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.

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.

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.

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.

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.

Non-cell autonomous and spatiotemporal signalling from a tissue organizer orchestrates root vascular development.

During plant development, a precise balance of cytokinin is crucial for correct growth and patterning, but it remains unclear how this is achieved across different cell types and in the context of a growing organ. Here we show that in the root apical meristem, the TMO5/LHW complex increases active cytokinin levels via two cooperatively acting enzymes. By profiling the transcriptomic changes of increased cytokinin at single-cell level, we further show that this effect is counteracted by a tissue-specific increase in CYTOKININ OXIDASE 3 expression via direct activation of the mobile transcription factor SHORTROOT. In summary, we show that within the root meristem, xylem cells act as a local organizer of vascular development by non-autonomously regulating cytokinin levels in neighbouring procambium cells via sequential induction and repression modules.

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.

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.

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.

A stochastic multicellular model identifies biological watermarks from disorders in self-organized patterns of phyllotaxis.

Exploration of developmental mechanisms classically relies on analysis of pattern regularities. Whether disorders induced by biological noise may carry information on building principles of developmental systems is an important debated question. Here, we addressed theoretically this question using phyllotaxis, the geometric arrangement of plant aerial organs, as a model system. Phyllotaxis arises from reiterative organogenesis driven by lateral inhibitions at the shoot apex. Motivated by recurrent observations of disorders in phyllotaxis patterns, we revisited in depth the classical deterministic view of phyllotaxis. We developed a stochastic model of primordia initiation at the shoot apex, integrating locality and stochasticity in the patterning system. This stochastic model recapitulates phyllotactic patterns, both regular and irregular, and makes quantitative predictions on the nature of disorders arising from noise. We further show that disorders in phyllotaxis instruct us on the parameters governing phyllotaxis dynamics, thus that disorders can reveal biological watermarks of developmental systems.

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.

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.

A modular analysis of the auxin signalling network.

Auxin is essential for plant development from embryogenesis onwards. Auxin acts in large part through regulation of transcription. The proteins acting in the signalling pathway regulating transcription downstream of auxin have been identified as well as the interactions between these proteins, thus identifying the topology of this network implicating 54 Auxin Response Factor (ARF) and Aux/IAA (IAA) transcriptional regulators. Here, we study the auxin signalling pathway by means of mathematical modeling at the single cell level. We proceed analytically, by considering the role played by five functional modules into which the auxin pathway can be decomposed: the sequestration of ARF by IAA, the transcriptional repression by IAA, the dimer formation amongst ARFs and IAAs, the feedback loop on IAA and the auxin induced degradation of IAA proteins. Focusing on these modules allows assessing their function within the dynamics of auxin signalling. One key outcome of this analysis is that there are both specific and overlapping functions between all the major modules of the signaling pathway. This suggests a combinatorial function of the modules in optimizing the speed and amplitude of auxin-induced transcription. Our work allows identifying potential functions for homo- and hetero-dimerization of transcriptional regulators, with ARF:IAA, IAA:IAA and ARF:ARF dimerization respectively controlling the amplitude, speed and sensitivity of the response and a synergistic effect of the interaction of IAA with transcriptional repressors on these characteristics of the signaling pathway. Finally, we also suggest experiments which might allow disentangling the structure of the auxin signaling pathway and analysing further its function in plants.

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.

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.

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.

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.

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.