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     

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.     

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.     

PHYLLOTAXIS AND VASCULAR ORGANIZATION OF THE CARPELS IN MICHELIA FUSCATA

Tucker , Shirley C. (U. Minnesota, Minneapolis.) Phyllotaxis and vascular organization of the carpels in Michelia fuscata. Amer. Jour. Bot. 48(1): 60–71. Illus. 1961.—Phyllotaxis pattern and vascular organization are closely related in the floral receptacle of Michelia fuscata (Magnoliaceae). The carpels arise in a spiral or helix. They are initiated alternately along each of 7, 8 or 10 helical parastichies according to a complex repetitive sequence. The pattern of the dorsal carpellary trace fusions is orderly for each of the 10 flowers investigated. The dorsal carpellary traces in each parastichy diverge from the same vascular sympodium. Among flowers one finds differing numbers of parastichies, differing angles of divergence, and varying sequences of parastichies which reflect the order of carpel initiation. The angle of divergence, although consistent for any 1 parastichy in a flower, can vary greatly between parastichies. The nature and importance of the organizers which determine appendage position at the apical meristem are considered. Changes in apical size, configuration, and activity are shown to be related to phyllotaxis.     

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.     

Phyllotaxis in the Oil Palm: Arrangement of Male/Female Florets along the Spikelets

The paper continues an earlier study of the geometry of inflorescencestructures in the oil palm in which geometry is measured interms of Equivalent Phyllotaxis Index (E.P.I.). In this casethe phyllotaxis of male and female florets along their respectivespikelets is considered. Regardless of the spikelet positionon the inflorescence or the palm age the very small male floretshave a higher E.P.I. than the large female structures and acompletely different apparent parastichy arrangement. TheseE.P.I. estimates seem to be independent of the age of palmsfrom which inflorescences and hence spikelets are sampled. However,there is considerable variation in phyllotaxis within bunches,E.P.I. being lower on spikelets sampled toward the base of theinflorescence and increasing in a more or less linear mannerin spikelets sampled at the tip; this pattern is not so definiteon male spikelets. The results are discussed in relation toother more simple measurements of spikelet architecture.     

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.     

PHOTOPERIOD INDUCED CHANGE IN PHYLLOTAXIS IN XANTHIUM

Photoperiodic floral induction in Xanthium, achieved by subjecting the plants to two long nights, is accompanied by a transient change of the phyllotaxis from the (2, 3) contact parastichy pattern of vegetative plants, to a (3, 5) pattern during the transition. To specify the phyllotaxis, two parameters were estimated from transverse sections of apical buds of control and treated plants: the divergence angle, α, and the plastochron ratio, a. The plastochron ratio decreased progressively during transition from the vegetative to the reproductive state of growth, from a = 1.48 initially to a = 1.15 six days after the beginning of induction. The divergence angle was not altered during the transition. This change in phyllotaxis is interpreted as a change in the relative positioning of leaf primordia on the transitional apex. This transient change appears to be identical with the previously described long-term change of the phyllotaxis of Xanthium brought about by treatment of plants with gibberellic acid.     

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.     

CAPITULUM PHYLLOTAXIS AND NUMERICAL CANALIZATION IN MICROSERIS PYGMAEA (ASTERACEAE: LACTUCEAE)

The positions at which floret primordia arise in developing capitulum buds of Microseris pygmaea D. Don have been mapped by computer-assisted light microscopy. The primordia can be assigned positions along a basic phyllotactic spiral with a divergence angle of about 137.5°. In addition, there are regular deviations from a spiral arrangement. Typically, the first 26 primordia in phyllotactic sequence are arranged in two concentric circles of 13 primordia with considerable deviations in the divergence angle and in the distances between primordia along a parastichy at positions 13 and 26. This arrangement can be simulated by geometric models that include nearest neighbor packing, together with spiral phyllotaxis. The circular arrangement of peripheral primordia at nearly equal radial distances from the center of the developing capitulum helps to explain the numerical constancy (canalization) of peripheral structures, especially the constant number of 13 inner phyllaries on heads with very different numbers of florets.     

Allometric relations in plant growth

Allometric relations Y = aX^b are shown to exist in phyllotaxis, under the form r = k log R = p ()(m + n) ^–2 , where r = Y is the normalized internode distance in the cylindrical representation of phyllotaxis and R is the plastochrone ratio in the centric representation, p()=a is a constant for every angle of intersection of the opposed parastichies of the visible pair (m, n), for every m and n, and for all possible limit divergence angles corresponding to the Fibonacci-type sequences..., m–n, n, m, m + n=X, 2m + n, ..., and where b=–2. Richards' phyllotaxis index will be deduced.This work was supported by the Natural Science and Engineering research Council Canada, grant A6240     

Phyllotaxis in the Oil Palm: Arrangement of Male/Female Spikelets on the Inflorescence Stalk

The geometry of spikelet arrangement along the massive maleand female inflorescences of this monoecious palm is investigatedthrough techniques of phyllotaxis measurement (Equivalent phyllotaxisindex or E.P.I. Essentially there is little apparent differencewhether E.P.I. values from male inflorescences or female areconsidered. In both sexes E.P.I. estimates vary in a curvilinearmanner along the inflorescence length, higher values being obtainedtoward the broad base. Based on this pattern one value-over-allinflorescence E.P.I. is suggested to compare bunches emanatingfrom different treatment or genetic sources. It is also shownthat over-all bunch (inflorescence) E.P.I. varies systematicallywith palm age, being low in young palms and rising to asymptoticlevels in c. 11-year-old palms. Some physiological explanationsfor this latter observation are put forward, and some possibleapplications in cultural and genetic studies are suggested.Attention is also given to describing the results in terms ofFibonacci contact parastichy systems.     

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.     

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.     

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.     

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.     

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.     

Computer models of auxin transport: a review and commentary

With the recent proliferation of computer models of auxin transport, it is important that plant biologists understand something about these techniques and how to evaluate them. The paper begins with a brief introduction to the parts of a computer model, followed by a discussion of the limitations of the most common auxin modelling technique. Lastly, several recent models of organ initiation in the shoot apical meristem (i.e. phyllotaxis) are reviewed. The cell and molecular biology of phyllotaxis is now understood well enough that computer models can go beyond a simple 'proof of principle' and start to provide insights into gene function.     

The theory of phyllotaxis. II. Crystallographic interpretation of the interrelation between lower and superior phyllotaxis forms

Cross-opposite phyllotaxis forms are defined as superior with respect to the alternate ones and verticillate phyllotaxis forms as superior with respect to the opposite ones. Different phyllotaxis forms can be interpreted as a result of stretching of crystal-like structures of the embryo formed by dense packing of rudiments. Based on hypothetical concepts of the properties of plant rudiments and embryos, possible mechanisms of the formation of superior phyllotaxis forms from the lower ones have been analyzed. It was shown that the superior phyllotaxis forms can be considered as the results of additive summation of the lower forms. The theoretical conclusions are confirmed by the examples of polymorphic phyllotaxis in conspecific plants and by the facts of accidental splitting of superior phyllotaxis forms into the corresponding lower forms in nature and in experiment. The mechanisms underlying the formation of multiple forms of helical phyllotaxis have been proposed. The concept of a new type of mixed hexagonal-tetragonal phyllotaxis has been formulated and the mechanism of its formation has been considered. The forms of corn grain packaging in the corncob and leaf arrangement on the strawberry tomato stem are given as examples of true hexagonal-tetragonal phyllotaxis in nature.