Canalization without flux sensors: a traveling-wave hypothesis

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.

Organ Patterning at the Shoot Apical Meristem (SAM): The Potential Role of the Vascular System

Auxin, which is transported in the outermost cell layer, is one of the major players involved in plant organ initiation and positioning at the shoot apical meristem (SAM). However, recent studies have recognized the role of putative internal signals as an important factor collaborating with the well-described superficial pathway of organogenesis regulation. Different internal signals have been proposed; however, their nature and transport route have not been precisely determined. Therefore, in this mini-review, we aimed to summarize the current knowledge regarding the auxin-dependent regulation of organ positioning at the SAM and to discuss the vascular system as a potential route for internal signals. In addition, as regular organ patterning is a universal phenomenon, we focus on the role of the vasculature in this process in the major lineages of land plants, i.e., bryophytes, lycophytes, ferns, gymnosperms, and angiosperms.

Scaling relations for auxin waves

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

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

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

Computational Models of Auxin-Driven Patterning in Shoots

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

Modeling Plant Tissue Development Using VirtualLeaf

Cell-based computational modeling and simulation are becoming invaluable tools in analyzing plant development. In a cell-based simulation model, the inputs are behaviors and dynamics of individual cells and the rules describing responses to signals from adjacent cells. The outputs are the growing tissues, shapes, and cell-differentiation patterns that emerge from the local, chemical, and biomechanical cell-cell interactions. In this updated and extended version of our previous chapter on VirtualLeaf (Merks and Guravage, Methods in Molecular Biology 959, 333-352), we present a step-by-step, practical tutorial for building cell-based simulations of plant development and for analyzing the influence of parameters on simulation outcomes by systematically changing the values of the parameters and analyzing each outcome. We show how to build a model of a growing tissue, a reaction-diffusion system on a growing domain, and an auxin transport model. Moreover, in addition to the previous publication, we demonstrate how to run a Turing system on a regular, rectangular lattice, and how to run parameter sweeps. The aim of VirtualLeaf is to make computational modeling more accessible to experimental plant biologists with relatively little computational background.

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.

How grass keeps growing: an integrated analysis of hormonal crosstalk in the maize leaf growth zone

We studied the maize leaf to understand how long-distance signals, auxin and cytokinin, control leaf growth dynamics. We constructed a mathematical model describing the transport of these hormones along the leaf growth zone and their interaction with the local gibberellin (GA) metabolism in the control of cell division. Assuming gradually declining auxin and cytokinin supply at the leaf base, the model generated spatiotemporal hormone distribution and growth patterns that matched experimental data. At the cellular level, the model predicted a basal leaf growth as a result of cell division driven by auxin and cytokinin. Superimposed on this, GA synthesis regulated growth through the control of the size of the region of active cell division. The predicted hormone and cell length distributions closely matched experimental data. To correctly predict the leaf growth profiles and final organ size of lines with reduced or elevated GA production, the model required a signal proportional to the size of the emerged part of the leaf that inhibited the basal leaf growth driven by auxin and cytokinin. Excision and shading of the emerged part of the growing leaf allowed us to demonstrate that this signal exists and depends on the perception of light intensity.

Mathematical Modelling of Auxin Transport in Plant Tissues: Flux Meets Signalling and Growth

Plant hormone auxin has critical roles in plant growth, dependent on its heterogeneous distribution in plant tissues. Exactly how auxin transport and developmental processes such as growth coordinate to achieve the precise patterns of auxin observed experimentally is not well understood. Here we use mathematical modelling to examine the interplay between auxin dynamics and growth and their contribution to formation of patterns in auxin distribution in plant tissues. Mathematical models describing the auxin-related signalling pathway, PIN and AUX1 dynamics, auxin transport, and cell growth in plant tissues are derived. A key assumption of our models is the regulation of PIN proteins by the auxin-responsive ARF-Aux/IAA signalling pathway, with upregulation of PIN biosynthesis by ARFs. Models are analysed and solved numerically to examine the long-time behaviour and auxin distribution. Changes in auxin-related signalling processes are shown to be able to trigger transition between passage- and spot-type patterns in auxin distribution. The model was also shown to be able to generate isolated cells with oscillatory dynamics in levels of components of the auxin signalling pathway which could explain oscillations in levels of ARF targets that have been observed experimentally. Cell growth was shown to have influence on PIN polarisation and determination of auxin distribution patterns. Numerical simulation results indicate that auxin-related signalling processes can explain the different patterns in auxin distributions observed in plant tissues, whereas the interplay between auxin transport and growth can explain the ‘reverse-fountain’ pattern in auxin distribution observed at plant root tips.

