Modeling Auxin Transport and Plant Development

The plant hormone auxin plays a critical role in plant development. Central to its function is its distribution in plant tissues, which is, in turn, largely shaped by intercellular polar transport processes. Auxin transport relies on diffusive uptake as well as carrier-mediated transport via influx and efflux carriers. Mathematical models have been used to both refine our theoretical understanding of these processes and to test new hypotheses regarding the localization of efflux carriers to understand auxin patterning at the tissue level. Here we review models for auxin transport and how they have been applied to patterning processes, including the elaboration of plant vasculature and primordium positioning. Second, we investigate the possible role of auxin influx carriers such as AUX1 in patterning auxin in the shoot meristem. We find that AUX1 and its relatives are likely to play a crucial role in maintaining high auxin levels in the meristem epidermis. We also show that auxin influx carriers may play an important role in stabilizing auxin distribution patterns generated by auxin-gradient type models for phyllotaxis.     

Computer simulation: The imaginary friend of auxin transport biology

Regulated transport of the plant hormone auxin is central to many aspects of plant development. Directional transport, mediated by membrane transporters, produces patterns of auxin distribution in tissues that trigger developmental processes, such as vascular patterning or leaf formation. Experimentation has produced many, largely qualitative, data providing strong evidence for multiple feedback systems between auxin and its transport. However, the exact mechanisms concerned remain elusive and the experiments required to evaluate alternative hypotheses are challenging. Because of this, computational modelling now plays an important role in auxin transport research. Here we review some current approaches and underlying assumptions of computational auxin transport models. We focus on self‐organising models for polar auxin transport and on recent attempts to unify conflicting mechanistic explanations. In addition, we discuss in general how these computer simulations are proving to be increasingly effective in hypothesis generation and testing, and how simulation can be used to direct future experiments. Editor's suggested further reading in BioEssays Local auxin production: a small contribution to a big field Abstract     

Dynamic integration of auxin transport and signalling

Recent years have seen rapid progress in our understanding of the mechanism of action of the plant hormone auxin. A major emerging theme is the central importance of the interplay between auxin signalling and the active transport of auxin through the plant to create dynamic patterns of auxin accumulation. Even in tissues where auxin distribution patterns appear stable, they are the product of standing waves, with auxin flowing through the tissue, maintaining local pockets of high and low concentration. The auxin distribution patterns result in changes in gene expression to trigger diverse, context-dependent growth and differentiation responses. Multi-level feedback loops between the signal transduction network and the auxin transport network provide self-stabilising patterns that remain sensitive to the external environment and to the developmental progression of the plant. The full biological implications of the behaviour of this system are only just beginning to be understood through a combination of experimental manipulation and mathematical modelling.     

Pattern formation of leaf veins by the positive feedback regulation between auxin flow and auxin efflux carrier

The generation of vascular pattern formation in plants is an interesting process of pattern formation in organisms. It is well known that the plant hormone auxin is involved in plant vascular differentiation and that the PIN1 protein, an auxin efflux carrier, localizes to one side of the cell membrane. Several hypotheses have been proposed to explain the formation of leaf venation. One is the canalization hypothesis that is based on the assumption that a positive feedback regulation exists between the flow of a signal molecule and the capacity of its flow. Here, we attempted to integrate the canalization hypothesis and experimental data. We investigated models of the positive feedback regulation between the auxin flow and PIN1 localization. Model 1, with conserved PIN1 amount in each cell, can generate a branching pattern similar to that of plant leaf venation. We introduced the diffusible enhancer "e" into the model as unknown factor. The obtained patterns show a quasi-periodic distribution of auxin flow paths, when the model dynamics includes domain growth. In order to understand the early initiation process that generates an inhomogeneity from an almost homogeneous distribution, we introduced model 2, a simplified version of model 1. Model 2 can generate inhomogeneity with a parameter dependency similar to that of model 1. To analyse parameter condition required for pattern development, approximated equations are obtained from model 2. The isocline analysis of the equations without spatial structure shows that the inhomogeneous distribution occurs from an almost homogeneous distribution. This parameter condition for generating inhomogeneity is consistent with the results of models 1 and 2.     

