Self-organization of the vascular system in plant leaves: Inter-dependent dynamics of auxin flux and carrier proteins

Connecting emerging with existing vasculature above and below ground

The vascular system was essential for plants to colonize land by facilitating the transport of water, nutrients, and minerals throughout the body. Our current knowledge on the molecular-genetic control of vascular tissue specification and differentiation is mostly based on studies in the Arabidopsis primary root. To what degree these regulatory mechanisms in the root meristem can be extrapolated to vascular tissue development in other organs is a question of great interest. In this review, we discuss the most recent progress on cotyledon vein formation, with a focus on polar auxin transport-dependent and -independent mechanisms. We also provide an overview of vasculature formation in postembryonic organs, namely lateral roots, which is more complex than anticipated as several tissues of the parent root must act in a spatio-temporally coordinated manner.

Computer models of cell polarity establishment in plants

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

Axes and polarities in leaf vein formation

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

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

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

Computational Models of Auxin-Driven Patterning in Shoots

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

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

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

What remains of the evidence for auxin feedback on PIN polarity patterns?

In light of recent findings, the feedback between auxin and PIN that plays a major role in models for self-organized auxin patterning requires revisiting.

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.

Shaping leaf vein pattern by auxin and mechanical feedback

This article comments on: Kneuper I, Teale W, Dawson JE, Tsugeki R, Katifori E, Palme K, Ditengou FA. 2021. Auxin biosynthesis and cellular efflux act together to regulate leaf vein patterning. Journal of Experimental Botany 72, 1151–1165.

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.

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.

Phenotypes of Vascular Flow Networks

Complex distribution networks are pervasive in biology. Examples include nutrient transport in the slime mold Physarum polycephalum as well as mammalian and plant venation. Adaptive rules are believed to guide development of these networks and lead to a reticulate, hierarchically nested topology that is both efficient and resilient against perturbations. However, as of yet, no mechanism is known that can generate such networks on all scales. We show how hierarchically organized reticulation can be constructed and maintained through spatially correlated load fluctuations on a particular length scale. We demonstrate that the network topologies generated represent a trade-off between optimizing transport efficiency, construction cost, and damage robustness and identify the Pareto-efficient front that evolution is expected to favor and select for. We show that the typical fluctuation length scale controls the position of the networks on the Pareto front and thus on the spectrum of venation phenotypes.

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.

Systems Biology Approach Pinpoints Minimum Requirements for Auxin Distribution during Fruit Opening

The phytohormone auxin is implied in steering various developmental decisions during plant morphogenesis in a concentration-dependent manner. Auxin maxima have been shown to maintain meristematic activity, for example, of the root apical meristem, and position new sites of outgrowth, such as during lateral root initiation and phyllotaxis. More recently, it has been demonstrated that sites of auxin minima also provide positional information. In the developing Arabidopsis fruit, auxin minima are required for correct differentiation of the valve margin. It remains unclear, however, how this auxin minimum is generated and maintained. Here, we employ a systems biology approach to model auxin transport based on experimental observations. This allows us to determine the minimal requirements for its establishment. Our simulations reveal that two alternative processes-which we coin "flux-barrier" and "flux-passage"-are both able to generate an auxin minimum, but under different parameter settings. Both models are in principle able to yield similar auxin profiles but present qualitatively distinct patterns of auxin flux. The models were tested by tissue-specific inducible ablation, revealing that the auxin minimum in the fruit is most likely generated by a flux-passage process. Model predictions were further supported through 3D PIN localization imaging and implementing experimentally observed transporter localization. Through such an experimental-modeling cycle, we predict how the auxin minimum gradually matures during fruit development to ensure timely fruit opening and seed dispersal.

