J. theor. Biol. (1996) 178, 255–274 Phyllotaxis as a Dynamical Self Organizing Process Part I: The Spiral Modes Resulting from Time-Periodic Iterations S. D Y. C Laboratoire de Physique Statistique, 24 rue Lhomond, 75231 Paris Cedex 05, France (Received on 12 January 1995, Accepted in revised form on 15 August 1995) This article is the first of a series of three in which the various phyllotaxic modes are shown to result from successive iterations of two possible simple dynamical systems. In this first part the hypotheses put forward by Hofmeister (1868) for the formation of primordia are re-examined and shown to form the rules of such a system. By means of a physics experiment and a numerical simulation, it is demonstrated that this system gives rise to the self organization of the spiral phyllotactic structures. The dynamic is controlled by only one parameter, G, equivalent to Richards’ plastochrone ratio (1951), which characterizes growth. The diagram representing the values of the divergence angle as a function of the plastochrone ratio resembles that obtained geometrically by van Iterson (1907). In the present results, however, only one branch of solutions is continuous for all values of G; this difference with the geometrical results is important. The predominance of Fibonacci order in botany is related to this continuity: during the ontogeny, because of the continuous decrease in G, only this branch is followed. In this framework the build-up of such complex structures as the inflorescence of compositae is described. 7 1996 Academic Press Limited 1. Introduction recognized in the history of phyllotactic studies, each of them having brought major contributions to their Phyllotaxic patterns are generated whenever a understanding. vascular plant repeatedly produces similar botanical elements at its tip (leaves, bractae, florets etc). These patterns are surprisingly regular, so regular in fact The first investigations resulted from the direct that a physicist can compare their order to that of observation of the arrangements of the elements on crystals. The observed organizations are classified in mature plants. Since the early works of Schimper only two categories. In the first, formed of the (1830), Braun (1831, 1835), Bravais & Bravais distichous and spiral modes, the leaves appear one at (1837a, b, 1839), Airy (1873) and van Iterson (1907), a time along the stem. The striking peculiarity of this the analysis of the geometry of the patterns has been family is that it is directly related to the Fibonacci aimed at a characterization of the observed spiral and series and the golden mean. In the second, constituted whorled organizations of mature stems. More recent of whorled modes, a constant number of leaves works in this domain were done by Erickson (1973), appear simultaneously at the same height on the stem Rivier (1988) and Rothen & Koch (1989). This type and form successive whorls. of approach provided an analysis of the very specific The investigation of phyllotactic patterns has a properties of these dispositions. In particular, it was long history and was undertaken from several found that two different descriptions of the spiral viewpoints. Recent reviews on this subject can be order could be given. If the elements are linked in found in Williams (1975), Jean (1984, 1994) Steeves & their order of appearance, the resulting curve forms Sussex (1989), Lyndon (1990), Sachs (1991) and a spiral (or a helix) called the generative spiral. Medford (1992). Three main lines of thought can be Projected in the plane perpendicular to the stem, the 0022–5193/96/030255+19 $18.00/0 255 7 1996 Academic Press Limited 256 . . divergence angle 8 between the base of two successive the apex, because the shoot grows with velocity U, the elements is constant and generally has a value close existing primordia drift away from the tip with a to F=2p(2−t)=137.5°, where t is the golden mean velocity V so that they leave room for the formation t=(−1+!5)/2. In the second description, two sets of new elements [Fig. 1(a)]. Hofmeister (1868) of intersecting spirals are defined by linking each proposed that this inception is periodic in time and element to its nearest (or contact) neighbours. The that the new primordium appears in the largest numbers i and j of these spirals (the parastichies) available space. The position of this primordium around the stem are generally two successive terms of around the apex thus results from the position of the the main Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, previous ones. Inspired by van Iterson’s work, Snow . , in which each term is the sum of the two previous & Snow (1952) tried to adapt Hofmeister’s dynamical ones (the first two terms being 1 and 1). The Bravais rules to the appearance of elements of a given finite brothers (1837a), showed that if the divergence angle size. Some of their