Turing's model and pattern adjustment after temperature shock, with application to Acetabularia whorls

Abstract

The giant single-celled alga Acetabularia repeatedly forms whorls of hairs during vegetative growth. The spacing between hair initials, measured when they first become visible, is temperature-sensitive. If the temperature is changed rapidly from 18°C to 29°C, the corresponding change in spacing takes place over several hours, roughly linearly. This functional form suggests that, if the pattern is readjusting from, typically, nine hairs to twelve, the intermediate values ten and eleven are favoured at intermediate times. Turing's reaction—diffusion model describes pattern determination in terms of the competitive growth and decay of patterns with different spacings at spacing-dependent rates. After the competitive rates have been changed by temperature shock, appearance of intermediate patterns is an expected feature of the Turing mechanism. Lacalli & Harrison 1979 indicated the ranges of relative values of rate constants in the Turing model which are appropriate to various kinds of morphogenetic behaviour. For any change which alters the ratio (system size)/(pattern spacing) = number of parts in pattern, the adjustment of pattern will be rapid only for a rather restricted range of values of the rate constants. A set of values proposed in this connection in the 1979 paper for pattern changes in Drosophila wing discs gives a good account of the changes in A. mediterranea whorls after temperature shock. The values required for diffusivities suggest that the morphogens are membrane-bound.