PIN and AUX/LAX proteins: their role in auxin accumulation

The flux-based PIN allocation mechanism can generate either canalyzed or diffuse distribution patterns depending on geometry and boundary conditions

Growth and morphogenesis in plants require controlled transport of the plant hormone auxin. An important participant is the auxin effluxing protein PIN, whose polarized subcellular localization allows it to effectively transport auxin large distances through tissues. The flux-based model, in which auxin flux through a wall stimulates PIN allocation to that wall, is a dominant contender among models determining where and in what quantity PIN is allocated to cell walls. In this paper we characterise the behaviour of flux-based PIN allocation models in various tissues of the shoot apical meristem. Arguing from both mathematical analysis and computer simulations, we describe the natural behaviours of this class of models under various circumstances. In particular, we demonstrate the important dichotomy between sink- and source- driven systems, and show that both diffuse and canalized PIN distributions can be generated simultaneously in the same tissue, without model hybridization or variation of PIN-related parameters. This work is performed in the context of the shoot apical and floral meristems and is applicable to the construction of a unified PIN allocation model.

Getting to the roots of it: Genetic and hormonal control of root architecture

Root system architecture (RSA) - the spatial configuration of a root system - is an important developmental and agronomic trait, with implications for overall plant architecture, growth rate and yield, abiotic stress resistance, nutrient uptake, and developmental plasticity in response to environmental changes. Root architecture is modulated by intrinsic, hormone-mediated pathways, intersecting with pathways that perceive and respond to external, environmental signals. The recent development of several non-invasive 2D and 3D root imaging systems has enhanced our ability to accurately observe and quantify architectural traits on complex whole-root systems. Coupled with the powerful marker-based genotyping and sequencing platforms currently available, these root phenotyping technologies lend themselves to large-scale genome-wide association studies, and can speed the identification and characterization of the genes and pathways involved in root system development. This capability provides the foundation for examining the contribution of root architectural traits to the performance of crop varieties in diverse environments. This review focuses on our current understanding of the genes and pathways involved in determining RSA in response to both intrinsic and extrinsic (environmental) response pathways, and provides a brief overview of the latest root system phenotyping technologies and their potential impact on elucidating the genetic control of root development in plants.

Getting to the roots of it: Genetic and hormonal control of root architecture

Root system architecture (RSA) - the spatial configuration of a root system - is an important developmental and agronomic trait, with implications for overall plant architecture, growth rate and yield, abiotic stress resistance, nutrient uptake, and developmental plasticity in response to environmental changes. Root architecture is modulated by intrinsic, hormone-mediated pathways, intersecting with pathways that perceive and respond to external, environmental signals. The recent development of several non-invasive 2D and 3D root imaging systems has enhanced our ability to accurately observe and quantify architectural traits on complex whole-root systems. Coupled with the powerful marker-based genotyping and sequencing platforms currently available, these root phenotyping technologies lend themselves to large-scale genome-wide association studies, and can speed the identification and characterization of the genes and pathways involved in root system development. This capability provides the foundation for examining the contribution of root architectural traits to the performance of crop varieties in diverse environments. This review focuses on our current understanding of the genes and pathways involved in determining RSA in response to both intrinsic and extrinsic (environmental) response pathways, and provides a brief overview of the latest root system phenotyping technologies and their potential impact on elucidating the genetic control of root development in plants.

Identification of quantitative trait loci controlling fibre length and lignin content in Arabidopsis thaliana stems

Fibre properties and the biochemical composition of cell walls are important traits in many applications. For example, the lengths of fibres define the strength and quality of paper, and lignin content is a critical parameter for the use of biomass in biofuel production. Identifying genes controlling these traits is comparatively difficult in woody species, because of long generation times and limited amenability to high-resolution genetic mapping. To address this problem, this study mapped quantitative trait loci (QTLs) defining fibre length and lignin content in the Arabidopsis recombinant inbred line population Col-4 × Ler-0. Adapting high-throughput phenotyping techniques for both traits for measurements in Arabidopsis inflorescence stems identified significant QTLs for fibre length on chromosomes 2 and 5, as well as one significant QTL affecting lignin content on chromosome 2. For fibre length, total variation within the population was 208% higher than between parental lines and the identified QTLs explained 50.58% of the observed variation. For lignin content, the values were 261 and 26.51%, respectively. Bioinformatics analysis of the associated intervals identified a number of candidate genes for fibre length and lignin content. This study demonstrates that molecular mapping of QTLs pertaining to wood and fibre properties is possible in Arabidopsis, which substantially broadens the use of Arabidopsis as a model species for the functional characterization of plant genes.

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.

