Andrea Schnepf

Modelling root–soil interactions using three–dimensional models of root growth, architecture and function

BackgroundThree–dimensional root architectural models emerged in the late 1980s, providing an opportunity to conceptualise and investigate that all important part of plants that is typically hidden and difficult to measure and study. These models have progressed from representing pre–defined root architectural arrangements, to simulating root growth in response to heterogeneous soil environments. This was done through incorporating soil properties and more complete descriptions of plant function, moving into the realm of functional-structural plant modelling. Modelling studies are often designed to investigate the relationship between root architectural traits and root distribution in soil, and the spatio–temporal variability of resource supply. Modelling root systems presents an opportunity to investigate functional tradeoffs between foraging strategies (i.e. shallow vs deep rooting) for contrasting resources (immobile versus mobile resources), and their dependence on soil type, rainfall and other environmental conditions. The complexity of the interactions between root traits and environment emphasises the need for models in which traits and environmental conditions can be independently manipulated, unlike in the real world.ScopeWe provide an overview of the development of three–dimensional root architectural models from their origins, to their place today in the world of functional–structural plant modelling. The uses and capability of root architectural models to represent virtual plants and soil environment are addressed. We compare features of six current models, RootTyp, SimRoot, ROOTMAP, SPACSYS, R-SWMS, and RootBox, and discuss the future development of functional-structural root architectural modelling.ConclusionFunctional-structural root architectural models are being used to investigate numerous root–soil interactions, over a range of spatial scales. They are not only providing insights into the relationships between architecture, morphology and functional efficiency, but are also developing into tools that aid in the design of agricultural management schemes and in the selection of root traits for improving plant performance in specific environments.

A dynamic root system growth model based on L-Systems Tropisms and coupling to nutrient uptake from soil

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.

Impact of growth and uptake patterns of arbuscular mycorrhizal fungi on plant phosphorus uptake—a modelling study

In this paper we present a mathematical model for estimating external mycelium growth of arbuscular mycorrhizal fungi and its effect on root uptake of phosphate (P). The model describes P transport in soil and P uptake by both root and fungi on the single root scale. We investigate differences in soil P depletion and overall P influx into a mycorrhizal root by assuming that different spatial regions of mycelia are active in P uptake. When all external hyphae contribute to P uptake, overall uptake is dominated by the fungus and the most effective growth pattern appears to be the one using a high level of anastomosis. The same is true when only the proportion of external hyphae assumed to be active contributes to uptake. When uptake is restricted to the tips, hyphal contribution to overall P uptake is less dominant; the most effective growth pattern appears to be the one characterised by nonlinear branching where branching stops at a given maximal hyphal tip density. Comparison to measured P depletion in the literature suggests that the scenario where active hyphae are contributing to P uptake is likely to fit the data best. These quantitative predictions promote our understanding of the mycorrhizal symbiosis and its role in plant P nutrition.

A dynamic model of nutrient uptake by root hairs.

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.

Mathematical models of plant–soil interaction

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.

Recovering Root System Traits Using Image Analysis Exemplified by Two-Dimensional Neutron Radiography Images of Lupine

Image-based parameterization of root architectural models is advanced by a new approach for the analysis of image sequences of plant root systems. 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.

Verification and intercomparison of reactive transport codes to describe root-uptake

Several mathematical models have been developed to simulate processes and inter- actions in the plant rhizosphere. Most of these models are based on a rather simplified descrip- tion of the soil chemistry and interactions of plant roots in the rhizosphere. In particular the feed- back loops between exudation, water and solute uptake are mostly not considered, although their importance in the bioavailability of mineral ele- ments for plants has been demonstrated. The aim of this work was to evaluate three existing coupled speciation-transport tools to model rhi- zosphere processes. In the field of hydrogeo- chemistry, such computational tools have been developed to describe acid-base and redox reac- tions, complexation and ion exchange, adsorption and precipitation of chemical species in soils and aquifers using thermodynamic and kinetic rela- tionships. We implemented and tested a simple rhizosphere model with three geochemical com- putational tools (ORCHESTRA, MIN3P, and PHREEQC). The first step was an accuracy

Root System Markup Language: toward a unified root architecture description language

Portability of root architecture data with the Root System Markup Language paves the way for central root phenotype repositories. The number of image analysis tools supporting the extraction of architectural features of root systems has increased in recent years. These tools offer a handy set of complementary facilities, yet it is widely accepted that none of these software tools is able to extract in an efficient way the growing array of static and dynamic features for different types of images and species. We describe the Root System Markup Language (RSML), which has been designed to overcome two major challenges: (1) to enable portability of root architecture data between different software tools in an easy and interoperable manner, allowing seamless collaborative work; and (2) to provide a standard format upon which to base central repositories that will soon arise following the expanding worldwide root phenotyping effort. RSML follows the XML standard to store two- or three-dimensional image metadata, plant and root properties and geometries, continuous functions along individual root paths, and a suite of annotations at the image, plant, or root scale at one or several time points. Plant ontologies are used to describe botanical entities that are relevant at the scale of root system architecture. An XML schema describes the features and constraints of RSML, and open-source packages have been developed in several languages (R, Excel, Java, Python, and C#) to enable researchers to integrate RSML files into popular research workflow.

Growth model for arbuscular mycorrhizal fungi

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; Glomus sp.; and Acaulospora laevis Gerdemann & Trappe associated with Trifolium subterraneum L. 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.

Modelling Nutrient Uptake by Individual Hyphae of Arbuscular Mycorrhizal Fungi: Temporal and Spatial Scales for an Experimental Design

Arbuscular mycorrhizas, associations between plant roots and soil fungi, are ubiquitous among land plants. Arbuscular mycorrhizas can be beneficial for plants by overcoming limitations in nutrient supply. Hyphae, which are long and thin fungal filaments extending from the root surface into the soil, increase the volume of soil accessible for plant nutrient uptake. However, no models so far specifically consider individual hyphae. We developed a mathematical model for nutrient uptake by individual fungal hyphae in order to assess suitable temporal and spatial scales for a new experimental design where fungal uptake parameters are measured on the single hyphal scale. The model was developed based on the conservation of nutrients in an artificial cylindrical soil pore (capillary tube) with adsorbing wall, and analysed based on parameter estimation and non-dimensionalisation. An approximate analytical solution was derived using matched asymptotic expansion. Results show that nutrient influx into a hypha from a small capillary tube is characterized by three phases: Firstly, uptake rapidly decreases as the hypha takes up nutrients, secondly, the depletion zone reaches the capillary wall and thus uptake is sustained by desorption of nutrients from the capillary wall, and finally, uptake goes to zero after nutrients held on the capillary wall have been completely depleted. Simulating different parameter regimes resulted in recommending the use of capillaries filled with hydrogel instead of water in order to design an experiment operating over measurable time scales.