A network of transcriptional repressors modulates auxin responses.

The regulation of signalling capacity, combined with the spatiotemporal distribution of developmental signals themselves, is pivotal in setting developmental responses in both plants and animals1. The hormone auxin is a key signal for plant growth and development that acts through the AUXIN RESPONSE FACTOR (ARF) transcription factors2-4. A subset of these, the conserved class A ARFs5, are transcriptional activators of auxin-responsive target genes that are essential for regulating auxin signalling throughout the plant lifecycle2,3. Although class A ARFs have tissue-specific expression patterns, how their expression is regulated is unknown. Here we show, by investigating chromatin modifications and accessibility, that loci encoding these proteins are constitutively open for transcription. Through yeast one-hybrid screening, we identify the transcriptional regulators of the genes encoding class A ARFs from Arabidopsis thaliana and demonstrate that each gene is controlled by specific sets of transcriptional regulators. Transient transformation assays and expression analyses in mutants reveal that, in planta, the majority of these regulators repress the transcription of genes encoding class A ARFs. These observations support a scenario in which the default configuration of open chromatin enables a network of transcriptional repressors to regulate expression levels of class A ARF proteins and modulate auxin signalling output throughout development.

Fluctuations in auxin levels depend upon synchronicity of cell divisions in a one-dimensional model of auxin transport.

Auxin is a well-studied plant hormone, the spatial distribution of which remains incompletely understood. Here, we investigate the effects of cell growth and divisions on the dynamics of auxin patterning, using a combination of mathematical modelling and experimental observations. In contrast to most prior work, models are not designed or tuned with the aim to produce a specific auxin pattern. Instead, we use well-established techniques from dynamical systems theory to uncover and classify ranges of auxin patterns as exhaustively as possible as parameters are varied. Previous work using these techniques has shown how a multitude of stable auxin patterns may coexist, each attainable from a specific ensemble of initial conditions. When a key parameter spans a range of values, these steady patterns form a geometric curve with successive folds, often nicknamed a snaking diagram. As we introduce growth and cell division into a one-dimensional model of auxin distribution, we observe new behaviour which can be explained in terms of this diagram. Cell growth changes the shape of the snaking diagram, and this corresponds in turn to deformations in the patterns of auxin distribution. As divisions occur this can lead to abrupt creation or annihilation of auxin peaks. We term this phenomenon 'snake-jumping'. Under rhythmic cell divisions, we show how this can lead to stable oscillations of auxin. We also show that this requires a high level of synchronisation between cell divisions. Using 18 hour time-lapse imaging of the auxin reporter DII:Venus in roots of Arabidopsis thaliana, we show auxin fluctuates greatly, both in terms of amplitude and periodicity, consistent with the snake-jumping events observed with non-synchronised cell divisions. Periodic signals downstream of the auxin signalling pathway have previously been recorded in plant roots. The present work shows that auxin alone is unlikely to play the role of a pacemaker in this context.

Modeling root loss reveals impacts on nutrient uptake and crop development.

Despite the widespread prevalence of root loss in plants, its effects on crop productivity are not fully understood. While root loss reduces the capacity of plants to take up water and nutrients from the soil, it may provide benefits by decreasing the resources required to maintain the root system. Here, we simulated a range of root phenotypes in different soils and root loss scenarios for barley (Hordeum vulgare), common bean (Phaseolus vulgaris), and maize (Zea mays) using and extending the open-source, functional-structural root/soil simulation model OpenSimRoot. The model enabled us to quantify the impact of root loss on shoot dry weight in these scenarios and identify in which scenarios root loss is beneficial, detrimental, or has no effect. The simulations showed that root loss is detrimental for phosphorus uptake in all tested scenarios, whereas nitrogen uptake was relatively insensitive to root loss unless main root axes were lost. Loss of axial roots reduced shoot dry weight for all phenotypes in all species and soils, whereas lateral root loss had a smaller impact. In barley and maize plants with high lateral branching density that were not phosphorus-stressed, loss of lateral roots increased shoot dry weight. The fact that shoot dry weight increased due to root loss in these scenarios indicates that plants overproduce roots for some environments, such as those found in high-input agriculture. We conclude that a better understanding of the effects of root loss on plant development is an essential part of optimizing root system phenotypes for maximizing yield.