Two-stage patterning dynamics in conifer cotyledon whorl morphogenesis

Background and Aims

Conifer embryos, unlike those of monocots or dicots, have variable numbers of cotyledons, even within the same species. Cotyledons form in a single whorl on a dome-shaped embryo. The closely spaced cotyledons are not found outside this ring, indicating a radial control on where they can form. Polar transport of the hormone auxin affects outgrowth of distinct cotyledons, but not the radial aspect of the whorl or the within-whorl spacing between cotyledons. A quantitative model of plant growth regulator patterning is needed to understand the dynamics of this complex morphogenetic process.

Methods

A two-stage reaction-diffusion model is developed for the spatial patterning of growth regulators on the embryo surface, with a radial pattern (P1) constraining the shorter-wavelength cotyledon pattern (P2) to a whorl. These patterns drive three-dimensional (3-D) morphogenesis by catalysing local surface growth.

Key Results

Growth driven by P2 generates single whorls across the experimentally observed range of two to 11 cotyledons, as well as the circularly symmetric response to auxin transport interference. These computations are the first corroboration of earlier theoretical proposals for hierarchical control of whorl formation. The model generates the linear relationship between cotyledon number and embryo diameter observed experimentally. This accounts for normal integer cotyledon number selection, as well as the less common cotyledon fusings and splittings observed experimentally. Flattening of the embryo during development may affect the upward outgrowth angle of the cotyledons.

Conclusions

Cotyledon morphogenesis is more complex geometrically in conifers than in angiosperms, involving 2-D patterning which deforms a surface in three dimensions. This work develops a quantitative framework for understanding the growth and patterning dynamics involved in conifer cotyledon development, and applies more generally to the morphogenesis of whorls with many primordia.

Coordination of cell polarity and the patterning of leaf vein networks

During development, the behavior of cells in tissues is coordinated along specific orientations or directions by coordinating the polar localization of components in those cells. The coordination of such cell polarity is perhaps nowhere more spectacular than in developing leaves, where the polarity of hundreds of cells is coordinated in the leaf epidermis and inner tissue to pattern vein networks. Available evidence suggests that the spectacular coordination of cell polarity that patterns vein networks is controlled by auxin transport and levels, and by genes that have been implicated in the polar localization of auxin transporters.

On genes and form

The mechanisms by which organisms acquire their sizes and shapes through growth was a major focus of D'Arcy Thompson's book On Growth and Form. By applying mathematical and physical principles to a range of biological forms, Thompson achieved fresh insights, such as the notion that diverse biological shapes could be related through simple deformations of a coordinate system. However, Thompson considered genetics to lie outside the scope of his work, even though genetics was a growing discipline at the time the book was published. Here, we review how recent advances in cell, developmental, evolutionary and computational biology allow Thompson's ideas to be integrated with genes and the processes they influence to provide a deeper understanding of growth and morphogenesis. We consider how genes interact with subcellular-, cellular- and tissue-level processes in plants to yield patterns of growth that underlie the developmental and evolutionary shape transformations Thompson so eloquently described.

Computational Modeling of Auxin: A Foundation for Plant Engineering

Since the development of agriculture, humans have relied on the cultivation of plants to satisfy our increasing demand for food, natural products, and other raw materials. As we understand more about plant development, we can better manipulate plants to fulfill our particular needs. Auxins are a class of simple metabolites that coordinate many developmental activities like growth and the appearance of functional structures in plants. Computational modeling of auxin has proven to be an excellent tool in elucidating many mechanisms that underlie these developmental events. Due to the complexity of these mechanisms, current modeling efforts are concerned only with single phenomena focused on narrow spatial and developmental contexts; but a general model of plant development could be assembled by integrating the insights from all of them. In this perspective, we summarize the current collection of auxin-driven computational models, focusing on how they could come together into a single model for plant development. A model of this nature would allow researchers to test hypotheses in silico and yield accurate predictions about the behavior of a plant under a given set of physical and biochemical constraints. It would also provide a solid foundation toward the establishment of plant engineering, a proposed discipline intended to enable the design and production of plants that exhibit an arbitrarily defined set of features.

