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Image based modeling of nutrient movement in and around the rhizosphere
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Author list: Keith R. Daly1, Samuel D. Keyes1, Shakil Masum2, Tiina Roose§1
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Author affiliations:
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University of Southampton, University Road, SO17 1BJ Southampton, United Kingdom,
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United Kingdom,
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§
Bioengineering Sciences Research Group, Faculty of Engineering and Environment,
School of Energy, Environment and Agrifood, Cranfield University, Cranfield, MK43 0AL,
Corresponding author: Faculty of Engineering and Environment, University of
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Southampton, Hampshire, SO17 1BJ. United Kingdom, email: t.roose@soton.ac.uk
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Summary
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Using rigorous mathematical techniques and image based modelling we quantify the effect of
root hairs on nutrient uptake. In addition we investigate how uptake is influenced by growing
root hairs.
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1. Abstract
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In this paper we develop a spatially explicit model for nutrient uptake by root hairs based on
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X-ray Computed Tomography images of rhizosphere soil structure. This work extends our
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previous study (Keyes et al., 2013) to larger domains and, hence, is valid for longer times.
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Unlike the model used in (Keyes et al., 2013), which considered only a small region of soil
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about the root, we consider an effectively infinite volume of bulk soil about the rhizosphere.
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We ask the question “at what distance away from root surfaces do the specific structural
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features of root-hair and soil aggregate morphology not matter because average properties
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start dominating the nutrient transport?” The resulting model captures bulk and rhizosphere
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soil properties by considering representative volumes of soil far from the root and adjacent to
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the root respectively. By increasing the size of the volumes we consider, the diffusive
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impedance of the bulk soil and root uptake are seen to converge. We do this for two different
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values of water content which provides two different geometries. We find that the size of
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region for which the nutrient uptake properties converge to a fixed value is dependent on the
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water saturation. In the fully saturated case the region of soil which we need to consider is
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only of radius 1.1 mm for poorly soil-mobile species such as phosphate. However, in the
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case of a partially saturated medium (relative saturation 0.3) we find that a radius of 1.4 mm
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is necessary. This suggests that, in addition to the geometrical properties of the rhizosphere,
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there is an additional effect on soil moisture properties which extends further from the root
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and may relate to other chemical changes in the rhizosphere which are not explicitly included
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in our model.
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Keywords:
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Rhizosphere, X-ray CT, structural imaging, phosphate, plant-soil interaction
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Abbreviations:
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CT – Computer Tomography
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Short title for page headings: Modelling nutrient uptake
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Introduction
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The uptake of phosphate and other low mobility nutrients is essential for plant growth and,
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hence, global food security (Barber, 1984; Nye and Tinker, 1977; Tinker and Nye, 2000).
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However, the overuse of inorganic fertilizers has caused an accumulation of phosphate in
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European soils which potentially causes eutrophication (Gahoonia and Nielsen, 2004).
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Hence, there is a need to optimize the efficiency of phosphate uptake not only to increase
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crop yields, but also to minimize the detrimental effects of excess phosphate on the
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environment.
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Plants are known to uptake nutrients both through their roots and root hairs (Datta et al.,
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2011). Root hairs are thought to play a significant role in the uptake of poorly soil-mobile
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nutrients such as phosphate, water uptake, plant stability and microbial interactions (Datta et
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al., 2011; Gahoonia and Nielsen, 2004). In order to quantify the role of root hairs in the
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uptake of poorly soil-mobile nutrients, such as phosphate, a detailed understanding of the
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root-hair morphology and the region of soil around the root, known as the rhizosphere
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(Hiltner, 1904), is required. This is because the properties of soil in the rhizosphere are
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thought to be significantly different due to soil microbial interactions and compaction of soil
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by the root (Aravena et al., 2010; Aravena et al., 2014; Daly et al., 2015; Dexter, 1987;
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Whalley et al., 2005).
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Early models for nutrient movement in soil treated the rhizosphere as a volume averaged
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continuum (Barber, 1984; Ma et al., 2001; Nye and Tinker, 1977). More recently models for
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nutrient movement have been derived and parametrized for dual porosity soil (Zygalakis et
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al., 2011) and for soil adjacent to cluster roots (Zygalakis and Roose, 2012). These models
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were based around the technique of homogenization (Hornung, 1997; Pavliotis and Stuart,
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2008), a multi-scale technique which enables field scale equations to be derived and
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parametrized based on underlying continuum models.
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To obtain a better understanding of processes occurring in the rhizosphere, in comparison to
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the surrounding bulk soil, non-invasive measurements of the plant root and soil structure are
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essential (George et al., 2014; Hallett et al., 2013). Three dimensional imaging of plant roots
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in-situ using X-ray Computed Tomography (CT) is a rapidly growing field (Aravena et al.,
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2010; Aravena et al., 2014; Hallett et al., 2013; Keyes et al., 2013; Tracy et al., 2010). Using
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these techniques sub-micron resolutions can be achieved enabling the root hair morphology
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to be visualized (Keyes et al., 2013).
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In addition to direct in-situ visualization of plant-soil interaction, the use of X-ray CT also
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provides the means to apply numerical models describing the diffusion of nutrients directly to
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the imaged geometries (Aravena et al., 2010; Aravena et al., 2014; Daly et al., 2015; Keyes et
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al., 2013; Tracy et al., 2015). This approach has been used to quantify the effectiveness of
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root hairs in saturated soil conditions (Keyes et al., 2013) using a diffusion model originally
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developed for mycorrihzal fungi (Schnepf et al., 2011). However, the study in (Keyes et al.,
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2013) considered a small volume of soil adjacent to the root. This volume extended c.
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300 μm from the root surface and had a zero flux boundary condition on the outer surface.
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Hence, once the phosphate immediately adjacent to the root was depleted the uptake of
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nutrients into the root stopped.
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considered, limited the time frame for which the model was applicable to c. 3 hours.
This, whilst accurate for the experimental situation
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In this paper we extend the work of (Keyes et al., 2013) to include both bulk and rhizosphere
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soil. The key aims of this study are to develop an image based modelling approach which is
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applicable to root hairs surrounded by a large/infinite bulk of soil, to understand how root
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hairs contribute to nutrient uptake at different soil water content and to estimate the effects of
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root hair growth on nutrient uptake. We have chosen to model a single root in a large/infinite
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volume of soil, rather than multiple roots competing for nutrients for a number of reasons.
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Firstly, we want to compare our model to established models for nutrient uptake which are
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applied in this geometry (Roose et al., 2001). Secondly, whilst some models consider root-
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root competition, (Barber, 1984), these models are applied in a cylindrical geometry. Hence,
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any competition is provided by a ring of roots at a distance from the root under consideration.
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In order to accurately capture roots at a given root density additional assumptions and
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approximations would be required. Finally, our aim is to study the role of root hairs on the
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uptake of a single root. Hence, we have chosen to study a geometry which captures and
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isolates this effect. The modelling strategy developed in this paper will provide a framework
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for future modelling studies to consider how different root hair morphologies influence
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uptake. In addition it will provide a new level of understanding in to the role or root hairs in
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nutrient uptake as the root system begins to develop.
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The bulk soil properties are derived using multi-scale homogenization combined with image
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based modelling in a similar manner to (Daly et al., 2015). The resulting equations are
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solved analytically and are patched to an explicit image based geometry for the rhizosphere
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using a time dependent boundary condition. Reducing the bulk soil to a single boundary
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condition allows us to capture the full geometry and topology of the plant-root-soil system
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without the need for excessive computational resources and has particular relevance to poorly
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soil-mobile species such as phosphate, potassium and zinc. The model describes phosphate
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uptake by roots and root hairs in an effectively infinite volume of bulk soil whilst capturing at
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high precision any changes in the soil adjacent to the plant roots. In this case an effectively
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infinite volume of soil corresponds to any volume of soil which is sufficiently large that the
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phosphate depletion region about the root does not reach the edge of the domain considered.
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Using this model we are able to parameterize upscaled models for nutrient motion in soil
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(Zygalakis et al., 2011).
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This paper is arranged as follows: in section 2 we describe the plant growth, imaging and
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modelling approaches, in section 3 we discuss the results of numerical simulation and show
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how these models can be used to perform an image based study of root hair growth, finally in
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section 4 we discuss our results and show how these models might be further developed. The
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technical description of the mathematical models used in this paper is provided in appendix A
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and appendix B.
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2. Methods
2.1. Plant growth
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We study a rice genotype Oryza Sativa cv. Varyla provided by the Japan International
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Research Centre for Agricultural Sciences (JIRCAS). The seeds were heat-treated at 50℃ for
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48h to break dormancy and standardise germination. Germination was carried out at 23±1℃
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for 2 days between moistened sheets of Millipore filter paper. The growth medium was a
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sand-textured Eutric Cambisol soil collected from a surface plot at Abergwyngregyn, North
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Wales (53o14’N, 4o01’W), the soil organic matter was 7%, full details are given in (Lucas
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and Jones, 2006), soil B. The soil was sieved to <5 mm, autoclaved and air dried at 23±1℃
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for 2 days.
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producing a well-aggregated, textured growth medium.
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Growth took place in a controlled growth environment (Fitotron SGR, Weiss-Gallenkamp,
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Loughborough, UK) for a period of 14 days. Growth conditions were 23±1℃ and 60 %
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humidity for 16 hours (day) and 18±1℃ and 55 % humidity for 8 hours (night) with both
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ramped over 30 minutes.
The dried soil was sieved to between bounds of 1680 μm and 1000 μm,
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The seminal roots were guided by a specially designed growth environment fabricated in
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ABS plastic using an UP! 3D printer (PP3DP, China). The resulting assay is shown in
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Supplementary Figure 1 and further details are provided in (Keyes et al., 2015) for details.
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The morphology of the growth environment was used to guide the roots into 7 syringe barrels
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(root chambers) which can then be detached for imaging once the growth stage is complete.
