Phyllotactic Pattern Generation: A Conceptual Model

A conceptual model is proposed here that shows how all typesof whorled and peculiar patterns in phyllotaxis derive straightforwardlyfrom normal and anomalous spiral patterns. This is a completemodel of phyllotaxis, integrating the author's interpretativemodel for generating spiral patterns. The paper underlines thata better understanding of the variety of phyllotactic patterns,and of the transitions between them, involves a phylogeneticperspective. It stresses the working hypothesis that spiralpatterns are primitive and that all other patterns, such aswhorled systems, are by-products of evolution from spirality.An important epistemological consequence on mathematical modellingis drawn out of this hypothesis, namely that models of knowledgeor interpretative models, able to take care of the spiral patterns,must be formulated and then followed by simulation, mechanisticor conceptual models that are able to reproduce the transitionsto all other types of patterns. Phyllotaxis, parastichy pair, shoot apex, multijugy, spirality, whorl, modelling, phylogeny, telome theory, Hofmeister's rule     

The diversity and ontogenetic changes of phyllotaxis in wild-types and two leaf form mutants of Antirrhinum majus L.

We have analysed the phyllotactic patterns of the main shoot in vegetative and generative phases of growth in wild type and mutant plants of Antirrhinum majus L. Wild types 'Sippe50' and 'W l08' were compared to mutants grminifolia and phanlastica . The normal vegetative phyllotaxis of the wild type plants is decussate, but the inflorescence phyllotaxis is spiral and of the Fibonacci type. The phyllotaxis patterns of the mutants differ strongly from that of the wild type. Besides decussate phyllotaxis, whorls of three or four elements as well as spiral patterns in vegetative phase were observed. The vegetative phyllotaxis in mutants is ontogenetically unstable with frequent transitions between patterns, including the reversion of chirality of spiral phyllotaxis. The number of transitions per plant was larger in graminifolia than in phantastica . The inflorescence phyllotaxis was more stable and occasional non-typical phyllotaxis patterns finally transformed to a Fibonacci pattern. The results suggest a possible role of genetic factors in determining the regularity of spatial arrangement of organs.     

Geometrical relationships specifying the phyllotactic pattern of aquatic plants

The complete range of various phyllotaxes exemplified in aquatic plants provide an opportunity to characterize the fundamental geometrical relationships operating in leaf patterning. A new polar-coordinate model was used to characterize the correlation between the shapes of shoot meristems and the arrangements of young leaf primordia arising on those meristems. In aquatic plants, the primary geometrical relationship specifying spiral vs. whorled phyllotaxis is primordial position: primordia arising on the apical dome (as defined by displacement angles θ ≤ 90° during maximal phase) are often positioned in spiral patterns, whereas primordia arising on the subtending axis (as defined by displacement angles of θ ≥ 90° during maximal phase) are arranged in whorled patterns. A secondary geometrical relationship derived from the literature shows an inverse correlation between the primordial size?:?available space ratio and the magnitude of the Fibonacci numbers in spiral phyllotaxis or the number of leaves per whorl in whorled phyllotaxis. The data available for terrestrial plants suggest that their phyllotactic patterning may also be specified by these same geometrical relationships. Major exceptions to these correlations are attributable to persistent embryonic patterning, leaflike structures arising from stipules, congenital splitting of young primordia, and/or non-uniform elongating of internodes. The geometrical analysis described in this paper provides the morphological context for interpreting the phenotypes of phyllotaxis mutants and for constructing realistic models of the underlying mechanisms responsible for generating phyllotactic patterns.     

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.     

灌木铁线莲（毛茛科）花器官的发生与发育

用扫描电子显微镜（SEM）对铁线莲属（Clematis L.）植物灌木铁线莲(C. fruticosa Turcz.)花的形态发生和发育过程进行了观察。灌木铁线莲花原基形成后,4枚萼片以交互对生的方式首先发生,呈轮状排列。最早的4枚雄蕊原基在4枚萼片交接的位置上近螺旋状发生,此后,随着雄蕊原基的向心发生和数目不断增多,其发生的螺旋状序列逐渐明显。雄蕊原基发生后,在花原基顶端,心皮原基沿着雄蕊原基的发生序列呈螺旋状发生。本文结果支持在原始被子植物花中螺旋状排列和轮状排列同时存在的观点。此外,本文也进一步证实了花萼与苞片的同源性。     

Formation of Pseudowhorls inPeperomia verticillata(L.) A. Dietr. Shoots Exhibiting Various Phyllotactic Patterns

