Modelling how plant cell-cycle progression leads to cell size regulation.

Populations of cells typically maintain a consistent size, despite cell division rarely being precisely symmetrical. Therefore, cells must possess a mechanism of "size control", whereby the cell volume at birth affects cell-cycle progression. While size control mechanisms have been elucidated in a number of other organisms, it is not yet clear how this mechanism functions in plants. Here, we present a mathematical model of the key interactions in the plant cell cycle. Model simulations reveal that the network of interactions exhibits limit-cycle solutions, with biological switches underpinning both the G1/S and G2/M cell-cycle transitions. Embedding this network model within growing cells, we test hypotheses as to how cell-cycle progression can depend on cell size. We investigate two different mechanisms at both the G1/S and G2/M transitions: (i) differential expression of cell-cycle activator and inhibitor proteins (with synthesis of inhibitor proteins being independent of cell size), and (ii) equal inheritance of inhibitor proteins after cell division. The model demonstrates that both these mechanisms can lead to larger daughter cells progressing through the cell cycle more rapidly, and can thus contribute to cell-size control. To test how these features enable size homeostasis over multiple generations, we then simulated these mechanisms in a cell-population model with multiple rounds of cell division. These simulations suggested that integration of size-control mechanisms at both G1/S and G2/M provides long-term cell-size homeostasis. We concluded that while both size independence and equal inheritance of inhibitor proteins can reduce variations in cell size across individual cell-cycle phases, combining size-control mechanisms at both G1/S and G2/M is essential to maintain size homeostasis over multiple generations. Thus, our study reveals how features of the cell-cycle network enable cell-cycle progression to depend on cell size, and provides a mechanistic understanding of how plant cell populations maintain consistent size over generations.

Elementary effects for models with dimensional inputs of arbitrary type and range: Scaling and trajectory generation.

The Elementary Effects method is a global sensitivity analysis approach for identifying (un)important parameters in a model. However, it has almost exclusively been used where inputs are dimensionless and take values on [0, 1]. Here, we consider models with dimensional inputs, inputs taking values on arbitrary intervals or discrete inputs. In such cases scaling effects by a function of the input range is essential for correct ranking results. We propose two alternative dimensionless sensitivity indices by normalizing the scaled mean or median of absolute effects. Testing these indices with 9 trajectory generation methods on 4 test functions (including the Penman-Monteith equation for evapotranspiration) reveals that: i) scaled elementary effects are necessary to obtain correct parameter importance rankings; ii) small step-size methods typically produce more accurate rankings; iii) it is beneficial to compute and compare both sensitivity indices; and iv) spread and discrepancy of the simulation points are poor proxies for trajectory generation method performance.

Gibberellin and abscisic acid transporters facilitate endodermal suberin formation in Arabidopsis.

The plant hormone gibberellin (GA) regulates multiple developmental processes. It accumulates in the root elongating endodermis, but how it moves into this cell file and the significance of this accumulation are unclear. Here we identify three NITRATE TRANSPORTER1/PEPTIDE TRANSPORTER (NPF) transporters required for GA and abscisic acid (ABA) translocation. We demonstrate that NPF2.14 is a subcellular GA/ABA transporter, presumably the first to be identified in plants, facilitating GA and ABA accumulation in the root endodermis to regulate suberization. Further, NPF2.12 and NPF2.13, closely related proteins, are plasma membrane-localized GA and ABA importers that facilitate shoot-to-root GA12 translocation, regulating endodermal hormone accumulation. This work reveals that GA is required for root suberization and that GA and ABA can act non-antagonistically. We demonstrate how the clade of transporters mediates hormone flow with cell-file-specific vacuolar storage at the phloem unloading zone, and slow release of hormone to induce suberin formation in the maturation zone.

Long-distance hormone transport via the phloem.

