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Schnepf et al., (2020) defined benchmark scenarios for root growth models, soil water flow models, root water flow models, and for water flow in the coupled soil-root system. All benchmarks and corresponding reference solutions were published in the form of Jupyter Notebooks on the GitHub repository https://github.com/RSAbenchmarks/collaborative-comparison. Several groups of functional-structural model developers have joined this benchmarking activity and provided the results of their individual implementations of the different scenarios. The focus of this contribution is on water uptake from a drying soil by a static root architecture. The numerical solutions of the different participating simulators as compared to the provided reference solution. The participating simulators are CPlantBox, DuMux, R-SWMS, OpenSimRoot and SRI. They have in common that they simulate water flow in the 3D soil domain, water flow inside the root system that is represented as a mathematical tree graph, and the coupling between the two domains in form of a volumetric sink term that describes the transfer of water between the two domains. The simulators differ in the numerical schemes used for solving the water flow equations in roots and soil domains, as well as in the way the sink term is formulated, in particular in the way the possibly increased rhizosphere resistance to water flow is accounted for. The results to the water flow in soil benchmarks show how the different simulators perform against the analytical solution to a problem of infiltration into an initially dry soil, as well as a problem of evaporation from initially moist soil. All of the simulators could accurately predict the infiltration front in different soil types as well as the actual evaporation curves. The coupled problem of root water uptake by a static root architecture from an initially already dry soil posed a bigger challenge to the different simulators and revealed some diversity between the different solutions. The Benchmark with an initially rather dry soil defined a potential transpiration that immediately induced water stress of the plant. The simulators had to simulate the consequent rhizosphere drying and associated increase in rhizosphere resistance. All of the soil simulators smoothed the gradients in the rhizosphere at the soil grid size such that root water uptake was significantly overestimated unless the rhizosphere resistance was explicitly accounted for in the root water uptake model. As a result, all simulators came close to the reference solution (that itself is a numerical solution, see Schnepf et al. 2020 for details). In this study, we showed that all simulators are generally able to solve the benchmark problems but minor differences occur amongst the simulators when simulating different soil types. Benchmarking led to model improvements and helped interpret model results in a more informed way. The availability of “reference solutions“ made modellers aware of the range of validity of their numerical solution and encouraged them to improve either their numerical solution or to introduce new processes Future efforts may aim to extend the benchmarks from water flow to further processes, such as solute transport or rhizodeposition. Schnepf et al., 2020, Front. Plant Sci.

Microorganisms possess the ability to adapt to different environmental conditions through the use of various strategies. This diversity in strategies allows us to categorize them based on their functions in the ecosystem. Copiotrophs have a fast growth rate but a low carbon use efficiency (CUE), while oligotrophs have a slow growth rate but a high CUE. In the rhizosphere, the effect of root exudation on different functional microbial groups is not well understood. Process-based modeling is a useful tool to analyze the complex feedback between roots and soil in the rhizosphere. Here, we present a rhizosphere model that explicitly considers two different microbial groups (oligotrophs and copiotrophs) classified based on their microbial traits that correlates each other due to physiological trade-offs and organic carbon accessibility (dissolved organic carbon, mucilage and sorbed carbon). The model is one-dimensional axisymmetric, simulating a soil cylinder around individual root segments. The model was conditioned using a novel constraint-based Markov chain Monte Carlo parameter sampling method. Applying this approach enabled the identification of parameter sets that led to plausible model results in agreement with experimental findings from a comprehensive literature review. The conditioned model predicts organic matter concentration curves from the root surface into the soil driven by root exudation. Our simulations show a decreasing pattern of dissolved organic carbon, which is utilized by oligotrophs and copiotrophs, away from the root surface. Furthermore, we observe a slightly higher proportion of copiotrophs than oligotrophs near the root surface and dominance of copiotrophic biomass at very high nutrient availability conditions as expected from ecological theory and experimental evidence. However, the model predictions are still highly uncertain. Thus, further experimental data and observations are required for model conditioning.

The rhizosphere, or the soil directly influenced by plant roots, is a complex and dynamic environment shaped by both plant and soil processes. Plant processes include root growth, rhizodeposition, root water and nutrient uptake or signalling; soil processes include water flow, reactive transport, organic matter decomposition or soil microbe and fauna-related processes. In this contribution, we focus on the soil-related aspects of modelling the interactions within the rhizosphere and how these interactions lead to the emergence of specific properties. Factors such as radial transport, root growth, and diurnal variation all play a role in the formation of patterns within the rhizosphere. However, modelling these processes is challenging due to their interconnected nature and the fact that they occur on multiple temporal and spatial scales. Recent research by Vetterlein et al. (2020) and Schnepf et al. (2022) have addressed these challenges and advances in our understanding of modelling the rhizosphere. For example can the effect of root elongation rate on the radial extension of the rhizosphere be quantified by means of the rhizosphere Péclet number, a dimensionless number that compares the importance of diffusive transport relative to root elongation rate. New findings of Kuppe et al. (2022), who have organized rhizosphere models within a collective framework that allows for the incorporation of microorganisms and their activity and motility, and Deckmyn et al. (2020), who combined soil carbon and food web ecosystem models, will further enhance a mechanistic description of the rhizosphere Deckmyn G, Flores O, Mayer M, Domene X, Schnepf A, Kuka K, Van Looy K, Rasse DP, Briones MJI, Barot S, Berg M, Vanguelova E, Ostonen I, Vereecken H, Suz LM, Frey B, Frossard A, Tiunov A, Frouz J, Grebenc T, Öpik M, Javaux M, Uvarov A, Vinduskova O, Henning Krogh P, Franklin O, Jiménez J, Curiel Yuste J. 2020. KEYLINK: towards a more integrative soil representation for inclusion in ecosystem scale models. I. review and model concept. PeerJ 8:e9750 DOI 10.7717/peerj.9750Kuppe CW, Schnepf A, von Lieres E, Watt M, Postma JA (2022) Rhizosphere models: their concepts and application to plant-soil ecosystems. Plant Soil 474, 17–55. doi: 10.1007/s11104-021-05201-7Schnepf A, Carminati A, Ahmed MA, Ani M, Benard P, Bentz J, Bonkowski M, Knott M, Diehl D, Duddek P, Kröner E, Javaux M, Landl M, Lehndorff E, Lippold E, Lieu A, Mueller CW, Oburger E, Otten W, Portell X, Phalempin M, Prechtel A, Schulz R, Vanderborght J, Vetterlein D (2022) Linking rhizosphere processes across scales: Opinion. Plant and Soil 478: 5-42. doi: 10.1007/s11104-022-05306-7.Vetterlein D, Carminati A, Kögel-Knabner I, Bienert GP, Smalla K, Oburger E, Schnepf A, Banitz T, Tarkka MT, Schlüter S (2020) Rhizosphere Spatiotemporal Organization–A Key to Rhizosphere Functions. Frontiers in Agronomy 2. doi: 10.3389/fagro.2020.00008.

