Noise and robustness in phyllotaxis

A striking feature of vascular plants is the regular arrangement of lateral organs on the stem, known as phyllotaxis. The most common phyllotactic patterns can be described using spirals, numbers from the Fibonacci sequence and the golden angle. This rich mathematical structure, along with the experimental reproduction of phyllotactic spirals in physical systems, has led to a view of phyllotaxis focusing on regularity. However all organisms are affected by natural stochastic variability, raising questions about the effect of this variability on phyllotaxis and the achievement of such regular patterns. Here we address these questions theoretically using a dynamical system of interacting sources of inhibitory field. Previous work has shown that phyllotaxis can emerge deterministically from the self-organization of such sources and that inhibition is primarily mediated by the depletion of the plant hormone auxin through polarized transport. We incorporated stochasticity in the model and found three main classes of defects in spiral phyllotaxis--the reversal of the handedness of spirals, the concomitant initiation of organs and the occurrence of distichous angles--and we investigated whether a secondary inhibitory field filters out defects. Our results are consistent with available experimental data and yield a prediction of the main source of stochasticity during organogenesis. Our model can be related to cellular parameters and thus provides a framework for the analysis of phyllotactic mutants at both cellular and tissular levels. We propose that secondary fields associated with organogenesis, such as other biochemical signals or mechanical forces, are important for the robustness of phyllotaxis. More generally, our work sheds light on how a target pattern can be achieved within a noisy background.     

Regulation of phyllotaxis

Plant architecture is characterized by a high degree of regularity. Leaves, flowers and floral organs are arranged in regular patterns, a phenomenon referred to as phyllotaxis. Regular phyllotaxis is found in virtually all higher plants, from mosses, over ferns, to gymnosperms and angiosperms. Due to its remarkable precision, its beauty and its accessibility, phyllotaxis has for centuries been the object of admiration and scientific examination. There have been numerous hypotheses to explain the nature of the mechanistic principle behind phyllotaxis, however, not all of them have been amenable to experimental examination. This is due mainly to the delicacy and small size of the shoot apical meristem, where plant organs are formed and the phyllotactic patterns are laid down. Recently, the combination of genetics, molecular tools and micromanipulation has resulted in the identification of auxin as a central player in organ formation and positioning. This paper discusses some aspects of phyllotactic patterns found in nature and summarizes our current understanding of the regulatory mechanism behind phyllotaxis.     

Morphodynamics of phyllotaxis

A morphodynamic model for phyllotaxis based upon an axiomatic approach is presented. We show that the helical forms of alternate phyllotaxis can be derived from the assumption of the rudiment's growth and movement on the cylindrical embryo surface in the absence of a longitudinal displacement. This leads to the repeating transition of tetragonal packaging of the rudiments into hexagonal packaging and vice versa. Under these conditions, sequences of rudiments produce either left-handed or right-handed helices, the number of which at the circumference of the cylinder corresponds to adjacent numbers of the Fibonacci series. Cross-opposite phyllotaxis forms are defined as superior with respect to the alternate ones, and verticillate phyllotaxis forms as superior with respect to the cross-opposite ones. Different phyllotaxis forms can be interpreted as a result of stretching of crystalline structures of the embryo formed by dense packing of rudiments. The superior phyllotaxis forms can be considered as the additive summation of lower forms. Morphodynamic mechanisms underlying the formation of multiple forms of helical phyllotaxis are discussed.     

Physical phenomenology of phyllotaxis

We propose an evolutionary mechanism of phyllotaxis, regular arrangement of leaves on a plant stem. It is shown that the phyllotactic pattern with the Fibonacci sequence has a selective advantage, for it involves the least number of phyllotactic transitions during plant growth.     

