How canalization can make loops: A new model of reticulated leaf vascular pattern formation

Multiscale Asymptotic Analysis Reveals How Cell Growth and Subcellular Compartments Affect Tissue-Scale Hormone Transport

Determining how cell-scale processes lead to tissue-scale patterns is key to understanding how hormones and morphogens are distributed within biological tissues and control developmental processes. In this article, we use multiscale asymptotic analysis to derive a continuum approximation for hormone transport in a long file of cells to determine how subcellular compartments and cell growth and division affect tissue-scale hormone transport. Focusing our study on plant tissues, we begin by presenting a discrete multicellular ODE model tracking the hormone concentration in each cell's cytoplasm, subcellular vacuole, and surrounding apoplast, represented by separate compartments in the cell-file geometry. We allow the cells to grow at a rate that can depend both on space and time, accounting for both cytoplasmic and vacuolar expansion. Multiscale asymptotic analysis enables us to systematically derive the corresponding continuum model, obtaining an effective reaction-advection-diffusion equation and revealing how the effective diffusivity, effective advective velocity, and the effective sink term depend on the parameters in the cell-scale model. The continuum approximation reveals how subcellular compartments, such as vacuoles, can act as storage vessels, that significantly alter the effective properties of hormone transport, such as the effective diffusivity and the induced effective velocity. Furthermore, we show how cell growth and spatial variance across cell lengths affect the effective diffusivity and the induced effective velocity, and how these affect the tissue-scale hormone distribution. In particular, we find that cell growth naturally induces an effective velocity in the direction of growth, whereas spatial variance across cell lengths induces effective velocity due to the presence of an extra compartment, such as the apoplast and the vacuole, and variations in the relative sizes between the compartments across the file of cells. It is revealed that hormone transport is faster across cells of decreasing lengths than cells with increasing lengths. We also investigate the effect of cell division on transport dynamics, assuming that each cell divides as soon as it doubles in size, and find that increasing the time between successive cell divisions decreases the growth rate, which enhances the effect of cell division in slowing hormone transport. Motivated by recent experimental discoveries, we discuss particular applications for transport of gibberellic acid (GA), an important growth hormone, within the Arabidopsis root. The model reveals precisely how membrane proteins that mediate facilitated GA transport affect the effective tissue-scale transport. However, the results are general enough to be relevant to other plant hormones, or other substances that are transported in a similar way in any type of cells.

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.

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.

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.

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.

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.

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.

Chemical control of xylem differentiation by thermospermine, xylemin, and auxin

The xylem conducts water and minerals from the root to the shoot and provides mechanical strength to the plant body. The vascular precursor cells of the procambium differentiate to form continuous vascular strands, from which xylem and phloem cells are generated in the proper spatiotemporal pattern. Procambium formation and xylem differentiation are directed by auxin. In angiosperms, thermospermine, a structural isomer of spermine, suppresses xylem differentiation by limiting auxin signalling. However, the process of auxin-inducible xylem differentiation has not been fully elucidated and remains difficult to manipulate. Here, we found that an antagonist of spermidine can act as an inhibitor of thermospermine biosynthesis and results in excessive xylem differentiation, which is a phenocopy of a thermospermine-deficient mutant acaulis5 in Arabidopsis thaliana. We named this compound xylemin owing to its xylem-inducing effect. Application of a combination of xylemin and thermospermine to wild-type seedlings negates the effect of xylemin, whereas co-treatment with xylemin and a synthetic proauxin, which undergoes hydrolysis to release active auxin, has a synergistic inductive effect on xylem differentiation. Thus, xylemin may serve as a useful transformative chemical tool not only for the study of thermospermine function in various plant species but also for the control of xylem induction and woody biomass production.

