Savriama Y and Klingenberg CP (2011), Beyond bilateral symmetry: geometric morphometr...

ECB-ART-42187

BMC Evol Biol 2011 Sep 29;11:280. doi: 10.1186/1471-2148-11-280.

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Beyond bilateral symmetry: geometric morphometric methods for any type of symmetry.

Savriama Y , Klingenberg CP .

Abstract
BACKGROUND: Studies of symmetric structures have made important contributions to evolutionary biology, for example, by using fluctuating asymmetry as a measure of developmental instability or for investigating the mechanisms of morphological integration. Most analyses of symmetry and asymmetry have focused on organisms or parts with bilateral symmetry. This is not the only type of symmetry in biological shapes, however, because a multitude of other types of symmetry exists in plants and animals. For instance, some organisms have two axes of reflection symmetry (biradial symmetry; e.g. many algae, corals and flowers) or rotational symmetry (e.g. sea urchins and many flowers). So far, there is no general method for the shape analysis of these types of symmetry. RESULTS: We generalize the morphometric methods currently used for the shape analysis of bilaterally symmetric objects so that they can be used for analyzing any type of symmetry. Our framework uses a mathematical definition of symmetry based on the theory of symmetry groups. This approach can be used to divide shape variation into a component of symmetric variation among individuals and one or more components of asymmetry. We illustrate this approach with data from a colonial coral that has ambiguous symmetry and thus can be analyzed in multiple ways. Our results demonstrate that asymmetric variation predominates in this dataset and that its amount depends on the type of symmetry considered in the analysis. CONCLUSIONS: The framework for analyzing symmetry and asymmetry is suitable for studying structures with any type of symmetry in two or three dimensions. Studies of complex symmetries are promising for many contexts in evolutionary biology, such as fluctuating asymmetry, because these structures can potentially provide more information than structures with bilateral symmetry.

PubMed ID: 21958045
PMC ID: PMC3209460
Article link: BMC Evol Biol
Grant support: [+]

Genes referenced: LOC100887844 LOC100893907 LOC575085 LOC583082

Article Images: [+] show captions

	Figure 1. Some types of symmetry found in the structure of living organisms. A. An alga (Micrasterias rotata) with two perpendicular axes of reflection symmetry (dashed lines; biradial symmetry or disymmetry). B. A flower, (Plumeria alba) that shows rotational symmetry (the axis of rotation is shown by the black dot at the centre of the flower and the angle of rotation is represented by the curved arrow). C. The arrangement of vertebrae of a zebrafish exhibits translational symmetry (the translation is indicated by the double-headed arrow). D. A cross-section of a nautilus shell (Nautilus pompilius) showing scale symmetry that is a combination of rotations, translations and dilations (the centre of rotation is represented by the black dot, the rotation by the curved arrow, and the translation by the straight arrow).
	Figure 2. The set of symmetry transformations that define the symmetry group of the equilateral triangle. This symmetry group includes six symmetry transformations: the identity, rotations of order 3, and combinations of reflection with rotations.
	Figure 3. The Procrustes fit of the transformed and relabelled copies of a single triangle to the symmetric consensus. The diagram shows the symmetric mean shape (bold solid triangle) and six copies of the triangle that have been transformed and relabelled using six symmetry transformations: the identity, rotations of order 3, and combinations of reflection with rotations (i.e. this is the same symmetry group as in Figure 2). Copies of the triangle for which the transformation does or does not include reflection about the vertical axis are distinguished by dashed and dotted lines.
	Figure 4. Sectors and types of landmarks for complex object symmetry with a rotation and reflection. To compute the dimensionalities of the different components of shape space, it is helpful to subdivide the configuration of landmarks into sectors and to distinguish different types of landmarks. The diagram shows an example of symmetry under rotation of order 4 and reflection. Therefore, the configuration can be divided into four sectors: the regions that correspond to each other when the rotation is applied (sector boundaries are indicated by solid black lines). If the symmetry also includes reflection, as in this example, the arrangement of landmark in each sector is also bilaterally symmetric about the midline or mid-plane of each sector (dashed lines). Several types of landmarks can be distinguished. There may be a landmark in the centre of rotation or, for 3D data, there may be multiple landmarks of the axis of rotation (c = 0, 1 for 2D data; c ≥ 0 for 3D data). Each sector contains k landmarks. If the order of rotation is denoted o, the total number of landmarks is therefore c + ko (in the diagram, c = 1, k = 5 and o = 4, so that there are 1 + 5 × 4 = 21 landmarks). If the symmetry group contains reflection as well as a rotation, the k landmarks of each sector can be subdivided into b landmarks on the sector boundary, m landmarks on the midline or mid-plane of the sector, and p pairs of corresponding landmarks on either side of the midline (therefore, k = b + m + 2p). We define the sector boundary as running through the axis or plane of reflection on at least one side of the centre or axis of rotation (if the order of rotation is even, two sector boundaries are in the axis or plane of reflection).
	Figure 5. Schematic representation of a corallite with the landmarks used in this study. The septa colored in dark grey belong to the first cycle, the ones in light grey to the second cycle, and those in black to the third cycle. The ring in white represents the mural structure of the corallite.
	Figure 6. Analysis 1: Decomposition of shape variation for symmetry with respect to reflection and rotation of order 2. This figure shows examples of PCs that account for the maximum of variance for each category of shape variation. Each diagram shows the symmetric consensus (open circles and dotted lines) and the differences between the consensus and the other configuration (solid circles and solid lines) represent the shape change associated with the respective PC by an arbitrary amount of + 0.1 units of Procrustes distance. The percentages represent the part of the total shape variation for which each PC accounts. A. Asymmetric component, symmetric under rotation of order 2. B. Symmetric component. C. Asymmetric component, symmetric relative to reflection about the horizontal axis. D. Asymmetric component, symmetric under reflection about the vertical axis.
	Figure 7. Analysis 2: Decomposition of shape variation for symmetry with respect to rotation of order 6. This figure shows examples of PCs that account for the maximum of variance for each category of shape variation. Each diagram shows the symmetric consensus (open circles and dotted lines) and the differences between the consensus and the other configuration (solid circles and solid lines) represent the shape change associated with the respective PC by an arbitrary amount of + 0.1 units of Procrustes distance. The percentages represent the part of the total shape variation for which each PC accounts. A. Asymmetric component, symmetric under rotation of order 2. B. Asymmetric component, symmetric under rotation of order 3. C. Symmetric component. D. Totally asymmetric component.
	Figure 8. Analysis 3: Decomposition of shape variation for symmetry under reflection and rotation of order 6. This figure shows examples of PCs that account for the maximum of variance for each category of shape variation. Each diagram shows the symmetric consensus (open circles and dotted lines) and the differences between the consensus and the other configuration (solid circles and solid lines) represent the shape change associated with the respective PC by an arbitrary amount of + 0.1 units of Procrustes distance. The percentages represent the part of the total shape variation for which each PC accounts. A. Asymmetric component, symmetric under reflection and rotation of order 2. B. Asymmetric component, symmetric under rotation of order 2 but not reflection. C. Asymmetric component, symmetric under reflection about the vertical axis and rotation of order 3. D. Asymmetric component, symmetric under reflection about the horizontal axis and rotation of order 3. E. Completely symmetric component. F. Asymmetric component, symmetric under reflection about the vertical axis. G. Asymmetric component, symmetric under reflection about the horizontal axis. H. Asymmetric component, symmetric under rotation of order 6.

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