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The lower portion including the root chambers was housed in a foil-wrapped 50 ml centrifuge
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tube which occludes light during the growth period.
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2.2. Imaging
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Imaging of root chambers was conducted at the TOMCAT beamline on the X02DA port of
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the Swiss Light Source, a 3rd generation synchrotron at the Paul Scherrer Institute, Villigen,
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Switzerland. The imaging resolution was 1.2 μm, and a monochromatic beam was employed
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with an energy of 19 kV. A 90 ms exposure time was employed to collect 1601 projections
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over 180 degrees with a total scan time of 2.4 minutes. The data were reconstructed to 16-bit
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volumes using a custom back-projection algorithm implementing a Parzen filter.
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resulting volume size was 2560x2560x2180 voxels.
The
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The reconstructed volumes were analysed using a multi-pass approach (Keyes et al., 2015).
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Whilst root hairs can be visualised in situ for dry soils they are challenging to distinguish
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from some background phases even at high resolution. Specifically, the portions of root hairs
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which traverse water filled regions of the pore space are indistinguishable from the water.
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The visible sections of the root hairs were extracted manually using a graphical input tablet
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(Cintiq 24, WACOM) and a software package that allowed interpolation between lofted
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cross-sections (Avizo FIRE 8, FEI Company, Oregon, USA). A number of these sections
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represent an incomplete root hair segmentation since proportions of the hair paths can be
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occluded by fluid rich phases and pore water. Full root hair paths were extrapolated from the
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segmented sections of partially visible hairs using an algorithm that simulates root hair
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growth (Keyes et al., 2015). The algorithm was parameterised from the root hairs of plants
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(Oryza Sativa cv. Varyla) which were grown in rhizo-boxes with dimensions 30x30x2 cm,
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constructed of transparent polycarbonate with a removable front-plate, and wrapped in foil to
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occlude light. Hydration during the growth period was maintained via capillary rise from a
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tray of water, with occasional top-watering. The plants were grown in a glasshouse, with
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temperature ranging from 20-32 ℃ over the growth period. The root systems of 1 month old
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specimens were gently freed from soil using running water. One fine primary and one coarse
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primary were randomly sampled before manual removal of the remaining soil from the roots.
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Each was cut into a number of subsections of 2 cm length for analysis, each being imaged
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using an Olympus BX50 optical microscope. After stitching all images together 5 sub-regions
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were randomly defined for each section. Each region was chosen to represent a distance of 1
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mm on the root surface. All hairs in sub-regions which were in sharp focus were measured in
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FIJI using a poly-line tool to generate a set of 220 hair lengths. A normal distribution was
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fitted to these data and used to parameterize the hair-growth algorithm. The fitted and
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sampled distributions are shown in Supplementary Figure 1. The mean of experimental
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lengths was 171.56 µm, compared to the mean fitted lengths: 173.67 µm. The standard
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deviation of the experimental lengths was 72.88 µm compared to the standard deviation of
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the fitted lengths 73.44 µm.
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We observe that root hair density measured via Synchrotron Radiation X-ray CT data is
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generally slightly higher than for microscopy, probably due to the lack of damage artefacts.
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For the data in this study, the density is ≈ 77 hairs mm−1 , compared to 64 hairs mm−1
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measured via bright-field microscopy in a study of similar rice varieties grown under similar
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conditions (Wissuwa and Ae, 2001). This discrepancy is, in part, because the in situ hairs are
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often part-occluded by fluid and colloidal phases, it is usually only possible to obtain partial
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measurements in the Synchrotron CT data. This is offset to some degree by the lack of
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damage artefacts when compared to root-washing and microscopy, but the length values are
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low compared to most literature sources. However, in this study we explore the implications
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of soil domain size on nutrient uptake and have used hair data attained via quantification of
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soil-grown roots using bright field microscopy.
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The root geometry was extracted in the same manner, manually defining the root cross-
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section from slices sampled (at a spacing of 120 µm) along the root growth axis. The soil
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minerals, pore gas and pore fluid were extracted using a trainable segmentation plugin in FIJI
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that implements the WEKA classifiers for feature detection using a range of local statistics as
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training parameters (Hall et al., 2009).
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The segmented geometry was used to produce 3D surface meshes, using the ScanIP software
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package (Simpleware Ltd., Exeter, United Kingdom). The root, hairs, gas fluid phase and
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soil minerals were exported as separate .STL files which provide the geometries on which the
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numerical models can be tested.
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2.3. Modelling
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We model the uptake of phosphate by a single root and root hairs in an unbounded partially
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saturated geometry, see Figure 1. In order to do this we extend the model described in
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(Keyes et al., 2013). In (Keyes et al., 2013) the authors considered a segment of root and
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surrounding soil. On the outer boundary of the segment a zero flux condition was applied.
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By definition this condition prevents any influx of nutrient from outside the domain
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considered. Hence, the total possible uptake is limited to the nutrient available from within
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the initial segment. This was adequate for the experimental situation considered, i.e., a pot of
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radius c. 300 μm, but is clearly not applicable for isolated roots in soil columns of radius
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larger than 300 μm . We note that this is a slightly different scenario from the one studied by
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Barber (Barber, 1984) in which the domain around the root was considered to be half the size
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of the inter-root spacing. In other words, Barbers model considered multiple roots effectively
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introducing root competition. In this work we consider a single isolated root and how that is
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affected by soil structural properties.
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In order to overcome this limitation we consider a single root in a large/infinite domain of
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soil. However, as considering an infinite domain explicitly is computationally unfeasible, we
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break the domain into two regions: the rhizosphere and bulk soil. We assume that the bulk
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soil, which is far from the root, may be considered as a homogeneous medium with an
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effective phosphate diffusion constant. The effective diffusion properties of phosphate in the
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bulk soil are derived from the segmented CT data using the method of homogenization
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(Pavliotis and Stuart, 2008; Zygalakis et al., 2011; Zygalakis and Roose, 2012). The nutrient
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flux in the bulk soil is then patched to the rhizosphere domain using a boundary condition
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which simulates an effective infinite region of bulk soil beyond the rhizosphere. In this
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section we introduce the equations and summarize the model which results from these
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assumptions. Full details of the derivation have been included in appendices A and B.
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We assume that phosphate can only diffuse in the fluid domain, i.e., no diffusion occurs in
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the air filled portion of the pore space. Therefore, we only present equations in the fluid
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region and its associated boundaries.
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geometrical details regarding the air filled portion of pore space. If, from the CT data, there
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is air in contact with soil or root material this simply acts to reduce the surface area over
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which phosphate can be exchanged. We define two regions of fluid filled pore space we
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denote the rhizosphere by Ω𝑟 as the region explicitly considered near to the root. Physically
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we may think of this as an approximation to the rhizosphere. The second region is Ω𝑏 , the
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region of bulk soil outside of the rhizosphere, see Figure 2. As different boundary conditions
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need to be applied on each of the pore water interfaces we adopt the following naming
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convention. We take the interface between the two regions, located at a distance 𝑟̃𝑏 from the
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centre of the root to be Γ𝑟𝑏 . The root and root-hair surfaces are defined as Γ0𝑟 , and Γ0ℎ
However, we make no assumptions about the
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respectively, the soil particle surface to be Γ𝑠𝑟 and Γ𝑠𝑏 in region Ω𝑟 and Ω𝑏 respectively.
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Finally we define the air water interfaces as Γ𝑎𝑟 and Γ𝑎𝑏 in region Ω𝑟 and Ω𝑏 respectively.
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The method we use to determine phosphate movement is different in each region. Phosphate
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movement in the rhizosphere is calculated based on a spatially explicit model obtained from
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CT data. Phosphate movement in the bulk soil is calculated analytically based on the solution
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to a spatially explicit numerical model in a representative volume of bulk soil. We present
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each of these models separately.
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2.3.1. Rhizosphere
In the rhizosphere we assume that phosphate moves by diffusion only
𝜕𝐶̃𝑟
̃∇
̃2 𝐶̃𝑟 ,
=𝐷
𝜕𝑡̃
𝒙 ∈ Ω𝑟 ,
(1)
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̃ is the diffusion constant of
where 𝐶̃𝑟 is the phosphate concentration in the soil solution and 𝐷
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phosphate in the soil solution. The phosphate is assumed to bind to the soil particles based on
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linear first order kinetics
̃𝒏
̃ 𝐶̃𝑟 = −𝑘̃𝑎 𝐶̃𝑟 + 𝑘̃𝑑 𝐶̃𝑎 ,
̂⋅𝛁
𝐷
𝒙 ∈ Γ𝑠𝑟 ,
(2)
𝜕𝐶̃𝑎
= 𝑘̃𝑎 𝐶̃𝑟 − 𝑘̃𝑑 𝐶̃𝑎 ,
𝜕𝑡̃
𝒙 ∈ Γ𝑠𝑟 ,
(3)
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where 𝑘̃𝑎 and 𝑘̃𝑑 are the adsorption and desorption rates respectively, 𝐶̃𝑎 is the nutrient
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̂ is the outward pointing surface normal vector. Here
concentration on the soil surface and 𝒏
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we have assumed that the nutrient concentration of the soil can be represented as a surface
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concentration and that any replenishment from within the soil aggregate is either so fast that
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it is captured by these equations or so slow that we can neglect it. We assume that there is no
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nutrient diffusion across the air water boundary
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̃𝒏
̃ 𝐶̃𝑟 = 0,
̂⋅𝛁
𝐷
𝒙 ∈ Γ𝑎𝑟 . (4)
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We consider uptake of phosphate from the fluid only. On the root surface this follows a
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linear uptake condition
̃𝒏
̃ 𝐶̃𝑟 = 𝜆̃𝐶̃𝑟 ,
̂⋅𝛁
𝐷
𝒙 ∈ Γ0𝑟 . (5)
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where λ̃ is the nutrient uptake rate on the root and root-hair surfaces. For comparison with
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𝐹̃
the paper by Keyes et al (Keyes et al., 2013) we have chosen the absorbing power as λ̃ = 𝐾̃𝑚 ,
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consistent with Nye and Tinker. Equation (5) is the small concentration equivalent of the
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̃𝑚 is the
Michaelis Menten condition, in which 𝐹̃𝑚 is the maximum rate of uptake and 𝐾
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concentration when the uptake is half the maximum.
