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

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Abstract

Background

Three–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.

Scope

We 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.

Conclusion

Functional-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.

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References

  • Akay G, Burke D (2012) Agro-process intensification through synthetic rhizosphere media for nitrogen fixation and yield enhancement in plants. Am J Agric Biol Sci 7:150–172

    Google Scholar 

  • Ao J, Fu J, Tian J, Yan X, Liao H (2010) Genetic variability for root morph-architecture traits and root growth dynamics as related to phosphorus efficiency in soybean. Funct Plant Biol 37:304–312

    Google Scholar 

  • Baldwin JP, Nye PH, Tinker PB (1973) Uptake of solutes by multiple root systems from soil. III a model for calculating the solute uptake by a randomly dispersed root system developing in a finite volume of soil. Plant Soil 38:621–635

    CAS  Google Scholar 

  • Bates TR, Lynch JP (2000) The efficiency of Arabidopsis thaliana (Brassicaceae) root hairs in phosphorus acquisition. Am J Bot 87:964–970

    CAS  PubMed  Google Scholar 

  • Bear J, Cheng A (2010) Modeling groundwater flow and contaminant transport. Springer

  • Bechtold M, Vanderborght J, Ippisch O, Vereecken H (2011) Efficient random walk particle tracking algorithm for advective-dispersive transport in media with discontinuous dispersion coefficients and water contents. Water Res 47:W10526

    Google Scholar 

  • Bengough AG (2012) Root elongation is restricted by axial but not radial pressures: so what happens in field soil? Plant Soil 360:15–18

    CAS  Google Scholar 

  • Bengough AG, Mackenzie CJ, Diggle AJ (1992) Relations between root length densities and root intersections with horizontal and vertical planes using root growth modelling in 3-dimensions. Plant Soil 145:245–252

    Google Scholar 

  • Bengough AG, McKenzie BM, Hallett PD, Valentine TA (2011) Root elongation, water stress, and mechanical impedance: a review of limiting stresses and beneficial root tip traits. J Exp Bot 62:59–68

    CAS  PubMed  Google Scholar 

  • Bernier J, Serraj R, Kumar A, Venuprasad R, Impa S, Gowda RPV, Oane R, Spaner D, Atlin G (2009) The large–effect drought–resistance QTL qtl12.1 increases water uptake in upland rice. Field Crop Res 110:139–146

    Google Scholar 

  • Bidel LPR, Pagès L, Riviere LM, Pelloux G, Lorendeau JY (2000) MassFlowDyn I: a carbon transport and partitioning model for root system architecture. Ann Bot 85:869–886

    CAS  Google Scholar 

  • Bingham IJ, Wu L (2011) Simulation of wheat growth using the 3D root architecture model SPACSYS: validation and sensitivity analysis. Eur J Agron 34:181–189

    Google Scholar 

  • Bingham IJ, Rees RM, Bengough AG (2009) Influence of soil compaction on the dynamics of root growth and mortality in spring barley. International symposium “root research and applications” RootRAP, 2–4 Sept 2009. Boku, Vienna, pp 148–4

    Google Scholar 

  • Bohn M, Novais J, Fonseca R, Tuberosa R, Grift TE (2006) Genetic evaluation of root complexity in maize. Acta Agron Hung 54:291–303

    CAS  Google Scholar 

  • Brady DJ, Wenzel CL, Fillery IRP, Gregory PJ (1995) Root growth and nitrate uptake by wheat (Triticum aestivum L.) following wetting of dry surface soil. J Exp Bot 46:557–564

    CAS  Google Scholar 

  • Brück HH, Becker HC, Sattelmacher B (1992) Phosphate efficiencies of two maize inbred lines. In: Kutschera L, Hüble E, Lichtenegger E, Persson H, Sobotik M (eds) Root ecology and its practical applications. 3rd ISRR Symposium, Vienna, Austria, pp 193–196

  • Burton A, Williams M, Lynch J, Brown K (2012) RootScan: software for high-throughput analysis of root anatomical traits. Plant Soil 357:189–203

    CAS  Google Scholar 

  • Burton AL, Brown KM, Lynch JP (2013) Phenotypic diversity of root anatomical and architectural traits in Zea species. Crop Sci. doi:10.2135/cropsci2012.07.0440

    Google Scholar 

  • Campbell BD, Grime JP, Mackey JML (1991) A trade–off between scale and precision in resource foraging. Oecol 87:532–538

    Google Scholar 

  • Chandra K, Kumar N, Chand G (2010) Studies on mycorrhizal inoculation on dry matter yield and root colonization of some medicinal plants grown in stress and forest soils. J Environ Biol 31:975–979

    CAS  PubMed  Google Scholar 

  • Chen W, Dunbabin V, Bell R, Brennan R, Bowden B (2008) Simulating and understanding root growth using ROOTMAP to guide phosphorus fertiliser placement in wide row lupin cropping systems. In: Lupins for health and wealth - Proceedings of the 12th International Lupin Conference, 14–18 Sept, 2008, Fremantle, WA. International Lupin Association, Canterbury, pp 368–372

    Google Scholar 

  • Chen YL, Dunbabin VM, Postma JA, Diggle AJ, Palta JA, Lynch JP, Siddique KHM, Rengel Z (2011) Phenotypic variability and modelling of root structure of wild Lupinus angustifolius genotypes. Plant Soil 348:345–364

    CAS  Google Scholar 

  • Clark RT, MacCurdy RB, Jung JK, Shaff JE, McCouch SR, Aneshansley DJ, Kochian LV (2011) Three-dimensional root phenotyping with a novel imaging and software platform. Plant Physiol 156:455–465

    CAS  PubMed  Google Scholar 

  • Clausnitzer V, Hopmans JW (1993) An algorithm for three–dimensional simultaneous modelling of root growth and transient water flow. Land, air and water resources paper no. 100022. Department of Land, Air and Water Resources, University of California, Davis, pp 1–108