Topological Phenotypes Constitute a New Dimension in the Phenotypic Space of Leaf Venation Networks

The leaves of angiosperms contain highly complex venation networks consisting of recursively nested, hierarchically organized loops. We describe a new phenotypic trait of reticulate vascular networks based on the topology of the nested loops. This phenotypic trait encodes information orthogonal to widely used geometric phenotypic traits, and thus constitutes a new dimension in the leaf venation phenotypic space. We apply our metric to a database of 186 leaves and leaflets representing 137 species, predominantly from the Burseraceae family, revealing diverse topological network traits even within this single family. We show that topological information significantly improves identification of leaves from fragments by calculating a “leaf venation fingerprint” from topology and geometry. Further, we present a phenomenological model suggesting that the topological traits can be explained by noise effects unique to specimen during development of each leaf which leave their imprint on the final network. This work opens the path to new quantitative identification techniques for leaves which go beyond simple geometric traits such as vein density and is directly applicable to other planar or sub-planar networks such as blood vessels in the brain.

Planar reticular networks are ubiquitous in nature and engineering, formed for instance by the arterial vasculature in the mammalian neocortex, urban street grids or the vascular network of plant leaves. We use a topological metric to characterize the way loops are nested in such networks and analyze a large database of 186 leaves and leaflets, revealing for the first time that the nesting of the networks’ cycles constitutes a distinct phenotypic trait orthogonal to previously used geometric features. Furthermore, we demonstrate that the information contained in the leaf topology can significantly improve specimen identification from fragments, and provide an empirical growth model that can explain much of the observed data. Our work can improve understanding of the functional significance of the various leaf vein architectures and their correlation with the environment. It can pave the way for similar analyses in diverse areas of research involving reticulate networks.

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.

Modelling hormonal response and development

As our knowledge of the complexity of hormone homeostasis, transport, perception, and response increases, and their outputs become less intuitive, modelling is set to become more important. Initial modelling efforts have focused on hormone transport and response pathways. However, we now need to move beyond the network scales and use multicellular and multiscale modelling approaches to predict emergent properties at different scales. Here we review some examples where such approaches have been successful, for example, auxin-cytokinin crosstalk regulating root vascular development or a study of lateral root emergence where an iterative cycle of modelling and experiments lead to the identification of an overlooked role for PIN3. Finally, we discuss some of the remaining biological and technical challenges.

Canalization-based vein formation in a growing leaf

Vein formation is an important process in plant leaf development. The phytohormone auxin is known as the most important molecule for the control of venation patterning; and the canalization model, in which cells experiencing higher auxin flux differentiate into specific cells for auxin transportation, is widely accepted. To date, several mathematical models based on the canalization hypothesis have been proposed that have succeeded in reproducing vein patterns similar to those observed in actual leaves. However, most previous studies focused on patterning in fixed domains, and, in a few exceptional studies, limited tissue growth - such as cell proliferation at leaf margins and small deformations without large changes in cell number - were dealt with. Considering that, in actual leaf development, venation patterning occurs in an exponentially growing tissue, whether the canalization hypothesis still applies is an important issue to be addressed. In this study, we first show through a pilot simulation that the coupling of chemical dynamics for canalization and tissue growth as independent models cannot reproduce normal venation patterning. We then examine conditions sufficient for achieving normal patterning in a growing leaf by introducing various constraints on chemical dynamics, tissue growth, and cell mechanics; in doing so, we found that auxin flux- or differentiation-dependent modification of the cell cycle and elasticity of cell edges are essential. The predictions given by our simulation study will serve as guideposts in experiments aimed at finding the key factors for achieving normal venation patterning in developing plant leaves.

Polar auxin transport: models and mechanisms

Spatial patterns of the hormone auxin are important drivers of plant development. The observed feedback between the active, directed transport that generates auxin patterns and the auxin distribution that influences transport orientation has rendered this a popular subject for modelling studies. Here we propose a new mathematical framework for the analysis of polar auxin transport and present a detailed mathematical analysis of published models. We show that most models allow for self-organised patterning for similar biological assumptions, and find that the pattern generated is typically unidirectional, unless additional assumptions or mechanisms are incorporated. Our analysis thus suggests that current models cannot explain the bidirectional fountain-type patterns found in plant meristems in a fully self-organised manner, and we discuss future research directions to address the gaps in our understanding of auxin transport mechanisms.

