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.     

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.     

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.     

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.     

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...     

The consequences of contact pressure in phyllotaxis

To determine the consequences of contact pressure in phyllotaxis, a mathematical model is constructed in which a leaf distribution is represented by a point lattice of n + 1 lattice points at equal intervals on a helix wound around a cylinder. The model is normalized by taking the girth of the cylinder as 1 and by measuring time T in plastochrones, so that n = [T]. r stands for the normalized internode distance (component of the distance between two consecutive lattice points that is parallel to the axis of the cylinder). d stands for the divergence (fraction of a turn between consecutive lattice points). It is assumed that r is a monotonic decreasing function of T such that r(T) → 0 as T → ∞. Contact pressure is represented by the assumption that the minimum geodesic distance between lattice points is maximized. It is shown that if (p, q), with p < q, is the contact phyllotaxis determined when contact pressure first becomes effective, then the continuation of contact pressure requires that the advance to higher phyllotaxis as r decreases must proceed via successive pairs of consecutive terms of the Fibonacci sequence generated by the numbers p and q, namely, p, q, p + q, p + 2q, 2p + 3q, …. The divergence, starting from some value $\text{d =}\text{1}\text{t}\text{+}\text{1}\text{a}{}_2\text{+ … +}\text{1}\text{(a}{}_{\mathrm{n}}\text{+ x)}$ determined by p and q converges to an ideal angle $\text{1}\text{t}\text{+}\text{1}\text{a}{}_2\text{+ … +}\text{1}\text{a}{}_{\mathrm{n}}\text{+}\text{1}\text{τ}$, where τ is the golden section. A necessary and sufficient condition for the ideal angle to be $\text{1}\text{2}\text{+}\text{1}\text{τ}\text{= τ}{}^{\mathrm{?2}}$ is that the p and q of the initial contact phyllotaxis be consecutive Fibonacci numbers of the sequence 1, 2, 3, 5, 8, …. It is proved that a sufficient condition for convergence to the ideal angle τ^?2 of normal phyllotaxis is that contact pressure begin before T = 5 or before $\text{r <}\text{3}\text{38}\text{1}\text{2}$ with d initially between $\text{1}\text{3}$ and $\text{1}\text{2}$.     

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.     

An index of lipid phase diagrams

There is a growing awareness of the utility of lipid phase behavior data in studies of membrane-related phenomena. Such miscibility information is commonly reported in the form of temperature-composition (T-C) phase diagrams. The current index is a conduit to the relevant literature. It lists lipid phase diagrams, their components and conditions of measurement, and complete bibliographic information. The main focus of the index is on lipids of membrane origin where water is the dispersing medium. However, it also includes records on acylglycerols, fatty acids, cationic lipids, and detergent-containing systems. The miscibility of synthetic and natural lipids with other lipids, with water, and with biomolecules (proteins, nucleic acids, carbohydrates, etc.) and non-biological materials (drugs, anesthetics, organic solvents, etc.) is within the purview of the index. There are 2188 phase diagram records in the index, the bulk (81%) of which refers to binary (two-component) T-C phase diagrams. The remainder is made up of more complex (ternary, quaternary) systems, pressure-T phase diagrams, and other more exotic miscibility studies. The index covers the period from 1965 through to July, 2001.     

Bijugate phyllotaxis in Rhizophoreae (Rhizophoraceae)

Phyllotaxis in the mangrove genera of Rhizophoraceae is shown to be bijugate, the angle between orthostichies always less than 90 and in some species close to one half the Fibonacci angle (68.8). The same leaf arrangement occurs in both orthotropic shoots with extended internodes and in the crowded terminal rosettes of those branches which develop plagiotropy by apposition growth, and can be interpreted as an adaptive architectural feature in relation to mutual leaf shading.     

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.     

Information theoretic interpretation of frequency domain connectivity measures

In order to provide adequate multivariate measures of information flow between neural structures, modified expressions of partial directed coherence (PDC) and directed transfer function (DTF), two popular multivariate connectivity measures employed in neuroscience, are introduced and their formal relationship to mutual information rates are proved.     

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.     

On the mystery of the golden angle in phyllotaxis

Phyllotaxis, the arrangement of leaves around a stem, shows in the vast majority of cases a regularity in the divergence angle of subsequent leaves which divide the whole circle into regular fractions. These are in most cases rational fractions of two Fibonacci numbers in an alternating series, converging towards the irrational limit of the golden section, corresponding to the golden divergence angle of 137.5 . . . degrees. This peculiarity was a long‐standing mystery. Here, it is related to the evolutionary pressure of optimal light capture for maximal photosynthetic activity. A model is established which relates minimal shadowing for the lower leaves to the divergence angle. Numerical results of this model agree well with semi‐empirical data on the dependence of light capture from the divergence angle. The basic shadow function of the model is also related with the demand of minimal shadowing for the angular separation of leaves and obtain, using elementary number theory, as solution the golden section. Further numerical studies show that the rational approach to the golden section (Schimper–Braun series) is related to the leaf width and the number of leaves of the plant.