Pattern Selection in Plants: Coupling Chemical Dynamics to Surface Growth in Three Dimensions

Abstract

A study is made by computation of the interplay between the pattern formation of growth catalysts on a plant surface and the expansion of the surface to generate organismal shape. Consideration is made of the localization of morphogenetically active regions, and the occurrence within them of symmetry-breaking processes such as branching from an initially dome-shaped tip or meristem. Representation of a changing and growing three-dimensional shape is necessary, as two-dimensional work cannot distinguish, for example, formation of an annulus from dichotomous branching. For the formation of patterns of chemical concentrations, the Brusselator reaction-diffusion model is used, applied on a hemispherical shell and generating patterns that initiate as surface spherical harmonics. The initial shape is hemispherical, represented as a mesh of triangles. These are combined into finite elements, each made up of all the triangles surrounding each node. Chemical pattern is converted into shape change by moving nodes outwards according to the concentration of growth catalyst at each, to relieve misfits caused by area increase of the finite element. New triangles are added to restore the refinement of the mesh in rapidly growing regions. The postulated mechanism successfully generates: tip growth (or stalk extension by an apical meristem) to ten times original hemisphere height; tip flattening and resumption of apical advance; and dichotomous branching and higher-order branching to make whorled structures. Control of the branching plane in successive dichotomous branchings is tackled with partial success and clarification of the issues. The representation of a growing plant surface in computations by an expanding mesh that has no artefacts constraining changes of shape and symmetry has been achieved. It is shown that one type of pattern-forming mechanism, Turing-type reaction-diffusion, acting within a surface to pattern a growth catalyst, can generate some of the most important types of morphogenesis in plant development.