Reviewing models of auxin canalization in the context of leaf vein pattern formation in Arabidopsis

A transcriptional repressor HVA regulates vascular bundle formation through auxin transport in Arabidopsis stem

Vascular bundles transport water and photosynthate to all organs, and increased bundle number contributes to crop lodging resistance. However, the regulation of vascular bundle formation is poorly understood in the Arabidopsis stem. We report a novel semi-dominant mutant with high vascular activity, hva-d, showing increased vascular bundle number and enhanced cambium proliferation in the stem. The activation of a C2H2 zinc finger transcription factor, AT5G27880/HVA, is responsible for the hva-d phenotype. Genetic, biochemical, and fluorescent microscopic analyses were used to dissect the functions of HVA. HVA functions as a repressor and interacts with TOPLESS via the conserved Ethylene-responsive element binding factor-associated Amphiphilic Repression motif. In contrast to the HVA activation line, knockout of HVA function with a CRISPR-Cas9 approach or expression of HVA fused with an activation domain VP16 (HVA-VP16) resulted in fewer vascular bundles. Further, HVA directly regulates the expression of the auxin transport efflux facilitator PIN1, as a result affecting auxin accumulation. Genetics analysis demonstrated that PIN1 is epistatic to HVA in controlling bundle number. This research identifies HVA as a positive regulator of vascular initiation through negatively modulating auxin transport and sheds new light on the mechanism of bundle formation in the stem.

Leaf Vein Patterning

Leaves form veins whose patterns vary from a single vein running the length of the leaf to networks of staggering complexity where huge numbers of veins connect to other veins at both ends. For the longest time, vein formation was thought to be controlled only by the polar, cell-to-cell transport of the plant hormone auxin; recent evidence suggests that is not so. Instead, it turns out that vein patterning features are best accounted for by a combination of polar auxin transport, facilitated auxin diffusion through plasmodesma intercellular channels, and auxin signal transduction-though the latter's precise contribution remains unclear. Equally unclear remain the sites of auxin production during leaf development, on which that vein patterning mechanism ought to depend. Finally, whether that vein patterning mechanism can account for the variety of vein arrangements found in nature remains unknown. Addressing those questions will be the exciting challenge of future research.

A CUC1/auxin genetic module links cell polarity to patterned tissue growth and leaf shape diversity in crucifer plants

How tissue-level information encoded by fields of regulatory gene activity is translated into the patterns of cell polarity and growth that generate the diverse shapes of different species remains poorly understood. Here, we investigate this problem in the case of leaf shape differences between Arabidopsis thaliana, which has simple leaves, and its relative Cardamine hirsuta that has complex leaves divided into leaflets. We show that patterned expression of the transcription factor CUP-SHAPED COTYLEDON1 in C. hirsuta (ChCUC1) is a key determinant of leaf shape differences between the two species. Through inducible genetic perturbations, time-lapse imaging of growth, and computational modeling, we find that ChCUC1 provides instructive input into auxin-based leaf margin patterning. This input arises via transcriptional regulation of multiple auxin homeostasis components, including direct activation of WAG kinases that are known to regulate the polarity of PIN-FORMED auxin transporters. Thus, we have uncovered a mechanism that bridges biological scales by linking spatially distributed and species-specific transcription factor expression to cell-level polarity and growth, to shape diverse leaf forms.

Bilevel Optimization for Traffic Mitigation in Optimal Transport Networks

Global infrastructure robustness and local transport efficiency are critical requirements for transportation networks. However, since passengers often travel greedily to maximize their own benefit and trigger traffic jams, overall transportation performance can be heavily disrupted. We develop adaptation rules that leverage optimal transport theory to effectively route passengers along their shortest paths while also strategically tuning edge weights to optimize traffic. As a result, we enforce both global and local optimality of transport. We prove the efficacy of our approach on synthetic networks and on real data. Our findings on the international European highways suggest that thoughtfully devised routing schemes might help to lower car-produced carbon emissions.

Computer models of cell polarity establishment in plants

Plant development is a complex task, and many processes involve changes in the asymmetric subcellular distribution of cell components that strongly depend on cell polarity. Cell polarity regulates anisotropic growth and polar localization of membrane proteins and helps to identify the cell's position relative to its neighbors within an organ. Cell polarity is critical in a variety of plant developmental processes, including embryogenesis, cell division, and response to external stimuli. The most conspicuous downstream effect of cell polarity is the polar transport of the phytohormone auxin, which is the only known hormone transported in a polar fashion in and out of cells by specialized exporters and importers. The biological processes behind the establishment of cell polarity are still unknown, and researchers have proposed several models that have been tested using computer simulations. The evolution of computer models has progressed in tandem with scientific discoveries, which have highlighted the importance of genetic, chemical, and mechanical input in determining cell polarity and regulating polarity-dependent processes such as anisotropic growth, protein subcellular localization, and the development of organ shapes. The purpose of this review is to provide a comprehensive overview of the current understanding of computer models of cell polarity establishment in plants, focusing on the molecular and cellular mechanisms, the proteins involved, and the current state of the field.

Axes and polarities in leaf vein formation

For multicellular organisms to develop, cells must grow, divide, and differentiate along preferential or exclusive orientations or directions. Moreover, those orientations, or axes, and directions, or polarities, must be coordinated between cells within and between tissues. Therefore, how axes and polarities are coordinated between cells is a key question in biology. In animals, such coordination mainly depends on cell migration and direct interaction between proteins protruding from the plasma membrane. Both cell movements and direct cell-cell interactions are prevented in plants by cell walls that surround plant cells and keep them apart and in place. Therefore, plants have evolved unique mechanisms to coordinate their cell axes and polarities. Here I will discuss evidence suggesting that understanding how leaf veins form may uncover those unique mechanisms. Indeed, unlike previously thought, the cell-to-cell polar transport of the plant hormone auxin along developing veins cannot account for many features of vein patterning. Instead, those features can be accounted for by models of vein patterning that combine polar auxin transport with auxin diffusion through plasmodesmata along the axis of developing veins. Though it remains unclear whether such a combination of polar transport and axial diffusion of auxin can account for the formation of the variety of vein patterns found in plant leaves, evidence suggests that such a combined mechanism may control plant developmental processes beyond vein patterning.

