Simplified dynamic model for the motility of irregular echinoids

Abstract

Inspired by the locomotion mechanism of sea urchins, we study the locomotion of an irregular echinoid by means of a simplified dynamical model. We prove that if two conjectures are assumed, the geometrical arrangement of the five ambulacral petals of irregular echinoids should form a eutactic star in order to optimize motility. We firstly propose an adequate "measure" of eutacticity that allows us to to verify the statistical tendency to such a property for a representative collection of fossil specimens. Next, regarding dynamics, the biological advantage of eutactic stars is addressed as a minimal path problem. Finally, we study the stability of some eutactic stars under small perturbations.

A measure of regularity for polygonal mosaics in biological systems

“…Here denotes the transpose of the matrix and is the n × n identity matrix. When experimental measurements are involved, a more suitable criterion of eutacticity has been proposed [14]: if we define then the star producing is eutactic if and only if …”

“…This property will be particularly useful in this work. Actually, it can be proved that the lower bound of ε is [14]. Since in this work we will be concerned with polygonal patterns in the plane, stars are defined in two dimensions ( n =2) and the lower bound of ε is in this particular case .…”

Influence of Modularity and Regularity on Disparity of Atelostomata Sea Urchins

“…A reliable numerical criterion for the eutacticity of a star, suitable for working with experimental measurements, was proposed in Ref. 16 and is as follows. Let A be the matrix whose N columns are the components of the vector composing a star S , with respect to a given fixed orthonormal basis of R 2 .…”

“…Let A be the matrix whose N columns are the components of the vector composing a star S , with respect to a given fixed orthonormal basis of R 2 . Then, a star S in R 2 , represented by the matrix A , is eutactic if and only if 16 where S = AA T ; Tr denotes the trace and the superindex T denotes the transpose. Notice that the parameter ∊ is capable f indicating the degree of eutacticity of the star represented by A : if it is not strictly eutactic (∊ = 1), then this quantity is closer to 1, the more eutactic the star is.…”

“…Let A be the matrix whose N columns are the components of the vector composing a star S , with respect to a given fixed orthonormal basis of R 2 . Then, a star S in R 2 , represented by the matrix A , is eutactic if and only if 16 …”