Linking rhizosphere processes across scales: Opinion

Abstract

Purpose
Simultaneously interacting rhizosphere processes determine emergent plant behaviour, including growth, transpiration, nutrient uptake, soil carbon storage and transformation by microorganisms. However, these processes occur on multiple scales, challenging modelling of rhizosphere and plant behaviour. Current advances in modelling and experimental methods open the path to unravel the importance and interconnectedness of those processes across scales.

Methods
We present a series of case studies of state-of-the art simulations addressing this multi-scale, multi-process problem from a modelling point of view, as well as from the point of view of integrating newly available rhizosphere data and images.

Results
Each case study includes a model that links scales and experimental data to explain and predict spatial and temporal distribution of rhizosphere components. We exemplify the state-of-the-art modelling tools in this field: image-based modelling, pore-scale modelling, continuum scale modelling, and functional-structural plant modelling. We show how to link the pore scale to the continuum scale by homogenisation or by deriving effective physical parameters like viscosity from nano-scale chemical properties. Furthermore, we demonstrate ways of modelling the links between rhizodeposition and plant nutrient uptake or soil microbial activity.

Conclusion
Modelling allows to integrate new experimental data across different rhizosphere processes and scales and to explore more variables than is possible with experiments. Described models are tools to test hypotheses and consequently improve our mechanistic understanding of how rhizosphere processes impact plant-scale behaviour. Linking multiple scales and processes including the dynamics of root growth is the logical next step for future research.