DEFECTIVELY ORGANIZED TRIBUTARIES 5 is not required for leaf venation patterning in Arabidopsis thaliana

The search for genetic regulators of leaf venation patterning started over 30 years ago, primarily focused on mutant screens in the eudicotyledon Arabidopsis thaliana. Developmental perturbations in either cotyledons or true leaves led to the identification of transcription factors required to elaborate the characteristic reticulated vein network. An ortholog of one of these, the C2H2 zinc finger protein DEFECTIVELY ORGANIZED TRIBUTARIES 5 (AtDOT5), was recently identified through transcriptomics as a candidate regulator of parallel venation in maize (Zea mays) leaves. To elucidate how AtDOT5 regulates vein patterning, we generated three independent loss-of-function mutations by gene editing in Arabidopsis. Surprisingly, none of them exhibited any obvious phenotypic perturbations. To reconcile our findings with earlier reports, we re-evaluated the original Atdot5-1 and Atdot5-2 alleles. By genome sequencing, we show that reported mutations at the Atdot5-1 locus are actually polymorphisms between Landsberg erecta and Columbia ecotypes, and that other mutations present in the background most likely cause the pleiotropic mutant phenotype observed. We further show that a T-DNA insertion in the Atdot5-2 locus has no impact on leaf venation patterns when segregated from other T-DNA insertions present in the original line. We thus conclude that AtDOT5 plays no role in leaf venation patterning in Arabidopsis.

Scaling relations for auxin waves

We analyze an 'up-the-gradient' model for the formation of transport channels of the phytohormone auxin, through auxin-mediated polarization of the PIN1 auxin transporter. We show that this model admits a family of travelling wave solutions that is parameterized by the height of the auxin-pulse. We uncover scaling relations for the speed and width of these waves and verify these rigorous results with numerical computations. In addition, we provide explicit expressions for the leading-order wave profiles, which allows the influence of the biological parameters in the problem to be readily identified. Our proofs are based on a generalization of the scaling principle developed by Friesecke and Pego to construct pulse solutions to the classic Fermi-Pasta-Ulam-Tsingou model, which describes a one-dimensional chain of coupled nonlinear springs.

Developmental regulation of leaf venation patterns: monocot versus eudicots and the role of auxin

Organisation and patterning of the vascular network in land plants varies in different taxonomic, developmental and environmental contexts. In leaves, the degree of vascular strand connectivity influences both light and CO₂ harvesting capabilities as well as hydraulic capacity. As such, developmental mechanisms that regulate leaf venation patterning have a direct impact on physiological performance. Development of the leaf venation network requires the specification of procambial cells within the ground meristem of the primordium and subsequent proliferation and differentiation of the procambial lineage to form vascular strands. An understanding of how diverse venation patterns are manifest therefore requires mechanistic insight into how procambium is dynamically specified in a growing leaf. A role for auxin in this process was identified many years ago, but questions remain. In this review we first provide an overview of the diverse venation patterns that exist in land plants, providing an evolutionary perspective. We then focus on the developmental regulation of leaf venation patterns in angiosperms, comparing patterning in eudicots and monocots, and the role of auxin in each case. Although common themes emerge, we conclude that the developmental mechanisms elucidated in eudicots are unlikely to fully explain how parallel venation patterns in monocot leaves are elaborated.

Computational Models of Auxin-Driven Patterning in Shoots

Auxin regulates many aspects of plant development and behavior, including the initiation of new outgrowth, patterning of vascular systems, control of branching, and responses to the environment. Computational models have complemented experimental studies of these processes. We review these models from two perspectives. First, we consider cellular and tissue-level models of interaction between auxin and its transporters in shoots. These models form a coherent body of results exploring different hypotheses pertinent to the patterning of new outgrowth and vascular strands. Second, we consider models operating at the level of plant organs and entire plants. We highlight techniques used to reduce the complexity of these models, which provide a path to capturing the essence of studied phenomena while running simulations efficiently.

A coupled mechano-biochemical model for cell polarity guided anisotropic root growth

Plants develop new organs to adjust their bodies to dynamic changes in the environment. How independent organs achieve anisotropic shapes and polarities is poorly understood. To address this question, we constructed a mechano-biochemical model for Arabidopsis root meristem growth that integrates biologically plausible principles. Computer model simulations demonstrate how differential growth of neighboring tissues results in the initial symmetry-breaking leading to anisotropic root growth. Furthermore, the root growth feeds back on a polar transport network of the growth regulator auxin. Model, predictions are in close agreement with in vivo patterns of anisotropic growth, auxin distribution, and cell polarity, as well as several root phenotypes caused by chemical, mechanical, or genetic perturbations. Our study demonstrates that the combination of tissue mechanics and polar auxin transport organizes anisotropic root growth and cell polarities during organ outgrowth. Therefore, a mobile auxin signal transported through immobile cells drives polarity and growth mechanics to coordinate complex organ development.

Of Cells, Strands, and Networks: Auxin and the Patterned Formation of the Vascular System

Throughout plant development, vascular cells continually form from within a population of seemingly equivalent cells. Vascular cells connect end to end to form continuous strands, and vascular strands connect at both or either end to form networks of exquisite complexity and mesmerizing beauty. Here we argue that experimental evidence gained over the past few decades implicates the plant hormone auxin-its production, transport, perception, and response-in all the steps that lead to the patterned formation of the plant vascular system, from the formation of vascular cells to their connection into vascular networks. We emphasize the organizing principles of the cell- and tissue-patterning process, rather than its molecular subtleties. In the picture that emerges, cells compete for an auxin-dependent, cell-polarizing signal; positive feedback between cell polarization and cell-to-cell movement of the polarizing signal leads to gradual selection of cell files; and selected cell files differentiate into vascular strands that drain the polarizing signal from the neighboring cells. Although the logic of the patterning process has become increasingly clear, the molecular details remain blurry; the future challenge will be to bring them into razor-sharp focus.

Age, Wound Size, and Position of Injury - Dependent Vascular Regeneration Assay in Growing Leaves

Recurring damage to the aerial organs of plants necessitates their prompt repair, particularly their vasculature. While vascular regeneration assays for aerial plant parts such as the stem and inflorescence stalk are well established, those for leaf vasculature remain unexplored. Recently, we established a new vascular regeneration assay in growing leaves and discovered the underlying molecular mechanism. Here, we describe the detailed stepwise method for the incision and regeneration assay used to study leaf vascular regeneration. By using a combination of micro-surgical perturbations, brightfield microscopy, and other experimental approaches, we further show that the age of the leaf as well as the position and size of the injury determine the overall success rate of regeneration. This easy-to-master vascular regeneration assay is an efficient and rapid method to study the mechanism of vascular regeneration in growing leaves. The assay can be readily combined with cellular and molecular biology techniques.