Morphogengineering roots: comparing mechanisms of morphogen gradient formation

BACKGROUND

In developmental biology, there has been a recent focus on the robustness of morphogen gradients as possible providers of positional information. It was shown that functional morphogen gradients present strong biophysical constraints and lack of robustness to noise. Here we explore how the details of the mechanism which underlies the generation of a morphogen gradient can influence those properties.

RESULTS

We contrast three gradient-generating mechanisms, (i) a source-decay mechanism; and (ii) a unidirectional transport mechanism; and (iii) a so-called reflux-loop mechanism. Focusing on the dynamics of the phytohormone auxin in the root, we show that only the reflux-loop mechanism can generate a gradient that would be adequate to supply functional positional information for the Arabidopsis root, for biophysically reasonable kinetic parameters.

CONCLUSIONS

We argue that traits that differ in spatial and temporal time-scales can impose complex selective pressures on the mechanism of morphogen gradient formation used for the development of the particular organism.

Growth-induced hormone dilution can explain the dynamics of plant root cell elongation

In the elongation zone of the Arabidopsis thaliana plant root, cells undergo rapid elongation, increasing their length by ∼10-fold over 5 h while maintaining a constant radius. Although progress is being made in understanding how this growth is regulated, little consideration has been given as to how cell elongation affects the distribution of the key regulating hormones. Using a multiscale mathematical model and measurements of growth dynamics, we investigate the distribution of the hormone gibberellin in the root elongation zone. The model quantifies how rapid cell expansion causes gibberellin to dilute, creating a significant gradient in gibberellin levels. By incorporating the gibberellin signaling network, we simulate how gibberellin dilution affects the downstream components, including the growth-repressing DELLA proteins. We predict a gradient in DELLA that provides an explanation of the reduction in growth exhibited as cells move toward the end of the elongation zone. These results are validated at the molecular level by comparing predicted mRNA levels with transcriptomic data. To explore the dynamics further, we simulate perturbed systems in which gibberellin levels are reduced, considering both genetically modified and chemically treated roots. By modeling these cases, we predict how these perturbations affect gibberellin and DELLA levels and thereby provide insight into their altered growth dynamics.

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.

Functions of ABC transporters in plants

ABC (ATP-binding cassette) proteins are ubiquitously found in prokaryotes and eukaryotes and generally serve as membrane-intrinsic primary active pumps. In higher plants, ABC proteins constitute a large family, grouped phylogenetically into eight clusters, subfamilies ABCA-ABCI (ABCH is not found in plants). ABC transporters shuttle substrates as diverse as lipids, phytohormones, carboxylates, heavy metals, chlorophyll catabolites and xenobiotic conjugates across a variety of biological membranes. To date, the largest proportions of characterized members have been localized to the plasma membrane and the tonoplast, with dominant implications in cellular secretion and vacuolar sequestration, but they are also found in mitochondrial, plastidal and peroxisomal membranes. Originally identified as tonoplast-intrinsic proteins that shuttle xenobiotic conjugates from the cytosol into the vacuole, thus being an integral part of the detoxification machinery, ABC transporters are now recognized to participate in a multitude of physiological processes that allow the plant to adapt to changing environments and cope with biotic and abiotic stresses.

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.

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.

Developmental analysis of a Medicago truncatula smooth leaf margin1 mutant reveals context-dependent effects on compound leaf development

Compound leaf development requires highly regulated cell proliferation, differentiation, and expansion patterns. We identified loss-of-function alleles at the SMOOTH LEAF MARGIN1 (SLM1) locus in Medicago truncatula, a model legume species with trifoliate adult leaves. SLM1 encodes an auxin efflux carrier protein and is the ortholog of Arabidopsis thaliana PIN-FORMED1 (PIN1). Auxin distribution is impaired in the slm1 mutant, resulting in pleiotropic phenotypes in different organs. The most striking change in slm1 is the increase in the number of terminal leaflets and a simultaneous reduction in the number of lateral leaflets, accompanied by reduced expression of SINGLE LEAFLET1 (SGL1), an ortholog of LEAFY. Characterization of the mutant indicates that distinct developmental domains exist in the formation of terminal and lateral leaflets. In contrast with the pinnate compound leaves in the wild type, the slm1 sgl1 double mutant shows nonpeltately palmate leaves, suggesting that the terminal leaflet primordium in M. truncatula has a unique developmental mechanism. Further investigations on the development of leaf serrations reveal different ontogenies between distal serration and marginal serration formation as well as between serration and leaflet formation. These data suggest that regulation of the elaboration of compound leaves and serrations is context dependent and tightly correlated with the auxin/SLM1 module in M. truncatula.

