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Image based modeling of nutrient movement in and around the rhizosphere
Author list: Keith R. Daly1, Samuel D. Keyes1, Shakil Masum2, Tiina Roose§1
Author affiliations:
1 Bioengineering Sciences Research Group, Faculty of Engineering and Environment,
University of Southampton, University Road, SO17 1BJ Southampton, United Kingdom,
2 School of Energy, Environment and Agrifood, Cranfield University, Cranfield, MK43 0AL,
United Kingdom,
§ Corresponding author: Faculty of Engineering and Environment, University of
Southampton, Hampshire, SO17 1BJ. United Kingdom, email: [email protected]
Summary
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
In this paper we develop a spatially explicit model for nutrient uptake by root hairs based on
X-ray Computed Tomography images of rhizosphere soil structure. This work extends our
previous study (Keyes et al., 2013) to larger domains and, hence, is valid for longer times.
Unlike the model used in (Keyes et al., 2013), which considered only a small region of soil
about the root, we consider an effectively infinite volume of bulk soil about the rhizosphere.
We ask the question “at what distance away from root surfaces do the specific structural
features of root-hair and soil aggregate morphology not matter because average properties
start dominating the nutrient transport?” The resulting model captures bulk and rhizosphere
soil properties by considering representative volumes of soil far from the root and adjacent to
the root respectively. By increasing the size of the volumes we consider, the diffusive
impedance of the bulk soil and root uptake are seen to converge. We do this for two different
values of water content which provides two different geometries. We find that the size of
region for which the nutrient uptake properties converge to a fixed value is dependent on the
water saturation. In the fully saturated case the region of soil which we need to consider is
only of radius 1.1 mm for poorly soil-mobile species such as phosphate. However, in the
case of a partially saturated medium (relative saturation 0.3) we find that a radius of 1.4 mm
is necessary. This suggests that, in addition to the geometrical properties of the rhizosphere,
there is an additional effect on soil moisture properties which extends further from the root
and may relate to other chemical changes in the rhizosphere which are not explicitly included
in our model.
Keywords:
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Rhizosphere, X-ray CT, structural imaging, phosphate, plant-soil interaction
Abbreviations:
CT – Computer Tomography
Short title for page headings: Modelling nutrient uptake
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Introduction
The uptake of phosphate and other low mobility nutrients is essential for plant growth and,
hence, global food security (Barber, 1984; Nye and Tinker, 1977; Tinker and Nye, 2000).
However, the overuse of inorganic fertilizers has caused an accumulation of phosphate in
European soils which potentially causes eutrophication (Gahoonia and Nielsen, 2004).
Hence, there is a need to optimize the efficiency of phosphate uptake not only to increase
crop yields, but also to minimize the detrimental effects of excess phosphate on the
environment.
Plants are known to uptake nutrients both through their roots and root hairs (Datta et al.,
2011). Root hairs are thought to play a significant role in the uptake of poorly soil-mobile
nutrients such as phosphate, water uptake, plant stability and microbial interactions (Datta et
al., 2011; Gahoonia and Nielsen, 2004). In order to quantify the role of root hairs in the
uptake of poorly soil-mobile nutrients, such as phosphate, a detailed understanding of the
root-hair morphology and the region of soil around the root, known as the rhizosphere
(Hiltner, 1904), is required. This is because the properties of soil in the rhizosphere are
thought to be significantly different due to soil microbial interactions and compaction of soil
by the root (Aravena et al., 2010; Aravena et al., 2014; Daly et al., 2015; Dexter, 1987;
Whalley et al., 2005).
Early models for nutrient movement in soil treated the rhizosphere as a volume averaged
continuum (Barber, 1984; Ma et al., 2001; Nye and Tinker, 1977). More recently models for
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
were based around the technique of homogenization (Hornung, 1997; Pavliotis and Stuart,
2008), a multi-scale technique which enables field scale equations to be derived and
parametrized based on underlying continuum models.
To obtain a better understanding of processes occurring in the rhizosphere, in comparison to
the surrounding bulk soil, non-invasive measurements of the plant root and soil structure are
essential (George et al., 2014; Hallett et al., 2013). Three dimensional imaging of plant roots
in-situ using X-ray Computed Tomography (CT) is a rapidly growing field (Aravena et al.,
2010; Aravena et al., 2014; Hallett et al., 2013; Keyes et al., 2013; Tracy et al., 2010). Using
these techniques sub-micron resolutions can be achieved enabling the root hair morphology
to be visualized (Keyes et al., 2013).
In addition to direct in-situ visualization of plant-soil interaction, the use of X-ray CT also
provides the means to apply numerical models describing the diffusion of nutrients directly to
the imaged geometries (Aravena et al., 2010; Aravena et al., 2014; Daly et al., 2015; Keyes et
al., 2013; Tracy et al., 2015). This approach has been used to quantify the effectiveness of
root hairs in saturated soil conditions (Keyes et al., 2013) using a diffusion model originally
developed for mycorrihzal fungi (Schnepf et al., 2011). However, the study in (Keyes et al.,
2013) considered a small volume of soil adjacent to the root. This volume extended c.
300 μm from the root surface and had a zero flux boundary condition on the outer surface.
Hence, once the phosphate immediately adjacent to the root was depleted the uptake of
nutrients into the root stopped. This, whilst accurate for the experimental situation
considered, limited the time frame for which the model was applicable to c. 3 hours.
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In this paper we extend the work of (Keyes et al., 2013) to include both bulk and rhizosphere
soil. The key aims of this study are to develop an image based modelling approach which is
applicable to root hairs surrounded by a large/infinite bulk of soil, to understand how root
hairs contribute to nutrient uptake at different soil water content and to estimate the effects of
root hair growth on nutrient uptake. We have chosen to model a single root in a large/infinite
volume of soil, rather than multiple roots competing for nutrients for a number of reasons.
Firstly, we want to compare our model to established models for nutrient uptake which are
applied in this geometry (Roose et al., 2001). Secondly, whilst some models consider root-
root competition, (Barber, 1984), these models are applied in a cylindrical geometry. Hence,
any competition is provided by a ring of roots at a distance from the root under consideration.
In order to accurately capture roots at a given root density additional assumptions and
approximations would be required. Finally, our aim is to study the role of root hairs on the
uptake of a single root. Hence, we have chosen to study a geometry which captures and
isolates this effect. The modelling strategy developed in this paper will provide a framework
for future modelling studies to consider how different root hair morphologies influence
uptake. In addition it will provide a new level of understanding in to the role or root hairs in
nutrient uptake as the root system begins to develop.
The bulk soil properties are derived using multi-scale homogenization combined with image
based modelling in a similar manner to (Daly et al., 2015). The resulting equations are
solved analytically and are patched to an explicit image based geometry for the rhizosphere
using a time dependent boundary condition. Reducing the bulk soil to a single boundary
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
soil-mobile species such as phosphate, potassium and zinc. The model describes phosphate
uptake by roots and root hairs in an effectively infinite volume of bulk soil whilst capturing at
high precision any changes in the soil adjacent to the plant roots. In this case an effectively
infinite volume of soil corresponds to any volume of soil which is sufficiently large that the
phosphate depletion region about the root does not reach the edge of the domain considered.
Using this model we are able to parameterize upscaled models for nutrient motion in soil
(Zygalakis et al., 2011).
