A new theoretical model for growth of the echinoid test

Abstract

A new developmental model for growth of the echinoid test is based on a review of the growth patterns in regular echinoids. Echinoids are structurally composed of a tessellation of hundreds to thousands of individual plates. The two major aspects of echinoid growth are treated separately. (1) New plates are added in accordance with the Ocular Plate Rule and plate addition is hypothesized to be constitutively active but inhibited by a morphogen originating in coronal plates. Morphogen production is modeled as an inverse function of plate size and the concentration of inhibiting morphogen at a plate nucleation point is inversely proportional to the distance from surrounding plate centers. Plate addition is triggered whenever the inhibiting morphogen concentration falls below a threshold value. (2) The growth of individual plates is described using the Bertalanffy growth equation to model change in plate perimeter. The geometric model is based on a spherical frame of reference, and all calculations of position and growth are modeled over the surface of the sphere (i.e., along geodesics). The data structure defined to maintain the geometric parameters is based on a spherical Delaunay triangulation of plate centers, and the edge geometry approximated by the dual Voronoi polygonalization. Echinoid plates are thus modeled as Voronoi polygons covering the sphere. Growth is modeled by the increasing radius of the sphere and the changing topology of the plates as new plates are added and existing plates grow. Final form of the complete test is generated by an affine deformation of the sphere. The growth model is implemented as the program EFORECHINOID, coded in the object-oriented programming language C++ with significant usage of the Standard Template Library (STL) for efficient coding and memory management. Most parameters are available to a user via a Graphical User Interface (GUI), and output of 3-dimensional simulations is via standard 3-D AutoCAD[trademark] Drawing Exchange Format (DXF) files. Program efficiency is O(nlogn) and reasonably parameterized growth simulations with several hundred time steps can be performed in a matter of minutes per run.