Phyllotaxis--a new chapter in an old tale about beauty and magic numbers

Phyllotaxis, the regular arrangement of leaves and flowers around the stem, is one of the most fascinating patterning phenomena in biology. Numerous theoretical models, that are based on biochemical, biophysical and other principles, have been proposed to explain the development of the patterns. Recently, auxin has been identified as the inducer of organ formation. An emerging model for phyllotaxis states that polar auxin transport in the plant apex generates local peaks in auxin concentration that determine the site of organ formation and thereby the different phyllotactic patterns found in nature. The PIN proteins play a primary role in auxin transport. These proteins are localized in a polar fashion, reflecting the directionality of polar auxin transport. Recent evidence shows that most aspects of phyllotaxis can be explained by the expression pattern and the dynamic subcellular localization of PIN1.     

Modelling of auxin transport affected by gravity and differential radial growth

When a tree stem deviates from verticality, as a result of different environmental factors, patterns of differential radial growth appear. Higher rates of wood production have been observed on the lower side of the tree and lower rates in the opposite side. Biological studies on plant hormones have shown that the concentration of auxin induces radial growth. They also have demonstrated the redistribution of auxin transport in response to gravity. Auxin is then designated as a mediator for differential growth. This paper presents a model for three-dimensional (3-D) auxin transport in conifer trees, which includes gravity dependence. We obtain realistic heterogeneous patterns of auxin distribution over the tree. Then, we propose a law of growth based on auxin concentration to simulate successive differential radial growths. The predicted growths are compared with experimental results of reconstruction of 3-D annual growth of Radiata pine.     

Auxin conjugated to fluorescent dyes – a tool for the analysis of auxin transport pathways

Auxin is a small molecule involved in most processes related to plant growth and development. Its effect usually depends on the distribution in tissues and the formation of concentration gradients. Until now there has been no tool for the direct tracking of auxin transport at the cellular and tissue level; therefore the majority of studies have been based on various indirect methods. However, due to their various restrictions, relatively little is known about the relationship between various pathways of auxin transport and specific developmental processes. We present a new research tool: fluorescently labelled auxin in the form of a conjugate with two different fluorescent tracers, FITC and RITC, which allows direct observation of auxin transport in plant tissues. Chemical analysis and biological tests have shown that our conjugates have auxin‐like biological activity and transport; therefore they can be used in all experimental systems as an alternative to IAA. In addition, the conjugates are a universal tool that can be applied in studies of all plant groups and species. The conjugation procedure presented in this paper can be adapted to other fluorescent dyes, which are constantly being improved. In our opinion, the conjugates greatly expand the possibilities of research concerning the role of auxin and its transport in different developmental processes in plants.     

SLOW MOTION Is Required for Within-Plant Auxin Homeostasis and Normal Timing of Lateral Organ Initiation at the Shoot Meristem in Arabidopsis

The regular arrangement of leaves and flowers around a plant''s stem is a fascinating expression of biological pattern formation. Based on current models, the spacing of lateral shoot organs is determined by transient local auxin maxima generated by polar auxin transport, with existing primordia draining auxin from their vicinity to restrict organ formation close by. It is unclear whether this mechanism encodes not only spatial information but also temporal information about the plastochron (i.e., the interval between the formation of successive primordia). Here, we identify the Arabidopsis thaliana F-box protein SLOW MOTION (SLOMO) as being required for a normal plastochron. SLOMO interacts genetically with components of polar auxin transport, and mutant shoot apices contain less free auxin. However, this reduced auxin level at the shoot apex is not due to increased polar auxin transport down the stem, suggesting that it results from reduced synthesis. Independently reducing the free auxin level in plants causes a similar lengthening of the plastochron as seen in slomo mutants, suggesting that the reduced auxin level in slomo mutant shoot apices delays the establishment of the next auxin maximum. SLOMO acts independently of other plastochron regulators, such as ALTERED MERISTEM PROGRAM1 or KLUH/CYP78A5. We propose that SLOMO contributes to auxin homeostasis in the shoot meristem, thus ensuring a normal rate of the formation of auxin maxima and organ initiation.     

