In 1978, Landsberg and Fowkes presented a solution of the water flow equation inside a root with uniform hydraulic properties. These properties are root radial conductivity and axial conductance, which control, respectively, the radial water flow between the root surface and xylem and the axial flow within the xylem. From the solution for the xylem water potential, functions that describe the radial and axial flow along the root axis were derived. These solutions can also be used to derive root macroscopic parameters that are potential input parameters of hydrological and crop models. In this paper, novel analytical solutions of the water flow equation are developed for roots whose hydraulic properties vary along their axis, which is the case for most plants. We derived solutions for single roots with linear or exponential variations of hydraulic properties with distance to root tip. These solutions were subsequently combined to construct single roots with complex hydraulic property profiles. The analytical solutions allow one to verify numerical solutions and to get a generalization of the hydric behaviour with the main influencing parameters of the solutions. The resulting flow distributions in heterogeneous roots differed from those in uniform roots and simulations led to more regular, less abrupt variations of xylem suction or radial flux along root axes. The model could successfully be applied to maize effective root conductance measurements to derive radial and axial hydraulic properties. We also show that very contrasted root water uptake patterns arise when using either uniform or heterogeneous root hydraulic properties in a soil–root model. The optimal root radius that maximizes water uptake under a carbon cost constraint was also studied. The optimal radius was shown to be highly dependent on the root hydraulic properties and close to observed properties in maize roots. We finally used the obtained functions for evaluating the impact of root maturation versus root growth on water uptake. Very diverse uptake strategies arise from the analysis. These solutions open new avenues to investigate for optimal genotype–environment–management interactions by optimization, for example, of plant-scale macroscopic hydraulic parameters used in ecohydrogolocial models.

Global crop production is negatively affected by drought, which is the most
significant stress in agriculture

Measurements of root hydraulic properties demonstrated that radial
conductivity and axial conductance both change with root tissue maturation
for a given plant genotype

Despite this evidence that uniform root properties are more the exception
than the rule, today a majority of models assume explicitly

An analytical solution for root water uptake and flow was developed by

Top: maize radial conductivity (left axis) and axial conductance
(right axis) of primary roots as measured in several studies. Bottom: maize
primary root cross sections obtained at different development stages. The
root cross sections are the work of

However, solving the water flow equation in the root system can
also be achieved using finite difference for any root hydraulic property
distribution

In this paper we show that uniform root property assumption may be relaxed
and yet analytical solutions of the water flow equation in roots are within
our reach. Our objective is to present novel mathematical solutions of the
water flow equation in roots with non-uniform radial and axial hydraulic
properties closer to reality and more efficient than current existing models.
We also developed solutions for growing roots at given elongation rates,
which make the uptake distribution time dependent. This widens the solution
of

In the following, we only consider single root, i.e. without laterals. Consequently we sometimes use simply the word roots for single roots. When used, the terms root stretch or root segment designate a portion of a single root characterized by specific root properties. We use the abbreviations L, T and P for length, time and pressure unit dimensions, respectively.

Assuming that root water content does not fluctuate, water mass balance in
infinitesimal root segments of a cylindrical root of radius

The differential Eq. (

For simple functions

Deriving the coefficients

Single root made of five stretches (the dashed vertical lines are
stretch boundaries). For root stretch

Only

The root macroscopic parameters consist in the root system effective
conductance

Local hydraulic properties

Linearly independent functions

Local hydraulic property function parameters, their units, the expression of combined parameters and their corresponding units.

We here analyse six cases of hydraulic conductance variations along a root
axis: the uniform root (already developed by

The parameters used in Table

These functions can be combined in complex roots with several root stretches
with Eqs. (

Figure

If the root is made of only one stretch, there is no need to calculate
intermediary effective root conductances. The solutions of Table

Flowchart of the water flow equation resolution in roots with heterogeneous hydraulic properties.

The water flow equation resolution derived in the previous sections was obtained for a root of a specific length. In the next sections, we show how to modify the solution when the root is growing and developing.

