Influence of Modularity and Regularity on Disparity of Atelostomata Sea Urchins

Abstract

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.

Spatial Organization of Five-Fold Morphology as a Source of Geometrical Constraint in Biology

“…In order to define particular values of this property, regarding spatial organization for statistical geometrical samples of several shapes we must build that universe of shapes or star morphospace. Random stars (n=10,000), with number of vectors N= 3, 4, 5, 6, 7, 8, 9 and 10 were generated according to a well established previous methodology reported in reference [24]. Once these sets of stars are generated eutacticity is measured in stars given eight particular statistical distributions ( Figure 6).…”

“…The algorithms used in this section are found at reference [24]. In this section, we will show that eutacticity is an important parameter measuring spatial organization.…”

“…A partition of the localities into sub-localities , , … , is proposed using Voronoi tessellations as proposed in Ref. [24]. The goal in Ref.…”

“…The goal in Ref. [24] was to verify the spatial distribution of areas inside localities by comparing stars with high and low values of eutacticity. In this way, two experimental groups can be distinguished;…”

“…According to Ref. [24], let us assume that the areas , associated to sub-localities of the two groups of stars ( , ) have two crucial components: a) The eutacticity of the star and b) a set of random points to the stars, where represents the particular sub-locality and is the set of random points = 1, … , . So , ≠ , for every ≠ .…”