Full text

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https://doi.org/10.1007/s11104-022-05306-7
OPINION PAPER
Linking rhizosphere processes acrossscales: Opinion
A.Schnepf· A.Carminati· M.A.Ahmed· M.Ani· P.Benard· J.Bentz· M.Bonkowski · M.Knott·
D.Diehl· P.Duddek· E.Kröner· M.Javaux· M.Landl· E.Lehndorff· E.Lippold· A.Lieu·
C.W.Mueller · E.Oburger· W.Otten· X.Portell· M.Phalempin· A.Prechtel· R.Schulz·
J.Vanderborght· D.Vetterlein
Received: 20 July 2021 / Accepted: 16 January 2022
© The Author(s) 2022
Methods We present a series of case studies of
state-of-the art simulations addressing this multi-
scale, multi-process problem from a modelling point
of view, as well as from the point of view of integrat-
ing newly available rhizosphere data and images.
Results Each case study includes a model that links
scales and experimental data to explain and predict
spatial and temporal distribution of rhizosphere com-
ponents. We exemplify the state-of-the-art modelling
tools in this field: image-based modelling, pore-scale mod-
elling, continuum scale modelling, and functional-struc-
tural plant modelling. We show how to link the pore scale
to the continuum scale by homogenisation or by deriving
effective physical parameters like viscosity from nano-
scale chemical properties. Furthermore, we demonstrate
ways of modelling the links between rhizodeposition
and plant nutrient uptake or soil microbial activity.
Abstract
Purpose Simultaneously interacting rhizosphere
processes determine emergent plant behaviour,
including growth, transpiration, nutrient uptake, soil
carbon storage and transformation by microorgan-
isms. However, these processes occur on multiple
scales, challenging modelling of rhizosphere and
plant behaviour. Current advances in modelling and
experimental methods open the path to unravel the
importance and interconnectedness of those processes
across scales.
Responsible Editor: Philip John White
A. Schnepf contributed equally to this work.
Supplementary Information The online version
contains supplementary material available at https:// doi.
org/ 10. 1007/ s11104- 022- 05306-7.
A.Schnepf(*)· E.Kröner· M.Javaux· M.Landl·
J.Vanderborght
IBG-3 (Agrosphäre), Forschungszentrum Jülich GmbH,
Wilhelm Johnen Str, 52428Jülich, Germany
e-mail: a.schnepf@fz-juelich.de
E. Kröner
e-mail: e.kroener@fz-juelich.de
M. Javaux
e-mail: m.javaux@fz-juelich.de
M. Landl
e-mail: m.landl@fz-juelich.de
J. Vanderborght
e-mail: j.vanderborght@fz-juelich.de
A.Carminati(*)· P.Benard· P.Duddek
Department ofEnvironmental Systems Science, ETH
Zürich, Universitätstr. 16, 8092Zürich, Switzerland
e-mail: andrea.carminati@usys.ethz.ch
P. Benard
e-mail: pascal.benard@usys.ethz.ch
P. Duddek
e-mail: patrick.duddek@usys.ethz.ch
A.Carminati· M.A.Ahmed· E.Lehndorff
Chair ofSoil Physics, Bayreuth Center ofEcology
andEnvironmental Research (BayCEER), University
ofBayreuth, Universitätsstr 30, 95447Bayreuth, Germany
/ Published online: 31 January 2022
Plant Soil (2022) 478:5–42
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Conclusion Modelling allows to integrate new
experimental data across different rhizosphere pro-
cesses and scales and to explore more variables than
is possible with experiments. Described models are
tools to test hypotheses and consequently improve our
mechanistic understanding of how rhizosphere pro-
cesses impact plant-scale behaviour. Linking multiple
scales and processes including the dynamics of root
growth is the logical next step for future research.
Keywords Rhizosphere· Modelling· Up- and
downscaling· Emergent behaviour
Introduction
The rhizosphere is one of the most complex and
vital interfaces on earth (Hinsinger et al. 2009). It
hosts myriads of microorganisms, and its properties
affect terrestrial fluxes of water and various elements
including carbon and nitrogen. Since soil water and
nutrients have to traverse the rhizosphere before being
taken up by the plant, the rhizosphere is considered
to be the critical interface governing plant produc-
tivity and consequently food, fuel and fibre produc-
tion. Understanding and engineering rhizosphere
e-mail: Mutez.Ahmed@uni-bayreuth.de; maaahmed@ucdavis.
edue-mail: Eva.Lehndorff@uni-bayreuth.de
e-mail: Mutez.Ahmed@uni-bayreuth.de
e-mail: maaahmed@ucdavis.edu
E. Lehndorff
e-mail: Eva.Lehndorff@uni-bayreuth.de
M.A.Ahmed
Department ofLand, Air andWater Resources, University
ofCalifornia Davis, Davis, CA95616, USA
M.Ani· J.Bentz· M.Knott· D.Diehl
University Koblenz-Landau, Institute forEnvironmental
Sciences, Fortstr. 7, 76829Landau, Germany
e-mail: ani.mina@uni-landau.de
J. Bentz
e-mail: bentz@uni-landau.de; j.bentz@fz-juelich.de
M. Knott
e-mail: brax@uni-landau.de
D. Diehl
e-mail: diehl@uni-landau.de
M.Bonkowski
Institute ofZoology, University ofCologne, Zülpicher Str.
47b, 50674Cologne, Germany
e-mail: m.bonkowski@uni-koeln.de
E.Kröner
Institute ofCrop Science andResource Conservation
(INRES), University Bonn, Karlrobert-Kreiten-Straße 13,
53115BonnBonn, Germany
M.Javaux
Université Catholique de Louvain, Earth andLife Institute,
Croix du Sud L7.05.02, B-1348Louvain-la-Neuve,
Belgium
E.Lehndorff
Soil Ecology, Bayreuth University,
Dr.-Hans-Frisch-Strasse 1-3, 95448Bayreuth, Germany
E.Lippold· M.Phalempin· D.Vetterlein
Department ofSoil System Science, Helmholtz Centre
forEnvironmental Research – UFZ, Halle, Germany
e-mail: eva.lippold@ufz.de
M. Phalempin
e-mail: maxime.phalempin@ufz.de
D. Vetterlein
e-mail: doris.vetterlein@ufz.de
A.Lieu· A.Prechtel· R.Schulz
Department ofMathematics, Friedrich-Alexander
University ofErlangen-Nürnberg, Cauerstr. 11,
91058Erlangen, Germany
e-mail: alice.lieu@fau.de
A. Prechtel
e-mail: prechtel@math.fau.de
R. Schulz
e-mail: raphael.schulz@math.fau.de
C.W.Mueller
Department ofGeosciences andNatural Resource
Management, University ofCopenhagen, Øster Voldgade
10, 1350CopenhagenK, Denmark
e-mail: cm@ign.ku.dk
E.Oburger
University ofNatural Resources andLife Sciences,
Institute ofSoil Research, Konrad Lorenz-Str. 24,
3430TullnanDerDonau, Austria
e-mail: eva.oburger@boku.ac.at
W.Otten· X.Portell
School ofWater, Energy andEnvironment, Cranfield
University, CranfieldMK430AL, Bedfordshire, UK
e-mail: Wilfred.Otten@cranfield.ac.uk
X. Portell
e-mail: xavier.portell@cranfield.ac.uk
D.Vetterlein
Soil Science, Martin-Luther-University Halle Wittenberg,
Von-Seckendorff-Platz 3, 06120Halle, Germany
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properties may then well be the key to promoting sus-
tainable agriculture and mitigating the effects of cli-
mate change (Ahkami etal. 2017; Ryan etal. 2009).
Engineering rhizosphere properties may open new
avenues for crop production management and con-
tribute to limiting the input of mineral fertilization
or increase the water use efficiency of crops (Ahmed
etal. 2018). The application of the acquired knowl-
edge in the field of rhizosphere research for practical
management purposes is however still in its infancy.
Rhizosphere properties are the result of manifold
biological, physical, and chemical processes that
ultimately impact plant growth and soil properties.
These processes include water and nutrient uptake,
rhizodeposition and microbial activity (Paterson etal.
2007), and rearrangement of soil particles by the root
as it grows (Lucas etal. 2019; Phalempin etal. 2021).
These processes interactively affect each other and
dynamically determine the rhizosphere properties.
They act on scales which span more than six orders of
magnitude—from the scale at which rhizosphere pro-
cesses take place (typically between 1µm and 1mm)
to the management scale (> 1m). This imposes chal-
lenges not only in the measurements of corresponding
quantities but also in the appropriate description and
conceptualisation of these processes, on the model-
ling, parametrization and linking of the scales.
Novel technologies and experimental approaches
have been developed recently, which allow one to
image or characterize rhizosphere-scale processes
with an unprecedented spatial resolution (Roose etal.
2016; Vetterlein etal. 2020). These technologies open
new opportunities to understand the complexity but
provide fragmented information on rhizosphere archi-
tecture and functions. In addition, new models have
been developed during the last decade, which allow
to integrate biological, chemical and physical pro-
cesses at different scales, from pore (microbial or soil
pore models) to plant scale. These models can help (i)
integrate different processes, (ii) explore more vari-
ables than is possible with experiments, (iii) scale up
from local, or small-scale observations to the whole
system, or the reverse if only bulk information is
available and local information is of interest, (iv) pro-
vide high temporal resolution even if measurements
could only be provided for a limited number of time
steps and finally (v) test if hypothesised co-occurring
mechanisms/processes could bring about spatial or
temporal patterns observed, in particular if they are
non-linear in nature. Note that observed behaviours
at larger scales are more than the average or sum of
small-scale processes due to the functional complex-
ity of the system—this phenomenon is termed emer-
gent behaviour in the theory of complex systems
(Camazine etal. 2001; Thurner etal. 2018; Vetterlein
etal. 2020).
There are a number of excellent recent reviews on
challenges in imaging and predictive modelling of
rhizosphere processes (Pot etal. 2021; Roose et al.
2016), challenges in modelling soil processes (Ver-
eecken et al. 2016), or plant-soil modelling (Ruiz
etal. 2020b). The objective of this opinion paper is
to present a series of modelling scenarios, which inte-
grate detailed rhizosphere processes into a plant-scale
and soil profile-scale modelling concept. It results
from research that has been funded in the priority
program “Rhizosphere Spatiotemporal Organisation
– a Key to Rhizosphere Functions” (PP 2089 of the
German Research Foundation DFG).
The working hypothesis of the proposed approach
is that the emergent behaviour at the plant scale is
determined by the combined interaction of rhizos-
phere processes (Vetterlein etal. 2020). The emergent
behaviour includes plant growth (e.g. biomass), bulk
soil properties (e.g. permeability), transpiration, car-
bon fluxes, nutrient uptake, plant health, soil aggrega-
tion, and carbon storage and transformation.
Rhizosphere processes include for instance root
exudation, microbial transformation and biodiver-
sity, water and nutrient uptake. We further distin-
guish betweeneffective properties, which result from
upscaling small-scale properties to larger scales,
and emergent behaviour, which develops from the
interactions between multiple small-scale processes
and their responses to local soil and rhizosphere con-
ditions. We focus on three spatial scales, namely the
plant scale > single root scale > pore scale (see Fig.1).
One aspect of our approach is to combine time-
series images down to the pore scale with detailed
models to define effective properties to be included
in single root scale and plant scale models. Another
aspect proposes to feed information from a larger
scale models into the smaller scale models addressing
longer temporal scales. Doing so, we propose a cross-
talk between detailed processes and emerging behav-
iour, with both up- and down-scaling. A sketch of
the scales and related processes we aim to addressis
shown in Fig.1. The objective ofupscalingmultiple
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small-scale processes isto determine effective prop-
erties and emergent behavioursand to predict behav-
iours and trends. The objective of downscaling is
to set boundary conditions for small-scale models,
which are used to resolve smaller scale structures and
evaluate their effect on emergent behaviours at larger
scales.
Examples ofcase studies oflinking mechanistic
models anddata acrossscales
The common theme of the five case studies in this
paper is plant-derived soil organic carbon, its spatial
and temporal pattern and effects on plant resource
acquisition as well as microbial activity and biodi-
versity. The different presented modelling approaches
involve processes taking place from pore to plant
spatial scales (ranging from nm-µm to dm-m scales)
and from second to week temporal scales. They all
address how existing data could be integrated into
the respective models. Figure 1 summarises these
case studies, provides their specific scales and inves-
tigated processes (in colour). Some of the case stud-
ies are based on published data sets while others use
yet unpublished work and are intended to serve as
showcases.
Case study 1 couples a 3D dynamic root archi-
tecture simulation with root exudation of individual
roots. The goal is to compute the rhizodeposition pat-
terns at the plant scale during plant development as
driven by multiple single root growth and rhizodepo-
sition properties. It shows how plant root develop-
ment might affect mm-scale carbon distribution.
Case study 2 quantifies the impact of root exuda-
tion by multiple growing roots of a plant to root phos-
phorus uptake. The overall plant phosphorus uptake
and carbon investment emerge from the interaction
Fig. 1 Linking rhizosphere processes across scales as illus-
trated by the case studies (CS) 1–5 presented in this opinion
paper. The processes investigated in each case study are shown
by the coloured cubes. The arrows illustrate the links between
spatial scales through up- and downscaling
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between the rhizosphere processes on the single root
scales. Again, this example bridges the gap between
plant-scale development and the rhizosphere release
of compounds and uptake processes. Case studies 1
and 2 require information on the diffusion coefficient
in the rhizosphere. Case study 3 and 4 show how this
diffusion coefficient is influenced by root hairs and
mucilage, respectively.
