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 r (L) and total length l (L) under uniform soil–root interface water potential yields : dJx(z)dz=-2πrkr(z)Ψx(z)-Ψsoil, where Jx(L3T-1) is the axial flow of water within the xylem in the root, kr(LT-1P-1) is the root radial conductivity, Ψx(P) is the xylem water potential, Ψsoil(P) is the uniform water potential at soil–root interface, and z(L) is the distance from the root tip along the root axis. The axis z is always chosen parallel to the root axis. Note that the right-hand-side term corresponds to root radial flow rate per unit root length qr(L2T-1): qr=-2πrkr(z)Ψx(z)-Ψsoil. Axial flow is driven by the water potential gradient in the xylem vessels: Jx(z)=-kx(z)dΨx(z)dz, with kx(L4T-1P-1) the axial conductance of the root. Combining Eqs. () and (), we obtain the following: ddzkx(z)dΨx(z)dz=2πrkr(z)Ψx(z)-Ψsoil, which is the general equation of water flow equation in roots.

The differential Eq. () can be solved for various distributions of root properties and boundary conditions. Since Eq. () is a second-order differential equation, its general solution is of the following form: Ψx(z)=Ψsoil+c1,if1,i(z)+c2,if2,i(z), where c1,i and c2,i are constants whose values depend on root hydraulic properties and boundary conditions at roots' ends, and f1,i and f2,i are differentiable functions of z whose type depends on the root hydraulic property profiles. The subscript i, as will be further explained, is used to distinguish root stretches. It can vary between 1 and N, the total number of stretches in the single root. The length between the root apex and a root stretch proximal part is called li. When i=N, li=l.

For simple functions kx(z) and kr(z) (i.e. constant, linear and exponential), analytical expressions for f1(z) and f2(z), c1,i and c2,i are derived in Appendix A. However, kx(z) and kr(z) profiles along a root generally correspond to piecewise collections of these simple functions (see for example ). Hence, we establish a procedure to compute analytical expressions of water flow in a single root with segments connected in series with contrasted hydraulic property profiles. Figure  presents a sketch of a single root made of five stretches delimited by dashed vertical lines.

Deriving the coefficients c1,i and c2,i in any root stretch i requires boundary conditions at the limits of each stretch (i.e. at z=li-1, the root stretch i's distal end and at z=li, its proximal end). The bottom flux boundary condition at the distal end of stretch i is called Ji-1(L3T-1), and the xylem water potential at the proximal end of stretch i is Ψproximal,i(P) as it appears in Fig.  and as stated in Eq. (): Jx(li-1)=Ji-1Ψx(li)=Ψproximal,i. Note that J0=0 (no axial flow at the root tip), and Ψproximal,N=Ψcollar (the xylem water potential at the proximal end of the last root stretch N is the plant collar potential).

Only J0 and Ψproximal,N are predefined root boundary conditions and none of the root stretches has both boundary conditions known (Ji-1 and Ψproximal,i), so the solution of the water flow equation is not straightforward unless the root is made of a single stretch. To solve this problem, the concept of root macroscopic parameters is used.

