A dynamical phyllotaxis model to determine floral organ number

Flower Development in the Solanaceae

Flower development is the process leading from a reproductive meristem to a mature flower with fully developed floral organs. This multi-step process is complex and involves thousands of genes in intertwined regulatory pathways; navigating through the FLOR-ID website will give an impression of this complexity and of the astonishing amount of work that has been carried on the topic (Bouché et al., Nucleic Acids Res 44:D1167-D1171, 2016). Our understanding of flower development mostly comes from the model species Arabidopsis thaliana, but numerous other studies outside of Brassicaceae have helped apprehend the conservation of these mechanisms in a large evolutionary context (Moyroud and Glover, Curr Biol 27:R941-R951, 2017; Smyth, New Phytol 220:70-86, 2018; Soltis et al., Ann Bot 100:155-163, 2007). Integrating additional species and families to the research on this topic can only advance our understanding of flower development and its evolution.In this chapter, we review the contribution that the Solanaceae family has made to the comprehension of flower development. While many of the general features of flower development (i.e., the key molecular players involved in flower meristem identity, inflorescence architecture or floral organ development) are similar to Arabidopsis, our main objective in this chapter is to highlight the points of divergence and emphasize specificities of the Solanaceae. We will not discuss the large topics of flowering time regulation, inflorescence architecture and fruit development, and we will restrict ourselves to the mechanisms included in a time window after the floral transition and before the fertilization. Moreover, this review will not be exhaustive of the large amount of work carried on the topic, and the choices that we made to describe in large details some stories from the literature are based on the soundness of the functional work performed, and surely as well on our own preferences and expertise.First, we will give a brief overview of the Solanaceae family and some of its specificities. Then, our focus will be on the molecular mechanisms controlling floral organ identity, for which extended functional work in petunia led to substantial revisions to the famous ABC model. Finally, after reviewing some studies on floral organ initiation and growth, we will discuss floral organ maturation, using the examples of the inflated calyx of the Chinese lantern Physalis and petunia petal pigmentation.

Modeling phyllotaxis: From the Inhibition Potential to the Real Plant

We developed a new parametrization of the classic Douady and Couder model of phyllotaxis based on the inhibition potential: it allowed us to accurately reproduce the vegetative meristem of Linum usitatissimum displaying Fibonacci phyllotaxis, and the reproductive meristem of Ranunculus repens. We calculated the inhibition potential within the meristem and the auxin concentration at the front. We show that phyllotaxis modes and the convergence of the divergence angles towards "noble" angles are the consequence of minimizing the inhibitory potential under the constraint of decreasing plastochron ratios. Our approach, which gives a physicochemical basis to the van Iterson diagram, produces the same results as approaches based on mechanical constraints, suggesting these are two facets of the same botanical reality.

Model-based plant phenomics on morphological traits using morphometric descriptors

The morphological traits of plants contribute to many important functional features such as radiation interception, lodging tolerance, gas exchange efficiency, spatial competition between individuals and/or species, and disease resistance. Although the importance of plant phenotyping techniques is increasing with advances in molecular breeding strategies, there are barriers to its advancement, including the gap between measured data and phenotypic values, low quantitativity, and low throughput caused by the lack of models for representing morphological traits. In this review, we introduce morphological descriptors that can be used for phenotyping plant morphological traits. Geometric morphometric approaches pave the way to a general-purpose method applicable to single units. Hierarchical structures composed of an indefinite number of multiple elements, which is often observed in plants, can be quantified in terms of their multi-scale topological characteristics using topological data analysis. Theoretical morphological models capture specific anatomical structures, if recognized. These morphological descriptors provide us with the advantages of model-based plant phenotyping, including robust quantification of limited datasets. Moreover, we discuss the future possibilities that a system of model-based measurement and model refinement would solve the lack of morphological models and the difficulties in scaling out the phenotyping processes.

