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Evol Bioinform Online
2014 Jun 29;10:97-105. doi: 10.4137/EBO.S14457.
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Influence of modularity and regularity on disparity of atelostomata sea urchins.
López-Sauceda J
,
Malda-Barrera J
,
Laguarda-Figueras A
,
Solís-Marín F
,
Aragón JL
.
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A modularity approach is used to study disparity rates and evolvability of sea urchins belonging to the Atelostomata superorder. For this purpose, the pentameric sea urchin architecture is partitioned into modular spatial components and the interference between modules is quantified using areas and a measurement of the regularity of the spatial partitions. This information is used to account for the variability through time (disparity) and potential for morphological variation and evolution (evolvability) in holasteroid echinoids. We obtain that regular partitions of the space produce modules with high modular integrity, whereas irregular partitions produce low modular integrity; the former ones are related with high morphological disparity (facilitation hypothesis). Our analysis also suggests that a pentameric body plan with low regularity rates in Atelostomata reflects a stronger modular integration among modules than within modules, which could favors bilaterality against radial symmetry. Our approach constitutes a theoretical platform to define and quantify spatial organization in partitions of the space that can be related to modules in a morphological analysis.
Figure 1. Apical disc (encircled) showing landmarks (numbered) at ocular plates. The star vector is formed by vectors pointing to these landmarks, with a common origin at the center of mass.
Figure 2. Algorithm used to associate modules to a given vector star. (A) – (D) correspond to steps 1, 2, 3, and 4, respectively, of the algorithm described in Section Methods.
Figure 3. Geometrical parameters required to replicate random bilateral stars. The pentagonal star is used as example but vector magnitudes (lowercase letters) a, b, c, etc., and angles (Greek letters) α, β, δ, γ, etc., may increase or decrease according to the number of vectors.
Figure 4. ANOVA of differences of area variability, that is
σ¯m (see Eq. 6), for regular and irregular partitions of the space. Partitions with (A) three modules, (b) four modules, (C) five modules, (D) six modules, and (e) seven modules, p-value thresholds of the ANOVA are in the top right panel.
Figure 5. (A) Mean regularity (third column in Table 1) plotted versus Eble’s disparity rates (second column in Table 1), for its corresponding stratigraphic intervals; a high positive correlation is found, displayed on top of the graph. (b) Regularity standard deviation (fourth column in Table 1) plotted versus Eble’s disparity rates, where a high negative correlation is found, also displayed on top of the graph.
Figure 6. Scheme for area variability of modules defined by regular stars (A) versus modules defined by irregular stars (b) ones. In case (A), low variability in areas (standard deviations) is depicted; this low variability implies non-significant overlap between modules, that is, high within-module integrity. On the contrary, in case (b) high variability in areas (standard deviations) is depicted; it implies significant overlap between modules and low within-module integrity.
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