The Possible and the Actual in Phyllotaxis: Bridging the Gap Between Empirical Observations and Iterative Models Scott Hotton1, Valerie Johnson2, Jessica Wilbarger 2, Kajetan Zwieniecki3, Pau Atela2, Christophe Golé2, Jacques Dumais3,* 1 UC Merced Center for Computational Biology, University of California, Merced CA 95344 2 Department of Mathematics, Smith College, Northampton MA 01063 3 Department of Organismic and Evolutionary Biology, Harvard University, Cambridge MA 02138 *Corresponding author; email [email protected], phone: 617 496-0751, fax: 617 496- 5945 ABSTRACT This paper presents new methods for the geometrical analysis of phyllotactic patterns and their comparison with patterns produced by simple, discrete dynamical systems. We introduce the concept of ontogenetic graph as a parsimonious and mechanistically relevant representation of a pattern. The ontogenetic graph is extracted from the local geometry of the pattern and does not impose large-scale regularity on it as for the divergence angle and other classical descriptors. We exemplify our approach by analyzing the phyllotaxis of two asteraceae in the light of a hard disk model. The simulated patterns offer a very good match to the observed patterns for over 150 iterations of the model. Key words: artichoke, Delaunay triangulation, dynamical system, shoot apical meristem, phyllotaxis, sunflower, Voronoi tessellation. INTRODUCTION Numerous patterns in Nature are made of identical units repeated regularly in space. Microtubules and viral capsids provide examples at the molecular level (Erickson 1973). Given that proteins and other macromolecules assemble over length scales that are only slightly larger than those of inorganic crystals it is not overly surprising that they offer similar patterns. However, when crystal-like regularity is found at the level of an entire living organism, one may justly be astonished. Yet, this is a common occurrence in plants where the arrangement of leaves and flowers around the stem, known as phyllotaxis, yields striking patterns (Figure 1). Figure 1. Crystal-like patterns in plants. (A) Bellis perennis, (B) broccoli romanesco (Brassica oleracea), (C) pine cone (Pinus sp). The symmetries found in crystals, biopolymers, and plants reflect two simple geometrical rules: i) equivalent or nearly equivalent units are added in succession and ii) the position of new units is determined by interactions with the units already in place. To visualize these rules in plants, one must focus on the shoot apical meristem where leaf and flower primordia are initiated in a stereotypical manner. It is commonly assumed that the divergence angle between successive primordia is close to the golden angle α = 360(2 − τ)o ≈137.5o where τ ≈1.608 is the golden mean (Figure 2). Repetition of the constant divergence angle over many primordia leads to the emergence of two families of spirals called parastichies (the two families are particularly clear in Figure 1C). The numbers of parastichies in these two families are often consecutive Fibonacci € numbers (1, 2, 3, 5, 8, 13, …). € Figure 2. Primordium initiation at the shoot apical meristem of Arabidopsis thaliana. Flower primordia are numbered according to their order of initiation and α is the divergence angle. The phenomenon of phyllotaxis raises at least two major questions - what is the nature of the interactions that determine the position of new primordia? and why, of all the possible patterns that can be created by repeating identical units in space, only a rather small set is ever seen in plants? The first question calls for experimental work to uncover the molecular, cellular and, potentially, mechanical factors that control the positioning of new primordia. The second question offers a different challenge since it presupposes some formal understanding of what patterns are possible whether or not these patterns are observed. To answer this question we need to build a bridge between the universe of observed phyllotactic patterns in plants and the more abstract universe of all patterns. We provide one of the first attempts to bridge the gap between the possible and the actual in phyllotaxis. The bridge rests on three pillars: i) a protocol to extract precise quantitative information from meristem images. ii) a dynamical system model that can be used to explore the universe of phyllotactic patterns. iii) a new concept, the ontogenetic graph, that provides the basis for comparing observed and simulated phyllotactic patterns. This paper is organized around these three contributions following a brief review of previous work in the area. For clarity, we discuss only patterns on a disk although our approach encompasses the whole breadth of phyllotactic patterns found in plants. PREVIOUS WORK The mechanistic basis of phyllotaxis has been reviewed by Jonsson and others (2006) in this issue; we therefore focus on the steps that have been made towards bridging the gap between the analysis of phyllotactic patterns in plants and their modeling via dynamical systems. Classical Geometric Models of Common Phyllotactic Patterns The classical geometric