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

The spiral arrangement (phyllotaxis) of leaves is a shared morphology in land plants, and exhibits diversity constrained to the Fibonacci sequence. Phyllotaxis in vascular plants is produced at a multicellular meristem, whereas bryophyte phyllotaxis emerges from a single apical stem cell (AC) that is embedded in a growing tip of the gametophyte. An AC is asymmetrically divided into itself and a single 'merophyte', producing a future leaf and a portion of the stem. Although it has been suggested that the arrangement of merophytes is regulated by a rotation of the division plane of an AC, the quantitative description of the merophyte arrangement and its regulatory mechanism remain unclear. To clarify them, we examined three moss species, Tetraphis pellucida, Physcomitrium patens, and Niphotrichum japonicum, which exhibit 1/3, 2/5, and 3/8 spiral phyllotaxis, respectively. We measured the angle between the centroids of adjacent merophytes relative to the AC centroid on cross-transverse sections. At the outer merophytes, this divergence angle converged to nearly 120[Formula: see text] in T. pellucida, 136[Formula: see text] in N. japonicum, and 141[Formula: see text] in P. patens, which was nearly consistent with phyllotaxis, whereas the divergence angle deviated from the converged angle at the inner merophytes near an AC. A mathematical model, which assumes scaling growth of AC and merophytes and a constant angle of division plane rotation, quantitatively reproduced the sequence of the divergence angles. This model showed that successive relocations of the centroid position of an AC upon its division inevitably result in the transient deviation of the divergence angle. As a result, the converged divergence angle was equal to the rotation angle, predicting that the latter is a major regulator of the spiral phyllotaxis diversity in mosses.

Developmental stochasticity and variation in floral phyllotaxis

Floral phyllotaxis is a relatively robust phenotype; trimerous and pentamerous arrangements are widely observed in monocots and core eudicots. Conversely, it also shows variability in some angiosperm clades such as 'ANA' grade (Amborellales, Nymphaeales, and Austrobaileyales), magnoliids, and Ranunculales. Regardless of the phylogenetic relationship, however, phyllotactic pattern formation appears to be a common process. What are the causes of the variability in floral phyllotaxis and how has the variation of floral phyllotaxis contributed to floral diversity? In this review, I summarize recent progress in studies on two related fields to develop answers to these questions. First, it is known that molecular and cellular stochasticity are inevitably found in biological systems, including plant development. Organisms deal with molecular stochasticity in several ways, such as dampening noise through gene networks or maintaining function through cellular redundancy. Recent studies on molecular and cellular stochasticity suggest that stochasticity is not always detrimental to plants and that it is also essential in development. Second, studies on vegetative and inflorescence phyllotaxis have shown that plants often exhibit variability and flexibility in phenotypes. Three types of phyllotaxis variations are observed, namely, fluctuation around the mean, transition between regular patterns, and a transient irregular organ arrangement called permutation. Computer models have demonstrated that stochasticity in the phyllotactic pattern formation plays a role in pattern transitions and irregularities. Variations are also found in the number and positioning of floral organs, although it is not known whether such variations provide any functional advantages. Two ways of diversification may be involved in angiosperm floral evolution: precise regulation of organ position and identity that leads to further specialization of organs and organ redundancy that leads to flexibility in floral phyllotaxis.

A design principle for floral organ number and arrangement in flowers with bilateral symmetry

The bilateral symmetry of flowers is a striking morphological achievement during floral evolution, providing high adaptation potential for pollinators. The symmetry can appear when floral organ primordia developmentally initiate. Primordia initiation at the ventral and dorsal sides of the floral bud is differentially regulated by several factors, including external organs of the flower and CYCLOIDEA (CYC) gene homologues, which are expressed asymmetrically on the dorso-ventral axis. It remains unclear how these factors control the diversity in the number and bilateral arrangement of floral organs. Here, we propose a mathematical model demonstrating that the relative strength of the dorsal-to-ventral inhibitions and the size of the floral stem cell region (meristem) determines the number and positions of the sepal and petal primordia. The simulations reproduced the diversity of monocots and eudicots, including snapdragon Antirrhinum majus and its cyc mutant, with respect to organ number, arrangement and initiation patterns, which were dependent on the inhibition strength. These theoretical results suggest that diversity in floral symmetry is primarily regulated by the dorso-ventral inhibitory field and meristem size during developmental evolution.