Case study 3, for a given snapshot in root archi-
tecture development, zooms in to simulate the
rhizodeposition around a single root affected by root
hairs. This example demonstrates the use of µCT
images for image-based modelling to investigate how
root scale processes (at mm scale) impact micro-dis-
tribution of carbon in the rhizosphere at (sub-) mm
scale in the rhizosphere and define the boundary and
initial conditions for molecular and pore scale studies
considered in case studies 4 and 5.
With case study 4, we explore pore scale processes
in the range between nm and µm. In this example, we
analyse the impact of the physico-chemical properties
of mucilage on pore scale water dynamics. An effec-
tive diffusion coefficient is derived based on µCT pore
scale images. The resulting diffusion coefficient can be
included in models such as those of study 1 and 2.
Case study 5 focuses also on the pore scale and
illustrates how microbe-driven processes emerge
from interactions among the biotic and abiotic com-
ponent using an individual-based approach describing
microbial dynamics with an explicit description of
the 3D soil structure, water, and carbon distributions.
Table 1 summarises the origins of the inputs
needed for the different case studies. Most of the
detailed 3D information can be provided through
imaging, but some may also originate from the output
of another model simulation.
All case studies are developed hereafter with the
same basic structure: question, scales, approach,
results, challenges, and open questions.
Models considering 3D root architecture and root
growth
Case study 1: Rhizodeposition byagrowing root
system
Question What is the effect of root architecture
development and exudate properties on the 3D distri-
bution of rhizodeposits in soil?
Scale mm-cm, days-weeks.
Approach
The mathematical model behind this example is
explained in Landl etal. (2021). Briefly, the 3D root
architecture model CPlantBox (Schnepf etal. 2018)
was coupled with a rhizodeposition model to inves-
tigate the spatio-temporal distribution patterns of
rhizodeposits in the soil as affected by root archi-
tecture development and rhizodeposit properties. To
simulate the 3D dynamic patterns of rhizodepos-
its in the soil, each growing root was considered to
be a moving point source or a moving line source.
Roots are considered moving point sources when
rhizodeposition occurs mainly at the root tip (e.g.
mucilage) and moving line sources when rhizodepos-
its are released over a certain length behind the root
tip (e.g. citrate). In the soil, rhizodeposits were sub-
ject to diffusion, sorption and decomposition. Micro-
organisms were not considered explicitly, but degra-
dation of rhizodeposits was included in form of linear
first order decay (Kirk etal. 1999).
Analytical solutions for moving point or line
sources in an infinite domain have long been availa-
ble (Carslaw and Jaeger 1959). Thus, for any point in
time or space, the analytical solution for rhizodeposit
concentration around a growing root can be computed
analytically. Since the underlying partial differen-
tial equation is linear, the concentration attributable
to multiple roots of a growing root system was cal-
culated as the sum of the concentrations attributable
to each root using the superposition principle. How-
ever, the fact that the solution is based on analytical
solutions does not imply a small computational time,
since it involves the evaluation of integrals. Unless
approximations are made that allow these integrals
to be solved analytically, they must be evaluated
numerically as many times as the spatial and tempo-
ral resolution requires. In this example, the simula-
tion time was set to 21days, simulation outputs were
generated every day. The size of the soil domain was
20 × 20 × 45 cm3, and concentrations were computed
at every mm3. Simulations were performed for two
rhizodeposits, mucilage and citrate, and the example
root system Vicia faba. Details can be found in Landl
etal. (2021).
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Table 1 Information about the origin of the input of the different case studies
Input Information Case study 1 Case study 2 Case study 3 Case study 4 Case study 5
Root Architecture Computed
Computation of a dynamic
3D root architecture with
the model CPlantBox that
was parameterized using
µCT images of soil grown
root systems
Computed
Computation of a dynamic
3D root architecture with
the model CPlantBox that
was parameterized using
rhizotron images of soil
grown root systems
X-ray CT
Segmented images from
3D soil subsamples
showing the root seg-
ments
X-ray CT
Segmented images from 3D
soil subsamples showing
the root segments
X-ray CT
Segmented images from
cylindrical 3D soil sub-
samples showing the root
segments
Soil architecture - - X-ray CT
Segmented images from
3D soil subsamples
showing pore and solid
phases
X-ray CT
Segmented images from 3D
soil subsamples showing
pore and solid phases
X-ray CT
Segmented images from
cylindrical 3D soil subsam-
ples showing pore and solid
phases
Distribution of water / air - - - Computed
For eff. diff.: water distribu-
tion around solid and
mucilage
Computed
Computed with a two-phase
single component two
relaxation times Lattice-
Boltzmann model
Distribution of rhizodeposits - - - SRXTM
Light microscope images
showing the mucilage
distribution
Measured
Radial gradients of 13C
detected with LA-IRMS
and NanoSIMS
Distribution of microbes - - - - Assumed
Spatial statistical model
distributes biomass within
the pore space
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Parameterisation oftheroot architecture model For
the simulation of root architectures, CPlantBox
requires a set of appropriate model input parameters.
In this example, root architecture parameters were
derived from µCT images of Vicia faba plants from a
lab experiment (Gao etal. 2019). In this experiment,
six replicates of Vicia faba plants were grown and each
of them scanned at 7, 11, and 15days after sowing. The
roots on the µCT images were manually traced in the
3D virtual reality system of the Supercomputing Cen-
tre of Forschungszentrum Jülich, resulting in a data
structure called root system markup language (RSML)
(Lobet etal. 2015). It stores the 3D coordinates of the
nodes along the center line of the root system and
their connections, as well as various properties for the
edges (root segments) connecting two nodes, such as
the radius or the age of the root segment. The age of
each root segment was linearly interpolated between
the root tip (age = 0) and the root collar (age = time
of measurement). Many model parameters could be
calculated directly from the RSML files, such as the
mean values and standard deviations of the radii, the
lengths of the apical and basal zone, the intermodal
distances and the branching angles. The remaining
parameters, such as the emergence time of basal roots,
were obtained by an inverse estimation that minimized
the difference between the observed and the simulated
total root length at the different measurement time
points.
Rhizodeposit parameters Rhizodeposition rates of
mucilage and citrate from the roots of Vicia faba were
derived from literature (Rangel etal. 2010; Zickenrott
etal. 2016). We assumed that mucilage was released
at the root tips, while citrate exudation occurred at a
length of 4cm behind the root tips. The diffusion coef-
ficients in water for mucilage and citrate were taken
from (Watt etal. 2006), the impedance factor from
Olesen etal. (2001), the soil buffer power for citrate
from Oburger et al. (2011) and the decomposition
rate constants for mucilage and citrate from Kirk etal.
Fig. 2 Flow chart outlining the approach of case study 1. The oval shapes illustrate start and end points of the workflow, the rectan-
gular boxes describe activities and the rhombs include intermediate outputs that serve as inputs for the next step in the workflow.
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(1999) and Nguyen etal. (2008). All values can be
found in Table1 of Landl etal. (2021).
Figure2 outlines the workflow of case study 1. An
important step in this case study was the development
of a model that can simultaneously simulate root
architecture development and root exudation as well
as the fate of root exudate within the soil. Data acqui-
sition included a plant growth experiment in which.
soil columns containing root systems were scanned
with µCT. The roots on the µCT images were seg-
mented and traced to derive the parameters of the
root architecture model CPlantBox. The output of this
case study is the 3D dynamic pattern of rhizodeposi-
tion concentration for different rhizodeposits around
the growing root system of Vicia faba.
Results
Figure 3(a, b) shows the distribution of citrate and
mucilage concentrations around the 3-week-old
root system of Vicia faba. Due to the differences in
rhizodeposit properties, the maximum mucilage con-
centrations were larger than the maximum citrate con-
centrations, while the extent of the mucilage enrich-
ment zone was smaller than the extent of the citrate
enrichment zone. Figure3(c, d) shows the distribution
of rhizodeposit hotspots, in which the rhizodeposit
concentration is above a defined threshold. For muci-
lage, we chose a value at which the mucilage con-
centration is high enough to significantly alter soil
hydraulic properties, i.e., 0.33mg g−1 dry soil (Car-
minati etal. 2016). For citrate, we chose a concen-
tration that is high enough to significantly mobilise
phosphate, i.e., 5 µmol g−1 soil (Gerke 2015). We
demonstrated that root growth rate had a significant
effect on the volume of rhizodeposit hotspots and that
the volume of rhizodeposit hotspots was greatest at
intermediate root growth rates. At slow root growth
rates, rhizodeposit concentrations were high, but the
soil volume containing these high concentrations
was low. At rapid root growth rates, the soil volume
containing rhizodeposits was high, but rhizodeposit
concentrations were mainly below the threshold con-
centration. Analysis also showed that the rhizodeposit
hotspot volume around the 3-week-old root system of
Vicia faba was significantly larger for citrate than for
mucilage. Due to the different release behaviour of
the rhizodeposits, high mucilage concentrations were
mainly located at the root tip, while high citrate con-
centrations were found closer to the root base. As a
result, additional hotspots were created for citrate by
overlapping the rhizospheres of individual roots. Our
analysis showed that rhizosphere overlap accounted
for more than half of the total citrate hotspot vol-
ume, whereas only about 10% of the total mucilage
Fig. 3 Vertical cut through the distribution of the rhizodeposit
concentrations around 3-week-old root systems of Vicia faba
(citrate (a), mucilage (b)); note that the colours are in logarith-
mic scale (from Landl et al., 2021). Distribution of rhizode-
posit hotspots (pink patches) of citrate and mucilage around a
21-day-old root system of Vicia faba (citrate (c), mucilage (d))
Plant Soil (2022) 478:5–4212
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hotspot volume was due to rhizosphere overlap. In
addition, we showed that long duration of rhizodepo-
sition hotspots is also strongly influenced by the
overlap of rhizospheres, which is why long duration
rhizodeposition hotspots occurred mainly in the root
branching zone. This example demonstrates how the
interaction between plant scale processes like root
architecture development with single root processes
like rhizodeposition defines the emergent spatial and
temporal distribution of these rhizodeposits. Such a
distribution could then be used to define time-varying
boundary conditions for smaller scale models, e.g. in
case study 5.
The importance ofroot growth fortheradial
extension oftherhizosphere
Kim and Silk (1999) demonstrated the importance
of root elongation rate for the extension of the radial
concentration profiles around roots. They defined a
root-related version of the Péclet number, a dimen-
sionless number usually used to quantify the impor-
tance of diffusive relative to convective transport.
Replacing convection with root elongation, the
“rhizosphere Péclet number” Perhizo can be written as
where v is the root elongation rate, L is the char-
acteristic length and D is the diffusion coefficient in
soil.
The rhizosphere Péclet number in soils is usually
greater than one, indicating that the elongation rate
has an important effect on the rhizosphere gradients.
In the above case study, the Perhizo values for citrate
and mucilage are 192.68 and 1.17, respectively, for a
root with a radius of 1mm and growing at 2 cm/d.
Thus, we can already predict that the rhizosphere
(1)
Pe
rhizo =
v
⋅
L
D,
Fig. 4 Concentration of citrate (I) and mucilage (II) depos-
its around a single root of 4cm length that was grown at an
elongation rate of(a) 0.1 cm day−1, (b) 0.5 cm day−1, (c) 1 cm
day−1, (d) 2 cm day−1. The black dotted line represents the
root. Note that the colours are in logarithmic scale.
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will be considerably narrower for mucilage than for
citrate, and this will vary with elongation rate. In
the case of mucilage, root elongation is dominantly
affecting the rhizosphere extent while for citrate, root
elongation and diffusion both have an equally strong
affect. Also, in this case, root elongation may not be
neglected! This is illustrated in Fig.4 that shows for
mucilage and citrate how the extent of the zone of
influence decreases with increasing root elongation
rate. Thus, root growth dynamics are important for
the investigation and analysis of rhizosphere gradi-
ents around individual roots. This may have implica-
tions on experimental designs such as the choice of
sampling locations.
Challenges andopen questions
The combined effects of root architecture develop-
ment and rhizodeposition in turn affect multiple
other relevant processes such as soil carbon turnover
