Collaborative Research: New Methods in Phyllotaxis Introduction Many patterns in Nature are made of units repeated regularly in space (Fig. 1). The symmetries found in these patterns reflect two simple geometrical rules: i) Equivalent or nearly equivalent units are added in succession. ii) The position of new units is determined by interactions with the units already in place. Figure 1. Lattice-like patterns in Nature. A) Hexamers in a polyhead bacteriophage T4 (Erickson, 1973). B) Plates of a fossil receptaculitids (Gould and Katz, 1975). C) The mineralized silica shell of a centric diatom (Bart, 2000). D) The pineapple fruit, with two parastichies of each family drawn. Parastichy numbers are (8, 13) in this case. E) A young artichoke head, with parastichy numbers (34, 55). That proteins can assemble to create patterns that emulate the beautiful regularity found in crystals may not come as a great surprise. When the same regularity is found at the level of an entire living organism, one may justly be astonished. This is, however, a common occurrence in plants where the arrangement of leaves and flowers around the stem, known as phyllotaxis, can be very regular (Fig. 1D and E). To understand the origin of phyllotactic configurations in plants one must focus on the shoot apex where these configurations are established. The shoot apex is composed of a group of slowly dividing cells, the meristem, whose activity generates leaves, flowers, and other lateral organs of the shoot as bulges of cells called primordia (Fig. 2). The angle between two successive primordia is called the divergence angle. The most common configuration is one where the elements are arranged in two families of spirals called parastichies (Figs. 1D and 1E). Moreover, the numbers of parastichies in these two families are most often successors in the Fibonacci sequence (1, 2, 3, 5, 8, 13, …) and the divergence angle between primordia is close to the golden angle α = 360(2 − τ)o ≈137.5o where τ ≈1.608 is the golden mean (Figs. 1 and 2). € € Figure 2. Whorled and spiral configurations at the shoot apical meristem. A) Whorl of four in the larch embryo (Harrison and von Aderkas, 2004). B) Inflorescence meristem of Arabidopsis 1 thaliana with primordia numbered according to their age. C) Spruce meristem showing a (8, 13) pattern (Rutishauser 1998). In B) and C), the divergence angle is close to the golden angle. D) Side view of the meristem in C). Our overall goal is to understand the universe of all possible phyllotactic configurations and to determine why, in plants, some configurations are more common than others. To fulfill this goal we take a multidisciplinary approach. On the mathematical side, it has been necessary to develop a geometric framework that encompasses all the phyllotactic configurations found in Nature. We propose a drastic extension of the classical framework of parastichies and lattices to the new paradigm of primordia front and multilattices. A subset of multilattices is an attractor in our dynamical systems model, and contains the classical phyllotactic spirals and whorls, and continua of configurations close to these. On the biology side, we developed techniques that allow us to monitor and manipulate pattern formation at the meristem. By perturbing phyllotactic development, it will be possible to explore the set of stable configurations that are accessible to plants. The proposal includes four objectives: Objective 1: Develop the geometric framework of primordia fronts and multilattices. Objective 2: Generate a comprehensive dataset of time-resolved phyllotactic configurations and calibrate the dynamical models according to these data. Objective 3: Study the topology of the attractor in various systems and its implications on the frequency of certain configurations (e.g., Fibonacci configurations) and on the transitions of patterns in plants. Objective 4: Provide biologists with conceptual and computational tools to study the dynamics of meristem development, concentrating on Arabidopsis thaliana. Background Biological Control of Phyllotaxis One central question for phyllotaxis is the nature of the interactions that determine the position of new primordia. Recent observations indicate that these interactions could involve molecular and mechanical signals. At the molecular level, polar transport of the plant hormone auxin is now emerging as one of the key players in determining the position of new primordia at the meristem (Reinhardt et al., 2000 and 2003a). Auxin is believed to be transported in and out of cells by influx and efflux carriers located in the cell membrane. Putative influx and efflux carriers, AUX1 and PIN1 respectively, in Arabidopsis are present in cells of young primordia and at the site where the next primordium forms. At least one of these membrane proteins, PIN1, shows a clear subcellular localization that is believed to be the basis for polar transport of auxin. One putative mechanism for the propagation of phyllotactic configurations posits that auxin