Mathematical model studies of the comprehensive generation of major and minor phyllotactic patterns in plants with a predominant focus on orixate phyllotaxis

Plant leaves are arranged around the stem in a beautiful geometry that is called phyllotaxis. In the majority of plants, phyllotaxis exhibits a distichous, Fibonacci spiral, decussate, or tricussate pattern. To explain the regularity and limited variety of phyllotactic patterns, many theoretical models have been proposed, mostly based on the notion that a repulsive interaction between leaf primordia determines the position of primordium initiation. Among them, particularly notable are the two models of Douady and Couder (alternate-specific form, DC1; more generalized form, DC2), the key assumptions of which are that each leaf primordium emits a constant power that inhibits new primordium formation and that this inhibitory effect decreases with distance. It was previously demonstrated by computer simulations that any major type of phyllotaxis can occur as a self-organizing stable pattern in the framework of DC models. However, several phyllotactic types remain unaddressed. An interesting example is orixate phyllotaxis, which has a tetrastichous alternate pattern with periodic repetition of a sequence of different divergence angles: 180°, 90°, −180°, and −90°. Although the term orixate phyllotaxis was derived from Orixa japonica, this type is observed in several distant taxa, suggesting that it may reflect some aspects of a common mechanism of phyllotactic patterning. Here we examined DC models regarding the ability to produce orixate phyllotaxis and found that model expansion via the introduction of primordial age-dependent changes of the inhibitory power is absolutely necessary for the establishment of orixate phyllotaxis. The orixate patterns generated by the expanded version of DC2 (EDC2) were shown to share morphological details with real orixate phyllotaxis. Furthermore, the simulation results obtained using EDC2 fitted better the natural distribution of phyllotactic patterns than did those obtained using the previous models. Our findings imply that changing the inhibitory power is generally an important component of the phyllotactic patterning mechanism.

Phyllotaxis, the beautiful geometry of plant-leaf arrangement around the stem, has long attracted the attention of researchers of biological-pattern formation. Many mathematical models, as typified by those of Douady and Couder (alternate-specific form, DC1; more generalized form, DC2), have been proposed for phyllotactic patterning, mostly based on the notion that a repulsive interaction between leaf primordia spatially regulates primordium initiation. In the framework of DC models, which assume that each primordium emits a constant power that inhibits new primordium formation and that this inhibitory effect decreases with distance, the major types (but not all types) of phyllotaxis can occur as stable patterns. Orixate phyllotaxis, which has a tetrastichous alternate pattern with a four-cycle sequence of the divergence angle, is an interesting example of an unaddressed phyllotaxis type. Here, we examined DC models regarding the ability to produce orixate phyllotaxis and found that model expansion by introducing primordial age-dependent changes of the inhibitory power is absolutely necessary for the establishment of orixate phyllotaxis. The simulation results obtained using the expanded version of DC2 (EDC2) fitted well the natural distribution of phyllotactic patterns. Our findings imply that changing the inhibitory power is generally an important component of the phyllotactic patterning mechanism.

Understanding the role of floral development in the evolution of angiosperm flowers: clarifications from a historical and physico-dynamic perspective

Flower morphology results from the interaction of an established genetic program, the influence of external forces induced by pollination systems, and physical forces acting before, during and after initiation. Floral ontogeny, as the process of development from a meristem to a fully developed flower, can be approached either from a historical perspective, as a "recapitulation of the phylogeny" mainly explained as a process of genetic mutations through time, or from a physico-dynamic perspective, where time, spatial pressures, and growth processes are determining factors in creating the floral morphospace. The first (historical) perspective clarifies how flower morphology is the result of development over time, where evolutionary changes are only possible using building blocks that are available at a certain stage in the developmental history. Flowers are regulated by genetically determined constraints and development clarifies specific transitions between different floral morphs. These constraints are the result of inherent mutations or are induced by the interaction of flowers with pollinators. The second (physico-dynamic) perspective explains how changes in the physical environment of apical meristems create shifts in ontogeny and this is reflected in the morphospace of flowers. Changes in morphology are mainly induced by shifts in space, caused by the time of initiation (heterochrony), pressure of organs, and alterations of the size of the floral meristem, and these operate independently or in parallel with genetic factors. A number of examples demonstrate this interaction and its importance in the establishment of different floral forms. Both perspectives are complementary and should be considered in the understanding of factors regulating floral development. It is suggested that floral evolution is the result of alternating bursts of physical constraints and genetic stabilization processes following each other in succession. Future research needs to combine these different perspectives in understanding the evolution of floral systems and their diversification.