Cytokinin signalling inhibitory fields provide robustness to phyllotaxis.

How biological systems generate reproducible patterns with high precision is a central question in science. The shoot apical meristem (SAM), a specialized tissue producing plant aerial organs, is a developmental system of choice to address this question. Organs are periodically initiated at the SAM at specific spatial positions and this spatiotemporal pattern defines phyllotaxis. Accumulation of the plant hormone auxin triggers organ initiation, whereas auxin depletion around organs generates inhibitory fields that are thought to be sufficient to maintain these patterns and their dynamics. Here we show that another type of hormone-based inhibitory fields, generated directly downstream of auxin by intercellular movement of the cytokinin signalling inhibitor ARABIDOPSIS HISTIDINE PHOSPHOTRANSFER PROTEIN 6 (AHP6), is involved in regulating phyllotactic patterns. We demonstrate that AHP6-based fields establish patterns of cytokinin signalling in the meristem that contribute to the robustness of phyllotaxis by imposing a temporal sequence on organ initiation. Our findings indicate that not one but two distinct hormone-based fields may be required for achieving temporal precision during formation of reiterative structures at the SAM, thus indicating an original mechanism for providing robustness to a dynamic developmental system.

Inference of the Arabidopsis lateral root gene regulatory network suggests a bifurcation mechanism that defines primordia flanking and central zones.

A large number of genes involved in lateral root (LR) organogenesis have been identified over the last decade using forward and reverse genetic approaches in Arabidopsis thaliana. Nevertheless, how these genes interact to form a LR regulatory network largely remains to be elucidated. In this study, we developed a time-delay correlation algorithm (TDCor) to infer the gene regulatory network (GRN) controlling LR primordium initiation and patterning in Arabidopsis from a time-series transcriptomic data set. The predicted network topology links the very early-activated genes involved in LR initiation to later expressed cell identity markers through a multistep genetic cascade exhibiting both positive and negative feedback loops. The predictions were tested for the key transcriptional regulator AUXIN RESPONSE FACTOR7 node, and over 70% of its targets were validated experimentally. Intriguingly, the predicted GRN revealed a mutual inhibition between the ARF7 and ARF5 modules that would control an early bifurcation between two cell fates. Analyses of the expression pattern of ARF7 and ARF5 targets suggest that this patterning mechanism controls flanking and central zone specification in Arabidopsis LR primordia.

The logic of the floral transition: Reverse-engineering the switch controlling the identity of lateral organs.

Much laboratory work has been carried out to determine the gene regulatory network (GRN) that results in plant cells becoming flowers instead of leaves. However, this also involves the spatial distribution of different cell types, and poses the question of whether alternative networks could produce the same set of observed results. This issue has been addressed here through a survey of the published intercellular distribution of expressed regulatory genes and techniques both developed and applied to Boolean network models. This has uncovered a large number of models which are compatible with the currently available data. An exhaustive exploration had some success but proved to be unfeasible due to the massive number of alternative models, so genetic programming algorithms have also been employed. This approach allows exploration on the basis of both data-fitting criteria and parsimony of the regulatory processes, ruling out biologically unrealistic mechanisms. One of the conclusions is that, despite the multiplicity of acceptable models, an overall structure dominates, with differences mostly in alternative fine-grained regulatory interactions. The overall structure confirms the known interactions, including some that were not present in the training set, showing that current data are sufficient to determine the overall structure of the GRN. The model stresses the importance of relative spatial location, through explicit references to this aspect. This approach also provides a quantitative indication of how likely some regulatory interactions might be, and can be applied to the study of other developmental transitions.

Epidermal identity is maintained by cell-cell communication via a universally active feedback loop in Arabidopsis thaliana.