Auxin biosynthesis and cellular efflux act together to regulate leaf vein patterning

Our current understanding of vein development in leaves is based on canalization of the plant hormone auxin into self-reinforcing streams which determine the sites of vascular cell differentiation. By comparison, how auxin biosynthesis affects leaf vein patterning is less well understood. Here, after observing that inhibiting polar auxin transport rescues the sparse leaf vein phenotype in auxin biosynthesis mutants, we propose that the processes of auxin biosynthesis and cellular auxin efflux work in concert during vein development. By using computational modeling, we show that localized auxin maxima are able to interact with mechanical forces generated by the morphological constraints which are imposed during early primordium development. This interaction is able to explain four fundamental characteristics of midvein morphology in a growing leaf: (i) distal cell division; (ii) coordinated cell elongation; (iii) a midvein positioned in the center of the primordium; and (iv) a midvein which is distally branched. Domains of auxin biosynthetic enzyme expression are not positioned by auxin canalization, as they are observed before auxin efflux proteins polarize. This suggests that the site-specific accumulation of auxin, as regulated by the balanced action of cellular auxin efflux and local auxin biosynthesis, is crucial for leaf vein formation.

Shaping leaf vein pattern by auxin and mechanical feedback

This article comments on: Kneuper I, Teale W, Dawson JE, Tsugeki R, Katifori E, Palme K, Ditengou FA. 2021. Auxin biosynthesis and cellular efflux act together to regulate leaf vein patterning. Journal of Experimental Botany 72, 1151–1165.

The canalization hypothesis - challenges and alternatives

The 'canalization hypothesis' was suggested 50 years ago by Tsvi Sachs to account for the formation of vascular strands in response to wounding or auxin application. The hypothesis proposes that positive feedback between auxin movement through a cell and the cell's auxin conductivity leads to the gradual selection of narrow 'canals' of polar auxin transport that will differentiate into vascular strands. Though the hypothesis has provided an invaluable conceptual framework to understand the patterned formation of vascular strands, evidence has been accumulating that seems to be incompatible with the hypothesis. We suggest that the challenging evidence is incompatible with current interpretations of the hypothesis but not with the concept at the core of the hypothesis' original formulation.

Strigolactones inhibit auxin feedback on PIN-dependent auxin transport canalization

Directional transport of the phytohormone auxin is a versatile, plant-specific mechanism regulating many aspects of plant development. The recently identified plant hormones, strigolactones (SLs), are implicated in many plant traits; among others, they modify the phenotypic output of PIN-FORMED (PIN) auxin transporters for fine-tuning of growth and developmental responses. Here, we show in pea and Arabidopsis that SLs target processes dependent on the canalization of auxin flow, which involves auxin feedback on PIN subcellular distribution. D14 receptor- and MAX2 F-box-mediated SL signaling inhibits the formation of auxin-conducting channels after wounding or from artificial auxin sources, during vasculature de novo formation and regeneration. At the cellular level, SLs interfere with auxin effects on PIN polar targeting, constitutive PIN trafficking as well as clathrin-mediated endocytosis. Our results identify a non-transcriptional mechanism of SL action, uncoupling auxin feedback on PIN polarity and trafficking, thereby regulating vascular tissue formation and regeneration.

Vein Distribution on the Deformation Behavior and Fracture Mechanisms of Typical Plant Leaves by Quasi In Situ Tensile Test under a Digital Microscope

Angiosperm leaf venation is based on two major patterns, typically dicotyledonous branching and monocotyledonous parallel veins. The influence of these patterns on deformation and fracture properties is poorly understood. In this paper, three species of dicotyledons with netted venation and three species of monocots with parallel venation were selected, and the effect of vein distribution of leaves on their mechanical properties and deformation behavior was investigated. Whole images of leaves were captured using a digital camera, and their vein traits were measured using the image processing software Digimizer. A self-developed mechanical testing apparatus with high precision and low load was used to measure the tensile properties of leaves. The deformation behavior of the leaf was captured using a digital microscope during the tensile test. Results showed that the vein architecture of monocots and dicots is different, which had a remarkable effect on their mechanical properties, deformation behavior, and crack propagation behavior. The greater the diameter and the more the number of veins parallel to the tensile direction, the higher the tensile force, tensile strength, and elastic modulus of the leaves. The netted venation leaves evinced the elastic-plastic fracture type, and the hierarchy venation provided resistance to fracture propagation of cracks in the leaves by lengthening the crack path. The leaves with parallel venation behaved in a predominantly brittle manner or elastic fracture type, and the parallel venation inhibited the initiation of cracks in the leaves by increasing the load at complete fracture of the leaves. The investigation provides reference for a stiffened plate/shell structure and bionic anticrack design.

Phenotypes of Vascular Flow Networks

Complex distribution networks are pervasive in biology. Examples include nutrient transport in the slime mold Physarum polycephalum as well as mammalian and plant venation. Adaptive rules are believed to guide development of these networks and lead to a reticulate, hierarchically nested topology that is both efficient and resilient against perturbations. However, as of yet, no mechanism is known that can generate such networks on all scales. We show how hierarchically organized reticulation can be constructed and maintained through spatially correlated load fluctuations on a particular length scale. We demonstrate that the network topologies generated represent a trade-off between optimizing transport efficiency, construction cost, and damage robustness and identify the Pareto-efficient front that evolution is expected to favor and select for. We show that the typical fluctuation length scale controls the position of the networks on the Pareto front and thus on the spectrum of venation phenotypes.

CRISPR/Cas9-Mediated Multiplex Genome Editing of JAGGED Gene in Brassica napus L

Pod shattering resistance is an essential component to achieving a high yield, which is a substantial objective in polyploid rapeseed cultivation. Previous studies have suggested that the Arabidopsis JAGGED (JAG) gene is a key factor implicated in the regulatory web of dehiscence fruit. However, its role in controlling pod shattering resistance in oilseed rape is still unknown. In this study, multiplex genome editing was carried out by the CRISPR/Cas9 system on five homoeologs (BnJAG.A02, BnJAG.C02, BnJAG.C06, BnJAG.A07, and BnJAG.A08) of the JAG gene. Knockout mutagenesis of all homoeologs drastically affected the development of the lateral organs in organizing pod shape and size. The cylindrical body of the pod comprised a number of undifferentiated cells like a callus, without distinctive valves, replum, septum, and valve margins. Pseudoseeds were produced, which were divided into two halves with an incomplete layer of cells (probably septum) that separated the undifferentiated cells. These mutants were not capable of generating any productive seeds for further generations. However, one mutant line was identified in which only a BnJAG.A08-NUB-Like paralog of the JAG gene was mutated. Knockout mutagenesis in BnJAG.A08-NUB gene caused significant changes in the pod dehiscence zone. The replum region of the mutant was increased to a great extent, resulting in enlarged cell size, bumpy fruit, and reduced length compared with the wild type. A higher replum-valve joint area may have increased the resistance to pod shattering by ~2-fold in JAG mutants compared with wild type. Our results offer a basis for understanding variations in Brassica napus fruit by mutating JAG genes and providing a way forward for other Brassicaceae species.