Spiral phyllotaxis underlies constrained variation in Anemone (Ranunculaceae) tepal arrangement

Stabilization and variation of floral structures are indispensable for plant reproduction and evolution; however, the developmental mechanism regulating their structural robustness is largely unknown. To investigate this mechanism, we examined positional arrangement (aestivation) of excessively produced perianth organs (tepals) of six- and seven-tepaled (lobed) flowers in six Anemone species (Ranunculaceae). We found that the tepal arrangement that occurred in nature varied intraspecifically between spiral and whorled arrangements. Moreover, among the studied species, variation was commonly limited to three types, including whorls, despite five geometrically possible arrangements in six-tepaled flowers and two types among six possibilities in seven-tepaled flowers. A spiral arrangement, on the other hand, was unique to five-tepaled flowers. A spiral phyllotaxis model with stochasticity on initiating excessive primordia accounted for these limited variations in arrangement in cases when the divergence angle between preexisting primordia was less than 144°. Moreover, interspecific differences in the frequency of the observed arrangements were explained by the change of model parameters that represent meristematic growth and differential organ growth. These findings suggest that the phyllotaxis parameters are responsible for not only intraspecific stability but interspecific difference of floral structure. Decreasing arrangements from six-tepaled to seven-tepaled Anemone flowers demonstrate that the stabilization occurs as development proceeds to increase the component (organ) number, in contrast from the intuition that the variation will be larger due to increasing number of possible states (arrangements).

Two-stage patterning dynamics in conifer cotyledon whorl morphogenesis

Background and Aims

Conifer embryos, unlike those of monocots or dicots, have variable numbers of cotyledons, even within the same species. Cotyledons form in a single whorl on a dome-shaped embryo. The closely spaced cotyledons are not found outside this ring, indicating a radial control on where they can form. Polar transport of the hormone auxin affects outgrowth of distinct cotyledons, but not the radial aspect of the whorl or the within-whorl spacing between cotyledons. A quantitative model of plant growth regulator patterning is needed to understand the dynamics of this complex morphogenetic process.

Methods

A two-stage reaction-diffusion model is developed for the spatial patterning of growth regulators on the embryo surface, with a radial pattern (P1) constraining the shorter-wavelength cotyledon pattern (P2) to a whorl. These patterns drive three-dimensional (3-D) morphogenesis by catalysing local surface growth.

Key Results

Growth driven by P2 generates single whorls across the experimentally observed range of two to 11 cotyledons, as well as the circularly symmetric response to auxin transport interference. These computations are the first corroboration of earlier theoretical proposals for hierarchical control of whorl formation. The model generates the linear relationship between cotyledon number and embryo diameter observed experimentally. This accounts for normal integer cotyledon number selection, as well as the less common cotyledon fusings and splittings observed experimentally. Flattening of the embryo during development may affect the upward outgrowth angle of the cotyledons.

Conclusions

Cotyledon morphogenesis is more complex geometrically in conifers than in angiosperms, involving 2-D patterning which deforms a surface in three dimensions. This work develops a quantitative framework for understanding the growth and patterning dynamics involved in conifer cotyledon development, and applies more generally to the morphogenesis of whorls with many primordia.

