Complex morphogenesis of surfaces: theory and experiment on coupling of reaction-diffusion patterning to growth.

Reaction-diffusion theory for pattern formation is considered in relation to processes of biological development in which there is continuous growth and shape change as each new pattern forms. This is particularly common in the plant kingdom, for both unicellular and multicellular organisms. In addition to the feedbacks in the chemical dynamics, there is then another loop linking size and shape changes with the reaction-diffusion patterning of growth controllers in the growing region. In studies by computation, the codes must incorporate, alongside the usual solvers of the partial differential dynamic equations, a versatile growth code, to express any kind of shape change. We have found that regulation of shape change in particular ways (e.g. to make narrow-angle branchings) demands new features in our chemical mechanisms. Our growth algorithm is for a surface growing tangentially, but moving outward and changing shape to accommodate the extra area. This is potentially applicable both to the tunica layer of multicellular plant meristems and to the growing tip of the cell surface, e.g. in the morphogenesis of single-celled chlorophyte algae which display branching processes: whorl formation in Acetabularia (Dasycladales) and repeated dichotomous branching in Micrasterias (Desmidiaceae). For computational studies, a hemispherical shell is a reasonable idealization of the initial shape. We describe results of two types of study: (1) Pattern formation by three reaction-diffusion models, with contrasted nonlinearities, on the hemispherical shell, particularly to find conditions for robust formation of annular pattern or pattern for dichotomous branching, both of which are common in plants. (2) Sequential dichotomous branchings in a system growing and changing in shape from the hemispherical start.