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

A new mathematical model of phyllotaxis to solve the genuine puzzle spiromonostichy

Arrangement of plant leaves around the stem, termed phyllotaxis, exhibits beautiful and mysterious regularities and has been one of the most attractive subjects of biological pattern formation. After the long history of studies on phyllotaxis, it is now widely accepted that the inhibitory effect of existing leaf primordia on new primordium formation determines phyllotactic patterning. However, costoid phyllotaxis unique to Costaceae of Zingiberales, displaying spiromonostichy characterized by a steep spiral with a small divergence angle, seems to disagree with the inhibitory effect-based mechanism and has remained as a "genuine puzzle". We developed a new mathematical model, hypothesizing that each leaf primordium exerts not only the inhibitory effect but also some inductive effect. Computer simulations with the new model successfully generated a spiromonostichous pattern when these two effects met a certain relationship. The obtained spiromonostichy matched the real costoid phyllotaxis observed with Costus megalobractea, particularly for the decrease of the divergence angle associated with the enlargement of the shoot apical meristem. The new model was also shown to be able to produce a one-sided distichous pattern that is seen in phyllotaxis of a few plants of Zingiberales and has never been addressed in the previous model studies. These results implicated inductive as well as inhibitory mechanisms in phyllotactic patterning, at least in Zingiberales.

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.

Symmetry and its transition in phyllotaxis

Symmetry is an important component of geometric beauty and regularity in both natural and cultural scenes. Plants also display various geometric patterns with some kinds of symmetry, of which the most notable example is the arrangement of leaves around the stem, i.e., phyllotaxis. In phyllotaxis, reflection symmetry, rotation symmetry, translation symmetry, corkscrew symmetry, and/or glide reflection symmetry can be seen. These phyllotactic symmetries can be dealt with the group theory. In this review, we introduce classification of phyllotactic symmetries according to the group theory and enumerate all types of phyllotaxis, not only major ones such as spiral and decussate but also minor ones such as orixate and semi-decussate, with their symmetry groups. Next, based on the mathematical model studies of phyllotactic pattern formation, we discuss transitions between phyllotaxis types different in the symmetry class with a focus on the transition into one of the least symmetric phyllotaxis, orixate, as a representative of the symmetry-breaking process. By changes of parameters of the mathematical model, the phyllotactic pattern generated can suddenly switch its symmetry class, which is not constrained by the group-subgroup relationship of symmetry. The symmetry-breaking path to orixate phyllotaxis is also accompanied by dynamic changes of the symmetry class. The viewpoint of symmetry brings a better understanding of the variety of phyllotaxis and its transition.

Phyllotaxis: from classical knowledge to molecular genetics

Plant organs are repetitively generated at the shoot apical meristem (SAM) in recognizable patterns. This phenomenon, known as phyllotaxis, has long fascinated scientists from different disciplines. While we have an enriched body of knowledge on phyllotactic patterns, parameters, and transitions, only in the past 20 years, however, have we started to identify genes and elucidate genetic pathways that involved in phyllotaxis. In this review, I first summarize the classical knowledge of phyllotaxis from a morphological perspective. I then discuss recent advances in the regulation of phyllotaxis, from a molecular genetics perspective. I show that the morphological beauty of phyllotaxis we appreciate is the manifestation of many regulators, in addition to the critical role of auxin as a patterning signal, exerting their respective effects in a coordinated fashion either directly or indirectly in the SAM.

Two-step mechanism of spiral phyllotaxis

Fibonacci numbers such as 5, 8, and 13 occur in the spiral arrangement of lateral organs at shoot tips in plants. While the cone scales of conifers are normally arranged in 5 and 8 (or 8 and 13) curved rows in opposite directions, other numbers such as 4 and 7 (or 7 and 11) are found anomalously. The observed numbers still obey the Fibonacci rule, with the next number being the sum of the preceding two. Although these observations have been made for centuries, the underlying mechanisms of the numerical relationship have not been investigated. Here, we show that this phenomenon is caused by a two-step mechanism: (1) maintenance of a constant angle between consecutive lateral organs and (2) strong canalization of this angle to a specific value. The first step of the mechanism precedes the second step of the mechanism because the Fibonacci-rule pattern is due to the first step, while the second step distinguishes normal, anomalous and unobserved types. The current dominance of the normal type is a result of the evolutionary process of the second step.

Molecular and Hormonal Regulation of Leaf Morphogenesis in Arabidopsis

Shoot apical meristems (SAM) are tissues that function as a site of continuous organogenesis, which indicates that a small pool of pluripotent stem cells replenishes into lateral organs. The coordination of intercellular and intracellular networks is essential for maintaining SAM structure and size and also leads to patterning and formation of lateral organs. Leaves initiate from the flanks of SAM and then develop into a flattened structure with variable sizes and forms. This process is mainly regulated by the transcriptional regulators and mechanical properties that modulate leaf development. Leaf initiation along with proper orientation is necessary for photosynthesis and thus vital for plant survival. Leaf development is controlled by different components such as hormones, transcription factors, miRNAs, small peptides, and epigenetic marks. Moreover, the adaxial/abaxial cell fate, lamina growth, and shape of margins are determined by certain regulatory mechanisms. The over-expression and repression of various factors responsible for leaf initiation, development, and shape have been previously studied in several mutants. However, in this review, we collectively discuss how these factors modulate leaf development in the context of leaf initiation, polarity establishment, leaf flattening and shape.

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.