To describe plant transpiration in drying soil, several models use ‘α-stress functions’, which represent the ratio of the maximal possible water uptake when the plant reaches the wilting point to the transpiration demand or potential transpiration, as a function of the soil water potential. Water potentials vary within the root zone, and the plant ‘senses’ with its root system an average root zone water potential and redistributes the uptake from drier to wetter zones in the root zone. This redistribution or root water uptake compensation is accounted for using an average stress index, ω, which is a weighted average of the local stress indices α at different depths in the root zone, and a critical stress index ωc (Jarvis, 2011; Simunek & Hopmans, 2009). When ω > ωc, root water uptake is equal to the potential root water uptake or the energy limited potential transpiration. The α-ω approach refers to a mechanistic description of water fluxes in the soil-root system but remains semi-empirical missing a direct link with soil and, especially, root hydraulic properties. In this contribution, we derive the α-ω approach starting from a mechanistic description of water flow in a hydraulic root architecture assuming that resistance to flow in the soil towards the soil-root interface can be neglected. In a second step, we include the non-linear soil resistance.For relatively wet soil conditions and neglecting the soil resistance, root water uptake functions can be cast in a form that is identical to the α-ω approach that was derived by Jarvis (2011), but for opposite conditions, i.e., Jarvis neglected the root resistance compared to soil resistance. Following Jarvis, the α-function should be interpreted as the ratio of the maximal possible uptake by the root system for a certain soil water potential to the maximal possible uptake by the system when the soil is fully saturated, which differs from its common interpretation. This means that the α-function is just a linear function that ranges from zero when the soil water potential is equal to the wilting point to 1 when the soil water potential is zero and that it is independent of the transpiration rate. Another outcome is that the critical stress level ωc is inverse proportional to the hydraulic conductance of the root system and is not a constant but a variable parameter that is proportional to the transpiration rate. For dry soil conditions, when soil resistance is important, we find that α and ω are non-linear functions of the soil water potential. Using α and ω functions that are derived from soil and root hydraulic properties, the uptake distributions can be calculated directly from the soil water potentials without solving a non-linear equation with iterations to derive water potentials in the plant. But, this approach is based on a simplification, which requires further testing.Jarvis, N. J. (2011). Hydrology and Earth System Sciences, 15(11), 3431-3446. doi:10.5194/hess-15-3431-2011Simunek, J., & Hopmans, J. W. (2009). Ecological Modelling, 220(4), 505-521. doi:10.1016/j.ecolmodel.2008.11.004

Plant growth and development involve the integration of numerous processes, influenced by both endogenous and exogenous factors. At any given time during a plant’s life cycle, the plant architecture is a readout of this continuous integration. However, untangling the individual factors and processes involved in the plant development and quantifying their influence on the plant developmental process is experimentally challenging.Here we used a combination of computational plant models to help understand experimental findings about how local phloem anatomical features influence the root system architecture. In particular, we simulated the mutual interplay between the root system architecture development and the carbohydrate distribution to provide a plausible mechanistic explanation for several experimental results.Our in silico study highlighted the strong influence of local phloem hydraulics on the root growth rates, growth duration and final length. The model result showed that a higher phloem resistivity leads to shorter roots due to the reduced flow of carbon within the root system. This effect was due to local properties of individual roots, and not linked to any of the pleiotropic effects at the root system level.Our results open the door to a better representation of growth processes in plant computational models.

Accurate prediction of root growth and related resource uptake is crucial to accurately simulate crop growth especially under unfavorable environmental conditions. We coupled a 1D field-scale crop-soil model running in the SIMPLACE modeling framework with the 3D architectural root model CRootbox on a daily time step and implemented a stress function to simulate root elongation as a function of soil bulk density and matric potential. The model was tested with field data collected during two growing seasons of spring barley and winter wheat on Haplic Luvisol. In that experiment, mechanical strip-wise subsoil loosening (30–60 cm) (DL treatment) was tested, and effects on root and shoot growth at the melioration strip as well as in a control treatment were evaluated. At most soil depths, strip-wise deep loosening significantly enhanced observed root length densities (RLDs) of both crops as compared to the control. However, the enhanced root growth had a beneficial effect on crop productivity only in the very dry season in 2018 for spring barley where the observed grain yield at the strip was 18% higher as compared to the control. To understand the underlying processes that led to these yield effects, we simulated spring barley and winter wheat root and shoot growth using the described field data and the model. For comparison, we simulated the scenarios with the simpler 1D conceptual root model. The coupled model showed the ability to simulate the main effects of strip-wise subsoil loosening on root and shoot growth. It was able to simulate the adaptive plasticity of roots to local soil conditions (more and thinner roots in case of dry and loose soil). Additional scenario runs with varying weather conditions were simulated to evaluate the impact of deep loosening on yield under different conditions. The scenarios revealed that higher spring barley yields in DL than in the control occurred in about 50% of the growing seasons. This effect was more pronounced for spring barley than for winter wheat. Different virtual root phenotypes were tested to assess the potential of the coupled model to simulate the effect of varying root traits under different conditions.

Exudation of organic carbon triggers complex spatial and temporal patterns of biophysical and biochemical processes in the root-influenced soil (rhizosphere). We use process-based modeling as a tool to gain insights into microbial interactions and carbon cycling in the rhizosphere. Here, we present a trait-based rhizosphere model that accounts for two different functional microbial groups (copiotrophs, oligotrophs) that differ according to life-history strategies, microbial physiology (e.g., dormancy) and carbon turnover (small and large polymers). The model is calibrated and validated against experimental data from the literature. We apply a parameter search algorithm that identifies plausible parameter spaces by conditioning model outputs to parameter and process constraints that reflect current ecological knowledge. We show the general concept of the model, first simulations after model conditioning, and a concept for coupling the rhizosphere model with the structural-functional plant model CPlantBox to cover the whole-plant scale.

Water and nutrient uptake are essential for plant productivity. Therefore, the development of precise functional-structural root models will enable better agricultural management in particular in resource limited environments. In such models water movement is of special importance since the rhizosphere's biochemical reactions are strongly influenced by water content and water movement. We present a general sink term for larger scale root water uptake models that includes the root-rhizosphere hydraulic architecture.We derive the new aggregated sink term from a more complex model that describes the rhizosphere around each root segment in dependence of a hydraulic root system model. We use CPlantBox (Schnepf et al. 2018) to represent root architecture, and calculate water movement within the root system using the hybrid analytical solution of Meunier et al. (2017). Around each root segment we represent water movement within the rhizosphere by a 1D axisymmetric model. Such models are flexible in the way the rhizosphere is represented (Mai et al. 2019). They are able to accurately describe water depletion and redistribution, but are computationally expensive. To simplify the model we use the analytical solution of the steady rate approximation following (Schröder et al. 2008) for water movement in the 1D axisymmetric models. The analytic solution depends on the matric potentials of the macroscopic soil (which is calculated in 1D, 2D or 3D) and the hydraulic root architecture model, root radial conductivity, and radius of the rhizosphere domain. We use fixed-point iteration to determine the matric potential at the soil root interface and store the solutions in a look up table for speedup. Moving to larger scales it is generally not useful to keep track of all root system architectures. Therefore, we aim for a coarser approximation of the root architecture by representing it as detached parallel root segments. Parallel segment conductivities are based on standard uptake fraction (SUF) and root system conductivity (Krs) of the original topology (Couvreur et al. 2012), which was shown by Vanderborght et al. (2021) to be a close approximation of the uptake by the original root topology. This approach makes the computation of the full root hydraulic architecture model superfluous, leading to a stable and performant sink term. This new sink term increases the accuracy of water uptake in a suite of larger scale models including crop modes, earth system models, and hydrological models. Using the presented approach, the sink term can be derived directly from 3D root hydraulic architecture. This avoids parameterizations based on proxy information about root system hydraulics and can acknowledge age dependent axial and radial root segment hydraulic conductances. Finally, information about rhizosphere hydraulic properties, which may differ from bulk soil hydraulic properties can be injected effectively in this sink term model. ReferencesSchnepf, A., et al. (2018) Annals of botany, 121(5) 1033-1053.Meunier, F. et al. Applied Mathematical Modelling, 52, 648-663.Mai, TH., et al. Plant and Soil, 439(1), 273-292.Schröder, et al. (2008) Vadose Zone Journal 7(3), 1089-1098.Couvreur, V., et al. (2012) Hydrology and Earth System Sciences, 16(8), 2957-2971.Vanderborght, J., et al. (2021) Hydrology and Earth System Sciences, 25(9), 4835-4860.

