Jean-Marc Custos
email: jm.custos@gmail.com

Christian Moyne
email: christian.moyne@univ-lorraine.fr

Thibault Sterckeman
email: thibault.sterckeman@univ-lorraine.fr

How root nutrient uptake affects rhizosphere pH: A modelling study

Keywords: soil solution, nutrient, root uptake, nitrate, sensitivity analysis

Rhizosphere pH is the consequence of numerous, complex processes and is a key factor of biogeochemistry at the soil-root interface. This work aimed at evaluating the impact of nutrient root absorption on rhizosphere pH using a mechanistic model formalizing major nutrient cation and anion transport, under the influence of concentration, electric potential gradients and of convection. The electrical neutrality of the system was maintained through a H + root efflux or influx (or equivalent opposite OH -fluxes). The model outputs were compared to the rhizosphere pH of maize (Zea mays L.), ryegrass (Lolium perenne L.) and Alpine pennycress (Noccaea caerulescens (J. & C. Presl) F. K. Mey.) grown in three different soils, determined with planar optodes. A global sensitivity analysis was carried out in order to highlight the dominating factors. The composition of 37 soil solutions from the literature was used to predict rhizosphere pH. The model most of the time predicts the alkalinisation of the rhizosphere by up to +3.3 pH units, given that in aerobic soils, nitrate is the most absorbed nutrient and its internalization is equilibrated by H + root absorption (or equivalent OH - excretion). This alkalinisation is close to that observed in the rhizosphere of ryegrass and Alpine pennycress. The rhizosphere pH predictions are very sensitive to the initial composition of the soil solution and quite insensitive to root absorption kinetics and soil buffer power for the various nutrients. Because it favours nitrate diffusion, water content is positively correlated with rhizosphere alkalinisation. Moreover, the lower the root density, the lower the effect of the root uptake on both ion concentrations and on rhizosphere pH.

Convection has little influence on rhizosphere pH. This study confirms the prominent role of mineral nutrition in the determination of rhizosphere pH. The model therefore constitutes a tool for understanding and predicting rhizosphere pH.

Introduction

It has long been established that soil pH in the vicinity of the roots may differ from that of the bulk soil, by up to two units. The sign of this difference, i.e. alkalinisation or acidification of the rhizosphere, and its intensity depends on plant and soil factors. Indeed, the root-mediated pH changes in the rhizosphere has been explained by cation-anion exchange between root and soil, root excretion of organic anions, root respiration (CO 2 production) and root-induced redox processes (O 2 emission). These processes may be affected by nutritional constraints, such as Fe and P deficiencies or Al toxicity, whose effect on rhizosphere pH varies according to the soil buffering capacity [START_REF] Hinsinger | Origins of root-mediated pH changes in the rhizosphere and their responses to environmental constraints: A review[END_REF][START_REF] Neumann | Rhizosphere chemistry in relation to plant nutrition[END_REF].

Of these different processes, the cation and anion exchange balance appears to be most influential, because soil solution pH depends on the net charge from ions which do not react to OH -and H + [START_REF] Sposito | The chemistry of soils[END_REF] and all living plants have to take up ions (mainly as nutrients) from this solution, modifying the net charge. The impact of plant mineral nutrition on rhizosphere pH has often been shown to be strong [START_REF] Haynes | Active ion uptake and maintenance of cation-anion balance: A critical examination of their role in regulating rhizosphere pH[END_REF][START_REF] Marschner | Root-induced changes in the rhizosphere: Importance for the mineral nutrition of plants[END_REF].

When more cations than anions are absorbed by the root cells, protons are released into the apoplast to compensate for the excess positive charges in the cell, thereby resulting in a decrease in the rhizosphere pH. However, when more anions are absorbed, either H + is taken up, or OH -or HCO 3 -(resulting from the reaction of OH -with CO 2 ) is excreted, thence leading to the alkalinisation of the rhizosphere [START_REF] Haynes | Active ion uptake and maintenance of cation-anion balance: A critical examination of their role in regulating rhizosphere pH[END_REF][START_REF] Riley | Bicarbonate accumulation and pH changes at the soybean (Glycine max (L.) Merr.) root-soil interface[END_REF][START_REF] Riley | Effect of ammonium and nitrate fertilization on phosphorus uptake as related to root-induced pH changes at the root-soil interface[END_REF]. The release or uptake of H + by roots is therefore equivalent to the cation-anion balance of root-absorbed ions [START_REF] Hinsinger | Origins of root-mediated pH changes in the rhizosphere and their responses to environmental constraints: A review[END_REF][START_REF] Rufyikiri | Effect of aluminium on bananas (Musa spp.) cultivated in acid solutions. II. Water and nutrient uptake[END_REF]. However, it has yet to be clearly established whether roots have a general tendency to acidify or to alkalinise their neighbourhood. Although [START_REF] Cunningham | Cation-anion relationships in crop nutrition: III. Relationships between the ratios of sum of the cations: sum of the anions and nitrogen concentrations in several plant species[END_REF] showed that on average, plants absorb more anions than cations, plants are often considered to acidify their rhizosphere, arguing that they absorb more cations than anions [START_REF] Sposito | The chemistry of soils[END_REF].

These modifications to the rhizosphere pH can have a feed-back effect on the mineral nutrition of plants. They can affect the roots' uptake properties through competition between H + and OH -for sorption sites in apoplast and cell membrane, through modification of the external electrochemical gradient or alterations to the root metabolism [START_REF] Chaignon | Copper availability and bioavailability are controlled by rhizosphere pH in rape grown in an acidic Cu-contaminated soil[END_REF][START_REF] Neumann | Rhizosphere chemistry in relation to plant nutrition[END_REF]. However, the main effects of the rhizosphere pH on mineral nutrition of plants occur through modification of the nutrient availability in soil. Rhizosphere acidification can be regarded as a process selectively evolved in some species or group of species to overcome P and Fe deficiencies. Indeed, most plant species, except Poaceae, can be gathered into the Strategy I group, which possesses an enhanced capacity to excrete H + and to reduce Fe(III) through a reductase system, in order to acquire Fe when it is poorly available. Similarly, increased proton exudation can also be triggered by P deficiency in various plant species [START_REF] Hinsinger | Origins of root-mediated pH changes in the rhizosphere and their responses to environmental constraints: A review[END_REF][START_REF] Neumann | Rhizosphere chemistry in relation to plant nutrition[END_REF]. Alternatively, an alkalinisation of the rhizosphere can reduce the availability of elements as demonstrated for P [START_REF] Riley | Effect of ammonium and nitrate fertilization on phosphorus uptake as related to root-induced pH changes at the root-soil interface[END_REF] and Cu [START_REF] Bravin | Rhizosphere pH gradient controls copper availability in a strongly acidic soil[END_REF][START_REF] Chaignon | Copper availability and bioavailability are controlled by rhizosphere pH in rape grown in an acidic Cu-contaminated soil[END_REF].

