arXiv:1301.4190v1 [math.AP] 17 Jan 2013 Phyllotaxis, Pushed Pattern-Forming Fronts and Optimal Packing Matthew Pennybacker and Alan C. Newell University of Arizona, Department of Mathematics, Tucson, Arizona 85721, USA (Dated: January 18, 2013) We demonstrate that the pattern forming partial differential equation derived from the auxin distribution model proposed by Meyerowitz, Traas and others gives rise to all spiral phyllotaxis properties observed on plants. We show how the advancing pushed pattern front chooses spiral families enumerated by Fibonacci sequences with all attendant self similar properties and connect the results with the optimal packing based algorithms previously used to explain phyllotaxis. Our results allow us to make experimentally testable predictions. PACS numbers: 87.18.Hf, 87.10.Ed, 02.60.Lj, 02.30.Jr INTRODUCTION Using a model derived from the pioneering ideas of Meyerowitz, Traas et al [1] (MT) on the role of PIN 1 proteins in creating an instability of uniform auxin con- centrations near a plant’s shoot apical meristem (SAM), this Letter reports on stunning new results which lend credence to the view that almost all of the features of phyllotactic configurations are the result of a pushed pat- tern forming front whose origin is the MT instability. The front leaves in its wake either whorls or Fibonacci spirals. The patterns we observe exhibit all known self-similar properties, reveal some new invariants and reproduce the well known van Iterson diagrams associated with discrete algorithms (Levitov, Douady and Couder, Atela, Gol´ e and Hotton [2]) which reflect optimal packing strategies based on ideas and observations of Hofmeister and Snow and Snow [3]. Further, the location of the maxima of the auxin fields coincide very closely with the point con- figurations generated by the discrete algorithms, which suggests that pattern forming systems may be a new tool for addressing optimal packing challenges. In short, in- stability generated patterns are the mechanism by which plants and other organisms can pursue optimal strate- gies. The equation we use [4] for the auxin concentration fluctuation about its mean derives from the continuum approximation to a discrete cell by cell description pro- posed in [1]. The resulting PDE is strikingly similar to those found in many pattern forming systems which sug- gests that Fibonacci spiral patterns should be observable in many physical contexts. The instability of the uniform auxin concentration state which gives rise to the pattern is primarily due to reverse diffusion. In most equilib- rium situations, inhomogeneities in chemical concentra- tions are smoothed by ordinary diffusion. But in plants the situation is not an equilibrium one. PIN 1 proteins in the interiors of cells move under the influence of an auxin gradient to the cell walls where they orient so as to drive auxin in the direction of its gradient. When the effect is sufficiently strong, the net diffusion is negative and an instability with a preferred length scale occurs. As the FIG. 1. A pseudocolor plot of u(x,t) on the inner sec- tion r < 89 of a pattern initiated at r = 233 with parastichy numbers (89, 144). A movie may be found at http://math.arizona.edu/ ~ pennybacker/media/sunflower/ . shapes initiated by the instability grow, nonlinear inter- actions determine the preferred planforms. The resulting PDE for the auxin fluctuation concentration u(x,t) turns out to be very close to a gradient flow, and is given by ∂u ∂t = − δE δu = µu − ( ∇ 2 +1 ) 2 u − β 3 ( |∇u| 2 +2u∇ 2 u ) − u 3 (1) E [u]= −  µ 2 u 2 − 1 2 ( u + ∇ 2 u ) 2 − β 3 u|∇u| 2 − 1 4 u 4 (2) where x =(x, y) or (r, θ) are horizontal coordinates on the tunica (plant skin) surface, t is time, the most lin- early unstable wavelength is 2π, and µ and β are the coefficients of the linear growth and quadratic terms re- spectively.