is actively transported up the stem to the apex where a high local level of auxin favors differentiation of a new primordium. On the other hand, young primordia act as auxin sinks thus lowering the level of auxin and preventing the differentiation of new primordia in their vicinity. The interaction between the apical flow of auxin and the activity of primordia as auxin sinks offers a possible molecular basis for the geometrical rules stated in the introduction. Tissue mechanics has also been implicated in the control of phyllotaxis. For example, it is possible to alter phyllotaxis by constraining the meristem mechanically during development (Hernández and 2 Green, 1993; Green, 1999). Another intriguing observation is the possibility of initiating primordia by local application or expression of the wall loosening molecule expansin (Fleming et al., 1997, Reinhardt et al., 1998; Pien et al., 2001). Based on these observations, it has been argued that “mechanical communication”, perhaps in the form of buckling of the meristem surface, offers another putative mechanism for the placement of new primordia at the meristem. Biologists have also attempted to identify the set of primordia directly involved in determining the location of new primordia. Through a series of ablation experiments, it was shown that only the set of most recently formed primordia around the circumference of the meristem plays a role in determining the position of the next primordium (Snow and Snow, 1932; Reinhardt et al., 2003b). Moreover, within this set some primordia seem to be playing a more important role although no quantitative analysis is currently available. From the results presented above it is clear that phyllotaxis is a dynamic process that evolves in space and time. This conclusion served as the impetus for the development of experimental approaches to observe meristem growth and the evolution of phyllotactic configurations over time (Hernández et al., 1991; Dumais and Kwiatkowska, 2002; Grandjean et al., 2004; Reddy et al., 2004). These experimental tools have achieved a high degree of sophistication but additional tools are needed to analyze quantitatively the massive amount of data that is being generated. Moreover, biologists are increasingly challenged by the complexity of processes such as phyllotaxis that evolve in space and time. There is therefore a need for a modeling environment that will help researchers test alternative hypotheses for the control of phyllotaxis. Classical Geometric Models of Common Phyllotactic Patterns The geometry of the surface on which the primordia configuration is formed can range anywhere from a cylinder to a dome or a flat disk. For simplicity, in this text we will concentrate on a cylinder of unit circumference. Since Bravais and Bravais (1837) it has been assumed that the regular configurations of cylindrical plant phyllotaxis are cylindrical lattices. A cylindrical lattice is obtained by placing primordia one at a time, at constant angular and height increments (x, y) along the cylinder (Figs. 3A and 3B). If one were to use the cylinder as a printing press and rolled it out on a plane, the successive primordia of a cylindrical lattice would appear as part of a straight line (which, rolled back on the cylinder, is called the “generative helix”). There are also other helices that are usually more visible in lattices: those going through nearest neighbors. These are called parastichies, and they often come in pairs of Fibonacci numbers in plants. Figure 3. Cylindrical lattices and multijugate configurations. The cylinder is flattened out on the plane. A) A lattice that is not phyllotactic: primordia are not equidistant to their nearest neighbors. B) A phyllotactic lattice with its generating vector (x,y). All other points can be obtained by adding integer multiples of this vector, and a multiple of the vector (1,0). This lattice 3 has parastichy numbers (3, 5): this is the number of distinct helices joining nearest neighbors, three parallel to (0, 3, 6, 9, …) and five to (0, 5, 10, 15, …). This is a common Fibonacci lattice. C) a multijugate (bijugate in this case) configuration that is comprised of two copies of the lattice in B) scaled by 1/2 and shifted by 1/2 turn from one another. This configuration has parastichy numbers (6,10). A common assumption is that points of lattices in nature are equidistant to their two nearest neighbors. Lattices satisfying this equidistant condition are called rhombic lattices and can be represented by configurations of tangent disks of same diameter D (Fig. 3). The parameter D, which biologically represents the ratio of the primordia diameter over the circumference of the meristem, is central to our study. Van Iterson (1907) charted out all possible values of (x,y) corresponding to rhombic lattices in a planar diagram (Fig. 4). This diagram is central in phyllotaxis, and an accepted measure of validity of a model of phyllotaxis is to be able to reproduce some of its qualitative and quantitative features.