Mechanistic modelling based on the diffusion-convection equation of solutes in the rhizosphere [START_REF] Barber | Soil nutrient bioavailability. A mechanistic approach[END_REF] has continuously developed in order to predict plant uptake of both nutrients [START_REF] Darrah | Modelling the rhizosphere: a review of methods for 'upscaling' to the whole-plant scale[END_REF][START_REF] Keyes | High resolution synchrotron imaging of wheat root hairs growing in soil and image based modelling of phosphate uptake[END_REF][START_REF] Leitner | A dynamic model of nutrient uptake by root hairs[END_REF][START_REF] Ptashnyk | Enhanced zinc uptake by rice through phytosiderophore secretion: a modelling study[END_REF] and of metallic contaminants [START_REF] Custos | Contribution of Cd-EDTA complexes to cadmium uptake by maize: a modelling approach[END_REF][START_REF] Lin | Can ligand addition to soil enhance Cd phytoextraction? A mechanistic model study[END_REF][START_REF] Mullins | Modelling the plant uptake of cadmium and zinc from soils treated with sewage sludge[END_REF][START_REF] Puschenreiter | Changes of Ni biogeochemistry in the rhizosphere of the hyperaccumulator Thlaspi goesingense: experimental evidence and modeling[END_REF][START_REF] Sterckeman | Applying a mechanistic model to cadmium uptake by Zea mays and Thlaspi caerulescens: Consequences for the assessment of the soil quantity and capacity factors[END_REF]. One approach to improving the solute uptake model would be to couple it to a model predicting rhizosphere pH [START_REF] Gérard | Modelling the interactions between root system architecture, root functions and reactive transport processes in soil[END_REF][START_REF] Szegedi | Modelling rhizosphere transport in the presence of goethite, including competitive uptake of phosphate and arsenate[END_REF]. [START_REF] Nye | The measurement and mechanism of ion diffusion in soils. VIII -A theory for the propagation of changes of pH in soils[END_REF][START_REF] Nye | Changes of pH across the rhizosphere induced by roots[END_REF] was the first author to model the changes in rhizosphere pH from a constant H + or HCO 3 -flux from the root surface and the diffusion of these species according to Fick's law, taking into account the pH buffer capacity of the soil. Given that plants, except legumes, on average absorb more anions than cations [START_REF] Cunningham | Cation-anion relationships in crop nutrition: III. Relationships between the ratios of sum of the cations: sum of the anions and nitrogen concentrations in several plant species[END_REF], root excretion of HCO 3 -was fixed at a value estimated from the literature. The proton emission from well-nodulated legume roots was fixed in a similar way. Close agreement between Nye's theory and experimental results was found in the case of rhizosphere acidification of submerged rice [START_REF] Kirk | Root-induced iron oxidation, pH changes and zinc solubilization in the rhizosphere of lowland rice[END_REF] or of the growth zone of maize root tip [START_REF] Schaller | pH-Änderungen in der Rhizosphäre von Mais-und Erdnußwurzeln[END_REF].

Nevertheless, these validations of Nye's model did not cover the case of a HCO 3 -efflux which often occurs above root tips, in well-aerated soils where nitrate is the main form of absorbed nitrogen [START_REF] Blossfeld | Rhizosphere pH dynamics in trace-metal-contaminated soils, monitored with planar pH optodes[END_REF][START_REF] Nye | Changes of pH across the rhizosphere induced by roots[END_REF]. Moreover, Nye's model is based on an a priori setting, namely an estimated H + or OH -(or HCO 3 -) proton efflux to fit a situation known to cause an acidification or alkalinisation of the root vicinity, respectively. This approach does not apply to the general case, when the efflux of acid-base species resulting from the plant mineral nutrition is unknown. Therefore, a model determining the root efflux of acid-base species from the balance of cation and anion root absorption would be an improvement of the rhizosphere pH prediction.

We consequently developed such a model, and used it to evaluate the effect of major nutrient root uptake on rhizosphere pH. The model describes the transport of major cations and anions under the influence of both a concentration and an electric potential gradient, as well as their reaction with the soil solid phase (soil buffer power). At the root surface, the elements are taken up according to Michaelis-Menten kinetics and the electrical neutrality of the system is maintained through a proton or hydroxyl efflux [START_REF] Bouldin | A multiple ion uptake model[END_REF]. The model was tested by comparing its outputs with the pH measured in the rhizosphere of three plants species (ryegrass, maize and Alpine pennycress) grown in three different soils, using a non-invasive optical method [START_REF] Blossfeld | Rhizosphere pH dynamics in trace-metal-contaminated soils, monitored with planar pH optodes[END_REF]. A global sensitivity analysis was then carried out in order to highlight the most influential factors on the rhizosphere pH. This approach was completed by simulations using the composition of 37 soil solutions from various origins, taken from the literature.

Materials and methods

Model development

The new and original model to compute the rhizosphere pH is derived rigorously from first principles. A simplified description of the model is given here, while an extensive version can be found in the Supplementary Material. The soil is divided into two compartments: the soil solution, where the electro-neutrality condition is verified locally and the solid phase, including the very thin electrical double layer (thickness  10 nm) at the surface of the solid phase. The electrical double layer is neglected because of its very small volume compared to that of the pore solution. Its influence is supposed to be part of the buffer power.

Transport in soil solution

The various ions contained in the soil solution are 3 HO  , OH  , 3 HCO  and n ionic species in a specific soil). Organic ions, such as organic or amino acids were not taken into account.

M i z i  of valence i z ( Ca 2+ , K + , Mg
The valence of the P ions is taken as -1.5, as in soil solution at pH around 7, phosphates in the soil solution are constituted in nearly equal parts of 24 H PO et 2 4

HPO [START_REF] Barber | Soil nutrient bioavailability. A mechanistic approach[END_REF]. The concentrations of the OH  and 3 HCO  ions are given from the 3 HO  concentration according to the following equilibrium conditions:

3 [H O ][OH ] e K  
(1)

2 3 3 CO [H O ][HCO ] k   (2)
where e K is the water ionic product and 2 CO k is assumed to be proportional to the (uniform) partial pressure of CO 2 ( 2 CO P ) in the soil. Micro-organism mediated processes and root emissions of CO 2 were not formalized.

As the objective is to compute the pH, the electro-neutrality of the soil solution

2 CO 33 1 3 [H O ] [M ] [OH ] [HCO ] [H O ] j n z e jj j Kk z             (3)
has to be verified rigorously. Using Eqs. (1-3) gives for a given partial ). Fulfilling the electro-neutrality condition (i.e. adding Eq. (3) to the transport equations) requires adding a supplementary unknown, the (electric) streaming potential . Therefore, the molar mass fluxes for the n ion species ( M i z  ) are given as the sum of a convective term due to the water movement with the velocity v , as well as a diffusive term written in the Nernst-Planck form to take into account the electric effects on the ions diffusive motion:

𝐉 M 𝑖 𝑧 𝑖 + = [M 𝑖 𝑧 𝑖 + ]𝐯 - M 𝑖 𝑧 𝑖 + (𝛁[M 𝑖 𝑧 𝑖 + ] + 𝐹 𝑧 𝑖 [M 𝑖 𝑧 𝑖 + ] 𝑅𝑇 𝛁𝜓) 1 ≤ 𝑖 ≤ 𝑛 (4) M M 13 i z i i z i in t          J     (
z i i n ci i Fz I J J J J              (7)
The electric current in the soil solution is divergence free. The convective part being necessarily 0 due to the electro-neutrality condition, c I only depends on the

J

for the n

M i z i
 species is consequently written:

M 1 M - M 1 j i z i i n z z i ij j j in J v           (8)
where the diffusion coefficient ij is given as a function of the molecular diffusive coefficients

M z j j  as   2 H M M CO 2 3H M 1 3 [M ] 1, [M ] [H O ] [H O ] k z j j z i i k z k k z i j k ij ij n e z kk k z z D i j n Kk zD                            (9)
where ij  is the Kronecker symbol and the diffusion coefficient H D is given by

2 2 3 3 CO CO H 2 2 2 H O OH HCO 3 3 3 1 [H O ] [H O ] [H O ] e e K k k K D               (10)

Finally, the proton concentration 3 [H O ]

 is computed from the electro-neutrality condition (Eq. 3).

