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.     

Finding the center of a phyllotactic pattern

The calculation of divergence angles between primordia in a plant apex depends on the point used as the center of the apex. In mathematically ideal phyllotactic patterns, the center is well defined but there has not been a precise definition for the center of naturally occurring phyllotactic patterns. A few techniques have been proposed for estimating the location of the center but without a precise definition for the center the accuracy of these methods cannot be known. This paper provides a precise definition that can be used as the center of a phyllotactic pattern and a numerical method which can accurately find it. These tools will make it easier to compare theory against experiment in phyllotaxis.     

Interrelations between Phyllotaxis, Leaf Development and the Primary-secondary Vascular Transition in Populus deltoides

The procambial system of Populus deltoides Bartr. ex Marsh.plants progressed from phyllotaxy in the cotyledon stage throughthe phyllotactic orders 3/2;5/13. The nodal position at whicheach of these phyllotactic transitions occured was determinedby anatomical analyses; they were found to be remarkably consistentin a large population of young plants. The data were used todiagrammatically reconstruct the procambial system of a typical16 leaf plant. Because all plant parts grew continuously anduninterruptedly, it was not possible to verify the positionsof the phyllotactic transitions by morphological criteria. However,several measured parameters (the number and lengths of primordiawithin the terminal bud, the plastochron interval, and the numberof leaf traces with birefringent xylem elcments) attained constantvalues following establishment of the 5/13 phyllotaxy, suggestingthis to be the stable phyllotactic order for the species. Althoughbud size continued to increase in plants exhibiting 5/13 phyllotaxy,it could be accounted for by the increased number and size ofbasipetal subsidiary bundles in the procambial leaf traces.It was suggested that these phyllotactic transitions in theprocambial system are programmed in the plant to occur at ratherspecific stages of ontogeny. The process is mediated by theolder leaves and it is therefore modified by plant vigour. Locationof the primary-secondary vascular transition zone was also relatedto the order of phyllotaxy. It advanced acropetally in the stemin close association with leaf maturation, but this associationwas further influenced by plant vigour. Populus deltoides Bartr. ex Marsh., cottonwood, vascular anatomy, phyllotaxis, leaf growth, xylem     

Phyllotactic transitions in the vascular system of Populus deltoides bartr. as determined by 14C labeling

Populus deltoides seedlings progress through 2/5, 3/8, and 5/13 orders of phyllotaxis in attaining Plastochron Index 16 (PI 16). The manner in which the vascular system was reoriented during these phyllotactic transitions was determined by anatomical analysis of serial microsections, whereas the positions of the transitions were determined by ¹⁴C labeling. The midvein at the tip of leaves representing plants of different PI and leaves of different Leaf Plastochron Index (LPI) was fed ¹⁴CO₂ photosynthetically, and primordia LPI 0 through LPI-9 were dissected from the buds and analyzed for ¹⁴C. By combining the labeling data with the anatomical observations it was possible to reconstruct the vascular system of a plant of PI 16 and to locate the phyllotactic transitions. Both the anatomical and the labeling data showed a high degree of reproducibility among plants suggesting that the phyllotactic pattern to which the vascular system conforms may be programmed in the plant and transmitted acropetally through the developing leaves and procambial strands. The origin of new primordia and the concepts of orthostichy, ontogenetic helix, and Fibonacci sequence are discussed as they apply to the vascular system of P. deltoides.     

Rice DECUSSATE controls phyllotaxy by affecting the cytokinin signaling pathway

Phyllotaxy is defined as the spatial arrangement of leaves on the stem. The mechanism responsible for this extremely regular pattern is one of the most fascinating enigmas in plant biology. In this study, we identified a gene regulating the phyllotactic pattern in rice. Loss‐of‐function mutants of the DECUSSATE (DEC) gene displayed a phyllotactic conversion from normal distichous pattern to decussate. The dec mutants had an enlarged shoot apical meristem with enhanced cell division activity. In contrast to the shoot apical meristem, the size of the root apical meristem in the dec mutants was reduced, and cell division activity was suppressed. These phenotypes indicate that DEC has opposite functions in the shoot apical meristem and root apical meristem. Map‐based cloning revealed that DEC encodes a plant‐specific protein containing a glutamine‐rich region and a conserved motif. Although its molecular function is unclear, the conserved domain is shared with fungi and animals. Expression analysis showed that several type A response regulator genes that act in the cytokinin signaling pathway were down‐regulated in the dec mutant. In addition, dec seedlings showed a reduced responsiveness to exogenous cytokinin. Our results suggest that DEC controls the phyllotactic pattern by affecting cytokinin signaling in rice.     

A stochastic approach to phyllotactic patterns analysis

A statistical method is presented to characterize the degree of order in phyllotactic systems. We developed equations allowing the theoretical estimation of the number of leaves regularly distributed (spiral or verticillate) in a partially random phyllotactic system. The equations are simple and accurate enough to make quantitative predictions concerning the organization of different phyllotactic patterns (verticillate, distichous, spiral and random). This method can bring out patterns that are not visible a priori on a planar representation of the shoot apex. As a case study, the method was applied to the quantitative analysis of the sho mutants recently produced by Itoh et al. [2000. SHOOT ORGANIZATION genes regulate shoot apical meristem organization and the pattern of leaf primordium initiation in Rice. Plant Cell 12, 2161-2174]. By using our method, it was possible to predict the number of leaves distributed in distichous or random patterns on these phyllotactic mutants.     

