Systems approaches reveal that ABCB and PIN proteins mediate co-dependent auxin efflux.

Members of the B family of membrane-bound ATP-binding cassette (ABC) transporters represent key components of the auxin efflux machinery in plants. Over the last two decades, experimental studies have shown that modifying ATP-binding cassette sub-family B (ABCB) expression affects auxin distribution and plant phenotypes. However, precisely how ABCB proteins transport auxin in conjunction with the more widely studied family of PIN-formed (PIN) auxin efflux transporters is unclear, and studies using heterologous systems have produced conflicting results. Here, we integrate ABCB localization data into a multicellular model of auxin transport in the Arabidopsis thaliana root tip to predict how ABCB-mediated auxin transport impacts organ-scale auxin distribution. We use our model to test five potential ABCB-PIN regulatory interactions, simulating the auxin dynamics for each interaction and quantitatively comparing the predictions with experimental images of the DII-VENUS auxin reporter in wild-type and abcb single and double loss-of-function mutants. Only specific ABCB-PIN regulatory interactions result in predictions that recreate the experimentally observed DII-VENUS distributions and long-distance auxin transport. Our results suggest that ABCBs enable auxin efflux independently of PINs; however, PIN-mediated auxin efflux is predominantly through a co-dependent efflux where co-localized with ABCBs.

Gene Regulatory Network Investigation Using Ordinary Differential Equations.

This chapter reviews mathematical models of gene regulation, either as "pure" gene regulatory networks, as signal transduction pathways or as combinations of these. The basic underlying methods are discussed from first principles, relying on rigorous mathematical concepts but with an aim to avoid technical details and focus on the intuitive aspects of this type of mathematical models. After reviewing the principles, some real biological examples are presented to illustrate the practice of modeling, using recent examples from the literature. The proposed examples all arise in the context of plant biology, either at the single cell scale, looking at auxin signaling, or at higher scales, looking at auxin active transport.

The Virtual Root : Mathematical Modeling of Auxin Transport in the Arabidopsis Root Tip Using the Open-Source Software SimuPlant.

Hormone signals like auxin play a critical role controlling plant growth and development. Determining the mechanisms that regulate auxin distribution in cells and tissues is a vital step in understanding this hormone's role during plant development. Recent mathematical models have enabled us to understand the essential role that auxin influx and efflux carriers play in auxin transport in the Arabidopsis root tip (Band et al., Plant Cell 26(3):862-875, 2014; Grieneisen et al., Nature 449(7165):1008-1013, 2007; van den Berg et al., Development 143(18):3350-3362, 2016). In this chapter, we describe SimuPlant: The Virtual Root (SimuPlant, University of Nottingham. https://www.simuplant.org/ . Accessed 20 Sept 2019); an open source software suite, built using the OpenAlea (Pradal et al., Funct Plant Biol 35(10):751-760, 2008) framework, that is designed to simulate vertex-based models in real plant tissue geometries. We provide guidance on how to install SimuPlant, run 2D auxin transport models in the Arabidopsis root tip, manipulate parameters, and visualize model outputs.SimuPlant features a graphical user interface (GUI) designed to allow users with no programming experience to simulate auxin dynamics within the Arabidopsis root tip. Within the user interface, users of SimuPlant can select from a range of model assumptions and can choose to manipulate model and simulation parameter values. Users can then investigate how their choices affect the predicted distribution of auxin in the Arabidopsis root tip. The results of the model simulations are shown visually within the root geometry and can be exported and saved as PNG image files.

Auxin fluxes through plasmodesmata modify root-tip auxin distribution.

Auxin is a key signal regulating plant growth and development. It is well established that auxin dynamics depend on the spatial distribution of efflux and influx carriers on the cell membranes. In this study, we employ a systems approach to characterise an alternative symplastic pathway for auxin mobilisation via plasmodesmata, which function as intercellular pores linking the cytoplasm of adjacent cells. To investigate the role of plasmodesmata in auxin patterning, we developed a multicellular model of the Arabidopsis root tip. We tested the model predictions using the DII-VENUS auxin response reporter, comparing the predicted and observed DII-VENUS distributions using genetic and chemical perturbations designed to affect both carrier-mediated and plasmodesmatal auxin fluxes. The model revealed that carrier-mediated transport alone cannot explain the experimentally determined auxin distribution in the root tip. In contrast, a composite model that incorporates both carrier-mediated and plasmodesmatal auxin fluxes re-capitulates the root-tip auxin distribution. We found that auxin fluxes through plasmodesmata enable auxin reflux and increase total root-tip auxin. We conclude that auxin fluxes through plasmodesmata modify the auxin distribution created by efflux and influx carriers.