Transport in soil

The transport equations in the soil solution are then up-scaled to the soil scale using a classical two-scale homogenization technique. At the interface between the soil solution and the solid phase (including the thin electric double layer), the where  is the volumetric liquid fraction of the soil (considered to be constant), and D v the Darcy velocity (water volumetric flux per unit surface area of the soil). The buffer power

M z i i b 
is calculated as the ratio of the concentration of solid-phase sorbed ion (i.e. exchangeable, mol m -3 soil) to that of the ion in solution (mol m -3 solution) [START_REF] Roose | A mathematical model of plant nutrient uptake[END_REF]. No feedback effect of pH on soil buffer power for ions was taken into account. The source term 𝑺 𝑴 𝒊 𝒛 𝒊 + is added in order to describe the production of nitrate from organic matter (Supplementary Material).

A simple approximation of the homogenized diffusion coefficient at the soil scale is provided by the relation:

* 3/2 ij ij   (12)

Boundary conditions

The model is solved in a cylindrical unit soil volume around the root. At the surface of the roots in cylindrical coordinates (

0 rr  ) the mass flux of species M i z i  (1 ≤ 𝑖 ≤ 𝑛) is given by a Michaelis-Menten relation (v 𝐷 < 0): 0 1 [M ] [M ] v [M ] [M ] j i i i z z n j z i i D i ij max z i j mi rr I rK             (13)
As with the interface conditions between the soil solution and the solid phase, a simultaneous flux of 3 HO  , OH  or 3 HCO  at the surface of the root is necessary to maintain the electroneutrality of the system, which is implicitly calculated (Eq. ( 14) in Supplementary Material).

Note that all fluxes are in the soil and not in the plant as some chemical reactions (including CO 2 release) may occur at the soil/plant interface, but with the constraint of charge conservation. The periodicity condition around each root is simulated by setting a null flux for each ionic species i at the boundary of the root zone of influence:

1 1 [M ] v [M ] 0 j i z n j z D i ij j rr r         (14)
The root absorption functions along the root, above the growth zone are assumed to be invariable in space and time for all the ions considered [START_REF] Barber | Soil nutrient bioavailability. A mechanistic approach[END_REF]. This corresponds to young active and growing roots, as those for which variations in the rhizosphere pH have been measured (e.g. [START_REF] Blossfeld | Rhizosphere pH dynamics in trace-metal-contaminated soils, monitored with planar pH optodes[END_REF]; [START_REF] Marschner | In vivo measurement of root-induced pH changes at the soil-root interface: Effect of plant species and nitrogen source[END_REF]; [START_REF] Marschner | Root-induced changes in the rhizosphere: Importance for the mineral nutrition of plants[END_REF]; [START_REF] Marschner | Rapid method for measuring changes in pH and reducing processes along roots of intact plants[END_REF]). The ion transport is supposed to be purely radial, in a zone sufficiently distant from the root apex for the growth effects (as described by [START_REF] Kim | A mathematical model for pH patterns in the rhizospheres of growth zones[END_REF]) to be negligible. The realism of the simulated uptakes according to the mean plant composition is discussed in Supplementary Material (2.2, Table S1).

Initial conditions

The initial soil composition is assumed to be uniform. The initial value of 2 CO P is deduced from the initial pH and composition of the soil solution, according to Eq. ( 3) and does not vary with time.

Model parameters and numerical solution

The model parameters concern self-ionization of water, the solubility of CO 2 in water, soil water content, soil buffer power for each ion, diffusion properties of the ions in water, root geometry and root ion absorption properties ( i m K and i max I

for each nutrient 1 ≤ 𝑖 ≤ 𝑛). These are given in Table 1. The root absorption parameters are set at the same level for the three plant species, based on the relatively little data available from the literature (see Supplementary Material).

The equations were solved using the pdepe function of Matlab®, using hundred space steps logarithmically distributed and an adaptive time step.

Model validation

The model was experimentally evaluated by comparing its output, i.e. the soil pH at the root

Plant cultivation

Plants were grown on three soils, designated as A, B and C, sampled in the Ap horizon of Cambisols (Table 2). Soils A and B were a loam with a pH (H 2 O) of 6.2 and a silt loam soil with a pH (H 2 O) of 8.1, respectively. Soil C was chosen because it was also a loam and its bulk soil pH fell between those of soils A and B. Soil C also enabled the measurement of pH in the rhizosphere of ryegrass on a slightly acidic soil, as this plant only rarely grew on soil A in pot cultivation, because of the toxicity of a highly available Zn content [START_REF] Blossfeld | Rhizosphere pH dynamics in trace-metal-contaminated soils, monitored with planar pH optodes[END_REF]. This Zn level, however, affected neither the growth of Alpine pennycress which is hypertolerant to this metal, nor that of maize, probably because this is still heterotrophic (see below).

Plants were transplanted to or sown in PVC rhizoboxes (height 300 mm, width 150 mm, depth 50 mm) that were filled with the soil. During cultivation, the rhizoboxes were placed on a rack inclined at 45° in order to force the roots to grow along the front plates. The transparent side being placed against the rack, the roots were protected from light. The plants were grown and investigated in a growth chamber The cultivation period of the three species in the rhizoboxes prior to the measurements differed due to the specific root growth of individual species, i.e. 6-8 weeks for alpine pennycress, 4-5 weeks for ryegrass and 8-9 days for maize.

Measurement of rhizosphere pH

Rhizosphere pH was measured using planar pH optodes, according to the device described by [START_REF] Blossfeld | Rhizosphere pH dynamics in trace-metal-contaminated soils, monitored with planar pH optodes[END_REF]. The optode, a 10 µm inert polymer film measuring a few square centimetres, was placed on a single root or on several roots, at the soil surface, behind the rhizobox glass. This film contained a dispersed and immobilized dye which reacts to protons.

The measurement was based on the fluorescence decay time of the dye and was carried out via an optical glass fibre from outside the rhizobox, which was connected to a light source and measuring device. The glass fibre was moved automatically by an x-y stepper motor device.

By this means, the pH value could be measured in a line scanning mode, at 2 to 3 mm steps and 4 s intervals. This enabled the evolution of the pH in the rhizosphere to be mapped, as illustrated in [START_REF] Blossfeld | Rhizosphere pH dynamics in trace-metal-contaminated soils, monitored with planar pH optodes[END_REF].

To evaluate the model, we used three types of measured data (Table 3):

 The average pH along the axis of a single root, which can be considered as the pH at the rhizoplane. This was available for all the plant species in the three soils, except ryegrass which did not grow in soil A.

 The average pH in a portion of the root network, a planar optode being placed on several young roots. This was not available for ryegrass in soil A.

 The pH of the bulk soil as measured with the optode.

The latter was used as the initial pH value, while the other two were compared to the model outputs. The pH measurements showed that the functioning of roots was rather uniform along their axes, except for maize, as will be shown and discussed later. ) and cations (Ca 2+ , K + , Mg 2+ , Na + , NH 4 + ) were determined using ionic chromatography (Dionex AS50/EG40/IC25, Sunnyvale, CA, USA) and a certified solution from Sigma-Aldrich (reference number 89886). Induced coupled plasma atomic emission spectrometry (iCAP 6300 DUO, Thermo Scientific, UK) was used to determine Al, Fe, Mn and Zn in the solution. In this case, three certified solutions from SCP Science were used (reference numbers 701-021-067, 701-021-432 and 140-025-038).

