Computer models of auxin transport: a review and commentary

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.

Electrophysiological study of Arabidopsis ABCB4 and PIN2 auxin transporters: Evidence of auxin activation and interaction enhancing auxin selectivity

Polar auxin transport through plant tissue strictly requires polarly localized PIN proteins and uniformly distributed ABCB proteins. A functional synergy between the two types of membrane protein where their localizations overlap may create the degree of asymmetric auxin efflux required to produce polar auxin transport. We investigated this possibility by expressing ABCB4 and PIN2 in human embryonic kidney cells and measuring whole-cell ionic currents with the patch-clamp technique and CsCl-based electrolytes. ABCB4 activity was 1.81-fold more selective for Cl- over Cs+ and for PIN2 the value was 2.95. We imposed auxin gradients and determined that ABCB4 and PIN2 were 12-fold more permeable to the auxin anion (IAA-) than Cl-. This measure of the intrinsic selectivity of the transport pathway was 21-fold when ABCB4 and PIN2 were co-expressed. If this increase occurs in plants, it could explain why asymmetric PIN localization is not sufficient to create polar auxin flow. Some form of co-action or synergy between ABCB4 and PIN2 that increases IAA- selectivity at the cell face where both occur may be important. We also found that auxin stimulated ABCB4 activity, which may contribute to a self-reinforcement of auxin transport known as canalization.

Seeing is better than believing: visualization of membrane transport in plants

Recently, the plant transport field has shifted their research focus toward a more integrative investigation of transport networks thought to provide the basis for long-range transport routes. Substantial progress was provided by of a series of elegant techniques that allow for a visualization or prediction of substrate movements in plant tissues in contrast to established quantitative methods offering low spatial resolution. These methods are critically evaluated in respect to their spatio-temporal resolution, invasiveness, dynamics and overall quality. Current limitations of transport route predictions-based on transporter locations and transport modeling are addressed. Finally, the potential of new tools that have not yet been fully implemented into plant research is indicated.

Virtual Plant Tissue: Building Blocks for Next-Generation Plant Growth Simulation

Motivation: Computational modeling of plant developmental processes is becoming increasingly important. Cellular resolution plant tissue simulators have been developed, yet they are typically describing physiological processes in an isolated way, strongly delimited in space and time. Results: With plant systems biology moving toward an integrative perspective on development we have built the Virtual Plant Tissue (VPTissue) package to couple functional modules or models in the same framework and across different frameworks. Multiple levels of model integration and coordination enable combining existing and new models from different sources, with diverse options in terms of input/output. Besides the core simulator the toolset also comprises a tissue editor for manipulating tissue geometry and cell, wall, and node attributes in an interactive manner. A parameter exploration tool is available to study parameter dependence of simulation results by distributing calculations over multiple systems. Availability: Virtual Plant Tissue is available as open source (EUPL license) on Bitbucket (https://bitbucket.org/vptissue/vptissue). The project has a website https://vptissue.bitbucket.io.

Modelling the emergence of polarity patterns for the intercellular transport of auxin in plants

The hormone auxin is actively transported throughout plants via protein machineries including the dedicated transporter known as PIN. The associated transport is ordered with nearby cells driving auxin flux in similar directions. Here, we provide a model of both the auxin transport and of the dynamics of cellular polarization based on flux sensing. Our main findings are: (i) spontaneous intracellular PIN polarization arises if PIN recycling dynamics are sufficiently nonlinear, (ii) there is no need for an auxin concentration gradient and (iii) ordered multi-cellular patterns of PIN polarization are favoured by molecular noise.

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.

Modelling the dynamics of polar auxin transport in inflorescence stems of Arabidopsis thaliana

The polar transport of the plant hormone auxin has been the subject of many studies, several involving mathematical modelling. Unfortunately, most of these models have not been experimentally verified. Here we present experimental measurements of long-distance polar auxin transport (PAT) in segments of inflorescence stems of Arabidopsis thaliana together with a descriptive mathematical model that was developed from these data. It is based on a general advection-diffusion equation for auxin density, as suggested by the chemiosmotic theory, but is extended to incorporate both immobilization of auxin and exchange with the surrounding tissue of cells involved in PAT, in order to account for crucial observations. We found that development of the present model assisted effectively in the analysis of experimental observations. As an example, we discuss the analysis of a quadruple mutant for all four AUX1/LAX1-LAX3 influx carriers genes. We found a drastic change in the parameters governing the exchange of PAT channels with the surrounding tissue, whereas the velocity was still of the order of magnitude of the wild type. In addition, the steady-state flux of auxin through the PAT system of the mutant did not exhibit a saturable component, as we found for the wild type, suggesting that the import carriers are responsible for the saturable component in the wild type. In the accompanying Supplementary data available at JXB online, we describe in more detail the data-driven development of the model, review and derive predictions from a mathematical model of the chemiosmotic theory, and explore relationships between parameters in our model and processes and parameters at the cellular level.