Combined in silico/in vivo analysis of mechanisms providing for root apical meristem self-organization and maintenance

BACKGROUND AND AIMS

The root apical meristem (RAM) is the plant stem cell niche which provides for the formation and continuous development of the root. Auxin is the main regulator of RAM functioning, and auxin maxima coincide with the sites of RAM initiation and maintenance. Auxin gradients are formed due to local auxin biosynthesis and polar auxin transport. The PIN family of auxin transporters plays a critical role in polar auxin transport, and two mechanisms of auxin maximum formation in the RAM based on PIN-mediated auxin transport have been proposed to date: the reverse fountain and the reflected flow mechanisms.

METHODS

The two mechanisms are combined here in in silico studies of auxin distribution in intact roots and roots cut into two pieces in the proximal meristem region. In parallel, corresponding experiments were performed in vivo using DR5::GFP Arabidopsis plants.

KEY RESULTS

The reverse fountain and the reflected flow mechanism naturally cooperate for RAM patterning and maintenance in intact root. Regeneration of the RAM in decapitated roots is provided by the reflected flow mechanism. In the excised root tips local auxin biosynthesis either alone or in cooperation with the reverse fountain enables RAM maintenance.

CONCLUSIONS

The efficiency of a dual-mechanism model in guiding biological experiments on RAM regeneration and maintenance is demonstrated. The model also allows estimation of the concentrations of auxin and PINs in root cells during development and under various treatments. The dual-mechanism model proposed here can be a powerful tool for the study of several different aspects of auxin function in root.

Computational models of plant development and form

The use of computational techniques increasingly permeates developmental biology, from the acquisition, processing and analysis of experimental data to the construction of models of organisms. Specifically, models help to untangle the non-intuitive relations between local morphogenetic processes and global patterns and forms. We survey the modeling techniques and selected models that are designed to elucidate plant development in mechanistic terms, with an emphasis on: the history of mathematical and computational approaches to developmental plant biology; the key objectives and methodological aspects of model construction; the diverse mathematical and computational methods related to plant modeling; and the essence of two classes of models, which approach plant morphogenesis from the geometric and molecular perspectives. In the geometric domain, we review models of cell division patterns, phyllotaxis, the form and vascular patterns of leaves, and branching patterns. In the molecular-level domain, we focus on the currently most extensively developed theme: the role of auxin in plant morphogenesis. The review is addressed to both biologists and computational modelers.

On evaluating models in Computational Morphodynamics

Recent advances in experimental plant biology have led to an increased potential to investigate plant development at a systems level. The emerging research field of Computational Morphodynamics has the aim to lead this development by combining dynamic spatial experimental data with computational models of molecular networks, growth, and mechanics in a multicellular context. The increased number of published models may lead to a diversification of our understanding of the systems, and methods for evaluating, comparing, and sharing models are main challenges for the future. We will discuss this problem using ideas originating from physics and use recent computational models of plant development as examples.

Quantitative analysis of venation patterns of Arabidopsis leaves by supervised image analysis

The study of transgenic Arabidopsis lines with altered vascular patterns has revealed key players in the venation process, but details of the vascularization process are still unclear, partly because most lines have only been assessed qualitatively. Therefore, quantitative analyses are required to identify subtle perturbations in the pattern and to test dynamic modeling hypotheses using biological measurements. We developed an online framework, designated Leaf Image Analysis Interface (LIMANI), in which venation patterns are automatically segmented and measured on dark-field images. Image segmentation may be manually corrected through use of an interactive interface, allowing supervision and rectification steps in the automated image analysis pipeline and ensuring high-fidelity analysis. This online approach is advantageous for the user in terms of installation, software updates, computer load and data storage. The framework was used to study vascular differentiation during leaf development and to analyze the venation pattern in transgenic lines with contrasting cellular and leaf size traits. The results show the evolution of vascular traits during leaf development, suggest a self-organizing mechanism for leaf venation patterning, and reveal a tight balance between the number of end-points and branching points within the leaf vascular network that does not depend on the leaf developmental stage and cellular content, but on the leaf position on the rosette. These findings indicate that development of LIMANI improves understanding of the interaction between vascular patterning and leaf growth.

