A synergic approach to plant pattern generation

The paper shows convergences between the results found in various models of phyllotaxis. It shows that a synergic approach is needed to deal with the problems of phyllotaxis. An algorithm, called the phi-model, based on the observation of the meaningful and symmetry-generating presence of the golden ratio phi in all types of spiral patterns, and consequently in all types of regular patterns in phyllotaxis, is proposed. The model is suggested by a property of the allometry-type model for pattern recognition in phyllotaxis. It extends recent morphological models developed around the idea of packing efficiency of plant primordia, models that yield the noble numbers, among which are the divergence angles of spiral patterns. The phi-model also gives the noble numbers and moreover orders them in a way that establishes connections with the morphogenetic principles used in models for pattern generation; the order has to do with the relative frequencies of the spiral patterns in nature. The phi-model is a link between the two entropy models in phyllotaxis and offers a nice correspondence with the minimal entropy model generated by a systemic and holistic approach. This latter type of approach is put forward as being able to give a general framework in which to organize the concepts, results, and models in phyllotaxis in a way that produces a synergy of efforts. The necessity of doing so is seen clearly when one considers that phyllotaxis-like patterns are encountered in other fields of research, so that the problem appears to transcend the strict botanical substratum.     

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.     

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.     

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.     

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.     

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     

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.     

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.     

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     

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.     

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.     

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.     

Leaf phyllotaxis: Does it really affect light capture?

The intriguing mathematical properties of leaf phyllotaxis still attract scientific attention after centuries of research. Phyllotaxis, and in particular the divergence angle between successive leaves, have been frequently interpreted in terms of maximization of light capture, although certain model simulations of light capture by vertical shoots revealed minor effects of phyllotaxis in comparison with the effect of other morphological features of the plant. However, these simulations assumed a number of simplifications, did not take into account diffuse light, and were not based on real plants with their natural range of morphological variation. This study was aimed at filling these gaps by examining the influence on light harvesting of shoot architecture and divergence angle in four species with spiral phyllotaxis (Quercus ilex, Arbutus unedo, Heteromeles arbutifolia and Daphne gnidium) with a realistic 3-D model (Y-plant). A wide range of divergence angles (from 100° to 154°) was observed within each species, with 144° being the most frequent one. These different divergence angles rendered very different vertical projections of the shoot due to contrasting patterns of leaf overlap as seen from above, but they rendered indistinguishable light interception efficiencies (Ea). Setting the leaves with an opposite-decussate phyllotaxis led, however, to a 40–50% decrease of Ea. The interplay of internode length, leaf size and shape, and leaf elevation angle led to significant species differences in Ea. Thus, only particular phyllotaxis (e.g., decussate) might be functionally inefficient under certain combinations of the various morphological variables that influence light capture of a shoot. This revised version was published online in June 2006 with corrections to the Cover Date.     

An interpretation of Fujita's frequency diagrams in phyllotaxis

Fujita's diagrams in phyllotaxis, showing the frequencies of divergence angles as a function of these angles for low phyllotactic patterns such as (2, 1) and (3, 2), which are approximately normal curves centered at the limitdivergence angle of 137.51°, are shown to be puzzling when compared to results and observations in the field. An analysis of these diagrams is proposed, in the context of Fujita's methodology, of data from other sources, of a mathematical theorem on lattices, and of the contact pressure theory of phyllotaxis.     

Theory of Phyllotaxis. 2. 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 hexagonal-tetragonal type of phyllotaxis was theoretically predicted and found in nature. The mechanism underlying the formation of multiple forms of the helical phyllotaxis was considered.     

ROLE OF PARENCHYMA IN LINUM USITATISSIMUM LEAF TRACE PATTERNS

A new notation for leaf trace patterns was developed which is consistent with contemporary contact parastichy phyllotaxis notation. New computer-aided methods for generating accurate stem tissue maps were developed. Application of these methods resulted in clarification of the role that parenchyma differentiation plays in delimiting the procambial template for Linum usitatissimum L. stem vasculature through ontogeny. Study of the tissue maps for the various leaf trace patterns exhibited by Linum stems through ontogeny generated a set of observations which permits more rigorous definition of the developmental rules for vascular pattern formation. Long-known geometric principles of phyllotaxis were found applicable to leaf trace patterns.     

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.