Auxin influx carriers control vascular patterning and xylem differentiation in Arabidopsis thaliana

Auxin is an essential hormone for plant growth and development. Auxin influx carriers AUX1/LAX transport auxin into the cell, while auxin efflux carriers PIN pump it out of the cell. It is well established that efflux carriers play an important role in the shoot vascular patterning, yet the contribution of influx carriers to the shoot vasculature remains unknown. Here, we combined theoretical and experimental approaches to decipher the role of auxin influx carriers in the patterning and differentiation of vascular tissues in the Arabidopsis inflorescence stem. Our theoretical analysis predicts that influx carriers facilitate periodic patterning and modulate the periodicity of auxin maxima. In agreement, we observed fewer and more spaced vascular bundles in quadruple mutants plants of the auxin influx carriers aux1lax1lax2lax3. Furthermore, we show AUX1/LAX carriers promote xylem differentiation in both the shoot and the root tissues. Influx carriers increase cytoplasmic auxin signaling, and thereby differentiation. In addition to this cytoplasmic role of auxin, our computational simulations propose a role for extracellular auxin as an inhibitor of xylem differentiation. Altogether, our study shows that auxin influx carriers AUX1/LAX regulate vascular patterning and differentiation in plants.

Physical forces regulate plant development and morphogenesis

Plant cells in tissues experience mechanical stress not only as a result of high turgor, but also through interaction with their neighbors. Cells can expand at different rates and in different directions from neighbors with which they share a cell wall. This in connection with specific tissue shapes and properties of the cell wall material can lead to intricate stress patterns throughout the tissue. Two cellular responses to mechanical stress are a microtubule cytoskeletal response that directs new wall synthesis so as to resist stress, and a hormone transporter response that regulates transport of the hormone auxin, a regulator of cell expansion. Shape changes in plant tissues affect the pattern of stresses in the tissues, and at the same time, via the cellular stress responses, the pattern of stresses controls cell growth, which in turn changes tissue shape, and stress pattern. This feedback loop controls plant morphogenesis, and explains several previously mysterious aspects of plant growth.

Plasma membrane H⁺ -ATPase regulation is required for auxin gradient formation preceding phototropic growth

Phototropism is a growth response allowing plants to align their photosynthetic organs toward incoming light and thereby to optimize photosynthetic activity. Formation of a lateral gradient of the phytohormone auxin is a key step to trigger asymmetric growth of the shoot leading to phototropic reorientation. To identify important regulators of auxin gradient formation, we developed an auxin flux model that enabled us to test in silico the impact of different morphological and biophysical parameters on gradient formation, including the contribution of the extracellular space (cell wall) or apoplast. Our model indicates that cell size, cell distributions, and apoplast thickness are all important factors affecting gradient formation. Among all tested variables, regulation of apoplastic pH was the most important to enable the formation of a lateral auxin gradient. To test this prediction, we interfered with the activity of plasma membrane H⁺-ATPases that are required to control apoplastic pH. Our results show that H⁺-ATPases are indeed important for the establishment of a lateral auxin gradient and phototropism. Moreover, we show that during phototropism, H⁺-ATPase activity is regulated by the phototropin photoreceptors, providing a mechanism by which light influences apoplastic pH.

Molecular control of cell specification and cell differentiation during procambial development

Land plants develop vascular tissues that enable the long-distance transport of water and nutrients in xylem and phloem, provide mechanical support for their vertical growth, and produce cells in radial growth. Vascular tissues are produced in many parts of the plant and during different developmental stages. Early vascular development is focused in procambial meristems, and in some species it continues during the secondary phase of plant development in cambial meristems. In this review, we highlight recent progress in understanding procambial development. This involves the analysis of stem cell-like properties of procambial tissues, specification of xylem and phloem, and differentiation of the conductive tissues. Several major plant hormones, small-RNA species, and transcriptional networks play a role in vascular development. We describe current approaches to integrating these networks as well as their potential role in explaining the diversity and evolution of plant vascular systems.