Polygonal planforms and phyllotaxis on plants

We demonstrate how phyllotaxis (the arrangement of leaves on plants) and the ribbed, hexagonal, or parallelogram planforms on plants can be understood as the energy-minimizing buckling pattern of a compressed sheet (the plant's tunica) on an elastic foundation. The key idea is that the elastic energy is minimized by configurations consisting of special triads of periodic deformations. We study the conditions that lead to continuous or discontinuous transitions between patterns, state testable predictions, and suggest experiments to test the theory.     

Symmetry and its transition in phyllotaxis

Journal of Plant Research - Symmetry is an important component of geometric beauty and regularity in both natural and cultural scenes. Plants also display various geometric patterns with some kinds...     

Structural theory of phyllotaxis. II. Relationship between lower and higher forms of phyllotaxis

Opposite phyllotaxis forms are defined as superior ones in relation to alternate phyllotaxis forms, and verticillate phyllotaxis forms are defined as superior ones in relation to opposite phyllotaxis forms. On the basis of hypothetical notions about the properties of plant bumps and embryos, the probable mechanisms of creation of superior phyllotaxis forms from the lower ones are analyzed. It is shown that superior phyllotaxis forms can be considered to result from the combination of lower ones and that the superior forms can be split into the corresponding lower ones under artificial or natural influences.     

Auxin Receptors and Auxin Binding Proteins

In the last few years, a large number of auxin-binding proteins (ABPs) have been reported. Implicitly or explicitly, interest in such proteins resides in their possible role as auxin receptors. Many of these proteins are characterized as ABPs solely by their susceptibility to covalent photolabeling by tritiated azido-indole-3-acetic acid. In most cases where the labeled polypeptides have been identified, they turn out to have roles unconnected with primary auxin perception. It seems likely that auxin is binding to sites of catholic specificity in these cases and the influence of experimental protocols on the data is discussed. Because the term ABP implies that auxin binding affects the function of that protein, the importance of establishing further criteria before photolabeled peptides can be termed ABPs is emphasized. Applying such criteria, only a very few ABPs are currently of interest and only one of these, maize ABP1, has been characterized in detail. This protein is located primarily within the lumen of the endoplasmic reticulum, although an important fraction appears to function on the outside of the plasma membrane. The protein has a wide species distribution and it now seems highly probable that it is a genuine auxin receptor, the only protein for which such a function has yet been established. This conclusion is based on three independent lines of electrophysiological evidence, together with confocal imaging of cytoplasmic pH changes.     

Theory of phyllotaxis. I. A geometric model for helical forms of consecutive phyllotaxis

We have developed a geometric model for helical forms of consecutive phyllotaxis on the basis of an axiomatic approach. It follows from the model that rudiment growth and the movement of the cylindrical rudiment surface in the absence of a displacement in the direction along the rudiment axis leads to a repeating transition of tetragonal packaging of the rudiment into hexagonal packaging and vice versa. Under these conditions, sequences of rudiments produce left-handed and right-handed helices, the number of which at the circumference of the cylinder corresponds to adjacent numbers of the Fibonacci series. We demonstrate that the left-handed and right-handed isomers of helical forms of the consecutive phyllotaxis appear as a result of the transition of an unstable symmetric structure of the embryo at early developmental stages into stable left-handed or right-handed structures.     

The theory of phyllotaxis. II. Crystallographic interpretation of the interrelation between lower and superior phyllotaxis forms

Cross-opposite phyllotaxis forms are defined as superior with respect to the alternate ones and verticillate phyllotaxis forms as superior with respect to the opposite ones. Different phyllotaxis forms can be interpreted as a result of stretching of crystal-like structures of the embryo formed by dense packing of rudiments. Based on hypothetical concepts of the properties of plant rudiments and embryos, possible mechanisms of the formation of superior phyllotaxis forms from the lower ones have been analyzed. It was shown that the superior phyllotaxis forms can be considered as the results of additive summation of the lower forms. The theoretical conclusions are confirmed by the examples of polymorphic phyllotaxis in conspecific plants and by the facts of accidental splitting of superior phyllotaxis forms into the corresponding lower forms in nature and in experiment. The mechanisms underlying the formation of multiple forms of helical phyllotaxis have been proposed. The concept of a new type of mixed hexagonal-tetragonal phyllotaxis has been formulated and the mechanism of its formation has been considered. The forms of corn grain packaging in the corncob and leaf arrangement on the strawberry tomato stem are given as examples of true hexagonal-tetragonal phyllotaxis in nature.     