A core mechanism for specifying root vascular patterning can replicate the anatomical variation seen in diverse plant species.

Pattern formation is typically controlled through the interaction between molecular signals within a given tissue. During early embryonic development, roots of the model plant Arabidopsis thaliana have a radially symmetric pattern, but a heterogeneous input of the hormone auxin from the two cotyledons forces the vascular cylinder to develop a diarch pattern with two xylem poles. Molecular analyses and mathematical approaches have uncovered the regulatory circuit that propagates this initial auxin signal into a stable cellular pattern. The diarch pattern seen in Arabidopsis is relatively uncommon among flowering plants, with most species having between three and eight xylem poles. Here, we have used multiscale mathematical modelling to demonstrate that this regulatory module does not require a heterogeneous auxin input to specify the vascular pattern. Instead, the pattern can emerge dynamically, with its final form dependent upon spatial constraints and growth. The predictions of our simulations compare to experimental observations of xylem pole number across a range of species, as well as in transgenic systems in Arabidopsis in which we manipulate the size of the vascular cylinder. By considering the spatial constraints, our model is able to explain much of the diversity seen in different flowering plant species.

Mobile PEAR transcription factors integrate positional cues to prime cambial growth.

Apical growth in plants initiates upon seed germination, whereas radial growth is primed only during early ontogenesis in procambium cells and activated later by the vascular cambium1. Although it is not known how radial growth is organized and regulated in plants, this system resembles the developmental competence observed in some animal systems, in which pre-existing patterns of developmental potential are established early on2,3. Here we show that in Arabidopsis the initiation of radial growth occurs around early protophloem-sieve-element cell files of the root procambial tissue. In this domain, cytokinin signalling promotes the expression of a pair of mobile transcription factors-PHLOEM EARLY DOF 1 (PEAR1) and PHLOEM EARLY DOF 2 (PEAR2)-and their four homologues (DOF6, TMO6, OBP2 and HCA2), which we collectively name PEAR proteins. The PEAR proteins form a short-range concentration gradient that peaks at protophloem sieve elements, and activates gene expression that promotes radial growth. The expression and function of PEAR proteins are antagonized by the HD-ZIP III proteins, well-known polarity transcription factors4-the expression of which is concentrated in the more-internal domain of radially non-dividing procambial cells by the function of auxin, and mobile miR165 and miR166 microRNAs. The PEAR proteins locally promote transcription of their inhibitory HD-ZIP III genes, and thereby establish a negative-feedback loop that forms a robust boundary that demarks the zone of cell division. Taken together, our data establish that during root procambial development there exists a network in which a module that links PEAR and HD-ZIP III transcription factors integrates spatial information of the hormonal domains and miRNA gradients to provide adjacent zones of dividing and more-quiescent cells, which forms a foundation for further radial growth.

OpenSimRoot: widening the scope and application of root architectural models.

OpenSimRoot is an open-source, functional-structural plant model and mathematical description of root growth and function. We describe OpenSimRoot and its functionality to broaden the benefits of root modeling to the plant science community. OpenSimRoot is an extended version of SimRoot, established to simulate root system architecture, nutrient acquisition and plant growth. OpenSimRoot has a plugin, modular infrastructure, coupling single plant and crop stands to soil nutrient and water transport models. It estimates the value of root traits for water and nutrient acquisition in environments and plant species. The flexible OpenSimRoot design allows upscaling from root anatomy to plant community to estimate the following: resource costs of developmental and anatomical traits; trait synergisms; and (interspecies) root competition. OpenSimRoot can model three-dimensional images from magnetic resonance imaging (MRI) and X-ray computed tomography (CT) of roots in soil. New modules include: soil water-dependent water uptake and xylem flow; tiller formation; evapotranspiration; simultaneous simulation of mobile solutes; mesh refinement; and root growth plasticity. OpenSimRoot integrates plant phenotypic data with environmental metadata to support experimental designs and to gain a mechanistic understanding at system scales.

Theoretical approaches to understanding root vascular patterning: a consensus between recent models.

The root vascular tissues provide an excellent system for studying organ patterning, as the specification of these tissues signals a transition from radial symmetry to bisymmetric patterns. The patterning process is controlled by the combined action of hormonal signaling/transport pathways, transcription factors, and miRNA that operate through a series of non-linear pathways to drive pattern formation collectively. With the discovery of multiple components and feedback loops controlling patterning, it has become increasingly difficult to understand how these interactions act in unison to determine pattern formation in multicellular tissues. Three independent mathematical models of root vascular patterning have been formulated in the last few years, providing an excellent example of how theoretical approaches can complement experimental studies to provide new insights into complex systems. In many aspects these models support each other; however, each study also provides its own novel findings and unique viewpoints. Here we reconcile these models by identifying the commonalities and exploring the differences between them by testing how transferable findings are between models. New simulations herein support the hypothesis that an asymmetry in auxin input can direct the formation of vascular pattern. We show that the xylem axis can act as a sole source of cytokinin and specify the correct pattern, but also that broader patterns of cytokinin production are also able to pattern the root. By comparing the three modeling approaches, we gain further insight into vascular patterning and identify several key areas for experimental investigation.