The root macroscopic parameters consist in the root system effective conductance Krs(L3T-1P-1) and the standard uptake density SUD(L-1) . These parameters are used in soil–root water transfer models that stem from principles of water flow in root hydraulic architecture but do not rely on an explicit geometrical description of root system . In contrast to Couvreur, revisited these definitions and applied them to any root or part of a root system. They are calculated in homogeneous soil conditions and are defined here as function of the root length li as follows. Krs,i=Jx(li)Ψsoil-Ψx(li)SUDi(z)=qr(z)Jx(li) These macroscopic parameters are always independent of the boundary conditions (Ψsoil and Ψcollar): they only depend on root geometric (radius and length) and hydraulic properties (kr(z) and kx(z)). Let us note that the Krs,i and SUDi as defined in Eq. () represent the root macroscopic parameters after addition of i stretches, thus ignoring the root stretches after the considered zone. The terms Krs and SUD are used instead of Krs,N and SUDN, respectively, and correspond to the macroscopic parameters of the entire root. Krs=Jx(l)Ψsoil-Ψx(l)SUD(z)=qr(z)Jx(l) The macroscopic parameters may be calculated by a recursive equation (see Appendix B for demonstration): Krs,i=kx(li)-Krs,i-1f2,i(li-1)+f2,i′(li-1)kx(li-1)f1,i′(li)+Krs,i-1f1,i(li-1)-f1,i′(li-1)kx(li-1)f2,i′(li)Krs,i-1f1,i(li-1)f2,i(li)-Krs,i-1f1,i(li)f2,i(li-1)-f1,i′(li-1)f2,i(li)kx(li-1)+f1,i(li)f2,i′(li-1)kx(li-1)SUDi(z)=2πrkr(z)kx(li)-Krs,i-1f2,i(li-1)+f2,i′(li-1)kx(li-1)f1,i(z)+Krs,i-1f1,i(li-1)-f1,i′(li-1)kx(li-1)f2,i(z)-Krs,i-1f2,i(li-1)+f2,i′(li-1)kx(li-1)f1,i′(li)+Krs,i-1f1,i(li-1)-f1,i′(li-1)kx(li-1)f2,i′(li) where Lagrange notation is used for derivative. The recursive formulation is useful to solve the water flow equation when dealing with multiply stretched roots. We start, in such a case, with the calculation of Krs,1 (i.e. the effective conductance of the most distal stretch) using Eq. () (with Krs,0=0 because the axial flow is null at root apex). The obtained Krs,1 is then used as Krs,i-1 to calculate the effective conductance of the distal part from the second stretch Krs,2 using Eq. (). This procedure is then used again to derive the Krs,i of all stretches until the root collar is reached. The obtained set of Krs,i or Krs,i-1 values are then used to calculate the coefficients c1,i and c2,i for the different stretches using the following: c1,i=Ψproximal,i-Ψsoil-Krs,i-1f2,i(li-1)+f2,i′(li-1)kx(li-1)Krs,i-1f1,i(li-1)f2,i(li)-Krs,i-1f1,i(li)f2,i(li-1)-f1,i′(li-1)f2,i(li)kx(li-1)+f1,i(li)f2,i′(li-1)kx(li-1)c2,i=Ψproximal,i-1-ΨsoilKrs,i-1f1,i(li-1)-f1,i′(li-1)kx(li-1)Krs,i-1f1,i(li-1)f2,i(li)-Krs,i-1f1,i(li)f2,i(li-1)-f1,i′(li-1)f2,i(li)kx(li-1)+f1,i(li)f2,i′(li-1)kx(li-1) as demonstrated in Appendix B. Since, in Eq. (), only the potential at the proximal part of the stretch is needed as a boundary condition (in addition to the distal effective root conductance), the calculation starts at the proximal end of the root system where the root collar potential Ψcollar is known. The obtained c1,i and c2,i for the proximal stretch are subsequently used in Eq. () to calculate the water potential at the distal part of the stretch. The water potential is then used to calculate the coefficients of the next stretch (towards the root tip). This procedure is used until the most distal stretch is reached. As SUD depends on the collar water flow, it can not be calculated for each zone. However, it can be derived at the end of the procedure when the xylem water potential is defined everywhere inside the root and when the effective root conductance has been already calculated.

We here analyse six cases of hydraulic conductance variations along a root axis: the uniform root (already developed by ), a single root with linear root hydraulic property profiles and a single root with exponential root hydraulic property profiles. For the two latter cases, the radial conductivity and the axial conductance may change alone or simultaneously. Table  summarizes the six cases with the corresponding local hydraulic properties. Note that the linear increase of axial conductance was already studied by . Table  summarizes the solutions of the root water flow equation obtained for the six considered cases. The resolution details are provided in Appendix A.

The parameters used in Table  are gathered as well as their units in Table  (two first columns). To simplify the solutions of the root water flow equation, some parameters are combined. The definition of combined parameters is also given in Table  with their corresponding units (two last columns).

These functions can be combined in complex roots with several root stretches with Eqs. (), () and the entire root boundary conditions: Ψcollar and J0=0.

Figure  shows the procedure to solve the water flow problem in a root with variable hydraulic properties. First we need to know whether the root is made of one or several stretches. Then we have to determine the coefficients c1,i and c2,i for each single zone, i∈[1N], by determining the type of root stretch we deal with and by using Eq. (). Thanks to these coefficients, we obtain the effective root conductance after addition of each stretch by applying Eq. (). Finally we calculate the xylem potential, the axial flow, the radial flow per root length and the macroscopic parameters using the appropriate equations and analytical functions given in Table . The corresponding equations are mentioned in the figure.

If the root is made of only one stretch, there is no need to calculate intermediary effective root conductances. The solutions of Table  are then used with the no-flux boundary condition coefficients to obtain the macroscopic parameters as well as the water xylem potential radial and axial water flow profiles along the root axis. This particular case is analysed in Appendix C.

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.

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 . We only give the general idea here. For more details, we refer to the explanations of .