𝑚
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The flux on the root hair surface is given by one of two different scenarios: the first is linear
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uptake
̃𝒏
̃ 𝐶̃𝑟 = 𝜆̃𝐶̃𝑟 ,
̂⋅𝛁
𝐷
𝒙 ∈ Γ0ℎ .
(6)
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The second case we consider is a pseudo time-dependent uptake which simulates linear
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uptake for a growing root-hair. We have termed this pseudo root-hair growth as we do not
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consider the geometrical and mechanical effects of root hair growth explicitly. Rather, we
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assume that at 𝑡̃ = 0 the root hairs are present, but do not contribute to uptake. As the
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simulation progresses the root hairs are assumed to grow at a certain rate. We model this by
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assuming an active region of root hair which grows outward from the root; this is illustrated
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in Figure 3. We write the root-hair uptake condition as
𝜆̃
̃𝒏
̃ 𝐶̃𝑟 = 𝐶̃𝑟 {1 + tanh[−𝛼̃(𝑟̃ − 𝑔̃𝑟 𝑡̃)]},
̂⋅𝛁
𝐷
2
𝒙 ∈ Γ0ℎ ,
(7)
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where 𝑟̃ is the axial distance along the root hair, 𝑔̃𝑟 is the root hair growth rate, 𝛼̃ is an
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empirical parameter which controls the sharpness of the transition between the root hairs
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being “on”, i.e., actively taking up nutrient and “off”, i.e., not actively taking up nutrient.
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The adsorption and desorption constants (𝑘̃𝑎 and k̃ 𝑑 ) are taken from (Keyes et al., 2013)
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where they were derived using standard soil tests (Giesler and Lundström, 1993; Murphy and
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Riley, 1962). The remaining parameters used in equations (1) to (5) were obtained from the
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literature (Barber, 1984; Tinker and Nye, 2000) and are summarized in Table 1.
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2.3.2. Rhizosphere bulk interface
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Due to the success of averaged equations in bulk soil, for example Darcy’s law and Richards’
311
equation (Van Genuchten, 1980), the precise details of the geometry are less important in
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determining the total phosphate movement and average properties are sufficient to be
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considered. Therefore, rather than consider the explicit details of the geometry we derive an
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effective diffusion constant based on a representative soil volume in absence of roots. This is
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achieved using the method of homogenization, (Pavliotis and Stuart, 2008), which is
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implemented as follows. First it is shown that, on the length scale of interest, the nutrient
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concentration is only weakly dependent on the precise structure of the soil. Secondly a set of
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318
equations, often called the cell problem, are derived which determine the local variation in
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concentration due to the representative soil volume. Finally an averaged equation is derived
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which describes the effective rate of diffusion in terms of an effective diffusion constant
321
̃𝑒𝑓𝑓 = 𝐷
̃ 𝐷𝑒𝑓𝑓 where 𝐷𝑒𝑓𝑓 is calculated from the bulk soil geometry (equations (A31) to
𝐷
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(A34) and (A58)) and describes the impedance to diffusion offered by the soil. Full details of
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how 𝐷𝑒𝑓𝑓 is derived are provided in Appendix A. Hence, in the bulk the diffusion of
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phosphate is described by
𝜕𝐶̃𝑏
̃𝑒𝑓𝑓 ∇
̃2 𝐶̃𝑏 ,
=𝐷
𝜕𝑡̃
(8)
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where 𝐶̃𝑏 is the nutrient concentration in the bulk soil. The advantage of equation (8) is that
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we do not need to explicitly consider the soil geometry in the bulk soil domain. In itself this
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significantly reduces the computational cost for finite but large domains. However, rather
329
than solving equation (8) numerically we find an approximate analytic solution for an infinite
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bulk soil domain subject to the conditions 𝐶̃𝑏 = 𝐶̃𝑟 (𝑡) for 𝒙 ∈ Γ𝑟𝑏 and 𝐶̃𝑏 → 𝐶̃∞ as 𝑟̃ → ∞.
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Using this solution we are able to define a relationship between concentration and flux at the
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edge of the rhizosphere. The result is a condition which simulates the presence of an infinite
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region of bulk soil at the rhizosphere soil domain boundary
̃𝒏
̃ 𝐶̃𝑟 = −
̂⋅𝛁
𝐷
̃𝑒𝑓𝑓 (𝐶̃∞ − 𝐶̃𝑟 )
2𝐷
,
̃𝑒𝑓𝑓 𝑡̃
4𝐷
𝑟̃𝑏 ln ( 2 −𝛾 )
𝑟̃𝑏 𝑒
𝒙 ∈ Γ𝑟𝑏 ,
(9)
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where 𝛾 = 0.57721 is the Euler–Mascheroni constant. Hence, the bulk soil is dealt with
336
entirely by condition (9). This significantly reduces the computational cost whilst allowing
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̃𝑒𝑓𝑓 . We
us to include the averaged geometric details of the bulk soil through the parameter 𝐷
338
note that the boundary condition (9) is singular at 𝑡̃ = 0 . This is regularized by the fact that
16
339
when 𝑡̃ = 0 we have 𝐶̃∞ = 𝐶̃𝑟 . To overcome the difficulties of implementing this we follow
340
the suggestion of (Roose et al., 2001) and modify equation (9) such that for small 𝑡̃ the
341
equation is non-singular, see Appendix B. To summarise the final set of equations we solve
342
in the rhizosphere are
𝜕𝐶̃𝑟
̃∇
̃2 𝐶̃𝑟 ,
=𝐷
𝜕𝑡̃
̃𝒏
̃ 𝐶̃𝑟 = −𝑘̃𝑎 𝐶̃𝑟 + 𝑘̃𝑑 𝐶̃𝑎 ,
̂⋅𝛁
𝐷
𝒙 ∈ Ω𝑟 ,
(10)
𝒙 ∈ Γ𝑠𝑟 ,
(11)
𝒙 ∈ Γ𝑠𝑟 ,
(12)
𝜕𝐶̃𝑎
= 𝑘̃𝑎 𝐶̃𝑟 − 𝑘̃𝑑 𝐶̃𝑎 ,
𝜕𝑡̃
̃𝒏
̃ 𝐶̃𝑟 = 0,
̂⋅𝛁
𝐷
𝒙 ∈ Γ𝑎𝑟 , (13)
̃𝒏
̃ 𝐶̃𝑟 = 𝜆̃𝐶̃𝑟 ,
̂⋅𝛁
𝐷
𝒙 ∈ Γ0𝑟 , (14)
̃𝑒𝑓𝑓 (𝐶̃∞ − 𝐶̃𝑟 )
𝒙 ∈ Γ𝑟𝑏 , (15)
−2𝐷
̃𝑒𝑓𝑓 𝑡̃/𝑟̃𝑏2 + 𝑒 𝛾 )]
2𝑟̃𝑏 + 𝑟̃𝑏 [𝛾 − ln(4𝐷
where, in the linear uptake case, the root hair boundary condition is given by
̃𝒏
̃ 𝐶̃𝑟 =
̂⋅𝛁
𝐷
343
̃𝒏
̃ 𝐶̃𝑟 = 𝜆̃𝐶̃𝑟 ,
̂⋅𝛁
𝐷
344
𝒙 ∈ Γ0ℎ . (16)
In the root hair growth scenario the root hair boundary condition is given by
𝜆̃
̃𝒏
̃ 𝐶̃𝑟 = 𝐶̃𝑟 {1 + tanh[−𝛼̃(𝑟̃ − 𝑔̃𝑟 𝑡̃)]},
̂⋅𝛁
𝐷
2
𝒙 ∈ Γ0ℎ , (17)
345
346
̃𝒏
̃ 𝐶̃𝑟 = 0 is applied on the remaining external boundaries, see Figure 2. Equations
̂⋅𝛁
and 𝐷
347
(10) to (17) describe the uptake of phosphate by roots and root hairs from an infinite bulk of
348
soil.
349
350
Using the STL files generated from the CT data a computational mesh is constructed using
351
the snappyHexMesh package. The mesh is generated from a coarse hexahedral mesh and the
352
STL surface meshes from the imaged geometry. The hexahedral mesh is shaped like a
17
353
segment of initial angle 𝜃̃, height ℎ̃ and radius 𝑟̃𝑏 corresponding to the size of the domain to
354
be modelled. Each of these parameters is increased until the root uptake properties converge.
355
Successive mesh refinements are made where the mesh intersects any geometrical feature
356
described by the STL files, i.e., the mesh is refined about the surfaces of interest. Once the
357
mesh has been refined to hexahedra of side length less than 1 μm about the STL surfaces the
358
regions outside of the water domain are removed. The remaining mesh is then deformed to
359
match to the STL surfaces. Finally the mesh is smoothed to produce a high quality mesh on
360
which the numerical models can be run.
361
solved in OpenFOAM, an open source finite volume code (Jasak et al., 2013), using a
362
modified version of the inbuilt LaplacianFOAM solver to couple the bulk concentration to
363
the sorbed concentration at the soil boundaries. Time stepping is achieved using an implicit
364
Euler method with a variable time stepping algorithm for speed and accuracy. All numerical
365
solutions were obtained using the Iridis 4 supercomputing cluster at the University of
366
Southampton.