This paper is arranged as follows: in section 2 we describe the plant growth, imaging and
modelling approaches, in section 3 we discuss the results of numerical simulation and show
how these models can be used to perform an image based study of root hair growth, finally in
section 4 we discuss our results and show how these models might be further developed. The
technical description of the mathematical models used in this paper is provided in appendix A
and appendix B.
2. Methods
2.1. Plant growth
We study a rice genotype Oryza Sativa cv. Varyla provided by the Japan International
Research Centre for Agricultural Sciences (JIRCAS). The seeds were heat-treated at 50℃ for 48h to break dormancy and standardise germination. Germination was carried out at
23±1℃ for 2 days between moistened sheets of Millipore filter paper. The growth medium
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was a sand-textured Eutric Cambisol soil collected from a surface plot at Abergwyngregyn,
North Wales (53o14’N, 4o01’W), the soil organic matter was 7%, full details are given in
(Lucas and Jones, 2006), soil B. The soil was sieved to <5 mm, autoclaved and air dried at
23±1℃ for 2 days. The dried soil was sieved to between bounds of 1680 μm and 1000 μm,
producing a well-aggregated, textured growth medium.
Growth took place in a controlled growth environment (Fitotron SGR, Weiss-Gallenkamp,
Loughborough, UK) for a period of 14 days. Growth conditions were 23±1℃ and 60 %
humidity for 16 hours (day) and 18±1℃ and 55 % humidity for 8 hours (night) with both
ramped over 30 minutes.
The seminal roots were guided by a specially designed growth environment fabricated in
ABS plastic using an UP! 3D printer (PP3DP, China). The resulting assay is shown in
Supplementary Figure 1 and further details are provided in (Keyes et al., 2015) for details.
The morphology of the growth environment was used to guide the roots into 7 syringe barrels
(root chambers) which can then be detached for imaging once the growth stage is complete.
The lower portion including the root chambers was housed in a foil-wrapped 50 ml centrifuge
tube which occludes light during the growth period.
2.2. Imaging
Imaging of root chambers was conducted at the TOMCAT beamline on the X02DA port of
the Swiss Light Source, a 3rd generation synchrotron at the Paul Scherrer Institute, Villigen,
Switzerland. The imaging resolution was 1.2 μm, and a monochromatic beam was employed
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
volumes using a custom back-projection algorithm implementing a Parzen filter. The
resulting volume size was 2560x2560x2180 voxels.
The reconstructed volumes were analysed using a multi-pass approach (Keyes et al., 2015).
Whilst root hairs can be visualised in situ for dry soils they are challenging to distinguish
from some background phases even at high resolution. Specifically, the portions of root hairs
which traverse water filled regions of the pore space are indistinguishable from the water.
The visible sections of the root hairs were extracted manually using a graphical input tablet
(Cintiq 24, WACOM) and a software package that allowed interpolation between lofted
cross-sections (Avizo FIRE 8, FEI Company, Oregon, USA). A number of these sections
represent an incomplete root hair segmentation since proportions of the hair paths can be
occluded by fluid rich phases and pore water. Full root hair paths were extrapolated from the
segmented sections of partially visible hairs using an algorithm that simulates root hair
growth (Keyes et al., 2015). The algorithm was parameterised from the root hairs of plants
(Oryza Sativa cv. Varyla) which were grown in rhizo-boxes with dimensions 30x30x2 cm,
constructed of transparent polycarbonate with a removable front-plate, and wrapped in foil to
occlude light. Hydration during the growth period was maintained via capillary rise from a
tray of water, with occasional top-watering. The plants were grown in a glasshouse, with
temperature ranging from 20-32 ℃ over the growth period. The root systems of 1 month old
specimens were gently freed from soil using running water. One fine primary and one coarse
primary were randomly sampled before manual removal of the remaining soil from the roots.
Each was cut into a number of subsections of 2 cm length for analysis, each being imaged
using an Olympus BX50 optical microscope. After stitching all images together 5 sub-regions
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
FIJI using a poly-line tool to generate a set of 220 hair lengths. A normal distribution was
fitted to these data and used to parameterize the hair-growth algorithm. The fitted and
sampled distributions are shown in Supplementary Figure 1. The mean of experimental
lengths was 171.56 µm, compared to the mean fitted lengths: 173.67 µm. The standard
deviation of the experimental lengths was 72.88 µm compared to the standard deviation of
the fitted lengths 73.44 µm.
We observe that root hair density measured via Synchrotron Radiation X-ray CT data is
generally slightly higher than for microscopy, probably due to the lack of damage artefacts.
For the data in this study, the density is ≈ 77 hairs mm−1, compared to 64 hairs mm−1 measured
via bright-field microscopy in a study of similar rice varieties grown under similar conditions
(Wissuwa and Ae, 2001). This discrepancy is, in part, because the in situ hairs are often part-
occluded by fluid and colloidal phases, it is usually only possible to obtain partial
measurements in the Synchrotron CT data. This is offset to some degree by the lack of
damage artefacts when compared to root-washing and microscopy, but the length values are
low compared to most literature sources. However, in this study we explore the implications
of soil domain size on nutrient uptake and have used hair data attained via quantification of
soil-grown roots using bright field microscopy.
The root geometry was extracted in the same manner, manually defining the root cross-
section from slices sampled (at a spacing of 120 µm) along the root growth axis. The soil
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
training parameters (Hall et al., 2009).
The segmented geometry was used to produce 3D surface meshes, using the ScanIP software
package (Simpleware Ltd., Exeter, United Kingdom). The root, hairs, gas fluid phase and
soil minerals were exported as separate .STL files which provide the geometries on which the
numerical models can be tested.
2.3. Modelling
We model the uptake of phosphate by a single root and root hairs in an unbounded partially
saturated geometry, see Figure 1. In order to do this we extend the model described in
(Keyes et al., 2013). In (Keyes et al., 2013) the authors considered a segment of root and
surrounding soil. On the outer boundary of the segment a zero flux condition was applied.
By definition this condition prevents any influx of nutrient from outside the domain
considered. Hence, the total possible uptake is limited to the nutrient available from within
the initial segment. This was adequate for the experimental situation considered, i.e., a pot of
radius c. 300 μm, but is clearly not applicable for isolated roots in soil columns of radius
larger than 300 μm . We note that this is a slightly different scenario from the one studied by
Barber (Barber, 1984) in which the domain around the root was considered to be half the size
of the inter-root spacing. In other words, Barbers model considered multiple roots effectively
introducing root competition. In this work we consider a single isolated root and how that is
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
soil. However, as considering an infinite domain explicitly is computationally unfeasible, we
break the domain into two regions: the rhizosphere and bulk soil. We assume that the bulk
soil, which is far from the root, may be considered as a homogeneous medium with an
effective phosphate diffusion constant. The effective diffusion properties of phosphate in the
bulk soil are derived from the segmented CT data using the method of homogenization
(Pavliotis and Stuart, 2008; Zygalakis et al., 2011; Zygalakis and Roose, 2012). The nutrient
flux in the bulk soil is then patched to the rhizosphere domain using a boundary condition
which simulates an effective infinite region of bulk soil beyond the rhizosphere. In this
section we introduce the equations and summarize the model which results from these
assumptions. Full details of the derivation have been included in appendices A and B.