Analysis of the soil solution and rhizosphere pH simulations

Using these soil solution compositions (Table 4), soil pH at the root surface or in the rhizosphere (mean value over r 1 ) was simulated for the three plant species (for Alpine pennycress, ryegrass and maize) in the three soils (A, B and C), fixing the initial pH at the value in the bulk soil and neglecting the nitrate source, so 𝑆 M 𝑖 𝑧 𝑖 + = 𝑆 NO 3 -= 0. The simulation conditions were adapted to the root age and density (Table 3). In total, the model outputs were compared to 15 measurements of the rhizosphere pH.

Sensitivity analysis

A sensitivity analysis was carried out using Matlab® in order to evaluate which of the 41 factors (including the nitrate source) had the greatest impact on the variation of the pH at the root surface (∆𝑝𝐻):

∆𝑝𝐻 = 𝑌 = 𝑝𝐻 𝑓 -𝑝𝐻 𝑖 (15)
where 𝑝𝐻 𝑖 is the initial pH, corresponding to that of the non-rhizospheric (bulk) soil and 𝑝𝐻 𝑓 is the soil pH at the end of the simulation.

Factors (𝑋 𝑗 ) were ranked according to standardized regression coefficients (SRC) (Faivre et al., 2013;[START_REF] Saltelli | Sensitivity analysis in practice. A guide to assessing scientific models[END_REF].

𝑆𝑅𝐶 𝑗 = 𝛽 𝑗 √ 𝑉𝑎𝑟(𝑋 𝑗 ) 𝑉𝑎𝑟(𝑌) (16)
in which 𝑉𝑎𝑟() is the variance, 𝛽 𝑗 is the regression coefficient of 𝑋 𝑗 , according to the correlation

𝑌 = ∑ 𝛽 𝑗 𝑁 𝑗=1 𝑋 𝑗 + 𝜀 (17)
where 𝑌 is the output of the model and 𝜀 the model error, N = 41.

The factors were sampled using random sampling of uniform distributions (Table S2). The initial pH (Figure S3) was calculated according to Eq. 3, randomly sampling 2 CO P , as follows.

After sampling all the concentrations of the transported ions, their charge balance (CB, mol c m -3 ) was calculated according to

𝐶𝐵 = ∑ 𝑧 𝑗 [M 𝑗 𝑧 𝑗 + ] 𝑛 𝑗=1 (18)
as well as the sum of absolute charges:

∑ |𝑧 𝑗 |[M 𝑗 𝑧 𝑗 + ] 𝑛 𝑗=1 (19)
Solution compositions were retained when the relative charge balance (CB r , dimensionless) satisfied the following condition:

0 < 𝐶𝐵 𝑟 = ∑ 𝑧 𝑗 [M 𝑗 𝑧 𝑗 + ] 𝑛 𝑗=1 ∑ |𝑧 𝑗 |[M 𝑗 𝑧 𝑗 + ] 𝑛 𝑗=1 < 0.2 (20)
Negative charge balances were discarded, as they led to very acidic initial pH, falling beyond of the range of natural pH. The upper limit of CB r (0.2) was chosen by rounding the mean relative charge balance measured on soils worldwide (see 2.4), which is 0.12.

CO

P was then randomly sampled between 3 10 3 ppmv and 10 5 ppmv [START_REF] Sposito | The chemistry of soils[END_REF], to calculate the corresponding solution pH according to Eq. ( 3). The final sample size was 20,000. The model was run for two durations: one and four weeks.

Simulating rhizosphere pH in 37 soils sampled worldwide

As the 20,000 random cases simulated in the sensitivity analysis could not all be representative of the reality of soil solution composition, we analysed the output of the model on a set of data originating from real soils. In Tables 7.2 and 7.3 of his book, [START_REF] Wolt | Soil solution chemistry. Applications to environmental science and agriculture[END_REF] summarized the solution composition of soil horizons sampled worldwide by various authors.

From this data set, we kept the solution composition from 35 surface horizons, for which our calculations based on Eq. 3 led to an initial pH close to that measured. Indeed, for some of them, it was impossible to reproduce the measured pH, probably because of errors or missing data in the reported solution composition. In most of the solutions, there was no phosphate content, probably because this ion was below the detection limits. The NH 4 + and the SO 4 2contents were not always available. However, as will be shown later, these ions have a limited influence on rhizosphere pH and the simulations could usefully be carried out with null values for P ions, NH 4 + and SO 4 2-

. The solution composition of soils used to validate the model, i.e. soils B and C were added to those from [START_REF] Wolt | Soil solution chemistry. Applications to environmental science and agriculture[END_REF]. Solutions from soil A were not used because of its exceptionally high Zn content. The 37 solutions (Table S3) were used to simulate ∆𝑝𝐻. For these simulations, nitrate source (𝑆 𝑁𝑂 3 -), soil buffer powers and root absorption parameters of the different ions were taken as averages of the distributions used in the sensitivity analysis (Table S2). The root radius (r 0 ), the mean distance between roots (r 1 )

and the volumetric water content (θ) were taken as 1 10 -4 m, 5.5 10 -3 m [START_REF] Lin | Ranking of mechanisms governing the phytoavailability of cadmium in agricultural soils using a mechanistic model[END_REF] and 0.25, respectively.

2 CO P was taken as 3 10 3 ppmv (around10 times the atmospheric pressure).

The water volumetric flux v 𝐷 was 2.5005 10 -8 m s -1 and 𝑆 NO 3 -was zero. The model also enabled a calculation of other variables, such as the final composition of the rhizosphere solution and the amounts of ions taken up by the root.

Results and discussion

Model validation

Ryegrass and Alpine pennycress

The simulated rhizosphere pH was compared to 15 measured values (Table 5). In the case of ryegrass and Alpine pennycress, along the surface of a single young root and in root network, an alkalinisation of the rhizosphere was systematically measured. In soil A, the alkalinisation ranged from 1.2 to 1.4 pH unit. In soil B, this ranged from 0.4 to 0.9 pH units, while in soil C, it varied between 0.3 and 1.7 pH units. It must be noted that pH measured in the bulk soil with the planar optodes was close to that determined with dry soil dispersed in a 0.01 M CaCl 2 solution, and was slightly lower than that measured in water (Table 2).

All the simulations predicted a clear alkalinisation of the rhizosphere. In soil A, the simulated pH at the root surface was 6.4, i.e. 0.8 units above the initial pH (= 5.6) and 0.4 units below the measured value (Table 5). In the root network, the simulated pH was 7.7, i.e. 0.7 unit above the measured one. In soil B, the simulation produced a rhizoplane alkalinisation of 0.2 pH unit. The rhizoplane pH was 0.2 units below that measured at the rhizoplane of ryegrass (7.6 against 7.8) and 0.8 units below that measured at the rhizoplane of Alpine pennycress (7.6 against 8.4). For root networks, the simulations (pH = 8.0) were closer to the measured values (pH = 7.9) in both ryegrass and Alpine pennycress. In soil C, the simulated rhizoplane pH was 0.7 pH units above that in the bulk soil (= 6.2). It was 0.9 units below that measured at the rhizoplane of Alpine pennycress (6.9 against 7.8), but 0.4 units above that measured at the rhizoplane of ryegrass (6.9 against 6.5). In all the simulations, the initial 2 CO P fell within the range that has been observed in soils [up to 100 times the atmospheric partial pressure [START_REF] Sposito | The chemistry of soils[END_REF]].