Racemose inflorescences of monocots: structural and morphogenetic interaction at the flower/inflorescence level

BACKGROUND

Understanding and modelling early events of floral meristem patterning and floral development requires consideration of positional information regarding the organs surrounding the floral meristem, such as the flower-subtending bracts (FSBs) and floral prophylls (bracteoles). In common with models of regulation of floral patterning, the simplest models of phyllotaxy consider only unbranched uniaxial systems. Racemose inflorescences and thyrses offer a useful model system for investigating morphogenetic interactions between organs belonging to different axes.

SCOPE

This review considers (1) racemose inflorescences of early-divergent and lilioid monocots and their possible relationship with other inflorescence types, (2) hypotheses on the morphogenetic significance of phyllomes surrounding developing flowers, (3) patterns of FSB reduction and (4) vascular patterns in the primary inflorescence axis and lateral pedicels.

CONCLUSIONS

Racemose (partial) inflorescences represent the plesiomorphic condition in monocots. The presence or absence of a terminal flower or flower-like structure is labile among early-divergent monocots. In some Alismatales, a few-flowered racemose inflorescence can be entirely transformed into a terminal 'flower'. The presence or absence and position of additional phyllomes on the lateral pedicels represent important taxonomic markers and key features in regulation of flower patterning. Racemose inflorescences with a single floral prophyll are closely related to thyrses. Floral patterning is either unidirectional or simultaneous in species that lack a floral prophyll or possess a single adaxial floral prophyll and usually spiral in the outer perianth whorl in species with a transversely oriented floral prophyll. Inhibitory fields of surrounding phyllomes are relevant but insufficient to explain these patterns; other important factors are meristem space economy and/or the inhibitory activity of the primary inflorescence axis. Two patterns of FSB reduction exist in basal monocots: (1) complete FSB suppression (cryptic flower-subtending bract) and (2) formation of a 'hybrid' organ by overlap of the developmental programmes of the FSB and the first abaxial organ formed on the floral pedicel. FSB reduction affects patterns of interaction between the conductive systems of the flower and the primary inflorescence axis.

Sequential induction of auxin efflux and influx carriers regulates lateral root emergence

Emergence of a new lateral root primordium through the outer layers of the parental root requires the sequential auxin-mediated induction of two auxin transporters. This positive feedback regulatory loop coordinates patterned gene expression in outer tissues.

The emergence of lateral roots through several tissues requires the precise regulation of gene expression in overlaying cells to trigger cell separation.

Auxin derived from new lateral root primordia induces a positive feedback loop in the outer tissues by promoting the expression of the auxin influx transporter LAX3.

A mathematical model based on realistic 3D geometries predicted the involvement of an auxin efflux carrier that was later identified to be PIN3.

The model also revealed that PIN3 must be expressed before LAX3 to ensure a ‘robust' pattern of LAX3 induction in just two overlaying cortical cell files, thereby delimiting cell separation.

In Arabidopsis, lateral roots originate from pericycle cells deep within the primary root. New lateral root primordia (LRP) have to emerge through several overlaying tissues. Here, we report that auxin produced in new LRP is transported towards the outer tissues where it triggers cell separation by inducing both the auxin influx carrier LAX3 and cell-wall enzymes. LAX3 is expressed in just two cell files overlaying new LRP. To understand how this striking pattern of LAX3 expression is regulated, we developed a mathematical model that captures the network regulating its expression and auxin transport within realistic three-dimensional cell and tissue geometries. Our model revealed that, for the LAX3 spatial expression to be robust to natural variations in root tissue geometry, an efflux carrier is required—later identified to be PIN3. To prevent LAX3 from being transiently expressed in multiple cell files, PIN3 and LAX3 must be induced consecutively, which we later demonstrated to be the case. Our study exemplifies how mathematical models can be used to direct experiments to elucidate complex developmental processes.

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.