We assume that phosphate can only diffuse in the fluid domain, i.e., no diffusion occurs in
the air filled portion of the pore space. Therefore, we only present equations in the fluid
region and its associated boundaries. However, we make no assumptions about the
geometrical details regarding the air filled portion of pore space. If, from the CT data, there
is air in contact with soil or root material this simply acts to reduce the surface area over
which phosphate can be exchanged. We define two regions of fluid filled pore space we
denote the rhizosphere by Ωr as the region explicitly considered near to the root. Physically
we may think of this as an approximation to the rhizosphere. The second region is Ωb, the
region of bulk soil outside of the rhizosphere, see Figure 2. As different boundary conditions
need to be applied on each of the pore water interfaces we adopt the following naming
convention. We take the interface between the two regions, located at a distance ~rb from the
centre of the root to be Γrb. The root and root-hair surfaces are defined as Γ0 r, and Γ0 h
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respectively, the soil particle surface to be Γ sr and Γ sb in region Ωr and Ωb respectively.
Finally we define the air water interfaces as Γ ar and Γ ab in region Ωr and Ωb respectively.
The method we use to determine phosphate movement is different in each region. Phosphate
movement in the rhizosphere is calculated based on a spatially explicit model obtained from
CT data. Phosphate movement in the bulk soil is calculated analytically based on the solution
to a spatially explicit numerical model in a representative volume of bulk soil. We present
each of these models separately.
2.3.1. Rhizosphere
In the rhizosphere we assume that phosphate moves by diffusion only
∂~C r
∂ ~t=~D~∇2~C r ,
x∈Ωr , (1)
where ~Cr is the phosphate concentration in the soil solution and ~D is the diffusion constant of
phosphate in the soil solution. The phosphate is assumed to bind to the soil particles based on
linear first order kinetics
~D n⋅~∇~C r=−~ka~Cr+
~kd~Ca , x∈ Γsr , (2)
∂~C a
∂ ~t=~ka
~C r−~kd
~Ca ,x∈ Γsr , (3)
where ~k a and ~k d are the adsorption and desorption rates respectively, ~Ca is the nutrient
concentration on the soil surface and n is the outward pointing surface normal vector. Here
we have assumed that the nutrient concentration of the soil can be represented as a surface
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
nutrient diffusion across the air water boundary
~D n⋅~∇~C r=0 , x∈ Γar . (4)
We consider uptake of phosphate from the fluid only. On the root surface this follows a
linear uptake condition
~D n⋅~∇~C r=~λ~C r , x∈ Γ0 r . (5)
where ~λ is the nutrient uptake rate on the root and root-hair surfaces. For comparison with
the paper by Keyes et al (Keyes et al., 2013) we have chosen the absorbing power as ~λ=~Fm~Km
,
consistent with Nye and Tinker. Equation (5) is the small concentration equivalent of the
Michaelis Menten condition, in which ~Fm is the maximum rate of uptake and ~Km is the
concentration when the uptake is half the maximum.
The flux on the root hair surface is given by one of two different scenarios: the first is linear
uptake
~D n⋅~∇~C r=~λ~C r , x∈ Γ0 h . (6)
The second case we consider is a pseudo time-dependent uptake which simulates linear
uptake for a growing root-hair. We have termed this pseudo root-hair growth as we do not
consider the geometrical and mechanical effects of root hair growth explicitly. Rather, we
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assume that at ~t=0 the root hairs are present, but do not contribute to uptake. As the
simulation progresses the root hairs are assumed to grow at a certain rate. We model this by
assuming an active region of root hair which grows outward from the root; this is illustrated
in Figure 3. We write the root-hair uptake condition as
~D n⋅~∇~C r=~λ2~C
r{1+ tanh [−~α (~r−~gr
~t ) ]}, x∈ Γ0h , (7)
where ~r is the axial distance along the root hair, ~gr is the root hair growth rate, ~α is an
empirical parameter which controls the sharpness of the transition between the root hairs
being “on”, i.e., actively taking up nutrient and “off”, i.e., not actively taking up nutrient.
The adsorption and desorption constants (~k a∧~k d) are taken from (Keyes et al., 2013) where
they were derived using standard soil tests (Giesler and Lundström, 1993; Murphy and Riley,
1962). The remaining parameters used in equations (1) to (5) were obtained from the
literature (Barber, 1984; Tinker and Nye, 2000) and are summarized in Table 1.
2.3.2. Rhizosphere bulk interface
Due to the success of averaged equations in bulk soil, for example Darcy’s law and Richards’
equation (Van Genuchten, 1980), the precise details of the geometry are less important in
determining the total phosphate movement and average properties are sufficient to be
considered. Therefore, rather than consider the explicit details of the geometry we derive an
effective diffusion constant based on a representative soil volume in absence of roots. This is
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
concentration is only weakly dependent on the precise structure of the soil. Secondly a set of
equations, often called the cell problem, are derived which determine the local variation in
concentration due to the representative soil volume. Finally an averaged equation is derived
which describes the effective rate of diffusion in terms of an effective diffusion constant
~Deff =~D D eff where Deff is calculated from the bulk soil geometry (equations (A31) to (A34)
and (A58)) and describes the impedance to diffusion offered by the soil. Full details of how
Deff is derived are provided in Appendix A. Hence, in the bulk the diffusion of phosphate is
described by
∂~Cb
∂ ~t=~Deff
~∇2~Cb ,(8)
where ~Cb is the nutrient concentration in the bulk soil. The advantage of equation (8) is that
we do not need to explicitly consider the soil geometry in the bulk soil domain. In itself this
significantly reduces the computational cost for finite but large domains. However, rather
than solving equation (8) numerically we find an approximate analytic solution for an infinite
bulk soil domain subject to the conditions ~Cb=~Cr ( t ) for x∈ Γrb and ~Cb →~C∞ as ~r →∞. Using
this solution we are able to define a relationship between concentration and flux at the edge
of the rhizosphere. The result is a condition which simulates the presence of an infinite
region of bulk soil at the rhizosphere soil domain boundary
~D n⋅~∇~C r=−2~Deff (
~C∞−~C r)
~rb ln ( 4~D eff~t
~rb2 e−γ )
,x∈ Γrb , (9)
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where γ=0.57721 is the Euler–Mascheroni constant. Hence, the bulk soil is dealt with
entirely by condition (9). This significantly reduces the computational cost whilst allowing
us to include the averaged geometric details of the bulk soil through the parameter ~Deff . We
note that the boundary condition (9) is singular at ~t=0. This is regularized by the fact that
when ~t=0 we have ~C∞=~C r. To overcome the difficulties of implementing this we follow the
suggestion of (Roose et al., 2001) and modify equation (9) such that for small ~t the equation
is non-singular, see Appendix B. To summarise the final set of equations we solve in the
rhizosphere are
∂~C r
∂ ~t=~D~∇2~C r ,
x∈Ωr , (10)
~D n⋅~∇~C r=−~ka~Cr+
~kd~Ca , x∈ Γsr , (11)
∂~Ca
∂ ~t=~ka
~C r−~kd
~Ca ,x∈ Γsr , (12)
~D n⋅~∇~C r=0 , x∈ Γar , (13)
~D n⋅~∇~C r=~λ~C r , x∈ Γ0 r , (14)
~D n⋅~∇~C r=−2~Deff (
~C∞−~C r)
2 ~rb+~rb [γ−ln (4~Deff~t /~r b
2+eγ ) ]x∈ Γrb, (15)
where, in the linear uptake case, the root hair boundary condition is given by
~D n⋅~∇~C r=~λ~C r , x∈ Γ0 h . (16)
In the root hair growth scenario the root hair boundary condition is given by
~D n⋅~∇~C r=~λ2~C
r{1+ tanh [−~α (~r−~gr
~t ) ]}, x∈ Γ0 h , (17)
and ~D n⋅~∇~C r=0 is applied on the remaining external boundaries, see Figure 2. Equations
(10) to (17) describe the uptake of phosphate by roots and root hairs from an infinite bulk of
soil.