Auxin transport model for leaf venation

The plant hormone auxin controls many aspects of the development of plants. One striking dynamical feature is the self-organization of leaf venation patterns which is driven by high levels of auxin within vein cells. The auxin transport is mediated by specialized membrane-localized proteins. Many venation models have been based on polarly localized efflux-mediator proteins of the PIN family. Here, we investigate a modelling framework for auxin transport with a positive feedback between auxin fluxes and transport capacities that are not necessarily polar, i.e. directional across a cell wall. Our approach is derived from a discrete graph-based model for biological transportation networks, where cells are represented by graph nodes and intercellular membranes by edges. The edges are not a priori oriented and the direction of auxin flow is determined by its concentration gradient along the edge. We prove global existence of solutions to the model and the validity of Murray's Law for its steady states. Moreover, we demonstrate with numerical simulations that the model is able connect an auxin source-sink pair with a mid-vein and that it can also produce branching vein patterns. A significant innovative aspect of our approach is that it allows the passage to a formal macroscopic limit which can be extended to include network growth. We perform mathematical analysis of the macroscopic formulation, showing the global existence of weak solutions for an appropriate parameter range.

Formal description of plant morphogenesis

Plant morphogenesis may be characterized by complex feedback mechanisms between signals specifying growth and by the growth of the plant body itself. Comprehension of such feedback mechanisms is an ongoing research task and can be aided with formal descriptions of morphogenesis. In this review, we present a number of established mathematical paradigms that are useful to the formal representation of plant shape, and of biomechanical and biochemical signaling. Specifically, we discuss work from a range of research areas including plant biology, material sciences, fluid dynamics, and computer graphics. Treating plants as organized systems of information processing allows us to compare these different mathematical methods in terms of their expressive power of biological hypotheses. This is an attempt to bring together a large number of computational modeling concepts and make them accessible to the analytical as well as empirical student of plant morphogenesis.

Toward a 3D model of phyllotaxis based on a biochemically plausible auxin-transport mechanism

Polar auxin transport lies at the core of many self-organizing phenomena sustaining continuous plant organogenesis. In angiosperms, the shoot apical meristem is a potentially unique system in which the two main modes of auxin-driven patterning—convergence and canalization—co-occur in a coordinated manner and in a fully three-dimensional geometry. In the epidermal layer, convergence points form, from which auxin is canalized towards inner tissue. Each of these two patterning processes has been extensively investigated separately, but the integration of both in the shoot apical meristem remains poorly understood. We present here a first attempt of a three-dimensional model of auxin-driven patterning during phyllotaxis. We base our simulations on a biochemically plausible mechanism of auxin transport proposed by Cieslak et al. (2015) which generates both convergence and canalization patterns. We are able to reproduce most of the dynamics of PIN1 polarization in the meristem, and we explore how the epidermal and inner cell layers act in concert during phyllotaxis. In addition, we discuss the mechanism by which initiating veins connect to the already existing vascular system.

The regularity of leaf arrangement around stems has long puzzled scientists. The key role played by the plant hormone auxin is now well established. On the surface of the tissue responsible for leaf formation, auxin accumulates at several points, from which new leaves eventually emerge. Auxin also guides the progression of new veins from the nascent leaves to the vascular system of the plant. Models of auxin transport have been developed to explain either auxin accumulation or auxin-driven venation. We propose the first three-dimensional model embracing both phenomena using a unifying mechanism of auxin transport. This integrative approach allows an assessment of our present knowledge on how auxin contributes to the early development of leaves. Our model reproduces many observations of auxin dynamics. It highlights how the inner and epidermal tissues act together to position new leaves. We also show that an additional, yet unknown, mechanism is required to attract new developing veins towards the main vasculature of the plant.

An Update on the Signals Controlling Shoot Branching

Many new questions on the regulation of shoot branching have been raised in recent years, prompting a review and reassessment of the role of each signal involved. Sugars and their signaling networks have been attributed a major role in the early events of axillary bud outgrowth, whereas cytokinin appears to play a critical role in the modulation of this process in response to the environment. Perception of the recently discovered hormone strigolactone is now quite well understood, while the downstream targets remain largely unknown. Recent literature has highlighted that auxin export from a bud is important for its subsequent growth.

The arrangement of lateral veins along the midvein of leaves is not related to leaf phyllotaxis

Positions of leaves along a stem usually adhere to a genetically determined, species-specific pattern known as a leaf phyllotaxis. We investigated whether the arrangement of lateral secondary veins along primary midveins adhered to a species-specific pattern that resembled an alternate or opposite phyllotaxis. We analyzed the venation of temperate dicotyledonous species from different taxonomic groups and chose 18 woody and 12 herbaceous species that have reticulated leaf venation. The arrangement of the lateral veins was neither alternate nor opposite for any of the species. Lateral vein arrangements were instead mixtures of symmetric and asymmetric patterns. Our results show that lateral vein arrangements are related neither to stem-level leaf phyllotaxis (alternate vs. opposite) nor to life form (woody vs. herbaceous). Our results are therefore generally consistent with the canalization hypothesis that the locations of lateral veins are not completely specified genetically prior to leaf formation.

Auxin-mediated regulation of vascular patterning in Arabidopsis thaliana leaves

The vascular system develops in response to auxin flow as continuous strands of conducting tissues arranged in regular spatial patterns. However, a mechanism governing their regular and repetitive formation remains to be fully elucidated. A model system for studying the vascular pattern formation is the process of leaf vascularization in Arabidopsis. In this paper, we present current knowledge of important factors and their interactions in this process. Additionally, we propose the sequence of events leading to the emergence of continuous vascular strands and point to significant problems that need to be resolved in the future to gain a better understanding of the regulation of the vascular pattern development.

Ectopic BASL Reveals Tissue Cell Polarity throughout Leaf Development in Arabidopsis thaliana

Tissue-wide polarity fields, in which cell polarity is coordinated across the tissue, have been described for planar organs such as the Drosophila wing and are considered important for coordinating growth and differentiation [1]. In planar plant organs, such as leaves, polarity fields have been identified for subgroups of cells, such as stomatal lineages [2], trichomes [3, 4], serrations [5], or early developmental stages [6]. Here, we show that ectopic induction of the stomatal protein BASL (BREAKING OF ASYMMETRY IN THE STOMATAL LINEAGE) reveals a tissue-wide epidermal polarity field in leaves throughout development. Ectopic GFP-BASL is typically localized toward the proximal end of cells and to one lobe of mature pavement cells, revealing a polarity field that aligns with the proximodistal axis of the leaf (base to tip). The polarity field is largely parallel to the midline of the leaf but diverges in more lateral positions, particularly at later stages in development, suggesting it may be deformed during growth. The polarity field is observed in the speechless mutant, showing that it is independent of stomatal lineages, and is observed in isotropic cells, showing that cell shape anisotropy is not required for orienting polarity. Ectopic BASL forms convergence and divergence points at serrations, mirroring epidermal PIN polarity patterns, suggesting a common underlying polarity mechanism. Thus, we show that similar to the situation in animals, planar plant organs have a tissue-wide cell polarity field, and this may provide a general cellular mechanism for guiding growth and differentiation.

