TURING CONDITIONS AND THE ANALYSIS OF MORPHOGENETIC MODELS

The conditions under which changes in parameters lead to changes in the pattern-generating behaviour of Turing's two-morphogen linear model can be expressed in terms of two reduced rate constants, k′1 and k′4 which represent autocatalytic and self-inhibitory rates in relation to cross-catalysis and cross-inhibition, and the ratio of the diffusivities for the two morphogens. This allows a new type of diagram to be drawn in which a two-dimensional k′1k′4 space is divided by Turing's conditions into regions where the various morphogenetic behaviours occur. An analysis using this type of diagram is applied to the linear limit of two non-linear models, those of Gierer and Meinhardt and of Tyson's modification of the Brusselator, and is used to clarify what is happening in their non-linear development. Several possible applications are mentioned; to the stability of a simple binary pattern in slime moulds, to the development and decay of a succession of patterns in the imaginal wing discs of Drosophila as treated by Kauffman et al., and to the apparent ease of disturbing the sea urchin blastula to produce a three-part, rather than a two-part pattern. © 1979.