Leaf venation, as a resistor, to optimize a switchable IR absorber

Leaf vascular patterns are the mechanisms and mechanical support for the transportation of fluidics for photosynthesis and leaf development properties. Vascular hierarchical networks in leaves have far-reaching functions in optimal transport efficiency of functional fluidics. Embedding leaf morphogenesis as a resistor network is significant in the optimization of a translucent thermally functional material. This will enable regulation through pressure equalization by diminishing flow pressure variation. This paper investigates nature's vasculature networks that exhibit hierarchical branching scaling applied to microfluidics. To enable optimum potential for pressure drop regulation by algorithm design. This code analysis of circuit conduit optimization for transport fluidic flow resistance is validated against CFD simulation, within a closed loop network. The paper will propose this self-optimization, characterization by resistance seeking targeting to determine a microfluidic network as a resistor. To advance a thermally function material as a switchable IR absorber.

Formation of polarity convergences underlying shoot outgrowths

The development of outgrowths from plant shoots depends on formation of epidermal sites of cell polarity convergence with high intracellular auxin at their centre. A parsimonious model for generation of convergence sites is that cell polarity for the auxin transporter PIN1 orients up auxin gradients, as this spontaneously generates convergent alignments. Here we test predictions of this and other models for the patterns of auxin biosynthesis and import. Live imaging of outgrowths from kanadi1 kanadi2 Arabidopsis mutant leaves shows that they arise by formation of PIN1 convergence sites within a proximodistal polarity field. PIN1 polarities are oriented away from regions of high auxin biosynthesis enzyme expression, and towards regions of high auxin importer expression. Both expression patterns are required for normal outgrowth emergence, and may form part of a common module underlying shoot outgrowths. These findings are more consistent with models that spontaneously generate tandem rather than convergent alignments.

DOI:http://dx.doi.org/10.7554/eLife.18165.001

Plants, unlike animals, are able to grow and develop throughout their lives. New leaves and flowers are made from outgrowths that constantly form at the tip of growing shoots. Groups of cells in the outer layer of the shoot tip arrange a protein called PIN1 so that it is more abundant on the cell surfaces that face towards the centre of the group. PIN1 transports a hormone called auxin out of plant cells and this “convergent” arrangement of PIN1 increases the levels of auxin in cells at the centre of the group, leading to the formation of a new outgrowth. However, it is not clear what causes these cells to position their PIN1 proteins in this way.

Several hypotheses have been proposed to explain how convergent patterns of PIN1 form. For example, according to the “up-the-gradient” hypothesis, PIN1 is allocated to the end of a cell that is next to a cell with a higher level of auxin. Abley et al. have now compared predictions from computer models with new experimental data from a plant called Arabidopsis to evaluate three hypotheses for how convergent PIN1 patterns form. A computer model based on the up-the-gradient hypothesis naturally creates convergent PIN1 patterns, even if each cell starts off with the same level of auxin. On the other hand, models based on two other hypotheses generate tandem alignments of PIN1 so that auxin is transported in the same direction along lines of cells.

Next, Abley et al. tested these models using mutant Arabidopsis plants that develop outgrowths from the lower surface of their leaves. These outgrowths form in a similar way to outgrowths at the growing shoot tip, but in a simpler context. The experiments show that the patterns of where auxin is produced in growing leaves were more compatible with the tandem alignment models than the up-the-gradient model. This suggests that plants use a tandem alignment mechanism to form convergences of PIN1 proteins that generate the local increases in auxin needed to make new outgrowths.

This study only examined a single layer of cells on the plant surface. Other cell layers also show highly organised patterns of PIN1 proteins, so a future challenge is to extend the approach to study the entire 3D structure of new shoot outgrowths.