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

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

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

Auxin-mediated regulation of vascular patterning in Arabidopsis thaliana leaves

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

A Critical View on ABC Transporters and Their Interacting Partners in Auxin Transport

Different subclasses of ATP-binding cassette (ABC) transporters have been implicated in the transport of native variants of the phytohormone auxin. Here, the putative, individual roles of key members belonging to the ABCB, ABCD and ABCG families, respectively, are highlighted and the knowledge of their assumed expression and transport routes is reviewed and compared with their mutant phenotypes. Protein-protein interactions between ABC transporters and regulatory components during auxin transport are summarized and their importance is critically discussed. There is a focus on the functional interaction between members of the ABCB family and the FKBP42, TWISTED DWARF1, acting as a chaperone during plasma membrane trafficking of ABCBs. Further, the mode and relevance of functional ABCB-PIN interactions is diagnostically re-evaluated. A new nomenclature describing precisely the most likely ABCB-PIN interaction scenarios is suggested. Finally, available tools for the detection and prediction of ABC transporter interactomes are summarized and the potential of future ABC transporter interactome maps is highlighted.

Immunogold-EM analysis reveal brefeldin a-sensitive clusters of auxin in Arabidopsis root apex cells

Immunogold electron microscopy (EM) study of Arabidopsis root apices analyzed using specific IAA antibody and high-pressure freeze fixation technique allowed, for the first time, vizualization of subcellular localization of IAA in cells assembled intactly within plant tissues. Our quantitative analysis reveals that there is considerable portion of IAA gold particles that clusters within vesicles and membraneous compartments in all root apex cells. There are clear tissue-specific and developmental differences of clustered IAA in root apices. These findings have significant consequences for our understanding of this small molecule which is controlling plant growth, development and behavior.

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

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

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

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

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.

MorphoGraphX: A platform for quantifying morphogenesis in 4D

Morphogenesis emerges from complex multiscale interactions between genetic and mechanical processes. To understand these processes, the evolution of cell shape, proliferation and gene expression must be quantified. This quantification is usually performed either in full 3D, which is computationally expensive and technically challenging, or on 2D planar projections, which introduces geometrical artifacts on highly curved organs. Here we present MorphoGraphX ( www.MorphoGraphX.org), a software that bridges this gap by working directly with curved surface images extracted from 3D data. In addition to traditional 3D image analysis, we have developed algorithms to operate on curved surfaces, such as cell segmentation, lineage tracking and fluorescence signal quantification. The software's modular design makes it easy to include existing libraries, or to implement new algorithms. Cell geometries extracted with MorphoGraphX can be exported and used as templates for simulation models, providing a powerful platform to investigate the interactions between shape, genes and growth.

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

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

A plant cell division algorithm based on cell biomechanics and ellipse-fitting

BACKGROUND AND AIMS

The importance of cell division models in cellular pattern studies has been acknowledged since the 19th century. Most of the available models developed to date are limited to symmetric cell division with isotropic growth. Often, the actual growth of the cell wall is either not considered or is updated intermittently on a separate time scale to the mechanics. This study presents a generic algorithm that accounts for both symmetrically and asymmetrically dividing cells with isotropic and anisotropic growth. Actual growth of the cell wall is simulated simultaneously with the mechanics.

METHODS

The cell is considered as a closed, thin-walled structure, maintained in tension by turgor pressure. The cell walls are represented as linear elastic elements that obey Hooke's law. Cell expansion is induced by turgor pressure acting on the yielding cell-wall material. A system of differential equations for the positions and velocities of the cell vertices as well as for the actual growth of the cell wall is established. Readiness to divide is determined based on cell size. An ellipse-fitting algorithm is used to determine the position and orientation of the dividing wall. The cell vertices, walls and cell connectivity are then updated and cell expansion resumes. Comparisons are made with experimental data from the literature.