Overall, these results show a reasonably good fit between the model predictions and the measured pH values at the root surface of ryegrass and Alpine pennycress. To interpret the differences between measurements and simulations, it is important to bear in mind the relationship between the solution pH, the ion charge balance (CB) and the CO 2 partial pressure ( 2 CO P ) (Eq. ( 3), Figure 1). There is a quasi-discontinuity in the function pH = f(CB)

at CB = 0. Therefore, when the charge balance is close to zero, the pH varies strongly with even a very small changes in charge balance. Moreover, in the case of a positive charge balance, the value at which Indeed, this law may transform small uncertainties in the composition of the soil solution and the CO 2 pressure, related to sampling and analysis, into wide variations in the pH calculation.

However, this does not prevent the model from being used. Indeed, instead of using the initial solution composition to predict the initial soil pH, the latter measured in the bulk soil can be imposed, thereby slightly adjusting the composition of the soil solution or of the partial CO 2 pressure. To illustrate this, slightly increasing the NO 3 -content of 0.08 mol m -3 in soil A, i.e.

+0.7% of the total charges in solution, gives a simulated alkalinisation of the rhizosphere of 1.3 pH units (corresponding to the measured alkalinisation), instead of 0.8 units.

Maize

This adequation was not observed in the case of maize, whose rhizosphere pH was clearly different from that of the two other species and from the simulation results. Indeed, when measured along a single young root, the rhizoplane pH decreased from 0.2 to 0.7 pH units (Table 5). At the rhizoplane of a single older root, this increased by only 0.1 to 0.3 pH units when measured in soils B and C, while it decreased by 0.1 units in soil A. None of the rhizoplane pH simulations led to an acidification, but did generate alkalinisation which most frequently, was above that of the single older root in soil B and C.

As already shown by [START_REF] Blossfeld | Rhizosphere pH dynamics in trace-metal-contaminated soils, monitored with planar pH optodes[END_REF] (particularly in Fig. 6 of their article), there was an acidification of the rhizosphere of maize, at about 15 to 25 mm from the apex. This phenomenon had already been observed by other authors [START_REF] Peters | The correlation of profiles of surface pH and elongation growth in maize roots[END_REF][START_REF] Pilet | Growth distribution and surface pH patterns along maize roots[END_REF]) [START_REF] Schaller | pH-Änderungen in der Rhizosphäre von Mais-und Erdnußwurzeln[END_REF]. This acidification could be explained by the acid growth theory [START_REF] Rayle | The acid growth theory of auxin-induced cell elongation is alive and well[END_REF], which states that growing cells excrete protons into the cell-wall at an enhanced rate, resulting in a cell-wall loosening to facilitate the root elongation. This acidification zone is alternatively explained by a higher nutrient uptake in the growth zone that would constitute a large nutrient sink, particularly for K + , whose root absorption would be equilibrated by proton excretion [START_REF] Kim | A mathematical model for pH patterns in the rhizospheres of growth zones[END_REF][START_REF] Taylor | Ammonium, nitrate, and proton fluxes along the maize root[END_REF].

On the other hand, the lower alkalinisation observed here in the root network or in older maize roots (i.e. away from the apex) could be due to a much lower nitrate uptake by the maize than by ryegrass and Alpine pennycress. Indeed, maize is completely heterotrophic from 0 to around 75 degree-days (about 7 days after germination) and completely autotrophic from around 250 degree-days [START_REF] Girardin | Ecophysiologie du maïs[END_REF][START_REF] Nadeem | Remobilization of seed phosphorus reserves and their role in attaining phosphorus autotrophy in maize (Zea mays L.) seedlings[END_REF]. Between 75 degree-days and 250 degree-days, maize becomes progressively autotrophic. Therefore, 8-9 day-old maize seedlings (that is 88 to 99 degree-days), as used in the present experiment, were only slightly autotrophic. That means that the seedling was mostly provided with mineral nutrients from the caryopsis reserves and that it absorbed only a minor part of these nutrients from the soil solution. [START_REF] Nadeem | Remobilization of seed phosphorus reserves and their role in attaining phosphorus autotrophy in maize (Zea mays L.) seedlings[END_REF] demonstrated that phosphorus was exclusively provided by the maize seed until 75 degree-days and was increasingly absorbed from the soil solution from that date until complete autotrophy was attained and the seed P reserves were exhausted.

Similar schemes could be drawn for the other nutrients, and particularly for nitrogen (Mollier, 2015, personal communication). Eight to nine day-old maize would therefore consume nitrogen mostly from the seed reserves and absorb fewer nitrates from the soil solution than ryegrass and Alpine pennycress, which are much older and completely autotrophic. Therefore, at the rhizoplane of maize behind the acidification zone, the sum of charges is only slightly higher, as is, consequently the pH of the solution.

Sensitivity analysis

Simulating root uptake over one week

Those ions which most frequently accumulated at the root surface, i.e. their concentrations at the rhizoplane was higher at the end of the simulations than at in the initial condition, were Ca 2+ , Mg 2+ , Na + , SO 4 -and Cl -. In contrast, K + , NH 4 + NO 3 The most influential factor on ΔpH was the initial nitrate concentration in the soil solution ([NO 3 -]) (Fig. 3) (SRC = 0.81). The more NO 3 -in the soil solution, the more the pH increased at the root surface. This is due to the root uptake of NO 3 -, as shown by the depletion of this ion in the rhizosphere in most of the simulations (Fig. S2), which is compensated for by the root absorption of H + (or equivalent excretion of OH -). Nitrate was the most taken-up ion, representing 64% of the absorbed ion absolute charges (Table 6).

The second factor was the initial Ca 2+ concentration ([Ca 2+ ]) in the soil solution (SRC = -0.63). This could be explained by the fact the higher [Ca 2+ ], the higher its uptake by the roots in exchange for H + , which tended to reduce ΔpH. However, this explanation would be contradictory to the fact that, Ca 2+ generally accumulated at the root surface (Fig. S2). The explanation can be found in another phenomenon. During the random sampling for the constitution of the simulation dataset, the initial solutions with a negative charge balance were discarded (Eq. ( 20)). This process favoured the higher [Ca 2+ ], as can be seen from the initial distribution of this factor (Fig. S3). Because the initial charge balance is a linear combination of the charges of each ion, there is therefore a positive correlation between the initial charge balance and the [Ca 2+ ]. Moreover, the relationship between the pH and the charge balance (Fig. 1) shows that the pH curve slope decreases as the charge balance increases. Therefore, as [Ca 2+ ] rises, the initial charge balance increases and the higher it is, the lower ΔpH will be.

The initial sulphate concentration had a positive effect on ΔpH (SRC = 0.55). One could expect that the root absorption of SO 4 2-would alkalinise the rhizosphere, as that of NO 3 -did.

However, SO 4 2-accumulated in the rhizosphere (Fig. S2). Again, the explanation of the [SO 4

2-] effect relies on the dataset sampling process and the calculation of the initial charge balance, which made the initial sulphate concentration negatively correlated to that of the initial charge balance (Fig. S3). The relationship between the pH and the charge balance (Fig. 1) shows that in the vicinity of a zero charge balance, the slope of the pH curve strongly increases as the charge balance decreases. It is this trend which explains that the higher the [SO 4

2-], the lower the initial charge balance and pH and the greater the variation in pH (ΔpH) will be.