    Google Scholar 

  • Clausnitzer V, Hopmans JW (1994) Simultaneous modeling of transient three–dimensional root growth and soil water flow. Plant Soil 164:299–314

    CAS  Google Scholar 

  • Collet C, Löf M, Pagès L (2006) Root system development of oak seedlings analysed using an architectural model. Effects of competition with grass. Plant Soil 279:367–383

    CAS  Google Scholar 

  • Couvreur V, Vanderborght J, Javaux M (2012) A simple three–dimensional macroscopic root water uptake model based on the hydraulic architecture approach. Hydrol Earth Syst Sci 16:2957–2971

    Google Scholar 

  • de Dorlodot S, Forster B, Pages L, Price A, Tuberosa R, Draye X (2007) Root system architecture: opportunities and constraints for genetic improvement of crops. Trends Plant Sci 12:474–481

    Google Scholar 

  • de Willigen P, van Dam JC, Javaux M, Heinen M (2012) Root water uptake as simulated by three soil water flow models. Vadose Zone J 11(3). doi:10.2136/vzj2012.0018

  • Deans JD, Ford ED (1983) Modelling root structure and stability. Plant Soil 71:189–195

    Google Scholar 

  • Diggle AJ (1988a) ROOTMAP – a model in three-dimensional coordinates of the growth and structure of fibrous root systems. Plant Soil 105:169–178

    Google Scholar 

  • Diggle AJ (1988b) ROOTMAP: a root growth model. Math Comput Simul 30:175–180

    Google Scholar 

  • Diggle AJ (1988c) ROOTMAP2.1 – a root growth simulation program. Department of Agriculture Western Australia, Western Australia

    Google Scholar 

  • Diggle AJ (1990) Interaction between mineral nitrogen and growth of wheat roots in a leaching environment. PhD, The University of Western Australia, Perth

    Google Scholar 

  • Diggle AJ (1996) Developments in architectural models of root systems and the potential for their application to intercropping. In: Ito O, Johansen C, Adu–Gyamfi JJ, Katayama K, Kumar Rao JVDK, Rego TJ (eds) Dynamics of roots and nitrogen in cropping systems of the semi–arid tropics. Japan International Research Center for Agricultural Sciences, Japan, pp 559–571

    Google Scholar 

  • Dinkelaker B, Hengeler C, Marschner H (1995) Distribution and function of proteoid roots and other root clusters. Bot Acta 108:183–200

    Google Scholar 

  • Douds D, Nagahashi G, Reider C, Hepperly P (2008) Choosing a mixture ratio for the on-farm production of AM fungus inoculum in mixtures of compost and vermiculite. Compost Sci Utiliz 16:52–60

    Google Scholar 

  • Doussan C, Pagès L, Vercambre G (1998) Modelling of the hydraulic architecture of root systems: an integrated approach to water absorption – model description. Ann Bot 81:213–223

    Google Scholar 

  • Doussan C, Pagès L, Pierret A (2003) Soil exploration and resource acquisition by plant roots: an architectural and modelling point of view. Agronomie 23:419–431

    Google Scholar 

  • Doussan C, Pierret A, Garrigues E, Pagès L (2006) Water uptake by plant roots: II – modelling of water transfer in the soil root–system with explicit account of flow within the root system – comparison with experiments. Plant Soil 283:99–117

    CAS  Google Scholar 

  • Draye X, Pagès L (2007) CrossTalk: a simulation platform for the linking of existing soil, plant and atmosphere models. In: Proceeding of international symposium on plant growth modeling and applications. IEEE Computer Society, Los Alamitos, pp 93–100

    Google Scholar 

  • Draye X, Kim Y, Lobet G, Javaux M (2010) Model–assisted integration of physiological and environmental constraints affecting the dynamic and spatial patterns of root water uptake from soils. J Exp Bot 61:2145–2155

    CAS  PubMed  Google Scholar 

  • Drew MC, Saker LR (1975) Nutrient supply and the growth of the seminal root system in barley. II. Localised compensatory increases in lateral root growth and rates of nitrate uptake when nitrate supply is restricted to only one part of the root system. J Exp Bot 26:79–90

    CAS  Google Scholar 

  • Drew MC, Saker LR, Ashley TW (1973) Nutrient supply and the growth of the seminal root system in barley. I. The effect of nitrate concentration on the growth of axes and laterals. J Exp Bot 24:1189–1202

    CAS  Google Scholar 

  • Dunbabin V (2007) Simulating the role of rooting traits in crop–weed competition. Field Crop Res 104:44–51

    Google Scholar 

  • Dunbabin V, Rengel Z, Diggle A (2001a) The root growth response to heterogeneous nitrate supply differs for Lupinus angustifolius and Lupinus pilosus. Australian Journal of Agricultural Research 52:495–503

    Google Scholar 

  • Dunbabin V, Rengel Z, Diggle A (2001b) Lupinus angustifolius has a plastic uptake response to heterogeneously supplied nitrate while Lupinus pilosus does not. Australian Journal of Agricultural Research 52:505–512

    Google Scholar 

  • Dunbabin V, Diggle A, Rengel Z, van Hugten R (2002a) Modelling the interactions between water and nutrient uptake and root growth. Plant Soil 239:19–38

    CAS  Google Scholar 

  • Dunbabin V, Diggle A, Rengel Z (2002b) Simulation of field data by a three–dimensional model of interactive root growth. Plant Soil 239:39–54

    CAS  Google Scholar 

  • Dunbabin V, Diggle A, Rengel Z (2003a) Is there an optimal root architecture for nitrate capture in leaching environments? Plant Cell Environ 26:835–844

    PubMed  Google Scholar 

  • Dunbabin V, Rengel Z, Diggle A (2003b) Root architecture and nutrient capture – the complex riddle of what constitutes optimality of root form and function. In: Lynch JM, Schepers JS, Unver I (eds) Innovative soil-plant systems for sustainable agricultural practices. Organisation for Economic Co–operation and Development (OECD), Paris, pp 2–16

