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

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          Abstract

          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.

          Author summary

          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.

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          Most cited references34

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          Local, efflux-dependent auxin gradients as a common module for plant organ formation.

          Plants, compared to animals, exhibit an amazing adaptability and plasticity in their development. This is largely dependent on the ability of plants to form new organs, such as lateral roots, leaves, and flowers during postembryonic development. Organ primordia develop from founder cell populations into organs by coordinated cell division and differentiation. Here, we show that organ formation in Arabidopsis involves dynamic gradients of the signaling molecule auxin with maxima at the primordia tips. These gradients are mediated by cellular efflux requiring asymmetrically localized PIN proteins, which represent a functionally redundant network for auxin distribution in both aerial and underground organs. PIN1 polar localization undergoes a dynamic rearrangement, which correlates with establishment of auxin gradients and primordium development. Our results suggest that PIN-dependent, local auxin gradients represent a common module for formation of all plant organs, regardless of their mature morphology or developmental origin.
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            Auxin regulates the initiation and radial position of plant lateral organs.

            Leaves originate from the shoot apical meristem, a small mound of undifferentiated tissue at the tip of the stem. Leaf formation begins with the selection of a group of founder cells in the so-called peripheral zone at the flank of the meristem, followed by the initiation of local growth and finally morphogenesis of the resulting bulge into a differentiated leaf. Whereas the mechanisms controlling the switch between meristem propagation and leaf initiation are being identified by genetic and molecular analyses, the radial positioning of leaves, known as phyllotaxis, remains poorly understood. Hormones, especially auxin and gibberellin, are known to influence phyllotaxis, but their specific role in the determination of organ position is not clear. We show that inhibition of polar auxin transport blocks leaf formation at the vegetative tomato meristem, resulting in pinlike naked stems with an intact meristem at the tip. Microapplication of the natural auxin indole-3-acetic acid (IAA) to the apex of such pins restores leaf formation. Similarly, exogenous IAA induces flower formation on Arabidopsis pin-formed1-1 inflorescence apices, which are blocked in flower formation because of a mutation in a putative auxin transport protein. Our results show that auxin is required for and sufficient to induce organogenesis both in the vegetative tomato meristem and in the Arabidopsis inflorescence meristem. In this study, organogenesis always strictly coincided with the site of IAA application in the radial dimension, whereas in the apical-basal dimension, organ formation always occurred at a fixed distance from the summit of the meristem. We propose that auxin determines the radial position and the size of lateral organs but not the apical-basal position or the identity of the induced structures.
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              Phyllotaxis and the fibonacci series.

              The principal conclusion is that Fibonacci phyllotaxis follows as a mathematical necessity from the combination of an expanding apex and a suitable spacing mechanism for positioning new leaves. I have considered an inhibitory spacing mechanism at some length, as it is a plausible candidate. However, the same treatment would apply equally well to depletion of, or competition for, a compound by developing leaves, and could no doubt accommodate other ingredients. The mathematical principles involved are clear when it is assumed that only two leaves (the contacts) position a new leaf. There is some experimental evidence for this assumption. Nonetheless, it is not a precondition for Fibonacci phyllotaxis, since a computer model shows that this pattern is generated even when many leaves contribute to inhibition at a given point. Indeed, the Fibonacci pattern seems to be a robust and stable mathematical phenomenon, a finding which goes some way to explaining its widespread occurrence throughout the plant kingdom.
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                Author and article information

                Contributors
                Role: ConceptualizationRole: Formal analysisRole: Funding acquisitionRole: InvestigationRole: MethodologyRole: ResourcesRole: Writing – original draftRole: Writing – review & editing
                Role: ResourcesRole: Writing – review & editing
                Role: Funding acquisitionRole: Writing – review & editing
                Role: ConceptualizationRole: MethodologyRole: Project administrationRole: SupervisionRole: Writing – original draftRole: Writing – review & editing
                Role: Editor
                Journal
                PLoS Comput Biol
                PLoS Comput. Biol
                plos
                ploscomp
                PLoS Computational Biology
                Public Library of Science (San Francisco, CA USA )
                1553-734X
                1553-7358
                6 June 2019
                June 2019
                : 15
                : 6
                : e1007044
                Affiliations
                [1 ] Botanical Gardens, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo, Japan
                [2 ] Department of Biology, Tokyo Gakugei University, Koganei, Tokyo, Japan
                [3 ] Astrobiology Center, National Institutes of Natural Sciences, Mitaka, Tokyo, Japan
                [4 ] National Institute for Basic Biology, National Institutes of Natural Sciences, Okazaki, Aichi, Japan
                Purdue University, UNITED STATES
                Author notes

                The authors have declared that no competing interests exist.

                [¤]

                Current address: Department of Biological Sciences, Faculty of Science, Kanagawa University, Hiratsuka, Kanagawa, Japan

                Author information
                http://orcid.org/0000-0001-5428-8212
                http://orcid.org/0000-0002-7050-8964
                Article
                PCOMPBIOL-D-18-01906
                10.1371/journal.pcbi.1007044
                6553687
                31170142
                17dfd063-2544-4bb9-b37d-f9cc234d115b
                © 2019 Yonekura et al

                This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

                History
                : 9 November 2018
                : 19 April 2019
                Page count
                Figures: 11, Tables: 0, Pages: 27
                Funding
                Funded by: funder-id http://dx.doi.org/10.13039/501100001691, Japan Society for the Promotion of Science;
                Award ID: Research Fellowship for Young Scientists No. 18J12309
                Award Recipient :
                Funded by: funder-id http://dx.doi.org/10.13039/501100001691, Japan Society for the Promotion of Science;
                Award ID: Grant-in-Aid for Scientific Research on Innovative Areas No. 16H01248
                Award Recipient :
                This work was supported in part by a Research Fellowship for Young Scientists (No. 18J12309 to TY) and by a Grant-in-Aid for Scientific Research on Innovative Areas (No. 16H01248 to HF) from the Japan Society for the Promotion of Science. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
                Categories
                Research Article
                Computer and Information Sciences
                Computer Modeling
                Biology and Life Sciences
                Biochemistry
                Hormones
                Plant Hormones
                Auxins
                Biology and Life Sciences
                Biochemistry
                Plant Biochemistry
                Plant Hormones
                Auxins
                Biology and Life Sciences
                Plant Science
                Plant Biochemistry
                Plant Hormones
                Auxins
                Biology and Life Sciences
                Plant Science
                Plant Anatomy
                Leaves
                Computer and Information Sciences
                Computerized Simulations
                Biology and Life Sciences
                Organisms
                Eukaryota
                Plants
                Flowering Plants
                Biology and Life Sciences
                Plant Science
                Plant Anatomy
                Buds
                Research and Analysis Methods
                Microscopy
                Electron Microscopy
                Scanning Electron Microscopy
                Research and Analysis Methods
                Mathematical and Statistical Techniques
                Mathematical Models
                Custom metadata
                All relevant data are within the manuscript and its Supporting Information files.

                Quantitative & Systems biology
                Quantitative & Systems biology

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