Seven things we think we know about auxin transport

Polar transport of the phytohormone auxin and the establishment of localized auxin maxima regulate embryonic development, stem cell maintenance, root and shoot architecture, and tropic growth responses. The past decade has been marked by dramatic progress in efforts to elucidate the complex mechanisms by which auxin transport regulates plant growth. As the understanding of auxin transport regulation has been increasingly elaborated, it has become clear that this process is involved in almost all plant growth and environmental responses in some way. However, we still lack information about some basic aspects of this fundamental regulatory mechanism. In this review, we present what we know (or what we think we know) and what we do not know about seven auxin-regulated processes. We discuss the role of auxin transport in gravitropism in primary and lateral roots, phototropism, shoot branching, leaf expansion, and venation. We also discuss the auxin reflux/fountain model at the root tip, flavonoid modulation of auxin transport processes, and outstanding aspects of post-translational regulation of auxin transporters. This discussion is not meant to be exhaustive, but highlights areas in which generally held assumptions require more substantive validation.

Regulation of solute flux through plasmodesmata in the root meristem

Plasmodesmata permit solutes to move between cells nonspecifically and without having to cross a membrane. This symplastic connectivity, while straightforward to observe using fluorescent tracers, has proven difficult to quantify. We use fluorescence recovery after photobleaching, combined with a mathematical model of symplastic diffusion, to assay plasmodesmata-mediated permeability in the Arabidopsis (Arabidopsis thaliana) root meristem in wild-type and transgenic lines, and under selected chemical treatments. The permeability measured for the wild type is nearly 10-times greater than previously reported. Plamodesmal permeability remains constant in seedlings treated with auxin (30 mM indoleacetic acid for 2 and 24 h; 100 nm indoleacetic acid for 2 h); however, permeability is diminished in two lines previously reported to have impaired plasmodesmal function as well as in wild-type seedlings treated for 24 h with 0.6 mM tryptophan. Moreover, plasmodesmal permeability is strongly altered by applied hydrogen peroxide within 2 h of treatment, being approximately doubled at a low concentration (0.6 mM) and nearly eliminated at a higher one (6 mM). These results reveal that the plasmodesmata in the root meristem carry a substantial flux of small molecules and that this flux is subject to rapid regulation.

Emergence of tissue polarization from synergy of intracellular and extracellular auxin signaling

Here, we provide a novel mechanistic framework for cell polarization during auxin-driven plant development that combines intracellular auxin signaling for regulation of expression of PINFORMED (PIN) auxin efflux transporters and the theoretical assumption of extracellular auxin signaling for regulation of PIN subcellular dynamics.

The competitive utilization of auxin signaling component in the apoplast might account for the elusive mechanism for cell-to-cell communication for tissue polarization.

Computer model simulations faithfully and robustly recapitulate experimentally observed patterns of tissue polarity and asymmetric auxin distribution during formation and regeneration of vascular systems, and during the competitive regulation of shoot branching by apical dominance.

Our model generated new predictions that could be experimentally validated, highlighting a mechanistically conceivable explanation for the PIN polarization and canalization of the auxin flow in plants.

A key question of developmental biology relates to a fundamental issue in cell and tissue polarities, namely, how an individual cell in a polarized tissue senses the polarities of its neighbors and its position within tissue. In plant development, this issue is of pronounced importance, because plants have a remarkable ability to redefine cell and tissue polarities in different developmental programs, such as embryogenesis, postembryonic organogenesis, vascular tissue formation, and tissue regeneration (Kleine-Vehn and Friml, 2008).

A polar, cell-to-cell transport of the small signaling molecule auxin in conjunction with local auxin biosynthesis determines auxin gradients during embryonic and postembryonic development, giving positional cues for primordia formation, organ patterning, and tropistic growth (Friml et al, 2002; Benková et al, 2003; Reinhardt et al, 2003; Heisler et al, 2005; Scarpella et al, 2006; Dubrovsky et al, 2008). Over the past decades, theoretical models proposed that auxin acts as a polarizing cue in the center of a positive feedback mechanisms for auxin transport that has a key role in synchronized polarity rearrangements. However, the mechanistic basis for such a feedback loop between auxin and its own transport remains to a large extent elusive.

The direction of auxin transport largely depends on the polar subcellular localization of PINFORMED (PIN) proteins at the plasma membrane (Petrášek et al, 2006; Wiśniewska et al, 2006). These proteins recycle between the plasma membrane and intracellular endosomal compartments (Geldner et al, 2001; Dhonukshe et al, 2007), and their recycling modulates PIN-dependent auxin efflux rates and enable rapid changes in PIN polarity (Dubrovsky et al, 2008; Kleine-Vehn et al, 2008a). Nevertheless, the molecular basis for PIN polarization in plants remains unknown.