    Google Scholar 

  • Dunbabin V, Rengel Z, Diggle A (2004) Simulating form and function of root systems: efficiency of nitrate uptake is dependent on root system architecture and the spatial and temporal variability of nitrate supply. Funct Ecol 18:204–211

    Google Scholar 

  • Dunbabin VM, McDermott S, Bengough AG (2006) Upscaling from rhizosphere to whole root system: modelling the effects of phospholipid surfactants on water and nutrient uptake. Plant Soil 283:57–72

    CAS  Google Scholar 

  • Dunbabin VM, Armstrong RD, Officer SJ, Norton RM (2009) Identifying fertiliser management strategies to maximize nitrogen and phosphorus acquisition by wheat in two contrasting soils from Victoria, Australia. Aust J Soil Res 47:74–90

    CAS  Google Scholar 

  • Dunbabin VM, Airey M, Diggle AJ, Renton M, Rengel Z, Armstrong R, Chen Y, Siddique KHM (2011) Simulating the interaction between plant root, soil water and nutrient flows, and barriers and objects in soil using ROOTMAP. In: Anderssen RS, Chan F, Marinova D (eds) 19th International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand, Dec 2011, pp 975–981. http://mssanz.org.au/modsim2011

  • Dupuy L, Gregory PJ, Bengough AG (2010) Root growth models: towards a new generation of continuous approaches. J Exp Bot 61(8):2131–2143

    CAS  PubMed  Google Scholar 

  • Eissenstat DM, Yanai RD (1997) The ecology of root lifespan. Adv Ecol Res 27:1–60

    Google Scholar 

  • Feddermann N, Finlay R, Boller T, Elfstrand M (2010) Functional diversity in arbuscular mycorrhiza– the role of gene expression, phosphorous nutrition and symbiotic efficiency. Fungal Ecol 3:1–8

    Google Scholar 

  • Fitter AH (1991) The ecological significance of root system architecture: an economic approach. In: Plant root growth – an ecological perspective. Blackwell Scientific Publications, Oxford, pp 229–243

    Google Scholar 

  • Fitter AH (2002) Characteristics and functions of root systems. In: Waisel Y, Eshel A, Kafkafi U (eds) Plant roots, the hidden half. Marcel Dekker Inc, New York, pp 15–32

    Google Scholar 

  • Fitter AH, Stickland TR, Harvey ML, Wilson GW (1991) Architectural analysis of plant root systems. 1. Architectural correlates of exploitation efficiency. New Phytol 118:375–382

    Google Scholar 

  • Flavel RJ, Guppy CN, Tighe M, Watt M, McNeill A, Young IM (2012) Non–destructive quantification of cereal roots in soil using high-resolution X-ray tomography. J Exp Bot 63:2503–2511

    CAS  PubMed  Google Scholar 

  • Forde BG (2009) Is it good noise? The role of developmental instability in the shaping of a root system. J Exp Bot 60:3989–4002

    CAS  PubMed  Google Scholar 

  • Gahoonia TS, Nielsen NE (2004) Root traits as tools for creating phosphorus efficient crop varieties. Plant Soil 260:47–57

    Google Scholar 

  • Garré S, Pagès L, Laloy E, Javaux M, Vanderborght J, Vereecken H (2012) Parameterizing the root system development of spring barley using minirhizotron data. Vadose Zone J 11(4). doi: 10.2136/vzj2011.0179

  • Ge ZY, Rubio G, Lynch JP (2000) The importance of root gravitropism for inter-root competition and phosphorus acquisition efficiency: results from a geometric simulation model. Plant Soil 218:159–171

    CAS  PubMed  Google Scholar 

  • Gilroy S, Jones DL (2000) Through form to function: root hair development and nutrient uptake. Trends Plant Sci 5:56–60

    CAS  PubMed  Google Scholar 

  • Godin C, Sinoquet H (2005) Functional–structural plant modelling. New Phytol 166:705–708

    PubMed  Google Scholar 

  • Gregory PJ, Bengough AG, Grinev D, Schmidt S, Thomas WTB, Wojciechowski T, Young I (2009) Root phenomics of crops: opportunities and challenges. Funct Plant Biol 36:922–929

    Google Scholar 

  • Grime JP, Campbell BD, Mackey JML, Crick JC (1991) Root plasticity, nitrogen capture and competitive ability. In: Atkinson D (ed) Plant root growth – an ecological perspective. Blackwell Scientific Publications, Oxford, pp 381–397

    Google Scholar 

  • Hammer GL, Woodruff DR, Robinson JB (1987) Effects of climatic variability and possible climatic change on reliability of wheat cropping – a modelling approach. Agr For Meteor 41:123–142

    Google Scholar 

  • Han L, Gresshoff PM, Hanan J (2011) A functional-structural modelling approach to auto regulation of nodulation. Ann Bot 107:855–863

    CAS  PubMed  Google Scholar 

  • Hinsinger P, Brauman A, Devau N, Gérard F, Jourdan C, Laclau J, Le Cadre E, Jaillard B, Plassard C (2011) Acquisition of phosphorus and other poorly mobile nutrients by roots. Where do plant nutrition models fail? Plant Soil 348:29–61

    CAS  Google Scholar 

  • Ho MD, Rosas JC, Brown KM, Lynch JP (2005) Root architectural tradeoffs for water and phosphorus acquisition. Funct Plant Biol 32:737–748

    CAS  Google Scholar 

  • Hutchinson JMC (2000) Three into two doesn’t go: two-dimensional models of bird eggs, snail shells and plant roots. Biol J Linn Soc 70:161–187

    Google Scholar 

  • Jahnke S, Menzel MI, Van Dusschoten D, Roeb GW, Bühler J, Minwuyelet S, Blümler P, Temperton VM, Hombach T, Streun M, Beer S, Khodaverdi M, Ziemons K, Coenen HH, Schurr U (2009) Combined MRI–PET dissects dynamic changes in plant structures and functions. Plant J 59:634–644