models of phyllotaxis all refer to lattices – either helices on a cylinder k (Atela and others 2002) or spirals on a disk. A spiral lattice is a set of points Pk = (ρ , kα), where α is the divergence angle and ρ is the plastochrone ratio (i.e. the ratio of radial distances for two successive primordia). The curve that joins these points ordered by k is called the ontogenetic spiral. The visible spirals in these patterns, joining each point to its nearest neighbors, are called parastichies. The parastichies are usually grouped in two families of parallel spirals (Figure 3). The number of parastichies in these families is the basis of the traditional classification of phyllotactic patterns. Figure 3. (A) A spiral lattice with parastichy numbers 11, 20. This is not a phyllotactic lattice as only one parastichy family can be formed with tangent disks. (B) A phyllotactic lattice, with parastichy numbers 13, 21. (C) The space of all lattice patterns as drawn by van Iterson in 1907. The tree structure is the subset of lattices corresponding to phyllotactic patterns. The longest branch shown corresponds to lattices with Fibonacci numbers. Lattices that are similar to patterns exhibited in plants are called phyllotactic lattices. Phyllotactic lattices can be represented by a collection of disks centered at the lattice points, with radii growing as βρk (for some β) and such that disks are tangent along two families of parastichies (compare Figure 3A and Figure 3B). The set of parameters yielding phyllotactic lattices was studied by van Iterson (1907) (Figure 3C) and forms a tree structure whose main branch contains all pairs (ρ, α) yielding lattices with Fibonacci parastichy numbers and converging toward the golden angle α ≈ 137.5o as ρ approaches 1. Dynamical Models of Phyllotaxis A large part of the literature offers geometric studies of models with no time evolution, involving only spiral configurations (Bravais and Bravais 1837; van Iterson 1907; Levitov 1991; Adler 1998). Unfortunately, they cannot explain the convergence to (and hence the stability of) regular spiral configurations from general initial conditions, nor can they account for plant patterns that are not lattices. In this paper, we use a geometric dynamical systems approach, as initiated in Atela and others (2002) (see also Kunz 1997; d’Ovidio and Mosekilde 2000). We were inspired by the iterative models of the physicists Douady and Couder (1996) (see also the related systems proposed by Schwabe and Clewer 1984; Koch and others 1998): they are sufficiently simple to be cast as discrete dynamical systems; yet, they are compatible with most of the biological mechanisms proposed such as buckling and auxin transport. These dynamical systems models are an implementation of the two rules stated in the introduction, as applied to phyllotaxis. More specifically: 1) Primordia are formed successively in time 2) Primordia are positioned in the “least crowded spot” at the edge of the meristem If one assumes that primordia form with fixed period, one obtains what Douady and Couder called the Hofmeister hypothesis. If one assumes that new primordia form when and where there is enough space, one obtains the so-called Snow hypothesis. The Snow hypothesis has the advantage of allowing simultaneous formation of several primordia (sometimes yielding multijugate configurations). Douady and Couder (1996) numerically reproduced many features of the van Iterson diagram. They also showed that, for a given set of parameters, configurations starting with different initial conditions could converge to either spiral or whorl configurations. This result suggests that the initial configuration of primordia found in the meristem of the plant seedling could play an important role in determining the phyllotaxis later in development. Mathematically, these models can be seen as discrete dynamical systems acting on points (R0,θ0 ),...,(RN ,θN ) of the disk (each point representing the center of a primordium in polar coordinates) with a transformation from one configuration to the next of the form: (R0,θ0,...,RN ,θN ) → ( f (R0,θ0,...,RN ,θN ),R0,θ0,...,RN−1,θN−1), where the function f determines the location of the new primordium that minimizes its interaction € with the existing ones. € The Hofmeister type models have pre-established plastochrone ratio ρ = Rk /Rk-1 so that R needs not be explicitly part of the transformation above. In the Snow-type models, f expresses the location and time at which the interaction W falls below a certain threshold. In either case, most authors consider an interaction of the form W (R,θ,R0,R0,...,RN ,θN ) = ∑u(R,θ,Rk,θk ), k where (R,θ) is the test location of the new primordium and u is a more or less explicit function of a the distance between (Rk,θk) and (R,θ). Douady and Couder used, among others, u = c dis where a and c are constants and dis means distance between (R ,θ ) and (R,θ). Our group has € k k € € proposed, in either the Hofmeister or Snow models (Atela and others 2002; Atela and Golé 2006) a radical simplification of the interaction as * W (R,θ,R0,θ0,...,RN ,θN ) = maxk u(R,θ,Rk,θk ).