Auxin-regulated cell polarity: an inside job?

Leaf Vein Patterning

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

Computer models of cell polarity establishment in plants

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

Axes and polarities in leaf vein formation

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

Computational Models of Auxin-Driven Patterning in Shoots

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

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

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

Signaling network regulating plant branching: Recent advances and new challenges

Auxin alone or supplemented with cytokinins and strigolactones were long considered as the main player(s) in the control of apical dominance (AD) and correlative inhibition of the lateral bud outgrowth, the processes that shape the plant phenotype. However, past decade data indicate a more sophisticated pathways of AD regulation, with the involvement of mobile carbohydrates which perform both signal and trophic functions. Here we provide a critical comprehensive overview of the current status of the AD problem. This includes insight into intimate mechanisms regulating directed auxin transport in axillary buds with participation of phytohormones and sugars. Also roles of auxin, cytokinin and sugars in the dormancy or sustained growth of the lateral meristems were assigned. This review not only provides the latest data on implicated phytohormone crosstalk and its relationship with the signaling of sugars and abscisic acid, new AD players, but also focuses on the emerging biochemical mechanisms, at first positive feedback loops involving both sugars and hormones, that ensure the sustained bud growth. Data show that sugars act in concert with cytokinins but antagonistically to strigolactone signaling. A complex bud growth regulating network is demonstrated and unresolved issues regarding the hormone-carbohydrate regulation of AD are highlighted.

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

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

The canalization hypothesis - challenges and alternatives

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

Auxin transport model for leaf venation

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

Auxin-driven patterning with unidirectional fluxes

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

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

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

Auxin perception is required for arbuscule development in arbuscular mycorrhizal symbiosis

Most land plant species live in symbiosis with arbuscular mycorrhizal fungi. These fungi differentiate essential functional structures called arbuscules in root cortical cells from which mineral nutrients are released to the plant. We investigated the role of microRNA393 (miR393), an miRNA that targets several auxin receptors, in arbuscular mycorrhizal root colonization. Expression of the precursors of the miR393 was down-regulated during mycorrhization in three different plant species: Solanum lycopersicum, Medicago truncatula, and Oryza sativa. Treatment of S. lycopersicum, M. truncatula, and O. sativa roots with concentrations of synthetic auxin analogs that did not affect root development stimulated mycorrhization, particularly arbuscule formation. DR5-GUS, a reporter for auxin response, was preferentially expressed in root cells containing arbuscules. Finally, overexpression of miR393 in root tissues resulted in down-regulation of auxin receptor genes (transport inhibitor response1 and auxin-related F box) and underdeveloped arbuscules in all three plant species. These results support the conclusion that miR393 is a negative regulator of arbuscule formation by hampering auxin perception in arbuscule-containing cells.

Novel ABP1-TMK auxin sensing system controls ROP GTPase-mediated interdigitated cell expansion in Arabidopsis

ROP GTPases (Rho-like GTPase from plants), plant counterparts of animal and fungal Rho-family GTPases, have recently been shown to be key components of a novel signaling pathway activated by the plant hormone auxin. Auxin (indole acetic acid) is a key regulator of virtually every aspect of plant growth and development, yet the molecular mechanisms of auxin responses remain largely unknown. AUXIN BINDING PROTEIN1 (ABP1) is an ancient protein that binds auxin and has been implied as a receptor for a number of auxin responses, but its precise mechanism remains unresolved. A paradox for ABP1's action is that it is predominantly found in the endoplasmic reticulum (ER) lumen, while it has been implicated as a cell surface auxin receptor, functionally distinct from the nuclear TIR1/AFB auxin receptor family that regulates transcriptional responses. Since our group reported that ABP1 is required for activating two antagonizing ROP signaling pathways involved in cytoskeletal reorganization and cell shape formation in Arabidopsis leaf pavement cells, we recently further showed that the plasma membrane-localized TMK receptor-like kinases functionally interact in a complex with ABP1 and are required for ABP1-dependent activation of ROP GTPases by auxin. The formation of this cell surface complex is induced by auxin and requires functional ABP1. These exciting findings provide convincing evidence for this novel auxin sensing system on the cell surface and suggest intriguing mechanisms for TMKs being functional partners of ABP1 to transmit extracellular auxin signal to intracellular ROP signaling module during polar cell expansion.