On the diffusion theory of phyllotaxis

An inhibitor diffusion theory of phyllotaxis is examined in the steady-state approximation for cylindrical shoot apex models. The model calculations give rise naturally to common patterns of spiral phyllotaxis, as well as to higher whorled patterns. The model also predicts commonly observed subpatterns of axillary organs superimposed on primary phyllotaxis patterns. Application of the model to phyllotaxis patterns in other organisms and in flowers is proposed.     

A contact pressure model for semi-decussate and related phyllotaxis

Semi-decussate phyllotaxis, in which leaves arise singly and the divergence angles between successive pairs of leaves alternate between approximately 90° and approximately 180°, is accounted for by a contact pressure model. It is assumed that leaf primordia are initiated at a divergence angle close to the Fibonacci angle of 137·5°, that the primordia move under contact pressure, and that when a primordium first experiences contact pressure all other primordia are fixed. Extensions of the model account for: psuedodecussate phyllotaxis, where the leaves appear to arise in pairs; semi-tricussate and pseudo-tricussate phyllotaxis, where the leaves are arranged in, respectively, dissolved or apparent trimerous whorls; and phyllotaxis of the 1,3 series, where the divergence angle is about 100°. The compatibility of the model with current theories of Fibonacci phyllotaxis is discussed.     

Ideal phyllotaxis on general surfaces of revolution

A mathematical model is presented for botanical features growing at an arbitrary rate on an arbitrary surface of revolution. At each point on the surface a lattice is defined, describing the phyllotaxis (that is, the arrangement of the features) there. It is shown how two parameters determine on which conspicuous spirals successive features are in contact at any point, whether the numbers of intersecting spirals change from point to point, and, if so, through what values. These parameters are the divergence δ, which is assumed to be constant, and a quantity ξ, which is the reciprocal of the normalized rise, and which in general varies from point to point. Finally, it is proved that Fibonacci phyllotaxis (in which the numbers of intersecting spirals are always Fibonacci numbers) produces greater packing efficiency than any other, provided that the lattice varies over the surface.     

Two notions of conspicuity and the classification of phyllotaxis

Invoking cylindrical Bravais lattices, Adler (1974, 1977) proposed a mathematically precise definition for the botanical classification of phyllotaxis. It is based on opposed pairs of parastichy families, that are conspicuous and visible. Jean (1988) generalized this concept to non-opposed pairs of parastichy families. In the present paper it is shown that this generalization implies a notion of conspicuity different from Adler's. This is made obvious by redefining the key terms of the two approaches. Both classifications are well defined. For Adler's, this is shown by presenting a general proof for his conjecture that conspicuous (in the sense of Adler) opposed pairs of parastichy families are visible. There are indications that in applications to models of phyllotaxis (van Iterson model, inhibitor models) their solutions are better characterized by Jean's classification. The differences between Adler's and Jean's classification show up only in very rare cases, so that the practice of pattern determination is only insignificantly touched by the present results. It turns out that the widely used contact parastichy method to determine phyllotactic patterns gives results according to Jean's classification rather than Adler's.     

Golden ratio and phyllotaxis,a clear mathematical link

Exploiting Markoff’s theory for rational approximations of real numbers, we explicitly link how hard it is to approximate a given number to an idealized notion of growth capacity for plants which we express as a modular invariant function depending on this number. Assuming that our growth capacity is biologically relevant, this allows us to explain in a satisfying mathematical way why the golden ratio occurs in nature.