can also be relevant in systematics. For example, Neustupa (2013) demonstrated the possible discrimination of different species of Micrasterias (single-cell green algae of the Desmidiales order) according to the share of symmetric and asymmetric components, relative to the two orthogonal planes of symmetry that characterize the cell shape.
Measures of fluctuating asymmetry are also used to infer patterns of developmental integration, in other words, the modular organization of phenotypes resulting from differential interactions between developmental processes (Klingenberg 2008).
Savriama et al. (2016) quantified the fluctuating “translational” asymmetry in order to assess the developmental cost of segmented modular organization in eight species of soil centipedes (Geophilomorpha). The results did not show any impact of the degree of modularity (number of segments) on developmental precision, rejecting the hypothesis of a “cost” of modularity.
Finally, the architecture of some organisms or organic structures may combine several hierarchically arranged patterns of symmetry. This is, for example, the case for Aristotle’s lantern, the masticatory apparatus of sea urchins, which, in regular sea urchins, combines bilateral and rotational symmetries (Savriama and Gerber 2018). The lantern exhibits the fifth-order rotational symmetry that is typical of echinoderms, and results from the repetition of a composite skeletal unit (hemipyramids + epiphyses) with bilateral symmetry. Analysis of the symmetric architecture of the lantern revealed a torsion (directional asymmetry) of the hemipyramids, contributing to the functioning of the lantern, and patterns of fluctuating asymmetry reflecting the spatialization of the skeletal precursors during the morphogenesis of the lantern.
1.8. Conclusion
The symmetry of biological forms, initially an object of curiosity and fascination, has become an important research topic in several branches of biological sciences in recent decades. The understanding of the developmental processes involved in the morphogenesis of symmetric phenotypes is a major issue in developmental and evolutionary biology (see Citerne et al. (2010), for example). In parallel to these genetic and developmental approaches, morphometrics has established a rigorous methodological framework for the analysis of symmetric shapes. Beyond the characterization of the symmetry of shapes, these approaches also quantify the deviations from the symmetric expectation. Their statistical analysis allows inferences to be made with regards to the architecture of complex phenotypes (genetic modularity, developmental modularity) and their variational properties (evolutionary modularity, evolvability). Coupled with molecular and developmental approaches, the recent generalization of the morphometric framework to all types of symmetry thus opens up new ways to describe, study and understand the origin and evolution of symmetries in the living world.
1.9. References
Adams, D.C., Rohlf, F.J., and Slice, D.E. (2004). Geometric morphometrics: Ten years of progress following the “revolution”. Italian Journal of Zoology, 71, 5–16.
Arthur, W. (2006). D’Arcy Thompson and the theory of transformations. Nature Reviews Genetics, 7, 401–406.
Bookstein, F.L. (1978). The Measurement of Biological Shape and Shape Change. Springer, New York.
Bookstein, F.L. (1989). Principal warps: Thin-plate splines and the decomposition of deformation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 567–585.
Bookstein, F.L. (1991). Morphometric Tools for Landmark Data: Geometry and Biology. Cambridge University Press, Cambridge.
Bookstein, F.L. (1996). Biometrics, biomathematics and the morphometric synthesis. Bulletin of Mathematical Biology, 58, 313–365.
Briscoe, J. and Kicheva, A. (2017). The physics of development 100 years after D’Arcy Thompson’s “On Growth and Form”. Mechanisms of Development, 145, 26–31.
Chaplain, M.A.J., Singh, G.D., and McLachlan, J.C. (1999). On Growth and Form: Spatio-temporal Pattern Formation in Biology. John Wiley & Sons, New York.
Citerne, H., Jabbour, F., Nadot, S., and Damerval, C. (2010). The evolution of floral symmetry. Advances in Botanical Research, 54, 85–137.
Coxeter, H.S.M. (1969). Introduction to Geometry. John Wiley & Sons, New York.
Dryden, I.L. and Mardia, K.V. (1998). Statistical Shape Analysis. John Wiley & Sons, New York.
Gómez, J.M. and Perfectti, F. (2010). Evolution of complex traits: The case of Erysimum corolla shape. International Journal of Plant Sciences, 171, 987–998.
Gould, S.J. (1971). D’Arcy Thompson and the science of form. New Literary History, 2, 229–258.
Gould, S.J. (2002). The Structure of Evolutionary Theory. Harvard University Press, Cambridge.
Graham, J.H., Freeman, D.C., and Emlen, J.M. (1993). Antisymmetry, directional asymmetry, and dynamic morphogenesis. Genetica, 89, 121–137.
Haeckel, E. (1904). Kunstformen der Natur. Bibliographischen Instituts, Leipzig.
Keller, E.F. (2018). Physics in biology – Has D’Arcy Thompson been vindicated? The Mathematical Intelligencer, 40, 33–38.
Kendall, D.G. (1984). Shape manifolds, procrustean metrics, and complex projective spaces. Bulletin of the London Mathematical Society, 16, 81–121.
Klingenberg, C.P. (2008). Morphological integration and developmental modularity. Annual Review of Ecology, Evolution, and Systematics, 39, 115–132.
Klingenberg, C.P. (2015). Analyzing fluctuating asymmetry with geometric morphometrics: Concepts, methods, and applications. Symmetry, 7, 843–934.
Klingenberg, C.P. and McIntyre, G.S. (1998). Geometric morphometrics of developmental instability: Analyzing patterns of fluctuating asymmetry with Procrustes methods. Evolution, 52, 1363–1375.
Klingenberg, C.P., Barluenga, M., and Meyer, A. (2002). Shape analysis of symmetric structures: Quantifying variation among individuals and asymmetry. Evolution, 56, 1909–1920.
Kolamunnage, R. and Kent, J.T. (2003). Principal component analysis for shape variation about an underlying symmetric shape. In Stochastic Geometry, Biological Structure and Images, Aykroyd, R.G., Mardia,