Morphological Plant Modeling: Unleashing Geometric and Topological Potential within the Plant Sciences

The geometries and topologies of leaves, flowers, roots, shoots, and their arrangements have fascinated plant biologists and mathematicians alike. As such, plant morphology is inherently mathematical in that it describes plant form and architecture with geometrical and topological techniques. Gaining an understanding of how to modify plant morphology, through molecular biology and breeding, aided by a mathematical perspective, is critical to improving agriculture, and the monitoring of ecosystems is vital to modeling a future with fewer natural resources. In this white paper, we begin with an overview in quantifying the form of plants and mathematical models of patterning in plants. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the movement of leaves in air streams. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

Morphological Plant Modeling: Unleashing Geometric and Topological Potential within the Plant Sciences

The geometries and topologies of leaves, flowers, roots, shoots, and their arrangements have fascinated plant biologists and mathematicians alike. As such, plant morphology is inherently mathematical in that it describes plant form and architecture with geometrical and topological techniques. Gaining an understanding of how to modify plant morphology, through molecular biology and breeding, aided by a mathematical perspective, is critical to improving agriculture, and the monitoring of ecosystems is vital to modeling a future with fewer natural resources. In this white paper, we begin with an overview in quantifying the form of plants and mathematical models of patterning in plants. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the movement of leaves in air streams. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

Characterizing Floral Symmetry in the Core Goodeniaceae with Geometric Morphometrics

Core Goodeniaceae is a clade of ~330 species primarily distributed in Australia. Considerable variation in flower morphology exists within this group and we aim to use geometric morphometrics to characterize this variation across the two major subclades: Scaevola sensu lato (s.l.) and Goodenia s.l., the latter of which was hypothesized to exhibit greater variability in floral symmetry form. We test the hypothesis that floral morphological variation can be adequately characterized by our morphometric approach, and that discrete groups of floral symmetry morphologies exist, which broadly correlate with subjectively determined groups. From 335 images of 44 species in the Core Goodeniaceae, two principal components were computed that describe >98% of variation in all datasets. Increasing values of PC1 ventralize the dorsal petals (increasing the angle between them), whereas increasing values of PC2 primarily ventralize the lateral petals (decreasing the angle between them). Manipulation of these two morphological “axes” alone was sufficient to recreate any of the general floral symmetry patterns in the Core Goodeniaceae. Goodenia s.l. exhibits greater variance than Scaevola s.l. in PC1 and PC2, and has a significantly lower mean value for PC1. Clustering clearly separates fan-flowers (with dorsal petals at least 120° separated) from the others, whereas the distinction between pseudo-radial and bilabiate clusters is less clear and may form a continuum rather than two distinct groups. Transitioning from the average fan-flower to the average non-fan-flower is described almost exclusively by PC1, whereas PC2 partially describes the transition between bilabiate and pseudo-radial morphologies. Our geometric morphometric method accurately models Core Goodeniaceae floral symmetry diversity.

Relationship between the species-representative phenotype and intraspecific variation in Ranunculaceae floral organ and Asteraceae flower numbers

BACKGROUND AND AIMS

Phenotypic variation in floral morphologies contributes to speciation by testing various morphologies that might have higher adaptivity, leading eventually to phylogenetic diversity. Species diversity has been recognized, however, by modal morphologies where the variation is averaged out, so little is known about the relationship between the variation and the diversity.

METHODS

We analysed quantitatively the intraspecific variation of the organ numbers within flowers of Ranunculaceae, a family which branched near the monocot-eudicot separation, and the numbers of flowers within the capitula of Asteraceae, one of the most diverse families of eudicots. We used four elementary statistical quantities: mean, standard deviation (s.d.), degree of symmetry (skewness) and steepness (kurtosis).

KEY RESULTS

While these four quantities vary among populations, we found a common relationship between s.d. and the mean number of petals and sepals in Ranunculaceae and number of flowers per capitulum in Asteraceae. The s.d. is equal to the square root of the difference between the mean and specific number, showing robustness: for example, 3 in Ficaria sepals, 5 in Ranunculus petals and Anemone tepals, and 13 in Farfugium ray florets. This square-root relationship was not applicable to Eranthis petals which show little correlation between the s.d. and mean, and the stamens and carpels of Ranunculaceae whose s.d. is proportional to the mean. The specific values found in the square-root relationship provide a novel way to find the species-representative phenotype among varied morphologies.

CONCLUSIONS

The representative phenotype is, in most cases, unique to the species or genus level, despite intraspecific differences of average phenotype among populations. The type of variation shown by the statistical quantities indicates not only the robustness of the morphologies but also how flowering plants changed during evolution among representative phenotypes that eventually led to phylogenetic diversification.