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Using the STL files generated from the CT data a computational mesh is constructed using
the snappyHexMesh package. The mesh is generated from a coarse hexahedral mesh and the
STL surface meshes from the imaged geometry. The hexahedral mesh is shaped like a
segment of initial angle ~θ, height ~h and radius ~rb corresponding to the size of the domain to
be modelled. Each of these parameters is increased until the root uptake properties converge.
Successive mesh refinements are made where the mesh intersects any geometrical feature
described by the STL files, i.e., the mesh is refined about the surfaces of interest. Once the
mesh has been refined to hexahedra of side length less than 1 μm about the STL surfaces the
regions outside of the water domain are removed. The remaining mesh is then deformed to
match to the STL surfaces. Finally the mesh is smoothed to produce a high quality mesh on
which the numerical models can be run. The equations are discretised on the mesh and
solved in OpenFOAM, an open source finite volume code (Jasak et al., 2013), using a
modified version of the inbuilt LaplacianFOAM solver to couple the bulk concentration to
the sorbed concentration at the soil boundaries. Time stepping is achieved using an implicit
Euler method with a variable time stepping algorithm for speed and accuracy. All numerical
solutions were obtained using the Iridis 4 supercomputing cluster at the University of
Southampton.
3. Results and discussion
Using the theory developed in section 2.3 we calculate the bulk soil effective diffusion
constant and nutrient uptake properties of a single root with hairs. We consider two different
water contents. The first of which is the case in which all the pore space is full of water, i.e.,
S=1 where S is the volumetric water content defined as the volume of water divided by the
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volume of pore space. Secondly we use the segmented CT image to obtain the water content
and air-water interface from the scanned soil. In this case the volumetric water content is
S=0.33. In addition to being able to parametrize existing models we also calculate the size
of the region which needs to be considered for the uptake predicted by our simulations to
converge. Finally we consider how a growing root-hair system affects the overall uptake
properties of the root and root-hairs.
3.1. Bulk soil properties
Before we consider the nutrient uptake properties of the plant root system we first find the
homogenized properties of the bulk soil. From the CT image, Figure 1, we select a cube of
soil of side length Lmax=2 mm. From this we sub sample a series of geometries of size
L=2−n/3 Lmax for n=0,1 , …8, i.e., we repeatedly half the volume of bulk soil considered, and
solve equations (A31) to (A34) and (A58) to obtain ~Deff . The key difficulty that arises in
solving these equations is that they require the geometry to be periodic, i.e, it is composed of
regularly repeating units. In reality this is not the case. Hence, we have to impose periodicity.
Following the method used in (Tracy et al., 2015) we impose periodicity by reflecting the
geometry about the three coordinate axes. We note that this reflection is achieved
mathematically through analysis of the symmetries of the problem rather than by physically
copying the meshes. This is discussed in more detail in the appendix and results in solving
equations (A35) to (A39). Computational resources ranged from 46 Gb across 2 nodes for 3
minutes for Saturated case with n=8 to 900Gb across 16 nodes for 45 minutes for the more
complex partially saturated case with n=0. As we increase the size of L we find that the
value of ~Deff converges to a fixed value once a sufficiently large sub sample volume is
included. Figure 4 shows these values for saturated and partially saturated soil. We see that
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for small L the effective diffusion coefficient is a function of L. However, as L is increased
the value is seen to level off and become effectively independent of our choice of L. We
interpret this to mean that ~Deff has converged and will not change if L is increased further.
As a measure of convergence we assume that ~Deff has converged if doubling the volume of
soil considered, corresponding to increasing n by 1, causes a change in ~Deff which is smaller
than 5% of its final value. The effective diffusion constant returns a converged value for n≤ 5,
corresponding to L=0.63 mm, see Figure 4. However, due to the symmetry reduction used in
deriving equations (A35) to (A39) this side length must be doubled to obtain the true
representative volume. Hence, the actual size of the representative volume is L=1.26 mm.
At this value the computational resources used were 75Gb across 2 nodes for 45 minutes in
the saturated case and 150 Gb across 4 nodes for 7 minutes in the partially saturated case.
The larger resource requirements are necessary for the partially saturated case due to the
increased resolution needed to capture thin water films. We note that, although a
representative side length of L=1.26 mm may seem small the soils used here were sieved to
between two relatively close tolerances 1.00 and 1.68 mm. Hence, these soils will be very
homogeneous and we would expect the final soil packing to be close to an ideal sphere
packing. The value of ~Deff is seen to converge to ~Deff ≈ 0.56 ×10−9m2 s−1 for the saturated soil
and ~Deff ≈ 0.15 ×10−9m2 s−1 for the partially saturated soil. No data is plotted for the partially
saturated case with n=8 because, for samples this small, the air films and soil provided a
complete barrier to diffusion and no effective diffusion coefficient could be calculated.
3.2. Nutrient uptake properties
Using the values obtained for ~Deff at full and partial saturation we now consider the uptake of
nutrient by the plant root system shown in Figure 4. We are interested in finding the size of
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the volume of soil about the root which we need to consider geometry explicitly in order to
accurately represent the uptake of phosphate by the root and root hairs. To determine the size
of this volume we consider a segment of angle ~θ, height ~h and radius ~rb centred on the root.
The maximum radius considered is ~rmax=1.76 mm, ~θmax=π and ~hmax=1.8 mm. The height,
radius and angle of the segment considered are increased from 20% of the maximum domain
size in steps of 20% till the maximum is reached corresponding to a total of 125 simulations
for each of the saturated and partially saturated cases. Computational requirements ranged
from 15 Gb across 2 nodes for 15 minutes in the simplest case to 300 Gb across 12 nodes for
16 hours in the most complex case.
Typical plots for the nutrient movement in the saturated and partially saturated cases are
shown in Figure 5 for a variety of ~rb values with the smallest ~θ and ~h values used. Initially
we expected that there would be two convergence criteria; firstly a sufficiently large root
surface area will need to be considered in order for the number of root hairs involved in
nutrient uptake to be representative, i.e., the density of root hairs in the sub volume is equal to
the density of root measured density of root hairs. Secondly the radius of the region
considered must be sufficiently large that all the rhizosphere soil is captured. However,
whilst there is some variance in the data with root surface area, Figure 7 and error bars in
Figure 6, it seems that the main convergence criteria is that a sufficiently large value of ~rb,
i.e., the location of the outer rhizosphere-bulk soil boundary, is used. Whilst this conclusion
is somewhat surprising it seems that only a relatively small value for ~θ and ~h are required to
capture enough of the root hair structure that it is representative. On inspection we see that
the ratio between the root hair area and root area quickly settles to ≈ 0.5 ± 0.05 for root hair
surface areas greater than 0.2 mm2. Hence, for root surfaces areas above this value the
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effective density of root hair surface area does not change and the uptake properties are not
expected to change significantly.