The fourth influential factor is [Mg 2+ ], with SRC = -0.26. Its effect can be explained in a similar manner to that of [Ca 2+ ], although being less intense, because the initial concentration of Mg 2+ is lower than that of Ca 2+ (Fig. S3).

The average distance between roots r 1 , also had a negative effect on ΔpH (Fig. 3, SRC = -0.15). This can be explained by the fact that the higher r 1 , the greater the amount of ions in the rhizosphere and as a consequence, the lower the effect of the root uptake on ion concentrations and therefore, on solution pH. The water content θ also had a positive effect on ΔpH (SRC = 0.12). This variable takes part in the calculation of the element content in soil

(term (𝜃 + 𝑏 M i z i + ) 𝜕[M i z i + ]
𝜕𝑡 in Eq. ( 11)) , although its contribution is very slight when compared to the buffer power 𝑏 𝑀 𝑖 𝑧 𝑖 + . 𝜃 is also involved in the calculation of the homogenized diffusion coefficient at the soil scale (Eq. ( 12)). Increasing θ enhances ion diffusion in general and NO 3 in particular, whose contribution dominates those of the other ions. Therefore, θ is positively correlated to ΔpH because it favours NO 3 -uptake.

It must be noted that although absorbed by the root in such a way that their concentrations decreased in the rhizosphere during uptake, K + , NH 4 + and P ions had no strong effect on ΔpH because their concentration variations (and those of the associated charges) was small compared to that of NO 3 -. Whatever the ion, the buffer power and the root absorption parameters had very little or no significant influence on the variation of pH in the rhizosphere.

The nitrate source term (𝑆 NO 3 -) had no significant influence: over one week, it represented on average only 2% of the initial stock of NO 3 -.

Simulating root uptake over four weeks

After a four-week uptake, ΔpH had a mean value of 0.45 and was therefore slightly more positive than after a one-week uptake (Fig. 2). This again corresponded to the positive final charge balance of the transported ions.

The four most influential factors were the same as for the one-week uptake simulations - ], this is negatively correlated to the initial charge balance. As after four weeks, 𝑆 𝑁𝑂 3 -represented only 8% of the initial NO 3 -stock, the nitrate source only slightly reduced the nitrate depletion and had nearly no effect on ΔpH.

Main lessons from the sensitivity analysis

The first conclusion which can be drawn from the sensitivity analysis is that the general trend to rhizosphere pH is alkalinisation. The second point is that the main factor affecting the alkalinisation level is by far the initial nitrate concentration in solution. The third conclusion is that the soil buffer power and the root absorption parameters (K m and I max ), whatever the ion, have little or no influence on the rhizosphere pH. This justifies the use of the same root absorption parameters for the three different plant species. The sensitivity analysis also showed that the root density (or r 1 ) and the soil humidity could both have an influence on the rhizosphere pH.

The final conclusion concerns the sensitivity analysis itself. Despite the conditions imposed for the sampling of the ion concentrations in the initial solution (Eq. ( 20)), there was a dependency of the initial charge balance on major ion concentrations (Ca 2+ , Mg 2+ , SO 4

), which artificially made the latter influential variables. This bias in the sensitivity analysis also

shows how strongly ΔpH depends on the initial ionic composition of the solution. This justifies a statistical analysis of the model using more realistic soil solution compositions.

Simulating rhizosphere pH in 37 soils

To test the model in more realistic conditions, the rhizosphere pH was simulated using a welldocumented series of various soil solutions taken from [START_REF] Wolt | Soil solution chemistry. Applications to environmental science and agriculture[END_REF].

General trend of rhizosphere pH and effect of solution composition

Let us consider first the results of the simulation of a four-week uptake of ions. The simulated initial pH was close to the measured one (Fig. S4). On average, the rhizosphere pH increased by 1.1 units (Table S4). For each soil, the sign of ΔpH corresponded to that of ΔCB, the variation in charge balance in the soil solution at the root surface being expressed as

∆𝐶𝐵 = 𝐶𝐵 𝑓 -𝐶𝐵 𝑖 (21)
with the subscript f for "final" (t = 1 or 4 week) and i for "initial" (t = 0). 𝐶𝐵 𝑓 results from ion root uptake and ion accumulation at the root surface through convection.

In 32 of the 37 soils (i.e. 86% of the soils), ∆𝑝𝐻 was positive. In 26 soils over the 32 where ∆𝑝𝐻 was positive, the charge balance of the ions absorbed by the roots (RCB, mol c m -1 ) was negative, indicating that the root absorbed mainly anions (Table S4). This absorption of anions caused the alkalinisation of the rhizosphere. This alkalinisation was not perturbed by the accumulation of ions due to convection, i.e. the convection leading to the accumulation of cations or the accumulation of anions was not sufficient to counteract anion root absorption.

In these 26 cases, NO 3 -uptake represented 80% (against an overall mean of 62%) of the total absolute charge uptake, showing that the rhizosphere alkalinisation was mainly the consequence of nitrate uptake. Fourteen soils had a particularly high ∆𝑝𝐻, ranging from nearly 1.7 to 3.3. These values explain the mean ∆𝑝𝐻 of 1.1, higher than that estimated during the sensitivity analysis (∆𝑝𝐻 = 0.45 after 4 weeks of uptake), the rest of the soils having a mean ∆𝑝𝐻 of 0.30. Thirteen of these high ∆𝑝𝐻 soils had an initial pH between ca 4.7 and 6.0, which corresponds to the pH zone, where a small change in charge balance can strongly affect the solution pH (Fig. 1). One of them (Tujunga), had an initial pH of 7.0, beyond the high variation zone of this parameter. However, this soil with high initial cation contents, showed a very positive charge balance (23.7 mol c m -3 versus 5.2 mol c m -3 on average) after 4 weeks, due to an elevated uptake of NO 3 -(representing 91% of the absorbed charges).

In 6 soils out of the 32 where ∆𝑝𝐻 was positive (for instance in soils Cecil unamended, Fyfield, and Grove), RCB was positive (i.e. dominated by cation absorption) (Table S4). In these cases, however, ∆𝐶𝐵 was still positive, leading to a positive ∆𝑝𝐻. Although the root absorption, after 4 weeks of uptake, tended to acidify the rhizosphere, this was not sufficient to significantly reduce the positive charge balance created at the rhizoplane after the initial uptake. This shows that the initial uptake plays an important role in rhizosphere pH, in determining a charge balance which will be slow to modify afterwards. At the beginning of uptake, nitrate is quickly absorbed, causing alkalinisation at the rhizoplane. Afterwards, although nitrate is less absorbed than cations, the absorption of cations is not quick enough to acidify the rhizoplane solution.

In the 5 soils where ∆𝑝𝐻 was negative, correlatively with ∆𝐶𝐵, RCB was positive, indicating that the plant predominantly absorbed cations, which was compensated for by an acidification at the rhizoplane. In these soils, nitrate represented only 3.6% of the absorbed charges, while K + represented 68%. This is due to a lack of NO 3 -in the initial solution (0.03 mol.m -3 against 8.3 mol.m -3 on average), and not to an excess of K + (0.25 mol.m -3 against 0.85 mol.m -3 on average).

