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.     

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.     

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.     

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.     

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.     

The origin of fibonacci phyllotaxis—an analysis of Adler's contact pressure model and Mitchison's expanding apex model

Adler's contact pressure model for Fibonacci phyllotaxis is examined theoretically. It is shown that the model, as it stands, does not account for Fibonacci phyllotaxis, since it requires, but does not provide, a mechanism for initiating new primordia with increasingly greater precision as phyllotaxis rises. Modifications are suggested which remedy this deficiency in the model; one of these modifications involves a combination of Adler's model with Mitchison's model.From a comparison of the ranges of divergence angles permitted by Adler's model against Fujita's measurements of divergence angles in plants with low phyllotaxis, it is shown that the modified contact pressure model, if based on the concept of mechanical pressures between primordia in contact, cannot account for the divergence angles found in low phyllotaxis systems. However it is shown that this deficiency can be overcome if the contact pressure effect is regarded as a chemical phenomenon, mediated by a growth inhibitor produced by the prirnordia and moving more readily in vertical directions than in other directions.Mitchison's model, which is based on the concepts of an expanding apex and primordium initiation by existing primordia, is shown to account for Fibonacci phyllotaxis only if phyllotaxis rises sufficiently slowly; to guarantee that an F_n + F_n+1 system can develop there must already be at least F_n+1 primordia present in an F_n?1 + F_n system, at least F_n primordia in an F_n?2 + F_n?1 system, and so on down to at least three primordia in a 1 + 2 system, making a total of at least F_n+3?5 primordia (where F_n = nth term of the Fibonacci series with F₁ = F₂ = 1). Adler's model, modified, requires only that F_{n + 1} primordia be present with divergence angles in the range 120–180° to guarantee that an F_n + F_{n + 1} system can develop.     

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.     

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 SHOOT APICAL ONTOGENY OF THE PICEA ABIES SEEDLING. I. ANATOMY,APICAL DOME DIAMETER,AND PLASTOCHRON DURATION

Developing embryos in immature Picea abies seeds already have well-delineated shoot apical meristems with clearly evident cytohistological zonation. During early seedling development the zonation characteristic of gymnospermous apical meristems is attained. Seedling development is also accompanied by an approximately threefold increase in apical dome diameter. The latter approaches a steady state about 140 days after germination. Seedlings display a spiral phyllotaxis consisting of a contact parastichy system, usually of the primary Fibonacci series. As the seedlings age and apical domes enlarge, higher Fibonacci number-pairs characterize their phyllotaxis. Mathematical analysis of the relation between cumulative leaf number and age revealed that the length of the plastochronic time interval declines from about 18.5 hr to 5.7 hr as seedling age increases from 20 to 140 days.     

Rotation angle of stem cell division plane controls spiral phyllotaxis in mosses

Journal of Plant Research - The spiral arrangement (phyllotaxis) of leaves is a shared morphology in land plants, and exhibits diversity constrained to the Fibonacci sequence. Phyllotaxis in...     

The CUP-SHAPED COTYLEDON2 and 3 genes have a post-meristematic effect on Arabidopsis thaliana phyllotaxis

Background and Aims The arrangement of flowers in inflorescence shoots of Arabidopsis thaliana represents a regular spiral Fibonacci phyllotaxis. However, in the cuc2 cuc3 double mutant, flower pedicels are fused to the inflorescence stem, and phyllotaxis is aberrant in the mature shoot regions. This study examined the causes of this altered development, and in particular whether the mutant phenotype is a consequence of defects at the shoot apex, or whether post-meristematic events are involved.Methods The distribution of flower pedicels and vascular traces was examined in cross-sections of mature shoots; sequential replicas were used to investigate the phyllotaxis and geometry of shoot apices, and growth of the young stem surface. The expression pattern of CUC3 was analysed by examining its promoter activity.Key Results Phyllotaxis irregularity in the cuc2 cuc3 double mutant arises during the post-meristematic phase of shoot development. In particular, growth and cell divisions in nodes of the elongating stem are not restricted in the mutant, resulting in pedicel–stem fusion. On the other hand, phyllotaxis in the mutant shoot apex is nearly as regular as that of the wild type. Vascular phyllotaxis, generated almost simultaneously with the phyllotaxis at the apex, is also much more regular than pedicel phyllotaxis. The most apparent phenotype of the mutant apices is a higher number of contact parastichies. This phenotype is associated with increased meristem size, decreased angular width of primordia and a shorter plastochron. In addition, the appearance of a sharp and deep crease, a characteristic shape of the adaxial primordium boundary, is slightly delayed and reduced in the mutant shoot apices.Conclusions The cuc2 cuc3 double mutant displays irregular phyllotaxis in the mature shoot but not in the shoot apex, thus showing a post-meristematic effect of the mutations on phyllotaxis. The main cause of this effect is the formation of pedicel–stem fusions, leading to an alteration of the axial positioning of flowers. Phyllotaxis based on the position of vascular flower traces suggests an additional mechanism of post-meristematic phyllotaxis alteration. Higher density of flower primordia may be involved in the post-meristematic effect on phyllotaxis, whereas delayed crease formation may be involved in the fusion phenotype. Promoter activity of CUC3 is consistent with its post-meristematic role in phyllotaxis.     