To check the convergence of the simulated phosphorous dynamics we compare the total flux
at the root and root hair surfaces as a function of time for the linear uptake conditions. These
data are presented in Figure 6 for both saturated and partially saturated conditions for the root
uptake and Figure 7 for the root-hair uptake. The simulated nutrient uptake is seen to settle to
a rate which is independent of ~rb for sufficiently large ~rb. We assume that the total uptake
has converged once increasing ~rb produces a change in uptake of less than 5% of the final
value. The radius at which this is seen to occur is ~rb=0.6 ~rmax=1.1 mm for the saturated case
and ~rb=0.8 ~rmax=1.4mm for the partially saturated case. In the saturated case at ~rb=0.2~rmax
the cumulative uptake of the root and root hair surfaces is 58.8 ×10−3 μmol mm−2, this
converges to 50.0 ×10−3 μmol mm−2± 10−3 μmol mm−2 for ~rb ≥ 0.6 ~rmax. In the partially saturated
case the cumulative uptake at ~rb=0.2~rmax is 62.4 × 10−3 μmol mm−2 . This converges to
31 ×10−3 μmolmm−2 ±10−3 μ mol mm−2 for ~rb ≥ 0.8 ~rmax. It is interesting at this point to
compare our model with the zero flux approximation used in (Keyes et al., 2013). In this
case the total uptake can easily be calculated as the total amount of phosphate available in the
geometry at time ~t=0. For the largest saturated domain we have considered this would be a
total uptake of 5.54 ×10−4 μmolmm−2, which is significantly less than the uptake measured
here (50.0 ×10−3 μmolmm−2). We note that the uptake by the root hairs converges in a
diminishing oscillatory manner, i.e., the uptake first increase with ~rb before decaying to a
steady state. This is because at the smallest value of ~rb the geometry does not contain the
entire length of the root hairs. Hence, the root hair uptake is noticeably lower than the cases
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~rb ≥ 0.4 ~rmax. Once the full hair length is taken into account the convergence is smooth and
behaves the same way as the convergence of the root uptake.
As with the effective diffusion constant we find that we do not need to consider a large
amount of soil in order to obtain converged properties. Hence, we can observe that the radius
required to capture the behaviour of the rhizosphere was dependent on the saturation of the
fluid. In the fully saturated case the radius simply needed to be large enough to slightly
greater than 100% of the root hair length, i.e., 1.1 mm. However, in the partially saturated
case a much larger radius (roughly double the root hair length) was required to capture the
corresponding effect on the nutrient motion. We observe that if we using this model we can
parameterise mathematically simpler models such as the one developed in (Roose et al.,
2001). By using the effective diffusion constant and additional uptake provided by the root
hairs to parameterize the model in (Roose et al., 2001) we obtain a cumulative uptake of
50.0 ×10−3 μmolmm−2 using ~Deff =0.56~D and ~λeff=1.3 ~λ for the uptake parameter in the
saturated case. In the partially saturated case we obtain a cumulative uptake of 31.2
×10−3 μmolmm−2 using ~Deff =0.15~D and ~λeff=0.75~λ. In both cases the model developed by
Roose can be agrees well with our predictions assuming the effective uptake parameter is a
function of saturation. In the fully saturated case the root hairs offer an increase in uptake of
30%. In the partially saturated case the effective uptake is decreased. This is in part due to
the decrease in root and root hair surface area which is in contact with water, 80% compared
to the fully saturated case. Hence, whilst the root hairs considered in this geometry do
increase root uptake when comparing image based simulations, they do not significantly alter
the uptake parameters in the conventional models.
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3.3. Root hair growth
We now consider the root hair growth scenario. We neglect the geometrical growth of the
root hairs and consider a growing region on which uptake occurs. We use a representative set
of values for ~θ, ~h and ~rb for which the simulation has converged. Specifically we take
~θ=0.6 ~θmax, ~h=0.6 ~hmax and ~rb=0.6 ~rmax for the fully saturated case and ~θ=0.6 ~θmax,
~h=0.6 ~hmax
and ~rb=0.8 ~rmax for the partially saturated case. The root hair growth is shown schematically
in Figure 8 for a range of different times. The hair growth parameter ~gr=5 ×10−9 ms−1 is
chosen to give the root hairs an effective growth period of two days and the empirical
parameter ~α=0.2× 106 m−1 is chosen such that the ~α−1, i.e., the distance between the active
and non-active sections of the root hair is of length 5 μm. Alongside the uptake results for the
saturated and partially saturated cases, shown in Figure 9, we have plotted the uptake results
for the standard fixed root hairs case. Interestingly, other than the magnitude of the flux,
there is little qualitative difference between the saturated and partially saturated cases.
The main difference between the root hair growth and the static root hair scenario is observed
towards the start of the growth period, between 0 and 48 hours. In the growth scenario, as
expected, the uptake is dominated by the root for short times. However, as time progresses
the root-hairs grow and provide the dominant contribution to nutrient uptake. Once the
nutrients in the region immediately adjacent to the root have been taken up the uptake in the
growing root hair quickly settles to match the uptake rate for the fixed root hair case. After
48 hours have passed and the root hairs are fully developed the fixed and growing root hair
scenarios have identical uptake power. Hence, after 48 hours, the difference in cumulative
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uptake between the two scenarios is less than 1%. This suggests that the fixed root hair
provides a good approximation for nutrient uptake and that more detailed modelling of root
hair growth may not be necessary for timescales longer than 48 hours and shorter than the
root hair lifetime.
4. Conclusions
In this paper we have extended the image based modelling in (Keyes et al., 2013) to consider
large/infinite volume of soil around the single hairy root. The bulk soil properties are
captured from an X-ray CT image based geometry using the method of homogenization to
transform the imaged geometry into a homogeneous medium with an effective diffusion
constant based on the geometrical impedance offered by the soil. The bulk soil region is then
patched onto the rhizosphere using a boundary condition which relates the concentration at
the surface of the rhizosphere to the flux into the rhizosphere. The advantage to image based
modelling of this type is that it can be used to answer specific questions on the movement of
nutrient based on the soil geometry and root hair morphology. This information can then be
used to parameterise simplified models and gain understanding of rhizosphere processes.