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Ectopic expression of BASL in Arabidopsis leaves reveals coordinated polarity

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The ectopic BASL polarity field is independent of the stomatal lineage

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The polarity field reorients around serrations, mirroring PIN1 polarity

Feedback from Lateral Organs Controls Shoot Apical Meristem Growth by Modulating Auxin Transport

Stem cells must balance self-renewal and differentiation; thus, their activities are precisely controlled. In plants, the control circuits that underlie division and differentiation within meristems have been well studied, but those that underlie feedback on meristems from lateral organs remain largely unknown. Here we show that long-distance auxin transport mediates this feedback in a non-cell-autonomous manner. A low-auxin zone is associated with the shoot apical meristem (SAM) organization center, and auxin levels negatively affect SAM size. Using computational model simulations, we show that auxin transport from lateral organs can inhibit auxin transport from the SAM through an auxin transport switch and thus maintain SAM auxin homeostasis and SAM size. Genetic and microsurgical analyses confirmed the model's predictions. In addition, the model explains temporary change in SAM size of yabby mutants. Our study suggests that the canalization-based auxin flux can be widely adapted as a feedback control mechanism in plants.

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.

MdPIN1b encodes a putative auxin efflux carrier and has different expression patterns in BC and M9 apple rootstocks

Lower promoter activity is closely associated with lower MdPIN1b expression in the M9 interstem, which might contribute to the dwarfing effect in apple trees. Apple trees grafted onto dwarfing rootstock Malling 9 (M9) produce dwarfing tree architecture with high yield and widely applying in production. Previously, we have reported that in Malus 'Red Fuji' (RF) trees growing on M9 interstem and Baleng Crab (BC) rootstock, IAA content was relatively higher in bark tissue of M9 interstem than that in scion or rootstock. As IAA polar transportation largely depends on the PIN-FORMED (PIN) auxin efflux carrier. Herein, we identify two putative auxin efflux carrier genes in Malus genus, MdPIN1a and MdPIN1b, which were closely related to the AtPIN1. We found that MdPIN1b was expressed preferentially in BC and M9, and the expression of MdPIN1b was significantly lower in the phloem of M9 interstem than that in the scion and rootstock. The distinct expression of MdPIN1b and IAA content were concentrated in the cambium and adjacent xylem or phloem, and MdPIN1b protein was localized on cell plasma membrane in onion epidermal cells transiently expressing 35S:MdPIN1b-GFP fusion protein. Interestingly, an MdPIN1b mutant allele in the promoter region upstream of M9 exhibited decreased MdPIN1b expression compared to BC. MdPIN1b over-expressing interstem in tobacco exhibited increased polar auxin transport. It is proposed that natural allelic differences decreased promoter activity is closely associated with lower MdPIN1b expression in the M9 interstem, which might limit the basipetal transport of auxin, and in turn might contribute to the dwarfing effect. Taken together, these results reveal allelic variation underlying an important apple rootstock trait, and specifically a novel molecular genetic mechanism underlying dwarfing mechanism.

Self-organizing periodicity in development: organ positioning in plants

Periodic patterns during development often occur spontaneously through a process of self-organization. While reaction-diffusion mechanisms are often invoked, other types of mechanisms that involve cell-cell interactions and mechanical buckling have also been identified. Phyllotaxis, or the positioning of plant organs, has emerged as an excellent model system to study the self-organization of periodic patterns. At the macro scale, the regular spacing of organs on the growing plant shoot gives rise to the typical spiral and whorled arrangements of plant organs found in nature. In turn, this spacing relies on complex patterns of cell polarity that involve feedback between a signaling molecule - the plant hormone auxin - and its polar, cell-to-cell transport. Here, we review recent progress in understanding phyllotaxis and plant cell polarity and highlight the development of new tools that can help address the remaining gaps in our understanding.

Coordination of cell polarity and the patterning of leaf vein networks

During development, the behavior of cells in tissues is coordinated along specific orientations or directions by coordinating the polar localization of components in those cells. The coordination of such cell polarity is perhaps nowhere more spectacular than in developing leaves, where the polarity of hundreds of cells is coordinated in the leaf epidermis and inner tissue to pattern vein networks. Available evidence suggests that the spectacular coordination of cell polarity that patterns vein networks is controlled by auxin transport and levels, and by genes that have been implicated in the polar localization of auxin transporters.

WRKY23 is a component of the transcriptional network mediating auxin feedback on PIN polarity

Auxin is unique among plant hormones due to its directional transport that is mediated by the polarly distributed PIN auxin transporters at the plasma membrane. The canalization hypothesis proposes that the auxin feedback on its polar flow is a crucial, plant-specific mechanism mediating multiple self-organizing developmental processes. Here, we used the auxin effect on the PIN polar localization in Arabidopsis thaliana roots as a proxy for the auxin feedback on the PIN polarity during canalization. We performed microarray experiments to find regulators of this process that act downstream of auxin. We identified genes that were transcriptionally regulated by auxin in an AXR3/IAA17- and ARF7/ARF19-dependent manner. Besides the known components of the PIN polarity, such as PID and PIP5K kinases, a number of potential new regulators were detected, among which the WRKY23 transcription factor, which was characterized in more detail. Gain- and loss-of-function mutants confirmed a role for WRKY23 in mediating the auxin effect on the PIN polarity. Accordingly, processes requiring auxin-mediated PIN polarity rearrangements, such as vascular tissue development during leaf venation, showed a higher WRKY23 expression and required the WRKY23 activity. Our results provide initial insights into the auxin transcriptional network acting upstream of PIN polarization and, potentially, canalization-mediated plant development.