Multiscale modelling of auxin transport in the plant-root elongation zone

In the root elongation zone of a plant, the hormone auxin moves in a polar manner due to active transport facilitated by spatially distributed influx and efflux carriers present on the cell membranes. To understand how the cell-scale active transport and passive diffusion combine to produce the effective tissue-scale flux, we apply asymptotic methods to a cell-based model of auxin transport to derive systematically a continuum description from the spatially discrete one. Using biologically relevant parameter values, we show how the carriers drive the dominant tissue-scale auxin flux and we predict how the overall auxin dynamics are affected by perturbations to these carriers, for example, in knockout mutants. The analysis shows how the dominant behaviour depends on the cells' lengths, and enables us to assess the relative importance of the diffusive auxin flux through the cell wall. Other distinguished limits are also identified and their potential roles discussed. As well as providing insight into auxin transport, the study illustrates the use of multiscale (cell to tissue) methods in deriving simplified models that retain the essential biology and provide understanding of the underlying dynamics.

Endogenous electromagnetic fields in plant leaves: a new hypothesis for vascular pattern formation

Electromagnetic (EM) phenomena have long been implicated in biological development, but few detailed, practical mechanisms have been put forth to connect electromagnetism with morphogenetic processes. This work describes a new hypothesis for plant leaf veination, whereby an endogenous electric field forming as a result of a coherent Frohlich process, and corresponding to an EM resonant mode of the developing leaf structure, is capable of instigating leaf vascularisation. In order to test the feasibility of this hypothesis, a three-dimensional, EM finite-element model (FEM) of a leaf primordium was constructed to determine if suitable resonant modes were physically possible for geometric and physical parameters similar to those of developing leaf tissue. Using the FEM model, resonant EM modes with patterns of relevance to developing leaf vein modalities were detected. On account of the existence of shared geometric signatures in a leaf's vascular pattern and the electric field component of EM resonant modes supported by a developing leaf structure, further theoretical and experimental investigations are warranted. Significantly, this hypothesis is not limited to leaf vascular patterning, but may be applicable to a variety of morphogenetic phenomena in a number of living systems.

The grass leaf developmental gradient as a platform for a systems understanding of the anatomical specialization of C(4) leaves

C(4) photosynthesis relies on spatial and quantitative specializations of common features of leaf anatomy, including venation pattern, bundle sheath cell and chloroplast differentiation, plasmodesmatal abundance, and secondary cell wall enhancement. It has thus far been challenging to dissect the molecular basis for these C(4)-specific alterations in spatial and quantitative patterns of regulation. The target downstream networks of genes and protein interactions that produce these fundamental anatomical features in both C(4) and C(3) species are poorly understood. The developing leaves of monocot grasses provide a base-to-tip gradient of developmental stages that can provide the platform for comprehensive molecular and anatomical data that can yield a better understanding both of the regulators and the targets that produce C(4) patterns, through a variety of gene discovery and systems analysis strategies.

Quantitative predictions on auxin-induced polar distribution of PIN proteins during vein formation in leaves

The dynamic patterning of the plant hormone auxin and its efflux facilitator the PIN protein are the key regulators for the spatial and temporal organization of plant development. In particular auxin induces the polar localization of its own efflux facilitator. Due to this positive feedback, auxin flow is directed and patterns of auxin and PIN arise. During the earliest stage of vein initiation in leaves auxin accumulates in a single cell in a rim of epidermal cells from which it flows into the ground meristem tissue of the leaf blade. There the localized auxin supply yields the successive polarization of PIN distribution along a strand of cells. We model the auxin and PIN dynamics within cells with a minimal canalization model. Solving the model analytically we uncover an excitable polarization front that triggers a polar distribution of PIN proteins in cells. As polarization fronts may extend to opposing directions from their initiation site, we suggest a possible resolution to the puzzling occurrence of bipolar cells, thus we offer an explanation for the development of closed, looped veins. Employing non-linear analysis, we identify the role of the contributing microscopic processes during polarization. Furthermore, we deduce quantitative predictions on polarization fronts establishing a route to determine the up to now largely unknown kinetic rates of auxin and PIN dynamics.

