A Numerical Study of Pattern Forming Fronts in Phyllotaxis

Abstract

Using a partial differential equation model derived from the ideas of the Meyerowitz and Traas groups on the role of the growth hormone auxin and those of Green and his group on the role compressive stresses can play in plants, we demonstrate how all features of spiral phyllotaxis can be recovered by the passage of a pushed pattern forming front. The front is generated primarily by a PIN1 mediated instability of a uniform auxin concentration and leaves in its wake an auxin fluctuation field at whose maxima new primordia are assumed to be initiated. Because it propagates through a slowly changing metric, the patterns have to make transitions between spirals enumerated by decreasing parastichy numbers. The point configurations of maxima coincide almost exactly with those configurations generated by the use of discrete algorithms based on optimal packing ideas which suggests that pushed pattern forming fronts may be a general mechanism by which natural organisms can follow optimal strategies. We also describe in detail a numerical method that is used to efficiently and accurately integrate the model equations while preserving the variational structure from which they are derived.