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.     

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.     

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.     

Implications of opposite phyllotaxis for light interception efficiency of Mediterranean woody plants

Opposite leaves lead to a greater leaf overlapping than leaves spirally arranged along a shoot, decreasing light interception efficiency (Ea, fraction of the light reaching the plant actually intercepted by the leaves) of the crown. However, Ea results from a whole suite of morphological traits. The interplay between phyllotaxis, crown architecture, leaf morphology and Ea was explored in 12 woody species from Mediterranean-type ecosystems, where the abundance of woody species with opposite phyllotaxis is unusually high. The three-dimensional model Y-plant was used to estimate Ea in unbranched, vertical shoots of each species encompassing the natural morphological variation found from moderate shade to open light environments. Ea exhibited significant interspecific differences, ranging from 0.25 in Daphne gnidium to 0.75 in Cistus ladanifer, Olea europaea and Salvia officinalis, decreasing with leaf inclination angle and leaf area ratio (LAR), and increasing with internode-to-leaf-length ratio and supporting biomass. Species with spiral vs. opposite phyllotaxis did not differ in their mean Ea. However, the former had higher Ea than the latter at short internode lengths. The natural range of variation in internode length had a larger effect on Ea than the natural range of leaf elevation angle. Principal component analysis segregated species with opposite phyllotaxis from those with spiral leaves because of their greater self-shading for high sun elevation angles (>45°); they were in turn distributed in two groups, one with high Ea, large investment in supporting biomass and long internodes, and another with low Ea and large LAR. Species with spiral phyllotaxis all had intermediate or low Ea and steep leaf elevation angles. Species with opposite phyllotaxis can compensate their less efficient leaf arrangement by decreasing leaf elevation angle and increasing internode length, but they may experience a real phylogenetic constraint for light interception when biomass allocation to supporting tissues (internodes and petioles) becomes very costly. This constraint could be involved in the shade intolerance of woody Mediterranean species exhibiting opposite phyllotaxis.     

A systemic model of growth in botanometry

This paper deals with the problem of phyllotaxis. It is well known that each type of phyllotactic arrangement of the primordia of plants leads to a recurring series of positive integers 〈H(k)〉, where H(k) = H(k?1)+H(k?2), closely related to a precise divergence angle between the primordia. We already introduced, in former articles, a mechanism using a concept of information, a bi-dimensional architectural concept of hierarchy, a principle of optimal design and specific languages to cope with the problem, and we drew preliminary conclusions. This setting is enhanced here as it is shown to give the different types of phyllotaxis, to explain their relative occurrences in Nature and to suggest verifiable predictions.     

Probability density functions for axial ratios of sectioning profiles of anisotropically arranged elliptical microvessels

The article theoretically regards probability density functions (PDFs) for axial ratio (X/Y) of sectioning profiles of elliptical microvessels (MVs) arranged with anisotropy in a biological tissue volume. A technique for the PDF_X/Y calculations in anisotropy of the elliptical MVs is described. The essence of this technique is introducing anisotropy in PDF(α,φ), i.e. the function of the joint distribution of the polar and planar angles α and φ, which define mutual orientation of the elliptical MVs and sectioning planes. With the aid of this technique, the anisotropy cases are studied with PDF(α,φ) given by pair combinations of the following distributions: (i) a uniform distribution of the angles α and/or φ, (ii) the angle α distribution with , and (iii) Gaussian distributions of the α or φ values. Specifically, PDF_X/Y curves are obtained for MVs with the true, or three-dimensional, axial ratio X₀/Y₀=2.0, and the anisotropy effects on the X/Y expected frequencies are analysed. Conclusions of this analysis, the PDF_X/Y calculation technique, and the PDF_X/Y curves obtained are useful for stereological reconstruction of anisotropically organised microcirculatory networks, with an ellipticity of their MVs being taken into consideration.     

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.     

Phyllotaxis in two species ofRubia,R. akane andR. sikkimensis

Phyllotaxis and vascular course in the vegetative shoots ofRubia akane andR. sikkimensis were studied. Each node of both species has a whorl of four leafy members among which two are true leaves. Arrangement of the true leaves is not decussate but bijugate, i.e., opposite leaves are arranged spirally. Bijugy was ascertained not only by gross morphology but also by arrangement of primordia around the shoot apex and vascular course through several internodes. Divergence angle differed widely with internodes even within a single shoot and with shoots even in the internodes which are separated by a same number of nodes from the apex. Mean divergence angles obtained for five youngest internodes of some shoots were between 49.4° and 61.8° inR. akane and between 53.6° and 59.4° inR. sikkimensis. Young seedlings ofR. akane showed decussate phyllotaxis in the lowermost several internodes. In the internodes near the lower end of the bijugate part, the divergence angle was wider than in the upper internodes. The directions of the phyllotactic spirals in the main axis and the lateral branches were either homodromous or antidromous, and those in the oppositely paired branches also were either homo- or antidromous.     