Several key plant hormones are synthesised in the shoot and are advected within the phloem to the root tip. In the root tip, these hormones regulate growth and developmental processes, and responses to environmental cues. However, we lack understanding of how environmental factors and biological parameters affect the delivery of hormones to the root tip. In this study, we build on existing models of phloem flow to develop a mathematical model of sugar transport alongside the transport of a generic hormone. We derive the equations for osmotically driven flow in a long, thin pipe with spatially varying membrane properties to capture the phloem loading and unloading zones. Motivated by experimental findings, we formulate solute membrane transport in terms of passive and active components, and incorporate solute unloading via bulk flow (i.e. advection with the water efflux) by including the Staverman reflection coefficient. We use the model to investigate the coupling between the sugar and hormone dynamics. The model predicts that environmental cues that lead to an increase in active sugar loading, an increase in bulk flow sugar unloading or a decrease in the relative root sugar concentration result in an increase in phloem transport velocity. Furthermore, the model reveals that such increases in phloem transport velocity result in an increase in hormone delivery to the root tip for passively loaded hormones.

Uncertainty and error in SARS-CoV-2 epidemiological parameters inferred from population-level epidemic models.

During the SARS-CoV-2 pandemic, epidemic models have been central to policy-making. Public health responses have been shaped by model-based projections and inferences, especially related to the impact of various non-pharmaceutical interventions. Accompanying this has been increased scrutiny over model performance, model assumptions, and the way that uncertainty is incorporated and presented. Here we consider a population-level model, focusing on how distributions representing host infectiousness and the infection-to-death times are modelled, and particularly on the impact of inferred epidemic characteristics if these distributions are mis-specified. We introduce an SIR-type model with the infected population structured by 'infected age', i.e. the number of days since first being infected, a formulation that enables distributions to be incorporated that are consistent with clinical data. We show that inference based on simpler models without infected age, which implicitly mis-specify these distributions, leads to substantial errors in inferred quantities relevant to policy-making, such as the reproduction number and the impact of interventions. We consider uncertainty quantification via a Bayesian approach, implementing this for both synthetic and real data focusing on UK data in the period 15 Feb-14 Jul 2020, and emphasising circumstances where it is misleading to neglect uncertainty. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

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.

Plasmodesmata play a key role in leaf vein patterning.

Leaf veins provide a vital transport route in plants. The formation of leaf vein patterns has fascinated many scientists. In PLOS Biology, Linh and Scarpella reveal that transport through plasmodesmata plays a key role in vein patterning.

Modeling reveals posttranscriptional regulation of GA metabolism enzymes in response to drought and cold.

The hormone gibberellin (GA) controls plant growth and regulates growth responses to environmental stress. In monocotyledonous leaves, GA controls growth by regulating division-zone size. We used a systems approach to investigate the establishment of the GA distribution in the maize leaf growth zone to understand how drought and cold alter leaf growth. By developing and parameterizing a multiscale computational model that includes cell movement, growth-induced dilution, and metabolic activities, we revealed that the GA distribution is predominantly determined by variations in GA metabolism. Considering wild-type and UBI::GA20-OX-1 leaves, the model predicted the peak in GA concentration, which has been shown to determine division-zone size. Drought and cold modified enzyme transcript levels, although the model revealed that this did not explain the observed GA distributions. Instead, the model predicted that GA distributions are also mediated by posttranscriptional modifications increasing the activity of GA 20-oxidase in drought and of GA 2-oxidase in cold, which we confirmed by enzyme activity measurements. This work provides a mechanistic understanding of the role of GA metabolism in plant growth regulation.

The cellulose synthase-like F3 (CslF3) gene mediates cell wall polysaccharide synthesis and affects root growth and differentiation in barley.