    CAS  PubMed  Google Scholar 

  • Janott M, Gayler S, Gessler A, Javaux M, Klier C, Priesack E (2011) A one-dimensional model of water flow in soil-plant systems based on plant architecture. Plant Soil 341:233–256

    CAS  Google Scholar 

  • Jansa J, Mozafar A, Frossard E (2005) Phosphorus acquisition strategies within arbuscular mycorrhizal fungal community of a single field site. Plant Soil 276:163–176

    CAS  Google Scholar 

  • Javaux M, Schröder T, Vanderborght J, Vereecken H (2008) Use of a three–dimensional detailed modeling approach for predicting root water uptake. Vadose Zone J 7:1079–1088

    Google Scholar 

  • Jones D, Nguyen C, Finlay R (2009) Carbon flow in the rhizosphere: carbon trading at the soil-root interface. Plant Soil 321:5–33

    CAS  Google Scholar 

  • Kalliokoski T, Sievanen R, Nygren P (2010) Tree roots s self-similar branching structures: axis differentiation and segment tapering in coarse roots of three boreal forest tree species. Trees 24:219–236

    Google Scholar 

  • Kurth W, Lanwert D (2011) Grammar-based models and fractals. In: Jopp F, Reuter H, Breckling B (eds) Modelling complex ecological dynamics. Springer, Berlin Heidelberg, pp 147–161

    Google Scholar 

  • Kutschera L (1960) Wurzelatlas mitteleuropäischer Ackerunkräuter und Kulturpflanzen. DLG Verlag, Frankfurt am main

    Google Scholar 

  • Lambers H, Shane MW, Cramer MD, Pearse SJ, Veneklaas EJ (2006) Root structure and functioning for efficient acquisition of phosphorus: matching morphological and physiological traits. Ann Bot 98:693–713

    PubMed  Google Scholar 

  • Lambers H, Raven JA, Shaver GR, Smith SE (2008) Plant nutrient–acquisition strategies change with soil age. Trends Ecol Evol 23:95–103

    PubMed  Google Scholar 

  • Lambers H, Finnegan P, Laliberté E, Pearse S, Ryan M, Shane M, Veneklaas E (2011) Phosphorus nutrition of Proteaceae in severely phosphorus–impoverished soils: are there lessons to be learned for future crops? Plant Physiol 156:1058–1066

    CAS  PubMed  Google Scholar 

  • Le Roux Y, Pagès L (1994) Développement et polymorphisme racinaire chez de jeunes semis d’hévéa (Hevea brasiliensis). Can J Bot 72:924–932

    Google Scholar 

  • Leitner D, Schnepf A (2012) Image analysis of 2-dimensional root system architecture. ALGORITMY 2012, 19th Conference on Scientific Computing, Vysoké Tatry – Podbanské, Slovakia, September 9 –14, 2012. http://www.iam.fmph.uniba.sk/algoritmy2012/

  • Leitner D, Klepsch S, Knieß A, Schnepf A (2010a) The algorithmic beauty of plant roots - an L-system model for dynamic root growth simulation. Math Comput Model Dyn 16:575–587

    Google Scholar 

  • Leitner D, Klepsch S, Bodner G, Schnepf A (2010b) A dynamic root system growth model based on L-systems. Tropisms and coupling to nutrient uptake from soil. Plant Soil 332:177–192

    CAS  Google Scholar 

  • Leitner D, Schnepf A, Klepsch S, Roose T (2010c) Water uptake by a maize root system – an explicit numerical 3-dimensional simulation. In: Geophysical Research Abstracts, Vol. 12, EGU2010-8473, 2010, EGU General Assembly 2010, 2–7 March, Vienna, Austria

  • Lendenmann M, Thonar C, Barnard R, Salmon Y, Werner R, Frossard E, Jansa J (2011) Symbiont identity matters: carbon and phosphorus fluxes between medicago truncatula and different arbuscular mycorrhizal fungi. Mycorrhiza 21:689–702

    CAS  PubMed  Google Scholar 

  • Liao H, Rubio G, Yan XL, Cao AQ, Brown KM, Lynch JP (2001) Effect of phosphorus availability on basal root shallowness in common bean. Plant Soil 232:69–79

    CAS  PubMed  Google Scholar 

  • Liao M, Palta JA, Fillery IRP (2006) Root characteristics of vigorous wheat improve early nitrogen uptake. Aust J Agric Res 57:1097–1107

    Google Scholar 

  • Lungley DR (1973) The growth of root systems – a numerical computer simulation model. Plant Soil 38:145–159

    Google Scholar 

  • Lynch JP (2005) Root architecture and nutrient acquisition. In: BassiriRad H (ed) Nutrient acquisition by plants -an ecological perspective. Springer–Verlag, Berlin, pp 147–183

    Google Scholar 

  • Lynch J (2007) Roots of the second green revolution. Aust J Bot 55:493–512

    Google Scholar 

  • Lynch JP (2011) Root phenes for enhanced soil exploration and phosphorus acquisition: tools for future crops. Plant Physiol 156:1041–1049

    CAS  PubMed  Google Scholar 

  • Lynch JP, Beebe SE (1995) Adaptation of beans (Phaseolus vulgaris L.) to low phosphorus availability. Hortscience 30:1165–1171

    CAS  Google Scholar 

  • Lynch JP, Brown KM (2001) Topsoil foraging – an architectural adaptation of plants to low phosphorus availability. Plant Soil 237:225–237

    CAS  Google Scholar 

  • Lynch JP, Brown KM (2008) Root strategies for phosphorus acquisition. In: White PJ, Hammond JP (eds) The ecophysiology of plant–phosphorus interaction. Springer, the Netherlands, pp 83–116

    Google Scholar 

  • Lynch JP, Brown KM (2012) New roots for agriculture: exploiting the root phenome. Phil Trans R Soc B 367:1598–1604

    PubMed  Google Scholar 

  • Lynch J, Ho M (2005) Rhizoeconomics: carbon costs of phosphorus acquisition. Plant Soil 269:45–56