The transcription factors ARABIDOPSIS THALIANA MERISTEM L1 (ATML1) and PROTODERMAL FACTOR2 (PDF2) are indispensable for epidermal cell-fate specification in Arabidopsis embryos. However, the mechanisms of regulation of these genes, particularly their relationship with cell-cell signalling pathways, although the subject of considerable speculation, remain unclear. Here we demonstrate that the receptor kinase ARABIDOPSIS CRINKLY4 (ACR4) positively affects the expression of ATML1 and PDF2 in seedlings. In contrast, ATML1- and PDF2-containing complexes directly and negatively affect both their own expression and that of ACR4. By modelling the resulting feedback loop, we demonstrate a network structure that is capable of maintaining robust epidermal cell identity post-germination. We show that a second seed-specific signalling pathway involving the subtilase ABNORMAL LEAFSHAPE1 (ALE1) and the receptor kinases GASSHO1 (GSO1) and GASSHO2 (GSO2) acts in parallel to the epidermal loop to control embryonic surface formation via an ATML1/PDF2-independent pathway. Genetic interactions between components of this linear pathway and the epidermal loop suggest that an intact embryo surface is necessary for initiation and/or stabilization of the epidermal loop, specifically during early embryogenesis.

Modeling regulatory networks to understand plant development: small is beautiful.

We now have unprecedented capability to generate large data sets on the myriad genes and molecular players that regulate plant development. Networks of interactions between systems components can be derived from that data in various ways and can be used to develop mathematical models of various degrees of sophistication. Here, we discuss why, in many cases, it is productive to focus on small networks. We provide a brief and accessible introduction to relevant mathematical and computational approaches to model regulatory networks and discuss examples of small network models that have helped generate new insights into plant biology (where small is beautiful), such as in circadian rhythms, hormone signaling, and tissue patterning. We conclude by outlining some of the key technical and modeling challenges for the future.

A data-driven integrative model of sepal primordium polarity in Arabidopsis.

Flower patterning is determined by a complex molecular network but how this network functions remains to be elucidated. Here, we develop an integrative modeling approach that assembles heterogeneous data into a biologically coherent model to allow predictions to be made and inconsistencies among the data to be found. We use this approach to study the network underlying sepal development in the young flower of Arabidopsis thaliana. We constructed a digital atlas of gene expression and used it to build a dynamical molecular regulatory network model of sepal primordium development. This led to the construction of a coherent molecular network model for lateral organ polarity that fully recapitulates expression and interaction data. Our model predicts the existence of three novel pathways involving the HD-ZIP III genes and both cytokinin and ARGONAUTE family members. In addition, our model provides predictions on molecular interactions. In a broader context, this approach allows the extraction of biological knowledge from diverse types of data and can be used to study developmental processes in any multicellular organism.

Systems biology approaches to understand the role of auxin in root growth and development.

The past decade has seen major advances in our understanding of auxin regulated root growth and developmental processes. Key genes have been identified that regulate and/or mediate auxin homeostasis, transport, perception and response. The molecular and biochemical reactions that underpin auxin signalling are non-linear, with feed-forward and feedback loops contributing to the robustness of the system. As our knowledge of auxin biology becomes increasingly complex and their outputs less intuitive, modelling is set to become much more important. For the last several decades modelling efforts have focused on auxin transport and, latterly, on auxin response. Recently researchers have employed multi-scale modelling approaches to predict emergent properties at the tissue and organ scales. Such innovative modelling approaches are proving very promising, revealing new mechanistic insights about how auxin functions within a multicellular context to control plant growth and development. In this review we initially describe examples of models capturing auxin transport and response pathways, and then discuss increasingly complex models that integrate multiple hormone response pathways, tissues and/or scales.

Quantitative imaging reveals the role of MpARF proteasomal degradation during gemma germination.