Investigating soil, roots and their interaction is important to optimize agricultural practices like irrigation and fertilization and therefore increase the sustainability and productivity of crop production. In this study, we are combining two methods to examine non-invasively, characterize and monitor the soil-root zone throughout crop growing seasons: crosshole ground penetrating radar (GPR) and root-images within horizontal mini-rhizotrons. Over three maize crop growing seasons, we acquired in-situ time-lapse crosshole ground penetrating radar data and time-lapse root images, at two mini-rhizotron facilities in Selhausen, Germany. These facilities allow to horizontally measure data at six different depths, ranging between 0.1 m - 1.2 m and below three different plots with varying agricultural treatments, such as irrigation, sowing density, sowing date and cultivars. The GPR measurements result in the dielectric permittivity slices by applying standard ray-based analysis to zero-offset measurements along a pair of rhizotubes. Such horizontal permittivity slices can be linked to soil water content using petro‑physical relationships. Additionally, the root images provide a root fraction per image, which is derived by using a workflow combining state-of-the-art software tools, deep neural networks and automated feature extraction. The dielectric permittivity slices suggest a permittivity variation along the horizontal and vertical axes, depending on atmospheric conditions, soil properties, and root architecture. To quantify the influence of the roots on the spatial and temporal distribution of dielectric permittivity, we used statistical methods to reduce the impacting factors like soil heterogeneity, tube deviations and changing atmospheric conditions, which results in the spatial and temporal variability. For verification these permittivity variabilities are compared to the root fraction values. In general, using the spatial and temporal permittivity variations, we can detect the presence of roots and additionally recognize a varying influence of the roots over the duration of the crop growing season. Using these first results, we demonstrate that GPR can be applied to improve the characterization of the root-soil system related to maize plants. This could be the first step towards developing proxies e.g. for irrigation and fertilization applications using this non-invasive method.

3D models of root growth, architecture and function are becoming important tools to aid the design of agricultural management schemes and the selection of beneficial root traits. While benchmarking is common for water and solute transport models in soil, 3D root-soil interaction models have not yet been systematically analysed. Several interacting processes might induce disagreement between models: root growth, sink term definitions of root water and solute uptake and representation of the rhizosphere. Schnepf et al. (2020) proposed a framework for quantitatively comparing such models. It builds upon benchmark scenarios that test individual components, followed by benchmark scenarios for the coupled root-rhizosphere-soil system.Here we present the results of benchmarking different well-known models (“simulators”) with respect to water flow in soil, water flow in roots, and water flow and root water uptake in a coupled soil-root system for the case of a given prescribed root architecture as observed from an MRI experiment. The participating simulators areCPlantBox and DuMux (Koch et al. 2021; Mai et al. 2019), R-SWMS (Javaux et al. 2008), OpenSimRoot (Postma et al. 2017) and ArchiSimple, RootTyp and SRI (Beudez et al. 2013; Pagès et al. 2014; Pagès et al. 2004).In the benchmark scenarios that represent individual modules, the different simulators solved the same mathematical model but with different numerical approaches; all perform well with respect to the given analytical reference solution. For the coupled problem of root water uptake from a drying soil, the different simulators make different choices for the coupling of the different sub-problems. Thus, the results of the different simulators show a larger heterogeneity amongst each other.We expect that this benchmarking will result in improved models, with which we can simulate various scenarios with greater confidence, avoiding that future work is based on accidental results caused by bugs, numerical errors or conceptual misunderstandings and will set a standard for model development.Beudez N, Doussan C, Lefeuve-Mesgouez G, Mesgouez A (2013) Procedia Environmental Sciences 19: 37-46. doi:Javaux M, Schröder T, Vanderborght J, Vereecken H (2008) Vadose Zone Journal 7: 1079-1088.Koch T, Wu H, Schneider M (2021) Journal of Computational Physics: 110823.Mai TH, Schnepf A, Vereecken H, Vanderborght J (2019) Plant and Soil 439: 273-292. doi: 10.1007/s11104-018-3890-4.Pagès L, Bécel C, Boukcim H, Moreau D, Nguyen C, Voisin A-S (2014) Ecological Modelling 290: 76-84.Pagès L, Vercambre G, Drouet J-L, Lecompte F, Collet C, Le Bot J (2004) Plant and Soil 258: 103-119.Postma JA, Kuppe C, Owen MR, Mellor N, Griffiths M, Bennett MJ, Lynch JP, Watt M (2017) New Phytologist 215: 1274-1286.Schnepf A, Black CK, Couvreur V, et al. (2020) Frontiers in Plant Science 11.