    CAS  Google Scholar 

  • Lynch J, van Beem J (1993) Growth and architecture of seedling roots of common bean genotypes. Crop Sci 33:1253–1257

    Google Scholar 

  • Lynch JP, Nielsen KL, Davis RD, Jablokow AG (1997) SimRoot: modelling and visualization of root systems. Plant Soil 188:139–151

    CAS  Google Scholar 

  • Ma Z, Walk TC, Marcus A, Lynch JP (2001) Morphological synergism in root hair length, density, initiation and geometry for phosphorus acquisition in Arabidopsis thaliana: a modelling approach. Plant Soil 236:221–235

    CAS  Google Scholar 

  • Marschner P, Crowley D, Rengel Z (2011) Rhizosphere interactions between microorganisms and plants govern iron and phosphorus acquisition along the root axis – model and research methods. Soil Biol Biochem 43:883–894

    CAS  Google Scholar 

  • Metselaar K, Van Lier QJ (2007) The shape of the transpiration reduction function under plant water stress. Vadose Zone J 6:124–139

    Google Scholar 

  • Meyer KM, Mooij WM, Vos M, Hol WHG, van der Putten WH (2009) The power of simulating experiments. Ecol Model 220:2594–2597

    Google Scholar 

  • Miguel MA, Postma JA, Lynch JP (2012) Functional role and synergystic effect of root traits for phosphorus acquisition efficiency and their genetic basis in common bean (Phaseolus vulgaris L.). PhD Thesis, The Pennsylvania State University, USA

  • Miller CR, Ochoa I, Nielsen KL, Beck D, Lynch JP (2003) Genetic variation for adventitious rooting in response to low phosphorus availability: potential utility for phosphorus acquisition from stratified soils. Funct Plant Biol 30:973–985

    CAS  Google Scholar 

  • Morris MD (1991) Factorial sampling plans for preliminary computational experiments. Techinometrics 33:161–174

    Google Scholar 

  • Mulia R, Dupraz C, van Noordwijk M (2010) Reconciling root plasticity and architectural ground rules in tree root growth models with voxel automata. Plant Soil 337:77–92

    CAS  Google Scholar 

  • Nielsen KL, Lynch JP, Jablokow AG, Curtis PS (1994) Carbon cost of root systems: an architectural approach. Plant Soil 165:161–169

    CAS  Google Scholar 

  • Nielsen KL, Miller CR, Beck D, Lynch JP (1998) Fractal geometry of root systems: field observations of contrasting genotypes of common bean (Phaseolus vulgaris L.) grown under different phosphorus regimes. Plant Soil 206:181–190

    CAS  Google Scholar 

  • Nye PH, Tinker PB (1977) Solute movements in the root-soil system. Blackwell, Oxford

    Google Scholar 

  • Oswald S, Menon M, Carminati A, Vontobel P, Lehmann E, Schulin R (2008) Quantitative imaging of infiltration, root growth, and root water uptake via neutron radiography. Vadose Zone J 7:1035–1047

    Google Scholar 

  • Ozdemir G, Akpinar C, Sabir A, Bilir H, Tangolar S, Ortas I (2010) Effect of inoculation with mycorrhizal fungi on growth and nutrient uptake of grapevine genotypes (vitis spp.). Eur J Hortic Sci 75:103–110

    Google Scholar 

  • Page ER, Gerwitz A (1974) Mathematical models, based on diffusion equations, to describe root systems of isolated plants, row crops, and swards. Plant Soil 41:243–254

    Google Scholar 

  • Pagès L (1999) Root system architecture: from its representation to the study of its elaboration. Agronomie 19:295–304

    Google Scholar 

  • Pagès L (2000) How to include organ interactions in models of the root system architecture? The concept of endogenous environment. Ann For Sci 57:535–541

    Google Scholar 

  • Pagès L (2011) Links between root development traits and foraging performance. Plant Cell Environ 42:1749–1760

    Google Scholar 

  • Pagès L, Aries F (1988) SARAH : modèle de simulation de la croissance, du développement, et de l’architecture des systèmes racinaires. Agronomie 8:889–896

    Google Scholar 

  • Pagès L, Bengough AG (1997) Modelling minirhizotron observations to test experimental procedures. Plant Soil 189:81–89

    Google Scholar 

  • Pagès L, Jordan MO, Picard D (1989) A simulation model of the three–dimensional architecture of the maize root system. Plant Soil 119:147–154

    Google Scholar 

  • Pagès L, Vercambre G, Drouet JL, Lecompte F, Collet C, Le Bot J (2004) Root Typ: a generic model to depict and analyse the root system architecture. Plant Soil 258:103–119

    Google Scholar 

  • Pierret A, Doussan C, Capowiez Y, Bastardie F, Pages L (2007) Root functional architecture: a framework for modeling the interplay between roots and soil. Vadose Zone J 6:269–281

    Google Scholar 

  • Pohlmeier A, Vanderborght J, Haber–Pohlmeier S, Wienke S, Vereecken H, Javaux M (2010) Root water uptake and tracer transport in a lupin root system: integration of magnetic resonance images and the numerical model RSWMS. In: EGU General Assembly Conference Abstracts, pp 7217

  • Poorter H, Bühler J, van Dusschoten D, Climent J, Postma JA (2012a) Pot size matters: a meta–analysis of the effects of rooting volume on plant growth. Funct Plant Biol. doi:10.1071/FP12049

    Google Scholar 

  • Poorter H, Niklas KJ, Reich PB, Oleksyn J, Poot P, Mommer L (2012b) Biomass allocation to leaves, stems and roots: meta–analyses of interspecific variation and environmental control. New Phytol 193:30–50

    CAS  PubMed  Google Scholar 

  • Postma JA, Lynch JP (2011a) Theoretical evidence for the functional benefit of root cortical aerenchyma in soils with low phosphorus availability. Ann Bot 107:829–841