Modelling polar auxin transport in developmental patterning

Auxin interacts with its own polar transport to influence cell polarity and tissue patterning. Research over the past decade has started to deliver new insights into the molecular mechanisms that drive and regulate polar auxin transport. The most prominent auxin efflux protein, PIN1, has subsequently become a crucial component of auxin transport models because it is now known to direct auxin flow and maintain local auxin gradients. Recent molecular and genetic experiments have allowed the formulation of conceptual models that are able to interpret the role of (i) auxin, (ii) its transport, and (iii) the dynamics of PIN1 in generating temporal and spatial patterns. Here we review the current mathematical models of patterning in two specific developmental contexts: lateral shoot and vein formation, focusing on how these models can help to untangle the details of auxin transport-mediated patterning.

FORKED1 encodes a PH domain protein that is required for PIN1 localization in developing leaf veins

The formation of Arabidopsis leaf veins is believed to require canalization of auxin into discrete and continuous cell files to generate a highly reproducible branched and reticulate pattern. During canalization, incipient veins become preferred routes for auxin transport through expression and asymmetric localization of the PINFORMED1 (PIN1) auxin efflux protein: PIN1 expression narrows from a group of cells to a single cell file, and localization of PIN1 protein becomes polarized to the cell membrane facing a previously formed vein. The shift in PIN1 localization is believed to require active vesicle cycling and be auxin-dependent, generating an autoregulatory loop. Previously, we have shown that fkd1 mutant leaves have an open vein pattern that lacks distal vein meeting. Here, we identify FKD1 as encoding a pleckstrin homology domain- and DUF828-containing protein. A fusion of the FKD1 promoter and the GUS reporter gene was expressed in vascular tissue throughout the plant, and its expression in incipient veins in leaves narrows in a manner similar to that of PIN1. FKD1 expression in roots and leaves can be altered by changes to auxin response and auxin transport. In the absence of FKD1, PIN1::GFP narrowing to incipient veins is delayed, and localization to the apical cell face is infrequent. The lack of apical PIN1 localization correlates with the failure of newly forming veins to connect distally with previously formed veins. Our data suggest that FKD1 influences PIN1 localization in an auxin-dependent manner, and we propose that it represents a key component of the auxin canalization pathway.

A plausible mechanism for auxin patterning along the developing root

Background

In plant roots, auxin is critical for patterning and morphogenesis. It regulates cell elongation and division, the development and maintenance of root apical meristems, and other processes. In Arabidopsis, auxin distribution along the central root axis has several maxima: in the root tip, in the basal meristem and at the shoot/root junction. The distal maximum in the root tip maintains the stem cell niche. Proximal maxima may trigger lateral or adventitious root initiation.

Results

We propose a reflected flow mechanism for the formation of the auxin maximum in the root apical meristem. The mechanism is based on auxin's known activation and inhibition of expressed PIN family auxin carriers at low and high auxin levels, respectively. Simulations showed that these regulatory interactions are sufficient for self-organization of the auxin distribution pattern along the central root axis under varying conditions. The mathematical model was extended with rules for discontinuous cell dynamics so that cell divisions were also governed by auxin, and by another morphogen Division Factor which combines the actions of cytokinin and ethylene on cell division in the root. The positional information specified by the gradients of these two morphogens is able to explain root patterning along the central root axis.

Conclusion

We present here a plausible mechanism for auxin patterning along the developing root, that may provide for self-organization of the distal auxin maximum when the reverse fountain has not yet been formed or has been disrupted. In addition, the proximal maxima are formed under the reflected flow mechanism in response to periods of increasing auxin flow from the growing shoot. These events may predetermine lateral root initiation in a rhyzotactic pattern. Another outcome of the reflected flow mechanism - the predominance of lateral or adventitious roots in different plant species - may be based on the different efficiencies with which auxin inhibits its own transport in different species, thereby distinguishing two main types of plant root architecture: taproot vs. fibrous.