(Hütsch etal. 2002) and microbial activity (Paterson
2003) or plant water and nutrient uptake (Carminati
etal. 2016; McKay Fletcher etal. 2020). Modelling
those interactions will enable us to assess the impact
of rhizodeposition on these processes at the root sys-
tem scale. Little information exists on changes in the
diurnal rhizodeposition and its dependence on light
quality and quantity (Kuzyakov 2002; Melnitchouck
etal. 2005). New experimental data sets on the dis-
tribution of rhizodeposits in soil are now available,
including zymography or co-registered Magnetic
Resonance Imaging and Positron Emission Tomogra-
phy (Koller etal. 2018; Spohn and Kuzyakov 2014).
Integrating those data into our model, we will be able
to increase our process understanding as well as gain
additional information on the location, intensity and
temporal dynamics of rhizodeposition rates using
inverse modelling. In case study 1, we have con-
sidered the root tip as a point. For detailed analysis
within the elongation zone, its spatial (in the order of
mm) and temporal (in the order of < 1 h) scales are
relevant (Baskin 2013; Silk 1984).
Case study 2: Phosphate uptake by a growing root
architecture as affected by citrate exudation
Question How is overall phosphate solubility and
consequently uptake affected by root exudates
released from a growing root system? How well
are diffusion, transport and reaction of nutrients
in the rhizosphere of a growing root architecture
understood?
Scale: mm-cm, days-weeks.
Approach
In this case study, the dynamic root architecture
model CPlantBox, formerly RootBox (Leitner etal.
2010a), was coupled with a model of phosphate trans-
port in soil and rhizosphere that takes into account
competitive sorption of phosphate and citrate in soil.
In particular, the sink term for phosphate uptake by
roots from soil was developed in a way to recognize
the dynamic development of the rhizosphere gradi-
ents around each individual root segment. For each
root segment of the root system, a 1D radially sym-
metric rhizosphere model was solved in each time
step and coupled to the macroscopic plant-scale
model in a mass-conservative way. Contrary to Case
study 1, the rhizosphere domain recognises the physi-
cal presence of the roots. The inner boundary of the
radially symmetric domain is at radius r = r0 while the
outer boundary is at radius r = r1, where r0 is the root
radius and r1 is the half mean inter-root distance
which is computed from the root length density RLD
as
r
1=
1
√RLD∙𝜋
. The model setup was a virtual repre-
sentation of a rhizobox experiment with oilseed rape
(Brassica napus L.) (see Schnepf et al. 2012 for
details). Briefly, oilseed rape was grown in a rhizo-
tron for 16days and kept at an inclination of 49°. The
simulations mimicked this experiment such that sim-
ulated root growth was confined inside the rhizotron
boundaries and had the same root length as observed,
namely 734 cm. The root architectural parameters
were obtained by inverse estimation such that the dif-
ference of observed and simulated root length densi-
ties was minimized. Root exudation was assumed to
occur at the whole root length with an exudation rate
of 3 × 10–6 µmol cm−2 s
−1 (Kirk et al. 1999). The
water content was set constant at 0.3 cm3 cm−3; the
effective diffusion coefficient in soil was computed
based on the Millington-Quirk model (Millington and
Quirk 1961). A competitive Langmuir sorption iso-
therm for citrate and phosphate was used to simulate
phosphate mobilisation due to citrate exudation. The
sorption parameters, microbial decomposition rate
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constant for citrate, and Michaelis Menten P uptake
parameters were derived from literature (see Tables1
and 2 in Schnepf etal. 2012). Initially, the soil had a
homogeneous initial P concentration and zero citrate
concentration. At the boundaries of the rhizotron, no-
flux boundary conditions were prescribed, so that the
only changes in citrate and phosphate concentration
in the rhizotron occurred through the root activities,
citrate exudation and P uptake. The Perhizo values for
P and citrate individually are 0.15 and 0.22, respec-
tively. The relatively smaller values compared to the
previous example are due to the fact that root radii of
oilseed rape in case study 2 are one order of magni-
tude smaller than those of bean in case study 1; i.e.,
the smaller the root radii, the smaller the rhizosphere
Péclet number. As the values are only slightly smaller
than one, still both root elongation and diffusion
affect the radial extension of the concentration pro-
files, with diffusion being the dominant process.
Figure 5 outlines the workflow of case study 2.
Model development resulted in a model that could
simultaneously simulate root growth, root exudation
and its effect on nutrient availability and uptake. Data
acquisition involved a plant growth experiment where
roots were washed from the soil, scanned, and the
root length determined with WinRhizo. This data was
used to inversely estimate root architectural parame-
ters for the CPlantBox model. Together with transport
properties and reaction rate parameters, the simula-
tion results in the 3D dynamic pattern of phosphate
and citrate concentration as well as cumulative nutri-
ent uptake and exudation.
Results
This model quantifies how two traits, root system
growth and exudation, affect plant phosphate uptake.
Figure 6 shows the simulated root age distribution
of roots grown inside the rhizotron, as well as the
total phosphate and citrate concentrations in soil as
affected by root exudation and phosphate uptake. The
highest citrate concentration and phosphate deple-
tion occurred in the region where the root system was
oldest. In this case study, performed for a soil with
medium sorption capacity, the model showed that
cumulative phosphate uptake was more than doubled
through root exudation of citrate (see Schnepf etal.
2012). This value would vary for soils with different
sorption behaviours. The case study shown here is
Fig. 5 Flow chart outlining the approach of case study 2. The
oval shapes illustrate start and end points of the workflow, the
rectangular boxes describe activities and the rhombs include
outputs of intermediate steps that serve as inputs for the next
step in the workflow.
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based on the assumption that root exudation occurs over
the whole root length, so that the citrate concentration
would be higher around older roots. This is different if the
root exudation occurs only near the root tips as we have
seen in case study 1, as then concentrations are highest
near the root tips. In both cases, overlapping accumulation
zones would further induce hotspot formation.
Challenges andopen questions
Functional-structural models of root water and nutri-
ent uptake may help to understand what the optimal
coordination would be between root growth and
rhizodeposition of different substances to obtain opti-
mal water and nutrient uptake by the root system.
The simulation results presented here made some
assumptions about the location and dynamics of exu-
date release along the root axes. Based on available
experimental data, a more accurate parameterisation
would be possible. Alternatively, whole-plant struc-
tural functional models are now being developed that
can model the flow of carbon from photosynthesis
to the different plant organs and release into the soil
(Zhou etal. 2020). Those model results may be used
as input for the model presented in case study 1.
As shown in case study 4, it is now possible to
derive an effective diffusion coefficient as a function
of radial distance from the root surface using math-
ematical homogenisation based on µCT-derived
soil structure and mucilage information. This is a
chance to test where the standard Millington-Quirk
model is valid and where new approaches to derive
the effective diffusion coefficient in soils and rhizo-
sphere are needed. By image based models it is also
possible to investigate the effect of root hairs on
element distribution (case study 3).
Another challenge is to compare the simulated
nutrient gradients to the gradients found through
chemical imaging methods at different positions
within the root system and to evaluate whether
the model can reproduce those observed gradi-
ents. If this is established, then observed element
distributions could serve as input for inverse
simulations for nutrient uptake and transport
parameters.
-5 05
-35
-30
-25
-20
-15
-10
-5
0
x (cm)
)mc( z
total P (µmolcm-3)
total C (µmolcm-3)
0
5
10
15
root system, age distributio nafter 16 days
Fig. 6 Left: Simulated root architecture of oilseed rape, col-
ours denote root segment age. Right: Total phosphate and cit-
rate concentrations in soil after 16 days of simulation when
root exudation occurs over the whole length of the root axes.
From Schnepf etal. (2012).
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Models computing the effect of root hairs and
mucilage on effective diffusion
The following case studies 3–5 are examples of
image-based modelling with a fixed soil and root
geometry as captured by µCT images. This is either a
limitation that could be accepted for the sake of sim-
plicity or limited to slowly growing or thin roots (case
studies 3 and 4), or required boundary conditions are
made dynamic, e.g. from outputs of a larger scale
model (as outlined in case study 5).
Case study 3: Root exudation andtherole ofroot
hairs onthesingle root scale
Question How do root hairs and the contact between
the root surface and the soil matrix affect rhizodepo-
sition and the spatial distribution of root exudates
from the root surface?
Scales Spatial scale: < 1 mm, temporal
scale: < 1day.
Approach
Data acquisition andprocessing Maize plants (Zea
mays L.) were grown in seedling holder microcosms
(Keyes etal., 2013; Koebernick etal., 2017) which
were filled with a sieved and fertilized loamy substrate
(Vetterlein etal. 2021). For image acquisition, roots of
the 14days old plants were scanned non-destructively
at an actual voxel size of 0.653µm3 using a synchro-
tron radiation X-ray CT (TOMCAT, Paul Scherrer
Institute Villigen, Switzerland). In order to segment
the scanned roots as well as soil aggregates, image
processing was performed in Avizo (Thermo Fisher
Scientific). 3D finite element meshes consisting of the
soil matrix and the root-soil contact, where the exuda-
tion boundary conditions are defined, were generated
in Gmsh. A more detailed description of the experi-
mental setup and data processing are available in the
supplementary information.
Modelling To assess the effect of root hairs on the
spatial distribution of root exudates, image-based mod-
elling on the pre-processed CT data was performed.
Diffusion simulations of carbon released by a root seg-
ment of approx. 1.4mm length into the rhizosphere
were carried out for two cases on one exemplary sam-
ple – a root with and without hairs. We considered the
root as well as its’ hairs to release carbon (Holz etal.
2017) treating their contact surface to soil aggregates
as inlet at a constant concentration. Carbon diffusion
was calculated in the partially saturated soil micropore
region with a no-flow boundary condition at the sur-
face of the soil aggregates. There was no carbon dif-
fusion in the air-filled macropores. For a root growing
at 2cm/d, the root radius measured in our images of
300µm, and a diffusion coefficient of carbon in soil
of
0.025cm2d−1
, the rhizosphere Péclet number in this
case study is
Pe
rhizo =
2
⋅
0.03
2.88∙10
−7=
2.41.
Thus, both root
elongation and diffusion affect the radial extent of the
rhizosphere.
However, the aim of our model was to illustrate the
effect of carbon released from root hairs on the car-
bon distribution at the time scale of 1h using image-
based modelling where adding root growth would
add too much complexity. Therefore, we neglect root
and root hair elongation and other dynamics such
as shrinkage of roots and root hairs. We justify our
assumption with the following considerations: root
hairs grow at an elongation rate of 0.144cm d−1 (Gri-
erson and Schiefelbein 2002). Applying the same
concept of rhizosphere Péclet number to a growing
root hair (instead of a growing root), the Péclet num-
ber is
Pe
hair,radial =
0.0018∙0.144
0.025
=0.01 ≪
1
due to the
small root hair radius of only 18µm. Thus, we can
neglect the effect of root hair growth on the radial
extend of the carbon concentration around the indi-
vidual root hairs. However, it is also relevant to com-
pare the characteristic time of root hair growth with
the diffusion time scale from the main root. Consid-
ering a characteristic length of the root radius plus
the root hair length of approximately 500 µm, we
obtain that hair growth and diffusion from the main
root are in the same order of magnitude. This means
that the root hair growth affects the radial extent of
the rhizosphere, and its role becomes increasingly
more important when the diffusion coefficient
decreases, for instance due to soil drying. It is
therefore important to include root hairs in the
estimation of exudate distribution as a function
of distance from the root surface. These consid-
erations are based on the implicit assumption that
all root surface exudes into the soil (i.e., perfect
contact between root and soil matrix) and that the
diffusion coefficient in the soil is homogeneous.
However, at the root hair scale, the soil matrix as
well as the root-soil contact are not homogeneous
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and their specific geometry needs to be explicitly
accounted for. For simplicity, our simulations are
built on a static and image-based geometry, rep-
resenting already grown root hairs.
Mathematically, the underlying problem is
described by the diffusion equation—a second order
parabolic partial differential equation:
where
c(