The equations are discretised on the mesh and
367
368
3. Results and discussion
369
Using the theory developed in section 2.3 we calculate the bulk soil effective diffusion
370
constant and nutrient uptake properties of a single root with hairs. We consider two different
371
water contents. The first of which is the case in which all the pore space is full of water, i.e.,
372
𝑆 = 1 where 𝑆 is the volumetric water content defined as the volume of water divided by the
373
volume of pore space. Secondly we use the segmented CT image to obtain the water content
374
and air-water interface from the scanned soil. In this case the volumetric water content is 𝑆 =
375
0.33. In addition to being able to parametrize existing models we also calculate the size of
376
the region which needs to be considered for the uptake predicted by our simulations to
18
377
converge. Finally we consider how a growing root-hair system affects the overall uptake
378
properties of the root and root-hairs.
379
380
3.1. Bulk soil properties
381
Before we consider the nutrient uptake properties of the plant root system we first find the
382
homogenized properties of the bulk soil. From the CT image, Figure 1, we select a cube of
383
soil of side length 𝐿𝑚𝑎𝑥 = 2 mm. From this we sub sample a series of geometries of size 𝐿 =
384
2−𝑛/3 𝐿𝑚𝑎𝑥 for 𝑛 = 0,1, … 8, i.e., we repeatedly half the volume of bulk soil considered, and
385
̃𝑒𝑓𝑓 . The key difficulty that arises in
solve equations (A31) to (A34) and (A58) to obtain 𝐷
386
solving these equations is that they require the geometry to be periodic, i.e, it is composed of
387
regularly repeating units. In reality this is not the case. Hence, we have to impose periodicity.
388
Following the method used in (Tracy et al., 2015) we impose periodicity by reflecting the
389
geometry about the three coordinate axes.
390
mathematically through analysis of the symmetries of the problem rather than by physically
391
copying the meshes. This is discussed in more detail in the appendix and results in solving
392
equations (A35) to (A39). Computational resources ranged from 46 Gb across 2 nodes for 3
393
minutes for Saturated case with 𝑛 = 8 to 900Gb across 16 nodes for 45 minutes for the more
394
complex partially saturated case with 𝑛 = 0. As we increase the size of 𝐿 we find that the
395
̃𝑒𝑓𝑓 converges to a fixed value once a sufficiently large sub sample volume is
value of 𝐷
396
included. Figure 4 shows these values for saturated and partially saturated soil. We see that
397
for small 𝐿 the effective diffusion coefficient is a function of 𝐿. However, as 𝐿 is increased
398
the value is seen to level off and become effectively independent of our choice of 𝐿. We
399
̃𝑒𝑓𝑓 has converged and will not change if 𝐿 is increased further.
interpret this to mean that 𝐷
400
̃𝑒𝑓𝑓 has converged if doubling the volume of
As a measure of convergence we assume that 𝐷
We note that this reflection is achieved
19
401
̃𝑒𝑓𝑓 which is smaller
soil considered, corresponding to increasing 𝑛 by 1, causes a change in 𝐷
402
than 5% of its final value. The effective diffusion constant returns a converged value for 𝑛 ≤
403
5, corresponding to 𝐿=0.63 mm, see Figure 4. However, due to the symmetry reduction used
404
in deriving equations (A35) to (A39) this side length must be doubled to obtain the true
405
representative volume. Hence, the actual size of the representative volume is 𝐿=1.26 mm. At
406
this value the computational resources used were 75Gb across 2 nodes for 45 minutes in the
407
saturated case and 150 Gb across 4 nodes for 7 minutes in the partially saturated case. The
408
larger resource requirements are necessary for the partially saturated case due to the increased
409
resolution needed to capture thin water films. We note that, although a representative side
410
length of 𝐿=1.26 mm may seem small the soils used here were sieved to between two
411
relatively close tolerances 1.00 and 1.68 mm. Hence, these soils will be very homogeneous
412
and we would expect the final soil packing to be close to an ideal sphere packing. The value
413
̃𝑒𝑓𝑓 is seen to converge to 𝐷
̃𝑒𝑓𝑓 ≈ 0.56 × 10−9 m2 s −1 for the saturated soil and 𝐷
̃𝑒𝑓𝑓 ≈
of 𝐷
414
0.15 × 10−9 m2 s−1 for the partially saturated soil.
415
saturated case with 𝑛 = 8 because, for samples this small, the air films and soil provided a
416
complete barrier to diffusion and no effective diffusion coefficient could be calculated.
No data is plotted for the partially
417
418
3.2. Nutrient uptake properties
419
̃𝑒𝑓𝑓 at full and partial saturation we now consider the uptake
Using the values obtained for 𝐷
420
of nutrient by the plant root system shown in Figure 4. We are interested in finding the size
421
of the volume of soil about the root which we need to consider geometry explicitly in order to
422
accurately represent the uptake of phosphate by the root and root hairs. To determine the size
423
of this volume we consider a segment of angle 𝜃̃, height ℎ̃ and radius 𝑟̃𝑏 centred on the root.
424
The maximum radius considered is 𝑟̃𝑚𝑎𝑥 = 1.76 mm, 𝜃̃𝑚𝑎𝑥 = 𝜋 and ℎ̃𝑚𝑎𝑥 = 1.8 mm. The
20
425
height, radius and angle of the segment considered are increased from 20% of the maximum
426
domain size in steps of 20% till the maximum is reached corresponding to a total of 125
427
simulations for each of the saturated and partially saturated cases.
428
requirements ranged from 15 Gb across 2 nodes for 15 minutes in the simplest case to 300 Gb
429
across 12 nodes for 16 hours in the most complex case.
Computational
430
431
Typical plots for the nutrient movement in the saturated and partially saturated cases are
432
shown in Figure 5 for a variety of 𝑟̃𝑏 values with the smallest 𝜃̃ and ℎ̃ values used. Initially
433
we expected that there would be two convergence criteria; firstly a sufficiently large root
434
surface area will need to be considered in order for the number of root hairs involved in
435
nutrient uptake to be representative, i.e., the density of root hairs in the sub volume is equal to
436
the density of root measured density of root hairs.
437
considered must be sufficiently large that all the rhizosphere soil is captured. However,
438
whilst there is some variance in the data with root surface area, Figure 7 and error bars in
439
Figure 6, it seems that the main convergence criteria is that a sufficiently large value of 𝑟̃𝑏 ,
440
i.e., the location of the outer rhizosphere-bulk soil boundary, is used. Whilst this conclusion
441
is somewhat surprising it seems that only a relatively small value for 𝜃̃ and ℎ̃ are required to
442
capture enough of the root hair structure that it is representative. On inspection we see that
443
the ratio between the root hair area and root area quickly settles to ≈ 0.5 ± 0.05 for root hair
444
surface areas greater than 0.2 mm2 . Hence, for root surfaces areas above this value the
445
effective density of root hair surface area does not change and the uptake properties are not
446
expected to change significantly.
Secondly the radius of the region
447
21
448
To check the convergence of the simulated phosphorous dynamics we compare the total flux
449
at the root and root hair surfaces as a function of time for the linear uptake conditions. These
450
data are presented in Figure 6 for both saturated and partially saturated conditions for the root
451
uptake and Figure 7 for the root-hair uptake. The simulated nutrient uptake is seen to settle to
452
a rate which is independent of 𝑟̃𝑏 for sufficiently large 𝑟̃𝑏 . We assume that the total uptake
453
has converged once increasing 𝑟̃𝑏 produces a change in uptake of less than 5% of the final
454
value. The radius at which this is seen to occur is 𝑟̃𝑏 = 0.6 𝑟̃𝑚𝑎𝑥 = 1.1 mm for the saturated
455
case and 𝑟̃𝑏 = 0.8 𝑟̃𝑚𝑎𝑥 = 1.4 mm for the partially saturated case. In the saturated case at
456
𝑟̃𝑏 = 0.2 𝑟̃𝑚𝑎𝑥 the cumulative uptake of the root and root hair surfaces is 58.8 ×
457
10−3 μmol mm−2 , this converges to 50.0 × 10−3 μmol mm−2 ± 10−3 μmol mm−2 for 𝑟̃𝑏 ≥
458
0.6 r̃max . In the partially saturated case the cumulative uptake at 𝑟̃𝑏 = 0.2 𝑟̃𝑚𝑎𝑥 is 62.4 ×
459
10−3 μmol mm−2 . This converges to 31 × 10−3 μmol mm−2 ± 10−3 μ mol mm−2 for 𝑟̃𝑏 ≥
460
0.8 r̃max. It is interesting at this point to compare our model with the zero flux approximation
461
used in (Keyes et al., 2013). In this case the total uptake can easily be calculated as the total
462
amount of phosphate available in the geometry at time 𝑡̃ = 0. For the largest saturated
463
domain we have considered this would be a total uptake of 5.54 × 10−4 μmol mm−2, which
464
is significantly less than the uptake measured here (50.0 × 10−3 μmol mm−2 ). We note that
465
the uptake by the root hairs converges in a diminishing oscillatory manner, i.e., the uptake
466
first increase with 𝑟̃𝑏 before decaying to a steady state. This is because at the smallest value
467
of 𝑟̃𝑏 the geometry does not contain the entire length of the root hairs. Hence, the root hair
468
uptake is noticeably lower than the cases 𝑟̃𝑏 ≥ 0.4 r̃max . Once the full hair length is taken
469
into account the convergence is smooth and behaves the same way as the convergence of the
470
root uptake.