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.     

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.     

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.     

Studies of aberrant phyllotaxy1 Mutants of Maize Indicate Complex Interactions between Auxin and Cytokinin Signaling in the Shoot Apical Meristem

One of the most fascinating aspects of plant morphology is the regular geometric arrangement of leaves and flowers, called phyllotaxy. The shoot apical meristem (SAM) determines these patterns, which vary depending on species and developmental stage. Auxin acts as an instructive signal in leaf initiation, and its transport has been implicated in phyllotaxy regulation in Arabidopsis (Arabidopsis thaliana). Altered phyllotactic patterns are observed in a maize (Zea mays) mutant, aberrant phyllotaxy1 (abph1, also known as abphyl1), and ABPH1 encodes a cytokinin-inducible type A response regulator, suggesting that cytokinin signals are also involved in the mechanism by which phyllotactic patterns are established. Therefore, we investigated the interaction between auxin and cytokinin signaling in phyllotaxy. Treatment of maize shoots with a polar auxin transport inhibitor, 1-naphthylphthalamic acid, strongly reduced ABPH1 expression, suggesting that auxin or its polar transport is required for ABPH1 expression. Immunolocalization of the PINFORMED1 (PIN1) polar auxin transporter revealed that PIN1 expression marks leaf primordia in maize, similarly to Arabidopsis. Interestingly, maize PIN1 expression at the incipient leaf primordium was greatly reduced in abph1 mutants. Consistently, auxin levels were reduced in abph1, and the maize PIN1 homolog was induced not only by auxin but also by cytokinin treatments. Our results indicate distinct roles for ABPH1 as a negative regulator of SAM size and a positive regulator of PIN1 expression. These studies highlight a complex interaction between auxin and cytokinin signaling in the specification of phyllotactic patterns and suggest an alternative model for the generation of altered phyllotactic patterns in abph1 mutants. We propose that reduced auxin levels and PIN1 expression in abph1 mutant SAMs delay leaf initiation, contributing to the enlarged SAM and altered phyllotaxy of these mutants.     

Phyllotaxis: cooperation and competition between mechanical and biochemical processes

Current theories and models of the formation of phyllotactic patterns at plant apical meristems center on either transport of the growth hormone auxin or the mechanical buckling of the plant tunica. By deriving a continuum approximation of an existing discrete biochemical model and comparing it with a mechanical model, we show that the model partial differential equations are similar in form. The implications of this universality in the form of the equations on interpreting the results of simulations are discussed. We develop a combined model that incorporates the coupling of biochemistry and mechanics. The combined model is accessible to analysis by reduction to a set of ordinary differential equations for the amplitudes of shapes associated with both the auxin concentration field and plant surface deformation. Analysis of these amplitude equations reveals the parameter choices under which the two mechanisms may cooperate in determining the pattern, or under which one or the other mechanism may dominate.     

Shoot apex and bulbil development in Dioscorea sansibarensis Pax

Plants of D. sansibarensis Pax bear opposite or alternate leaves, very rarely both on the same plant. The shoot apex shows distinct histological zonations irrespective of the phyllotactic variation. Three or more buds are produced in almost every axil, starting from the third node. The largest (axillary) bud is nearest the stem and always develops into a lateral branch. The 2–4 accessory buds develop conjunctly into a bulbil. Bulbil growth is traced and allometric relationships are shown to exist between the different parts of the bulbils. Histogenesis and organogenesis in bulbils are explained.     

Application of Two Mathematical Models to the Araceae, a Family of Plants with Enigmatic Phyllotaxis

In this paper we show that two mathematical models can be ofgreat help in the analysis of observational data, in this casethe difficult and little studied phyllotactic phenomena thatoccur in the Araceae family. We apply the Fundamental Theoremof Phyllotaxis, together with an explanatory model of phyllotaxis,to plant specimens of this family, to obtain phyllotactic parametersand information that cannot be otherwise obtained. Most significantis the fact that the two models show evidence of regularitiesin the overwhelming diversity of the patterns observed in theAraceae (essentiallyDracontium and Anthurium) characterizedby discontinuous transitions. In particular, this work revealsthe regularity of the behaviour of the divergence angle in thespecimens analysed. Features of the inflorescences ofDracontium, especially the presence of whorls, are compared to those observedin inflorescences ofAnthurium (characterized by the absenceof whorls), and in the capitulae of Compositae (characterizedby continuous transition). We question the possible meaningat the genetic level of the diversity of patterns observed atthe macroscopic level. Copyright 2001 Annals of Botany Company Phyllotaxis, Araceae, mathematical models, inflorescence, development     

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.     

