Meeting Abstract
The coronal skeletons of modern sea urchins comprise 20 columns of plates, of two basic types (ambulacral and interambulacral), in five paired sets. Following an initial plate configuration derived from the rudiment, plates are added to each column over the lifetime of the animal. The growth of individual plates of a sea urchin can be accurately modeled by logistic functions. The addition and growth of plates in a single column can be envisioned as a stage-structured density-dependent Markov chain, where stages represent plate cohorts. The parameters of the Markov model, represented on a life cycle graph, can be used to define a tridiagonal transition matrix. Inclusion of density-dependent factors results in a model of logistic growth. The model has the general form X(t+1) = QAX(t) where X(t) is an n-dimensional size vector at age t, A is an n×n tridiagonal transition matrix, and Q is an n×n density-dependent damping matrix. The model thus assumes that the growth of a plate at a given stage is dependent on the state of the previous plate stage and some function of plate density. The transition matrix and damping coefficients are unknown a priori and require solution of an inverse model using a Monte Carlo approach. The inverse models are derived from estimated plate growth series (beginning with the classic Deutler model of Echinus esculentus) and adult plate size distributions measured from 3D scans of species representative of major clades of regular sea urchins. The Markov model is shown to generate the stable plate size distributions of these different clades of sea urchins. This offers a new approach to quantifying the growth of these organisms.