Pseudowhorls are composed of leaves attached at almost equallevels and separated by single fully elongated internodes. InPeperomiaverticillata, pseudowhorls form regularly in shoots exhibitingboth spiral and truly whorled patterns of phyllotaxis. In spiralsystems, they are composed of successive leaves positioned onthe ontogenetic helix. In whorled phyllotaxis, leaves of twoadjacent whorls occur at almost the same level and this wayform a pseudowhorl. The number of leaves per pseudowhorl dependson the type of phyllotactic pattern and also the system of primordiapacking. In all the shoots, regardless of the type of phyllotaxis,the number of leaves per pseudowhorl equals the number of leafprimordia in physical contact with the apical dome. It is thesame as the higher number in contact parastichy pairs in spiralpatterns or the number of orthostichies in whorled phyllotaxis.The pseudowhorled pattern is already manifested in the arrangementof leaf primordia. In spiral and whorled phyllotaxis the plastochronratio calculated for primordia or whorls belonging to adjacentpseudowhorls is always higher than that calculated for membersof one pseudowhorl. Moreover, angular distances between primordiaof one pseudowhorl in spiral patterns are more uniform thanexpected in Fibonacci phyllotaxis. These observations were madeon plants both growing in pots and culturedin vitro. 6-Benzylaminopurine,a synthetic cytokinin, added to the medium increases the meannumber of leaves per pseudowhorl. It seems that this effectis indirect: phyllotaxis changes first rather than the destinyof a particular internode in a process of selective elongation.Copyright1999 Annals of Botany Company Peperomia verticillata, pseudowhorls, phyllotaxis, shoot apex.     

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.     

Phyllotaxis--a new chapter in an old tale about beauty and magic numbers

Phyllotaxis, the regular arrangement of leaves and flowers around the stem, is one of the most fascinating patterning phenomena in biology. Numerous theoretical models, that are based on biochemical, biophysical and other principles, have been proposed to explain the development of the patterns. Recently, auxin has been identified as the inducer of organ formation. An emerging model for phyllotaxis states that polar auxin transport in the plant apex generates local peaks in auxin concentration that determine the site of organ formation and thereby the different phyllotactic patterns found in nature. The PIN proteins play a primary role in auxin transport. These proteins are localized in a polar fashion, reflecting the directionality of polar auxin transport. Recent evidence shows that most aspects of phyllotaxis can be explained by the expression pattern and the dynamic subcellular localization of PIN1.     

Floral phyllotaxis in basal angiosperms: development and evolution

To date, molecular developmental studies have focused on vegetative rather than floral phyllotaxis because vegetative shoot apices are technically more tractable than floral apices in model plants. In contrast to evolutionary changes in the phyllotaxis of vegetative shoots, however, changes in floral phyllotaxis appear to have played a major role in angiosperm evolution. Consolidation of a whorled floral phyllotaxis in derived groups allowed synorganization of floral organs and further adaptive radiations. In basal angiosperms, floral phyllotaxis is more flexible. To study these phenomena, we need clarification of the complex relations of both spiral and whorled phyllotaxis with divergence angles, plastochrons, spiral versus simultaneous initiation of organs, parastichies, orthostichies, organ series, and whorls. Improved resolution of phylogenetic relationships and increased knowledge of the diversity of floral phyllotaxis will allow us to trace evolutionary changes in floral phyllotaxis in ever more detail. Already, such surveys have confirmed that floral phyllotaxis was unusually labile early in angiosperm evolution. Whether the original floral phyllotaxis in angiosperms was spiral or whorled is equivocal, but it appears that spiral floral phyllotaxis in Magnoliales and Laurales is derived rather than primitive.     

Do Fibonacci numbers reveal the involvement of geometrical imperatives or biological interactions in phyllotaxis?