471
22
472
As with the effective diffusion constant we find that we do not need to consider a large
473
amount of soil in order to obtain converged properties. Hence, we can observe that the radius
474
required to capture the behaviour of the rhizosphere was dependent on the saturation of the
475
fluid. In the fully saturated case the radius simply needed to be large enough to slightly
476
greater than 100% of the root hair length, i.e., 1.1 mm. However, in the partially saturated
477
case a much larger radius (roughly double the root hair length) was required to capture the
478
corresponding effect on the nutrient motion. We observe that if we using this model we can
479
parameterise mathematically simpler models such as the one developed in (Roose et al.,
480
2001). By using the effective diffusion constant and additional uptake provided by the root
481
hairs to parameterize the model in (Roose et al., 2001) we obtain a cumulative uptake of
482
̃𝑒𝑓𝑓 = 0.56 𝐷
̃ and 𝜆̃𝑒𝑓𝑓 = 1.3𝜆̃ for the uptake parameter in
50.0 × 10−3 μmol mm−2 using 𝐷
483
the saturated case. In the partially saturated case we obtain a cumulative uptake of 31.2×
484
10−3 μmol mm−2 using
485
developed by Roose can be agrees well with our predictions assuming the effective uptake
486
parameter is a function of saturation. In the fully saturated case the root hairs offer an
487
increase in uptake of 30%. In the partially saturated case the effective uptake is decreased.
488
This is in part due to the decrease in root and root hair surface area which is in contact with
489
water, 80% compared to the fully saturated case. Hence, whilst the root hairs considered in
490
this geometry do increase root uptake when comparing image based simulations, they do not
491
significantly alter the uptake parameters in the conventional models.
̃𝑒𝑓𝑓 = 0.15 𝐷
̃ and 𝜆̃𝑒𝑓𝑓 = 0.75𝜆̃ .
𝐷
In both cases the model
492
493
494
3.3. Root hair growth
23
495
We now consider the root hair growth scenario. We neglect the geometrical growth of the
496
root hairs and consider a growing region on which uptake occurs. We use a representative set
497
of values for 𝜃̃, ℎ̃ and 𝑟̃𝑏 for which the simulation has converged. Specifically we take 𝜃̃ =
498
0.6 𝜃̃𝑚𝑎𝑥 , ℎ̃ = 0.6ℎ̃𝑚𝑎𝑥 and 𝑟̃𝑏 = 0.6𝑟̃𝑚𝑎𝑥 for the fully saturated case and 𝜃̃ = 0.6 𝜃̃𝑚𝑎𝑥 , ℎ̃ =
499
0.6ℎ̃𝑚𝑎𝑥 and 𝑟̃𝑏 = 0.8𝑟̃𝑚𝑎𝑥 for the partially saturated case. The root hair growth is shown
500
schematically in Figure 8 for a range of different times. The hair growth parameter 𝑔̃𝑟 = 5 ×
501
10−9 m s−1 is chosen to give the root hairs an effective growth period of two days and the
502
empirical parameter 𝛼̃ = 0.2 × 106 m−1 is chosen such that the 𝛼̃ −1 , i.e., the distance
503
between the active and non-active sections of the root hair is of length 5 μm. Alongside the
504
uptake results for the saturated and partially saturated cases, shown in Figure 9, we have
505
plotted the uptake results for the standard fixed root hairs case. Interestingly, other than the
506
magnitude of the flux, there is little qualitative difference between the saturated and partially
507
saturated cases.
508
509
The main difference between the root hair growth and the static root hair scenario is observed
510
towards the start of the growth period, between 0 and 48 hours. In the growth scenario, as
511
expected, the uptake is dominated by the root for short times. However, as time progresses
512
the root-hairs grow and provide the dominant contribution to nutrient uptake. Once the
513
nutrients in the region immediately adjacent to the root have been taken up the uptake in the
514
growing root hair quickly settles to match the uptake rate for the fixed root hair case. After
515
48 hours have passed and the root hairs are fully developed the fixed and growing root hair
516
scenarios have identical uptake power. Hence, after 48 hours, the difference in cumulative
517
uptake between the two scenarios is less than 1%. This suggests that the fixed root hair
518
provides a good approximation for nutrient uptake and that more detailed modelling of root
24
519
hair growth may not be necessary for timescales longer than 48 hours and shorter than the
520
root hair lifetime.
521
522
4. Conclusions
523
In this paper we have extended the image based modelling in (Keyes et al., 2013) to consider
524
large/infinite volume of soil around the single hairy root. The bulk soil properties are
525
captured from an X-ray CT image based geometry using the method of homogenization to
526
transform the imaged geometry into a homogeneous medium with an effective diffusion
527
constant based on the geometrical impedance offered by the soil. The bulk soil region is then
528
patched onto the rhizosphere using a boundary condition which relates the concentration at
529
the surface of the rhizosphere to the flux into the rhizosphere. The advantage to image based
530
modelling of this type is that it can be used to answer specific questions on the movement of
531
nutrient based on the soil geometry and root hair morphology. This information can then be
532
used to parameterise simplified models and gain understanding of rhizosphere processes.
533
534
The method is tested for two different soil water saturation values by considering bulk and
535
rhizosphere soil samples of different sizes which are increased until the effective transport
536
and uptake properties of the two regions are seen to converge. We found that the key
537
criterion for convergence of nutrient uptake simulations is that a sufficiently large radius of
538
soil about the root is considered. However, we emphasize that this is likely to be dependent
539
on root hair morphology, moisture content and soil type. Hence, convergence checks should
540
be carried out on a smaller scale for different geometries. It is interesting to note that the
541
radius of the segment about the root needed for convergence to occur is dependent on the
25
542
saturation considered. Specifically we need to consider a larger region of soil about the root
543
for lower saturation values 1.4 mm of soil compared to the 1.1 mm of soil for the saturated
544
case. This observation is attributed to chemical effects present in the rhizosphere which
545
cause variation in the fluid properties (Gregory, 2006). In our CT scans these changes would
546
be picked up as a geometrical variation either in the fluid location or the contact angle at the
547
air water interface. Hence, in the fully saturated case we would not see these effects as the
548
specific water location is neglected. By dividing the CT image into regions 𝑟̃ < 0.2𝑟̃𝑚𝑎𝑥 ,
549
0.2𝑟̃𝑚𝑎𝑥 < 𝑟̃ < 0.4𝑟̃𝑚𝑎𝑥 , 0.4𝑟̃𝑚𝑎𝑥 < 𝑟̃ < 0.6𝑟̃𝑚𝑎𝑥 , 0.6𝑟̃𝑚𝑎𝑥 < 𝑟̃ < 0.8𝑟̃𝑚𝑎𝑥 and 0.8𝑟̃𝑚𝑎𝑥 < 𝑟̃ <
550
𝑟̃𝑚𝑎𝑥 and calculating the saturation in each case we see a noticeable decrease in saturation for
551
0.4𝑟̃𝑚𝑎𝑥 < 𝑟̃ < 0.6𝑟̃𝑚𝑎𝑥 , (𝑆 ≈ 0.5 compared to 𝑆 > 0.7 for all other regions). Therefore, in
552
order to capture the geometric impedance created by this region we require 𝑟̃𝑏 > 0.6𝑟̃𝑚𝑎𝑥 .
553
554
In agreement with the paper by (Keyes et al., 2013) we can observe that in both cases the root
555
hairs contribute less than the root surfaces to the nutrient uptake. However, this difference is
556
small and, in terms of order of magnitude, both the hairs and the roots contribute equally to
557
the nutrient uptake. A major advantage to this type of modelling is that these simulations can
558
be used to parameterise existing models which may be computationally less challenging, for
559
example the one in (Roose et al., 2001). In this case we see that the root hairs contribute
560
approximately an extra 30% to the root uptake, an important consideration which will
561
increase the predictive ability of the model developed in (Roose et al., 2001). Again, it may
562
be that in poorly saturated soils, i.e., those with much lower moisture content than the 33%,
563
there will be less water in contact with the root and the root hairs, which can penetrate into
564
the smaller pores within the soil, may provide a more significant contribution. Further
565
investigation across a range of saturation values either based on imaging of samples at
26
566
different water content, or application of a spatially explicit moisture content model such as
567
the one developed in (Daly and Roose, 2015), would be needed to verify this.
568
569
Using the minimum value of ℎ̃, 𝑟̃𝑏 and 𝜃̃ obtained for the two cases we have considered the
570
effect of pseudo root-hair growth on nutrient uptake neglecting the geometrical and
571
mechanical effects of root-hair growth. As the non-active root hairs are still geometrically
572
represented in the domain they may act to weakly impede the diffusion from the soil into the
573
root itself. The root hair growth modelling showed that, although there are differences in the
574
ratio of root and hair uptake rates towards the start of the growth period, i.e., two days.
575
However, on the long timescale, i.e., times larger than two days, the rate of uptake is
576
unaffected by the growing root hairs, see Figure 9.
577
modelling of the root hair growth might not yield different results and the fixed root hair
578
approximation may provide a detailed enough picture for further investigation into the effects
579
of root hairs.
This suggests that more detailed
580
581
The method developed here provides a powerful framework to study properties of the
582
rhizosphere. We have tested the model for a specific geometry obtained from X-ray CT,
583
however we emphasise that the results obtained are suggestive of certain trends they will be
584
dependent on the precise soil type, water content and root hair morphology studied. Hence,
585
studying different root hair distributions and soil structures will allow a greater understanding
586
of how root hairs uptake nutrients to be developed. Further development of this method to
587
couple a spatially explicit water model to the nutrient uptake will allow similar understanding
588
to be developed for water distribution and uptake.
27
589
590
5. Acknowledgements
591
The authors acknowledge the use of the IRIDIS High Performance Computing Facility, and
592
associated support services at the University of Southampton, in the completion of this work.