KEY RESULTS

The generic plant cell division algorithm has been implemented successfully. It can handle both symmetrically and asymmetrically dividing cells coupled with isotropic and anisotropic growth modes. Development of the algorithm highlighted the importance of ellipse-fitting to produce randomness (biological variability) even in symmetrically dividing cells. Unlike previous models, a differential equation is formulated for the resting length of the cell wall to simulate actual biological growth and is solved simultaneously with the position and velocity of the vertices.

CONCLUSIONS

The algorithm presented can produce different tissues varying in topological and geometrical properties. This flexibility to produce different tissue types gives the model great potential for use in investigations of plant cell division and growth in silico.

Plant cells use auxin efflux to explore geometry

Cell movement is the central mechanism for animal morphogenesis. Plant cell development rather relies on flexible alignment of cell axis adjusting cellular differentiation to directional cues. As central input, vectorial fields of mechanical stress and gradients of the phytohormone auxin have been discussed. In tissue contexts, mechanical and chemical signals will always act in concert; experimentally it is difficult to dissect their individual roles. We have designed a novel approach, based on cells, where directionality has been eliminated by removal of the cell wall. We impose a new axis using a microfluidic set-up to generate auxin gradients. Rectangular microvessels are integrated orthogonally with the gradient. Cells in these microvessels align their new axis with microvessel geometry before touching the wall. Auxin efflux is necessary for this touch-independent geometry exploration and we suggest a model, where auxin gradients can be used to align cell axis in tissues with minimized mechanical tensions.

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.

Canalization: what the flux?

Polarized transport of the hormone auxin plays crucial roles in many processes in plant development. A self-organizing pattern of auxin transport--canalization--is thought to be responsible for vascular patterning and shoot branching regulation in flowering plants. Mathematical modeling has demonstrated that membrane localization of PIN-FORMED (PIN)-family auxin efflux carriers in proportion to net auxin flux can plausibly explain canalization and possibly other auxin transport phenomena. Other plausible models have also been proposed, and there has recently been much interest in producing a unified model of all auxin transport phenomena. However, it is our opinion that lacunae in our understanding of auxin transport biology are now limiting progress in developing the next generation of models. Here we examine several key areas where significant experimental advances are necessary to address both biological and theoretical aspects of auxin transport, including the possibility of a unified transport model.

Root apex transition zone as oscillatory zone

Root apex of higher plants shows very high sensitivity to environmental stimuli. The root cap acts as the most prominent plant sensory organ; sensing diverse physical parameters such as gravity, light, humidity, oxygen, and critical inorganic nutrients. However, the motoric responses to these stimuli are accomplished in the elongation region. This spatial discrepancy was solved when we have discovered and characterized the transition zone which is interpolated between the apical meristem and the subapical elongation zone. Cells of this zone are very active in the cytoskeletal rearrangements, endocytosis and endocytic vesicle recycling, as well as in electric activities. Here we discuss the oscillatory nature of the transition zone which, together with several other features of this zone, suggest that it acts as some kind of command center. In accordance with the early proposal of Charles and Francis Darwin, cells of this root zone receive sensory information from the root cap and instruct the motoric responses of cells in the elongation zone.

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.

An intracellular partitioning-based framework for tissue cell polarity in plants and animals

Tissue cell polarity plays a major role in plant and animal development. We propose that a fundamental building block for tissue cell polarity is the process of intracellular partitioning, which can establish individual cell polarity in the absence of asymmetric cues. Coordination of polarities may then arise through cell-cell coupling, which can operate directly, through membrane-spanning complexes, or indirectly, through diffusible molecules. Polarity is anchored to tissues through organisers located at boundaries. We show how this intracellular partitioning-based framework can be applied to both plant and animal systems, allowing different processes to be placed in a common evolutionary and mechanistic context.

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.

Building simulation models of developing plant organs using VirtualLeaf

Cell-based computational modeling and simulation are becoming invaluable tools in analyzing plant -development. In a cell-based simulation model, the inputs are behaviors and dynamics of individual cells and the rules describe responses to signals from adjacent cells. The outputs are the growing tissues, shapes and cell-differentiation patterns that emerge from the local, chemical and biomechanical cell-cell interactions. Here, we present a step-by-step, practical tutorial for building cell-based simulations of plant development with VirtualLeaf, a freely available, open-source software framework for modeling plant development. We show how to build a model of a growing tissue, a reaction-diffusion system on a growing domain, and an auxin transport model. The aim of VirtualLeaf is to make computational modeling better accessible to experimental plant biologists with relatively little computational background.