Auxin transport at cellular level: new insights supported by mathematical modelling

The molecular basis of cellular auxin transport is still not fully understood. Although a number of carriers have been identified and proved to be involved in auxin transport, their regulation and possible activity of as yet unknown transporters remain unclear. Nevertheless, using single-cell-based systems it is possible to track the course of auxin accumulation inside cells and to specify and quantify some auxin transport parameters. The synthetic auxins 2,4-dichlorophenoxyacetic acid (2,4-D) and naphthalene-1-acetic acid (NAA) are generally considered to be suitable tools for auxin transport studies because they are transported specifically via either auxin influx or efflux carriers, respectively. Our results indicate that NAA can be metabolized rapidly in tobacco BY-2 cells. The predominant metabolite has been identified as NAA glucosyl ester and it is shown that all NAA metabolites were retained inside the cells. This implies that the transport efficiency of auxin efflux transporters is higher than previously assumed. By contrast, the metabolism of 2,4-D remained fairly weak. Moreover, using data on the accumulation of 2,4-D measured in the presence of auxin transport inhibitors, it is shown that 2,4-D is also transported by efflux carriers. These results suggest that 2,4-D is a promising tool for determining both auxin influx and efflux activities. Based on the accumulation data, a mathematical model of 2,4-D transport at a single-cell level is proposed. Optimization of the model provides estimates of crucial transport parameters and, together with its validation by successfully predicting the course of 2,4-D accumulation, it confirms the consistency of the present concept of cellular auxin transport.

Modeling a cortical auxin maximum for nodulation: different signatures of potential strategies

Lateral organ formation from plant roots typically requires the de novo creation of a meristem, initiated at the location of a localized auxin maximum. Legume roots can form both root nodules and lateral roots. From the basic principles of auxin transport and metabolism only a few mechanisms can be inferred for increasing the local auxin concentration: increased influx, decreased efflux, and (increased) local production. Using computer simulations we investigate the different spatio-temporal patterns resulting from each of these mechanisms in the context of a root model of a generalized legume. We apply all mechanisms to the same group of preselected cells, dubbed the controlled area. We find that each mechanism leaves its own characteristic signature. Local production by itself can not create a strong auxin maximum. An increase of influx, as is observed in lateral root formation, can result in an auxin maximum that is spatially more confined than the controlled area. A decrease of efflux on the other hand leads to a broad maximum, which is more similar to what is observed for nodule primordia. With our prime interest in nodulation, we further investigate the dynamics following a decrease of efflux. We find that with a homogeneous change in the whole cortex, the first auxin accumulation is observed in the inner cortex. The steady state lateral location of this efflux reduced auxin maximum can be shifted by slight changes in the ratio of central to peripheral efflux carriers. We discuss the implications of this finding in the context of determinate and indeterminate nodules, which originate from different cortical positions. The patterns we have found are robust under disruption of the (artificial) tissue layout. The same patterns are therefore likely to occur in many other contexts.

Models of long-distance transport: how is carrier-dependent auxin transport regulated in the stem?

• This paper presents two models of carrier-dependent long-distance auxin transport in stems that represent the process at different scales. • A simple compartment model using a single constant auxin transfer rate produced similar data to those observed in biological experiments. The effects of different underlying biological assumptions were tested in a more detailed model representing cellular and intracellular processes that enabled discussion of different patterns of carrier-dependent auxin transport and signalling. • The output that best fits the biological data is produced by a model where polar auxin transport is not limited by the number of transporters/carriers and hence supports biological data showing that stems have considerable excess capacity to transport auxin. • All results support the conclusion that auxin depletion following apical decapitation in pea (Pisum sativum) occurs too slowly to be the initial cause of bud outgrowth. Consequently, changes in auxin content in the main stem and changes in polar auxin transport/carrier abundance in the main stem are not correlated with axillary bud outgrowth.

AUX/LAX family of auxin influx carriers-an overview

Auxin regulates several aspects of plant growth and development. Auxin is unique among plant hormones for exhibiting polar transport. Indole-3-acetic acid (IAA), the major form of auxin in higher plants, is a weak acid and its intercellular movement is facilitated by auxin influx and efflux carriers. Polarity of auxin movement is provided by asymmetric localization of auxin carriers (mainly PIN efflux carriers). PIN-FORMED (PIN) and P-GLYCOPROTEIN (PGP) family of proteins are major auxin efflux carriers whereas AUXIN1/LIKE-AUX1 (AUX/LAX) are major auxin influx carriers. Genetic and biochemical evidence show that each member of the AUX/LAX family is a functional auxin influx carrier and mediate auxin related developmental programmes in different organs and tissues. Of the four AUX/LAX genes, AUX1 regulates root gravitropism, root hair development and leaf phyllotaxy whereas LAX2 regulates vascular development in cotyledons. Both AUX1 and LAX3 have been implicated in lateral root (LR) development as well as apical hook formation whereas both AUX1 and LAX1 and possibly LAX2 are required for leaf phyllotactic patterning.