Stochastic and deterministic multiscale models for systems biology: an auxin-transport case study

BACKGROUND

Stochastic and asymptotic methods are powerful tools in developing multiscale systems biology models; however, little has been done in this context to compare the efficacy of these methods. The majority of current systems biology modelling research, including that of auxin transport, uses numerical simulations to study the behaviour of large systems of deterministic ordinary differential equations, with little consideration of alternative modelling frameworks.

RESULTS

In this case study, we solve an auxin-transport model using analytical methods, deterministic numerical simulations and stochastic numerical simulations. Although the three approaches in general predict the same behaviour, the approaches provide different information that we use to gain distinct insights into the modelled biological system. We show in particular that the analytical approach readily provides straightforward mathematical expressions for the concentrations and transport speeds, while the stochastic simulations naturally provide information on the variability of the system.

CONCLUSIONS

Our study provides a constructive comparison which highlights the advantages and disadvantages of each of the considered modelling approaches. This will prove helpful to researchers when weighing up which modelling approach to select. In addition, the paper goes some way to bridging the gap between these approaches, which in the future we hope will lead to integrative hybrid models.

Auxin biosynthesis and its role in plant development

Indole-3-acetic acid (IAA), the main auxin in higher plants, has profound effects on plant growth and development. Both plants and some plant pathogens can produce IAA to modulate plant growth. Although the genes and biochemical reactions for auxin biosynthesis in some plant pathogens are well understood, elucidation of the mechanisms by which plants produce auxin has proven to be difficult. So far, no single complete pathway of de novo auxin biosynthesis in plants has been firmly established. However, recent studies have led to the discoveries of several genes in tryptophan-dependent auxin biosynthesis pathways. Recent findings have also determined that local auxin biosynthesis plays essential roles in many developmental processes including gametogenesis, embryogenesis, seedling growth, vascular patterning, and flower development. In this review, I summarize the recent advances in dissecting auxin biosynthetic pathways and how the understanding of auxin biosynthesis provides a crucial angle for analyzing the mechanisms of plant development.

Computational morphodynamics: a modeling framework to understand plant growth

Computational morphodynamics utilizes computer modeling to understand the development of living organisms over space and time. Results from biological experiments are used to construct accurate and predictive models of growth. These models are then used to make novel predictions that provide further insight into the processes involved, which can be tested experimentally to either confirm or rule out the validity of the computational models. This review highlights two fundamental challenges: (a) to understand the feedback between mechanics of growth and chemical or molecular signaling, and (b) to design models that span and integrate single cell behavior with tissue development. We review different approaches to model plant growth and discuss a variety of model types that can be implemented to demonstrate how the interplay between computational modeling and experimentation can be used to explore the morphodynamics of plant development.

Shifts in leaf vein density through accelerated vein formation in C4 Flaveria (Asteraceae)

BACKGROUND AND AIMS

Leaf venation in many C(4) species is characterized by high vein density, essential in facilitating rapid intercellular diffusion of C(4) photosynthetic metabolites between different tissues (mesophyll, bundle sheath). Greater vein density has been hypothesized to be an early step in C(4) photosynthesis evolution. Development of C(4) vein patterning is thought to occur from either accelerated or prolonged procambium formation, relative to ground tissue development.

METHODS

Cleared and sectioned tissues of phylogenetically basal C(3) Flaveria robusta and more derived C(4) Flaveria bidentis were compared for vein pattern in mature leaves and vein pattern formation in developing leaves.

KEY RESULTS

In mature leaves, major vein density did not differ between C(3) and C(4) Flaveria species, whereas minor veins were denser in C(4) species than in C(3) species. The developmental study showed that both major and minor vein patterning in leaves of C(3) and C(4) species were initiated at comparable stages (based on leaf length). An additional vein order in the C(4) species was observed during initiation of the higher order minor veins compared with the C(3) species. In the two species, expansion of bundle sheath and mesophyll cells occurred after vein pattern was complete and xylem differentiation was continuous in minor veins. In addition, mesophyll cells ceased dividing sooner and enlarged less in C(4) species than in C(3) species.

