A MECHANICAL MODEL OF GROWTH IN REGULAR SEA-URCHINS - PREDICTIONS OF SHAPE AND A DEVELOPMENTAL MORPHOSPACE

The shapes of several urchins are correctly predicted by a model that uses only measured height and diameter as fitted variables. The predic ted shape is based on the engineering theory of thin shells, which is conventionally used to calculate the shapes of a fluid droplet on a ho rizontal plane, and of 'buckle-free' engineered domes. The magnitudes of forces theoretically required to generate urchin shapes at realisti c sizes are similar to forces typically exerted on the skeleton by sel f-weight, podia and coelomic pressure. An urchin's shape, despite comp lex details of plate growth, is thus determined by a force balance at each point in the skeleton. Despite the skeleton's apparent rigidity, over developmental time it must deform in a manner similar to a stretc hy balloon. This membrane model of morphogenesis specifies two develop mental shape parameters: (i) the apical curvature; and (ii) a ratio of the mimicked vertical gradient of pressure (podial forces, etc.) to t he internal coelomic pressure. An ontogenetic series of urchins is rep resented as a curved line in a two-dimensional, developmental morphosp ace. This morphospace, which is useful for studying developmental cons traints and macroevolutionary dynamics, explains observed patterns of allometry in height and diameter in urchins.