A computational model of auxin and pH dynamics in a single plant cell

Directed cell-to-cell movement of the plant growth hormone auxin is often referred to as polar auxin transport, and has gained much interest since its discovery at the beginning of the 20th century, both by biologists and theoreticians. Computational modelling of auxin transport at tissue and whole plant scales has given valuable insights into the feedback dynamics between auxin and its transport, which often leads to cell polarisation. However, one cellular feedback mechanism that has been overlooked so far in previous models is the interplay between auxin and pH during auxin transport, even though this is well known from biology. We propose a kinetic model of such a feedback mechanism, linking knowledge about auxin-induced acidification of cell wall compartments to the chemiosmotic hypothesis of auxin transport. Our results suggest that proton fluxes may play a significant role in auxin transport. Since active auxin transport relies on the proton motive force over the cellular membrane, allocation of auxin is linked to its effects on compartmental pH. Our auxin/pH feedback model predicts enhanced accumulation of auxin in cells and increases in both auxin influx and efflux when this feedback is in effect. These results were robust in all simulations and consistent with biological evidence, thus providing a framework for generating and testing hypotheses of auxin-related polarisation events at a cellular level.

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.

Cell polarity in plants: Linking PIN polarity generation mechanisms to morphogenic auxin gradients

Auxin efflux carrier PIN proteins have been intensively investigated as they are the first polar cargos to be identified in plants with a direct relevance for plant patterning. Based on their polar localization; PIN proteins direct the intercellular flow of signaling molecule auxin and thus bear a rate limiting effect on the formation of auxin activity gradients. With this influence on directionality and extent of auxin transport PINs play crucial roles in plant body organization. Many factors such as vesicle trafficking regulator ARF-GEF GNOM, a kinase PINOID, a retromer complex and membrane sterol composition influence polar PIN localization. Recent work uncovers the mechanism that generates default PIN polarity. Real time PIN tracking reveals that PIN polarity is generated from initially non-polar secretion via endocytosis and subsequent polar recycling. In addition, the Rab5 endocytic pathway emerges to be important for polar PIN localization as Rab5 interference causes non-polar distribution of PINs. This non-polar distribution of PINs during embryogenesis transiently alters auxin activity gradients and changes organ identity by transforming embryonic leaf cells to root fates. These findings for the first time link PIN polarity-based auxin activity gradient to cell fate decisions and thus demonstrate morphogen (a substance influencing cell fates on its concentration gradient) characters of auxin. They also suggest an auxin activity distribution-dependent sensing module that executes differential apical and basal developmental program during plant embryogenesis.

Feedback models for polarized auxin transport: an emerging trend

The phytohormone auxin is vital to plant growth and development. A unique property of auxin among all other plant hormones is its cell-to-cell polar transport that requires activity of polarly localized PIN-FORMED (PIN) auxin efflux transporters. Despite the substantial molecular insight into the cellular PIN polarization, the mechanistic understanding for developmentally and environmentally regulated PIN polarization is scarce. The long-standing belief that auxin modulates its own transport by means of a positive feedback mechanism has inspired both experimentalists and theoreticians for more than two decades. Recently, theoretical models for auxin-dependent patterning in plants include the feedback between auxin transport and the PIN protein localization. These computer models aid to assess the complexity of plant development by testing and predicting plausible scenarios for various developmental processes that occur in planta. Although the majority of these models rely on purely heuristic principles, the most recent mechanistic models tentatively integrate biologically testable components into known cellular processes that underlie the PIN polarity regulation. The existing and emerging computational approaches to describe PIN polarization are presented and discussed in the light of recent experimental data on the PIN polar targeting.

Plant systems biology: network matters

Systems biology is all about networks. A recent trend has been to associate systems biology exclusively with the study of gene regulatory or protein-interaction networks. However, systems biology approaches can be applied at many other scales, from the subatomic to the ecosystem scales. In this review, we describe studies at the sub-cellular, tissue, whole plant and crop scales and highlight how these studies can be related to systems biology. We discuss the properties of system approaches at each scale as well as their current limits, and pinpoint in each case advances unique to the considered scale but representing potential for the other scales. We conclude by examining plant models bridging different scales and considering the future prospects of plant systems biology.