The barley cellulose synthase-like F (CslF) genes encode putative cell wall polysaccharide synthases. They are related to the cellulose synthase (CesA) genes involved in cellulose biosynthesis, and the CslD genes that influence root hair development. Although CslD genes are implicated in callose, mannan and cellulose biosynthesis, and are found in both monocots and eudicots, CslF genes are specific to the Poaceae. Recently the barley CslF3 (HvCslF3) gene was shown to be involved in the synthesis of a novel (1,4)-ß-linked glucoxylan, but it remains unclear whether this gene contributes to plant growth and development. Here, expression profiling using qRT-PCR and mRNA in situ hybridization revealed that HvCslF3 accumulates in the root elongation zone. Silencing HvCslF3 by RNAi was accompanied by slower root growth, linked with a shorter elongation zone and a significant reduction in root system size. Polymer profiling of the RNAi lines revealed a significant reduction in (1,4)-ß-linked glucoxylan levels. Remarkably, the heterologous expression of HvCslF3 in wild-type (Col-0) and root hair-deficient Arabidopsis mutants (csld3 and csld5) complemented the csld5 mutant phenotype, in addition to altering epidermal cell fate. Our results reveal a key role for HvCslF3 during barley root development and suggest that members of the CslD and CslF gene families have similar functions during root growth regulation.

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.

Integrated root phenotypes for improved rice performance under low nitrogen availability.

Greater nitrogen efficiency would substantially reduce the economic, energy and environmental costs of rice production. We hypothesized that synergistic balancing of the costs and benefits for soil exploration among root architectural phenes is beneficial under suboptimal nitrogen availability. An enhanced implementation of the functional-structural model OpenSimRoot for rice integrated with the ORYZA_v3 crop model was used to evaluate the utility of combinations of root architectural phenes, namely nodal root angle, the proportion of smaller diameter nodal roots, nodal root number; and L-type and S-type lateral branching densities, for plant growth under low nitrogen. Multiple integrated root phenotypes were identified with greater shoot biomass under low nitrogen than the reference cultivar IR64. The superiority of these phenotypes was due to synergism among root phenes rather than the expected additive effects of phene states. Representative optimal phenotypes were predicted to have up to 80% greater grain yield with low N supply in the rainfed dry direct-seeded agroecosystem over future weather conditions, compared to IR64. These phenotypes merit consideration as root ideotypes for breeding rice cultivars with improved yield under rainfed dry direct-seeded conditions with limited nitrogen availability. The importance of phene synergism for the performance of integrated phenotypes has implications for crop breeding.

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.

Characterising the processes that control auxin dynamics is essential to understanding how auxin regulates plant development. Over recent years, several studies have investigated auxin diffusion through plasmodesmata, characterising this cell-to-cell diffusion and demonstrating that it affects auxin distributions. Furthermore, studies have shown that plasmodesmatal auxin diffusion affects developmental processes, including phototropism, lateral root emergence and leaf hyponasty. This short Tansley Insight review describes how these studies have contributed to our understanding of auxin dynamics and discusses potential future directions in this area.

Differential biosynthesis and cellular permeability explain longitudinal gibberellin gradients in growing roots.

Control over cell growth by mobile regulators underlies much of eukaryotic morphogenesis. In plant roots, cell division and elongation are separated into distinct longitudinal zones and both division and elongation are influenced by the growth regulatory hormone gibberellin (GA). Previously, a multicellular mathematical model predicted a GA maximum at the border of the meristematic and elongation zones. However, GA in roots was recently measured using a genetically encoded fluorescent biosensor, nlsGPS1, and found to be low in the meristematic zone grading to a maximum at the end of the elongation zone. Furthermore, the accumulation rate of exogenous GA was also found to be higher in the elongation zone. It was still unknown which biochemical activities were responsible for these mobile small molecule gradients and whether the spatiotemporal correlation between GA levels and cell length is important for root cell division and elongation patterns. Using a mathematical modeling approach in combination with high-resolution GA measurements in vivo, we now show how differentials in several biosynthetic enzyme steps contribute to the endogenous GA gradient and how differential cellular permeability contributes to an accumulation gradient of exogenous GA. We also analyzed the effects of altered GA distribution in roots and did not find significant phenotypes resulting from increased GA levels or signaling. We did find a substantial temporal delay between complementation of GA distribution and cell division and elongation phenotypes in a GA deficient mutant. Together, our results provide models of how GA gradients are directed and in turn direct root growth.