To gain new mechanistic insights in the hypothetical feedback mechanisms governing PIN polarization, several theoretical studies (Mitchison, 1980; Sachs, 1981; Rolland-Lagan and Prusinkiewicz, 2005; Jönsson et al, 2006; Smith et al, 2006; Merks et al, 2007; Bayer et al, 2009; Kramer, 2009) have been carried out. These models suggest that auxin promotes its own transport by modulating the amount of PIN proteins at the plasma membrane by incorporating either not yet identified flux gradient-based component (Mitchison, 1980; Rolland-Lagan and Prusinkiewicz, 2005; Bayer et al, 2009; Kramer, 2009) or an unknown short-range intercellular signal-transmitting auxin concentrations of its direct neighbors (Jönsson et al, 2006; Smith et al, 2006; Merks et al, 2007; Bayer et al, 2009; Sahlin et al, 2009).

Here, we propose a feedback driven, biologically plausible model for PIN polarization and auxin transport that introduces the combination of intracellular and extracellular auxin signaling pathways as a unified approach for tissue polarization in plants. Our computer model is based on chemiosmotic hypothesis (Goldsmith et al, 1981; Figure 1A) and integrates up-to-date experimental data, such as auxin feedback on PIN expression (Peer et al, 2004; Heisler et al, 2005) via a nuclear auxin signaling pathway (Chapman and Estelle, 2009; Figure 1B), auxin carrier recycling auxin (Dubrovsky et al, 2008; Kleine-Vehn et al, 2008a; Figure 1C), and auxin feedback on PIN endocytosis (Paciorek et al, 2005) via novel hypothetical, yet plausible, assumption of extracellular auxin perception (Figure 1D).

The heart of our extracellular receptor-based polarization (ERP) mechanism is the competitive utilization of auxin receptors in the intercellular space that allows a direct and simple cell-to-cell communication scheme. In our model, auxin binds to its extracellular receptor in the concentration-dependent manner and induces signal to modulate PIN protein abundance at the plasma membrane (Figure 1D). The direct mode of the signal transfer involves temporal immobilization of recruited receptors to the plasma membrane, which is reflected by reduced diffusion of receptors involved in auxin signaling (Figure 1D). This competitive utilization mechanism enables cell-to-cell communication in our model, leading to receptor enrichment at the site of higher auxin concentration (Figure 1D). The PIN polarization and polar auxin transport in our model both depend on and contribute to the establishment of differential auxin signaling in the cell wall. This feedback loop leads ultimately to the alignment of PIN polarization within a tissue.

We demonstrated the plausibility of the ERP model for various processes, including de novo vascularization, venation patterning, and tissue regeneration in computer simulations performed with only minimal initial assumptions, a discrete auxin source, and a distal sink. The ERP model reproduces the very detailed PIN polarization events that occur during primary vein initiation (Scarpella et al, 2006), such as basal PIN1 polarity in provascular cells, transient adverse PIN1 polarization in neighboring cells during the alignment of tissue polarization, and inner-lateral polarity displayed by the tissues surrounding a conductive auxin channel (Figure 3). Additionally, the ERP model generates high auxin concentration and high auxin flux simultaneously in emerging veins, revising the classical canalization models (Mitchison, 1980; Rolland-Lagan and Prusinkiewicz, 2005). Importantly, all our model simulations support the claim that the ERP model represents the first single approach that faithfully reproduces PIN polarization, both with the auxin gradient (basal PIN1 polarity in provascular cells) and against the auxin gradient (transient adverse PIN1 polarization in neighboring cells surrounding the provascular bundle), as well as producing the corresponding auxin distribution patterns during auxin canalization.

The proposed model introduces the extracellular auxin signaling pathway, which is crucial to account for coordinated PIN polarization and auxin distribution during venation patterning in plants. The putative candidate for extracellular auxin receptor is auxin-binding protein 1 (ABP1), which resides in the lumen of the endoplasmic reticulum and is secreted to the cell wall (Napier et al, 2002; Tromas et al, 2009) where it is physiologically active (Leblanc et al, 1999; Steffens et al, 2001). Additionally, auxin inhibits clathrin-dependent PIN internalization via binding to ABP1 (Robert et al, 2010). Thus, we speculate that the extracellular fraction of ABP1 (or additionally yet to be identified ABPs) could correspond to the common pool of extracellular auxin receptors in the ERP model. A future challenge will be to test whether the ERP model unifies complex PIN polarization and auxin distribution patterns in embryogenesis, root system maintenance, and de novo organ formation.