A division in PIN-mediated auxin patterning during organ initiation in grasses

The hormone auxin plays a crucial role in plant morphogenesis. In the shoot apical meristem, the PIN-FORMED1 (PIN1) efflux carrier concentrates auxin into local maxima in the epidermis, which position incipient leaf or floral primordia. From these maxima, PIN1 transports auxin into internal tissues along emergent paths that pattern leaf and stem vasculature. In Arabidopsis thaliana, these functions are attributed to a single PIN1 protein. Using phylogenetic and gene synteny analysis we identified an angiosperm PIN clade sister to PIN1, here termed Sister-of-PIN1 (SoPIN1), which is present in all sampled angiosperms except for Brassicaceae, including Arabidopsis. Additionally, we identified a conserved duplication of PIN1 in the grasses: PIN1a and PIN1b. In Brachypodium distachyon, SoPIN1 is highly expressed in the epidermis and is consistently polarized toward regions of high expression of the DR5 auxin-signaling reporter, which suggests that SoPIN1 functions in the localization of new primordia. In contrast, PIN1a and PIN1b are highly expressed in internal tissues, suggesting a role in vascular patterning. PIN1b is expressed in broad regions spanning the space between new primordia and previously formed vasculature, suggesting a role in connecting new organs to auxin sinks in the older tissues. Within these regions, PIN1a forms narrow canals that likely pattern future veins. Using a computer model, we reproduced the observed spatio-temporal expression and localization patterns of these proteins by assuming that SoPIN1 is polarized up the auxin gradient, and PIN1a and PIN1b are polarized to different degrees with the auxin flux. Our results suggest that examination and modeling of PIN dynamics in plants outside of Brassicaceae will offer insights into auxin-driven patterning obscured by the loss of the SoPIN1 clade in Brassicaceae.

Computational models and functional studies using the plant Arabidopsis thaliana have led to competing models for how the PIN-FORMED1 (PIN1) auxin transporter polarizes in the cell to create both the maxima required for organ initiation and the narrow streams required for vein patterning. Here we identify a previously uncharacterized PIN protein most closely related to PIN1 that is present in all flowering plants but lost in the Brassicaceae, including Arabidopsis. We localized this protein, here termed Sister-of-PIN1 (SoPIN1), along with duplicate members of PIN1 (PIN1a and PIN1b), in two grass species. Our localization data provide striking evidence for a spatial and temporal split between SoPIN1 and the two PIN1s during organ initiation in grasses. Based on our localization results we created a computational model showing that the observed patterns of expression and polarization of the grass PINs can emerge assuming SoPIN1 polarizes up the gradient of auxin concentration while the PIN1 members polarize with the auxin flux. This model reveals a minimal framework of necessary functions involved in auxin-transport-mediated patterning in the shoot and demonstrates that work outside of Arabidopsis is essential to understanding how auxin-transport mediates patterning in most flowering plants.

Polarity, continuity, and alignment in plant vascular strands

Plant vascular cells are joined end to end along uninterrupted lines to connect shoot organs with roots; vascular strands are thus polar, continuous, and internally aligned. What controls the formation of vascular strands with these properties? The "auxin canalization hypothesis"-based on positive feedback between auxin flow through a cell and the cell's capacity for auxin transport-predicts the selection of continuous files of cells that transport auxin polarly, thus accounting for the polarity and continuity of vascular strands. By contrast, polar, continuous auxin transport-though required-is insufficient to promote internal alignment of vascular strands, implicating additional factors. The auxin canalization hypothesis was derived from the response of mature tissue to auxin application but is consistent with molecular and cellular events in embryo axis formation and shoot organ development. Objections to the hypothesis have been raised based on vascular organizations in callus tissue and shoot organs but seem unsupported by available evidence. Other objections call instead for further research; yet the inductive and orienting influence of auxin on continuous vascular differentiation remains unique.

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.

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.

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.