Unbranched intact roots were around 35 cm long when first connected to the root pressure probe. A series of pressure increase steps were carried out (3–5 steps) to induce a flow of water into the root after a stable value of root pressure was reached, which took between 30 and 120 min. Increments of 20 to 50 kPa were used for successive pressure steps. Pressure was held constant at each increment for 10 to 120 s. The distance that the metal rod (meniscus) inside the capillary had moved was used to calculate the water volume and as a consequence the flow rate, by dividing by the duration of the step. The linear regression of flow rate plotted against the applied pressure difference led to the effective root conductance for this specific length. The root was then cut towards its tip and reconnected to the root pressure probe to repeat the measurement until the root end was reached.

Axial hydraulic conductance was determined after cutting the root connected to the root pressure probe with a razor blade at a distance of 2 cm from its proximal end. Similar to the measurement of effective conductance, a series of pressure clamps were carried out to induce water flow into the small root segment. The local axial conductance was calculated as the product of the segment length by the slope of the linear regression of flow rate plotted against the applied pressure difference.

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 . In total six scenarios were considered: uniform radial conductivity and axial conductances alongside a linear stepwise distribution for the axial conductance (that corresponded to the observations; see results section) combined with five different root radial conductivity distributions. The latter are linear, exponential, 3-stepwise, 2-step linear piecewise and 3-step linear piecewise distributions. Together they represent the complete set of solutions that were presented in this study.

The uniform and heterogeneous hydraulic profiles that best fitted the measurements of the previous experiments were then tested to compute the macroscopic parameters, Krs and SUD, for any root length. These calculations allowed us to simulate the water flow in the soil–root continuum. To do so, we used the model developed by and , which only requires the macroscopic parameters (in terms of root parameters) to predict the water flow from the bulk soil to the root collar. The soil sink term is calculated as the sum of the local radial flow to the root which, in turn, is given by qr(z)=TactSUD(z)+KrsΨsr(z)-ΨseqSUD(z), with Tact(L3T-1) the proximal end water flow, Ψsr(z)(P) the soil–root interface potential and Ψseq=∫0lSUD(z)Ψsr(z)dz(P) the equivalent soil potential. In the simulations, the root were supposed to be vertical in a 3-D loamy soil box of 10×10×30 cm3 with a constant total water uptake of 1 cm3d-1 and a constant elongation rate of 1 cm (root initial length of 1 cm, final length of 21 cm). When the root collar potential reached -1.5 MPa, the root-top boundary condition was switched from flux to pressure head. A constant pressure was then maintained as a top boundary condition and the total uptake was consequently reduced . We here considered root growth to assess how it impacts the sink location and hence the plant water status. The coupling between the root and soil water flow was achieved following , i.e. the soil water flow is obtained by solving the Richards equation, the soil–root interface potential is the distance-weighted soil root potential, and soil and root equations are solved iteratively until convergence. For more details, see and .

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 Krs of uniform roots and compare the results with those of roots with varying hydraulic property profiles. Using a carbon cost as a constraint, it is written as follows: maximize Krs(r,l) subject to V0=πr2l, where V0(L3) is the volume constraint. To solve this optimization problem, Lagrange multipliers λ are useful. We define a new function L: L(r,l,λ)=Krs(r,l)+λπr2l-V0, whose maximum is found when ∇r,l,λL=0. The previous equation is equivalent to the following. ∂Krs∂r+2λπrl=0∂Krs∂l+λπr2=0V0=πr2l Using appropriate equations to calculate the effective conductance, we find an optimal radius and length that maximize root water uptake. As an illustration, we compare the optimal root radius of a root with constant hydraulic properties, a root whose constant root properties depend on the root radius and a root with heterogeneous root properties as observed in the pressure probe experiment. Mathematically, it is written as follows. Constantkr and kx:kr(z)=kr0 and kx(z)=kx0Uniformkr(r) and kx(r):kr(r)=kr0r0r and kx(r)=b*r5Heterogeneouskr and kx:kr(z)=f(z) and kx(z)=g(z) The parametrization for the heterogeneous root and the uniform root are the best heterogeneous and uniform profiles derived from the pressure probe experiment for brace roots. For the root whose hydraulic properties depend on the root radius, we used the parameters of for maize (r0(L), kr0(L4P-1T-1) and b*(L-1P-1T-1)). The constraint volume V0 was the mean observed volume for the maize brace roots; f(z) and g(z) are observed functions for the radial and axial hydraulic property, respectively.