The plant hormone auxin belongs to the major plant-specific developmental regulators. It mediates or modifies almost all aspects of plant life. One of the fascinating features of the auxin action is its directional movement between cells, whose direction can be regulated by auxin signaling itself. This plant-specific feedback regulation has been proposed decades ago and allows for the self-organizing formation of distinct auxin channels shown to be crucial for processes, such as the regular pattern formation of leaf venation, organ formation, and regeneration of plant tissues. Despite the prominent importance of this so called auxin canalization process, the insight into the underlying molecular mechanism is very limited. Here, we identified a number of genes that are transcriptionally regulated and act downstream of the auxin signaling to mediate the auxin feedback on the polarized auxin transport. One of them is the WRKY23 transcription factor that has previously been unsuspected to play a role in this process. Our work provides the first insights into the transcriptional regulation of the auxin canalization and opens multiple avenues to further study this crucial process.

On genes and form

The mechanisms by which organisms acquire their sizes and shapes through growth was a major focus of D'Arcy Thompson's book On Growth and Form. By applying mathematical and physical principles to a range of biological forms, Thompson achieved fresh insights, such as the notion that diverse biological shapes could be related through simple deformations of a coordinate system. However, Thompson considered genetics to lie outside the scope of his work, even though genetics was a growing discipline at the time the book was published. Here, we review how recent advances in cell, developmental, evolutionary and computational biology allow Thompson's ideas to be integrated with genes and the processes they influence to provide a deeper understanding of growth and morphogenesis. We consider how genes interact with subcellular-, cellular- and tissue-level processes in plants to yield patterns of growth that underlie the developmental and evolutionary shape transformations Thompson so eloquently described.

Cloning and characterization of auxin efflux carrier genes EcPIN1a and EcPIN1b from finger millet Eleusine coracana L

Auxin signaling events in plants play important role in developmental regulation as well as gravitropic responses and plays crucial role in the development of root, lateral root and root hairs. The gene that is known to be most important in the development of root, lateral root and root hairs is commonly known as auxin efflux carrier (PIN). Being commonly known as orphan plant, the genome sequence of Eleusine coracana is not known yet, and hence it was very difficult to conduct advanced research in root development in this plant. As PIN gene plays crucial role in root development, to have some advanced study we proposed to clone the PIN genes from E. coracana. We cloned two PIN genes in E. coracana and named them as EcPIN1a and EcPIN1b. The coding sequence (CDS) of EcPIN1a was 1779 bp and EcPIN1b was 1788 bp long that encodes for 593 and 596 amino acids, respectively. In-silico analysis shows the presence of transmembrane domain in EcPIN1a and EcPIN1b protein. Multiple sequence alignment of EcPIN1a and EcPIN1b protein shows the presence of several conserved motifs. Phylogenetic analysis of EcPIN1a and EcPIN1b grouped with the PIN gene of monocot plant Oryza sativa. This shows that EcPIN genes were monocot specific, and closely match with the PIN genes of O. sativa. The transcript analysis of EcPIN1a gene in leaf tissue shows gradual up-regulation from 7th to 28th days of developmental time period while the transcript level was found to be lower in root tissue. The transcript abundance of EcPIN1b was not detected. Gradual up-regulation of EcPIN1a gene in developmental stages signifies its important role in root development in E. coracana.

Computational Modeling of Auxin: A Foundation for Plant Engineering

Since the development of agriculture, humans have relied on the cultivation of plants to satisfy our increasing demand for food, natural products, and other raw materials. As we understand more about plant development, we can better manipulate plants to fulfill our particular needs. Auxins are a class of simple metabolites that coordinate many developmental activities like growth and the appearance of functional structures in plants. Computational modeling of auxin has proven to be an excellent tool in elucidating many mechanisms that underlie these developmental events. Due to the complexity of these mechanisms, current modeling efforts are concerned only with single phenomena focused on narrow spatial and developmental contexts; but a general model of plant development could be assembled by integrating the insights from all of them. In this perspective, we summarize the current collection of auxin-driven computational models, focusing on how they could come together into a single model for plant development. A model of this nature would allow researchers to test hypotheses in silico and yield accurate predictions about the behavior of a plant under a given set of physical and biochemical constraints. It would also provide a solid foundation toward the establishment of plant engineering, a proposed discipline intended to enable the design and production of plants that exhibit an arbitrarily defined set of features.

Integrating Physiology and Architecture in Models of Fruit Expansion

Architectural properties of a fruit, such as its shape, vascular patterns, and skin morphology, play a significant role in determining the distributions of water, carbohydrates, and nutrients inside the fruit. Understanding the impact of these properties on fruit quality is difficult because they develop over time and are highly dependent on both genetic and environmental controls. We present a 3D functional-structural fruit model that can be used to investigate effects of the principle architectural properties on fruit quality. We use a three step modeling pipeline in the OpenAlea platform: (1) creating a 3D volumetric mesh representation of the internal and external fruit structure, (2) generating a complex network of vasculature that is embedded within this mesh, and (3) integrating aspects of the fruit's function, such as water and dry matter transport, with the fruit's structure. We restrict our approach to the phase where fruit growth is mostly due to cell expansion and the fruit has already differentiated into different tissue types. We show how fruit shape affects vascular patterns and, as a consequence, the distribution of sugar/water in tomato fruit. Furthermore, we show that strong interaction between tomato fruit shape and vessel density induces, independently of size, an important and contrasted gradient of water supply from the pedicel to the blossom end of the fruit. We also demonstrate how skin morphology related to microcracking distribution affects the distribution of water and sugars inside nectarine fruit. Our results show that such a generic model permits detailed studies of various, unexplored architectural features affecting fruit quality development.

Global Optimization, Local Adaptation, and the Role of Growth in Distribution Networks

Highly optimized complex transport networks serve crucial functions in many man-made and natural systems such as power grids and plant or animal vasculature. Often, the relevant optimization functional is nonconvex and characterized by many local extrema. In general, finding the global, or nearly global optimum is difficult. In biological systems, it is believed that such an optimal state is slowly achieved through natural selection. However, general coarse grained models for flow networks with local positive feedback rules for the vessel conductivity typically get trapped in low efficiency, local minima. In this work we show how the growth of the underlying tissue, coupled to the dynamical equations for network development, can drive the system to a dramatically improved optimal state. This general model provides a surprisingly simple explanation for the appearance of highly optimized transport networks in biology such as leaf and animal vasculature.

Vascular cambium regeneration and vessel formation in wounded inflorescence stems of Arabidopsis

Synchronized tissue polarization during regeneration or de novo vascular tissue formation is a plant-specific example of intercellular communication and coordinated development. According to the canalization hypothesis, the plant hormone auxin serves as polarizing signal that mediates directional channel formation underlying the spatio-temporal vasculature patterning. A necessary part of canalization is a positive feedback between auxin signaling and polarity of the intercellular auxin flow. The cellular and molecular mechanisms of this process are still poorly understood, not the least, because of a lack of a suitable model system. We show that the main genetic model plant, Arabidopsis (Arabidopsis thaliana) can be used to study the canalization during vascular cambium regeneration and new vasculature formation. We monitored localized auxin responses, directional auxin-transport channels formation, and establishment of new vascular cambium polarity during regenerative processes after stem wounding. The increased auxin response above and around the wound preceded the formation of PIN1 auxin transporter-marked channels from the primarily homogenous tissue and the transient, gradual changes in PIN1 localization preceded the polarity of newly formed vascular tissue. Thus, Arabidopsis is a useful model for studies of coordinated tissue polarization and vasculature formation after wounding allowing for genetic and mechanistic dissection of the canalization hypothesis.