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     

Quantitative f torsion angle analysis in Desulfovibrio vulgaris flavodoxin based on six f related 3 J couplings

Quantitative φ-dihedral angle determinations of non-glycine and non-proline residues in Desulfovibrio vulgaris flavodoxin are carried out on the exclusive basis of ³ J coupling constants. In total 124 ³ JHNH _α , 123 ³ JHNC _′i , 118 ³ JHNC _β , 117 ³ JC′ _i–1Hα , 109 ³ JC′ _i–1C′i , and 103 ³ JC′ _i–1Hβ values form the experimental basis for translating J coupling data into geometry information using various combinations of Karplus parameters given in the literature. In addition, each backbone torsional angle φ is adjusted assuming different models of local geometry, either a rigid torsion, a Gaussian distribution centered at a distinct angle, or a two-site jump model. Numerical optimization is followed by a statistical significance evaluation to assess the results. It is found that experimental coupling constants of most of the residues involved in secondary structure elements agree best with those predicted from rigid local conformations. For dihedral angles in loop regions, mobility effects are not negligible, and a single torsion (Glu 42) is likely to adopt two distinct adjustments. However, α-helix conformations with –60° < φ < –45° give rise to an alternate solution with φ≈+170° with similar statistical significance when using the four traditionally determined proton-involved ³ J couplings. This ambiguity is efficiently avoided only when taking advantage of the complete data set comprising six available experimental ³ J coupling constants and of the degeneracy intrinsic to the Karplus relation. The optimized φ conformations are compared with reference values from the crystal structure of flavodoxin.     

PHYLLOTACTIC CHANGE INDUCED BY GIBBERELLIC ACID IN XANTHIUM SHOOT APICES

Gibberellic acid (GA) treatment of vegetative shoots of Xanthium leads to a change in phyllotaxis as diagnosed in transverse sections of apical buds. A method of analysis is proposed for estimating the phyllotactic parameters, the plastochron ratio, a, and the divergence angle, α, from measurements of the angular and radial positions of leaf primordia in sections. GA treatment significantly decreases the plastochron ratio, a, from 1.35 in controls, to 1.19 in GA-treated plants, as shown by an analysis of variance, but has no significant effect on the divergence angle. The estimates of a and α are compared with the parameters of theoretical phyllotaxis models, leading to the designation (2, 3) for controls, and (3, 5) for GA-treated plants, where the integers 2, 3, and 5 designate sets of contact parastichies. The change in a is interpreted as indicating a change in the relative position at which leaf primordia are initiated in the apical meristem, and this effect is discussed in relation to theories of leaf initiation.     

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

Phyllotaxis, the regular arrangement of leaves or flowers around a plant stem, is an example of developmental pattern formation and organogenesis. Phyllotaxis is characterized by the divergence angles between the organs, the most common angle being 137.5 degrees , the golden angle. The quantitative aspects of phyllotaxis have stimulated research at the interface between molecular biology, physics and mathematics. This review documents the rich history of different approaches and conflicting hypotheses, and then focuses on recent molecular work that establishes a novel patterning mechanism based on active transport of the plant hormone auxin. Finally, it shows how computer simulations can help to formulate quantitative models that in turn can be tested by experiment. The accumulation of ever increasing amounts of experimental data makes quantitative modeling of interest for many developmental systems.     

Microsurgical and laser ablation analysis of leaf positioning and dorsoventral patterning in tomato

Leaves are arranged according to regular patterns, a phenomenon referred to as phyllotaxis. Important determinants of phyllotaxis are the divergence angle between successive leaves, and the size of the leaves relative to the shoot axis. Young leaf primordia are thought to provide positional information to the meristem, thereby influencing the positioning of new primordia and hence the divergence angle. On the contrary, the meristem signals to the primordia to establish their dorsoventral polarity, which is a prerequisite for the formation of a leaf blade. These concepts originate from classical microsurgical studies carried out between the 1920s and the 1970s. Even though these techniques have been abandoned in favor of genetic analysis, the resulting insights remain a cornerstone of plant developmental biology. Here, we employ new microsurgical techniques to reassess and extend the classical studies on phyllotaxis and leaf polarity. Previous experiments have indicated that the isolation of an incipient primordium by a tangential incision caused a change of divergence angle between the two subsequent primordia, indicating that pre-existing primordia influence further phyllotaxis. Here, we repeat these experiments and compare them with the results of laser ablation of incipient primordia. Furthermore, we explore to what extent the different pre-existing primordia influence the size and position of new organs, and hence phyllotaxis. We propose that the two youngest primordia (P1 and P2) are sufficient for the approximate positioning of the incipient primordium (I1), and therefore for the perpetuation of the generative spiral, whereas the direct contact neighbours of I1 (P2 and P3) control its delimitation and hence its exact size and position. Finally, we report L1-specific cell ablation experiments suggesting that the meristem L1 layer is essential for the dorsoventral patterning of leaf primordia.     