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

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

Complex regulation of Arabidopsis AGR1/PIN2-mediated root gravitropic response and basipetal auxin transport by cantharidin-sensitive protein phosphatases

Polar auxin transport, mediated by two distinct plasma membrane-localized auxin influx and efflux carrier proteins/complexes, plays an important role in many plant growth and developmental processes including tropic responses to gravity and light, development of lateral roots and patterning in embryogenesis. We have previously shown that the Arabidopsis AGRAVITROPIC 1/PIN2 gene encodes an auxin efflux component regulating root gravitropism and basipetal auxin transport. However, the regulatory mechanism underlying the function of AGR1/PIN2 is largely unknown. Recently, protein phosphorylation and dephosphorylation mediated by protein kinases and phosphatases, respectively, have been implicated in regulating polar auxin transport and root gravitropism. Here, we examined the effects of chemical inhibitors of protein phosphatases on root gravitropism and basipetal auxin transport, as well as the expression pattern of AGR1/PIN2 gene and the localization of AGR1/PIN2 protein. We also examined the effects of inhibitors of vesicle trafficking and protein kinases. Our data suggest that protein phosphatases, sensitive to cantharidin and okadaic acid, are likely involved in regulating AGR1/PIN2-mediated root basipetal auxin transport and gravitropism, as well as auxin response in the root central elongation zone (CEZ). BFA-sensitive vesicle trafficking may be required for the cycling of AGR1/PIN2 between plasma membrane and the BFA compartment, but not for the AGR1/PIN2-mediated root basipetal auxin transport and auxin response in CEZ cells.     

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.     

Pattern formation during T-cell adhesion

T cells form intriguing patterns during adhesion to antigen-presenting cells. The patterns are composed of two types of domains, which either contain short TCR/MHCp receptor-ligand complexes or the longer LFA-1/ICAM-1 complexes. The final pattern consists of a central TCR/MHCp domain surrounded by a ring-shaped LFA-1/ICAM-1 domain, whereas the characteristic pattern formed at intermediate times is inverted with TCR/MHCp complexes at the periphery of the contact zone and LFA-1/ICAM-1 complexes in the center. Several mechanisms have been proposed to explain the T-cell pattern formation. Whereas biologists have emphasized the role of active cytoskeletal processes, previous theoretical studies suggest that the pattern evolution may be caused by spontaneous self-assembly processes alone. Some of these studies focus on circularly symmetric patterns and propose a pivot mechanism for the formation of the intermediate inverted pattern. Here, we present a statistical-mechanical model which includes thermal fluctuations and the full range of spatial patterns. We confirm the observation that the intermediate inverted pattern may be formed by spontaneous self-assembly. However, we find a different self-assembly mechanism in which numerous TCR/MHCp microdomains initially nucleate throughout the contact zone. The diffusion of free receptors and ligands into the contact zone subsequently leads to faster growth of peripheral TCR/MHCp microdomains and to a closed ring for sufficiently large TCR/MHCp concentrations. At smaller TCR/MHCp concentrations, we observe a second regime of pattern formation with characteristic multifocal intermediates, which resemble patterns observed during adhesion of immature T cells or thymozytes. In contrast to other theoretical models, we find that the final T-cell pattern with a central TCR/MHCp domain is only obtained in the presence of active cytoskeletal transport processes.     

Reviewing models of auxin canalization in the context of leaf vein pattern formation in Arabidopsis

In both plants and animals vein networks play an essential role in transporting nutrients. In plants veins may also provide mechanical support. The mechanism by which vein patterns are formed in a developing leaf remains largely unresolved. According to the canalization hypothesis, a signal inducing vein differentiation is transported in a polar manner and is channeled into narrow strands. Since inhibition of auxin transport affects venation patterns, auxin is likely to be part of the signal involved. However, it is not clear whether the canalization hypothesis, initially formulated over 25 years ago, is compatible with recent experimental data. In this paper we focus on three aspects of this question, and show that: (i) canalization models can account for an acropetal development of the midvein if vein formation is sink-driven; (ii) canalization models are in agreement with venation patterns resulting from inhibited auxin transport and (iii) loops and discontinuous venation patterns can be obtained assuming proper spacing of discrete auxin sources.     