The auxin signaling molecule controls a variety of growth and developmental processes in land plants. Auxin regulates gene expression through a nuclear auxin signaling pathway (NAP) consisting of a ubiquitin ligase auxin receptor TIR1/AFB, its Aux/IAA degradation substrate, and DNA-binding ARF transcription factors. While extensive qualitative understanding of the pathway and its interactions has been obtained, mostly by studying the flowering plant Arabidopsis thaliana, it is so far unknown how these translate to quantitative system behaviour in vivo, a problem that is confounded by large NAP gene families in most species. Here we used the minimal NAP of the liverwort Marchantia polymorpha to quantitatively map NAP protein accumulation and dynamics in vivo through the use of knock-in fluorescent fusion proteins. Beyond revealing the dynamic native accumulation profile of the entire NAP protein network, we discovered that the two central ARFs, MpARF1 and MpARF2, are proteasomally degraded. This auxin-independent degradation tunes ARF protein stoichiometry to favor gene activation, thereby reprogramming auxin response during developmental progression. Thus, quantitative analysis of the entire NAP allowed us to identify ARF degradation and stoichiometries of activator and repressor ARFs as a potential mechanism for controlling gemma germination.

Pattern identification and characterization reveal permutations of organs as a key genetically controlled property of post-meristematic phyllotaxis.

In vascular plants, the arrangement of organs around the stem generates geometric patterns called phyllotaxis. In the model plant, Arabidopsis thaliana, as in the majority of species, single organs are initiated successively at a divergence angle from the previous organ close to the canonical angle of 137.5°, producing a Fibonacci spiral. Given that little is known about the robustness of these geometric arrangements, we undertook to characterize phyllotaxis by measuring divergence angles between organs along the stems of wild-type and specific mutant plants with obvious defects in phyllotaxis. Sequences of measured divergence angles exhibit segments of non-canonical angles in both genotypes, albeit to a far greater extent in the mutant. We thus designed a pipeline of methods for analyzing these perturbations. The latent structure models used in this pipeline combine a non-observable model representing perturbation patterns (either a variable-order Markov chain or a combinatorial model) with von Mises distributions representing divergence angle uncertainty. We show that the segments of non-canonical angles in both wild-type and mutant plants can be explained by permutations in the order of insertion along the stem of two or three consecutive organs. The number of successive organs between two permutations reveals specific patterns that depend on the nature of the preceding permutation (2- or 3-permutation). We also highlight significant individual deviations from 137.5° in the level of baseline segments and a marked relationship between permutation of organs and defects in the elongation of the internodes between these organs. These results demonstrate that permutations are an intrinsic property of spiral phyllotaxis and that their occurrence is genetically regulated.

Probabilistic approach for predicting periodic orbits in piecewise affine differential models.

Piecewise affine models provide a qualitative description of the dynamics of a system, and are often used to study genetic regulatory networks. The state space of a piecewise affine system is partitioned into hyperrectangles, which can be represented as nodes in a directed graph, so that the system's trajectories follow a path in a transition graph. This paper proposes and compares two definitions of probability of transition between two nodes A and B of the graph, based on the volume of the initial conditions on the hyperrectangle A whose trajectories cross to B. The parameters of the system can thus be compared to the observed transitions between two hyperrectangles. This property may become useful to identify sets of parameters for which the system yields a desired periodic orbit with a high probability, or to predict the most likely periodic orbit given a set of parameters, as illustrated by a gene regulatory system composed of two intertwined negative loops.

The auxin signalling network translates dynamic input into robust patterning at the shoot apex.

The plant hormone auxin is thought to provide positional information for patterning during development. It is still unclear, however, precisely how auxin is distributed across tissues and how the hormone is sensed in space and time. The control of gene expression in response to auxin involves a complex network of over 50 potentially interacting transcriptional activators and repressors, the auxin response factors (ARFs) and Aux/IAAs. Here, we perform a large-scale analysis of the Aux/IAA-ARF pathway in the shoot apex of Arabidopsis, where dynamic auxin-based patterning controls organogenesis. A comprehensive expression map and full interactome uncovered an unexpectedly simple distribution and structure of this pathway in the shoot apex. A mathematical model of the Aux/IAA-ARF network predicted a strong buffering capacity along with spatial differences in auxin sensitivity. We then tested and confirmed these predictions using a novel auxin signalling sensor that reports input into the signalling pathway, in conjunction with the published DR5 transcriptional output reporter. Our results provide evidence that the auxin signalling network is essential to create robust patterns at the shoot apex.