In this last example, we again consider a single growing root whose development and elongation rates can both vary. We varied the root parameters m (development) and v0lmax (growth) between 0.05 and 0.5 d-1 and between 0.01 and 0.1 d-1, respectively, to investigate the impact of relative growth and maturation processes on macroscopic parameters and uptake patterns.

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. a and d). A perfect fit of the xylem conductance profile could be obtained with a linear piecewise function (black dashed line, panel a). Panel (b) reveals the best radial conductivity profiles for some of the tested functions: linear (orange), exponential (yellow), 3-stepwise (mauve), 2-step linear piecewise (light blue), 3-step linear piecewise (green). The resulting effective conductance profiles are represented with the same legend in panel (d) with contrasting performances (the adjusted coefficients of determination are indicated as well). Note that the y scales of the panels are all in log scale, and this explains why the linear stepwise curves are not straight and why the exponential scenario is a straight line. Interestingly the best linear scenario (orange) is the constant function as it also can be seen from panel (b). In panel (c), radial to axial hydraulic properties ratios are plotted as a function of the distance from root tip. The best scenario is the stepwise function for the radial conductivity profile coupled to a piecewise function for the axial conductance profile (mauve scenario). The last scenario that is represented in Fig.  is an axially and radially uniform root (dark blue). If, with such profiles, it is possible to reasonably fit the effective conductance profile (adjusted r2=0.81), then of course the modelled xylem conductance profile does not fit at all, since high variations were observed between both root extremities.

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 kr, and as a consequence the ratio of radial conductivity to axial conductance, are similar in these best scenarios. It implies that the experiment is sensitive to the changes of root radial conductivity even though more knowledge and measurements of local variations of root conductance would be needed to know accurately the root radial conductivity of the brace roots.

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 .

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. . Here the roots are growing vertically in a homogeneous soil to assess the impact of water uptake location on plant water status. In all panels of this figure, the uniform and heterogeneous roots are represented with dashed and solid lines, respectively. The uniform root decreases its uptake after 6 days because the collar potential reaches the threshold at that time. At the very end of the simulation (20 days), this root had transpired 15 % less water than the heterogeneous root, in total. This can be explained by the location of the root water uptake: while the uniform root always takes up water at the top, the heterogeneous root keeps looking for water at its tip. While, as a consequence, the heterogeneous root never feels a too low potential, the top water potential rapidly decreases for the uniform root and critical potential are reached within the xylem vessels that force the root reducing its water uptake. Interestingly for the heterogeneous root both the soil equivalent and collar potentials increase after 5 days because, as the water uptake is mainly located at the root tip in such a case, new wet soil regions are explored by the active root parts (in terms of water uptake) and this impacts the root water status.

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 xy-averaged soil potentials look similar for both roots, they diverge at the end of the simulation (see Fig. 10c and the video).

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. : the dashed line is the uniform root, the solid line is the uniform root whose hydraulic properties depend on the root radius and the dotted line is the heterogeneous root observed with the pressure probe experiment. The red stars denote the optimal radii at which the conductances are maximal for the three scenarios. The black star is the observed pair radius conductance of maize brace root.

Unlike previous approaches we consider here that the hydraulic properties may be functions of the root tissues and do not necessarily depend on the root radius. As highlighted by the red stars, the optimal root radii vary considerably when integrating this concept. We find that optimal root radius is closer with stepwise functions to the observed one than with other conductance variations.

Figure a shows the integral of Krs over time when simulating 40 days of growth and maturation of a root whose lmax is 10 cm. Four extreme situations are represented: fast growth and slow maturation (case 1), slow growth and maturation (case 2), fast growth and maturation (case 3), and slow growth and fast maturation (case 4). The effective conductances of these particular roots are plotted as a function of time in Fig. b. They exhibit very contrasted strategies in terms of effective conductance and uptake distributions. When the root growth is fast, the Krs quickly reaches a high value. If the root maturation process is rapid, the final conductance is low. It can also be observed that the root properties change slowly when the root growth is low and rapidly when the root elongation rate is high. Figure c represents the relative SUD(z) (i.e. multiplied by root length) as a function of the relative root position z (i.e. divided by root length) after 20 days of growth. Again contrasted strategies appear with a maximal water uptake location at the root tip for cases 3 and 4 or at the root collar for cases 1 and 2. It is worth noting that a constant elongation rate means a very small ratio of elongation rate to maximal root length. So depending on the maturation rate, the constant elongation case rate is similar to case 2 or 4. This illustrates that plants might control their uptake patterns not only by changing their local conductances but also by adapting their growth rate.