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Sci Rep
2018 Dec 21;81:18055. doi: 10.1038/s41598-018-36147-z.
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Geometric morphometrics of nested symmetries unravels hierarchical inter- and intra-individual variation in biological shapes.
Savriama Y
,
Gerber S
.
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Symmetry is a pervasive feature of organismal shape and the focus of a large body of research in Biology. Here, we consider complex patterns of symmetry where a phenotype exhibits a hierarchically structured combination of symmetries. We extend the Procrustes ANOVA for the analysis of nested symmetries and the decomposition of the overall morphological variation into components of symmetry (among-individual variation) and asymmetry (directional and fluctuating asymmetry). We illustrate its use with the Aristotle''s lantern, the masticatory apparatus of ''regular'' sea urchins, a complex organ displaying bilateral symmetry nested within five-fold rotational symmetry. Our results highlight the importance of characterising the full symmetry of a structure with nested symmetries. Higher order rotational symmetry appears strongly constrained and developmentally stable compared to lower level bilateral symmetry. This contrast between higher and lower levels of asymmetry is discussed in relation to the spatial pattern of the lantern morphogenesis. This extended framework is applicable to any biological object exhibiting nested symmetries, regardless of their type (e.g., bilateral, rotational, translational). Such cases are extremely widespread in animals and plants, from arthropod segmentation to angiosperm inflorescence and corolla shape. The method therefore widens the research scope on developmental instability, canalization, developmental modularity and morphological integration.
Figure 1. (a) CT-Scan of the Aristotle’s lantern of the ‘regular’ sea urchin P. lividus showing its nested symmetries. The lower level bilateral object symmetry of the pyramids is nested within the higher level rotational matching symmetry of order five characteristics of echinoderms (see text for details). This hierarchical architecture rests on symmetry transformations applied to sets of hemipyramids and epiphyses (highlighted). Scale bar 5 mm. (b) The configuration of homologous anatomical landmarks used to capture the geometry of the pyramids and epiphyses.
Figure 2. Alternative options for the analysis of the symmetric architecture of the Aristotle’s lantern. (a) rotational matching symmetry of order five only (symmetry group C5), (b) bilateral object symmetry only (symmetry group C1v), and (c) bilateral object symmetry nested within rotational matching symmetry of order five (symmetry group C5v).
Figure 3. Analysis 1: rotational matching symmetry of order five (C5). Principal components describing the patterns of the lantern’s rotational symmetric shape variation, rotational FA, and measurement error shown as lollipop graphs. Open circles represent the consensus configuration and the thick black lines illustrate the magnitude and direction of the vectors of shape change over the first two principal components (PC).
Figure 4. Analysis 2: bilateral object symmetry (C1v). Bilateral object directional asymmetry (DA) of the pyramids, measured as the difference between the averages of original and reflected configurations (ten-fold amplification).
Figure 5. Analysis 2: bilateral object symmetry (C1v). Principal components describing the patterns of pyramid symmetric variation, bilateral FA, and measurement error. Legend as in Fig. 3. Note that for the measurement error panels, the symmetry or asymmetry of the residual vectors with respect to the plane of symmetry depend on the way the symmetric and asymmetric components of the total error variation are distributed along principal components. Here PC1 captures a part of the symmetric variation while PC2 captures an asymmetric part.
Figure 6. Analysis 3: bilateral object symmetry nested within rotational matching symmetry of order five (C5v). Principal components describing the patterns of variation for nested bilateral object FA, and measurement error. The patterns of lantern variation and rotational FA are not shown here since they are similar to those shown in Fig. 3. Legend as in Fig. 3.
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