593
KRD is funded by BB/J000868/1 and by ERC consolidation grant 646809. SDK is funded by
594
and EPSRC Doctoral Prize fellowship, ERC consolidation grant 646809 and funding proved
595
by the Institute for Life Sciences at the University of Southampton. TR would like to
596
acknowledge the receipt of the following funding: Royal Society University Research
597
Fellowship, BBSRC grants BB/C518014, BB/I024283/1, BB/J000868/1, BB/J011460/1,
598
BB/L502625/1, BB/L026058/1, NERC grant NE/L000237/1, EPSRC grant EP/M020355/1
599
and by ERC consolidation grant 646809. All data supporting this study are openly available
600
from the University of Southampton repository at http://eprints.soton.ac.uk/id/eprint/384502
601
Data Accessibility
602
All reconstructed scan data will be available on request by emailing krd103@soton.ac.uk
603
604
A. Diffusion in bulk soil
605
As in the rhizosphere transport is assumed to be by diffusion only
𝜕𝐶̃𝑏
̃∇
̃2 𝐶̃𝑏 ,
=𝐷
𝜕𝑡̃
𝒙 ∈ Ω𝑏 ,
(A1)
606
607
where 𝐶̃𝑏 is the phosphate concentration in the soil solution for the bulk soil. We assume that
608
the concentration and concentration flux are continuous across the rhizosphere-bulk soil
609
boundary
28
𝐶̃𝑟 = 𝐶̃𝑏 ,
𝒙 ∈ Γ𝑟𝑏 , (A2)
̃𝒏
̃ 𝐶̃𝑟 = 𝐷
̃𝒏
̃ 𝐶̃𝑏 ,
̂⋅𝛁
̂⋅𝛁
𝐷
𝒙 ∈ Γ𝑟𝑏 , (A3)
610
611
and consider a linear reaction on the soil surface
̃𝒏
̃ 𝐶̃𝑏 = −𝑘̃𝑎 𝐶̃𝑏 + 𝑘̃𝑑 𝐶̃𝑎 ,
̂⋅𝛁
𝐷
𝒙 ∈ Γ𝑠𝑏 , (A4)
𝜕𝐶̃𝑎
= 𝑘̃𝑎 𝐶̃𝑏 − 𝑘̃𝑑 𝐶̃𝑎 ,
𝜕𝑡̃
𝒙 ∈ Γ𝑠𝑏 , (A5)
612
613
and zero flux on the air water boundary
̃𝒏
̃ 𝐶̃𝑏 = 0,
̂⋅𝛁
𝐷
𝒙 ∈ Γ𝑎𝑏 , (A6)
614
615
In keeping with (Keyes et al., 2013) we non-dimensionalize with the following variables:
616
̃ = [𝐿]𝒙 (for all lengths) and use [𝐶𝑟 ] =
𝐶̃𝑎 = [𝐶𝑎 ]𝐶𝑎 , 𝐶̃𝑟 = [𝐶𝑟 ]𝐶𝑟 , 𝐶̃𝑏 = [𝐶𝑏 ]𝐶𝑏 , 𝑡̃ = [𝑡]𝑡, 𝒙
617
̃𝑚 , [𝐶𝑎 ] = 𝐾𝑚 𝑘𝑎 , [𝑓] = 𝐷𝐾𝑚 and [𝑡] = 𝐿 to obtain
[𝐶𝑏 ] = 𝐾
̃
̃
𝑘
𝐿
𝐷
̃ ̃
̃̃
2
𝑑
𝜕𝐶𝑏
= ∇2 𝐶𝑏 ,
𝜕𝑡
𝐶𝑟 = 𝐶𝑏 ,
𝒙 ∈ Ω𝑏 ,
(A7)
𝒙 ∈ Γ𝑟𝑏 ,
(A8)
̂ ⋅ 𝛁𝐶𝑟 = 𝒏
̂ ⋅ 𝛁Cb ,
𝒏
𝒙 ∈ Γ𝑟𝑏 ,
(A9)
̂ ⋅ 𝛁𝐶𝑏 = −𝛿1 (𝐶𝑏 − 𝐶𝑎 ),
𝒏
𝒙 ∈ Γ𝑠𝑏 ,
(A10)
𝜕𝐶𝑎
= 𝛿2 (𝐶𝑏 − 𝐶𝑎 ),
𝜕𝑡
̂ ⋅ 𝛁𝐶𝑏 = 0,
𝒏
𝒙 ∈ Γ𝑠𝑏 ,
(A11)
𝒙 ∈ Γ𝑎𝑏 .
(A12)
618
29
𝐹̃𝑚 𝐿
,
̃𝑚 𝐷
̃
𝐾
𝑔̃𝑟 𝐿2
̃
𝐷
619
̃ , 𝛿2 = 𝑘̃𝑑 𝐿2 /𝐷
̃, 𝜆 =
Here 𝛿1 = 𝑘̃𝑎 𝐿/𝐷
620
and 𝐿 is a typical length scale of the domain we are interested in. Throughout this paper we
621
take 𝐿 = 10−3 m, i.e. the average radius of the rice/wheat/corn fine roots.
𝛼 = 𝐿𝛼̃ and 𝑔̃𝑟 =
are summarized in Table 2
622
623
Before we derive the boundary condition we consider the homogenization problem in the
624
bulk soil. The key assumption of the homogenization method is that the two length scales, 𝐿𝑥
625
and 𝐿, may be dealt with independently. Therefore, we can expand the vector gradient
626
operator as 𝛁 = 𝜖𝛁𝒙 + 𝛁𝒚 , where 𝛁𝒙 and 𝛁𝒚 are the vector gradient operators on the scale 𝐿𝑥
627
and 𝐿 respectively, 𝜖 = 𝐿 = 10−2 , corresponding to 𝐿𝑥 = 10−1 m . We also define the
628
concentrations in terms of an expansion of the form
𝐿
𝑥
𝐶𝑏 = 𝐶𝑏,0 + 𝜖𝐶𝑏,1 + 𝜖 2 𝐶𝑏,2 + 𝜖 3 𝐶𝑏,3 + 𝑂(𝜖 4 )
(A13)
𝐶𝑎 = 𝐶𝑎,0 + 𝜖𝐶𝑎,1 + 𝜖 2 𝐶𝑎,2 + 𝜖 3 𝐶𝑎,3 + 𝑂(𝜖 4 ).
(A14)
629
630
Physically 𝐶𝑏,0 and 𝐶𝑎,0 can be thought of as the average concentrations on the soil surface
631
and in the solution within the bulk soil domain. The remaining parameters 𝐶𝑏,𝑖 and 𝐶𝑎,𝑖 are
632
corrections to the average concentration which capture small scale variations within the soil
633
and the effects of the linear reactions on the soil surface. We rescale time to capture diffusion
634
over the distance 𝐿𝑥 (𝑡𝑥 = 𝜖 2 𝑡, 𝛿1 = 𝜖𝛿1̅ and 𝛿2 = 𝛿2̅ ) and obtain the scaled equations in the
635
bulk soil
𝜖2
∂Cb
= ∇2 𝐶𝑏 ,
∂t x
̂ ⋅ 𝛁𝐶𝑏 = −𝜖𝛿1̅ (𝐶𝑏 − 𝐶𝑎 ),
𝒏
𝒚 ∈ Ω𝑏 ,
(A15)
𝒙 ∈ Γ𝑠𝑏 ,
(A16)
30
𝜖2
𝜕𝐶𝑎
= 𝛿2̅ (𝐶𝑏 − 𝐶𝑎 ),
𝜕𝑡𝑥
̂ ⋅ 𝛁𝐶𝑏 = 0
𝒏
𝒙 ∈ Γ𝑠𝑏 ,
(A17)
𝒙 ∈ Γ𝑎𝑏 ,
(A18)
periodic in 𝒚.
(A19)
636
637
Substituting equations (A and (A14) into equation (A15) to (A19) and collecting terms in
638
ascending orders of 𝜖 we obtain a cascade of problems for 𝐶𝑏 and 𝐶𝑎 .
639
640
641
A.1 𝑂(𝜖 0 ):
The equations at 𝑂(𝜖 0 ) are
∇2𝑦 𝐶𝑏,0 = 0,
𝒚 ∈ Ω𝑏 ,
(A20)
̂ ⋅ 𝛁𝐲 𝐶𝑏,0 = 0,
𝒏
𝒙 ∈ Γ𝑠𝑏 ,
(A21)
𝛿2̅ (𝐶𝑏,0 − 𝐶𝑎,0 ) = 0,
𝒙 ∈ Γ𝑠𝑏 ,
(A22)
̂ ⋅ 𝛁𝐶𝑏,0 = 0
𝒏
𝒙 ∈ Γ𝑎𝑏 ,
(A23)
periodic in 𝒚.
(A24)
642
643
which have solution 𝐶𝑎,0 = 𝐶𝑏,0 = 𝐶0 (𝒙). Hence, on the timescale of diffusion the linear
644
reactions have already happened and the surface concentration and the bulk concentration are
645
equal and the leading order concentration is independent of the microscale structure.
646
647
648
A.2 𝑂(𝜖 1 ):
Expanding to 𝑂(𝜖 1 ) and using the results from the 𝑂(𝜖 0 ) expansion we obtain
31
∇2𝑦 𝐶𝑏,1 = 0,
𝒚 ∈ Ω𝑏 ,
(A25)
̂ ⋅ 𝛁𝐲 𝐶𝑏,1 + 𝒏
̂ ⋅ 𝛁𝐱 𝐶𝑏,0 = 0,
𝒏
𝒙 ∈ Γ𝑠𝑏 ,
(A26)
𝛿2̅ (𝐶𝑏,1 − 𝐶𝑎,1 ) = 0,
𝒙 ∈ Γ𝑠𝑏 ,
(A27)
̂ ⋅ 𝛁𝐲 𝐶𝑏,1 + 𝒏
̂ ⋅ 𝛁𝐱 𝐶𝑏,0 = 0
𝒏
𝒙 ∈ Γ𝑎𝑏 ,
(A28)
periodic in 𝒚.