Orthostichy, Parastichy and Plastochrone Ratio in a Central Theory of Phyllotaxis

A view of phyllotaxis theory is proposed which combines theessentials of the orthostichy, parastichy and plastochrone ratioideas. The advocated thesis takes cognisance of the biologicallyexact tangential divergence angles between nodes or primordiaand this is in contrast with other major theories which haveassumed the attainment of mathematically ‘ideal’divergence angles. Rectiserial orthostichy lines are demonstratedto be present in both 'spiral' and ‘non-spiral’systems. Basically similar mathematical laws govern the definitionof such systems whether the phyllotaxis designation be Fibonacci,anomalous or multijugate. The inter-relationships of the variousangular and other measures associated with plastochrone ratio,orthostichy lines and parastichy curves are dealt with in detail.     

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.     

Phyllotaxis. I. A Mechanistic Model

A quantitative two-dimensional model for phyllotaxis is described.The model is based on the production, diffusion and degradationof a morphogen, and it is assumed that primordial initiationcan only take place in competent tissue when the morphogen concentrationdrops to a certain critical level. The model predicts the anglesbetween successive primordia; under appropriate conditions asteady state is achieved where the divergence angle is constant;and for limiting values of the parameters, the predicted steady-statedivergence angle approaches the Fibonacci angle. To accountfor non-spiral phyllotaxis, an additional hypothesis, calledspatial competence, is introduced. As well as a morphogen levelbelow the threshold level, and for competence in the usual sense,this assumes that particular spatial demands must be satisfiedbefore primordial initiation can proceed.     

Phyllotaxis, phenology and architecture in Cephalotaxus, Torreya and c(Coniferales)

In Amentotaxus, Cephalotaxus and Torreya there is a regular seasonal alternation of foliage leaves and bud-scales, with foliage leaves largely preformed, i.e. initiated in the season before they expand. On most plagiotropic shoots phyllotaxis in the production of foliage leaves may be either bijugate ( Cephulotaxus, Torreya ) or decussate ( Amentotaxus ). In bijugate phyllotaxis successive leaf pairs originate at an angle of about 68° to each other, i.e. approximately one-half of the 'ideal' or Fibonacci angle of 137.5°. Secondary leaf orientation in Cephulotaxus and Torreya , by twisting of the leaf base, produces the dorsiventrality of plagiotropic shoots, whereas in Amentotaxus secondary orientation involves a twisting of the stemc as well as the leaf base. In Cephalotaxus cc condition is constant in the production of the numerous but imprecise number of bud-scales and in the production of foliage leaves. However, in Torreya the phyllotaxis changes from bijugate in the production of foliage leaves to decussate in the production of bud-scales, which are constant in number (about eight pairs). This allows a precise analysis of the biphasic production of leaf primordia in the seasonal cycle. The phyllotactic change in Torreya may not be the result of reported changes in shoot apex dimensions since Cephalotaxus , with its constant phyllotaxis, has a comparable seasonal change in apex dimensions. Information on architecture, chirality and cone morphology is also included.     

Disruption of a DNA topoisomerase I gene affects morphogenesis in Arabidopsis

The genesis of phyllotaxis, which often is associated with the Fibonacci series of numbers, is an old unsolved puzzle in plant morphogenesis. Here, we show that disruption of an Arabidopsis topoisomerase (topo) I gene named TOP1alpha affects phyllotaxis and plant architecture. The divergence angles and internode lengths between two successive flowers were more random in the top1alpha mutant than in the wild type. The top1alpha plants sporadically produced multiple flowers from one node, and the number of floral organ primordia often was different. The mutation also caused the twisting of inflorescences and individual flowers and the serration of leaf margins. These morphological abnormalities indicate that TOP1alpha may play a critical role in the maintenance of a regular pattern of organ initiation. The top1alpha mutant transformed with the RNA interference construct for TOP1beta, another topo I gene arrayed tandemly with TOP1alpha, was found to be lethal at young seedling stages, suggesting that topo I activity is essential in plants.     

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.     

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.