r,t
)
denotes the concentration of the
diffusing material at location

r=(x,y,z)
and time
t
.D
(
c
(

r,t
)
,

r
)
represents the diffusion coefficient
for a concentration
c
at a location

r
and
� �
∇
the nabla
operator.
Assuming a constant diffusion coefficient, our
model is formulated as follows:
(2)
𝜕
𝜕t
c


r,t

=� �
∇⋅

D

c


r,t

,
r

� �
∇c


r,t
,
(3)
𝜕
𝜕t
c
(

r,t
)
=DΔc
(

r,t
)
inΩ × Σ
,
with
The domain obtained from CT images is denoted
by
Ω
,
Σ=[0, 3600s]
is the simulated time interval.
𝜈
represents the unit outer normal and
Δ
the Laplacian
operator. The inlet carbon concentration
cin
was taken
to be a fixed value from Holz etal. (2018) and it is
imposed as constant boundary condition. The diffu-
sion equation was discretized and solved by the “sca-
larTransportFoam” solver of OpenFoam – an open
(4)
c=cinonΓD⊆𝜕Ω,
(5)
� �
∇
c
∙
𝜈
=
0
onΓN=
𝜕
Ω�ΓD,
(6)
c=0att=0,
D=2.88 ∙10−7cm2s−1,
cin =
1037.52
μgcm−3
Fig. 7 Flow chart outlining the approach of case study 3. The
oval shapes illustrate start and end points of the workflow, the
rectangular boxes describe activities and the rhombs include
outputs of intermediate steps that serve as inputs for the next
step in the workflow.
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source CFD software package (Weller et al. 1998).
Further details are provided in the supplementary
information.
Figure 7 outlines the workflow of case study 3.
Data acquisition includes conducting an experiment
of plant growth in a soil column and the imaging of
this soil column in a synchrotron facility. The raw
images are processes in a way that results in a 3D
finite element mesh of the soil domain on which dif-
fusion of root-derived exudates was performed using
image-based modelling techniques. The outcome is
the spatio-temporal pattern of exudate concentration.
Results
We simulated the diffusion of root exudates, in par-
ticular carbon, into soil for one hour and one illus-
trative sample. The selected region of interest of
approx.
1.8mmx1.3mmx1.4mm
contained a soil
aggregate volume of
1.3mm3
, an air-filled macropore
volume of
1.1mm3
, a root volume of
0.4mm3
and an
air volume of
0.4mm3
that surrounded the sample
tube (see Fig. 8a). The epidermis surface area was
3.9mm2
and a fraction of
6.1%
was in contact to soil.
Fig. 8 Results of image
analysis and simulations.
(a) 3D rendering of the seg-
mented synchrotron X-ray
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The corresponding soil contact fraction of the root
including hairs was approximately three times bigger
(17.7%).
Figure8b shows the simulated carbon concentra-
tion in the soil surrounding a hairless root (left) and
a root with hairs (right). Regarding the hairless root,
a total carbon mass of
0.067μg
diffused into the soil
after 1h, whereas this value was three times higher
for the root with hair (
0.204μg
). The comparison of
carbon distributions within the rhizosphere revealed a
lower radial concentration gradient for the hair case
resulting in a right shift of the concentration data
(Fig. 8c). In the hairless case, carbon concentration
dropped below 1% at a distance of 0.29mm from the
root surface whereas the same value was reached in
the hair-case at 0.74mm.
CT sample consisting of the root with its elon-
gated hairs in yellow and soil particles in grey; scale
bar = 500µm. (b) Illustrative comparison of the simu-
lated carbon diffusion within the soil domain for the
hairless root (left) and the root with hair (right) after
a simulation time of 1h. (c) Spatial carbon distribu-
tion within the rhizosphere represented by the carbon
concentration regarding the radial distance from the
root-surface.
Challenges andopen questions
This illustrative case study shows how root hairs can
increase the total amount of root exudates and their
diffusional distance into the soil when we assume a
constant concentration at the root and root hair sur-
faces. Explicit image-based simulations of root exu-
dation including information of root-soil contact and
the spatial distribution of root hairs allows estimating
the importance of root hairs for rhizodeposition. This
case study shows good agreement with experimental
measurements by Holz etal. (2017).
Note that we assumed the same constant exudate
concentration boundary condition for the two geno-
types. Our simulations are based on the grid of a sin-
gle wild-type sample before and after removing its’
root hairs. As demonstrated before, in order to capture
system dynamics, root growth needs to be taken into
account. However, the focus of this case study lies on
exudate diffusion from a static (non-growing) root.
Nevertheless, the outcome of our simulation provides
value to more complex simulations considering a root
elongation rate. Particularly, the results can be used
to define the boundary conditions of the simulations
performed in case study 5 that is described below.
One open question is how to implicitly account
for hairs when their spatial distribution, and in par-
ticular as the fraction of their contact with the soil
matrix surrounding the root, are not known and how
to include such implementations in root system mod-
els. The problem of the partial contact between roots
and soil has been solved for the case of a single roots
by de Willigen etal. (2018). The solution should be
extended to the case of a single root with hairs pos-
sibly having different fraction of contact with the soil
matrix. The outcome of our simulation may be used
to define the boundary conditions of the simulations
performed in case study 5. An additional open ques-
tion is the representativity of the selected volume.
Indeed, soil porosity and root hair density might be
highly variable in space (at the scale of 1 mm3) and
time. Temporal information is needed to cover the
lifetime of root hairs. Additionally, rhizosphere bac-
teria were shown to contribute to mucilage produc-
tion, forming jointly with root mucilage a rhizosheath
in maize whose extent is determined by soil mois-
ture (Watt etal. 1994, 1993). Therefore different soil
water contents and soil textures should be simulated
in order to have a representative picture of the role of
root hairs on rhizodeposition for variable conditions.
Finally, we did not consider microbial degradation of
exudates, which is discussed in case study 5.
In summary, the biggest challenges are related
to the representativity of the imaged soil volume
and how to integrate these simulations in models
that consider root growth and other interacting pro-
cesses, such as root water uptake (affecting soil water
and the contacts) and microbial degradation of the
rhizodeposits.
Case study 4: Mucilage and hydraulic properties
Question How does mucilage affect pore scale distri-
bution and dynamics of water when soil dries? How
does the altered liquid configuration affect effective
diffusion?
Scales Spatial scale: from molecular to nm, µm
and to mm, temporal scale: < 1day.
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Approach
Background
In the previous studies we evaluated the impact of
root architecture and growth and of the impact of root
hairs on the spatio-temporal distribution of mucilage,
and on phosphate uptake. Although both approaches
represent excellent tools to study rhizosphere dynam-
ics, their predictive power depends on the quality of
provided input parameters. While the effect of muci-
lage on soil properties has been observed and dis-
cussed multiple times, little is known about how it
affects relevant pore-scale mechanisms.
In the following study, we examine how mucilage
alters the physicochemical properties of the soil solu-
tion, how this modification affects liquid connectiv-
ity at the pore scale and show how its potential impact
on nutrient diffusion can be evaluated by mathematical
homogenization based on high resolution images. For
the sake of simplicity, we neglect temporal changes of
mucilage properties resulting from processes like micro-
bial degradation or aging of the polymer network and
focus on liquid distribution in a simplified rhizosphere.
Root mucilage is primarily released at root tips and
mainly composed of polysaccharides, proteins and
some lipids. In contact with water, these polymeric
blends swell and form a 3D gel network. Gels pos-
sess specific properties, such as water holding capac-
ity, the ability to swell and shrink, and viscoelasticity,
which affect soil functions, such as water retention
(Ahmed etal. 2014; Benard etal. 2019; Kroener etal.
2018; Naveed etal. 2019). In contrast to pure water
whose distribution in soil is primarily controlled by
surface tension and capillary forces, the liquid config-
uration of mucilage is affected by its chemical hydro-
gel properties and its physical stability, which depend
on structure and arrangement of contained polysac-
charide units.
Gel properties vary for mucilage located in soil
pores in contrast to “free” mucilage (Brax etal. 2020,
2019; van Veelen etal. 2018). The polymers grip to
the soil particle surface and the network has thus an
increased strength compared to the “free” gel. The
confinement by the pore walls leads to an inhomo-
geneous distribution of the polymer network in the
pore during swelling and shrinking (Marcombe etal.
2010) and to pore-size specific organization of the
polymeric network.
Upon drying, the concentration of polysaccha-
rides within the liquid phase increases and the inter-
nal structural units of mucilage become more and
more important in controlling its physical properties.
Thereby, mucilage goes through a glassy transition,
passing from a liquid to a hydrogel and finally to a
rather solid structure (Carminati etal. 2017; Williams
etal. 2021).
This leads to characteristic mucilage drying pat-
terns that depend on intrinsic physical mucilage prop-
erties like viscosity and surface tension. Note that
chemical properties of mucilage vary with plant spe-
cies (Brax etal. 2020; Naveed etal. 2019, 2017) and
environmental conditions (pH, ionic strength, cations,
surfactants). For instance, viscosity of maize root
mucilage is higher than that of wheat.
The different viscosities of wheat and maize muci-
lage explain different patterns formed at the nano-
scale by the respective mucilage upon drying on a
flat surface (Fig.9a,b). For wheat, the nanostructure
appears as a network characterized by thin branches
smaller than 30nm in width. Maize mucilage builds
a more connected coating with larger hole-structures.
The thin threads of wheat mucilage suggest less and
weaker interactions between polymers in wheat com-
pared to maize mucilage, which is consistent with the
differences in viscosity (Fig.9c).
Analogue pictures are visible in 3D porous media.
Figure9d shows that upon drying in a porous medium
mucilage forms an interconnected surface that spans
through multiple pores, maintaining the liquid phase
connected during the drying process. This configu-
ration of the liquid phase upon severe drying is very
distinct from that of water, whose comparably high
surface tension and low viscosity causes the breakup
of liquid bridges (Carminati et al. 2017; Ohnesorge
1936; Williams etal. 2021). Improved liquid connec-
tivity on the pore scale was proposed as a mechanism
utilized by soil organisms to maintain and enhance
nutrient diffusion in dry soil (Benard etal. 2019). In
fact, addition of different kinds of mucilage lead to an
increase in diffusion coefficient by a factor 3 under
dry soil conditions for both glucose (Chenu and Rob-
erson 1996) and 137Cs (Zarebanadkouki etal. 2019).
Nevertheless, the impact of liquid connectivity at the
pore scale on effective diffusion has not been evalu-
ated systematically and remains unclear. Conven-
tional models, as e.g., in Millington and Quirk (1961),
relate diffusivity to the porosity or the water content,
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see e.g., (Chou etal. 2012) for a comparison. How-
ever, they do not take into account anisotropies, dif-
ferent tortuosities, or the explicit connected pathways
as can be done by mathematical homogenization (see,
e.g., (Ray etal. 2018). For this reason, we chose to
evaluate the impact of liquid connectivity at the pore
scale on diffusion using two simple examples. First,
we illustrate the effect of mucilage on connectivity by
simulating the drying of a liquid bridge between two
spherical particles. Second, we demonstrate how the
impact of increased liquid connectivity observed in
dry sand on the effective diffusion coefficient can be
quantified.
Spatial configuration
To simulate the interplay between molecules of the
polymer network and water, modelling tools are
needed that can describe both, dynamics of the poly-
mer network and the liquid within. Lattice-Boltzmann
methods are common tools to simulate pore scale
dynamics of liquids (Pot etal. 2015; Richefeu etal.
2016; Sukop and Or 2004; Tuller and Or 2005). Dis-
crete element methods are tools to describe defor-
mation and rupture processes of solids (Bobet etal.
2009) and have been used to simulate fracture of
hydrogels (Kimber et al. 2012; Yang et al. 2018).
While most hydrogel simulations consider free hydro-
gels, here, we study hydrogel deformation confined by
the pore space and attached to soil particle surfaces.
Fig. 9 a Atomic force microscopy height image of (a) wheat
and (b) maize root mucilage dried on flat mineral surface taken
in Peak-Force Quantitative Nanoscale Mechanical (PFQNM)
mode; (c) flow curves of maize and wheat root mucilage at
7 mg/mL measured for three replicates; (d) mucilage (red)
from maize nodal roots let dry in glass beads (100–200 µm
in diameter; blue) imaged via X-ray CT and segmented. The
content of mucilage was 8mg g−1 (weight of dry mucilage per
weight of dry particles). An interconnected 2D surface of dry
mucilage was deposited through multiple pores upon drying
(Benard etal. 2019)
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Transport properties
Both, phase distributions and connectivity as well
as intrinsic mucilage properties will finally affect
hydraulic properties, gas diffusion and nutrient trans-
port. Macroscale models cannot take into account the
explicit geometries and properties of phase distribu-
tions at the pore scale. The pore scale model however
is not amenable to large-scale computations because
of its high complexity. Information from the micro-
scale can be incorporated to the single root scale
using mathematical upscaling or homogenisation
techniques (Hornung 1996). These methods allow,
e.g., for the computation of, potentially anisotropic,
effective diffusion coefficient tensors requiring only
the geometric information on the microscale within
representative elementary volumes. They have been
used, e.g., for nutrient diffusion with regular geom-
etries (Leitner etal. 2010b), but have been extended
recently to irregular, evolving structures (Ray etal.
2018). Although the setting is periodic at the bound-
aries of the investigated domain, the underlying
geometries can be arbitrarily complex, and are not
restricted to idealised settings. This means that the
effect of concrete phase distributions of water and
mucilage in a realistic pore space can be quantified
given the respective diffusivities, and a sufficiently
large domain on the microscale. Compared to arti-
ficially constructed regular domains, CT images are
now being used to obtain the real micro-structure of
a given soil and serve as unit cell for the homogeni-
sation approach. We obtain the periodicity inherent
in the method of homogenisation (Hornung 1996) by
translation, i.e. the considered domain is surrounded
by identical samples (Guibert etal. 2016; Whitaker
1986). This method does not affect the geometrical
features of the domain, however, in particular in 2D
and unsaturated situations may decrease the connec-
tivity of the fluid phase (Guibert etal. 2016). Another
common approach is periodisation by symmetry (e.g.,
applied to CT images in Tracy etal. 2015), which
mirrors the domain in all directions. This increases
the computational domain and changes the properties
of the medium. In particular, it removes anisotropies
and therefore has been criticised as being unphysical
for pore scale flow properties in comparative studies
of different boundary conditions (Gerke etal. 2019).
We present results for both approaches on one exem-
plary CT image(for details on the original image see
Benard etal. 2019)being aware that the presented 2D
problem may be too small for representative conclu-
sions. We do not claim that this one CT image rep-
resents the real soil as there is a need to determine
the size of the soil’s REV in relation to the CT image
size (Auriault et al. 2010), and the connectivity is
most likely too low due to the 2D representation and
the discontinuities across boundaries. Note that con-
nectivity in CT images is related to resolution. With
smaller samples sizes there is a gain in resolution and
hence pore sizes which can be analysed, resulting in
higher connectivity. However, the soil volume ana-
lysed might become so small that larger structures are
not adequately represented any more. For a detailed
discussion see Lucas etal. (2020).
However, this CT image allowed to directly
observe the distribution of mucilage structures (see
Fig.11b) and to evaluate the impact of increased liq-
uid connectivity on effective diffusion at the same
water content. Liquid distribution was simulated
without (water only) and with observed mucilage
structures to illustrate the impact of the presence of
mucilage at the microscale – which can reduce or
enhance the effective diffusion coefficient depending
on the pore scale configuration and water (and muci-
lage) content.