(A29)
649
650
This is the standard correction to a diffusion only problem, it has solution 𝐶𝑏,1 = 𝐶𝑎,1 =
651
𝐶1 (𝒙) + 𝐶1̅ (𝒙, 𝒚), where 𝐶1 (𝒙) is a function of 𝒙 only which will be defined at higher order
652
and
3
𝒙 ∈ Γ0 ,
(A30)
∇2𝑦 𝜒𝑝 = 0,
𝒚 ∈ Ω𝑏 ,
(A31)
̂ ⋅ (𝛁𝑦 𝜒𝑝 + 𝒆̂𝑝 ) = 0,
𝒏
𝒚 ∈ Γ𝑠𝑏 ,
(A32)
̂ ⋅ (𝛁𝑦 𝜒𝑝 + 𝒆̂𝑝 ) = 0,
𝒏
𝒚 ∈ Γ𝑎𝑏 , (A33)
𝐶1̅ = ∑ 𝜒𝑝 𝜕𝑥𝑝 𝐶0 ,
𝑝=1
653
654
and 𝜒𝑝 satisfies the cell problem
𝜒𝑝 periodic in 𝒚.
(A34)
655
656
Equations (A31) to (A40) assume that the soil sample is geometrically periodic, i.e., it is
657
composed of regularly repeating units. In reality this is not the case so these equations cannot
658
directly be applied. To overcome this we impose a mathematical periodicity by reflecting the
659
geometry along the three coordinate axes. Reflecting about the 𝑝-th coordinate axis using the
660
substitution 𝒚𝑝 → −𝒚𝑝 in equations (A31) to (A40) we observe that, in order for the
32
661
equations to remain unchanged, we must have 𝜒𝑝 → −𝜒𝑝 . Similarly by reflecting about the
662
𝑞-th coordinate axis using the substitution 𝒚𝑞 → −𝒚𝑞 , for 𝑞 ≠ 𝑝 in equations (A31) to (A40)
663
we see the equations remain unchanged. Hence we infer that 𝜒𝑝 is odd when reflected about
664
the 𝑝 -th coordinate axis and even otherwise. As a result we are able to solve a set of
665
equations on the original domain which respect these symmetries
∇2𝑦 𝜒𝑝 = 0,
𝒚 ∈ Ω𝑏 ,
(A35)
̂ ⋅ (𝛁𝑦 𝜒𝑝 + 𝒆̂𝑝 ) = 0,
𝒏
𝒚 ∈ Γ𝑠𝑏 ,
(A36)
̂ ⋅ (𝛁𝑦 𝜒𝑝 + 𝒆̂𝑝 ) = 0,
𝒏
𝒚 ∈ Γ𝑎𝑏 ,
(A37)
𝜒𝑝 = 0,
𝒆̂𝑞 ⋅ 𝛁𝐲 𝜒𝑝 = 0,
𝒆̂𝑝 ⋅ 𝒚 = ±
1
2
(A38)
1
(A39)
𝒆̂𝑞 ⋅ 𝒚 = ± , 𝑞 ≠ 𝑝
2
666
667
A direct result of this analysis is that the dimensional volume used in equations (A35) to
668
(A39) represents one eighth of the total representative volume as the total volume can be
669
reconstructed by mirroring the result along the three axes.
670
671
A.3 𝑂(𝜖 2 ):
Expanding to 𝑂(𝜖 2 ) and using the previous solutions we obtain
672
∂Cb,0
= ∇2𝑦 𝐶𝑏,2 + 2𝛁𝒙 ⋅ 𝛁𝒚 𝐶𝑏,1 + ∇2𝑥 𝐶𝑏,0 ,
∂t x
𝒚 ∈ Ω𝑏 ,
(A40)
̂ ⋅ 𝛁𝐲 𝐶𝑏,2 + 𝒏
̂ ⋅ 𝛁𝑥 𝐶𝑏,1 = 0,
𝒏
𝒙 ∈ Γ𝑠𝑏 ,
(A41)
𝜕𝐶𝑎,0
= 𝛿2̅ (𝐶𝑏,2 − 𝐶𝑎,2 ),
𝜕𝑡
𝒙 ∈ Γ𝑠𝑏 ,
(A42)
33
̂ ⋅ 𝛁𝐲 𝐶𝑏,2 + 𝒏
̂ ⋅ 𝛁𝑥 𝐶𝑏,1 = 0,
𝒏
𝒙 ∈ Γ𝑎𝑏 ,
periodic in 𝒚.
(A43)
(A44)
673
674
Before we solve for 𝐶𝑏,2 and 𝐶𝑎,2 we require that a solution exists. Integrating and using the
675
divergence theorem we obtain
||Ω𝑏 ||
𝜕𝐶0
= 𝛁𝒙 ⋅ ∫ [𝛁𝑦 𝜒𝑝 ⊗ 𝒆̂𝑝 + 𝐼]𝑑𝑦 𝛁𝒙 𝐶0 ,
𝜕𝑡𝑥
Ω𝑏
(A45)
676
677
where ||Ω𝑏 || = ∫Ω 1 𝑑𝑦. As in the 𝑂(𝜖 1 ) case we obtain a cell problem for 𝐶𝑎,2 and 𝐶𝑏,2 .
678
𝐶𝑏,2 = 𝐶2 (𝒙) + 𝐶2̅ (𝒙, 𝒚), where 𝐶2 (𝒙) is an arbitrary function of 𝒙 which will be defined at
679
higher order and
𝑏
3
3
3
(A46)
𝐶2̅ = ∑ 𝜒𝑝 𝜕𝑥𝑝 𝐶1 + ∑ ∑ 𝜉𝑝𝑞 𝜕𝑥𝑝 𝜕𝑥𝑞 𝐶0 .
𝑝=1
𝑝=1 𝑞=1
680
681
Here 𝜉𝑝𝑞 satisfies
682
||Ωb ||[∇2𝑦 𝜉𝑝𝑞 + 2𝒆̂𝑝 ⋅ 𝛁𝒚 𝜒𝑞 + 𝛿𝑝𝑞 ] = 𝒆̂𝑝 ⋅ 𝐷𝑒𝑓𝑓 𝒆̂𝑞 ,
𝒚 ∈ Ω𝑏 ,
(A47)
̂ ⋅ (𝛁𝑦 𝜉𝑝𝑞 + 𝒆̂𝑝 𝜒𝑞 ) = 0,
𝒏
𝒚 ∈ Γ𝑠𝑏 ,
(A48)
̂ ⋅ (𝛁𝑦 𝜉𝑝𝑞 + 𝒆̂𝑝 𝜒𝑞 ) = 0,
𝒏
𝒚 ∈ Γ𝑎𝑏 , (A49)
𝜒𝑝 periodic in 𝒚
(A50)
683
684
and we have already solved the cell problem for 𝜒𝑞 .
34
685
686
A.4 𝑂(𝜖 3 ):
687
In order to capture the effect of the linear reaction on the soil surface we expand to 𝑂(𝜖 3 ) to
688
obtain
689
∂Cb,1
= ∇2𝑦 𝐶𝑏,3 + 2𝛁𝒙 ⋅ 𝛁𝒚 𝐶𝑏,2 + ∇2𝑥 𝐶𝑏,1 ,
∂t x
𝒚 ∈ Ω𝑏 ,
(A51)
̂ ⋅ 𝛁𝐲 𝐶𝑏,3 + 𝒏
̂ ⋅ 𝛁𝑥 𝐶𝑏,2 = −𝛿1̅ (𝐶𝑏,2 − 𝐶𝑎,2 ),
𝒏
𝒙 ∈ Γ𝑠𝑏 ,
(A52)
𝜕𝐶𝑎,1
= 𝛿2̅ (𝐶𝑏,3 − 𝐶𝑎,3 ),
𝜕𝑡
𝒙 ∈ Γ𝑠𝑏 ,
(A53)
̂ ⋅ 𝛁𝐲 𝐶𝑏,3 + 𝒏
̂ ⋅ 𝛁𝑥 𝐶𝑏,2 = 0
𝒏
periodic in 𝒚.
𝒙 ∈ Γ𝑎𝑏 , (A54)
(A55)
690
691
Applying the solvability condition and using the results from previous expansions we find
692
||Ω𝑏 ||
𝜕𝐶1
= 𝛁𝒙 ⋅ ∫ [𝛁𝑦 𝜒𝑝 ⊗ 𝒆̂𝑝 + 𝐼]𝑑𝑦 𝛁𝒙 𝐶1 +
𝜕𝑡𝑥
Ω𝑏
𝛁𝒙 ⋅ ∫ 𝛁𝑦 𝜉𝑝𝑞 𝑑𝑦𝝏𝑥𝑝 𝜕𝑥𝑞 𝐶0 −
Ω𝑏
(A56)
||Γ||𝛿1̅ 𝜕𝐶0
,
𝜕𝑡𝑥
𝛿2̅
693
694
where ||Γ|| = ∫Γ 1 𝑑𝑦. By writing 𝐶𝑎𝑣 = 𝐶0 + 𝜖𝐶1 we can combine the results at 𝑂(𝜖 0 ) and
695
𝑂(𝜖 1 ) to obtain
𝜕𝐶𝑎𝑣
= 𝛁𝒙 ⋅ 𝐷𝑒𝑓𝑓 𝛁𝒙 𝐶𝑎𝑣 + ϵ𝛁𝒙 ⋅ 𝐻𝑝𝑞 𝜕𝑥𝑝 𝜕𝑥𝑞 𝐶𝑎𝑣 ,
𝜕𝑡𝑥
(A57)
35
696
697
where we have neglected terms ∼ 𝑂(𝜖 2 ). The effective parameters in equation (A57) are
𝐷𝑒𝑓𝑓 = ∫
[𝛁𝜒𝑝 ⊗ 𝒆̂𝑝 + 𝐼]
Ω𝑏 ||Ω𝑏 ||
+ ||Γ||(𝛿1 /𝛿2 )
𝑑𝑦
(A58)
698
699
and
𝐻𝑝𝑞 = ∫
Ω𝑏 ||Ω𝑏 ||
𝛁𝜉𝑝𝑞
+ ||Γ||(𝛿1 /𝛿2 )
𝑑𝑦.