Increased photosynthetic activities and thermostability of photosystem II with leaf development of elm seedlings (Ulmus pumila) probed by the fast fluorescence rise OJIP

Experiments were conducted to investigate the photosynthetic activity and thermostability of photosystem II (PSII) in elm seedling (Ulmus pumila) leaves from initiation to full expansion. During leaf development, photosynthesis, measured as CO₂ fixation, increased gradually and reached a maximum value when leaves were fully developed. In parallel with the increase of carbon assimilation, chlorophyll content increased. The chlorophyll a fluorescence measurements showed that the maximum quantum yield of PSII primary photochemistry (φ_po), the efficiency with which the energy of trapped excitons is converted into the electron transport beyond Q_A (Ψ_o) and the quantum yield of electron transport beyond Q_A (φ_Eo) increased gradually. The low light experiments confirmed these results independently. When subjected to heat stress, young leaves exhibited progressively lower φ_po and maximal fluorescence (F_m) values with considerably higher minimal fluorescence (F_o) than mature leaves, demonstrating that PSII in newly initiating leaves is more sensitive to heat stress. Further analysis revealed that PSII structure in newly initiating leaves showed a robust alteration under heat stress, which was reflected by the clear K phase in the OJIP curves. Therefore, we suggest that the enhanced thermostability of PSII in the case of leaf growth might be associated with an improvement of the stability of the oxygen-evolving complex (OEC) to heat stress during leaf development.     

Photomorphogenesis and phyllotaxis during vegetative growth in Sinapis alba and Xanthium strumarium

Abstract The effect of light on the rate of formation of leaf primordia was investigated at the apex of seedlings of Sinapis alba and Xanthium strumarium. It was found that light accelerates this rate. On the other hand, no significant light effect was found on the angles of divergence of successive leaves during the transition from the almost decussate leaf position of the cotyledons to the helical phyllotaxis of the stem leaves. In fact, light and dark grown plants use the same leaves for the transition from decussate to helical phyllotaxis. Thus, if time is plotted in ‘biological units’ (number of primordia) there is no difference between light and dark grown plants. Using scanning electron microscope techniques it was found that the ‘primordia free apical area’ enlarges during development. The rate of enlargement is accelerated by light. However, if time is expressed in biological units (number of primordia) no difference between light and dark grown plants exists. It is concluded that light accelerates the realization of the apical pattern without interfering with the specification of the pattern. In other words, light accelerates the development of an apex without affecting the temporal and spatial coordination of the events.     

Helix — coil transition of poly(γ-bensyl L-glutamate) in dioxan — dichloracetic acid mixtures

Measurements have been made on solutions of poly(γ-benzyl-l-glutamate) in solvents comprising dichloracetic acid (volume fraction φ₁) and dioxan over the whole range of φ₁. The transition point at φ₁ = 0.91 found from intrinsic viscosity is close to that obtained by others via dielectric measurements. However, the specific refractive index increments at constant composition and constant chemical potential as well as the selective adsorption coefficient of dichloracetic acid all exhibit sharp changes at about φ₁ = 0.75, and the partial specific volume of the polymer increases when φ₁ >0.75. An unusual minimum in the dependence of the refractive index increment on solvent composition is attributed to a small change in the refractive index of the polymer. This corresponds to a decrease of about 2% in the molecular polarizability of the polypeptide during the helix → coil transition.     

The consequences of contact pressure in phyllotaxis

To determine the consequences of contact pressure in phyllotaxis, a mathematical model is constructed in which a leaf distribution is represented by a point lattice of n + 1 lattice points at equal intervals on a helix wound around a cylinder. The model is normalized by taking the girth of the cylinder as 1 and by measuring time T in plastochrones, so that n = [T]. r stands for the normalized internode distance (component of the distance between two consecutive lattice points that is parallel to the axis of the cylinder). d stands for the divergence (fraction of a turn between consecutive lattice points). It is assumed that r is a monotonic decreasing function of T such that r(T) → 0 as T → ∞. Contact pressure is represented by the assumption that the minimum geodesic distance between lattice points is maximized. It is shown that if (p, q), with p < q, is the contact phyllotaxis determined when contact pressure first becomes effective, then the continuation of contact pressure requires that the advance to higher phyllotaxis as r decreases must proceed via successive pairs of consecutive terms of the Fibonacci sequence generated by the numbers p and q, namely, p, q, p + q, p + 2q, 2p + 3q, …. The divergence, starting from some value $\text{d =}\text{1}\text{t}\text{+}\text{1}\text{a}{}_2\text{+ … +}\text{1}\text{(a}{}_{\mathrm{n}}\text{+ x)}$ determined by p and q converges to an ideal angle $\text{1}\text{t}\text{+}\text{1}\text{a}{}_2\text{+ … +}\text{1}\text{a}{}_{\mathrm{n}}\text{+}\text{1}\text{τ}$, where τ is the golden section. A necessary and sufficient condition for the ideal angle to be $\text{1}\text{2}\text{+}\text{1}\text{τ}\text{= τ}{}^{\mathrm{?2}}$ is that the p and q of the initial contact phyllotaxis be consecutive Fibonacci numbers of the sequence 1, 2, 3, 5, 8, …. It is proved that a sufficient condition for convergence to the ideal angle τ^?2 of normal phyllotaxis is that contact pressure begin before T = 5 or before $\text{r <}\text{3}\text{38}\text{1}\text{2}$ with d initially between $\text{1}\text{3}$ and $\text{1}\text{2}$.