The method is tested for two different soil water saturation values by considering bulk and
rhizosphere soil samples of different sizes which are increased until the effective transport
and uptake properties of the two regions are seen to converge. We found that the key
criterion for convergence of nutrient uptake simulations is that a sufficiently large radius of
soil about the root is considered. However, we emphasize that this is likely to be dependent
on root hair morphology, moisture content and soil type. Hence, convergence checks should
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be carried out on a smaller scale for different geometries. It is interesting to note that the
radius of the segment about the root needed for convergence to occur is dependent on the
saturation considered. Specifically we need to consider a larger region of soil about the root
for lower saturation values 1.4 mm of soil compared to the 1.1 mm of soil for the saturated
case. This observation is attributed to chemical effects present in the rhizosphere which
cause variation in the fluid properties (Gregory, 2006). In our CT scans these changes would
be picked up as a geometrical variation either in the fluid location or the contact angle at the
air water interface. Hence, in the fully saturated case we would not see these effects as the
specific water location is neglected. By dividing the CT image into regions ~r<0.2 ~rmax,
0.2 ~rmax<~r<0.4 ~rmax, 0.4 ~r max<~r<0.6 ~rmax, 0.6 ~rmax <~r<0.8 ~rmax and 0.8 ~rmax <~r<~rmax and
calculating the saturation in each case we see a noticeable decrease in saturation for
0.4 ~r max<~r<0.6 ~rmax, (S ≈ 0.5 compared to S>0.7 for all other regions). Therefore, in order to
capture the geometric impedance created by this region we require ~rb>0.6~rmax.
In agreement with the paper by (Keyes et al., 2013) we can observe that in both cases the root
hairs contribute less than the root surfaces to the nutrient uptake. However, this difference is
small and, in terms of order of magnitude, both the hairs and the roots contribute equally to
the nutrient uptake. A major advantage to this type of modelling is that these simulations can
be used to parameterise existing models which may be computationally less challenging, for
example the one in (Roose et al., 2001). In this case we see that the root hairs contribute
approximately an extra 30% to the root uptake, an important consideration which will
increase the predictive ability of the model developed in (Roose et al., 2001). Again, it may
be that in poorly saturated soils, i.e., those with much lower moisture content than the 33%,
there will be less water in contact with the root and the root hairs, which can penetrate into
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the smaller pores within the soil, may provide a more significant contribution. Further
investigation across a range of saturation values either based on imaging of samples at
different water content, or application of a spatially explicit moisture content model such as
the one developed in (Daly and Roose, 2015), would be needed to verify this.
Using the minimum value of ~h, ~rb and ~θ obtained for the two cases we have considered the
effect of pseudo root-hair growth on nutrient uptake neglecting the geometrical and
mechanical effects of root-hair growth. As the non-active root hairs are still geometrically
represented in the domain they may act to weakly impede the diffusion from the soil into the
root itself. The root hair growth modelling showed that, although there are differences in the
ratio of root and hair uptake rates towards the start of the growth period, i.e., two days.
However, on the long timescale, i.e., times larger than two days, the rate of uptake is
unaffected by the growing root hairs, see Figure 9. This suggests that more detailed
modelling of the root hair growth might not yield different results and the fixed root hair
approximation may provide a detailed enough picture for further investigation into the effects
of root hairs.
The method developed here provides a powerful framework to study properties of the
rhizosphere. We have tested the model for a specific geometry obtained from X-ray CT,
however we emphasise that the results obtained are suggestive of certain trends they will be
dependent on the precise soil type, water content and root hair morphology studied. Hence,
studying different root hair distributions and soil structures will allow a greater understanding
of how root hairs uptake nutrients to be developed. Further development of this method to
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couple a spatially explicit water model to the nutrient uptake will allow similar understanding
to be developed for water distribution and uptake.
5. Acknowledgements
The authors acknowledge the use of the IRIDIS High Performance Computing Facility, and
associated support services at the University of Southampton, in the completion of this work.
KRD is funded by BB/J000868/1 and by ERC consolidation grant 646809. SDK is funded by
and EPSRC Doctoral Prize fellowship, ERC consolidation grant 646809 and funding proved
by the Institute for Life Sciences at the University of Southampton. TR would like to
acknowledge the receipt of the following funding: Royal Society University Research
Fellowship, BBSRC grants BB/C518014, BB/I024283/1, BB/J000868/1, BB/J011460/1,
BB/L502625/1, BB/L026058/1, NERC grant NE/L000237/1, EPSRC grant EP/M020355/1
and by ERC consolidation grant 646809. All data supporting this study are openly available
from the University of Southampton repository at http://eprints.soton.ac.uk/id/eprint/384502
Data Accessibility
All reconstructed scan data will be available on request by emailing [email protected]
A. Diffusion in bulk soil
As in the rhizosphere transport is assumed to be by diffusion only
∂~C b
∂ ~t=~D~∇2~Cb ,
x∈Ωb , (A1)
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where ~Cb is the phosphate concentration in the soil solution for the bulk soil. We assume that
the concentration and concentration flux are continuous across the rhizosphere-bulk soil
boundary
~Cr=~Cb, x∈ Γrb , (A2)
~D n⋅~∇~C r=~D n ⋅~∇~Cb , x∈ Γrb , (A3)
and consider a linear reaction on the soil surface
~D n⋅~∇~Cb=−~ka~Cb+
~k d~Ca , x∈ Γsb , (A4)
∂~C a
∂ ~t=~ka
~Cb−~kd
~Ca ,x∈ Γsb , (A5)
and zero flux on the air water boundary
~D n⋅~∇~C b=0 , x∈ Γab , (A6)
In keeping with (Keyes et al., 2013) we non-dimensionalize with the following variables:
~Ca=[C a ]C a, ~Cr=[C r ]C r,
~Cb=[Cb ]Cb, ~t=[ t ] t , ~x= [ L ] x (for all lengths) and use [C r ]=[Cb ]=~K m,
[Ca ]=~Km
~ka~kd
, [ f ]=~D~Km
L and [ t ]=L2
~D to obtain
∂C b
∂ t=∇2Cb ,
x∈Ωb , (A7)
C r=C b , x∈ Γrb , (A8)
n ⋅∇C r=n ⋅∇Cb , x∈ Γrb , (A9)
n ⋅∇Cb=−δ1 (Cb−Ca ) , x∈ Γsb , (A10)
∂C a
∂ t=δ2 ( Cb−Ca ) ,
x∈ Γsb , (A11)
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n ⋅∇Cb=0 , x∈ Γab . (A12)
Here δ 1=~k a L/~D , δ 2=
~k d L2 /~D , λ=~Fm L~K m
~D, α=L~α and ~gr=
~gr L2
~D are summarized in Table 2
and L is a typical length scale of the domain we are interested in. Throughout this paper we
take L=10−3m, i.e. the average radius of the rice/wheat/corn fine roots.
Before we derive the boundary condition we consider the homogenization problem in the
bulk soil. The key assumption of the homogenization method is that the two length scales, Lx
and L, may be dealt with independently. Therefore, we can expand the vector gradient
operator as ∇=ϵ∇x+∇y, where ∇x and ∇ y are the vector gradient operators on the scale L x
and L respectively, ϵ=LLx
=10−2, corresponding to Lx=10−1 m. We also define the
concentrations in terms of an expansion of the form
Cb=Cb ,0+ϵ C b ,1+ϵ 2Cb , 2+ϵ3 Cb ,3+O (ϵ 4 ) (A13)
Ca=Ca ,0+ϵ C a ,1+ϵ 2Ca , 2+ϵ3 Ca ,3+O (ϵ 4 ) . (A14)
Physically Cb ,0 and Ca ,0 can be thought of as the average concentrations on the soil surface
and in the solution within the bulk soil domain. The remaining parameters Cb ,i and Ca ,i are
corrections to the average concentration which capture small scale variations within the soil
and the effects of the linear reactions on the soil surface. We rescale time to capture diffusion
over the distance L x (t x=ϵ2 t, δ 1=ϵ δ 1 and δ 2=δ 2) and obtain the scaled equations in the bulk
soil
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ϵ 2 ∂ Cb
∂ t x=∇2Cb ,
y∈Ωb , (A15)
n ⋅∇Cb=−ϵ δ1 (Cb−Ca ) , x∈ Γsb , (A16)
ϵ 2 ∂ Ca
∂ t x=δ
2(C b−Ca ) ,
x∈ Γsb , (A17)
n ⋅∇Cb=0 x∈ Γab , (A18)
periodic in y . (A19)
Substituting equations (A and (A14) into equation (A15) to (A19) and collecting terms in
ascending orders of ϵ we obtain a cascade of problems for Cb and Ca.