Alignment between PIN1 polarity and microtubule orientation in the shoot apical meristem reveals a tight coupling between morphogenesis and auxin transport

Morphogenesis during multicellular development is regulated by intercellular signaling molecules as well as by the mechanical properties of individual cells. In particular, normal patterns of organogenesis in plants require coordination between growth direction and growth magnitude. How this is achieved remains unclear. Here we show that in Arabidopsis thaliana, auxin patterning and cellular growth are linked through a correlated pattern of auxin efflux carrier localization and cortical microtubule orientation. Our experiments reveal that both PIN1 localization and microtubule array orientation are likely to respond to a shared upstream regulator that appears to be biomechanical in nature. Lastly, through mathematical modeling we show that such a biophysical coupling could mediate the feedback loop between auxin and its transport that underlies plant phyllotaxis.     

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

The vegetative hormone Auxin is involved in vascular tissues formation throughout the plant. Trans-membrane carrier proteins transporting auxin from cell to cell and distributed asymmetrically around each cell give to auxin a polarized movement in tissues, creating streams of auxin that presume future vascular bundles. According to the canalization hypothesis, auxin transport ability of cells is thought to increase with auxin flux, resulting in the self-enhancement of this flux along auxin paths. In this study we evaluate a series of models based on canalization hypothesis using carrier proteins, under different assumptions concerning auxin flux formation and carrier protein dynamics. Simulations are run on a hexagonal lattice with uniform auxin production. A single cell located in the margin of the lattice indicates the petiole, and acts as an auxin sink. The main results are: (1) We obtain branching auxin distribution patterns. (2) The type of self-enhancement described by the functional form of the carrier proteins regulation responding to the auxin flux intensity in different parts of a cell, has a strong effect on the possibility of generating the branching patterns. For response functions with acceleration in the increase of carrier protein numbers compared to the auxin flux, branching patterns are likely to be generated. For linear or decelerating response functions, no branching patterns are formed. (3) When branching patterns are formed, auxin distribution greatly differs between the case in which the number of carrier proteins in different parts of a cell are regulated independently, and the case in which different parts of a cell compete for a limited number of carrier proteins. In the former case, the auxin level is lower in veins than in the surrounding tissue, while in the latter, the auxin is present in greater abundance in veins. These results suggest that canalization is a good candidate for describing plant vein pattern formation.     

Morphomechanics: goals, basic experiments and models

Morphomechanics is a branch of developmental biology, studying the generation, space-time patterns and morphogenetic role of mechanical stresses (MS) which reside in embryonic tissues. All the morphogenetically active embryonic tissues studied in this respect have been shown to bear substantial mechanical stresses of tension or pressure. MS are indispensable for organized cell movements, expression of a number of developmentally important genes and the very viability of cells. Even a temporary relaxation of MS leads to an increase in the morphological variability and asymmetry of embryonic rudiments. Moreover, MS may be among the decisive links of morphogenetic feedback required for driving forth embryonic development and providing its regular space-time patterns. We hypothesize that one such feedback is based upon the tendency of cells and tissues to hyperrestore (restore with an overshoot) their MS values after any deviations, either artificial or produced by neighboring morphogenetically active tissues. This idea is supported by a number of observations and experiments performed on the tissue and individual cell levels. We describe also the models demonstrating that a number of biologically realistic stationary shapes and propagating waves can be generated by varying the parameters of the hyperrestoration feedback loop. Morphomechanics is an important and rapidly developing branch of developmental and cell biology, being complementary to other approaches.     

Combined in silico/in vivo analysis of mechanisms providing for root apical meristem self-organization and maintenance

Background and Aims

The root apical meristem (RAM) is the plant stem cell niche which provides for the formation and continuous development of the root. Auxin is the main regulator of RAM functioning, and auxin maxima coincide with the sites of RAM initiation and maintenance. Auxin gradients are formed due to local auxin biosynthesis and polar auxin transport. The PIN family of auxin transporters plays a critical role in polar auxin transport, and two mechanisms of auxin maximum formation in the RAM based on PIN-mediated auxin transport have been proposed to date: the reverse fountain and the reflected flow mechanisms.