Three-dimensional models of root system development and functioning have evolved as important tools that aid designing agricultural management schemes for improved resource use efficiency and selecting root traits for optimizing plant performance in specific environments (Benes et al., 2020). For their reliable application, benchmarking of such so-called functional-structural root architecture models (FSRM) is urgently needed. Similar relevant benchmarking initiatives have been performed for crop models (AgMIP), reactive transport models (Steefel et al., 2015), or models of water flow and solute transport in soils (Vanderborght et al., 2005). FSRMs generally solve flow and transport equations in the soil and in root system, and couple them via different approaches. Differences between different models' outputs might arise from differences in mathematical formulation of the processes and their coupling, in the numerical scheme, but also from coding errors. Consequently, potential errors might propagate into the plant and soil interaction simulations relying on an accurate simulation of root architecture development for describing root water and solute uptake processes. This Research Topic set out to shed some light on the extent of potential uncertainty due to these different factors. Benchmarking is an emerging procedure to measure performance of models against a set of defined standards (Luo et al., 2012). In this issue, Schnepf et al. announced a “Call for Participation: Collaborative Benchmarking of Functional-Structural Root Architecture Models. The Case of Root Water Uptake”. They designed benchmark problems for root growth models, soil water flow models, root water flow models, and for water flow in the coupled soil-root system. All the benchmarks and corresponding reference solutions were published in the form of Jupyter Notebooks on the GitHub repository https://github.com/RSA-benchmarks/collaborative-comparison. Several groups that develop such functional-structural root architecture models have contributed with their solutions to the benchmark problems on this GitHub repository, and it may provide orientation for future model developments as well. The benchmarks follow a multi-step approach with growing level of complexity regarding both the number of processes accounted for and the dimension of the system. The first set of benchmarks is about individual modules only. The scenarios are simple, potentially solvable with analytical solutions, and the goal is to build trust in accuracy of the individual models. The second set of benchmarks is about the fully coupled models, with a focus on comparison of numerical representation of agreed-upon equations and process representations. In the third set of benchmarks, models do not have to have the same process representations. Evaluations of those are only possible against available data sets and by comparing the different model outputs. In this issue, Khare et al. further extended the benchmark problem Schnepf et al. which is about root water uptake from a drying soil. They showed in a grid convergence study that the additional resistance to water flow toward the root surface caused by a dry rhizosphere must be considered for dry soil or else root water uptake is significantly overestimated. Simulations were performed with dumux-rosi. Solutions to the problem of dry rhizosphere are presented in Khare et al., but also in (Schröder et al., 2009a,b; Beudez et al., 2013; Mai et al., 2019; Koch et al., 2021). All of those solutions include a way to determine sub-resolution scale (with respect to the soil grid) rhizosphere water potential gradients in a computationally efficient way. The alternative is grid refinement but this comes at high computational costs as also discussed in Khare et al.. As part of this issue, soil compaction due to agricultural traffic and resulting mechanical and hydric stresses and their effect on root water uptake were simulated by de Moraes et al. using CRootBox (Schnepf et al., 2018) coupled with a 1-dimension soil water flow model. The model simulations could elucidate the feedback between root function and local soil stresses at the field scale for a Brazilian Oxisol. Here, the reference is not a mathematical reference solution but a reference data set. For field-scale simulations, FSRMs are often coupled to models that have 1-dimensional soil modules, e.g., crop models (Wu et al., 2015; Seidel et al., 2022). From a known 3D root hydraulic architecture, 1-dimensional sink terms can be derived (e.g. as shown in Vanderborght et al., 2021). Model simulations may elucidate the contributions of different root types to overall plant nutrient uptake. Using OpenSimRoot (Postma et al., 2017), Gonzalez et al. indicated in this issue that nodal roots contribute most to P uptake by rice plants, followed by L-type lateral roots, S-type laterals and root hairs, but these strategies have different carbon costs. Implications for improving adaption to P deficiency in rice breeding are discussed. These results have to be also seen in light of the respective soil P and water content (De Bauw et al., 2020). The longitudinal pattern of root aerenchyma formation modeled by the Ti-Gompertz model helped to deeply understand the relationship between the anatomical traits and physiological function in rice adventitious roots (Chen et al., as part of this issue). Such data will help to further develop models that include information on the root anatomy such as MECHA (Heymans et al., 2021) and GRANAR (Heymans et al., 2019). Through this Research Topic, we continue to provide the opportunity to participate in the development and application of suitable benchmarks. This exercise allows us to point out sources of inaccuracies, knowledge gaps and to pin-point current challenges in mathematical model development of FSRM's.

Root systems of crops play a significant role in agroecosystems. The root system is essential for water and nutrient uptake, plant stability, symbiosis with microbes, and a good soil structure. Minirhizotrons have shown to be effective to noninvasively investigate the root system. Root traits, like root length, can therefore be obtained throughout the crop growing season. Analyzing datasets from minirhizotrons using common manual annotation methods, with conventional software tools, is time-consuming and labor-intensive. Therefore, an objective method for high-throughput image analysis that provides data for field root phenotyping is necessary. In this study, we developed a pipeline combining state-of-the-art software tools, using deep neural networks and automated feature extraction. This pipeline consists of two major components and was applied to large root image datasets from minirhizotrons. First, a segmentation by a neural network model, trained with a small image sample, is performed. Training and segmentation are done using “RootPainter.” Then, an automated feature extraction from the segments is carried out by “RhizoVision Explorer.” To validate the results of our automated analysis pipeline, a comparison of root length between manually annotated and automatically processed data was realized with more than 36,500 images. Mainly the results show a high correlation ( r = 0.9 ) between manually and automatically determined root lengths. With respect to the processing time, our new pipeline outperforms manual annotation by 98.1-99.6%. Our pipeline, combining state-of-the-art software tools, significantly reduces the processing time for minirhizotron images. Thus, image analysis is no longer the bottle-neck in high-throughput phenotyping approaches.

In this chapter, we present the Root and Soil Water Movement and Solute transport model R-SWMS, which can be used to simulate flow and transport in the soil-plant system. The equations describing water flow in soil-root systems are presented and numerical solutions are provided. An application of R-SWMS is then briefly discussed, in which we combine in vivo and in silico experiments in order to decrypt water flow in the soil-root domain. More precisely, light transmission imaging experiments were conducted to generate data that can serve as input for the R-SWMS model. These data include the root system architecture, the soil hydraulic properties and the environmental conditions (initial soil water content and boundary conditions, BC). Root hydraulic properties were not acquired experimentally, but set to theoretical values found in the literature. In order to validate the results obtained by the model, the simulated and experimental water content distributions were compared. The model was then used to estimate variables that were not experimentally accessible, such as the actual root water uptake distribution and xylem water potential.

In this chapter, we present the Root and Soil Water Movement and Solute transport model R-SWMS, which can be used to simulate flow and transport in the soil-plant system. The equations describing water flow in soil-root systems are presented and numerical solutions are provided. An application of R-SWMS is then briefly discussed, in which we combine in vivo and in silico experiments in order to decrypt water flow in the soil-root domain. More precisely, light transmission imaging experiments were conducted to generate data that can serve as input for the R-SWMS model. These data include the root system architecture, the soil hydraulic properties and the environmental conditions (initial soil water content and boundary conditions, BC). Root hydraulic properties were not acquired experimentally, but set to theoretical values found in the literature. In order to validate the results obtained by the model, the simulated and experimental water content distributions were compared. The model was then used to estimate variables that were not experimentally accessible, such as the actual root water uptake distribution and xylem water potential.

Purpose Simultaneously interacting rhizosphere processes determine emergent plant behaviour, including growth, transpiration, nutrient uptake, soil carbon storage and transformation by microorganisms. However, these processes occur on multiple scales, challenging modelling of rhizosphere and plant behaviour. Current advances in modelling and experimental methods open the path to unravel the importance and interconnectedness of those processes across scales. Methods We present a series of case studies of state-of-the art simulations addressing this multi-scale, multi-process problem from a modelling point of view, as well as from the point of view of integrating newly available rhizosphere data and images. Results Each case study includes a model that links scales and experimental data to explain and predict spatial and temporal distribution of rhizosphere components. We exemplify the state-of-the-art modelling tools in this field: image-based modelling, pore-scale modelling, continuum scale modelling, and functional-structural plant modelling. We show how to link the pore scale to the continuum scale by homogenisation or by deriving effective physical parameters like viscosity from nano-scale chemical properties. Furthermore, we demonstrate ways of modelling the links between rhizodeposition and plant nutrient uptake or soil microbial activity. Conclusion Modelling allows to integrate new experimental data across different rhizosphere processes and scales and to explore more variables than is possible with experiments. Described models are tools to test hypotheses and consequently improve our mechanistic understanding of how rhizosphere processes impact plant-scale behaviour. Linking multiple scales and processes including the dynamics of root growth is the logical next step for future research., Plant and Soil, 478, ISSN:0032-079X, ISSN:1573-5036