TCR beta allelic exclusion in dynamical models of V(D)J recombination based on allele independence.

Allelic exclusion represents a major aspect of TCRbeta gene assembly by V(D)J recombination in developing T lymphocytes. Despite recent progress, its comprehension remains problematic when confronted with experimental data. Existing models fall short in terms of incorporating into a unique distribution all the cell subsets emerging from the TCRbeta assembly process. To revise this issue, we propose dynamical, continuous-time Markov chain-based modeling whereby essential steps in the biological procedure (D-J and V-DJ rearrangements and feedback inhibition) evolve independently on the two TCRbeta alleles in every single cell while displaying random modes of initiation and duration. By selecting parameters via fitting procedures, we demonstrate the capacity of the model to offer accurate fractions of all distinct TCRbeta genotypes observed in studies using developing and mature T cells from wild-type or mutant mice. Selected parameters in turn afford relative duration for each given step, hence updating TCRbeta recombination distinctive timings. Overall, our dynamical modeling integrating allele independence and noise in recombination and feedback-inhibition events illustrates how the combination of these ingredients alone may enforce allelic exclusion at the TCRbeta locus.

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 effect of renewable energy incorporation on power grid stability and resilience.

Contemporary proliferation of renewable power generation is causing an overhaul in the topology, composition, and dynamics of electrical grids. These low-output, intermittent generators are widely distributed throughout the grid, including at the household level. It is critical for the function of modern power infrastructure to understand how this increasingly distributed layout affects network stability and resilience. This paper uses dynamical models, household power consumption, and photovoltaic generation data to show how these characteristics vary with the level of distribution. It is shown that resilience exhibits daily oscillations as the grid's effective structure and the power demand fluctuate. This can lead to a substantial decrease in grid resilience, explained by periods of highly clustered generator output. Moreover, the addition of batteries, while enabling consumer self-sufficiency, fails to ameliorate these problems. The methodology identifies a grid's susceptibility to disruption resulting from its network structure and modes of operation.

In silico evidence for the utility of parsimonious root phenotypes for improved vegetative growth and carbon sequestration under drought.

Drought is a primary constraint to crop yields and climate change is expected to increase the frequency and severity of drought stress in the future. It has been hypothesized that crops can be made more resistant to drought and better able to sequester atmospheric carbon in the soil by selecting appropriate root phenotypes. We introduce OpenSimRoot_v2, an upgraded version of the functional-structural plant/soil model OpenSimRoot, and use it to test the utility of a maize root phenotype with fewer and steeper axial roots, reduced lateral root branching density, and more aerenchyma formation (i.e. the 'Steep, Cheap, and Deep' (SCD) ideotype) and different combinations of underlying SCD root phene states under rainfed and drought conditions in three distinct maize growing pedoclimatic environments in the USA, Nigeria, and Mexico. In all environments where plants are subjected to drought stress the SCD ideotype as well as several intermediate phenotypes lead to greater shoot biomass after 42 days. As an additional advantage, the amount of carbon deposited below 50 cm in the soil is twice as great for the SCD phenotype as for the reference phenotype in 5 out of 6 simulated environments. We conclude that crop growth and deep soil carbon deposition can be improved by breeding maize plants with fewer axial roots, reduced lateral root branching density, and more aerenchyma formation.

Cauliflower fractal forms arise from perturbations of floral gene networks.

Throughout development, plant meristems regularly produce organs in defined spiral, opposite, or whorl patterns. Cauliflowers present an unusual organ arrangement with a multitude of spirals nested over a wide range of scales. How such a fractal, self-similar organization emerges from developmental mechanisms has remained elusive. Combining experimental analyses in an Arabidopsis thaliana cauliflower-like mutant with modeling, we found that curd self-similarity arises because the meristems fail to form flowers but keep the "memory" of their transient passage in a floral state. Additional mutations affecting meristem growth can induce the production of conical structures reminiscent of the conspicuous fractal Romanesco shape. This study reveals how fractal-like forms may emerge from the combination of key, defined perturbations of floral developmental programs and growth dynamics.