Biosensors in plants

Biosensors come in an increasing array of forms and their development is defining the rate of advance for our understanding of many natural processes. Developmental biology is increasingly using mathematical models and yet few of these models are based on quantitative recordings. In particular, we know comparatively little about the endogenous concentrations or fluxes of signalling molecules such as the phytohormones, an area of great potential for new biosensors. There are extremely useful biosensors for some signals, but most remain qualitative. Other qualities sought in biosensors are temporal and spatial resolution and, usually, an ability to use them without significantly perturbing the system. Currently, the biosensors with the best properties are the genetically encoded optical biosensors based on FRET, but each sensor needs extensive specific effort to develop. Sensor technologies using antibodies as the recognition domain are more generic, but these tend to be more invasive and there are few examples of their use in plant biology. By capturing some of the opportunities appearing with advances in platform technologies it is hoped that more biosensors will become available to plant scientists.

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.

Expression of a gymnosperm PIN homologous gene correlates with auxin immunolocalization pattern at cotyledon formation and in demarcation of the procambium during Picea abies somatic embryo development and in seedling tissues

In seed plants, the body organization is established during embryogenesis and is uniform across gymnosperms and angiosperms, despite differences during early embryogeny. Evidence from angiosperms implicates the plant hormone auxin and its polar transport, mainly established by the PIN family of auxin efflux transporters, in the patterning of embryos. Here, PaPIN1 from Norway spruce (Picea abies [L.] Karst.), a gene widely expressed in conifer tissues and organs, was characterized and its expression and localization patterns were determined with reverse transcription polymerase chain reaction and in situ hybridization during somatic embryo development and in seedlings. PaPIN1 shares the predicted structure of other PIN proteins, but its central hydrophilic loop is longer than most PINs. In phylogenetic analyses, PaPIN1 clusters with Arabidopsis thaliana (L.) Heynh. PIN3, PIN4 and PIN7, but its expression pattern also suggests similarity to PIN1. The PaPIN1 expression signal was high in the protoderm of pre-cotyledonary embryos, but not if embryos were pre-treated with the auxin transport inhibitor N-1-naphthylphthalamic acid (NPA). This, together with a high auxin immunolocalization signal in this cell layer, suggests a role of PaPIN1 during cotyledon formation. At later stages, high PaPIN1 expression was observed in differentiating procambium, running from the tip of incipient cotyledons down through the embryo axis and to the root apical meristem (RAM), although the mode of RAM specification in conifer embryos differs from that of most angiosperms. Also, the PaPIN1 in situ signal was high in seedling root tips including root cap columella cells. The results thus suggest that PaPIN1 provides an ancient function associated with auxin transport and embryo pattern formation prior to the separation of angiosperms and gymnosperms, in spite of some morphological differences.

Auxin: a major regulator of organogenesis

Plant development is characterized by the continuous initiation of tissues and organs. The meristems, which are small stem cell populations, are involved in this process. The shoot apical meristem produces lateral organs at its flanks and generates the growing stem. These lateral organs are arranged in a stereotyped pattern called phyllotaxis. Organ initiation in the peripheral zone of the meristem involves accumulation of the plant hormone auxin. Auxin is transported in a polar way by influx and efflux carriers located at cell membranes. Polar localization of the PIN1 efflux carrier in meristematic cells generates auxin concentration gradients and PIN1 localization depends, in turn, on auxin gradients: this feedback loop generates a dynamic auxin distribution which controls phyllotaxis. Furthermore, PIN-dependent local auxin gradients represent a common module for organ initiation, in the shoot and in the root.

The AUX1 LAX family of auxin influx carriers is required for the establishment of embryonic root cell organization in Arabidopsis thaliana

BACKGROUND AND AIMS

The root meristem of the Arabidopsis thaliana mature embryo is a highly organized structure in which individual cell shape and size must be regulated in co-ordination with the surrounding cells. The objective of this study was to determine the role of the AUX1 LAX family of auxin import carriers during the establishment of the embryonic root cell pattern.

METHODS

The radicle apex of single and multiple aux1 lax mutant mature embryos was used to evaluate the effect of this gene family upon embryonic root organization and root cap size, cell number and cell size.