(A59)
700
701
Note, we have added terms at 𝑂(𝜖 2 ) to simplify the form of the final equation. The can be
702
formally calculated by expanding to 𝑂(𝜖 4 ). However, as we are only really interested in
703
knowing the approximate averaged behaviour of diffusion in the soil we neglect the final
704
term in equation (A57).
705
706
707
B. Resupply boundary condition
708
We assume that equation (A57) is valid far from the root and root hair. We now consider an
709
infinite volume of soil surrounding the root and derive an approximate boundary “resupply”
710
condition at the interface between the Rhizosphere and Bulk soil regions to capture the influx
711
of nutrient from outside the explicit domain of interest. We assume that the main direction of
712
diffusion is towards the root. Hence, we can reduce the diffusion equation far from the root
36
713
to a one spatial dimension equation with an isotropic diffusion constant 𝐷𝑒𝑓𝑓 . In the bulk soil
714
we consider the radial diffusion equation
𝜕𝐶𝑏 𝐷𝑒𝑓𝑓 𝜕
𝜕𝐶𝑏
=
(𝑟𝑥
),
𝜕𝑡𝑥
𝑟𝑥 𝜕𝑟𝑥
𝜕𝑟𝑥
(B1)
715
716
subject to the boundary conditions
𝐶𝑏 (𝑟𝑏 , 𝑡𝑥 ) = 𝐶𝑟
(B2)
717
718
and
̂ ⋅ 𝛁𝒙 𝐶𝑟 = 𝐷𝑒𝑓𝑓
𝒏
𝜕𝐶𝑏
.
𝜕𝑟𝑥
(B3)
719
720
We add the condition that 𝐶𝑏 (∞, 𝑡𝑥 ) = 𝐶∞ . We follow the approach of (Roose et al., 2001)
721
and solve equation (B1) in regions close to the root and far from the root. Matching these
722
solutions together at 𝑟𝑥 = 𝑟𝑏 we derive an expression for the flux into the rhizosphere. We
723
assume that, near the rhizosphere boundary, i.e., 𝑟𝑥 ≪ 1 the concentration will approximately
724
be given by the steady state solution
𝐶𝑟 ≈ 𝐶1 + 𝐹𝑟𝑏 ln(𝑟𝑥 )
(B4)
725
726
where 𝐶1 is an arbitrary constant and 𝐹 is the, as yet undefined, flux into the rhizosphere. Far
727
from the rhizosphere we use the similarity solution, see for example (King et al., 2003; Roose
728
et al., 2001),
37
𝑟𝑥2
𝐶𝑏 = 𝐶∞ − 𝐵E1 (
)
4𝐷𝑒𝑓𝑓 𝑡𝑥
(B5)
729
∞ 𝑒 −𝑦
730
where E1 (x) = ∫𝑥
731
𝑟𝑥 = 𝑟𝑏 and assuming continuity of concentration on the rhizosphere-bulk soil boundary we
732
find
𝑦
𝑑𝑦 is the exponential integral. By equating equations (B4) and (B5) at
𝐹=−
2(𝐶∞ − 𝐶𝑟 )
𝑟𝑏 [𝛾 −
(B6)
ln(4𝐷𝑒𝑓𝑓 𝑡𝑥 /𝑟𝑏2 )]
733
734
where 𝛾 = 0.57721 is the Euler–Mascheroni constant. We note that in deriving equation
735
(B6) we have implicitly assumed that 𝐹 does not vary significantly with time. The logarithm
736
in the denominator of equation (B6) is singular at 𝑡𝑥 = 0, hence, for numerical stability we
737
modify the equation slightly.
738
accurate for small time. However, the timescale over which this occurs is short and the effect
739
on the uptake properties is small. Following the method of (Roose et al., 2001) we modify
740
equation (B6) such that 𝐹 = 0 at 𝑡𝑥 = 0,
𝐹=−
This approach means that the boundary condition is not
2(𝐶∞ − 𝐶𝑟 )
(B7)
2 + 𝑟𝑏 [𝛾 − ln(4𝐷𝑒𝑓𝑓 𝑡𝑥 /𝑟𝑏2 + 𝑒 𝛾 )]
741
742
Equation (B7) is used to parameterize the bulk soil and feeds directly into the model for
743
nutrient movement in the rhizosphere in the main body of the paper.
2. References
38
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40
Constant
̃
𝐷
̃
𝐾𝑚
𝐹̃𝑚
𝑘̃𝑑
𝑘̃𝑎
𝑔̃𝑟
𝛼̃
Value
Units
−9
10
6 × 10−3
5 × 10−9
2.569 × 10−4
3.73 × 10−9
5 × 10−9
0.2 × 106
2
−1
m s
mol m−3
mol m−2 s−1
s−1
m s−1
m s−1
m−1
Description
Diffusion constant
Uptake constant
Uptake rate
Desorption rate
Adsorption rate
Root hair growth rate
Root hair cut off rate
Table 1: Physical parameters used in nutrient model.
Parameter
𝛿1
𝛿2
𝜆
𝑔𝑟
𝛼
Expression
k̃ 𝑎 𝐿
̃
𝐷
k̃ d 𝐿2
̃
𝐷
̃F𝑚 𝐿
̃𝑚 𝐷
̃
𝐾
𝑔̃𝑟 𝐿
̃
𝐷
𝛼̃𝐿
Values
3.73 × 10−3
0.2569
5/6
5 × 10−3
200
Description
Dimensionless flux
coefficient
Dimensionless desorption
rate
Dimensionless uptake rate
Dimensionless root hair
growth rate
Dimensionless root hair
cut off rate
Table 2: Dimensionless parameters used in rescaled diffusion model.
41
Figure captions
Figure 1: Image based geometry: (A) X-ray CT image of the roots, root hairs, soil and pore water. (B) shows one of the
rhizosphere segments considered with size described by the angle 𝜽, height h and radius r. (C) shows a cube of bulk soil
of side length L.
Figure 2: Schematic of the different regions used in the mathematical formulation of the problem. The bulk and
rhizosphere regions are highlighted with the modelling domains 𝛀𝒓 , 𝚪𝟎𝒓 , 𝚪𝟎𝒉 , 𝚪𝒔𝒓 , 𝚪𝒂𝒓 and 𝚪𝒓𝒃 labeled for the rhizosphere
and 𝛀𝒃 , 𝚪𝒔𝒃 and 𝚪𝒂𝒃 labeled for the bulk soil.
Figure 3: Schematic for root hair growth. The root hair geometry remains unchanged throughout the simulation. As the
simulation progresses the active portion of the root hair grows. The streamlines show an estimation of the nutrient
transport as affected by the presence of the root hair. At t=0 the root hair is non-active and acts only to geometrically
impede the transport of nutrient. At t=1 day the root hair has grown to half its final length and acts partly to take up
nutrients and partly as a geometrical impedance. Finally at t=2 days the root hair is fully grown.
Figure 4: Effective diffusion coefficient as a function of L , the side length of the cube. The results are plotted for the
saturated case, S=1, corresponding to the pore space being completely full of water and partially saturated, S=0.33,
corresponding to 1/3 of the pore space being occupied by water. As L is increased the saturated and unsaturated
effective diffusion properties are seen to converge to a steady value corresponding to the bulk diffusion coefficient.
Figure 5: Typical plots obtained from simulation for, (1A) to (1E). the saturated case and, (2A) to (2E), the partially
saturated case. The five images in each case correspond to 𝜽 = 𝟎. 𝟐 𝜽𝒎𝒂𝒙 = 𝝅/𝟓 and 𝒉 = 𝟎. 𝟐𝒉𝒎𝒂𝒙 = 𝟎. 𝟑𝟔 𝐦𝐦 for (A)
𝒓 = 𝟎. 𝟐𝒓𝒎𝒂𝒙 = 𝟎. 𝟑𝟓𝟐 𝐦𝐦 , (B) 𝒓 = 𝟎. 𝟒𝒓𝒎𝒂𝒙 = 𝟎. 𝟕𝟎𝟒 𝐦𝐦 , (C) 𝒓 = 𝟎. 𝟔𝒓𝒎𝒂𝒙 = 𝟏. 𝟎𝟓𝟔 𝐦𝐦 , (D) 𝒓 = 𝟎. 𝟖𝒓𝒎𝒂𝒙 =
𝟏. 𝟒𝟎𝟖 𝐦𝐦 and (E) 𝒓 = 𝟏. 𝟎𝒓𝒎𝒂𝒙 = 𝟏. 𝟕𝟔 𝐦𝐦 . The streamlines show the effective diffusion paths with red
corresponding to high nutrient concentration and blue corresponding to lower nutrient concentration.
Figure 6: Log-linear plot of flux into (A) the root hairs in the saturated case, (B) the roots in the saturated case, (C) the
root hairs in the unsaturated case and (D) the roots in the unsaturated case. The lines show the average flux over a
period of one week for the five different radii considered up to a maximum of r_max=1.76 mm. As r is increased the
uptake profile is seen to converge. The figure insets show a zoomed in section of the same curve with error bars
removed for clarity.
Figure 7: Total uptake into the root and root hair system averaged over all time points. Plot shows average uptake over
the different radii considered with standard deviation error bars as a function of root surface area.
Figure 8: Root growth scenario for four different times: The top left shows the initial condition ( 0 hours), the top right
shows the hair growth after 16 hours, the bottom left shows hair growth after 32 hours and the bottom right shows hair
growth after 48 hours.
Figure 9: Flux at the root and hair surfaces for the root growth scenario in comparison to the fixed root hair scenario for
the saturated and partially saturated geometries (logarithmic scale). The combined uptake of the root and hair is also
plotted.
42