Plant development is exceptionally flexible as manifested by its potential for organogenesis and regeneration, which are processes involving rearrangements of tissue polarities. Fundamental questions concern how individual cells can polarize in a coordinated manner to integrate into the multicellular context. In canalization models, the signaling molecule auxin acts as a polarizing cue, and feedback on the intercellular auxin flow is key for synchronized polarity rearrangements. We provide a novel mechanistic framework for canalization, based on up-to-date experimental data and minimal, biologically plausible assumptions. Our model combines the intracellular auxin signaling for expression of PINFORMED (PIN) auxin transporters and the theoretical postulation of extracellular auxin signaling for modulation of PIN subcellular dynamics. Computer simulations faithfully and robustly recapitulated the experimentally observed patterns of tissue polarity and asymmetric auxin distribution during formation and regeneration of vascular systems and during the competitive regulation of shoot branching by apical dominance. Additionally, our model generated new predictions that could be experimentally validated, highlighting a mechanistically conceivable explanation for the PIN polarization and canalization of the auxin flow in plants.

The vascular cambium: molecular control of cellular structure

Indeterminate growth and the production of new organs in plants require a constant supply of new cells. The majority of these cells are produced in mitotic regions called meristems. For primary or tip growth of the roots and shoots, the meristems are located in the apices. These apical meristems have been shown to function as developmentally regulated and environmentally responsive stem cell niches. The principle requirements to maintain a functioning meristem in a dynamic system are a balance of cell division and differentiation and the regulation of the planes of cell division and expansion. Woody plants also have secondary indeterminate mitotic regions towards the exterior of roots, stems and branches that produce the cells for continued growth in girth. The chief secondary meristem is the vascular cambium (VC). As its name implies, cells produced in the VC contribute to the growth in girth via the production of secondary vascular elements. Although we know a considerable amount about the cellular and molecular basis of the apical meristems, our knowledge of the cellular basis and molecular functioning of the VC has been rudimentary. This is now changing as a growing body of research shows that the primary and secondary meristems share some common fundamental regulatory mechanisms. In this review, we outline recent research that is leading to a better understanding of the molecular forces that shape the cellular structure and function of the VC.

Control of leaf and vein development by auxin

Leaves are the main photosynthetic organs of vascular plants and show considerable diversity in their geometries, ranging from simple spoon-like forms to complex shapes with individual leaflets, as in compound leaves. Leaf vascular tissues, which act as conduits of both nutrients and signaling information, are organized in networks of different architectures that usually mirror the surrounding leaf shape. Understanding the processes that endow leaves and vein networks with ordered and closely aligned shapes has captured the attention of biologists and mathematicians since antiquity. Recent work has suggested that the growth regulator auxin has a key role in both initiation and elaboration of final morphology of both leaves and vascular networks. A key feature of auxin action is the existence of feedback loops through which auxin regulates its own transport. These feedbacks may facilitate the iterative generation of basic modules that underlies morphogenesis of both leaves and vasculature.

Modeling auxin-regulated development

The phytohormone auxin plays an essential role in many aspects of plant growth and development. Its patterning, intercellular transport, and means of signaling have been extensively studied both in experiments and computational models. Here, we present a review of models of auxin-regulated development in different plant tissues. This includes models of organ initiation in the shoot apical meristem, development of vascular strands in leafs and stems, and auxin-related functioning in roots. The examples show how mathematical modeling can help to examine expected and unexpected behavior of the system, challenge our knowledge and hypotheses, obtain quantitative results, or suggest new experiments and ways to approach a problem.

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.

Gravitropism of Arabidopsis thaliana roots requires the polarization of PIN2 toward the root tip in meristematic cortical cells

In the root, the transport of auxin from the tip to the elongation zone, referred to here as shootward, governs gravitropic bending. Shootward polar auxin transport, and hence gravitropism, depends on the polar deployment of the PIN-FORMED auxin efflux carrier PIN2. In Arabidopsis thaliana, PIN2 has the expected shootward localization in epidermis and lateral root cap; however, this carrier is localized toward the root tip (rootward) in cortical cells of the meristem, a deployment whose function is enigmatic. We use pharmacological and genetic tools to cause a shootward relocation of PIN2 in meristematic cortical cells without detectably altering PIN2 polarization in other cell types or PIN1 polarization. This relocation of cortical PIN2 was negatively regulated by the membrane trafficking factor GNOM and by the regulatory A1 subunit of type 2-A protein phosphatase (PP2AA1) but did not require the PINOID protein kinase. When GNOM was inhibited, PINOID abundance increased and PP2AA1 was partially immobilized, indicating both proteins are subject to GNOM-dependent regulation. Shootward PIN2 specifically in the cortex was accompanied by enhanced shootward polar auxin transport and by diminished gravitropism. These results demonstrate that auxin flow in the root cortex is important for optimal gravitropic response.