Figure10 outlines the workflow of case study 4.
It is based on the same experimental setup as case
study 3, but uses additional modelling approaches to
(a) simulate the water distribution at the pore scale
for different water contents and mucilage concentra-
tions using Lattice-Boltzmann and Discrete Element
methods, and (b) use amended images for math-
ematical homogenisation to compute effective dif-
fusion tensors of a specific sample. Systematically
evaluated on a larger data base those may be used
in continuum-scale simulations such as case studies
1 and 2.
Results
Spatial configuration
Simulations based on Lattice Boltzmann meth-
ods and discrete element methods, respectively,
show the deformation of a liquid bridge between
two soil particles upon drying (Fig. 11a). In the
case of highly concentrated mucilage, hollow
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structures can form that have also been observed
in experiments (Benard etal. 2018). Due to inter-
nal polymeric structural units, hydrated mucilage
structures can remain connected at a water con-
tent at which water bridges would break. Similar
results were obtained in a theoretical study (Car-
minati etal. 2017) where the increased viscosity
of mucilage was held responsible for the damping
effect on motion.
Transport properties
Here we show how small changes in the spatial
configuration of the liquid phase induced by muci-
lage can affect the effective diffusion of solutes (e.g.
nutrients) across the rhizosphere. We deal with the
effect of mucilage on the effective diffusion coeffi-
cient, assuming that the presence of mucilage locally
decreases the diffusion coefficient of a given chemical
species. The species does not diffuse in the gas nor
in the solid phases. The diffusion in water is set to a
reference value of 1. In the mucilage, the diffusion is
assumed to be Dmuc/Dref = 0.5, exemplarily.
We present a scenario in a sandy soil with a volu-
metric water content of 0.19 cm3cm−3. Figure 11b
shows the 2D solid phase geometries of particles
derived from a cross-section of an X-ray CT used for
two different scenarios. In the first scenario, the liquid
phase consists of pure water whereas the second sce-
nario includes pure water and liquid bridges induced
by drying mucilage, which have been identified.
on images of a dry soil. In this particular case, the
mucilage makes up only 10% of the liquid phase. The
computed effective diffusion tensors differ signifi-
cantly for the “periodisation by translation” case and
read:
where
D1
and
D2
correspond to the matrices in
the scenarios without and with mucilage respectively.
The corresponding tensor for the fully saturated situ-
ation is
Dsat
.
For both scenarios, the first diagonal entry is non-
zero implying that the diffusion is achievable in the
domain horizontally. However, in the vertical direc-
tion, we notice that the diffusion is impeded though
not impossible in the first scenario (≈ 10–11). In the
D
1=
(
2.2663 0.00
0.00 0.00
)
⋅10−2,D2=
(
3.3411 0.7551
0.7551 1.8052
)
⋅10−2,Dsat =
(
1.2100 0.010
0.010 1.0130
)
⋅10−1
,
Fig. 10 Flow chart outlining the approach of case study 4. The
oval shapes illustrate start and end points of the workflow, the
rectangular boxes describe activities and the rhombs include
outputs of intermediate steps that serve as inputs for the next
step in the workflow
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where
D1
and
D2
correspond to the matrices in
the scenarios without and with mucilage respectively.
The corresponding tensor for the fully saturated situ-
ation is
Dsat
.
For both scenarios, the first diagonal entry is non-
zero implying that the diffusion is achievable in the
domain horizontally. However, in the vertical direc-
tion, we notice that the diffusion is impeded though
not impossible in the first scenario (≈ 10–11). In the
D1=(2.2663 0.00
0.00 0.00 )⋅10−2,D2=(3.3411 0.7551
0.7551 1.8052 )⋅10−2,Dsat =(1.2100 0.010
0.010 1.0130 )⋅10−1,
second scenario the mucilage bridges keep the liq-
uid phase more connected, creating new paths for the
species to diffuse in the vertical direction. Note in
particular the bridges (dark blue) at the top left and
bottom left part of the right image in Fig. 11b that
connect previously disconnected areas. For each case,
the main directions of diffusion are plotted in Fig.11.
For comparison, we also calculated the effective dif-
fusion tensors for the mirrored domain D
1,sym
and
D
2,
sym
. There every path at the boundary is perfectly
Fig. 11 a Simulation of drying dynamics of a liquid bridge
between two soil particles (modified from Haupenthal et al.
2021). Top: pure water simulated using Lattice Boltzmann
methods. Bottom: mucilage at high concentration simulated
using the discrete element method; (b) 2D geometry used for
the evaluation of the effective diffusion. Derived from X-ray
computed tomography [courtesy of M. Zarebanadkouki, Uni-
versity of Bayreuth, and P.Benard, A. Carminati, ETH Zurich]
of a sandy soil with porosity 41%. Artificial water distribution
with water content of 0.19 cm−3 cm3 without (left) and with
mucilage (right). Sand particles (brown); air (grey); water
(light blue); hydrated mucilage (dark blue). The arrows repre-
sent the eigenvectors and eigenvalues (length) of the effective
diffusion tensor of the “periodicity by translation” case, show-
ing, in each case, the main directions of diffusion
D
1,sym =
(
4.0818 0.00
0.00 3.1234
)
⋅10−2,D2,sym =
(
4.2290 0.00
0.00 3.1110
)
⋅10−
2
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continued, which may be an overestimation of the
real situation. Nevertheless we also see an enhanced
diffusivity due to mucilage in one direction.
The Millington-Quirk approximation which is a
function of water content and porosity only would
result in scalar values of
2.2628 ⋅10−1
for the saturated
case, and
2.3457 ⋅10−2
for the unsaturated case. It has
been reported e.g. in Chou etal. (2012) that it overes-
timates the diffusivity in various soils.
In this numerical experiment, the presence of
mucilage affects the diffusion paths and hence
increases the effective diffusion despite the reduced
solute diffusion in mucilage. Although this is a 2D
numerical study, it demonstrates that small changes in
the connectivity of the liquid phase can imply large
changes in the effective diffusivity, and thus 3D pore
scale information needs to be evaluated systemati-
cally in many samples and configurations to quantify
the effect of mucilage on solute diffusion.
Challenges andopen questions
We have demonstrated that the interaction between
the mucilage polymer networks, water and soil par-
ticles increases the connectivity of the liquid phase
across the rhizosphere. Upon drying, mucilage may be
deposited as two-dimensional surfaces forming hol-
low cylinders or interconnected surfaces across the
pore space. Although the solute diffusion coefficient
in mucilage is smaller than in pure water, the positive
impact of mucilage on liquid connectivity can result in
an enhanced effective diffusion coefficient in dry soils.
These physical mechanisms, which have been
addressed by the models and the experimental obser-
vations, qualitatively match. However, several param-
eters and processes are still not yet considered in these
models. Physical mucilage properties affecting the
contribution of mucilage to the maintenance of liquid
connectivity in drying soil comprise its water hold-
ing capacity and its viscosity, which are properties
that are methodically challenging to measure on the
relevant pore scale. They are determined by chemical
properties such as content of high molecular weight
material and the length and ramification of the poly-
mers, and can vary in the chemical environment, e.g.
with pH, absence or presence of mono- and multiva-
lent cations or of organic surfactants in soil solution.
As soil texture, structure, and drying rate are likely to
affect the spatial distribution of mucilage in soils, meas-
urements of mucilage deposition for varying soil particle
size and shape and drying rate are needed to improve our
understanding of mechanistic pore scale mucilage dry-
ing processes. When combined with measured model
input parameters, such as viscosity, surface tension
and elasticity of the polymeric structure, it may help to
advance pore scale modelling of spatial liquid distribu-
tions. Concerning the transport properties, simulations
need to be extended from two to three dimensions where
the effect of connectivity on the diffusion coefficient is
qualitatively similar, but quantitatively different. A chal-
lenge is the difficulty to image mucilage in soils while
it still contains a significant amount of water. In Benard
et al. (2019), the samples were scanned air-dry and
only the final mucilage distribution was imaged. Scan-
ning wet samples damaged the polymer network while
imaging dry mucilage structures is restricted to compara-
bly coarse soil due to the high resolution required (a few
micrometre). Nevertheless, improved diffusion reported
by Chenu and Roberson (1996) in xanthan-amended kao-
linite can hardly be compared to the observations of Zare-
banadkouki etal. (2019) in chia mucilage-amended sand
as both used different polymeric substances and concen-
trations. Overall, the impact of particle size and specific
soil surface on liquid connectivity and other soil hydrau-
lic properties remains indistinct. Image-based models
and simulations can nevertheless help to quantify possi-
ble pore scale effects and bridge the scales in these cases
(Roose etal. 2016).
From a computational point of view, dynamic interac-
tions between mucilage deformation, drying and solute
transport within the network need to be implemented.
Finally, the impact of ageing, microbial degradation and
transformation of mucilage (see case study 5) and the
impact of mucilage on microbial activity need to be inter-
connected. Beside the complexity of such computational
challenges, the experimental knowledge of such feedback
between mucilage properties, transport processes and
microbial activity is still in its infancy.
Case study 5: Microbial activity based on
C-distributions: using modelling to integrate
rhizosphere processes
Question Can we identify the key drivers of micro-
bial activity and C distribution in the rhizosphere
considering a holistic approach?
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Scale from nm to cm; seconds to days/weeks.
Approach
A modelling framework is proposed that can predict
microbial dynamics and processes resulting from
interactions in the soil surrounding roots.
We conjecture that the rhizosphere as we measure
it emerges from interactions occurring at the pore-
scale around roots. The complexity of the rhizos-
phere demonstrated by the case studies may appear
overwhelming at first sight, but we propose that it is
possible to fully embrace this by a new generation
of mechanistic, spatially explicit pore-scale models,
building on recent advances in this area. For instance,
using a mechanistic pore-scale fungal model, it has
been shown that fungal growth in soil is non-linear
and that, for a given volume and a given nutrient con-
tent, it depends on the micrometre scale distribution
of nutrients and microbes (Falconer et al. 2015a).
These micro-scale heterogeneities explain nonlineari-
ties in the temporal evolution of fungal biomass, car-
bon degradation and CO2 flux observed experimen-
tally. Focusing on soil bacterial dynamics, recently,
it has been demonstrated how heterogeneity at the
micro-scale can favour poor bacterial competitors,
suggesting that pore geometry is a driver of soil bio-
diversity (Portell etal. 2018). These results indicate
that mechanistic (first-principle) models are essential
for upscaling microbial driven processes and that this
approach can help to bridge the gap between pore-
scale and the continuum-scale description of the system
(Vetterlein etal. 2020). In this section, we propose the
modification of an existing pore-scale microbial soil
model as a first step to integrate rhizosphere experimen-
tal data and highlight the benefits and limitations of this
approach and the path ahead for further work.
Proposed modelling framework forrhizosphere
behaviour: anindividual based approach.
Our starting point can be a spatially explicit, pore-
scale model accounting for the activity of soil bacte-
ria (Portell etal. 2018). Briefly, Portell etal. (2018)
assumes a cubic lattice describing explicitly the soil
architecture where particulate organic matter (POM)
and microorganisms are located. POM hydrolyses
over time creates dissolved organic carbon (DOC)
that is released to the water phase where it becomes
available for bacterial growth or diffuses away to
more distant areas as determined by a lattice-Boltz-
mann model component. Bacterial position and phys-
iology are controlled by an individual-based approach
accounting for single bacteria that divide when the
cell attain a critical mass.
In the rhizosphere, organic matter distribution can
be seen as the result of the distribution of two main
components that can already be mapped to organic
matter pools considered by Portell etal. (2018). The
dominating fraction is a large and dynamic C fraction
that is continuously exuded to the soil from the region
between root tip and root hair zone and that can be
mapped to the DOC pool of the model. The distribu-
tion of “dynamic C” (i.e., root-exuded C) follows a
gradient into the soil and can be traced using stable
C isotope labelling. Labelled C exuded by the plant
root can be followed and quantified on the millimetre-
scale using an adequate sampling technique and can be
visualized in 2D at the micro-scale when using resin
embedded root-soil sections and nano-scale second-
ary ion mass spectrometry (NanoSIMS) (Vidal etal.
2018) or laser ablation-isotope ratio mass spectrom-
etry (LA-IRMS) (Rodionov etal. 2019). This informa-
tion can be used following two different approaches. In
the first approach, a boundary condition is placed on
the root surface of the living roots obtained from CT
images (Ruiz etal. 2020a) to account for the exudation
of carbon from the root. Following this approach, the
gradients measured will be used to calibrate the exu-
dation parameter of the model or, alternatively, use the
boundary conditions obtained from models such as the
one of case study 1. In a second approach when exper-
imental information is available, we assume a spatially
static gradient of exuded C.
The second fraction of C in the rhizosphere is a
rather static, discontinuous fraction made of dead
plant residues and microbial necromass and mineral-
associated organic matter (MAOM) (e.g., Liang etal.
2019) that can be mapped to the POM considered
by Portell etal. (2018). This fraction, which we can
expect to be located more irregularly in areas with
decreased accessibility by the microorganisms (Rodi-
onov etal. 2019; Totsche etal. 2018) can also be esti-
mated using NanoSIMS and LA-IRMS.
In bulk soil, bacteria invest energy to produce
enzymes that hydrolyse POM over time, thereby
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releasing DOC to the water phase where it becomes
available for bacterial growth or diffuses away to
more distant areas (as dictated by the lattice-Boltz-
mann component of the model). It must be noted that
most soil microorganisms has limited access to avail-
able carbon and resides in a dormant state most of the
time. In the immediate vicinity of the growing root
tip however, the release of DOC via exudation tran-
siently lifts the carbon limitation of microbial growth
in soil and favours fast-growing copiotrophic taxa
(Bonkowski etal. 2021; Rüger et al. 2021). Model-
ling these processes requires accounting explicitly for
trade-offs associated with the release of enzymes by
microorganism, a function currently assumed ubiqui-
tous in the model, as well as the diffusion of released
enzymes through the water phase from where they
can reach and control hydrolysis of distant volumes,
where it might initiate a priming effect on POM.
Experimentally it has been observed that the
microbial growth rate is a function of the availabil-
ity of C-containing exudates, whose diffusion rates
differ among molecule classes (Jones et al. 2004),
therefore showing effects dependent on the distance
and location of the root surface in relation to the point
in space considered. As a first approximation, mod-
els could be based on a uniform source of DOC. At
suboptimal DOC levels, carbon has been shown to
be respired without any microbial growth (Ander-
son and Domsch 1985), while higher DOC levels,
especially in the close vicinity of roots stimulate
microbial growth. High DOC levels may also favour
microbial enzyme production that subsequently leads
to enhanced hydrolysis of carbon and nutrients from
POM, a self-enhancing positive feedback mechanism
on microbial growth, known as priming effect (Kumar
etal. 2016; Mo etal. 2021). Assuming the individual-
based model approach, this can be simulated with the
implementation of appropriate rules ensuring that the
growth and enzyme production is only possible when
microbial maintenance requirements are covered.
In addition, current root growth models suggest that
microbial attachment to roots is a key strategy to gain
maximum access to rhizodeposition (Dupuy and Silk
2016). Therefore, next to the growth rate, future mod-
els must consider microbial motility. Microbial motil-
ity is disregarded in the work of Portell etal. (2018),
although the model structure is designed to allow the
implementation of such modification.
Recently, Dupuy and Silk (2016) studied the coloni-
sation of root surfaces resulting from the growth of root