DOI:http://dx.doi.org/10.7554/eLife.18165.002

Mechanical Stress Induces Remodeling of Vascular Networks in Growing Leaves

Differentiation into well-defined patterns and tissue growth are recognized as key processes in organismal development. However, it is unclear whether patterns are passively, homogeneously dilated by growth or whether they remodel during tissue expansion. Leaf vascular networks are well-fitted to investigate this issue, since leaves are approximately two-dimensional and grow manyfold in size. Here we study experimentally and computationally how vein patterns affect growth. We first model the growing vasculature as a network of viscoelastic rods and consider its response to external mechanical stress. We use the so-called texture tensor to quantify the local network geometry and reveal that growth is heterogeneous, resembling non-affine deformations in composite materials. We then apply mechanical forces to growing leaves after veins have differentiated, which respond by anisotropic growth and reorientation of the network in the direction of external stress. External mechanical stress appears to make growth more homogeneous, in contrast with the model with viscoelastic rods. However, we reconcile the model with experimental data by incorporating randomness in rod thickness and a threshold in the rod growth law, making the rods viscoelastoplastic. Altogether, we show that the higher stiffness of veins leads to their reorientation along external forces, along with a reduction in growth heterogeneity. This process may lead to the reinforcement of leaves against mechanical stress. More generally, our work contributes to a framework whereby growth and patterns are coordinated through the differences in mechanical properties between cell types.

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.

Modeling plant development: from signals to gene networks

Mathematical modeling has become a common tool in plant developmental biology. Indeed, it allows for the prediction of complex and often unintuitive dynamics of the molecular networks driving plant development. This has enabled the test of their possible involvement in robust and specific developmental processes. Modeling has also been fruitful in predicting new interactions within gene networks, such as the Arabidopsis circadian clock. A new challenge is to integrate patterning issues with tissue growth and biomechanics. The development of new tools to gain resolution in data collection as well as new frameworks to confront models and data might provide even more robust predictions.

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.

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

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

Modelling the development and arrangement of the primary vascular structure in plants

BACKGROUND AND AIMS

The process of vascular development in plants results in the formation of a specific array of bundles that run throughout the plant in a characteristic spatial arrangement. Although much is known about the genes involved in the specification of procambium, phloem and xylem, the dynamic processes and interactions that define the development of the radial arrangement of such tissues remain elusive.

METHODS

This study presents a spatially explicit reaction-diffusion model defining a set of logical and functional rules to simulate the differentiation of procambium, phloem and xylem and their spatial patterns, starting from a homogeneous group of undifferentiated cells.

KEY RESULTS

Simulation results showed that the model is capable of reproducing most vascular patterns observed in plants, from primitive and simple structures made up of a single strand of vascular bundles (protostele), to more complex and evolved structures, with separated vascular bundles arranged in an ordered pattern within the plant section (e.g. eustele).

CONCLUSIONS

The results presented demonstrate, as a proof of concept, that a common genetic-molecular machinery can be the basis of different spatial patterns of plant vascular development. Moreover, the model has the potential to become a useful tool to test different hypotheses of genetic and molecular interactions involved in the specification of vascular tissues.

Modelling hormonal response and development

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

Canalization-based vein formation in a growing leaf

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

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.

Molecular control of cell specification and cell differentiation during procambial development

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

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.

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.

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.

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.

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.

Signaling and gene regulatory programs in plant vascular stem cells

A key question about the development of multicellular organisms is how they precisely control the complex pattern formation during their growth. For plants to grow for many years, a tight balance between pluripotent dividing cells and cells undergoing differentiation should be maintained within stem cell populations. In this process, cell-cell communication plays a central role by creating positional information for proper cell type patterning. Cell-type specific gene regulatory networks govern differentiation of cells into particular cell types. In this review, we will provide a comprehensive overview of emerging key signaling and regulatory programs in the stem cell population that direct morphogenesis of plant vascular tissues.

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.

Simulating Self-Reproduction of Cells in a Two-Dimensional Cellular Automaton

Clarifying generalized self-reproduction is basic to applications in fields such as molecular machine production in nanotechnology and synthetic biology. The two-dimensional cellular automaton model we developed simulated cellular self-reproduction using a few state transition rules.

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.

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.