In this section we introduce the root elongation. As in many studies, the
properties are measured as a function of emerging time instead of distance to
tip, and we provide here a tool to switch from one to another. Basically an
equation of the root elongation rate is required. We consider here an
instantaneous growth rate

When roots get older, their macroscopic hydraulic parameters vary not only
because they grow but also because of root tissue maturation. Maturation is
defined here as an evolution of root hydraulic properties as a function of
root age (and not of distance to root tip as done in the previous sections).
This process is modelled by introducing root hydraulic properties depending
on root age, such as the following:

In this section, we highlight the potential of the new functions with a modelling exercise. We start with simple theoretical illustrations of the model. We then explain how pressure probe measurements can be used with modelling to derive local hydraulic properties. These results are then inserted first in a soil–root model to test the uptake efficiency of a heterogeneous root compared to a uniform one, and then in an optimization algorithm to assess an optimal radius of the analysed roots. Finally we show how root versus development rates can reveal very contrasted uptake strategies. Another added value of analytical solutions is their potential use to verify the current numerical solution. One example is given in Appendix E.

First, three theoretical roots were simulated. Figure

Variation with distance to root tip of radial conductivity

The water flow equation is then solved for these three roots and the xylem
potential as well as the radial and axial flow is computed to assess their
divergences. Xylem water potential is obtained thanks to Eq. (

We also define two root parameters combining root hydraulic and geometric
properties.

Here, we compare single roots with complex hydraulic profiles (as they have
been observed) and constant root hydraulic properties in terms of
distribution of water xylem potential, radial and axial flows and macroscopic
parameters. Lateral maize root properties from

A root pressure probe was used to measure root effective conductance as well
as axial conductance of unbranched brace maize roots. To do so, nine maize
plants were grown in aluminium containers filled with silty soil. When plants were
7 weeks old, the containers were opened, roots were carefully washed from
the soil and selected maize brace roots were excised from the stem. They were
then connected to a pressure probe in order to derive their hydraulic
properties in an experiment similar to the one first presented by

Unbranched intact roots were around 35

Axial hydraulic conductance was determined after cutting the root connected
to the root pressure probe with a razor blade at a distance of 2

The profile of axial conductance was first fitted using piecewise functions.
Absolute values and transition positions of axial conductances were both optimized. When obtained, further optimization was required to derive the
profile of radial conductivity using the effective root conductance.
Several scenarios, including uniform, single- and multi-stretch hydraulic
properties, were tested. Since the number of fitting parameters was not
constant between scenarios, an adjusted coefficient of determination allowed us
to discriminate the best scenario. A uniform root (both radially and
axially) was also tested to compare the results with the solution of

The uniform and heterogeneous hydraulic profiles that best fitted the
measurements of the previous experiments were then tested to compute the
macroscopic parameters,

The new solutions of the water flow equation are key to estimating optimal
geometric properties of roots. As an illustration we may want to maximize the
effective conductance

In this last example, we again consider a single growing root whose
development and elongation rates can both vary. We varied the root parameters

Solutions of the water flow equation are shown in Fig.

Solutions of the root water flow equation: xylem water potential

Even if the collar axial flow is identical for each root (because they have
the same effective conductance), their xylem water potential and root water
uptake profiles differ. The potential drop is more homogeneously distributed
along the roots with heterogeneous properties. Furthermore, it is observed
that

Root parameters

Variation along a single root of root (top line) and macroscopic
(bottom line) hydraulic parameters:

The local hydraulic property distribution for the lateral roots of maize, as
derived by

Distributions of radial conductivity

We used Eq. (

Macroscopic parameters: final SUD

Although

The analytical solutions developed here may also be used for obtaining local root
hydraulic properties distributions along root axes (see below)
experimental measurements of the observed axial and effective conductances as
measured by the root pressure probe (represented with black markers in of Fig.

Hydraulic properties of the maize brace roots: axial and effective
conductance of different root types as function of distance from root tip
(dark symbols, panels

Clearly it appears that there are different solutions possible for the
radial conductivity variations (light blue and mauve are equivalently good).
However, as shown in panels (b) and (c), the orders of magnitude of

The combination of pressure probe experiments with the newly developed
solutions of the water flow equation in the routine would allow us to derive the
local hydraulic properties of roots that are critical for root system water
uptake and plant performance

Root performances in a coupled soil–root model: changes of
instantaneous and cumulative water uptake

Effective conductance as a function of root radius under a volume constraint for a uniform root whose hydraulic properties depend on the root radius (solid line), a uniform root (dashed line) and a root with observed heterogeneous conductivity profiles (dotted line). The red stars denote the optimal radii (i.e. that maximize the effective conductance), the black star the observed one.