tips in bulk soil. These authors modelled root grow-
ing through a homogeneous (continuous) soil fraction
and used a population-based approach to account for
microbial growth and colonization dynamics. Dupuy
and Silk (2016) found that the root elongation rate
was a key trait for successful establishment of bacte-
ria on the root surface. This suggest that in addition to
microbial growth and motility, future models should
consider root growth in order to reflect more realistic
colonisation dynamics. When structural information of
the solid, water and air phases of the soil are taken into
account, as we are suggesting here, the introduction
of growing roots in these models is challenging. In cer-
tain situations, time-lapse imaging could be used to fol-
low root growth and its modifications of soil structure in
order to implement this into complementary modelling
scenarios. To our knowledge, this has not been achieved
to date. Instead, root (or root hair) growth is tackled by
introducing time dependent boundary conditions. In this
approach, a common technique is to use a fully grown
geometry and to activate the appropriate boundary condi-
tions according to a (measured) growth rate (e.g. McKay
Fletcher etal. 2020). Given the impossibility of follow-
ing root growth in the field, we advocate the use of an
equilibrium approach and static roots as a first approxi-
mation. Given the complexity of the rhizosphere, this is
also advisable. We need to understand rhizosphere mech-
anisms around static (or slowly growing roots) before
embracing the full complexity of the system.
Simulation scenarios and model initialisation
Structural information supplemented with existing
knowledge will describe the physico-chemical state
of the soil microhabitats around roots and be used as
initial condition to conduct spatially explicit simula-
tions with the pore-scale model Fig.12a).
A number of steps are required to set the physical
environment surrounding roots. The soil architecture
of the rhizosphere, including root positions, will be
obtained with segmented X-ray CT images. Smaller
sub-domains (e.g., 1283 voxel size) will be obtained
from different distances from the root surface
(Fig.12b). The first challenge is to simulate the dis-
tribution of water and air within the pore space of the
subdomains as described in the case study 4 above.
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Root derived C has been shown to be distributed at
least 100µm distant from the root plane (Rodionov etal.
2019), but more likely extends in the mm range (Jones
etal. 2004). Our first working assumption will be to assim-
ilate 13-C labelled C to diffusible dissolved organic matter
(DOC, i.e., readily available C for microorganisms) and all
non-labelled C within a 100–1000µm-rhizosphere-range
as particulate organic matter (POM, that can be mineral-
ized under the action of microbial enzymes).
The initial physico-chemical environment
described so far will be the set up where the microor-
ganisms evolve as controlled by the individual-based
bacterial model. The 3D distribution of microorgan-
isms has been obtained in bare soil (Eickhorst and
Tippkötter 2008; Juyal etal. 2019; Nunan etal. 2001),
but is largely unknown to date for the rhizosphere.
To account for this lack of experimental data, our
approach will use a preliminary simulation initialised
using the bacterial numbers and distributions found
in the bare soils. Since the microbial biomass and
activity is modulated by microbial traits and the car-
bon available in the media, the microbial numbers
reached in the preliminary simulation will depend
on the biochemical conditions of the media. Spatial
distributions of bacteria in soil thin sections have
been approached using a 2D spatial statistical model
(Raynaud and Nunan 2014) that can be expanded to
a 3D soil structure. Bacterial biomass and respira-
tion will be monitored in all spatial grid elements and
plotted as a function of the distance to the root plane.
The microbial model can be calibrated using quantita-
tive measures of microbial respiration (i.e., minerali-
zation) of different quantities of root exudates on bare
soil such as the measurements shown in Fig.13.
Model outputs will be qualitatively validated using
microbial biomass and respiration measured at 3mm
Fig. 12 Scheme of the simulation scenarios (a) and detail of
a 1283 sub-domain showing segmented roots (green) and the
water (cyan) and air (grey) phases computed using the lattice-
Boltzmann model described by Genty and Pot (2013) assum-
ing a water saturation level of 50% (b)
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intervals from the rhizosphere plane (Alphei et al.
1996).
A flow chart outlining steps from experimenta-
tion to the generation of model outputs can be found
Fig. 13 Microbial respira-
tion in soil amended with
increasing quantities of
exudates (Ex 1- Ex 4). Note
that the lowest exudate
quantities (Ex1, Ex 2) only
led to a short-term stimula-
tion of microbial respiration
(but not growth), while res-
piration curves with higher
concentrations (Ex 3, Ex
4) show typical microbial
growth dynamics
Fig. 14 Diagrammatic flow chart of case study 5. In the fig-
ure, CT stands for Computed Tomography, LBM stands for lat-
tice-Boltzmann model, µIbM stands for microbial individual-
based model, LA-IRMS stands for laser ablation-isotope ratio
monitoring, and NanoSIMS stands for nanoscale secondary
ion mass spectrometry. The oval shapes illustrate start and end
points of the workflow, the rectangular boxes describe activi-
ties and the rhombs include outputs of intermediate steps that
serve as inputs for the next step in the workflow
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in Fig.14. It combines the workflow of case study 4
(resulting in water distribution in 3D pore space) with
carbon distribution as measured from resin-impreg-
nated soil samples derived from the same experimen-
tal setup and the initial bacterial spatial distribution
as simulated from a bacterial spatial statistical model.
Together with the appropriate boundary conditions,
transport properties and microbial behaviours and
properties the simulation results in a 3D dynamic
system where biotic and abiotic components interact.
Results can be aggregated to provide continuum-scale
properties such as bacterial mass or respiration per
soil control element as a function of time.
Results
The approach highlighted here is a first step towards
the development of mechanistic models accounting
for C distribution, and C transformation by micro-
bial activity around roots. Assuming an equilib-
rium approach and one static root allows us to inte-
grate the various experimental data collected and
use them to predict the shape and distribution of the
microbial activity in the soil surrounding roots and,
more importantly, allows identification of the abi-
otic components or gradients measured driving the
rhizosphere shape and extend. The approach is also
a steppingstone towards the development of the much
needed, fully fletched pore-scale rhizosphere model
(see challenges and open questions section below).
The approach discussed in this section offers spa-
tially explicit information of the bacterial biomass,
the carbon decomposition, and the evolution of bac-
terial respiration over time. This allows estimating
microbial activity namely biomass, respiration and
spread rate as a function of the distance from the root
surface. The use of an individual-based approach
allows also to tackle fundamental and applied ques-
tion related to the maintenance and development of
microbial diversity in the rhizosphere.
Challenges andopen questions
Rhizosphere complexity requires accounting for more
reactive chemical species. For instance, recent publi-
cations highlight the important role of volatile carbon
signals that expand the rhizosphere in the cm range
(de la Porte et al. 2020). In addition to dissolved
organic carbon (e.g., glucose), inclusion of nitro-
gen containing organic compounds (peptides, amino
acids), CO2, O2, and signalling with phytochemicals
such as lipo-chitooligosacharides (Venturi and Keel
2016) would allow to study a number of important
mechanisms underlying microbial colonisation and
establishment in the rhizosphere.
Further differentiation of DOM into specific root
exudates, microbial biomass, microbial necromass
and POM (Angst et al. 2016; Baumert et al. 2021)
can be envisaged and tackled with the same model-
ling approach for an increased accuracy of the model
outputs. This approach may even allow the charac-
terization of priming effects in different aggregate size
classes in the rhizosphere based on microbial stoichi-
ometry (Mo etal. 2021; Wang etal. 2020). Molecular
marker analyses including a suite of exudate-, bacte-
ria- and fungi-specific substances has high potential to
shed more light on rhizosphere gradients and hot spots
of OM enrichment (time-integrated signals/stable on
medium-term in contrast to microbiological param-
eters) (Baumert etal. 2021). As a first step 13C labelled
carbon fraction which can be imaged in 2D with Nano-
Sims or LA-IRMS is used to represent the soluble root
derived carbon pool (exudates) and complement the
C initialisation described and/or to assist validation of
experimental results, especially in view of stoichiomet-
ric constraints for plant and microbial processes (Clode
etal. 2009; Gorka etal. 2019; Vidal etal. 2018).
This first adaptation of existing spatial-explicit
mechanistic models for the rhizosphere follows an equi-
librium approach and one static root. A fully fletched
pore-scale rhizosphere model requires the explicit rep-
resentation of a dynamic root capable of simulating
uptake and release of water and elements from (to) the
soil. In addition to a static water flow scheme, water
dynamic can be also taken into account using LBM
approaches in both saturated and unsaturated porous
media (Ginzburg 2008; Zhang etal. 2016).
Another fundamental issue comes from the huge,
and largely unknown, microbial variability existing
in soil. We advocate for the use of individual-based
modelling of soil microorganisms in conjunction with
a trait-based approach coupling bacterial (Portell etal.
2018) and fungal (Falconer etal. 2015a) models. Bac-
teria and fungi act at different scales and prefer differ-
ent soil phases (i.e., water against air filled porosity),
which suggest the need of this first division. Although
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the whole soil biodiversity is intractable using a
species-centric approach, a trait-based approach can
be used to identify trade-offs in microbial traits and
strategies that lead to an improved rhizosphere colo-
nisation. Such information has practical implications
by extrapolation to specific species. For bacteria, we
suggest accounting for the following minimum set
of individual traits: lag phase length, specific growth
rate, ability to grow anaerobically, mortality rate (by
predators), motility, antibiotic production, root attach-
ment behaviour, and C specificity (DOC, Mucilage,
and plant signalling molecule –e.g., malic acid like).
The consideration at the single-cell level of these
traits would provide insights on beneficial traits and
trait combinations for rhizosphere colonisation and
establishment on root surfaces. For fungi, physiological
trait combinations representative of functional groups
proposed for arbuscular mycorrhizal fungi (Chagnon
etal. 2013): competitors, stress tolerators, and ruderals,
can be adopted. The first challenge is the identification
of model parameters describing these experimentally
unknown behaviours. Relevant questions are: what are
the C costs vs. nutrient and/or water gains associated
with the establishment and maintenance of mycorrhiza
by roots? The adoption of an approach describing indi-
vidual fungi with different properties and behaviours
(Falconer etal. 2015b) will also allow to model inter-
actions between fungal species such as the competition
among symbionts (benefiting from root C) and necro-
trophic pathogens that gain access to plant C only after
killing parts of the root (Sarkar etal. 2019).
Synthesis andOutlook
Detailed pore scale and single root scale information
to predict plant scale emerging behaviour
We have presented a series of case studies illustrat-
ing the effect of rhizosphere scale properties affect-
ing transport across the root-soil interface and micro-
bial activity. We have chosen the overall theme of
rhizodeposition because of its relevance for water
and nutrient uptake and microbial activity. We have
presented two case studies, one in which we have cal-
culated the spatial extent of different rhizodeposits
across the rhizosphere and along the root system, and
one in which we simulated the effect of root exudation
on root nutrient uptake. For both examples, it became
apparent that for growing roots, the elongation rate
was important for the radial extent of the rhizosphere
and the formation of hotspot volumes, with rhizos-
phere Péclet numbers larger or only slightly smaller
than one. This should be considered in experimen-
tal designs such as the choice of sampling locations.
Using image-based modelling for a non-growing root,
we have illustrated how root hairs, at a given exuda-
tion rate, can increase the total exudation as well as
the spatial extent of rhizodeposits into the soil. We
have then discussed the mechanisms by which the
polymers present in mucilage increase the viscosity of
the soil solution and thus alter the spatial configura-
tion of the liquid phase. Specifically, the high viscos-
ity mucilage prevents the break-up of liquid bridges
between soil particles and maintains the diffuse path-
way for solute across the rhizosphere. The effect of
root hairs and mucilage on solute transport across the
rhizosphere could then be implicitly described with
effective diffusion coefficients, as we have shown for
the case of mucilage. Such effective diffusion coeffi-
cients can be implemented in root architecture mod-
els, where the effect of local rhizosphere properties
(defined at the scale of root segments) can be inves-
tigated at the root system case. Finally, we have pre-
sented a pore-scale case study of a model of microbial
activity, which takes explicitly into account the spatial
distribution of soil particles, water, carbon, and micro-
organisms. Such a model allows accounting for emer-
gent microbial driven processes emerging from large
number of biotic and abiotic interactions occurring at
the microscale, informing models using coarser spatial
resolutions. At the same time, this model would ben-
efit from the other case studies described as it depends
on the connectivity of the liquid and gas phases as
well as on the spatial distribution of rhizodeposits.
The potential links between the different case studies
are outlined in Fig.15.
Our objective was not to present a complete and
integrated multi-scale, multi-process model, but
rather to show what ingredients (constitutive equa-
tions, parameters and boundary conditions) should
be implemented in such model. The case studies
described in this contribution show how rhizosphere
traits (such as root hairs and mucilage) impact the
properties (e.g. diffusion) of the rhizosphere at the
single root scale, and how such properties could be
implemented into a root architecture model to inves-
tigate root system scale output (e.g. total uptake
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of nutrients). This is an example of a bottom-up
approach. Such upscaling is done by means of defin-
ing effective properties, as we have illustrated for
mucilage, and as can be done for root hairs using the
homogenization method (Leitner et al. 2010b; Zyg-
alakis etal. 2011). Effective properties depend on the
spatial arrangement of soil particles and of the liquid
phase, which could be imaged in situ, for instance
using X-ray CT. The price to reach a spatial resolu-
tion needed to resolve the pore space is that the field
of view of such images is limited and might be below
the REV (representative elementary volume) needed
to define properties (such as diffusion) at the continu-
ous scale. However, we argue that the spatial extent of
some rhizosphere properties, for instance the spread-
ing of mucilage in soils is smaller than the REV
– i.e. it might cover only a few layers of soil particles.
Therefore, it is allowed to define effective properties
for such processes while having in mind that such
transport properties cannot be used for longer dis-
tance transport. We illustrate this concept with an
example related to water flow. Consider a 2D hetero-
geneous medium composed of elements with varying
conductivity distributed randomly and with the con-
ductivity lognormally distributed. If the medium is
large enough compared to the size of its elements, the
effective conductivity of the discrete medium is the
geometric mean of the conductivities of the elements.
However, if the domain is thin, or the flow process
takes place only in a thin section of the domain, then
the effective conductivity that should be taken con-
verges to the arithmetic mean of the conductivities
(von Jeetze etal. 2020). For the rhizosphere, different
effective diffusion properties should then be defined
Fig. 15 Relations between the different case studies with regard to spatial scale, processes considered and parameter types. The
arrows show which model outputs can be used as inputs or parameters for another case study
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depending on the spatial extent of the process. To our
knowledge, this problem has not been addressed. Next