Computational models of plant development and form

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

On evaluating models in Computational Morphodynamics

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

Simulation of organ patterning on the floral meristem using a polar auxin transport model

An intriguing phenomenon in plant development is the timing and positioning of lateral organ initiation, which is a fundamental aspect of plant architecture. Although important progress has been made in elucidating the role of auxin transport in the vegetative shoot to explain the phyllotaxis of leaf formation in a spiral fashion, a model study of the role of auxin transport in whorled organ patterning in the expanding floral meristem is not available yet. We present an initial simulation approach to study the mechanisms that are expected to play an important role. Starting point is a confocal imaging study of Arabidopsis floral meristems at consecutive time points during flower development. These images reveal auxin accumulation patterns at the positions of the organs, which strongly suggests that the role of auxin in the floral meristem is similar to the role it plays in the shoot apical meristem. This is the basis for a simulation study of auxin transport through a growing floral meristem, which may answer the question whether auxin transport can in itself be responsible for the typical whorled floral pattern. We combined a cellular growth model for the meristem with a polar auxin transport model. The model predicts that sepals are initiated by auxin maxima arising early during meristem outgrowth. These form a pre-pattern relative to which a series of smaller auxin maxima are positioned, which partially overlap with the anlagen of petals, stamens, and carpels. We adjusted the model parameters corresponding to properties of floral mutants and found that the model predictions agree with the observed mutant patterns. The predicted timing of the primordia outgrowth and the timing and positioning of the sepal primordia show remarkable similarities with a developing flower in nature.

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.

Positional information by differential endocytosis splits auxin response to drive Arabidopsis root meristem growth

In the Arabidopsis root meristem, polar auxin transport creates a transcriptional auxin response gradient that peaks at the stem cell niche and gradually decreases as stem cell daughters divide and differentiate [1-3]. The amplitude and extent of this gradient are essential for both stem cell maintenance and root meristem growth [4, 5]. To investigate why expression of some auxin-responsive genes, such as the essential root meristem growth regulator BREVIS RADIX (BRX) [6], deviates from this gradient, we combined experimental and computational approaches. We created cellular-level root meristem models that accurately reproduce distribution of nuclear auxin activity and allow dynamic modeling of regulatory processes to guide experimentation. Expression profiles deviating from the auxin gradient could only be modeled after intersection of auxin activity with the observed differential endocytosis pattern and positive autoregulatory feedback through plasma-membrane-to-nucleus transfer of BRX. Because BRX is required for expression of certain auxin response factor targets, our data suggest a cell-type-specific endocytosis-dependent input into transcriptional auxin perception. This input sustains expression of a subset of auxin-responsive genes across the root meristem's division and transition zones and is essential for meristem growth. Thus, the endocytosis pattern provides specific positional information to modulate auxin response.

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.

AuxV: a database of auxin transport velocities

One of the most widely used techniques to quantify polar auxin transport is the measurement of auxin speed. To date there have been more than 90 published reports of auxin speed in 44 species. We have collected available speed measurements into a database, along with information on plant growth conditions and growth rate. Measured auxin speeds have a range of 1.2-18 mm/h, and show notable correlations with organ type, growth rate, and plant clade.

Is ABP1 an auxin receptor yet?

AUXIN BINDING PROTEIN 1 (ABP1) has long been proposed as an auxin receptor to regulate cell expansion. The embryo lethality of ABP1-null mutants demonstrates its fundamental role in plant development, but also hinders investigation of its involvement in post-embryonic processes and its mode of action. By taking advantage of weak alleles and inducible systems, several recent studies have revealed a role for ABP1 in organ development, cell polarization, and shape formation. In addition to its role in the regulation of auxin-induced gene expression, ABP1 has now been shown to modulate non-transcriptional auxin responses. ABP1 is required for activating two antagonizing ROP GTPase signaling pathways involved in cytoskeletal reorganization and cell shape formation, and participates in the regulation of clathrin-mediated endocytosis to subsequently affect PIN protein distribution. These exciting discoveries provide indisputable evidence for the auxin-induced signaling pathways that are downstream of ABP1 function, and suggest intriguing mechanisms for ABP1-mediated polar cell expansion and spatial coordination in response to auxin.

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.

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.

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.

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.