Dynamic regulation of auxin oxidase and conjugating enzymes AtDAO1 and GH3 modulates auxin homeostasis.

The hormone auxin is a key regulator of plant growth and development, and great progress has been made understanding auxin transport and signaling. Here, we show that auxin metabolism and homeostasis are also regulated in a complex manner. The principal auxin degradation pathways in Arabidopsis include oxidation by Arabidopsis thaliana gene DIOXYGENASE FOR AUXIN OXIDATION 1/2 (AtDAO1/2) and conjugation by Gretchen Hagen3s (GH3s). Metabolic profiling of dao1-1 root tissues revealed a 50% decrease in the oxidation product 2-oxoindole-3-acetic acid (oxIAA) and increases in the conjugated forms indole-3-acetic acid aspartic acid (IAA-Asp) and indole-3-acetic acid glutamic acid (IAA-Glu) of 438- and 240-fold, respectively, whereas auxin remains close to the WT. By fitting parameter values to a mathematical model of these metabolic pathways, we show that, in addition to reduced oxidation, both auxin biosynthesis and conjugation are increased in dao1-1 Transcripts of AtDAO1 and GH3 genes increase in response to auxin over different timescales and concentration ranges. Including this regulation of AtDAO1 and GH3 in an extended model reveals that auxin oxidation is more important for auxin homoeostasis at lower hormone concentrations, whereas auxin conjugation is most significant at high auxin levels. Finally, embedding our homeostasis model in a multicellular simulation to assess the spatial effect of the dao1-1 mutant shows that auxin increases in outer root tissues in agreement with the dao1-1 mutant root hair phenotype. We conclude that auxin homeostasis is dependent on AtDAO1, acting in concert with GH3, to maintain auxin at optimal levels for plant growth and development.

GH3-Mediated Auxin Conjugation Can Result in Either Transient or Oscillatory Transcriptional Auxin Responses.

The conjugation of the phytohormone auxin to amino acids via members of the gene family GH3 is an important component in the auxin-degradation pathway in the model plant species Arabidopsis thaliana, as well as many other plant species. Since the GH3 genes are themselves up-regulated in response to auxin, providing a negative feedback on intracellular auxin levels, it is hypothesised that the GH3s have a role in auxin homoeostasis. To investigate this, we develop a mathematical model of auxin signalling and response that includes the auxin-inducible negative feedback from GH3 on the rate of auxin degradation. In addition, we include a positive feedback on the rate of auxin input via the auxin influx transporter LAX3, shown previously to be expressed in response to auxin and to have an important role during lateral root emergence. In the absence of the LAX3 positive feedback, we show that the GH3 negative feedback suffices to generate a transient transcriptional response to auxin in the shape of damped oscillations of the model system. When LAX3 positive feedback is present, sustained oscillations of the system are possible. Using steady-state analyses, we identify and discuss key parameters affecting the oscillatory behaviour of the model. The transient peak of auxin and subsequent transcriptional response caused by the up-regulation of GH3 represents a possible protective homoeostasis mechanism that may be used by plant cells in response to excess auxin.

Modelling of Arabidopsis LAX3 expression suggests auxin homeostasis.

Emergence of new lateral roots from within the primary root in Arabidopsis has been shown to be regulated by the phytohormone auxin, via the expression of the auxin influx carrier LAX3, mediated by the ARF7/19 IAA14 signalling module (Swarup et al., 2008). A single cell model of the LAX3 and IAA14 auxin response was formulated and used to demonstrate that hysteresis and bistability may explain the experimentally observed 'all-or-nothing' LAX3 spatial expression pattern in cortical cells containing a gradient of auxin concentrations. The model was tested further by using a parameter fitting algorithm to match model output with qRT-PCR mRNA expression data following exogenous auxin treatment. It was found that the model is able to show good agreement with the data, but only when the exogenous auxin signal is degraded over time, at a rate higher than that measured in the experimental medium, suggesting the triggering of an endogenous auxin homeostasis mechanism. Testing the model over a more physiologically relevant range of extracellular auxin shows bistability and hysteresis still occur when using the optimised parameters, providing the rate of LAX3 active auxin transport is sufficiently high relative to passive diffusion.

Integration of hormonal signaling networks and mobile microRNAs is required for vascular patterning in Arabidopsis roots.