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.

Double or Nothing? Cell Division and Cell Size Control.

Size is a fundamental property that must be tightly regulated to ensure that cells and tissues function efficiently. Dynamic size control allows unicellular organisms to adapt to environmental changes, but cell size is also integral to multicellular development, affecting tissue size and structure. Despite clear evidence for homeostatic cell size maintenance, we are only now beginning to understand cell size regulation in the actively dividing meristematic tissues of higher plants. We discuss here how coupled advances in live cell imaging and modelling are uncovering dynamic mechanisms for size control mediated at the cellular level. We argue that integrated models of cell growth and division will be necessary to predict cell size and fully understand multicellular growth and development.

Parameter inference to motivate asymptotic model reduction: An analysis of the gibberellin biosynthesis pathway.

Developing effective strategies to use models in conjunction with experimental data is essential to understand the dynamics of biological regulatory networks. In this study, we demonstrate how combining parameter estimation with asymptotic analysis can reveal the key features of a network and lead to simplified models that capture the observed network dynamics. Our approach involves fitting the model to experimental data and using the profile likelihood to identify small parameters and cases where model dynamics are insensitive to changing particular individual parameters. Such parameter diagnostics provide understanding of the dominant features of the model and motivate asymptotic model reductions to derive simpler models in terms of identifiable parameter groupings. We focus on the particular example of biosynthesis of the plant hormone gibberellin (GA), which controls plant growth and has been mutated in many current crop varieties. This pathway comprises two parallel series of enzyme-substrate reactions, which have previously been modelled using the law of mass action (Middleton et al., 2012). Considering the GA20ox-mediated steps, we analyse the identifiability of the model parameters using published experimental data; the analysis reveals the ratio between enzyme and GA levels to be small and motivates us to perform a quasi-steady state analysis to derive a reduced model. Fitting the parameters in the reduced model reveals additional features of the pathway and motivates further asymptotic analysis which produces a hierarchy of reduced models. Calculating the Akaike information criterion and parameter confidence intervals enables us to select a parsimonious model with identifiable parameters. As well as demonstrating the benefits of combining parameter estimation and asymptotic analysis, the analysis shows how GA biosynthesis is limited by the final GA20ox-mediated steps in the pathway and generates a simple mathematical description of this part of the GA biosynthesis pathway.

Coordination of meristem and boundary functions by transcription factors in the SHOOT MERISTEMLESS regulatory network.

The Arabidopsis homeodomain transcription factor SHOOT MERISTEMLESS (STM) is crucial for shoot apical meristem (SAM) function, yet the components and structure of the STM gene regulatory network (GRN) are largely unknown. Here, we show that transcriptional regulators are overrepresented among STM-regulated genes and, using these as GRN components in Bayesian network analysis, we infer STM GRN associations and reveal regulatory relationships between STM and factors involved in multiple aspects of SAM function. These include hormone regulation, TCP-mediated control of cell differentiation, AIL/PLT-mediated regulation of pluripotency and phyllotaxis, and specification of meristem-organ boundary zones via CUC1. We demonstrate a direct positive transcriptional feedback loop between STM and CUC1, despite their distinct expression patterns in the meristem and organ boundary, respectively. Our further finding that STM activates expression of the CUC1-targeting microRNA miR164c combined with mathematical modelling provides a potential solution for this apparent contradiction, demonstrating that these proposed regulatory interactions coupled with STM mobility could be sufficient to provide a mechanism for CUC1 localisation at the meristem-organ boundary. Our findings highlight the central role for the STM GRN in coordinating SAM functions.