Complex biological patterns are often governed by simple mathematical rules. A favourite botanical example is the apparent relationship between phyllotaxis (i.e. the arrangements of leaf homologues such as foliage leaves and floral organs on shoot axes) and the intriguing Fibonacci number sequence (1, 2, 3, 5, 8, 13 . . .). It is frequently alleged that leaf primordia adopt Fibonacci-related patterns in response to a universal geometrical imperative for optimal packing that is supposedly inherent in most animate and inanimate structures. This paper reviews the fundamental properties of number sequences, and discusses the under-appreciated limitations of the Fibonacci sequence for describing phyllotactic patterns. The evidence presented here shows that phyllotactic whorls of leaf homologues are not positioned in Fibonacci patterns. Insofar as developmental transitions in spiral phyllotaxis follow discernible Fibonacci formulae, phyllotactic spirals are therefore interpreted as being arranged in genuine Fibonacci patterns. Nonetheless, a simple modelling exercise argues that the most common spiral phyllotaxes do not exhibit optimal packing. Instead, the consensus starting to emerge from different subdisciplines in the phyllotaxis literature supports the alternative perspective that phyllotactic patterns arise from local inhibitory interactions among the existing primordia already positioned at the shoot apex, as opposed to the imposition of a global imperative of optimal packing.  © 2006 The Linnean Society of London, Botanical Journal of the Linnean Society , 2006, 150 , 3–24.     

Floral development and floral phyllotaxis in Anaxagorea (Annonaceae)

Background and Aims Anaxagorea is the phylogenetically basalmost genus in the large tropical Annonaceae (custard apple family) of Magnoliales, but its floral structure is unknown in many respects. The aim of this study is to analyse evolutionarily interesting floral features in comparison with other genera of the Annonaceae and the sister family Eupomatiaceae. Methods Live flowers of Anaxagorea crassipetala were examined in the field with vital staining, liquid-fixed material was studied with scanning electron microscopy, and microtome section series were studied with light microscopy. In addition, herbarium material of two other Anaxagorea species was cursorily studied with the dissecting microscope. Key Results Floral phyllotaxis in Anaxagorea is regularly whorled (with complex whorls) as in all other Annonaceae with a low or medium number of floral organs studied so far (in those with numerous stamens and carpels, phyllotaxis becoming irregular in the androecium and gynoecium). The carpels are completely plicate as in almost all other Annonaceae. In these features Anaxagorea differs sharply from the sister family Eupomatiaceae, which has spiral floral phyllotaxis and ascidiate carpels. Flat stamens and the presence of inner staminodes differ from most other Annonaceae and may be plesiomorphic in Anaxagorea. However, the inner staminodes appear to be non-secretory in most Anaxagorea species, which differs from inner staminodes in other families of Magnoliales (Eupomatiaceae, Degeneriacae, Himantandraceae), which are secretory. Conclusions Floral phyllotaxis in Anaxagorea shows that there is no signature of a basal spiral pattern in Annonaceae and that complex whorls are an apomorphy not just for a part of the family but for the family in its entirety, and irregular phyllotaxis is derived. This and the presence of completely plicate carpels in Anaxagorea makes the family homogeneous and distinguishes it from the closest relatives in Magnoliales.     

Organization of the vascular system in the stems of Diplazium and Blechnum (Filicales)

Detailed analysis of the three-dimensional vascular organization in species of Diplazium and Blechnum indicates that these ferns possess reticulate (dictyostelic) vascular systems that closely reflect the helical phyllotaxis of the shoot. In each species, the vascular pattern shows a specific relationship to the phyllotaxis, so that the phyllotactic fraction can be determined by examination of the number of cauline vascular bundles (meristeles) in cross section of the stem. The number of meristeles in a cross section equals the denominator of the phyllotactic fraction, i.e., the number of foliar orthostichies on the stem. The same numerical relationship also exists in the eusteles of seed plants between the number of axial (sympodial) stem bundles and the phyllotaxis. There is a further parallel between the three-dimensional reticulate pattern of fern dictyosteles and the reticulate patterns that characterize some herbaceous dicotyledons. However, the hypothesized separate origins of seed plant eusteles and fern dictyosteles from protostelic precursors preclude any direct homologies between these similar patterns. The parallel evolution of presumably more physiologically efficient reticulate systems in herbaceous seed plants and in ferns that have only a primary plant body is noteworthy. The similar relationships between the primary stem vascular patterns and phyllotaxy in both ferns and seed plants further emphasize the likely similarity of the morphogenetic events that occur at the shoot apex in these taxonomically disparate groups.     

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.     

Sequences of symmetry-breaking in phyllotactic transitions

This paper studies the transition of phyllotactic patterns by a group-theoretic approach. Typical phyllotactic patterns are represented here as dotted patterns on a cylinder, where the cylinder is regarded as the stem of a plant and the dots are points where leaves branch from the stem. We can then classify the symmetries of the alternate and opposite phyllotaxis into four types of groups, and clarify sequences of symmetry-breaking among these groups. The sequences turn out to correspond to transition paths of phyllotactic patterns found in the wild. This result shows the usefulness of classification of phyllotactic patterns based on their group symmetries. Moreover, the breaking of reflection symmetry is found to be an important rule for real phyllotactic transitions.     