Effect of uptake duration

The results obtained after a one-week uptake simulation were close to those of a four-week uptake. ∆𝑝𝐻 after a one-week uptake was on average 0.2 unit lower than that after a fourweek uptake (Table 7). Up to 1 week of uptake, concentrations of Ca 2+ , Cl -, Mg 2+ , Na + and SO 4 2-in the soil solution increased slightly because of convection and then stabilized over time, probably because of an equilibrium between convection and retro-diffusion.

Concentrations of K + , NH 4 + , NO 3 -and P ions decreased because of root absorption. The decrease rate of NO 3 -was much higher than that of the other ions (Table 7). The charge balance in the rhizoplane solution regularly increased from +1.20 mol c m -3 to +8.12 mol c m -3 after 12 weeks. The proportion of Ca 2+ , Mg 2+ , K + , P ions and SO 4 2-in the total absolute charge uptake continuously increased, while that of NO 3 -strongly decreased from 69.8% to 51.7% as did that of NH 4 + also, but to a far lesser extent (Table 8).

As mentioned above, the first week of uptake determined most of ∆𝑝𝐻 at the rhizoplane even for the consecutive weeks and on average, increasing the simulation time to 8 and 12 weeks only slightly increased ∆𝑝𝐻, by a few hundredths of pH units. This is due to the fact that within the first week, there is a strong decrease in anion (i.e. mainly nitrate) concentration in the solution, and therefore an increase in the charge balance and in the pH. This initial increase in rhizosphere pH is sharp, because of the discontinuity in pH = f(CB) (Figure 1).

However, after this increase, the same function shows that pH only slightly increases with CB. In some cases, ∆𝑝𝐻 after four weeks was slightly lower than those after one week (Table S4), indicating the beginning of a depletion of the rhizosphere solution with the ions dominating the initial root uptake. This was particularly the case in the Xanthozem soil, which ∆𝑝𝐻 was clearly positive after one week (+2.5), decreased to 0.6 after 4 weeks and became negative (-0.55) after 8 weeks of uptake. In this exceptional soil, nitrate uptake was relatively low after 8 weeks, representing only 15% of the absorbed charges, while it was 50% after 1 week. The uptake of K + after 8 weeks was higher, with 24% of the absorbed charges, and NH 4 + exceptionally high, with 53%. This is the consequence of an exceptionally high ammonium concentration in the soil solution (4.03 mol.m -3 against 0.75 mol.m -3 on average).

Simulations taking into account a nitrate source (𝑆 𝑁𝑂 3 -) in the model were also carried out.

After a one-week uptake, with the nitrate source, ∆𝑝𝐻 was on average 2.3% lower than without a source. After 12 weeks, the difference was 0.6% (Table S4). This confirms the sensitivity analysis which showed that nitrification had nearly no effect on rhizosphere pH variation.

Conclusions

Many simplifications have been made during the modelling, because of a lack of knowledge and for the sake of mathematical and numerical simplicity: the same absorption parameters were used for the three plant species, soil water content was considered to be constant, microorganism mediated processes and root emissions of O 2 and CO 2 were not formalized, no feedback effect of pH on soil buffer power for ions was taken into account and organic ion release and uptake was neglected. However, the model predicted the same alkalinisation trend as that measured with the planar optodes. The simplifications therefore appear reasonable and the result supports the theory that the nutrient root uptake, in aerobic soils, is the important process responsible for the modification of the pH in the root vicinity [START_REF] Hinsinger | Origins of root-mediated pH changes in the rhizosphere and their responses to environmental constraints: A review[END_REF].

The simulations also suggest that among the nutrients, nitrogen is the one which most influences the rhizosphere pH. In aerobic soils without ammonium fertilization, nitrate is around 5 to 400 times more absorbed than the other ions. As its depletion in the root vicinity is compensated for by the root absorption of H + (or the equivalent excretion of OH -), the rhizosphere of non-leguminous crops in non-flooded soils might generally be more alkaline than the bulk soil.

This work also showed that the variations in the rhizosphere pH were dependent on the initial pH, more than on its evolution through root ion absorption. It highlighted how extremely this initial pH was sensitive to the initial composition of the soil solution, in such a way that small errors in analyses would produce a strong uncertainty about the calculated initial pH.

Therefore, it seems nearly impossible to precisely predict the initial pH from the initial solution composition. This is contrary to the pH at the rhizoplane which after a few weeks of root uptake, should generally stabilize from then on. However, it is the charge balance (CB) which determines the trend of the rhizosphere pH and generally, the plant roots increase CB, as they take up nitrates more than the other ions. Therefore, there is no doubt that most frequently the rhizosphere pH is higher than that of the bulk soil. Moreover, despite the uncertainty on the initial pH prediction, the model can be used to predict the rhizosphere pH, fixing the initial pH (which can be measured in 0.01 CaCl 2 ).

The model considers that the partial pressure of carbon dioxide is constant, being adjusted to the initial solution composition. However, in presence of roots and micro-organisms, 2 CO P probably vary, influencing both the pH and the solution composition. Therefore, it would be logical that 2 CO P should be measured, in the rhizosphere and CO 2 concentration taken into account in the model. Indeed, excretion of CO 2 could for instance limit the extent of the alkalinisation zone. 16) and ( 17)). Variables are described in Table 1. 16) and ( 17)). Variables are described in Table 1.

Figure captions

  5) Similar equations are written for the molar flux of 3 HO  , OH  and 3 HCO  . The introduction of the diffusion equation in the Nernst-Planck form allows a rigorous determination of the evolution of the rhizosphere pH from the properties of the soil and the mineral nutrition of the plant for the mineral nutrients [M ] 𝑖 ≤ 𝑛) without any additional hypotheses. Due to the electro-neutrality condition, adding the n transport equations for the nutrients

  surface (rhizoplane), to pH measured in the rhizosphere of ryegrass (Lolium perenne L., Prana cultivar), maize (Zea mays L. cv INRA MB862) and the Zn and Cd hyperaccumulator Alpine pennycress (Noccaea caerulescens (J. & C. Presl) F. K. Mey., Viviez population) grown in three different soils. For this, data previously produced by Blossfeld et al. (2010) were used (see 2.2.2).

-

  Because the initial concentrations of anions and cations in the soil solutions were not determined in the work of[START_REF] Blossfeld | Rhizosphere pH dynamics in trace-metal-contaminated soils, monitored with planar pH optodes[END_REF], these were measured de novo for the validation of the model, in the following manner. Approximately one kilogram of soil was put in a pot and maintained close to the water holding capacity (WHC) for 21 days by regular watering from the bottom. A microsuction cup (Rhizon soil moisture sampler, Eijkelkamp, Giesbeek, The Netherlands), placed in the centre of the soil, enabled us to sample the soil solution. The day before the solution sampling, the soil was watered to WHC. Less than 30 minutes were required to sample around 10 ml of soil solution. Anions (Cl -, NO 3

2 COP

 2 is fixed also has a considerable impact on the calculated pH, as shown by the curves for different 2 CO P . The characteristics of the function determining the pH explain the variability of the deviations between measured and simulated rhizospheric pH.

-

  and P ions -were clearly depleted (Fig S2). The sum of these balances led to a positive final charge balance of the transported ions, resulting from the mostly negative initial and final charge uptakes (Fig 2). As a consequence, in the vast majority of the cases, ΔpH was positive, reflecting an alkalinisation of the rhizosphere, whose mean value was 0.36 (Fig 2).