    CAS  PubMed  Google Scholar 

  • Postma JA, Lynch JP (2011b) Root cortical aerenchyma enhances the growth of maize on soils with suboptimal availability of nitrogen, phosphorus, and potassium. Plant Physiol 156:1190–1201

    CAS  PubMed  Google Scholar 

  • Postma JA, Lynch JP (2012) Complementarity in root architecture for nutrient uptake in ancient maize/bean and maize/bean/squash polycultures. Ann Bot 110:521–534

    CAS  PubMed  Google Scholar 

  • Postma JA, Jaramillo RE, Lynch JP (2008) Towards modeling the function of root traits for enhancing water acquisition by crops. In: Ahuja LR, Reddy VR, Saseendran SA, Yu Q (eds) Advances in agricultural systems modelling -response of crops to limited water: understanding and modelling water stress effects on plant growth processes. ASA-CSSA-SSSA, Madison, pp 251–276

    Google Scholar 

  • Pradal C, Dufour-Kowalski S, Boudon F, Fournier C, Godin C (2008) OpenAlea: a visual programming and component–based software platform for plant modelling. Funct Plant Biol 35:751–760

    Google Scholar 

  • Pradal C, Boudon F, Nouguier C, Chopard J, Godin C (2009) PlantGL: a python–based geometric library for 3D plant modelling at different scales. Graph Models 71:1–21

    Google Scholar 

  • Prusinkiewicz P (2004) Modeling plant growth and development. Curr Opin Plant Biol 7:79–83

    CAS  PubMed  Google Scholar 

  • Prusinkiewicz P, Lindenmayer A (1990) The algorithmic beauty of plants. Springer, Berlin

    Google Scholar 

  • Ptashnyk M (2010) Derivation of a macroscopic model for nutrient uptake by hairy-roots. Nonlinear Anal: Real World Appl 11:4586–4596

    CAS  Google Scholar 

  • Richardson AE, Barea JM, McNeill AM, Prigent-Combaret C (2009) Acquisition of phosphorus and nitrogen in the rhizosphere and plant growth promotion by microorganisms. Plant Soil 321:305–339

    CAS  Google Scholar 

  • Robinson D (1988) Optimal relations between root length and nutrient inflow rate in plant root systems. J Theo Biol 135:359–370

    Google Scholar 

  • Robinson D (1994) The responses of plants to non–uniform supplies of nutrients. New Phytol 127:635–674

    CAS  Google Scholar 

  • Robinson D (1996) Variation, co–ordination and compensation in root systems in relation to soil variability. Plant Soil 187:57–66

    CAS  Google Scholar 

  • Robinson D (2005) Integrated root responses to variations in nutrient supply. In: BassiriRad H (ed) Nutrient acquisition by plants - an ecological perspective. Springer–Verlag, Berlin, pp 43–62

    Google Scholar 

  • Roose T, Fowler A, Darrah P (2001) A mathematical model of plant nutrient uptake. Journal of Mathematical Biology 42:347–360

    Google Scholar 

  • Roose T, Fowler A (2004) A mathematical model for water and nutrient uptake by plant root systems. J Theor Biol 228:173–184

    CAS  PubMed  Google Scholar 

  • Roose T, Schnepf A (2008) Mathematical models of plant-soil interaction. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366:4597–4611

    Google Scholar 

  • Roose T, Kirk G (2009) The solution of convection-diffusion equations for solute transport to plant roots. Plant Soil 316:257–264

    CAS  Google Scholar 

  • Rose WC, Chichester FW, Williams JR, Ritchie JT (1982a) A contribution to simplified models of field solute transport. J Environ Qual 11:146–151

    Google Scholar 

  • Rose WC, Chichester FW, Williams JR, Ritchie JT (1982b) Application of an approximate analytic method of computing solute profiles with dispersion in soils. J Environ Qual 11:151–155

    Google Scholar 

  • Rose TJ, Bingah H, Rengel Z (2010) Wheat, canola and grain legume access to soil phosphorus fractions differs in soils with contrasting phosphorus dynamics. Plant Soil 326:159–170

    CAS  Google Scholar 

  • Rubio G, Walk T, Ge Z, Yan X, Liao H, Lynch JP (2001) Root gravitropism and below-ground competition among neighbouring plants: a modelling approach. Ann Bot 88:929–940

    Google Scholar 

  • Ryan PR, Delhaize E, Jones DL (2001) Function and mechanism of organic anion exudation from plant roots. Annu Rev Plant Physiol Plant MolBiol 52:527–560

    CAS  Google Scholar 

  • Saltelli A, Scott EM, Chan K (2000) Sensitivity analysis. John Wiley & Sons, Ltd, Chichester

    Google Scholar 

  • Schneider CL, Attinger S, Delfs JO, Hildebrandt A (2010) Implementing small scale processes at the soil-plant interface - the role of root architectures for calculating root water uptake profiles. Hydrol Earth Syst Sci 14:279–289

    Google Scholar 

  • Schnepf A, Roose T, Schweiger P (2008) Growth model for arbuscular mycorrhizal fungi. J R Soc Interface 5:773–784

    CAS  PubMed  Google Scholar 

  • Schnepf A, Leitner D, Klepsch S, Pellerin S, Mollier A (2011) Modelling phosphorus dynamics in the soil-plant system. In: Bünemann E, Oberson A, Frossard E (eds) Phosphorus in action: biological processes in soil phosphorus cycling. Springer-Verlag Berlin, Heidelberg, pp 113–133

    Google Scholar 

  • Schnepf A, Leitner D, Klepsch S (2012) Modeling phosphorus uptake by a growing and exuding root system. Vadose Zone J 11:318–327

    Google Scholar 

  • Schröder T, Javaux M, Vanderborght J, Korfgen B, Vereecken H (2008) Effect of local soil hydraulic conductivity drop using a three–dimensional root water uptake model. Vadose Zone J 7:1089–1098

    Google Scholar 

  • Schröder T, Javaux M, Vanderborght J, Koerfgen B, Vereecken H (2009a) Implementation of a microscopic soil-root hydraulic conductivity drop function in a soil-root architecture water transfer model. Vadose Zone J 8(3):783–792. doi:10.2136/vzj2008.0116