We developed root scale models that simulate water and nutrient uptake by crops. Water flow in the soil and root systems was linked in order to describe root water uptake as a function of root properties and distributions, soil and leaf water potentials. One of the underlying motivations is to predict the crop water stress level and its impact on transpiration and growth. The mechanistic description of water fluxes resulted in models that were sensitive to hydraulic properties of the root system, including root density, and root distribution with depth. These sensitivities improved predictions of crop water uptake and water stress in different soils and for different water treatments. Crucial was the correct representation of the root system and its response to different ‘treatments’. Thus, in order to predict the impact of water stress on growth, the growth response to the water stress must be predicted. So far, these response functions and especially the distribution of carbon within the plant to the different plant organs are empirical functions. A coupled carbon and water flow model within the plant is a way forward to more mechanistic descriptions of these responses. A similar storyline can be developed for nutrient uptake. Mechanistic nutrient uptake models do not consider nutrient transport within the root system but focus on transport towards the root surface. Multi-scale flow and transport simulations demonstrated that small scale transport towards growing root tips and root system scale water and nutrient distributions controlled nutrient uptake. These simulations predicted the observed interaction between water and phosphate uptake of an upland rice crop. However, here again, simulated uptake depended on the root development in response to nutrient and water stress. Mechanistic descriptions of root growth response to nutrients require a further understanding of plant physiological processes that cause these responses.

The rhizosphere shows complex spatial and temporal patterns of biophysical and biochemical processes. Process-based modeling that accounts for functional microbial traits provides a tool to gain a better understanding of microbial interactions involved in carbon cycling in the rhizosphere. Here, we present a trait-based rhizosphere model that accounts for microbial life-history strategies (copiotrophs, oligotrophs), microbial physiology (e.g., dormancy), and organic carbon bioaccessibility (small and large polymers). The model reflects the mm-scale microbial and carbon dynamics around a cylindrical root segment and will be linked with a structural-functional soil-plant model (CPlantBox), which enables to connect water, carbon and nitrogen dynamics in the rhizosphere to plant and bulk soil dynamics. We show the concept of trait-based rhizosphere modeling, first simulations, and our model coupling approach to CPlantBox.

Plant transpiration and root water uptake are dependent on multiple traits that interact with site soil characteristics and environmental factors such as radiation, atmospheric temperature, relative humidity, and soil-moisture content. Models of root architecture and functions are increasingly employed to simulate root-soil interactions. Root water uptake is thereby affected by the root hydraulic architecture, soil moisture conditions, soil hydraulic properties, and the transpiration demand as controlled by atmospheric conditions. Stomatal conductance plays a vital role in regulating transpiration in plants. We performed simulations of plant water uptake for plants having different mechanisms to control transpiration, spanned by isohydric/anisohydric spectrum. Isohydric plants follow the strategy to close their stomata in order to maintain the leaf water potential at a constant level, while anisohydric plants leave their stomata open when leaf water potentials fall due to drought stress. Modelling the stomatal regulation effectively will result in a more reliable model that will regulate the excessive loss of water. We implemented hydraulic and chemical stomatal controlof root water uptake following the current approach where stomatal control is regulated by simulated water potential and/or chemical signal concentration. In order to maintain water uptake from dry soil, low plant water potentials are required, which may lead to reversible or permanent cavitation. We parameterise our model with field data, including climate data and soil hydraulic properties under different tillage conditions. This helps us to understand the behaviour of different crops under drought conditions and predict at which growing stage the stress hits the plant. We conducted the simulations for different scenarios to study the effect of hydraulic and chemical regulation on root system performance under drought stress.

The combination of functional-structural root-system models with root architectures derived from non-invasive imaging is a promising approach for gaining a better understanding of root-soil interaction processes. However, root architectures can often not be fully recovered using imaging, which subsequently affects the assessment of function via the functional-structural root models. In this study, we explored theoretical and actual possibilities of root system reconstruction from MRI and X-ray CT images. Experiments with water-filled capillaries showed the same minimum detectable diameter for both MRI and X-ray CT for the used parameter setup. Experiments with soil-grown lupine roots, however, showed significantly lower root system recovery fractions for MRI than for X-ray CT, from which most roots thicker than 0.2 mm could be recovered. MRI allowed root signal detection below voxel resolution; however, the connection of this signal to a continuous root structure proved difficult for large, crowded root systems. Furthermore, soil moisture levels >30% hampered root system recovery from MRI scans in experiments with pure sand. To overcome the problem of low root system recovery fractions, we developed a new method that uses incomplete root systems as a scaffold onto which missing roots are simulated using information from WinRhizo measurements. Comparisons of root length within subsamples of semi-virtual root systems and root systems derived from X-ray CT scans showed good agreement. Evaluation of hydraulic root architecture measures of incomplete root system scaffolds and semi-virtual root systems proved the importance of using complete root system reconstructions to simulate root water uptake. Semi-virtual root reconstruction thus appears to be a promising technique to complete root systems for subsequent use in functional-structural root models.

In the course of climate change, the occurrence of extreme weather events is expected to increase. Drought tolerance of crops and careful irrigation management are becoming key factors for global food security and the sustainable resource use of water in agriculture. Root water uptake plays a vital role in drought tolerance. It is influenced by root architecture, plant and soil water status and their respective hydraulic properties. Models of said factors aid in organizing the current state of knowledge and enable a deeper understanding of their respective influence on crop performance. Water uptake by roots leads to a decrease in soil moisture and may cause the formation of soil water potential gradients between the bulk soil and the soil-root interface. Although the Richards equation in theory takes these gradients into account, a very fine discretization of the soil domain is necessary to capture these gradients in simulations. However, especially during drought stress, the drop in hydraulic conductivity in the rhizosphere could have a major impact on the overall water uptake of the root system. In order to investigate computationally feasible alternative approaches for simulations with source terms that take these hydraulic conductivity drops into account, we conducted experiments with lupine plants. The root architecture of the growing plants was measured several times using an MRI. Subsequently, these MRI images were used in a holobench for manual tracing of the roots. We were able to mimic the root growth between the measurement dates using linear interpolation. In addition to root architecture, soil water contents and transpiration rates were monitored. We then used this data to systematically compare the computational effort of different approaches to consider the hydraulic conductivity drop near roots in terms of accuracy and computational cost. Eventually we aim at using these results to improve existing root water uptake models for the presence of hydraulic conductivity drops in the rhizosphere in an efficient and accurate way.