Formation of polarity convergences underlying shoot outgrowths

The development of outgrowths from plant shoots depends on formation of epidermal sites of cell polarity convergence with high intracellular auxin at their centre. A parsimonious model for generation of convergence sites is that cell polarity for the auxin transporter PIN1 orients up auxin gradients, as this spontaneously generates convergent alignments. Here we test predictions of this and other models for the patterns of auxin biosynthesis and import. Live imaging of outgrowths from kanadi1 kanadi2 Arabidopsis mutant leaves shows that they arise by formation of PIN1 convergence sites within a proximodistal polarity field. PIN1 polarities are oriented away from regions of high auxin biosynthesis enzyme expression, and towards regions of high auxin importer expression. Both expression patterns are required for normal outgrowth emergence, and may form part of a common module underlying shoot outgrowths. These findings are more consistent with models that spontaneously generate tandem rather than convergent alignments.

DOI:http://dx.doi.org/10.7554/eLife.18165.001

Plants, unlike animals, are able to grow and develop throughout their lives. New leaves and flowers are made from outgrowths that constantly form at the tip of growing shoots. Groups of cells in the outer layer of the shoot tip arrange a protein called PIN1 so that it is more abundant on the cell surfaces that face towards the centre of the group. PIN1 transports a hormone called auxin out of plant cells and this “convergent” arrangement of PIN1 increases the levels of auxin in cells at the centre of the group, leading to the formation of a new outgrowth. However, it is not clear what causes these cells to position their PIN1 proteins in this way.

Several hypotheses have been proposed to explain how convergent patterns of PIN1 form. For example, according to the “up-the-gradient” hypothesis, PIN1 is allocated to the end of a cell that is next to a cell with a higher level of auxin. Abley et al. have now compared predictions from computer models with new experimental data from a plant called Arabidopsis to evaluate three hypotheses for how convergent PIN1 patterns form. A computer model based on the up-the-gradient hypothesis naturally creates convergent PIN1 patterns, even if each cell starts off with the same level of auxin. On the other hand, models based on two other hypotheses generate tandem alignments of PIN1 so that auxin is transported in the same direction along lines of cells.

Next, Abley et al. tested these models using mutant Arabidopsis plants that develop outgrowths from the lower surface of their leaves. These outgrowths form in a similar way to outgrowths at the growing shoot tip, but in a simpler context. The experiments show that the patterns of where auxin is produced in growing leaves were more compatible with the tandem alignment models than the up-the-gradient model. This suggests that plants use a tandem alignment mechanism to form convergences of PIN1 proteins that generate the local increases in auxin needed to make new outgrowths.

This study only examined a single layer of cells on the plant surface. Other cell layers also show highly organised patterns of PIN1 proteins, so a future challenge is to extend the approach to study the entire 3D structure of new shoot outgrowths.

DOI:http://dx.doi.org/10.7554/eLife.18165.002

Phyllotaxis: from patterns of organogenesis at the meristem to shoot architecture

The primary architecture of the aerial part of plants is controlled by the shoot apical meristem, a specialized tissue containing a stem cell niche. The iterative generation of new aerial organs, (leaves, lateral inflorescences, and flowers) at the meristem follows regular patterns, called phyllotaxis. Phyllotaxis has long been proposed to self-organize from the combined action of growth and of inhibitory fields blocking organogenesis in the vicinity of existing organs in the meristem. In this review, we will highlight how a combination of mathematical/computational modeling and experimental biology has demonstrated that the spatiotemporal distribution of the plant hormone auxin controls both organogenesis and the establishment of inhibitory fields. We will discuss recent advances showing that auxin likely acts through a combination of biochemical and mechanical regulatory mechanisms that control not only the pattern of organogenesis in the meristem but also postmeristematic growth, to shape the shoot. WIREs Dev Biol 2016, 5:460-473. doi: 10.1002/wdev.231 For further resources related to this article, please visit the WIREs website.

Mechanical Stress Induces Remodeling of Vascular Networks in Growing Leaves

Differentiation into well-defined patterns and tissue growth are recognized as key processes in organismal development. However, it is unclear whether patterns are passively, homogeneously dilated by growth or whether they remodel during tissue expansion. Leaf vascular networks are well-fitted to investigate this issue, since leaves are approximately two-dimensional and grow manyfold in size. Here we study experimentally and computationally how vein patterns affect growth. We first model the growing vasculature as a network of viscoelastic rods and consider its response to external mechanical stress. We use the so-called texture tensor to quantify the local network geometry and reveal that growth is heterogeneous, resembling non-affine deformations in composite materials. We then apply mechanical forces to growing leaves after veins have differentiated, which respond by anisotropic growth and reorientation of the network in the direction of external stress. External mechanical stress appears to make growth more homogeneous, in contrast with the model with viscoelastic rods. However, we reconcile the model with experimental data by incorporating randomness in rod thickness and a threshold in the rod growth law, making the rods viscoelastoplastic. Altogether, we show that the higher stiffness of veins leads to their reorientation along external forces, along with a reduction in growth heterogeneity. This process may lead to the reinforcement of leaves against mechanical stress. More generally, our work contributes to a framework whereby growth and patterns are coordinated through the differences in mechanical properties between cell types.

Localized auxin peaks in concentration-based transport models of the shoot apical meristem

We study the formation of auxin peaks in a generic class of concentration-based auxin transport models, posed on static plant tissues. Using standard asymptotic analysis, we prove that, on bounded domains, auxin peaks are not formed via a Turing instability in the active transport parameter, but via simple corrections to the homogeneous steady state. When the active transport is small, the geometry of the tissue encodes the peaks' amplitude and location: peaks arise where cells have fewer neighbours, that is, at the boundary of the domain. We test our theory and perform numerical bifurcation analysis on two models that are known to generate auxin patterns for biologically plausible parameter values. In the same parameter regimes, we find that realistic tissues are capable of generating a multitude of stationary patterns, with a variable number of auxin peaks, that can be selected by different initial conditions or by quasi-static changes in the active transport parameter. The competition between active transport and production rate determines whether peaks remain localized or cover the entire domain. In particular, changes in the auxin production that are fast with respect to the cellular life cycle affect the auxin peak distribution, switching from localized spots to fully patterned states. We relate the occurrence of localized patterns to a snaking bifurcation structure, which is known to arise in a wide variety of nonlinear media, but has not yet been reported in plant models.