Modeling of Branching Patterns in Plants

A major determinant of plant architecture is the arrangement of branches around the stem, known as phyllotaxis. However, the specific form of branching conditions is not known. Here we discuss this question and suggest a branching model which seems to be in agreement with biological observations. Recently, a number of models connected with the genetic network or molecular biology regulation of the processes of pattern formation appeared. Most of these models consider the plant hormone, auxin, transport and distribution in the apical meristem as the main factors for pattern formation and phyllotaxis. However, all these models do not take into consideration the whole plant morphogenesis, concentrating on the events in the shoot or root apex. On the other hand, other approaches for modeling phyllotaxis, where the whole plant is considered, usually are mostly phenomenological, and due to it, do not describe the details of plant growth and branching mechanism. In this work, we develop a mathematical model and study pattern formation of the whole, though simplified, plant organism where the main physiological factors of plant growth and development are taken into consideration. We model a growing plant as a system of intervals, which we will consider as branches. We assume that the number and location of the branches are not given a priori, but appear and grow according to certain rules, elucidated by the application of mathematical modeling. Four variables are included in our model: concentrations of the plant hormones auxin and cytokinin, proliferation and growth factor, and nutrients—we observe a wide variety of plant forms and study more specifically the involvement of each variable in the branching process. Analysis of the numerical simulations shows that the process of pattern formation in plants depends on the interaction of all these variables. While concentrations of auxin and cytokinin determine the appearance of a new bud, its growth is determined by the concentrations of nutrients and proliferation factors. Possible mechanisms of apical domination in the frame of our model are discussed.     

Simulation of organ patterning on the floral meristem using a polar auxin transport model

An intriguing phenomenon in plant development is the timing and positioning of lateral organ initiation, which is a fundamental aspect of plant architecture. Although important progress has been made in elucidating the role of auxin transport in the vegetative shoot to explain the phyllotaxis of leaf formation in a spiral fashion, a model study of the role of auxin transport in whorled organ patterning in the expanding floral meristem is not available yet. We present an initial simulation approach to study the mechanisms that are expected to play an important role. Starting point is a confocal imaging study of Arabidopsis floral meristems at consecutive time points during flower development. These images reveal auxin accumulation patterns at the positions of the organs, which strongly suggests that the role of auxin in the floral meristem is similar to the role it plays in the shoot apical meristem. This is the basis for a simulation study of auxin transport through a growing floral meristem, which may answer the question whether auxin transport can in itself be responsible for the typical whorled floral pattern. We combined a cellular growth model for the meristem with a polar auxin transport model. The model predicts that sepals are initiated by auxin maxima arising early during meristem outgrowth. These form a pre-pattern relative to which a series of smaller auxin maxima are positioned, which partially overlap with the anlagen of petals, stamens, and carpels. We adjusted the model parameters corresponding to properties of floral mutants and found that the model predictions agree with the observed mutant patterns. The predicted timing of the primordia outgrowth and the timing and positioning of the sepal primordia show remarkable similarities with a developing flower in nature.     

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.     

The common patterns of nature

We typically observe large‐scale outcomes that arise from the interactions of many hidden, small‐scale processes. Examples include age of disease onset, rates of amino acid substitutions and composition of ecological communities. The macroscopic patterns in each problem often vary around a characteristic shape that can be generated by neutral processes. A neutral generative model assumes that each microscopic process follows unbiased or random stochastic fluctuations: random connections of network nodes; amino acid substitutions with no effect on fitness; species that arise or disappear from communities randomly. These neutral generative models often match common patterns of nature. In this paper, I present the theoretical background by which we can understand why these neutral generative models are so successful. I show where the classic patterns come from, such as the Poisson pattern, the normal or Gaussian pattern and many others. Each classic pattern was often discovered by a simple neutral generative model. The neutral patterns share a special characteristic: they describe the patterns of nature that follow from simple constraints on information. For example, any aggregation of processes that preserves information only about the mean and variance attracts to the Gaussian pattern; any aggregation that preserves information only about the mean attracts to the exponential pattern; any aggregation that preserves information only about the geometric mean attracts to the power law pattern. I present a simple and consistent informational framework of the common patterns of nature based on the method of maximum entropy. This framework shows that each neutral generative model is a special case that helps to discover a particular set of informational constraints; those informational constraints define a much wider domain of non‐neutral generative processes that attract to the same neutral pattern.