Home-field advantage (HFA) encompasses all the processes leading to faster litter decomposition in the ‘home’ environment compared to that of ‘away’ environments. To determine the occurrence of HFA in a forest and adjacent clear-cut, we set up a reciprocal litter decomposition experiment within the forest and clear-cut for two soil types (Cambisols and Gleysols) in temperate Germany. The forest was dominated by Norway spruce (Picea abies), whereas forest regeneration of European Beech (Fagus sylvatica) after clearcutting was encouraged. Our observation that Norway spruce decomposed faster than European beech in 70-yr-old spruce forest was most likely related to specialized litter-soil interaction under existing spruce, leading to an HFA. Elevated soil moisture and temperature, and promoted litter N release, indicated the rapid change of soil-litter affinity of the original spruce forest even after a short-term regeneration following clearcutting, resulting in faster beech decomposition, particularly in moisture- and nutrient-deficient Cambisols. The divergence between forest and clear-cut in the Cambisol of their litter δ15N values beyond nine months implied litter N decomposition was only initially independent of soil and residual C status. We conclude that clearcutting modifies the litter-field affinity and helps promote the establishment or regeneration of European beech in this and similar forest mountain upland areas.

Although modelling of water and nutrient uptake by root systems has advanced considerably in recent years, steep local gradients of nutrient concentration near the root-soil interface in the rhizosphere are still a central challenge for accurate simulation of water and nutrient uptake at the root system scale. Conventionally, mesh refinement is used to resolve these gradients. However, it results in excessive computational costs. The object of the study is to present a multiscale approach which resolves the steep gradient of nutrient concentrations at rhizosphere scale and simulates nutrient and water fluxes within the entire root zone at macroscale scale in a computationally efficient way. We developed a 3D water and nutrient transport model of the root-soil system with explicit consideration of the 3D root architecture. To capture the nutrient gradients at root surfaces, 1D axisymmetric soil models at rhizosphere scale were constructed and coupled to the coarse 3D root-system-scale simulations using a mass conservative approach. The multiscale model was investigated under different scenarios for water and potassium (K+) uptake of a single root, multiple roots, and whole 3D architecture of a Zea mays L. root system in conditions of dynamic soil water and different soil buffer capacity of K+. The steep gradients of K+ concentrations were efficiently resolved in the multiscale simulations thanks to the 1D model at the rhizosphere scale. In comparison with the refinement method, the multiscale model achieved a significant accuracy of K+ uptake prediction with a relative error below 5%. Meanwhile, the simulation at macroscale with coarse mesh could overestimate the K+ uptake in one order of magnitude. Moreover, the computational cost of multiscale simulations was decreased considerably by using coarse soil mesh. The newly developed model can describe the effect of the drying and nutrient transport in the root zone on nutrient uptake. It also allows to simulate processes in larger and complex root systems because of the considerable reduction in computational cost.

Root elongation is generally limited by a combination of mechanical impedance and water stress in most arable soils. However, dynamic changes of soil penetration resistance with soil water content are rarely included in models for predicting root growth. Better modelling frameworks are needed to understand root growth interactions between plant genotype, soil management, and climate. Aim of paper is to describe a new model of root elongation in relation to soil physical characteristics like penetration resistance, matric potential, and hypoxia. A new diagrammatic framework is proposed to illustrate the interaction between root elongation, soil management, and climatic conditions. The new model was written in Matlab®, using the root architecture model RootBox and a model that solves the 1D Richards equations for water flux in soil. Inputs: root architectural parameters for Soybean; soil hydraulic properties; root water uptake function in relation to matric flux potential; root elongation rate as a function of soil physical characteristics. Simulation scenarios: (a) compact soil layer at 16 to 20 cm; (b) test against a field experiment in Brazil during contrasting drought and normal rainfall seasons. (a) Soil compaction substantially slowed root growth into and below the compact layer. (b) Simulated root length density was very similar to field measurements, which was influenced greatly by drought. The main factor slowing root elongation in the simulations was evaluated using a stress reduction function. The proposed framework offers a way to explore the interaction between soil physical properties, weather and root growth. It may be applied to most root elongation models, and offers the potential to evaluate likely factors limiting root growth in different soils and tillage regimes.

Background and aimsThe main difficulty in the use of 3D root architecture models is correct parameterization. We evaluated distributions of the root traits inter-branch distance, branching angle and axial root trajectories from contrasting experimental systems to improve model parameterization.MethodsWe analyzed 2D root images of different wheat varieties (Triticum aestivum) from three different sources using automatic root tracking. Model input parameters and common parameter patterns were identified from extracted root system coordinates. Simulation studies were used to (1) link observed axial root trajectories with model input parameters (2) evaluate errors due to the 2D (versus 3D) nature of image sources and (3) investigate the effect of model parameter distributions on root foraging performance.ResultsDistributions of inter-branch distances were approximated with lognormal functions. Branching angles showed mean values

Macropores may be preferential root pathways into the subsoil. We hypothesised that the presence of macropores promotes P-uptake from subsoil, particularly at limited water supply in surface soil. We tested this hypothesis in a rhizotron experiment with spring wheat (Triticum aestivum cv. Scirocco) under variation of fertilisation and irrigation. Rhizotrons were filled with compacted subsoil (bulk density 1.4 g cm−3), underneath a P-depleted topsoil. In half of these rhizotrons the subsoil contained artificial macropores. Spring wheat was grown for 41 days with and without irrigation and 31P–addition. Also, a 33P–tracer was added at the soil surface to trace P-distribution in plants using liquid scintillation counting and radioactive imaging. Fertilisation and irrigation promoted biomass production and plant P-uptake. Improved growing conditions resulted in a higher proportion of subsoil roots, indicating that the topsoil root system additionally promoted subsoil nutrient acquisition. The presence of macropores did not improve plant growth but tended to increase translocation of 33P into both above- and belowground biomass. 33P–imaging confirmed that this plant-internal transport of topsoil-P extended into subsoil roots. The lack of penetration resistance in macropores did not increase plant growth and nutrient uptake from subsoil here; however, wheat specifically re-allocated topsoil-P for subsoil root growth.

Structural reconstruction of plant roots from MRI is challenging, because of low resolution and low signal-to-noise ratio of the 3D measurements which may lead to disconnectivities and wrongly connected roots. We propose a two-stage approach for this task. The first stage is based on semantic root vs. soil segmentation and finds lowest-cost paths from any root voxel to the shoot. The second stage takes the largest fully connected component generated in the first stage and uses 3D skeletonization to extract a graph structure. We evaluate our method on 22 MRI scans and compare to human expert reconstructions., Comment: Accepted final version. In 25th International Conference on Pattern Recognition (ICPR2020)

The use of standard dynamic root architecture models to simulate root growth in soil containing macropores failed to reproduce experimentally observed root growth patterns. We thus developed a new, more mechanistic model approach for the simulation of root growth in structured soil. In our alternative modelling approach, we distinguish between, firstly, the driving force for root growth, which is determined by the orientation of the previous root segment and the influence of gravitropism and, secondly, soil mechanical resistance to root growth. The latter is expressed by its inverse, soil mechanical conductance, and treated similarly to hydraulic conductivity in Darcy’s law. At the presence of macropores, soil mechanical conductance is anisotropic, which leads to a difference between the direction of the driving force and the direction of the root tip movement. The model was tested using data from the literature, at pot scale, at macropore scale, and in a series of simulations where sensitivity to gravity and macropore orientation was evaluated. Qualitative and quantitative comparisons between simulated and experimentally observed root systems showed good agreement, suggesting that the drawn analogy between soil water flow and root growth is a useful one.