    Google Scholar 

  • Schröder T, Tang L, Javaux M, Vanderborght J, Körfgen B, Vereecken H (2009b) A grid refinement approach for a three–dimensional soil–root water transfer model. Water Resour Res 45, W10412

    Google Scholar 

  • Schröder N, Javaux M, Vanderborght J, Steffen B, Vereecken H (2012) Effect of root water and solute uptake on apparent soil dispersivity: a simulation study. Vadose Zone J. doi:10.2136/vzj2012.0009

    Google Scholar 

  • Schulz H, Postma JA, van Dusschoten D, Scharr H, Behnke S (2012) 3D reconstruction of plant roots from MRI images. Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP), Rome, February 2012

  • Shibusawa S (1994) Modelling the branching growth fractal pattern of the maize root system. Plant Soil 165:339–347

    CAS  Google Scholar 

  • Shu L, Shen J, Rengel Z, Tang C, Zhang F, Cawthray GR (2007) Formation of cluster roots and citrate exudation by Lupinus albus in response to localized application of different phosphorus sources. Plant Sci 172:1017–1024

    CAS  Google Scholar 

  • Simunek J, Huang K, Van Genuchten M Th (1995) The SWMS_3D code for simulating water flow and solute transport in three-dimensional variably-saturated media. Version 1.0. Research Report n°139. Riverside, California. U. S. Salinity Laboratory. Agricultural Research Service. U.S. Department of Agriculture

  • Smith S, De Smet I (2012) Root system architecture: insights from Arabidopsis and cereal crops. Phil Trans R Soc B 367:1441–1452

    CAS  PubMed  Google Scholar 

  • Somma F, Clausnitzer V, Hopmans JW (1997) An algorithm for three-dimensional, simultaneous modeling of root growth, transient soil water flow, and solute transport and uptake. Version 2.1. Paper No. 100034. Dept of Land, Air, and Water Resources, University of California

  • Somma F, Hopmans JW, Clausnitzer V (1998) Transient three-dimensional modeling of soil water and solute transport with simultaneous root growth, root water and nutrient uptake. Plant Soil 202:281–293

    CAS  Google Scholar 

  • Spek LY (1997) Generation and visualization of root-like structures in a three-dimensional space. Plant Soil 197:9–18

    CAS  Google Scholar 

  • Steele KA, Virk DS, Kumar R, Prasad SC, Witcombe JR (2007) Field evaluation of upland rice lines selected for QTLs controlling root traits. Field Crops Res 101:180–186

    Google Scholar 

  • Stingaciu L, Schulz H, Pohlmeier A, Behnke S, Zilken H, Javaux M, Vereecken H (2013) In situ root system architecture extraction from magnetic resonance imaging for application to water uptake modeling. Vadose Zone J. doi:10.2136/vzj2012.0019

    Google Scholar 

  • Subbaiah R, Rao KA (1993) Root growth simulation model under specified environment. J Irrig Drain Eng 119:898–904

    Google Scholar 

  • Tajini F, Suriyakup P, Vailhe H, Jansa J, Drevon JJ (2009) Assess suitability of hydroaeroponic culture to establish tripartite symbiosis between different amf species, beans, and rhizobia. BMC Plant Biol 9:73

    PubMed  Google Scholar 

  • Tardieu F (2010) Why work and discuss the basic principles of plant modelling 50 years after the first plant models? J Exp Bot 61:2039–2041

    CAS  PubMed  Google Scholar 

  • Thaler P, Pagès L (1998) Modelling the influence of assimilate availability on root growth and architecture. Plant Soil 201:307–320

    CAS  Google Scholar 

  • Thonar C, Schnepf A, Frossard E, Roose T, Jansa J (2011) Traits related to differences in function among three arbuscular mycorrhizal fungi. Plant Soil 339:231–245

    CAS  Google Scholar 

  • Thorup-Kristensen K (2001) Are differences in root growth of nitrogen catch crops important for their ability to reduce soil nitrate–N content, and how can this be measured. Plant Soil 230:185–195

    CAS  Google Scholar 

  • Tsegaye T, Mullins CE, Diggle AJ (1995) Modelling pea (Pisum sativum) root growth in drying soil. A comparison between observations and model predictions. New Phytol 131:179–189

    Google Scholar 

  • Van Lier QJ, Metselaar K, Van Dam JC (2006) Root water extraction and limiting soil hydraulic conditions estimated by numerical simulation. Vadose Zone J 5:1264–1277

    Google Scholar 

  • Vance CP, Uhde-Stone C, Allen DL (2003) Phosphorus acquisition and use: critical adaptations by plants for securing a non–renewable resource. New Phytol 157:423–447

    CAS  Google Scholar 

  • Vercambre G, Pagès L, Doussan C, Habib R (2003) Architectural analysis and synthesis of the plum tree root system in an orchard using a quantitative modelling approach. Plant Soil 51:1–11

    Google Scholar 

  • Vos J, Evers JB, Buck–Sorlin GH, Andrieu B, Chelle M, de Visser PHB (2010) Functional-structural plant modelling: a new versatile tool in crop science. J Exp Bot 61:2101–2115

    CAS  PubMed  Google Scholar 

  • Wagner B, Gärtner H, Ingensand H, Santini S (2010) Incorporating 2D tree-ring data in 3D laser scans of coarse–root systems. Plant Soil 334:175–187

    CAS  Google Scholar 

  • Wagner B, Ingensand H, Santini S, Gärtner H (2011) A tool to model 3D coarse-root development with annual resolution. Plant Soil 346:79–96

    CAS  Google Scholar 

  • Walk TC, van Erp E, Lynch JP (2004) Modelling applicability of fractal analysis to efficiency of soil exploration by roots. Ann Bot 94:119–128