The march of the PINs: developmental plasticity by dynamic polar targeting in plant cells

Development of plants and their adaptive capacity towards ever-changing environmental conditions largely depend on the spatial distribution of the plant hormone auxin. At the cellular level, various internal and external signals are translated into specific changes in the polar, subcellular localization of auxin transporters from the PIN family thereby directing and redirecting the intercellular fluxes of auxin. The current model of polar targeting of PIN proteins towards different plasma membrane domains encompasses apolar secretion of newly synthesized PINs followed by endocytosis and recycling back to the plasma membrane in a polarized manner. In this review, we follow the subcellular march of the PINs and highlight the cellular and molecular mechanisms behind polar foraging and subcellular trafficking pathways. Also, the entry points for different signals and regulations including by auxin itself will be discussed within the context of morphological and developmental consequences of polar targeting and subcellular trafficking.

Modelling polar auxin transport in developmental patterning

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

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

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

AUXIN BINDING PROTEIN 1: functional and evolutionary aspects

In this review, we examine the role of AUXIN BINDING PROTEIN 1 (ABP1) in mediating growth and developmental responses. ABP1 is involved in a broad range of cellular responses to auxin, acting either as the main regulator of the response, such as seen for entry into cell division or, as a fine-tuning device as for the regulation of expression of early auxin response genes. Phylogenetic analysis has revealed that ABP1 is an ancient protein that was already present in various algae and has acquired a motif of retention in the endoplasmic reticulum only recently. An evaluation of the evidence for ABP1 function according to its cellular localization supports the plasma membrane as a starting point for ABP1-mediated auxin signaling.

Recent progress in auxin biology

Like animals, plants have evolved into complex organisms. Developmental cohesion between tissues and cells is possible due to signaling molecules (messengers) like hormones. The first hormone discovered in plants was auxin. This phytohormone was first noticed because of its involvement in the response to directional light. Nowadays, auxin has been established as a central key player in the regulation of plant growth and development and in responses to environmental changes. At the cellular level, auxin controls division, elongation, and differentiation as well as the polarity of the cell. Auxin, to integrate so many different signals, needs to be regulated at many different levels. A tight regulation of auxin synthesis, activity, degradation as well as transport has been demonstrated. Another possibility to modulate auxin signaling is to modify the capacity of response of the cells by expressing differentially the signaling components. In this review, we provide an overview of the present knowledge in auxin biology, with emphasis on root development.

Auxin-binding proteins without KDEL sequence in the moss Funaria hygrometrica

Whereas the important plant growth regulator auxin has multiple effects in flowering plants, it induces a specific cell differentiation step in the filamentous moss protonema. Here, we analyse the presence of classical auxin-binding protein (ABP1) homologues in the moss Funaria hygrometrica. Microsomal membranes isolated from protonemata of F. hygrometrica have specific indole acetic acid-binding sites, estimated to be about 3-5 pmol/mg protein with an apparent dissociation constant (K (d)) between 3 and 5 microM. Western analyses with anti-ABP1 antiserum detected the canonical endoplasmic reticulum (ER)-localised 22-24 kDa ABP1 in Zea mays, but not in F. hygrometrica. Instead, polypeptides of 31-33 and 46 kDa were labelled in the moss as well as in maize. In F. hygrometrica these proteins were found exclusively in microsomal membrane fractions and were confirmed as ABPs by photo-affinity labelling with 5-azido-[7-(3)H]-indole-3-acetic acid. Unlike the classical corn ABP1, these moss ABPs did not contain the KDEL ER retention sequence. Consistently, the fully sequenced genome of the moss Physcomitrella patens, a close relative of F. hygrometrica, encodes an ABP1-homologue without KDEL sequence. Our study suggests the presence of putative ABPs in F. hygrometrica that share immunological epitopes with ABP1 and bind auxin but are different from the classical corn ABP1.

Control of bud activation by an auxin transport switch

In many plant species only a small proportion of buds yield branches. Both the timing and extent of bud activation are tightly regulated to produce specific branching architectures. For example, the primary shoot apex can inhibit the activation of lateral buds. This process is termed apical dominance and is dependent on the plant hormone auxin moving down the main stem in the polar auxin transport stream. We use a computational model and mathematical analysis to show that apical dominance can be explained in terms of an auxin transport switch established by the temporal precedence between competing auxin sources. Our model suggests a mechanistic basis for the indirect action of auxin in bud inhibition and captures the effects of diverse genetic and physiological manipulations. In particular, the model explains the surprising observation that highly branched Arabidopsis phenotypes can exhibit either high or low auxin transport.

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.