Methods

The two mechanisms are combined here in in silico studies of auxin distribution in intact roots and roots cut into two pieces in the proximal meristem region. In parallel, corresponding experiments were performed in vivo using DR5::GFP Arabidopsis plants.

Key Results

The reverse fountain and the reflected flow mechanism naturally cooperate for RAM patterning and maintenance in intact root. Regeneration of the RAM in decapitated roots is provided by the reflected flow mechanism. In the excised root tips local auxin biosynthesis either alone or in cooperation with the reverse fountain enables RAM maintenance.

Conclusions

The efficiency of a dual-mechanism model in guiding biological experiments on RAM regeneration and maintenance is demonstrated. The model also allows estimation of the concentrations of auxin and PINs in root cells during development and under various treatments. The dual-mechanism model proposed here can be a powerful tool for the study of several different aspects of auxin function in root.     

Modeling of Branching Patterns in Plants

A major determinant of plant architecture is the arrangement of branches around the stem, known as phyllotaxis. However, the specific form of branching conditions is not known. Here we discuss this question and suggest a branching model which seems to be in agreement with biological observations. Recently, a number of models connected with the genetic network or molecular biology regulation of the processes of pattern formation appeared. Most of these models consider the plant hormone, auxin, transport and distribution in the apical meristem as the main factors for pattern formation and phyllotaxis. However, all these models do not take into consideration the whole plant morphogenesis, concentrating on the events in the shoot or root apex. On the other hand, other approaches for modeling phyllotaxis, where the whole plant is considered, usually are mostly phenomenological, and due to it, do not describe the details of plant growth and branching mechanism. In this work, we develop a mathematical model and study pattern formation of the whole, though simplified, plant organism where the main physiological factors of plant growth and development are taken into consideration. We model a growing plant as a system of intervals, which we will consider as branches. We assume that the number and location of the branches are not given a priori, but appear and grow according to certain rules, elucidated by the application of mathematical modeling. Four variables are included in our model: concentrations of the plant hormones auxin and cytokinin, proliferation and growth factor, and nutrients—we observe a wide variety of plant forms and study more specifically the involvement of each variable in the branching process. Analysis of the numerical simulations shows that the process of pattern formation in plants depends on the interaction of all these variables. While concentrations of auxin and cytokinin determine the appearance of a new bud, its growth is determined by the concentrations of nutrients and proliferation factors. Possible mechanisms of apical domination in the frame of our model are discussed.     

The Dynamics of Turing Patterns for Morphogen-Regulated Growing Domains with Cellular Response Delays

Since its conception in 1952, the Turing paradigm for pattern formation has been the subject of numerous theoretical investigations. Experimentally, this mechanism was first demonstrated in chemical reactions over 20 years ago and, more recently, several examples of biological self-organisation have also been implicated as Turing systems. One criticism of the Turing model is its lack of robustness, not only with respect to fluctuations in the initial conditions, but also with respect to the inclusion of delays in critical feedback processes such as gene expression. In this work we investigate the possibilities for Turing patterns on growing domains where the morphogens additionally regulate domain growth, incorporating delays in the feedback between signalling and domain growth, as well as gene expression. We present results for the proto-typical Schnakenberg and Gierer–Meinhardt systems: exploring the dynamics of these systems suggests a reconsideration of the basic Turing mechanism for pattern formation on morphogen-regulated growing domains as well as highlighting when feedback delays on domain growth are important for pattern formation.     

Phyllotaxis: cooperation and competition between mechanical and biochemical processes

Current theories and models of the formation of phyllotactic patterns at plant apical meristems center on either transport of the growth hormone auxin or the mechanical buckling of the plant tunica. By deriving a continuum approximation of an existing discrete biochemical model and comparing it with a mechanical model, we show that the model partial differential equations are similar in form. The implications of this universality in the form of the equations on interpreting the results of simulations are discussed. We develop a combined model that incorporates the coupling of biochemistry and mechanics. The combined model is accessible to analysis by reduction to a set of ordinary differential equations for the amplitudes of shapes associated with both the auxin concentration field and plant surface deformation. Analysis of these amplitude equations reveals the parameter choices under which the two mechanisms may cooperate in determining the pattern, or under which one or the other mechanism may dominate.