Leaf asymmetry as a developmental constraint imposed by auxin-dependent phyllotactic patterning

In a majority of species, leaf development is thought to proceed in a bilaterally symmetric fashion without systematic asymmetries. This is despite the left and right sides of an initiating primordium occupying niches that differ in their distance from sinks and sources of auxin. Here, we revisit an existing model of auxin transport sufficient to recreate spiral phyllotactic patterns and find previously overlooked asymmetries between auxin distribution and the centers of leaf primordia. We show that it is the direction of the phyllotactic spiral that determines the side of the leaf these asymmetries fall on. We empirically confirm the presence of an asymmetric auxin response using a DR5 reporter and observe morphological asymmetries in young leaf primordia. Notably, these morphological asymmetries persist in mature leaves, and we observe left-right asymmetries in the superficially bilaterally symmetric leaves of tomato (Solanum lycopersicum) and Arabidopsis thaliana that are consistent with modeled predictions. We further demonstrate that auxin application to a single side of a leaf primordium is sufficient to recapitulate the asymmetries we observe. Our results provide a framework to study a previously overlooked developmental axis and provide insights into the developmental constraints imposed upon leaf morphology by auxin-dependent phyllotactic patterning.     

Patterns of arrangement of lateral appendages on axes of Stigmaria ficoides (Sternberg) Brongniart

The arrangement of lateral insertions has been examined on axes of Stigmaria ficoides . The arrangement can be considered as a phyllotactic pattern made up of parastichies and orthostichies. Orthodox phyllotactic patterns, based on parastichy numbers from the Fibonacci or common accessory series, do not occur. Instead spiral (or multijugate) arrangements occur based on pairs of parastichy numbers such as x and x + 1, x and x + 2, x and x + 3, etc. and x ranges from 12 to 19. Whorled arrangements occur relatively infrequently. Individual axes commonly show frequent changes in pattern. The observations are used to make deductions about the growth and homology of stigmarian axes.     

The Hierarchical Control of Phyllotaxis

Bolle's phyllotactic theory is enhanced here in the light ofrecent works which underline the importance of vascular organizationin the determination of phyllotactic patterns. It is emphasizedthat non-vascular plants, such as Fucus spiralis, can revealhow normal phyllotaxis originated. These two approaches to phyllotaxis,with others put forward here, present the problem of phyllotaxisas a matter of hierarchical control, which produces the integratedand simple behaviour of the primordia of growing plants. phyllotaxis, hierarchy, brown algae, evolution, vascularity, systems theory, control     

THE ABPHYL SYNDROME IN ZEA MAYS. I. ARRANGEMENT. NUMBER AND SIZE OF LEAVES

A range of phyllotactic variation in Zea mays has been obtained from progeny derived from a single ‘opposite-leaved’ plant. Some segregants exhibit a form of alternate phyllotaxy with poor separation between nodes, while others duplicate the original decussate condition. Numerous intermediate examples have also been observed. Some have both spiral and opposite arrangements on the same plant; others, which begin their ontogeny with the normal distichous arrangement, switch at different stages of maturity to spiral or decussate arrangement. Leaves from ABPHYL plants are up to one-half the width of comparable normal leaves although their length is similar. Since ABPHYL plants may have twice as many leaves as do normal siblings, total area and dry weight of leaf blades per ABPHYL plant are greater than in normal siblings. Leaf width of both ABPHYL and normal plants correlates well with the number rather than the width of epidermal cells.     

THE ROLE OF PHYLLOTACTIC PATTERN AS A “DEVELOPMENTAL CONSTRAINT” ON THE INTERCEPTION OF LIGHT BY LEAF SURFACES

Computer simulations of model plants are used to assess the influence of leaf shape, size, and pattern of arrangement (= phyllotaxy) on the direct solar radiation intercepted by leaf surfaces. Changes in phyllotaxy significantly influence light interception (and, by inference, net assimilation rate) for rosette growth habits. However, changes in leaf shape and orientation and in stem length can compensate for the negative effects of leaf overlap produced by phyllotactic patterns. Phyllotaxy is viewed as a developmental limiting factor in photobiology that may necessitate compensatory changes in other morphological features not directly controlled by patterns of leaf initiation. This distinguishes it from functioning as a “developmental constraint” sensu stricto and may provide a paradigm for other features in plant evolution.     

On the relationship between phyllotaxy and vasculature: a synthesis

There is a definite relationship between the phyllotactic fraction and the sympodia uniting median leaf traces in a stem. The denominator of the phyllotactic fraction is the number of sympodia in the stem, and the numerator is the number of sympodia counted in passing from the sympodium of one leaf to that of an adjacent leaf on the genetic helix. This relationship holds for species with closed as well as open vascular systems. Of the 100 shoots (93 species) whose vasculature has been reviewed from the literature only one shows no apparent relationship between the phyllotactic fraction and the vasculature. Shoots for 87 species show the relationship described above while shoots of five species have both irregular phyllotaxis and vasculature. The mathematical constraints on this relationship are shown to depend on the divergence angle and the fact that sympodia do not cross one another. That there are biological controls on this relationship in addition to the purely mathematical ones is shown by the fact that sympodial connections are almost universally made along orthostichies. These controls most likely operate on factors that influence the formation of orthostichies such as the relationship between leaf-shape and size of the apical dome.