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 transporters--why so many?

Interacting and coordinated auxin transporter actions in plants underlie a flexible network that mobilizes auxin in response to many developmental and environmental changes encountered by these sessile organisms. The independent but synergistic activity of individual transporters can be differentially regulated at various levels. This invests auxin transport mechanisms with robust functional redundancy and added auxin flow capacity when needed. An evolutionary perspective clarifies the roles of the different transporter groups in plant development. Mathematical and functional analysis of elements of auxin transport makes it possible to rationalize the relative contributions of members of the respective transporter classes to the localized auxin transport streams that then underlie both preprogrammed developmental changes and reactions to environmental stimuli.

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.

Coordination of plant cell division and expansion in a simple morphogenetic system

Morphogenesis in plants arises from the interplay of genetic and physical interactions within a growing network of cells. The physical aspects of cell proliferation and differentiation are genetically regulated, but constrained by mechanical interactions between the cells. Higher plant tissues consist of an elaborate three-dimensional matrix of active cytoplasm and extracellular matrix, where it is difficult to obtain direct measurements of geometry or cell interactions. To properly understand the workings of plant morphogenesis, it is necessary to have biological systems that allow simple and direct observation of these processes. We have adopted a highly simplified plant system to investigate how cell proliferation and expansion is coordinated during morphogenesis. Coleocheate scutata is a microscopic fresh-water green alga with simple anatomical features that allow for accurate quantification of morphogenetic processes. Image analysis techniques were used to extract precise models for cell geometry and physical parameters for growth. This allowed construction of a deformable finite element model for growth of the whole organism, which incorporated cell biophysical properties, viscous expansion of cell walls, and rules for regulation of cell behavior. The study showed that a simple set of autonomous, cell-based rules are sufficient to account for the morphological and dynamic properties of Coleochaete growth. A variety of morphogenetic behavior emerged from the application of these local rules. Cell shape sensing is sufficient to explain the patterns of cell division during growth. This simplifying principle is likely to have application in modeling and design for engineering of higher plant tissues.

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.

Measurement of auxin transport in Arabidopsis thaliana

This protocol allows the measurement of auxin transport in roots, hypocotyls and inflorescences of Arabidopsis thaliana plants by examining transport of radiolabeled auxin or movement of an auxin-induced gene expression signal. The protocol contains four stages: seedling growth, auxin application, a transport period of variable length, and quantification of auxin movement or reporter expression. Beyond the time for plant growth, the transport assay can be completed within 4-18 h. Auxin is applied to seedlings in agar cylinders or droplets, which does not require specialized liquid-handling equipment or micromanipulators, in contrast with methods that apply auxin in liquid droplets. Spatial control of auxin application is reduced, but this method has the advantages of being technically more feasible for most laboratories and allowing agar containing radioactive auxin to be removed for pulse chase assays that determine transport rates. These methods allow investigation of genetic and environmental factors that control auxin transport.

An auxin gradient and maximum in the Arabidopsis root apex shown by high-resolution cell-specific analysis of IAA distribution and synthesis

Local concentration gradients of the plant growth regulator auxin (indole-3-acetic acid [IAA]) are thought to instruct the positioning of organ primordia and stem cell niches and to direct cell division, expansion, and differentiation. High-resolution measurements of endogenous IAA concentrations in support of the gradient hypothesis are required to substantiate this hypothesis. Here, we introduce fluorescence-activated cell sorting of green fluorescent protein-marked cell types combined with highly sensitive mass spectrometry methods as a novel means for analyses of IAA distribution and metabolism at cellular resolution. Our results reveal the presence of IAA concentration gradients within the Arabidopsis thaliana root tip with a distinct maximum in the organizing quiescent center of the root apex. We also demonstrate that the root apex provides an important source of IAA and that cells of all types display a high synthesis capacity, suggesting a substantial contribution of local biosynthesis to auxin homeostasis in the root tip. Our results indicate that local biosynthesis and polar transport combine to produce auxin gradients and maxima in the root tip.

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.

Altered ARA2 (RABA1a) expression in Arabidopsis reveals the involvement of a Rab/YPT family member in auxin-mediated responses

Ras super family proteins serve as molecular switches regulating many different cellular processes. However, given the large number of family members, sequence information has provided little insight into the function of individual proteins. This study examined phenotypic alterations in an Arabidopsis ara2 mutant, in which a Ras super family member-encoding gene is disrupted by a T-DNA insertion. Although one mutant line (Salk_013811) was hypersensitive to auxin, its T-DNA insertion was in the 5'-UTR of ARA2. Thus, we examined a true ARA2 knock-out mutant (Salk_077747) which contains an insertion in the first exon of ARA2. We found that ARA2 expression is responsive to auxin and at low concentrations, ara2 mutant plants exhibit increased numbers of lateral roots. ARA2 overexpression causes plants to exhibit hypersensitivity to auxin, due to altered expression of auxin-responsive genes, and these plants exhibited reduced numbers of lateral roots. A GFP-ARA2 fusion protein localized to the endosomes, suggesting that ARA2 may play a role in vesicle trafficking of components involved in polar auxin transport. Taken together, these results show that ARA2 is an essential component of a pathway that couples auxin signaling to plant growth and development.