A.1 O(ϵ 0):
The equations at O(ϵ 0) are
∇ y2 Cb , 0=0 , y∈Ωb , (A20)
n ⋅∇ y Cb , 0=0 , x∈ Γsb , (A21)
δ 2 (Cb , 0−Ca ,0 )=0 , x∈ Γsb , (A22)
n ⋅∇Cb , 0=0 x∈ Γab , (A23)
periodic in y . (A24)
which have solution Ca ,0=Cb , 0=C 0(x). Hence, on the timescale of diffusion the linear
reactions have already happened and the surface concentration and the bulk concentration are
equal and the leading order concentration is independent of the microscale structure.
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A.2 O(ϵ 1):
Expanding to O(ϵ 1) and using the results from the O(ϵ 0) expansion we obtain
∇ y2 Cb , 1=0 , y∈Ωb , (A25)
n ⋅∇ y Cb , 1+ n⋅ ∇x Cb ,0=0 , x∈ Γsb , (A26)
δ 2 (Cb , 1−Ca , 1 )=0 , x∈ Γsb , (A27)
n ⋅∇ y Cb , 1+ n⋅ ∇x Cb ,0=0 x∈ Γab , (A28)
periodic in y. (A29)
This is the standard correction to a diffusion only problem, it has solution
Cb ,1=Ca ,1=C1 ( x )+C1(x , y ), where C1 ( x ) is a function of x only which will be defined at
higher order and
C1=∑p=1
3
χ p ∂xpC0 ,
x∈ Γ0 , (A30)
and χ p satisfies the cell problem
∇ y2 χ p=0 , y∈Ωb , (A31)
n ⋅(∇¿¿ y χ p+ ep)=0 ,¿ y∈Γ sb , (A32)
n ⋅(∇¿¿ y χ p+ ep)=0 ,¿ y∈Γ ab , (A33)
χ p periodic in y . (A34)
Equations (A31) to (A40) assume that the soil sample is geometrically periodic, i.e., it is
composed of regularly repeating units. In reality this is not the case so these equations cannot
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directly be applied. To overcome this we impose a mathematical periodicity by reflecting the
geometry along the three coordinate axes. Reflecting about the p-th coordinate axis using the
substitution y p →− y p in equations (A31) to (A40) we observe that, in order for the equations
to remain unchanged, we must have χ p →− χ p. Similarly by reflecting about the q-th
coordinate axis using the substitution yq →− yq, for q ≠ p in equations (A31) to (A40) we see
the equations remain unchanged. Hence we infer that χ p is odd when reflected about the p-th
coordinate axis and even otherwise. As a result we are able to solve a set of equations on the
original domain which respect these symmetries
∇ y2 χ p=0 , y∈Ωb , (A35)
n ⋅(∇¿¿ y χ p+ ep)=0 ,¿ y∈Γ s b , (A36)
n ⋅(∇¿¿ y χ p+ ep)=0 ,¿ y∈Γ ab , (A37)
χ p= 0, e p⋅ y=± 12
(A38)
eq ⋅∇ y χ p= 0, eq ⋅ y=± 12
, q ≠ p (A39)
A direct result of this analysis is that the dimensional volume used in equations (A35) to
(A39) represents one eighth of the total representative volume as the total volume can be
reconstructed by mirroring the result along the three axes.
A.3 O(ϵ 2):
Expanding to O(ϵ2) and using the previous solutions we obtain
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∂C b , 0
∂ t x=∇ y
2 Cb ,2+2∇x ⋅∇ y Cb ,1+∇x2 Cb , 0 ,
y∈Ωb , (A40)
n ⋅∇ y Cb , 2+ n⋅ ∇x Cb ,1=0 , x∈ Γsb , (A41)
∂Ca , 0
∂ t=δ
2(Cb ,2−Ca , 2) ,
x∈ Γsb , (A42)
n ⋅∇ y Cb , 2+ n⋅ ∇x Cb ,1=0 , x∈ Γab , (A43)
periodic in y . (A44)
Before we solve for C b ,2 and Ca ,2 we require that a solution exists. Integrating and using the
divergence theorem we obtain
||Ωb||∂C0
∂ t x=∇x ⋅∫
Ωb
[∇ y χ p⊗ e p+ I ]dy∇x C0 ,(A45)
where ¿|Ωb|∨¿∫Ωb
1dy. As in the O(ϵ 1) case we obtain a cell problem for Ca ,2 and Cb ,2.
Cb ,2=C2 (x )+C2(x , y ), where C2 ( x ) is an arbitrary function of x which will be defined at
higher order and
C2=∑p=1
3
χ p ∂xpC1+∑
p=1
3
∑q=1
3
ξ pq ∂x p∂xq
C0 .(A46)
Here ξ pq satisfies
¿|Ωb|∨[∇¿¿ y2 ξ pq+2 e p⋅ ∇ y χq+δ pq ]=e p⋅Deff eq ,¿ y∈Ωb , (A47)
n ⋅(∇¿¿ y ξ pq+e p χ q)=0 ,¿ y∈Γ sb , (A48)
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n ⋅(∇¿¿ y ξ pq+e p χ q)=0 ,¿ y∈Γ ab , (A49)
χ p periodic in y (A50)
and we have already solved the cell problem for χq.
A.4 O(ϵ 3):
In order to capture the effect of the linear reaction on the soil surface we expand to O(ϵ 3) to
obtain
∂Cb , 1
∂ t x=∇ y
2 C b ,3+2∇x ⋅ ∇y Cb ,2+∇x2Cb , 1 ,
y∈Ωb , (A51)
n ⋅∇ y Cb , 3+ n⋅ ∇x Cb ,2=−δ1 (Cb ,2−Ca , 2) , x∈ Γsb , (A52)
∂C a , 1
∂ t=δ
2(Cb ,3−Ca ,3 ) ,
x∈ Γsb , (A53)
n ⋅∇ y Cb , 3+ n⋅ ∇x Cb ,2=0 x∈ Γab , (A54)
periodic in y . (A55)
Applying the solvability condition and using the results from previous expansions we find
||Ωb||∂C1
∂ t x=∇x ⋅∫
Ωb
[∇y χ p⊗ ep+ I ] dy∇x C 1+¿(A56)
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∇x ⋅∫Ωb
∇ y ξ pq dy ∂x p∂xq
C0−||Γ||δ 1
δ 2
∂ C0
∂ t x,
where ¿|Γ|∨¿∫Γ
1 dy. By writing Cav=C0+ϵ C1 we can combine the results at O(ϵ 0) and
O(ϵ 1) to obtain
∂C av
∂ t x=∇x ⋅Deff ∇x Cav+ϵ ∇x ⋅H pq ∂x p
∂xqCav ,
(A57)
where we have neglected terms ∼O(ϵ 2). The effective parameters in equation (A57) are
Deff =∫Ωb
[∇ χ p⊗ ep+ I ]||Ωb||+¿|Γ|∨(δ 1/δ2)
dy(A58)
and
H pq=∫Ωb
∇ ξpq
||Ωb||+¿|Γ|∨(δ1/δ 2)dy .