CONCLUSIONS

Leaf vein pattern characteristic to C(4) Flaveria was achieved primarily through accelerated and earlier offset of higher order vein formation, rather than other modifications in the timing of vein pattern formation, as compared with C(3) species. Earlier cessation of mesophyll cell division and reduced expansion also contributed to greater vein density in the C(4) species. The relatively late expansion of bundle sheath and mesophyll cells shows that vein patterning precedes ground tissue development in C(4) species.

Auxin transport-feedback models of patterning in plants

Many patterning events in plants are regulated by the phytohormone auxin. In fact, so many things are under the influence of auxin that it seems difficult to understand how a single hormone can do so much. Auxin moves throughout the plant via a network of specialized membrane-bound import and export proteins, which are often differentially expressed and polarized depending on tissue type. Here, we review simulation models of pattern formation that are based on the control of these transporters by auxin itself. In these transport-feedback models, diversity in patterning comes not from the addition of more morphogens, but rather by varying the mechanism that regulates the transporters.

Describing long-range patterns in leaf vasculature by metaphoric fields

Some patterns in dicotyledonous leaf vasculature depict rather precise, long-range structural features. This work identifies and quantifies these previously unrecognized features in terms of an empirically derived mathematical formalism that generates wave-like spatial patterns referred to as metaphoric fields. These patterns were used to specify regularities in the long-range structure of dicot leaf vasculature, and were found to account significantly for the predominant features of all 27 dicot species studied. The conserved features of these metaphoric fields are discussed in terms of existing models for leaf pattern formation based on efflux-protein mediated auxin transport in a developing cellular field. This work highlights the complex, regular, long-range structures existing in leaf vascular patterns, and provides a means for specifying and identifying the inherent global features of vascular patterns which must be accounted for in functional developmental models.

In silico leaf venation networks: growth and reorganization driven by mechanical forces

Development commonly involves an interplay between signaling, genetic expression and biophysical forces. However, the relative importance of these mechanisms during the different stages of development is unclear. Leaf venation networks provide a fitting context for the examination of these questions. In mature leaves, venation patterns are extremely diverse, yet their local structure satisfies a universal property: at junctions between veins, angles and diameters are related by a vectorial equation analogous to a force balance. Using a cell proliferation model, we reproduce in silico the salient features of venation patterns. Provided that vein cells are given different mechanical properties, tensile forces develop along the veins during growth, causing the network to deform progressively. Our results suggest that the local structure of venation networks results from a reorganization driven by mechanical forces, independently of how veins form. This conclusion is supported by recent observations of vein development in young leaves and by the good quantitative agreement between our simulations and data from mature leaves.

Auxin-regulated cell polarity: an inside job?

Auxin is now known to be a key regulator of polar events in plant cells. The mechanism by which auxin conveys a polar signal to the cell is unknown, but one well-known hypothesis is that the auxin flux across the plasma membrane regulates vesicle trafficking. This hypothesis remains controversial because of its reliance on an as-yet-undiscovered membrane flux sensor. In this article I suggest instead that the polar signal is the auxin gradient within the cell cytoplasm. A computer model of vascular development is presented that demonstrates the plausibility of this scenario. The auxin-binding protein ABP1 might be the receptor for the auxin gradient.

The role of elastic stresses on leaf venation morphogenesis

We explore the possible role of elastic mismatch between epidermis and mesophyll as a driving force for the development of leaf venation. The current prevalent ‘canalization’ hypothesis for the formation of veins claims that the transport of the hormone auxin out of the leaves triggers cell differentiation to form veins. Although there is evidence that auxin plays a fundamental role in vein formation, the simple canalization mechanism may not be enough to explain some features observed in the vascular system of leaves, in particular, the abundance of vein loops. We present a model based on the existence of mechanical instabilities that leads very naturally to hierarchical patterns with a large number of closed loops. When applied to the structure of high-order veins, the numerical results show the same qualitative features as actual venation patterns and, furthermore, have the same statistical properties. We argue that the agreement between actual and simulated patterns provides strong evidence for the role of mechanical effects on venation development.