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.

Root system architecture from coupling cell shape to auxin transport

Lateral organ position along roots and shoots largely determines plant architecture, and depends on auxin distribution patterns. Determination of the underlying patterning mechanisms has hitherto been complicated because they operate during growth and division. Here, we show by experiments and computational modeling that curvature of the Arabidopsis root influences cell sizes, which, together with tissue properties that determine auxin transport, induces higher auxin levels in the pericycle cells on the outside of the curve. The abundance and position of the auxin transporters restricts this response to the zone competent for lateral root formation. The auxin import facilitator, AUX1, is up-regulated by auxin, resulting in additional local auxin import, thus creating a new auxin maximum that triggers organ formation. Longitudinal spacing of lateral roots is modulated by PIN proteins that promote auxin efflux, and pin2,3,7 triple mutants show impaired lateral inhibition. Thus, lateral root patterning combines a trigger, such as cell size difference due to bending, with a self-organizing system that mediates alterations in auxin transport.

Auxin transport through non-hair cells sustains root-hair development

The plant hormone auxin controls root epidermal cell development in a concentration-dependent manner 1-3. Root hairs are produced on a subset of epidermal cells as they increase in distance from the root tip. Auxin is required for their initiation 4-7 and continued growth 8-11, but little is known about its distribution in this region of the root. Counter to the expectation that hair cells might require active auxin influx to ensure auxin supply, we did not detect the auxin-influx transporter AUX1 in root-hair cells. A high level of AUX1 expression was detected in adjacent non-hair cell files. Non-hair cells were necessary to achieve wild-type root-hair length, although an auxin response was not required in these cells. 3D modelling of auxin flow in the root tip suggests that AUX1-dependent transport through non-hair cells maintains an auxin supply to developing hair cells as they increase in distance from the root tip and sustains root-hair outgrowth. Experimental data support the hypothesis that, instead of moving uniformly though the epidermal cell layer 3,12, auxin is mainly transported through canals that extend longitudinally into the tissue.

Auxin influx carriers stabilize phyllotactic patterning

One of the most striking features of plant architecture is the regular arrangement of leaves and flowers around the stem, known as phyllotaxis. Peaks in concentration of the plant hormone auxin, generated by the polar localization of the PIN1 auxin efflux carrier, provide the instructive signal for primordium initiation. This mechanism generates the spacing between neighboring primordia, which results in regular phyllotaxis. Studies of the role of auxin transport in phyllotactic patterning have focused on PIN1-mediated efflux. Recent computer simulations indicate an additional role for transporter-mediated auxin uptake. Mutations in the AUX1 auxin influx carrier have not, however, been reported to cause an aerial phenotype. Here, we study the role of AUX1 and its paralogs LAX1, LAX2, and LAX3. Analysis of the quadruple mutant reveals irregular divergence angles between successive primordia. A highly unusual aspect of the phenotype is the occurrence of clusters of primordia, in violation of classical theory. At the molecular level, the sharp peaks in auxin levels and coordinated PIN polarization are reduced or lost. In addition, the increased penetrance of the phenotype under short-day conditions suggests that the AUX LAX transporters act to buffer the PIN-mediated patterning mechanism against environmental or developmental influences.

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.

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.

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.

New insight into the biochemical mechanisms regulating auxin transport in plants

The transport of the plant hormone auxin has been under intense investigation since its identification 80 years ago. Studies have gradually refined our understanding of the importance of auxin transport in many aspects of plant signalling and development, and the focus has intensified in recent years towards the identification of the proteins involved in auxin transport and their functional mechanism. Within the past 18 months, the field has progressed rapidly, with confirmation that several distinct classes of proteins, previously dubbed as 'putative auxin permeases' or 'auxin transport facilitators', are bona fide transporters of IAA (indol-3-ylacetic acid). In this review we will appraise the recent transport data and highlight likely future research directions, including the characterization of auxiliary proteins necessary for the regulation of auxin transporters.