When inserted in a soil–root model, the maize brace uniform root and the one
with the best scenario of hydraulic properties present contrasted uptake patterns
and performances, as revealed in Fig.

The uptake location can be seen from the clipped domain where water
velocities at the end of the simulations are shown. A video of the change in
the main water uptake depth is also provided in the supplementary material.
While after 5 days, the

Integral of the effective root conductance over time when changing
the elongation rate

This simulation underlines how critical the root water uptake location is for the root performance. The environment (that was not changed in the presented simulation) is, of course, important for the overall plant transpiration. The soil hydraulic properties (that redistribute more or less water) and the climatic demand (that is more or less severe) strongly influence the results. However, we show that the water flow equation solutions, through the macroscopic parameters, can be inserted in water flow models, i.e. in heterogeneous environments to predict how efficient a particular root is.

In this section, we look for optimal root traits that would maximize the root
water uptake. As an illustration, we compare the effective conductances of a
uniform single root, a uniform single root with hydraulic properties varying
with its radius, and a single root with exponential hydraulic property
profiles. The effective root conductances of the three cases are shown in
Fig.

Unlike previous approaches

Figure

Six new solutions of the water flow equation in single roots with different
hydraulic property distributions are presented in addition to the uniform
hydraulic property solution of

This enabled us to investigate the effects of root maturation or root tissue development and differentiation on root water uptake. This gave interesting perspectives to evaluate both growth and maturation and their combined effects on root water uptake. We demonstrated how combinations of different maturation and growth functions lead to different strategies of water uptake.

These solutions were also used to revisit optimal root geometrical parameters
for water uptake. Indeed, the

The new models can be used to derive local hydraulic properties of roots or be
combined as building blocks to generate complete root system hydraulic
architectures defining plant genotypes in order to compare plant performances
in contrasted environments using soil–plant models such as R-SWMS

The code is available upon request and can be freely shared.

In this Appendix, we provide detailed solutions of the water flow equation
when the root hydraulic properties are constant

The solution of the water flow equation in a uniform root has been proposed
by

The simplest root is made of constant root properties, axial conductance

The coefficients are obtained for the bottom no-flux boundary conditions in
the case of single stretch root (see Appendix C and particularly
Eq.

In the case of roots with linear hydraulic property profiles, different cases are distinguished and are investigated successively.

Equation (

Equation (

If the root axial conductance varies while the radial conductivity is constant, i.e.

We assume now a linear relation between the hydraulic properties and the distance to the
tip.

Let us finally consider a root whose root hydraulic properties vary exponentially along the root axis:

The water flow equation becomes

Solutions of this differential equation are of the following type:

The cases of mixed constant and exponential hydraulic property may be easily solved using the same methodology.

To calculate the general form of the macroscopic parameters defined as
follows.

If the root system consist in a single stretch, the distal boundary condition
is no-flux and the proximal boundary conditions corresponds to the root
collar water potential

The macroscopic parameters of the single-stretch root are finally given by
the following (using the no-flux coefficients, Eq.

Let us consider a growing single root with an initial elongation rate

Inverting Eq. (

One of the main advantages of analytical solutions is their possible use to
verify the accuracy of numerical algorithms. All the developed solutions
should be asymptotic solutions provided by numerical algorithms for
infinitely small root segments. In Fig.

Numerical approximation (blue dashed line) versus analytical solution (dark solid line). The smaller the segment size, the better the numerical accuracy.

As the root segment size decreases, the numerical solution tends towards the analytical solution in terms of both xylem potential and axial flow.

FM solved the water flow equation with the help of VC and developed the model used in the simulations. JV, VC, MJ and X suggested simulations to illustrate the model and helped in the model conception. MZ performed the experiment, treated the data and helped in the analysis of the results. FM prepared the paper with contributions from all co-authors.

The authors declare that they have no conflict of interest.

During the preparation of this paper, Félicien Meunier was supported by the Fonds National de la Recherche Scientifique of Belgium (FNRS) as Research Fellow and is grateful to the organization for its support. This work was also supported by the Belgian French community ARC 16/21-075 project. Valentin Couvreur was supported by the Interuniversity Attraction Poles Programme – Belgian Science Policy (grant IAP7/29). Edited by: Anas Ghadouani Reviewed by: two anonymous referees