The flow of carbon (C) through soil is inherently complex due to the many thousands of different chemical transformations occurring simultaneously within the soil microbial community. The accurate modelling of this C flow therefore represents a major challenge. In response to this, isotopic tracers (e.g. 13 C, 14 C) are commonly used to experimentally parameterise models describing the fate and residence time of individual C compounds within soil. In this study, we critically evaluated the combined use of experimental 14 C labelling and mathematical modelling to estimate C turnover times in soil. We applied 14 C-labelled alanine and glucose to an agricultural soil and simultaneously measured their loss from soil solution alongside the rate of microbial C immobilization and mineralization. Our results revealed that chloroform fumigation-extraction (CFE) cannot be used to reliably quantify the amount of isotopically labelled 13 C/ 14 C immobilised by the microbial biomass. This is due to uncertainty in the extraction efficiency values ( k ec ) within the CFE methodology which are both substrate and incubation time dependent. Further, the traditional mineralization approach (i.e. measuring 14/13 CO 2 evolution) provided a poor estimate of substrate loss from soil solution and mainly reflected rates of internal microbial C metabolism after substrate uptake from the soil. Therefore, while isotope addition provides a simple mechanism for labelling the microbial biomass it provides limited information on the behaviour of the substrate itself. We used our experimental data to construct a new empirical model to describe the simultaneous flow of substrate-C between key C pools in soil. This model provided a superior estimate of microbial substrate use and microbial respiration flux in comparison to traditional first order kinetic modelling approaches. We also identify a range of fundamental problems associated with the modelling of isotopic-C in soil, including issues with variation in C partitioning within the community, model pool connectivity and variation in isotopic pool dilution, which make interpretation of any C isotopic flux data difficult. We conclude that while convenient, the use of isotopic data ( 13 C, 14 C, 15 N) has many potential pitfalls necessitating a critical evaluation of both past and future studies.

The functioning of ectomycorrhizal (ECM) symbioses is closely related to the development of the soil mycelial phase the ECM fungi. The properties and spatiotemporal dynamics of such mycelia in ecosystems is, however, poorly understood. Here we show, using a soil colony of summer truffle (Tuber aestivum) as a model, that an ECM mycelium may only grow and colonize newly-opened soil patches when soil temperatures rise above certain threshold, in this case +10 °C, provided other requirements such as sufficient soil moisture are fulfilled. Extension rates of truffle mycelium in the fields was recorded as >0.3 μm min−1, several-fold greater than that predicted from laboratory cultures. Further, we demonstrated that there was a consistent spatial differentiation in mycelium growth patterns within the fungal colony on a decimeter scale, changing from “diffusion” type of growth at the colony margin to “colony-front” pattern further away from the colony margin. This change was clearly accompanied by shifting structure of soil microbial communities with Terrimonas sp. and another unidentified bacterium correlating with the “colony-front” mycelium growth pattern, and Sphingomonas sp. and Lysobacter brunnescens with the “diffusion” type of mycelium growth. Possible implications of the observed truffle colony differentiation are discussed for processes like fruit-body formation and dispersal of this ECM fungus. Our data indicate that the thallus of T. aestivum has to be considered as a principally variable (“mutabilis”) being in space and time, whose behavior correlates with conditions in ever changing soil environment (“in mutabili”).

How much and where water is taken up by roots from the soil profile are important questions that need to be answered to close the soil water balance equation and to describe water fluxes in the soil–plant–atmosphere system. Physically-based root water uptake (RWU) models that relate RWU to transpiration, root density, and water potential distributions have been developed but far less used or tested. This study aims at evaluating the simulated RWU of winter wheat by the empirical Feddes–Jarvis (FJ) model and the physically-based Couvreur (C) model for different soil water conditions and soil textures against sap flow measurements. Soil water content (SWC), water potential, and root development were monitored non-invasively at six soil depths in two rhizotron facilities that were constructed in two soil textures: stony vs. silty with each three water treatments: sheltered, rainfed, and irrigated. Soil and root parameters of the two models were derived from inverse modeling and simulated RWU was compared with sap flow measurements for validation. The different soil types and water treatments resulted in different crop biomass, root densities and root distributions with depth. The two models simulated the lowest RWU in the sheltered plot of the stony soil where RWU was also lower than the potential RWU. In the silty soil, simulated RWU was equal to the potential uptake for all treatments. The variation of simulated RWU among the different plots agreed well with measured sap flow but the C model predicted the ratios of the transpiration fluxes in the two soil types slightly better than the FJ model. The root hydraulic parameters of the C model could be constrained by the field data but not the water stress parameters of the FJ model. This was attributed to differences in root densities between the different soils and treatments which are accounted for by the C model whereas the FJ model only considers normalized root densities. The impact of differences in root density on RWU could be accounted for directly by the physically-based RWU model but not by empirical models that use normalized root density functions.

Water stress is among the dominant yield limiting factors in global crop production. Better drought resistance is therefore a key challenge for breeding and crop management. Avoidance of water stress by effective root water uptake is considered a promising approach to yield stability in water limiting environments. Water uptake efficiency is the result of multiple plant root traits that dynamically interact with site hydrology. Root models are therefore an essential tool to identify key root traits for water efficient crops in a certain target cropping environment.

Root system traits are important in view of current challenges such as sustainable crop production with reduced fertilizer input or in resource-limited environments. We present a novel approach for recovering root architectural parameters based on image-analysis techniques. It is based on a graph representation of the segmented and skeletonized image of the root system, where individual roots are tracked in a fully automated way. Using a dynamic root architecture model for deciding whether a specific path in the graph is likely to represent a root helps to distinguish root overlaps from branches and favors the analysis of root development over a sequence of images. After the root tracking step, global traits such as topological characteristics as well as root architectural parameters are computed. Analysis of neutron radiographic root system images of lupine (Lupinus albus) grown in mesocosms filled with sandy soil results in a set of root architectural parameters. They are used to simulate the dynamic development of the root system and to compute the corresponding root length densities in the mesocosm. The graph representation of the root system provides global information about connectivity inside the graph. The underlying root growth model helps to determine which path inside the graph is most likely for a given root. This facilitates the systematic investigation of root architectural traits, in particular with respect to the parameterization of dynamic root architecture models.

The functional role of organic acid anions (e.g. citrate, oxalate, malonate, etc) in soil has been intensively investigated with special focus either on (i) microbial respiration and soil carbon dynamics, (ii) nutrient solubilization, or (iii) metal detoxification. Considering the potential impact of sorption processes on the functional significance of these effects, comparatively little is known about the adsorption and desorption dynamics of organic acid anions in soils. The aim of this study therefore was to experimentally characterize the adsorption and desorption dynamics of organic acid anions in different soils using citrate as a model carboxylate. Results showed that both adsorption and desorption processes were fast, reaching a steady state equilibrium solution concentration within approximately 1 hour. However, for a given total soil citrate concentration(ctot) the steady state value obtained was critically dependent on the starting conditions of the experiment (i.e. whether most of the citrate was initially present in solution (cl) or held on the solid phase (cs)). Specifically, desorption-led processes resulted in significantly lower equilibrium solution concentrations than adsorption led processes indicating time-dependent sorption hysteresis. As it is not possible to experimentally distinguish between different sorption pools in soil (i.e. fast, slow, irreversible adsorption/desorption), a new dynamic hysteresis model was developed that relies only on measured soil solution concentrations. The model satisfactorily explained experimental data and was able to predict dynamic adsorption and desorption behaviour. To demonstrate its use we applied the model to two relevant scenarios (exudation and microbial degradation), where the dynamic sorption behaviour of citrate occurs. Overall, this study highlights the complex nature of citrate sorption in soil and concludes that existing models need to incorporate both a temporal and sorption hysteresis component to realistically describe the role and fate of organic acids in soil processes.