Modelling auxin efflux carrier phosphorylation and localization

Regulation of the activity and localization of PIN-FORMED (PIN) membrane proteins, which facilitate efflux of the plant hormone auxin from cells, is important for plants to respond to environmental stimuli and to develop new organs. The protein kinase PINOID (PID) is involved in regulating PIN phosphorylation, and this is thought to affect PIN localization by biasing recycling towards shootwards (apical) (rather than rootwards (basal)) membrane domains. PID has been observed to undergo transient internalization following auxin treatment, and it has been suggested that this may be a result of calcium-dependent sequestration of PID by the calcium-binding protein TOUCH3 (TCH3). We present a mathematical formulation of these processes and examine the resulting steady-state and time-dependent behaviours in response to transient increases in cytosolic calcium. We further combine this model with one for the recycling of PINs in polarized cells and also examine its behaviour. The results provide insight into the behaviour observed experimentally and provide the basis for subsequent studies of the tissue-level implications of these subcellular processes for phenomena such as gravitropism.

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

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

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

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

Towards mechanistic models of plant organ growth

Modelling and simulation are increasingly used as tools in the study of plant growth and developmental processes. By formulating experimentally obtained knowledge as a system of interacting mathematical equations, it becomes feasible for biologists to gain a mechanistic understanding of the complex behaviour of biological systems. In this review, the modelling tools that are currently available and the progress that has been made to model plant development, based on experimental knowledge, are described. In terms of implementation, it is argued that, for the modelling of plant organ growth, the cellular level should form the cornerstone. It integrates the output of molecular regulatory networks to two processes, cell division and cell expansion, that drive growth and development of the organ. In turn, these cellular processes are controlled at the molecular level by hormone signalling. Therefore, combining a cellular modelling framework with regulatory modules for the regulation of cell division, expansion, and hormone signalling could form the basis of a functional organ growth simulation model. The current state of progress towards this aim is that the regulation of the cell cycle and hormone transport have been modelled extensively and these modules could be integrated. However, much less progress has been made on the modelling of cell expansion, which urgently needs to be addressed. A limitation of the current generation models is that they are largely qualitative. The possibilities to characterize existing and future models more quantitatively will be discussed. Together with experimental methods to measure crucial model parameters, these modelling techniques provide a basis to develop a Systems Biology approach to gain a fundamental insight into the relationship between gene function and whole organ behaviour.

Computational models of plant development and form

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

On evaluating models in Computational Morphodynamics

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

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

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

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.

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.

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

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

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.

VirtualLeaf: an open-source framework for cell-based modeling of plant tissue growth and development

Plant organs, including leaves and roots, develop by means of a multilevel cross talk between gene regulation, patterned cell division and cell expansion, and tissue mechanics. The multilevel regulatory mechanisms complicate classic molecular genetics or functional genomics approaches to biological development, because these methodologies implicitly assume a direct relation between genes and traits at the level of the whole plant or organ. Instead, understanding gene function requires insight into the roles of gene products in regulatory networks, the conditions of gene expression, etc. This interplay is impossible to understand intuitively. Mathematical and computer modeling allows researchers to design new hypotheses and produce experimentally testable insights. However, the required mathematics and programming experience makes modeling poorly accessible to experimental biologists. Problem-solving environments provide biologically intuitive in silico objects ("cells", "regulation networks") required for setting up a simulation and present those to the user in terms of familiar, biological terminology. Here, we introduce the cell-based computer modeling framework VirtualLeaf for plant tissue morphogenesis. The current version defines a set of biologically intuitive C++ objects, including cells, cell walls, and diffusing and reacting chemicals, that provide useful abstractions for building biological simulations of developmental processes. We present a step-by-step introduction to building models with VirtualLeaf, providing basic example models of leaf venation and meristem development. VirtualLeaf-based models provide a means for plant researchers to analyze the function of developmental genes in the context of the biophysics of growth and patterning. VirtualLeaf is an ongoing open-source software project (http://virtualleaf.googlecode.com) that runs on Windows, Mac, and Linux.

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.

A plausible mechanism for auxin patterning along the developing root

Background

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

Results

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

Conclusion

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

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

BACKGROUND

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

RESULTS

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

CONCLUSIONS

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

Auxin 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.