experiments [Snow & Snow (1935)] 8 was assumed to be equal to the ideal angle F, the also showed that there was a connection between the parastichies were then necessarily two consecutive Fibonacci numbers and that their particular values only depended on the longitudinal elongation of the stem. Airy (1873) was the first to consider the piling of elements of finite size instead of point like particles. In order to study the effect of a compression on a cylindrical piling he implemented an original system. He glued identical spheres onto an extended rubber band alternately on each side of the band (i.e. in a distichous disposition) and then slowly released the tension. As a result he observed transitions to the spiral modes (1, 2) and (2, 3). Inspired by this work, van Iterson (1907) systematically investigated the phyllotactic patterns obtained by paving a surface with solid elements, e.g. equal disks on a cylinder. He thus introduced a new parameter, the ratio b of the diameter of the disks over the perimeter of the cylinder. Without any other assumption he showed that, for a given value of b, spiral organizations were possible, corresponding to a series of well-defined values of 8. These values are a function of b, and the number of different possibilities increases with decreasing b, so that the graph 8(b) has the shape of a complex tree. A different approach was initiated when the apical region, where the formation of the primordia takes place, was observed using microscopic techniques. This led to consideration of the problem of how the phyllotactic structures appeared. Hofmeister (1868) was the first to systematically observe the apical region and to state general dynamical principles that, F. 1. Definition of the parameters of the apical growth. (a) he suggested, led to the formation of the observed Scheme of the botanical situation: given the growth of the shoot patterns. A smooth region of undifferentiated tissues, at velocity U, the resulting velocity at which the primordia move the apex is situated at the extremity of the shoot. away from the tip is V. The incipient primordium is represented in black. It forms at the periphery of the apex of radius Ro , shown Around it specific mitotic activity creates small in grey. (b) The plane model of the apex. The position of the new protrusions, the primordia, which already form primordium of appearance number n is defined relative to the conspicuous parastichies and will evolve into various previous one (of number n−1) by the divergence angle 8 and by the ratio of their distances to the centre: a=rn−1/rn (Richards’ types of botanical elements: leaves, bracts, sepals, plastochrone ratio). The radial motion of each particle V(r) and the petals, stamens, florets etc. In the reference frame of field lines of its repulsive potential E(d) are indicated. — 257 whorled phyllotactic modes and the spiral ones. They dynamics per se. This is our limited aim: we are thus suggested (Snow & Snow, 1962) removing the concerned here with the minimal dynamical hypoth- imposed periodicity of the process: a new primordium eses necessary to give rise to the phyllotactic patterns. simply forms where and when there is enough space In this first article we thus return to the hypotheses for its formation. of Hofmeister (1868) and in the second and third An important step in relating quantitatively the (Douady & Couder, 1996a, b) we will revisit those put dynamics of the growth to the geometry of the forward by Snow & Snow (1952). It must be kept in resulting spirals was taken by Richards (1951) who mind, however, that even if we are successful in introduced the plastochrone ratio a, a parameter of realistically reproducing the build-up of the phyllo- fundamental importance to quantify the growth and taxic patterns, the underlying physiological processes which will be described below. will still have to be identified. Finally a completely different set of hypotheses was put forward by Plantefol (1948), in which he gave a specific role to one of the set of parastichies that he 2. The First Iterative Model called foliar helices. These hypotheses, however, did The remarkable feature of the dynamical hypoth- not deal with the fact that the number of parastichies eses is that they do not have any specific effect directly follow the Fibonacci series; they cannot, therefore, be caused by the biological nature of the system. Rather, used here. In fact, as we will see, they do not seem to they define the principles of an iterative system, in be compatible with the minimal hypotheses necessary which the repeated application of the same rule gives to get this type of ordering (see Section 5). rise to the successive states. Our aim was to see whether these principles were able, by themselves, to produce the spiral organizations hitherto only A third direction for research was aimed at the observed in plants.