Development of arbuscular mycorrhizal fungal colonization of roots and the surrounding soil is the central process of mycorrhizal symbiosis, important for ecosystem functioning and commercial inoculum applications. To improve mechanistic understanding of this highly spatially and temporarily dynamic process, we developed a three-dimensional model taking into account growth of the roots and hyphae. It is for the first time that infection within the root system is simulated dynamically and in a spatially resolved way. Comparison between data measured in a calibration experiment and simulated results showed a good fit. Our simulations showed that the position of the fungal inoculum affects the sensitivity of hyphal growth parameters. Variation in speed of secondary infection and hyphal lifetime had a different effect on root infection and hyphal length, respectively, depending on whether the inoculum was concentrated or dispersed. For other parameters (branching rate, distance between entry points), the relative effect was the same independent of inoculum placement. The model also indicated that maximum root colonization levels well below 100%, often observed experimentally, may be a result of differential spread of roots and hyphae, besides intrinsic plant control, particularly upon localized placement of inoculum and slow secondary infection.

Phosphorus is a major limiting factor in plant growth and crop production. Phosphorus solubilisation, uptake by plant roots and efflux lead to complex, dynamic cycling of P in the vicinity of plant roots. However, direct observation of P dynamics in the rhizosphere at relevant spatial scales (sub-mm) is still lacking. Chemical imaging of the dissolved P concentration around Brassica napus roots was accomplished using diffusive gradients in thin films (DGT) coupled with laser-ablation inductively coupled plasma mass spectrometry (LA-ICP-MS). Mathematical simulations served for investigating the capabilities of the chemical imaging technique. We show here, for the first time, localised P release along root axes and at root apices. Our results point at differential P uptake efficiencies of the two investigated B. napus cultivars. This study advances the current understanding of P dynamics in the rhizosphere and thus of plant P nutrition. This technique can serve to unravel the genotypic effects on rhizosphere mobilisation of P and hence assist breeding of highly P efficient crop cultivars.

Understanding the impact of root architecture on plant resource efficiency is important, in particular, in the light of upcoming shortages of mineral fertilizers and changed environmental conditions. In the 1950s, a great number of root systems of European cultivated plants were excavated and studied by L. Kutschera (1960). Her work gave enormous insight into the variety of root system architectures and helped to realize the importance of belowground processes to plant productivity. We analysed the resulting hand drawings by using mathematical modelling and found root system parameters for a newly developed parametric L-System model. In this way we were able to first reproduce the illustrations, second computationally analyse root system traits and finally access the dynamic root architecture development.

We present a new numerical approach describing nutrient uptake in three dimensions. Dynamic boundary conditions are considered at the individual root surfaces within a root system. As an example, we compare the three‐dimensional simulation results of phosphate uptake by a young maize root system to the corresponding effective solution. We show that the two solutions are similar concerning phosphate uptake and the size of the depletion zones. The presented approach makes it possible to verify simplifications that are made in the development of effective models. Furthermore, it is possible to extend existing models by including spatial heterogeneities that will increase our understanding of rhizosphere processes.

Understanding the impact of roots and rhizosphere traits on plant resource efficiency is important, in particular in the light of upcoming shortages of mineral fertilizers and climate change with increasing frequency of droughts. We developed a modular approach to root growth and architecture modelling with a special focus on soil root interactions. The dynamic three-dimensional model is based on L-Systems, rewriting systems well-known in plant architecture modelling. We implemented the model in Matlab in a way that simplifies introducing new features as required. Different kinds of tropisms were implemented as stochastic processes that determine the position of the different roots in space. A simulation study was presented for phosphate uptake by a maize root system in a pot experiment. Different sink terms were derived from the root architecture, and the effects of gravitropism and chemotropism were demonstrated. This root system model is an open and flexible tool which can easily be coupled to different kinds of soil models.

Root hairs are known to be important in the uptake of sparingly soluble nutrients by plants, but quantitative understanding of their role in this is weak. This limits, for example, the breeding of more nutrient-efficient crop genotypes. We developed a mathematical model of nutrient transport and uptake in the root hair zone of single roots growing in soil or solution culture. Accounting for root hair geometry explicitly, we derived effective equations for the cumulative effect of root hair surfaces on uptake using the method of homogenization. Analysis of the model shows that, depending on the morphological and physiological properties of the root hairs, one of three different effective models applies. They describe situations where: (1) a concentration gradient dynamically develops within the root hair zone; (2) the effect of root hair uptake is negligibly small; or (3) phosphate in the root hair zone is taken up instantaneously. Furthermore, we show that the influence of root hairs on rates of phosphate uptake is one order of magnitude greater in soil than solution culture. The model provides a basis for quantifying the importance of root hair morphological and physiological properties in overall uptake, in order to design and interpret experiments in different circumstances.

In this paper, we set out to illustrate and discuss how mathematical modelling could and should be applied to aid our understanding of plants and, in particular, plant–soil interactions. Our aim is to persuade members of both the biological and mathematical communities of the need to collaborate in developing quantitative mechanistic models. We believe that such models will lead to a more profound understanding of the fundamental science of plants and may help us with managing real-world problems such as food shortages and global warming. We start the paper by reviewing mathematical models that have been developed to describe nutrient and water uptake by a single root. We discuss briefly the mathematical techniques involved in analysing these models and present some of the analytical results of these models. Then, we describe how the information gained from the single-root scale models can be translated to root system and field scales. We discuss the advantages and disadvantages of different mathematical approaches and make a case that mechanistic rather than phenomenological models will in the end be more trustworthy. We also discuss the need for a considerable amount of effort on the fundamental mathematics of upscaling and homogenization methods specialized for branched networks such as roots. Finally, we discuss different future avenues of research and how we believe these should be approached so that in the long term it will be possible to develop a valid, quantitative whole-plant model.

In order to quantify the contribution of arbuscular mycorrhizal (AM) fungi to plant phosphorus nutrition, the development and extent of the external fungal mycelium and its nutrient uptake capacity are of particular importance. We develop and analyse a model of the growth of AM fungi associated with plant roots, suitable for describing mechanistically the effects of the fungi on solute uptake by plants. The model describes the development and distribution of the fungal mycelium in soil in terms of the creation and death of hyphae, tip–tip and tip–hypha anastomosis, and the nature of the root–fungus interface. It is calibrated and corroborated using published experimental data for hyphal length densities at different distances away from root surfaces. A good agreement between measured and simulated values was found for three fungal species with different morphologies:Scutellospora calospora(Nicol. & Gerd.) Walker & Sanders;Glomussp.; andAcaulospora laevisGerdemann & Trappe associated withTrifolium subterraneumL. The model and findings are expected to contribute to the quantification of the role of AM fungi in plant mineral nutrition and the interpretation of different foraging strategies among fungal species.