Leaf venation patterns of most angiosperm plants are hierarchical structures that develop during leaf growth. A remarkable characteristic of these structures is the abundance of closed loops: the venation array divides the leaf surface into disconnected polygonal sectors. The initial vein generations are repetitive within the same species, while high-order vein generations are much more diverse but still show preserved statistical properties. The accepted view of vein formation is the auxin canalization hypothesis: a high flow of the hormone auxin triggers cell differentiation to form veins. Although the role of auxin in vein formation is well established, some issues are difficult to explain within this model, in particular, the abundance of loops of high-order veins. In this work, we explore the previously proposed idea that elastic stresses may play an important role in the development of venation patterns. This appealing hypothesis naturally explains the existence of hierarchical structures with abundant closed loops. To test whether it can sustain a quantitative comparison with actual venation patterns, we have developed and implemented a numerical model and statistically compare actual and simulated patterns. The overall similarity we found indicates that elastic stresses should be included in a complete description of leaf venation development.

Auxin: the looping star in plant development

The phytohormone auxin is a key factor in plant growth and development. Forward and reverse genetic strategies have identified important molecular components in auxin perception, signaling, and transport. These advances resulted in the identification of some of the underlying regulatory mechanisms as well as the emergence of functional frameworks for auxin action. This review focuses on the feedback loops that form an integrative part of these regulatory mechanisms.

Computer models of auxin transport: a review and commentary

With the recent proliferation of computer models of auxin transport, it is important that plant biologists understand something about these techniques and how to evaluate them. The paper begins with a brief introduction to the parts of a computer model, followed by a discussion of the limitations of the most common auxin modelling technique. Lastly, several recent models of organ initiation in the shoot apical meristem (i.e. phyllotaxis) are reviewed. The cell and molecular biology of phyllotaxis is now understood well enough that computer models can go beyond a simple 'proof of principle' and start to provide insights into gene function.

Canalization without flux sensors: a traveling-wave hypothesis

In 1969, Tsvi Sachs published his seminal hypothesis of vascular development in plants: the canalization hypothesis. A positive feedback loop between the flux of the phytohormone auxin and the cells' auxin transport capacity would canalize auxin progressively into discrete channels, which would then differentiate into vascular tissues. Recent experimental studies confirm the central role of polar auxin flux in plant vasculogenesis, but it is unclear if and by which mechanism plant cells could respond to auxin flux. In this Opinion article, we review auxin perception mechanisms and argue that these respond more likely to auxin concentrations than to auxin flux. We propose an alternative mechanism for polar auxin channeling, which is more consistent with recent molecular observations.

Patterning the axis in plants – auxin in control

Axis formation and patterning are fundamental processes establishing the body organization of multicellular organisms. In plants, patterning is not confined to embryogenesis but continues to produce new structures--lateral organs--along the growing primary body axis and also initiates secondary body axes. The signalling molecule auxin has been identified as a key player in plant axial patterning. The shoot and root sections of the axis seem to produce lateral organs in different ways. However, very recent findings suggest a general mechanism of branching triggered by local accumulation of auxin in a 'zone of competence' at the margin of stem-cell systems. How the general auxin signal is converted into organ-specific developmental programs remains a major challenge for the future.

Towards the systems biology of auxin-transport-mediated patterning

Polar auxin transport intimately connects plant cell polarity and multicellular patterning. Through the transport of the small molecule indole-3-acetic acid, plant cells integrate their polarities and communicate the degree of their polarization. In this way, they generate an apical-basal axis that serves as a positional reference anchoring subsequent patterning events. Research in recent years has brought the molecular mechanisms underlying auxin perception and auxin transport to light. This knowledge has been used to derive spectacular molecular visualization tools and animated computer simulations, which are now allied in a joint systems biology effort towards a mathematical description of auxin-transport-mediated patterning processes.