Theory and experiment in morphogenetic modelling: an examination of MacWilliams' data on hydra

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Abstract
A simple mathematical treatment is provided for H. K. MacWilliams' computer model for hydra regeneration (1982. J. Theor. Biol. 99: 681–703) as a practical demonstration of the value of mathematical analysis in tests of morphogenetic theory. MacWilliams uses a modified version of the Gierer–Meinhardt model, analysed here in terms of Turing's conditions on the linearized rate constants and n, the ratio of morphogen diffusivities, using a method previously described by T. C. Lacalli and L. G. Harrison (1979. J. Theor. Biol. 76: 419–436). The analysis justifies MacWilliams' reliance on "intuitive parameters" in testing his model, and the experimental values he obtains for these satisfy Turing's conditions. For the experiments, it is clear that n for hydra must be large, and this has interesting consequences. In particular, there is decreasing dependence on the autocatalytic features of the model with increasing n so that morphogen distributions approach the limit of exponential decay with distance from a fixed source. Examining the expression for λm, the chemical wavelength, shows explicitly how this happens. MacWiliams' work also gives an approximately correct value for the size limit below which regeneration cannot occur, which is related to λm in a well defined way.