Quantifying morphogenesis in plants in 4D

Plant development occurs in 3D space over time (4D). Recent advances in image acquisition and computational analysis are now enabling development to be visualized and quantified in its entirety at the cellular level. The simultaneous quantification of reporter abundance and 3D cell shape change enables links between signaling processes and organ morphogenesis to be accomplished organ-wide and at single cell resolution. Current work to integrate this quantitative 3D image data with computational models is enabling causal relationships between gene expression and organ morphogenesis to be uncovered. Further technical advances in imaging and image analysis will enable this approach to be applied to a greater diversity of plant organs and will become a key tool to address many questions in plant development.

Control of vein network topology by auxin transport

Background

Tissue networks such as the vascular networks of plant and animal organs transport signals and nutrients in most multicellular organisms. The transport function of tissue networks depends on topological features such as the number of networks’ components and the components’ connectedness; yet what controls tissue network topology is largely unknown, partly because of the difficulties in quantifying the effects of genes on tissue network topology. We address this problem for the vein networks of plant leaves by introducing biologically motivated descriptors of vein network topology; we combine these descriptors with cellular imaging and molecular genetic analysis; and we apply this combination of approaches to leaves of Arabidopsis thaliana that lack function of, overexpress or misexpress combinations of four PIN-FORMED (PIN) genes—PIN1, PIN5, PIN6, and PIN8—which encode transporters of the plant signal auxin and are known to control vein network geometry.

Results

We find that PIN1 inhibits vein formation and connection, and that PIN6 acts redundantly to PIN1 in these processes; however, the functions of PIN6 in vein formation are nonhomologous to those of PIN1, while the functions of PIN6 in vein connection are homologous to those of PIN1. We further find that PIN8 provides functions redundant and homologous to those of PIN6 in PIN1-dependent inhibition of vein formation, but that PIN8 has no functions in PIN1/PIN6-dependent inhibition of vein connection. Finally, we find that PIN5 promotes vein formation; that all the vein-formation-promoting functions of PIN5 are redundantly inhibited by PIN6 and PIN8; and that these functions of PIN5, PIN6, and PIN8 are independent of PIN1.

Conclusions

Our results suggest that PIN-mediated auxin transport controls the formation of veins and their connection into networks.

Electronic supplementary material

The online version of this article (doi:10.1186/s12915-015-0208-3) contains supplementary material, which is available to authorized users.

Auxin-driven patterning with unidirectional fluxes

The plant hormone auxin plays an essential role in the patterning of plant structures. Biological hypotheses supported by computational models suggest that auxin may fulfil this role by regulating its own transport, but the plausibility of previously proposed models has been questioned. We applied the notion of unidirectional fluxes and the formalism of Petri nets to show that the key modes of auxin-driven patterning-the formation of convergence points and the formation of canals-can be implemented by biochemically plausible networks, with the fluxes measured by dedicated tally molecules or by efflux and influx carriers themselves. Common elements of these networks include a positive feedback of auxin efflux on the allocation of membrane-bound auxin efflux carriers (PIN proteins), and a modulation of this allocation by auxin in the extracellular space. Auxin concentration in the extracellular space is the only information exchanged by the cells. Canalization patterns are produced when auxin efflux and influx act antagonistically: an increase in auxin influx or concentration in the extracellular space decreases the abundance of efflux carriers in the adjacent segment of the membrane. In contrast, convergence points emerge in networks in which auxin efflux and influx act synergistically. A change in a single reaction rate may result in a dynamic switch between these modes, suggesting plausible molecular implementations of coordinated patterning of organ initials and vascular strands predicted by the dual polarization theory.

Self-organization of plant vascular systems: claims and counter-claims about the flux-based auxin transport model

The plant hormone auxin plays a central role in growth and morphogenesis. In shoot apical meristems, auxin flux is polarized through its interplay with PIN proteins. Concentration-based mathematical models of the flux can explain some aspects of phyllotaxis for the L1 surface layer, where auxin accumulation points act as sinks and develop into primordia. The picture differs in the interior of the meristem, where the primordia act as auxin sources, leading to the initiation of the vascular system. Self-organization of the auxin flux involves large numbers of molecules and is difficult to treat by intuitive reasoning alone; mathematical models are therefore vital to understand these phenomena. We consider a leading computational model based on the so-called flux hypothesis. This model has been criticized and extended in various ways. One of the basic counter-arguments is that simulations yield auxin concentrations inside canals that are lower than those seen experimentally. Contrary to what is claimed in the literature, we show that the model can lead to higher concentrations within canals for significant parameter regimes. We then study the model in the usual case where the response function Φ defining the model is quadratic and unbounded, and show that the steady state vascular patterns are formed of loopless directed trees. Moreover, we show that PIN concentrations can diverge in finite time, thus explaining why previous simulation studies introduced cut-offs which force the system to have bounded PIN concentrations. Hence, contrary to previous claims, extreme PIN concentrations are not due to numerical problems but are intrinsic to the model. On the other hand, we show that PIN concentrations remain bounded for bounded Φ, and simulations show that in this case, loops can emerge at steady state.

Mathematical study for the mechanism of vascular and spot patterns by auxin and pin dynamics in plant development

Inhomogeneous distribution of auxin is essential in various differentiation processes of plant development. Auxin transfer between cells by efflux carrier protein called PINFORMED (PIN) has been considered to be responsible for inhomogeneous distribution of auxin. Two major types of auxin distribution patterns are "spot" patterns and "passage" patterns, which are responsible for determining the position of the primordia of a leaf or flower in shoot apical meristem and formation of leaf veins, respectively. In this study, we studied the pattern formation of auxin distribution mediated by polarization of PIN using mathematical methods. We developed several different models which show possible interaction mechanisms between auxin and PIN on 2-dimentional hexagonal cellular lattice, (1) Basic auxin flux model, (2) auxin-dependent PIN degradation model and (3) auxin self-feedback model. We analyzed these models by numerical calculation and mathematical analysis. From intensive numerical calculations under different conditions, we found that some models show three different types of pattern formations in dynamics, (a) homogeneous, (b) passage and (c) spot pattern depending on parameter condition. We analyzed these models mathematically using approximation of 1-dimensional periodic space. We determined the conditions that passage and spot patterns are generated in each model, respectively. After these analyses, we propose possible mechanisms by which plants switch passage and spot patterns in different organs by small modification.