    PubMed  Google Scholar 

  • Walk TC, Jaramillo R, Lynch JP (2006) Architectural tradeoffs between adventitious and basal roots for phosphorus acquisition. Plant Soil 279:347–366

    CAS  Google Scholar 

  • Wallstrom T, Christie M, Durlofsky L, Sharp D (2002) Effective flux boundary conditions for upscaling porous media equations. Transp Porous Media 46:139–153

    CAS  Google Scholar 

  • Wiesler F, Horst WJ (1994) Root growth and nitrate utilization of maize cultivars under field conditions. Plant Soil 163:267–277

    CAS  Google Scholar 

  • Wu L, Bingham IJ (2009) Using modelling as a tool to explore resource use efficiency by crops. Asp Appl Biol 93:257–261

    Google Scholar 

  • Wu L, Shepherd A (2011) Special features of the SPACSYS modeling package and procedures for parameterization and validation. In: Ahuja LR, Ma L (eds) Methods of introducing system models into agricultural research. ASA, CSSA & SSSA, Madison, pp 117–154

    Google Scholar 

  • Wu L, Baddeley JA, Watson CA (2005) Designer root systems – the value of modelling to determine optimum root systems for different end uses. Asp Appl Biol 71:11–16

    Google Scholar 

  • Wu L, McGechan MB, McRoberts N, Baddeley JA, Watson CA (2007) SPACSYS: Integration of a 3D root architecture component to carbon, nitrogen and water cycling - Model description. Ecol Model 200:343–359

    Google Scholar 

  • Wu L, Bingham IJ, Baddeley JA, Watson CA (2008) The importance of 3D root architecture when simulating plant N uptake. In: Ma L, Ahuja LR, Bruulsema TW (eds) Quantifying and understanding plant nitrogen uptake systems modelling (9). CRC Press, Boca Raton, pp 197–218

    Google Scholar 

  • Xu L, Henke M, Zhu J, Kurth W, Buck-Sorlin G (2011) A functional-structural model of rice linking quantitative genetic information with morphological development and physiological processes. Ann Bot 107:817–828

    CAS  PubMed  Google Scholar 

  • Yang HS, Janssen BH (2000) A mono-component model of carbon mineralization with a dynamic rate constant. European Journal of Soil Science 51:517–529

    Google Scholar 

  • Zhu J, Lynch JP (2004) The contribution of lateral rooting to phosphorus acquisition efficiency in maize (Zea mays L.) seedlings. Funct Plant Biol 31:949–958

    CAS  Google Scholar 

  • Zhu J, Kaeppler S, Lynch JP (2005) Topsoil foraging and phosphorus acquisition efficiency in maize (Zea mays L.). Funct Plant Biol 32:749–762

    CAS  Google Scholar 

  • Zhu J, Brown KM, Lynch JP (2010a) Root cortical aerenchyma improves the drought tolerance of maize (Zea mays L.). Plant Cell Environ 33:740–749

    PubMed  Google Scholar 

  • Zhu J, Zhang C, Lynch JP (2010b) The utility of phenotypic plasticity of root hair length for phosphorus acquisition. Funct Plant Biol 37:313–322

    Google Scholar 

  • Zygalakis K, Roose T (2012) A mathematical model for investigating the effect of cluster roots on plant nutrient uptake. Eur Phys J Spec Top 204:103–118

    CAS  Google Scholar 

  • Zygalakis K, Kirk G, Jones D, Wissuwa M, Roose T (2011) A dual porosity model of nutrient uptake by root hairs. New Phytol 192:676–688

    CAS  PubMed  Google Scholar 

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Acknowledgments

The authors would like to thank Dr Ian Bingham and Dr Davide Tarsitano SRUC, UK, for providing additional information on the SPACSYS model, and Prof Jonathan Lynch, Penn State University for providing additional information on the SimRoot model. This research was supported by the Australian Research Council and by the Austrian Science Fund FWF (Grant No.: V220-N13). Andrea Schnepf is an Elise-Richter research fellow. Daniel Leitner is recipient of an APART-fellowship of the Austrian Academy of Sciences at the Computational Science Center, University of Vienna.

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  1. Andrea Schnepf

    Present address: Forschungszentrum Jülich GmbH, Agrosphere (IBG-3), D-52425, Jülich, Germany

Authors and Affiliations

  1. Tasmanian Institute of Agriculture, The University of Tasmania, Private Bag 54, Hobart, TAS, 7000, Australia

    Vanessa M. Dunbabin

  2. IBG–2: Plant Sciences, Forschungszentrum Jülich, 52425, Jülich, Germany

    Johannes A. Postma

  3. Institute of Hydraulics and Rural Water Management, Department of Water, Air and Environment, BOKU - University of Natural Resources and Life Sciences, Vienna, Muthgasse 18, 1190, Vienna, Austria

    Andrea Schnepf

  4. INRA, UR 1115 PSH, Site Agroparc, 84914, Avignon cedex 9, France

    Loïc Pagès

  5. Université catholique de Louvain, Croix du Sud 2, L7.05.02, 1348, Louvain-la-Neuve, Belgium

    Mathieu Javaux

  6. Sustainable Soils and Grassland Systems Department, Rothamsted Research, North Wyke, Okehampton, Devon, EX20 2SB, UK

    Lianhai Wu

  7. Computational Science Center, University of Vienna, Nordbergstrasse 15, 1090, Vienna, Austria

    Daniel Leitner

  8. Soil Science and Plant Nutrition, School of Earth and Environment, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Ying L. Chen & Zed Rengel

  9. The UWA Institute of Agriculture, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Ying L. Chen & Zed Rengel

  10. The Department of Agriculture and Food, Western Australia, Locked Bag 4, Bentley, WA, 6983, Australia

    Art J. Diggle

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Correspondence to Vanessa M. Dunbabin.

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Dunbabin, V.M., Postma, J.A., Schnepf, A. et al. Modelling root–soil interactions using three–dimensional models of root growth, architecture and function. Plant Soil 372, 93–124 (2013). https://doi.org/10.1007/s11104-013-1769-y

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