Turing's model and branching tip growth: relation of time and spatial scales in morphogenesis, with application to Micrasterias

Abstract

Morphogenesis following cell division in Micrasterias rotata is by outgrowth and repeated branching of a series of semicell lobes. Though successive branching events are qualitatively similar, they display changes in time and space scales, and these can be quantitated with the aid of autoradiographic patterns of labelled wall precursors that appear late in morphogenesis but which seem to represent its history. This enables us to consider branching as the conversion of a single centre of growth activity into two and to attempt to locate these centres precisely, in terms of both position and time of establishment. Temporal and spatial scales both decrease, by 75%, through a sequence of five branching events, in linear functional relationship to each other. This correlation points toward kinetic control of morphogenesis, i.e., the involvement of something like a reaction–diffusion mechanism. We analyse this possibility in terms of available reaction–diffusion theory to show how, after various simplifying assumptions, and if the time and space scales of branch formation are known, an effective diffusivity, [Formula: see text], for the patterning mechanism can be estimated. For M. rotata we obtain orders of magnitude: [Formula: see text], with an upper limit on the diffusivity of the faster diffusing of the two morphogenetic substances in the mechanism of ca. 1 × 10−7 cm2/s. These values implicate the cell membrane as the most probable site of pattern formation.