to the REV, the non-periodic nature of CT images of
the pore space requires extended homogenization
methods on 3D rather than 2D images that are able to
study the influence of potential boundary effects.
Beside the complexity of defining and determining
effective properties, the advantage of such properties
is that they can then be implemented in larger scale
continuous models in which root architecture and
root growth can be explicitly modelled. This bottom-
up approach allows testing scenarios and qualitative
behaviours. For instance, it can be used to investigate
in what conditions and for what emergent behaviour
(e.g., water uptake, P uptake, etc.) processes that
affect the local transport properties in the rhizosphere
are relevant on the plant scale. Such analysis can
be done only at the plant scale where the appropri-
ate boundary conditions and vertical gradients in soil
variables are explicitly simulated. A problem with
posing this question is that the meaning of ‘relevant’
needs to be defined. An at-hand definition would be
that using bulk soil properties instead of considering
the rhizosphere properties for predicting processes at
the plant level might lead to incorrect estimates. This
means that plant system scale processes are sensitive
to rhizosphere properties. These sensitivities may
also be dependent on the environmental conditions.
For instance, in wet soils, the impact of mucilage
on water uptake is not important whereas it is in dry
soils. If some roots of the root system still have access
to wet soil, root water uptake by the entire root sys-
tem will hardly be affected by the impact of the muci-
lage around roots in the drier soil layers. When the
sensitivity question is answered positively, the next
question is whether the same prediction of processes
at the entire plant system scale could be obtained by
using different bulk soil properties or using different
root system properties. If the answer to this question
is yes, then an improper representation of rhizosphere
processes or properties could be simply ‘compen-
sated’ by adjusting root or soil properties. The ques-
tion translates to whether sensitivities of the behav-
iour at the plant scale to rhizosphere properties, root
system properties, and soil properties are correlated
or not. It implies that when processes are observed
at the plant system scale, rhizosphere properties can-
not be derived (e.g., by inverse modelling) without
knowing the other correlated properties. In order to
unravel the ‘relevance’ of rhizosphere properties and
processes for plant system scale processes, sensitiv-
ity analyses with multiscale models could be carried
out and used to identify interaction effects (i.e., when
the change in the larger scale process is larger than
the sum of the changes due to changes in rhizosphere
properties, bulk soil properties and root properties).
Conditions when such interaction effects occur could
then be used to infer rhizosphere properties from the
observed emergent behaviour at the larger scale.
However, upscaled simulations require not only the
effective properties in the rhizosphere at a given point
in space, but also information on how these properties
evolve over time and along the root system. This requires
then a top-down approach and a general cross talk of
models at different scales: the large scale to define the
boundary conditions and state variables for the small-
scale process, and the small-scale to estimate the effective
properties for the large scale model. The behaviour of the
system emerges from the interactions between the scales.
The importance ofinteractions betweendifferent
processes
In addition to the interactions between scales, the emerg-
ing rhizosphere behaviour is also the result of the inter-
actions between different simultaneously occurring
processes. Here we have focussed on the example of
rhizodeposition, and its interaction with root growth, soil
water flow, nutrient uptake, and microbial growth and
respiration. However, these are still limited examples
that could be extended. More generally, nutrient uptake
affects the plant status and most likely the quality and
quantity of exudation, resulting in a feedback loop. Root
water uptake takes also part to such feedback, due to
convective fluxes affecting solute distribution and to the
gradients in soil water potential and soil moisture around
the roots, which in turn affect diffusion. De Bauw etal.
(2020) illustrated that interactions between water uptake
and nutrient uptake at the plant level scale could be
linked to a combination of rhizosphere transport and
root system scale water uptake. Understanding and prop-
erly implementing into models interactions between
these processes, and their feedback with plant growth
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is a key to predict the emergent effect of rhizosphere
properties on the plant scale. While there is an understand-
ing on the effects of certain drivers such as drought on indi-
vidual rhizosphere processes, the effects of these drivers on
multiple, simultaneously occurring processes are still poorly
understood. This requires the extension of existing models
to a larger number of rhizosphere components.
How can measurements be used inmodelling?
State oftheart andchallenges
In the frame of a current priority program focusing
on rhizosphere spatiotemporal organization (PP 2089,
https:// www. ufz. de/ spp- rhizo sphere/) it was hypoth-
esized that spatiotemporal patterns in the rhizosphere
are formed as a result of local interactions and numer-
ous feed-back loops (self-organisation processes) and
that these patterns result in emerging system proper-
ties for the soil–plant system. Scientist were encour-
aged to work with the same experimental platforms
with a very reduced number of drivers (two textures,
two genotypes, four growth stages, few compart-
ments) in order to provide as many inputs for the dif-
ferent models operating at different scales. Can this
be a successful strategy? The following sub-sections
will tackle this question.
Parameters forbottom‑up aswell astop‑down
modelling approaches
The focus on joint experimental platforms with a few
drivers only, provides for the same system informa-
tion at a range of scales. If we take carbon flow as an
example, the experimental data range from changes
in standing above and below ground biomass in the
field during the whole growth season, to root system
exudation rates and their change with growth stage, to
spatial distribution of recently assimilated carbon in
the vicinity of roots at the µm scale. Thus, data from
different scales can be used as input parameters and
for validation, respectively. For all these scales, spa-
tially resolved, quantitative and qualitative informa-
tion is available; however, the level of detail changes
with the scale and the units cannot always be con-
verted directly into each other.
Experimenter can never provide all therequired
data – bottom‑up
In describing the rhizosphere, we are forced into
high-resolution data acquisition because some of the
players are very small and some of the gradients are
very steep. This is complicated by the fact that the
closer we look, the more complex things can become.
While we move from the continuum scale to the pore
scale for soil, we must move from the single root
scale down to the tissue scale for roots, and seem-
ingly well-established knowledge, such as smooth
gradients extending from the root surface into the
root (what is happening in the plant), are challenged
by questions such as where along the radial gradient
is uptake or release actually taking place within the
root (Sakurai etal. 2015), where along the root do we
have to measure, which is the tissue concentration rel-
evant for calculating 13C diffusion into the soil – 13C
concentration of tangential walls of the endodermis
or rather the mean 13C concentration of grind up root
tissue from the whole root system? What adds to the
experimenter’s dilemma is that the higher the resolu-
tion, the more cumbersome it is to work with a rea-
sonable number of replicates. That is, while we can
now derive very detailed information, the modeller
and experimenter must jointly develop strategies to
test representativeness or plausibility. From a model-
ling point of view, the need of accounting for what
happens in the plant suggest that strategies to couple
rhizosphere pore-scale models to single-cell models
of the root physiology and development (e.g., Dupuy
etal. 2008) should be investigated in the future.
Labelling experiments with radioactive 11C carbon
(or 14C) provide information where recent assimi-
lates are transported in relation to root type and
position along the root (Schulte et al., in prepara-
tion; Holz etal. 2018), but such information is only
available for distinct time points and for very young
plants. Likewise, stable carbon isotopes (13C) can be
used as a tracer for recent assimilates and the radial
spread of 13C across the root tissue into the soil can
be mapped in relation to tissue type and particle dis-
tribution, however, only for a small number of sam-
ples. Chemical quality of rhizodeposits can also be
mapped and quantified with a resolution of 20µm as
recently shown for individual disaccharides (Lohse
etal. 2021). With similar resolution, radial gradients
of elements around roots can be mapped in soil using
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different microscopy techniques (Vetterlein etal. 2020).
Only a few of these techniques, e.g., µXRF, have so far
shown the potential to process a larger number of repli-
cates (Lippold etal. in preparation). Methods to meas-
ure exudation rates from soil-grown plants (Oburger and
Jones 2018) are available and have great potential also
for model parameterisation (see case studies 1 and 2).
For structural gradients the available tools are
already more powerful. With a spatial resolution in the
range of 10–20µm bulk density gradients around roots
were investigated and statistically evaluated in relation
to a range of different drivers (Phalempin etal. 2021).
It is a major advance that approaches have now been
developed for chemical and structural parameters that
truly measure the magnitude of radial gradients, in con-
trast to measurements in linearized (compartmentalized)
or pseudo-linearized (rhizoboxes) systems. That we tend
to see narrower zones with these methods (Lohse etal.
2021, Lippold et al. in preparation) is consistent with
model predictions (Vetterlein etal. 2020, Fig.4).
Microbiota inmodels
A very special case are the data on rhizosphere
microbiota, the very tiny amounts of sample required
for sequencing studies enables investigation of com-
munity composition down to the single aggregate
level(Szoboszlay and Tebbe 2021) and network anal-
yses provides details on the interaction of the differ-
ent species and how sensitively this is controlled by
external drivers. Yet we lack quantitative data on
functional properties or activities at the same resolu-
tion. There is a scarcity of data regarding the explicit
distribution of microbes in general and of active ones
in particular. Likewise, their habitat demands in rela-
tion to soil structure and resulting water, air filled
pore space and carbon distribution can be addressed
by modelling approaches, but validation of model
output is challenging. Given the complexity and
difficulty of obtaining non-destructive measures,
confidence in models can be increased using simul-
taneously many measurable outputs related to the
process(es) of interest as advocated by the so called
pattern-oriented modelling approach (Grimm et al.
2005). In our opinion, validation using data from dif-
ferent levels of organisation such as emerging proper-
ties like respiration, change of C content and alike, or
less available spatially explicit data (e.g., distribution
of microorganisms on the root surface) can effec-
tively test the assumptions included in the models,
even when the data are not measured in the same
experiment. Direct imaging of microbes in the pore
space is possible with unspecific stains or Card-FISH
(Catalysed Reporter Deposition-Fluorescence in situ
Hybridisation), combined with stable isotope label-
ling, but the procedures are so tedious that it will be
long until large datasets will be available. In view of
the dynamic changes of microbiota along roots as they
grow (Bonkowski etal. 2021) it is an open question
whether focussing on specific microbes (CARD-FISH)
or rather unspecific approaches are more promising for
a given purpose. Each approach is able to tackle differ-
ent questions. Targeted approaches can deal with bio-
diversity related questions or to focus on species with
practical interest. An unspecific approach is more ame-
nable to address question where the biodiversity is not
the target but rather questions of CO2 release or total
microbial biomass around roots.
The art ofchoosing theright drivers
Allowing for a few drivers only (in the priority pro-
gram texture and genotype) bears the risk that impor-
tant ones are missed out, or the chosen ones prove to
be of no importance for the parameter in question. It
turned out that both the soil- and the plant-oriented
driver led to very interesting, sometimes surprising
results. Two such opposing drivers have also served
the function of motivating representatives of different
disciplines to work together: i.e. plant scientists and
microbiologist realized that their results are strongly
dependent on texture, while physicists were chal-
lenged by the fact that plant’s feedback mechanisms
are so smart that irrespective of substrate properties
resources were exploited completely (Jorda etal. in
preparation, Vetterlein etal. in preparation).
Core messages
To summarise the results of this opinion paper, we
formulated the following core messages:
– Rhizosphere processes occur at multiple spatial
and temporal scales. There is a large potential that
models at different scales inform each other.
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– Despite increasing complexity of rhizosphere
models, feedback loops are still underrepresented.
– Modelling reveals that 3D architecture of roots,
root hairs and soil (soil structure) and mucilage
and their temporal dynamics have a strong impact
on emerging properties/efficiency of processes.
– Pore-scale modelling has the potential to capture
system behaviours emerging from the myriad of
microscale biotic and abiotic interactions, comple-
menting existing continuous models.
– Continuum-scale models can be informed about
pore-scale processes through the derivation of
effective parameters or through more complex
upscaling approaches.
– For the first time, an approach combining model-
ling at different scales and multimodal imaging of
the rhizosphere allowing to integrate outputs from
the microscale to the whole root system has been
outlined. Thereby, the modeller and experimenter
must jointly develop strategies to test representa-
tiveness or plausibility.
Further challenges andpath ahead
Linking between more than two spatial scales or pro-
cesses is still a challenge. Linking multiple scales and
processes is the logical next step for future research
and needs the combined use of experimental and
modelling approaches.
So far, the impact of rhizosphere properties and
processes have been discussed from the soil perspec-
tive. Except for water flow, a link between processes
within the plant and how these react to or are coor-
dinated with rhizosphere processes is still missing.
Understanding which within-plant mechanisms con-
trol C-exudation is important to understand inter-
actions between exudation and rhizosphere condi-
tions. How local conditions influence growth such as
mechanical rhizosphere properties but also transport
properties that influence transport of signalling sub-
stances, e.g. ethylene, and thereby growth are exam-
ples of two-way feedbacks between growth and rhizo-
sphere properties and processes.
Another aspect is the temporal scale of the rhizos-
phere organisation. In this paper (and the priority pro-
gram), we focussed on the dynamics around a grow-
ing root system of an annual plant (maize). Whether
such a rhizosphere system has a legacy after the plant
died off that is of benefit for subsequent generations
or whether it rather has negative effects requires fur-
ther research. This would call for investigating multi-
annual self-organisation effects of the rhizosphere in
annual cropping systems. Understanding these effects
is important to design and manage crop rotations and
no-till systems.
Author contributions Our opinion is the result of many dis-
cussions we have had during meetings and workshops within
the framework of the German priority program “Rhizos-
phere Spatiotemporal Organisation – a Key to Rhizosphere
Functions”.
Funding Open Access funding enabled and organized by
Projekt DEAL. This project was carried out in the framework
of the priority programme 2089 Rhizosphere spatiotempo-
ral organization-a key to rhizosphere functions funded by the
German Research Foundation DFG under the project numbers
403633986, 403635931, 403640293, 403640522, 403641034,
403668613, 403660839, 403670197, 403670844, 403801423,
403803214. This work has partially been funded by the Ger-
man Research Foundation under Germany’s Excellence
Strategy, EXC-2070 – 390732324 – PhenoRob. XP and WO
acknowledge funding from the Natural Environment Research
Council (NE/S004920/1).
Data availability All data are available on request from the
authors.
Code availability Code is available on request from the
authors.
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