Measurement of diffusion within the cell wall in living roots of Arabidopsis thaliana

To quantify the diffusion constant of small molecules in the plant cell wall, fluorescence from carboxyfluorescein (CF) in the intact roots of Arabidopsis thaliana was recorded. Roots were immersed in a solution of the fluorescent dye and viewed through a confocal fluorescence microscope. These roots are sufficiently transparent that much of the apoplast can be imaged. The diffusion coefficient, D(cw), of CF in the cell wall was probed using two protocols: fluorescence recovery after photobleaching and fluorescence loss following perfusion with dye-free solution. Diffusion coefficients were obtained from the kinetics of the fluorescent transients and modelling apoplast geometry. Apoplastic diffusion constants varied spatially in the root. In the elongation zone and mature cortex, D(cw)=(3.2+/-1.4)x10(-11) m(2) s(-1), whereas in mature epidermis, D(cw)=(2.5+/-0.7)x10(-12) m(2) s(-1), at least an order of magnitude lower. Relative to the diffusion coefficient of CF in water, these represent reductions by approximately 1/15 and 1/195, respectively. The low value for mature epidermis is correlated with a suberin-like permeability barrier that was detected with either autofluorescence or berberine staining. This study provides a quantitative estimate of the permeability of plant cell walls to small organic acids-a class of compounds that includes auxin and other plant hormones. These measurements constrain models of solute transport, and are important for quantitative models of hormone signalling during plant growth and development.

PINOID positively regulates auxin efflux in Arabidopsis root hair cells and tobacco cells

Intercellular transport of auxin is mediated by influx and efflux carriers in the plasma membrane and subjected to developmental and environmental regulation. Here, using the auxin-sensitive Arabidopsis thaliana root hair cell system and the tobacco (Nicotiana tabacum) suspension cell system, we demonstrate that the protein kinase PINOID (PID) positively regulates auxin efflux. Overexpression of PID (PIDox) or the auxin efflux carrier component PINFORMED3 (PIN3, PIN3ox), specifically in the root hair cell, greatly suppressed root hair growth. In both PIDox and PIN3ox transformants, root hair growth was nearly restored to wild-type levels by the addition of auxin, protein kinase inhibitors, or auxin efflux inhibitors. Localization of PID or PIN3 at the cell boundary was disrupted by brefeldin A and staurosporine. A mutation in the kinase domain abrogated the ability of PID to localize at the cell boundary and to inhibit root hair growth. These results suggest that PIDox- or PIN3ox-enhanced auxin efflux results in a shortage of intracellular auxin and a subsequent inhibition of root hair growth. In an auxin efflux assay using transgenic tobacco suspension cells, PIDox or PIN3ox also enhanced auxin efflux. Collectively, these results suggest that PID positively regulates cellular auxin efflux, most likely by modulating the trafficking of PIN and/or some other molecular partners involved in auxin efflux.

A constant production hypothesis guides leaf venation patterning

We propose a theoretical mechanism that enables the elaboration of veins to supply distant cells during leaf development. In contrast to the more standard view that a signal (e.g., auxin) is produced at isolated sites to stimulate growth, we determine the consequences of the hypothesis that auxin is produced at a constant rate in every cell. High concentration sites for auxin emerge naturally in a reaction-diffusion model, together with global information about leaf shape and existing venation. Because the global information is encoded as auxin concentration and its gradient, those signals provide individual cells with sufficient information to determine their own fate. Unlike other models, a single substance suffices for the reaction-diffusion at early, but not initial, stages of development. Neither complex interactions nor predetermination are necessary. We predict angiosperm areolation patterns in simulation, and our model further implies the Sachs Canalization Hypothesis and resolves a dilemma regarding the role of auxin in cell growth.

Control of leaf vascular patterning by polar auxin transport

The formation of the leaf vascular pattern has fascinated biologists for centuries. In the early leaf primordium, complex networks of procambial cells emerge from homogeneous subepidermal tissue. The molecular nature of the underlying positional information is unknown, but various lines of evidence implicate gradually restricted transport routes of the plant hormone auxin in defining sites of procambium formation. Here we show that a crucial member of the AtPIN family of auxin-efflux-associated proteins, AtPIN1, is expressed prior to pre-procambial and procambial cell fate markers in domains that become restricted toward sites of procambium formation. Subcellular AtPIN1 polarity indicates that auxin is directed to distinct "convergence points" in the epidermis, from where it defines the positions of major veins. Integrated polarities in all emerging veins indicate auxin drainage toward pre-existing veins, but veins display divergent polarities as they become connected at both ends. Auxin application and transport inhibition reveal that convergence point positioning and AtPIN1 expression domain dynamics are self-organizing, auxin-transport-dependent processes. We derive a model for self-regulated, reiterative patterning of all vein orders and postulate at its onset a common epidermal auxin-focusing mechanism for major-vein positioning and phyllotactic patterning.