(A59)
Note, we have added terms at O(ϵ2) to simplify the form of the final equation. The can be
formally calculated by expanding to O(ϵ 4). However, as we are only really interested in
knowing the approximate averaged behaviour of diffusion in the soil we neglect the final
term in equation (A57).
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B. Resupply boundary condition
We assume that equation (A57) is valid far from the root and root hair. We now consider an
infinite volume of soil surrounding the root and derive an approximate boundary “resupply”
condition at the interface between the Rhizosphere and Bulk soil regions to capture the influx
of nutrient from outside the explicit domain of interest. We assume that the main direction of
diffusion is towards the root. Hence, we can reduce the diffusion equation far from the root
to a one spatial dimension equation with an isotropic diffusion constant Deff . In the bulk soil
we consider the radial diffusion equation
∂Cb
∂t x=
Deff
r x
∂∂ r x
(r x
∂ Cb
∂ r x), (B1)
subject to the boundary conditions
Cb ( rb ,t x )=Cr (B2)
and
n ⋅∇x Cr=Deff
∂Cb
∂ rx.
(B3)
We add the condition that Cb ( ∞, t x )=C∞. We follow the approach of (Roose et al., 2001) and
solve equation (B1) in regions close to the root and far from the root. Matching these
solutions together at r x=rb we derive an expression for the flux into the rhizosphere. We
assume that, near the rhizosphere boundary, i.e., r x≪1 the concentration will approximately
be given by the steady state solution
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C r ≈C1+F rb ln (r x ) (B4)
where C1 is an arbitrary constant and F is the, as yet undefined, flux into the rhizosphere. Far
from the rhizosphere we use the similarity solution, see for example (King et al., 2003; Roose
et al., 2001),
Cb=C∞−B E1( r x2
4 D eff t x) (B5)
where E1 ( x )=∫x
∞ e− y
ydy is the exponential integral. By equating equations (B4) and (B5) at
r x=rb and assuming continuity of concentration on the rhizosphere-bulk soil boundary we
find
F=−2(C∞−C r)
rb [ γ−ln (4 D eff t x/rb2 ) ]
(B6)
where γ=0.57721 is the Euler–Mascheroni constant. We note that in deriving equation (B6)
we have implicitly assumed that F does not vary significantly with time. The logarithm in
the denominator of equation (B6) is singular at t x=0, hence, for numerical stability we
modify the equation slightly. This approach means that the boundary condition is not
accurate for small time. However, the timescale over which this occurs is short and the effect
on the uptake properties is small. Following the method of (Roose et al., 2001) we modify
equation (B6) such that F=0 at t x=0,
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F=−2(C∞−C r)
2+rb [γ−ln (4 D eff t x/rb2+eγ ) ]
(B7)
Equation (B7) is used to parameterize the bulk soil and feeds directly into the model for
nutrient movement in the rhizosphere in the main body of the paper.
2. References
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Aravena, J. E., Berli, M., Ruiz, S., Suárez, F., Ghezzehei, T. A. and Tyler, S. W. (2014). Quantifying coupled deformation and water flow in the rhizosphere using X-ray microtomography and numerical simulations. Plant and Soil 376, 95-110.
Barber, S. A. (1984). Soil Nutrient Bioavailability: A Mechanistic Approach: Wiley-Blackwell.
Daly, K. R., Mooney, S., Bennett, M., Crout, N., Roose, T. and Tracy, S. (2015). Assessing the influence of the rhizosphere on soil hydraulic properties using X-ray Computed Tomography and numerical modelling. Journal of experimental botany 66, 2305-2314.
Daly, K. R. and Roose, T. (2015). Homogenization of two fluid flow in porous media.
Datta, S., Kim, C. M., Pernas, M., Pires, N. D., Proust, H., Tam, T., Vijayakumar, P. and Dolan, L. (2011). Root hairs: development, growth and evolution at the plant-soil interface. Plant and Soil 346, 1-14.
Dexter, A. (1987). Compression of soil around roots. Plant and Soil 97, 401-406.Gahoonia, T. S. and Nielsen, N. E. (2004). Root traits as tools for creating
phosphorus efficient crop varieties. Plant and Soil 260, 47-57.George, T. S., Hawes, C., Newton, A. C., McKenzie, B. M., Hallett, P. D. and
Valentine, T. A. (2014). Field phenotyping and long-term platforms to characterise how crop genotypes interact with soil processes and the environment. Agronomy 4, 242-278.
Giesler, R. and Lundström, U. (1993). Soil solution chemistry: effects of bulking soil samples. Soil science society of America journal 57, 1283-1288.
Gregory, P. (2006). Roots, rhizosphere and soil: the route to a better understanding of soil science? European Journal of Soil Science 57, 2-12.
Hall, M., Frank, E., Holmes, G., Pfahringer, B., Reutemann, P. and Witten, I. H. (2009). The WEKA data mining software: an update. ACM SIGKDD explorations newsletter 11, 10-18.
Hallett, P. D., Karim, K. H., Glyn Bengough, A. and Otten, W. (2013). Biophysics of the vadose zone: From reality to model systems and back again. Vadose Zone Journal 12.
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Roose, T., Fowler, A. and Darrah, P. (2001). A mathematical model of plant nutrient uptake. Journal of mathematical biology 42, 347-360.
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Constant Value Units Description~D 10−9 m2 s−1 Diffusion constant~Km 6×10−3 molm−3 Uptake constant~Fm 5 ×10−9 mol m−2 s−1 Uptake rate~k d 2.569 ×10−4 s−1 Desorption rate~k a 3.73 ×10−9 m s−1 Adsorption rate~gr 5 ×10−9 m s−1 Root hair growth rate~α 0.2 ×106 m−1 Root hair cut off rate
Table 1: Physical parameters used in nutrient model.
Parameter Expression Values Descriptionδ 1
~ka L~D
3.73 ×10−3 Dimensionless flux coefficient
δ 2~kd L2
~D0.2569 Dimensionless desorption
rateλ ~Fm L
~Km~D
5/6 Dimensionless uptake rate
gr~gr L~D
5 ×10−3 Dimensionless root hair growth rate
α ~α L 200 Dimensionless root hair cut off rate
Table 2: Dimensionless parameters used in rescaled diffusion model.
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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 , Γ0 r, Γ0h, Γ sr, Γ ar and Γrb labeled for the
rhizosphere and Ωb, Γ sb and Γ ab 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=0.2θmax=π /5 and h=0.2 hmax=0.36 mm for (A)
r=0.2 rmax=0.352mm, (B) r=0.4 rmax=0.704 mm, (C) r=0.6 rmax=1.056 mm, (D)
r=0.8 rmax=1.408mm and (E) r=1.0 rmax=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.
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