KEY RESULTS

It was demonstrated here that mutations within the AUX1 LAX family are associated with changes in cell pattern establishment in the embryonic quiescent centre and columella. aux1 lax mutants have a larger radicle root cap than the wild type and this is associated with a significant increase in the root-cap cell number, average cell size, or both. Extreme disorganization of the radicle apex was observed among quadruple aux1 lax1 lax2 lax3 mutant embryos, but not in single aux1 null or in lax1, lax2 and lax3 single mutants, indicating redundancy within the AUX1 LAX family.

CONCLUSIONS

It was determined that the AUX1 LAX family of auxin influx facilitators participates in the establishment of cell pattern within the apex of the embryonic root in a gene-redundant fashion. It was demonstrated that aux1 lax mutants are affected in cell proliferation and cell growth within the radicle tip. Thus AUX1 LAX auxin importers emerge as new players in morphogenetic processes involved in patterning during embryonic root formation.

Back to the future: evolution of computational models in plant morphogenesis

There has been a recent surge of studies in plant biology that combine experimental data with computational modeling. Here, we categorize a diversity of theoretical models and emphasize the need to tailor modeling approaches to the questions at hand. Models can start from biophysical or purely heuristic basic principles, and can focus at several levels of biological organization. Recent examples illustrate that this entire spectrum can be useful to understand plant development, and point to a future direction where more approaches are combined in fruitful ways--either by proving the same result with different basic principles or by exploring interactions across levels, in the so-called multilevel models.

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.

Heterologous expression of a membrane-spanning auxin importer: implications for functional analyses of auxin transporters

Biochemical studies of plant auxin transporters in vivo are made difficult by the presence of multiple auxin transporters and auxin-interacting proteins. Furthermore, the expression level of most such transporters in plants is likely to be too low for purification and downstream functional analysis. Heterologous expression systems should address both of these issues. We have examined a number of such systems for their efficiency in expressing AUX1 from Arabidopsis thaliana. We find that a eukaryotic system based upon infection of insect cells with recombinant baculovirus provides a high level, easily scalable expression system capable of delivering a functional assay for AUX1. Furthermore, a transient transfection system in mammalian cells enables localization of AUX1 and AUX1-mediated transport of auxin to be investigated. In contrast, we were unable to utilise P. pastoris or L. lactis expression systems to reliably express AUX1.

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.

Strigolactone acts downstream of auxin to regulate bud outgrowth in pea and Arabidopsis

During the last century, two key hypotheses have been proposed to explain apical dominance in plants: auxin promotes the production of a second messenger that moves up into buds to repress their outgrowth, and auxin saturation in the stem inhibits auxin transport from buds, thereby inhibiting bud outgrowth. The recent discovery of strigolactone as the novel shoot-branching inhibitor allowed us to test its mode of action in relation to these hypotheses. We found that exogenously applied strigolactone inhibited bud outgrowth in pea (Pisum sativum) even when auxin was depleted after decapitation. We also found that strigolactone application reduced branching in Arabidopsis (Arabidopsis thaliana) auxin response mutants, suggesting that auxin may act through strigolactones to facilitate apical dominance. Moreover, strigolactone application to tiny buds of mutant or decapitated pea plants rapidly stopped outgrowth, in contrast to applying N-1-naphthylphthalamic acid (NPA), an auxin transport inhibitor, which significantly slowed growth only after several days. Whereas strigolactone or NPA applied to growing buds reduced bud length, only NPA blocked auxin transport in the bud. Wild-type and strigolactone biosynthesis mutant pea and Arabidopsis shoots were capable of instantly transporting additional amounts of auxin in excess of endogenous levels, contrary to predictions of auxin transport models. These data suggest that strigolactone does not act primarily by affecting auxin transport from buds. Rather, the primary repressor of bud outgrowth appears to be the auxin-dependent production of strigolactones.

Local auxin production: a small contribution to a big field

Auxin is a plant growth regulator involved in diverse fundamental developmental responses. Much is now known about auxin transport, via influx and efflux carriers, and about auxin perception and its role in gene regulation. Many developmental processes are dependent on peaks of auxin concentration and, to date, attention has been directed at the role of polar auxin transport in generating and maintaining auxin gradients. However, surprisingly little attention has focussed on the role and significance of auxin biosynthesis, which should be expected to contribute to active auxin pools. Recent reports on the function of the YUCCA flavin monooxygenases and a tryptophan aminotransferase in Arabidopsis have caused us to look again at the importance of local biosynthesis in developmental processes. Many alternative and redundant pathways of auxin synthesis exist in many plants and it is emerging that they may function in response to environmental cues.