As multicellular organisms grow, positional information is continually needed to regulate the pattern in which cells are arranged. In the Arabidopsis root, most cell types are organized in a radially symmetric pattern; however, a symmetry-breaking event generates bisymmetric auxin and cytokinin signaling domains in the stele. Bidirectional cross-talk between the stele and the surrounding tissues involving a mobile transcription factor, SHORT ROOT (SHR), and mobile microRNA species also determines vascular pattern, but it is currently unclear how these signals integrate. We use a multicellular model to determine a minimal set of components necessary for maintaining a stable vascular pattern. Simulations perturbing the signaling network show that, in addition to the mutually inhibitory interaction between auxin and cytokinin, signaling through SHR, microRNA165/6, and PHABULOSA is required to maintain a stable bisymmetric pattern. We have verified this prediction by observing loss of bisymmetry in shr mutants. The model reveals the importance of several features of the network, namely the mutual degradation of microRNA165/6 and PHABULOSA and the existence of an additional negative regulator of cytokinin signaling. These components form a plausible mechanism capable of patterning vascular tissues in the absence of positional inputs provided by the transport of hormones from the shoot.

Sequential induction of auxin efflux and influx carriers regulates lateral root emergence.

In Arabidopsis, lateral roots originate from pericycle cells deep within the primary root. New lateral root primordia (LRP) have to emerge through several overlaying tissues. Here, we report that auxin produced in new LRP is transported towards the outer tissues where it triggers cell separation by inducing both the auxin influx carrier LAX3 and cell-wall enzymes. LAX3 is expressed in just two cell files overlaying new LRP. To understand how this striking pattern of LAX3 expression is regulated, we developed a mathematical model that captures the network regulating its expression and auxin transport within realistic three-dimensional cell and tissue geometries. Our model revealed that, for the LAX3 spatial expression to be robust to natural variations in root tissue geometry, an efflux carrier is required--later identified to be PIN3. To prevent LAX3 from being transiently expressed in multiple cell files, PIN3 and LAX3 must be induced consecutively, which we later demonstrated to be the case. Our study exemplifies how mathematical models can be used to direct experiments to elucidate complex developmental processes.

Reduction of off-flavor generation in soybean homogenates: a mathematical model.

UNLABELLED: The generation of off-flavors in soybean homogenates such as n-hexanal via the lipoxygenase (LOX) pathway can be a problem in the processed food industry. Previous studies have examined the effect of using soybean varieties missing one or more of the 3 LOX isozymes on n-hexanal generation. A dynamic mathematical model of the soybean LOX pathway using ordinary differential equations was constructed using parameters estimated from existing data with the aim of predicting how n-hexanal generation could be reduced. Time-course simulations of LOX-null beans were run and compared with experimental results. Model L(2), L(3), and L(12) beans were within the range relative to the wild type found experimentally, with L(13) and L(23) beans close to the experimental range. Model L(1) beans produced much more n-hexanal relative to the wild type than those in experiments. Sensitivity analysis indicates that reducing the estimated K(m) parameter for LOX isozyme 3 (L-3) would improve the fit between model predictions and experimental results found in the literature. The model also predicts that increasing L-3 or reducing L-2 levels within beans may reduce n-hexanal generation. PRACTICAL APPLICATION: This work describes the use of mathematics to attempt to quantify the enzyme-catalyzed conversions of compounds in soybean homogenates into undesirable flavors, primarily from the compound n-hexanal. The effect of different soybean genotypes and enzyme kinetic constants was also studied, leading to recommendations on which combinations might minimize off-flavor levels and what further work might be carried out to substantiate these conclusions.

Controls on soil carbon sequestration and dynamics: lessons from land-use change.

Soil carbon (C) dynamics and sequestration are controlled by interactions of chemical, physical and biological factors. These factors include biomass quantity and quality, physical environment and the biota. Management can alter these factors in ways that alter C dynamics. We have focused on a range of managed sites with documented land use change from agriculture or grassland to forest. Our results suggest that interactions of soil type, plant and environment impact soil C sequestration. Above and below ground C storage varied widely across sites. Results were related to plant type and calcium on sandy soils in our Northern sites. Predictors of sequestration were more difficult to detect over the temperature range of 12.4°C in the present study. Accrual of litter under pines in the moist Mississippi site limited C storage in a similar manner to our dry Nebraska site. Pre-planting heterogeneity of agricultural fields such as found in Illinois influences C contents. Manipulation of controls on C sequestration such as species planted or amelioration of soil quality before planting within managed sites could increase soil C to provide gains in terrestrial C storage. Cost effective management would also improve soil C pools positively affecting soil fertility and site productivity.