Integration of transport-based models for phyllotaxis and midvein formation

The plant hormone auxin mediates developmental patterning by a mechanism that is based on active transport. In the shoot apical meristem, auxin gradients are thought to be set up through a feedback loop between auxin and the activity and polar localization of its transporter, the PIN1 protein. Two distinct molecular mechanisms for the subcellular polarization of PIN1 have been proposed. For leaf positioning (phyllotaxis), an "up-the-gradient" PIN1 polarization mechanism has been proposed, whereas the formation of vascular strands is thought to proceed by "with-the-flux" PIN1 polarization. These patterning mechanisms intersect during the initiation of the midvein, which raises the question of how two different PIN1 polarization mechanisms may work together. Our detailed analysis of PIN1 polarization during midvein initiation suggests that both mechanisms for PIN1 polarization operate simultaneously. Computer simulations of the resulting dual polarization model are able to reproduce the dynamics of observed PIN1 localization. In addition, the appearance of high auxin concentration in our simulations throughout the initiation of the midvein is consistent with experimental observation and offers an explanation for a long-standing criticism of the canalization hypothesis; namely, how both high flux and high concentration can occur simultaneously in emerging veins.

Quantifying leaf venation patterns: two-dimensional maps

The leaf vasculature plays crucial roles in transport and mechanical support. Understanding how vein patterns develop and what underlies pattern variation between species has many implications from both physiological and evolutionary perspectives. We developed a method for extracting spatial vein pattern data from leaf images, such as vein densities and also the sizes and shapes of the vein reticulations. We used this method to quantify leaf venation patterns of the first rosette leaf of Arabidopsis thaliana throughout a series of developmental stages. In particular, we characterized the size and shape of vein network areoles (loops), which enlarge and are split by new veins as a leaf develops. Pattern parameters varied in time and space. In particular, we observed a distal to proximal gradient in loop shape (length/width ratio) which varied over time, and a margin-to-center gradient in loop sizes. Quantitative analyses of vein patterns at the tissue level provide a two-way link between theoretical models of patterning and molecular experimental work to further explore patterning mechanisms during development. Such analyses could also be used to investigate the effect of environmental factors on vein patterns, or to compare venation patterns from different species for evolutionary studies. The method also provides a framework for gathering and overlaying two-dimensional maps of point, line and surface morphological data.

Polar targeting and endocytic recycling in auxin-dependent plant development

Plant development is characterized by a profound phenotypic plasticity that often involves redefining of the developmental fate and polarity of cells within differentiated tissues. The plant hormone auxin and its directional intercellular transport play a major role in these processes because they provide positional information and link cell polarity with tissue patterning. This plant-specific mechanism of transport-dependent auxin gradients depends on subcellular dynamics of auxin transport components, in particular on endocytic recycling and polar targeting. Recent insights into these cellular processes in plants have revealed important parallels to yeast and animal systems, including clathrin-dependent endocytosis, retromer function, and transcytosis, but have also emphasized unique features of plant cells such as diversity of polar targeting pathways; integration of environmental signals into subcellular trafficking; and the link between endocytosis, cell polarity, and cell fate specification. We review these advances and focus on the translation of the subcellular dynamics to the regulation of whole-plant development.

Auxin transport is sufficient to generate a maximum and gradient guiding root growth

The plant growth regulator auxin controls cell identity, cell division and cell expansion. Auxin efflux facilitators (PINs) are associated with auxin maxima in distal regions of both shoots and roots. Here we model diffusion and PIN-facilitated auxin transport in and across cells within a structured root layout. In our model, the stable accumulation of auxin in a distal maximum emerges from the auxin flux pattern. We have experimentally tested model predictions of robustness and self-organization. Our model explains pattern formation and morphogenesis at timescales from seconds to weeks, and can be understood by conceptualizing the root as an 'auxin capacitor'. A robust auxin gradient associated with the maximum, in combination with separable roles of auxin in cell division and cell expansion, is able to explain the formation, maintenance and growth of sharply bounded meristematic and elongation zones. Directional permeability and diffusion can fully account for stable auxin maxima and gradients that can instruct morphogenesis.