  = 0.70) and[Mg 2+ ] (SRC = -0.32)and ΔpH were affected in the same sense, but a little more intensively, as reflected by higher SRC(Fig 4). Compared to the one-week simulations, r 1 regressed to the seventh place (SRC = -0.08), indicating that the buffer effect of the rhizosphere volume would decrease with uptake duration, because of the depletion of nitrate in the rhizosphere solution. In contrast, θ gained one place compared to the one-week uptake, as did [Cl -], which appears to positively affect ΔpH because, as with [SO 4 2-

Fig. 1 .

 1 Fig. 1. Relationship between pH, partial CO 2 pressure (𝑃 𝐶𝑂 2 ) and sum of ion charges

Fig. 2 .

 2 Fig. 2. Global sensitivity analysis of the model. Frequency distribution of a) ∆𝑝𝐻 and b) of

Fig. 3 .

 3 Fig. 3. Global sensitivity analysis of the model. Ranking of the model variables based on their

Fig. 4 .

 4 Fig. 4. Global sensitivity analysis of the model. Ranking of the model variables based on their

  2+

Table 1 .

 1 Model variables.

	777
	Symbol	Definition	Value
	𝐾 𝑒 𝑘 CO 2	Water self-ionization constant CO 2 dissolution constant	10 -14 mol 2 L -2 3.7 10 -8 𝑃 CO 2 ln 10 ⁄	in mol 2 L -2 †
	𝒟 H 3 O + Diffusion coefficient of H 3 O + in water	9.31 10 -9 m 2 s -1 ‡
	-in water Diffusion coefficient of OH -in water 𝒟 HCO 3 -Diffusion coefficient of HCO 3 𝒟 OH -Diffusion coefficient of Ca 2+ in water 𝒟 Ca 2+ 𝒟 K + Diffusion coefficient of K + in water	1.18 10 -9 m 2 s -1 ‡ 5.27 10 -9 m 2 s -1 ‡ 7.93 10 -10 m 2 s -1 ‡ 1.96 10 -9 m 2 s -1 ‡
	𝒟 Mg 2+ Diffusion coefficient of Mg 2+ in water	7.05 10 -10 m 2 s -1 ‡
	𝒟 Zn 2+ Diffusion coefficient of Zn 2+ in water 𝒟 Na + Diffusion coefficient of Na + in water	7.15 10 -10 m 2 s -1 1.13 10 -9 m 2 s -1 ‡
	𝒟 NH 4 + 𝒟 Cl -	Diffusion coefficient of NH 4 Diffusion coefficient of Cl -in water + in water	1.98 10 -9 m 2 s -1 ‡ 2.03 10 -9 m 2 s -1 ‡
	𝒟 NO 3 -	Diffusion coefficient of NO 3	-in water	1.90 10 -9 m 2 s -1 ‡
	2-in water Diffusion coefficient of P ions in water 𝒟 SO 4 2-Diffusion coefficient of SO 4 𝒟 P	1.07 10 -9 m 2 s -1 ‡ 7.9 10 -10 m 2 s -1
	𝜃	Volumetric water content		0.24 m 3 m -3
	𝑏 𝑀 𝑖 𝑧 𝑖 +	Buffer power		See Table 2
	Mean root radius Thickness of the rhizosphere Maximum root influx of Ca 2+ Michaelis constant for Ca 2+ Maximum root influx of K + Michaelis constant for K + Maximum root influx of Mg 2+ Michaelis constant for Mg 2+ Maximum root influx of Na + Michaelis constant for Na + Maximum root influx of NH 4 + Michaelis constant for NH 4 + Zn-Nc Maximum root influx of Zn 2+ in N. caerulescens 1.66 10 -12 mol m -2 s -1 10 -4 m § ¶ 𝑟 0 2.5 10 -2 m § 𝑟 1 Ca 1.042 10 -9 mol m -2 s -1 # 𝐼 𝑚𝑎𝑥 Ca 3.9 10 -2 mol m -3 𝐾 𝑚 K 5.02 10 -8 mol m -2 s -1 𝐼 𝑚𝑎𝑥 K 3.2 10 -2 mol m -3 𝐾 𝑚 Mg 6.5 10 -9 mol m -2 s -1 𝐼 𝑚𝑎𝑥 Mg 0.44 mol m -3 𝐾 𝑚 Na 0 𝐼 𝑚𝑎𝑥 Na -𝐾 𝑚 NH4 1.0 10 -7 mol m -2 s -1 𝐼 𝑚𝑎𝑥 NH4 5.4 10 -2 mol m -3 𝐾 𝑚𝑎𝑥 𝐼 𝑚𝑎𝑥 Zn-Nc Michaelis constant for Zn 2+ in N. caerulescens 2.83 10 -3 mol m -3 𝐾 𝑚 Zn Maximum root influx of Zn 2+ in other species 9 10 -13 mol m -2 s -1 𝐼 𝑚𝑎𝑥 Zn Michaelis constant for Zn 2+ in other species 2.8 10 -3 mol m -3 𝐾 𝑚 Cl Maximum root influx of Cl -0 𝐼 𝑚𝑎𝑥 Cl Michaelis constant for Cl --𝐾 𝑚 NO 3 𝐶𝐻 Constitutive maximum root influx of NO 3 -3.75 10 -8 mol m -2 s -1 𝐼 𝑚𝑎𝑥 NO 3 𝐶𝐻 Constitutive Michaelis constant for NO 3 -1.6 10 -2 mol m -3 𝐾 𝑚 𝐼 𝑚𝑎𝑥 NO 3 𝐼𝐻 Induced maximum root influx of NO 3 -1.9 10 -7 mol NO 3 -m -2 s -1

Table 2 .

 2 Main characteristics of the soils used to validate the model.

	Soil parameter		Soil A	Soil B	Soil C
		Clay	208	162	204
	Particle size distribution (g kg -1 )	Silt	469	602	454
		Sand	323	236	342
	pH (H 2 O)		6.2	8.1	6.3
	pH (CaCl 2 )		5.7	7.4	5.9
	CaCO 3 (g kg -1 )		0	13	<1
	Organic C (g kg -1 )		26.4	16.9	15.7
	P Olsen (mg P kg -1 )		4.6	34.1	5.5
	C/N ratio		16.8	12.9	8.9
	CEC cmol c kg -1		11.8	12.0	11.0
		Ca 2+	7.1	12.6	9.0
	Exchangeable cations (cmol c kg -	K +	0.6	0.5	0.3
	1 )
		Mg 2+	1.2	0.4	3.5
		Ca 2+	30.3	33.5	14.6
		Mg 2+	22.4	20.9	8.5
		K +	23.1	23.7	48.3
	𝑧 𝑖 + Buffer power, 𝑏 𝑀 𝑖	Na + Zn 2+	23.0 300	23.0 2760	23.0 270
		Cl -	0.4	0.4	0.4
		NO 3 - †	0.4	0.4	0.4
		SO 4 2- †	6.0	6.0	6.0
	† Source: Barber (1995)

Table 3 .

 3 Conditions of the model validation. The pH measurement conditions refer to[START_REF] Blossfeld | Rhizosphere pH dynamics in trace-metal-contaminated soils, monitored with planar pH optodes[END_REF].

Table 4 .

 4 Ionic composition of the soil solutions of soils A, B and C.

	Ion	Unit	Soil A	Soil B	Soil C
	Ca 2+		1.638	2.632	4.322
	K +		0.363	0.295	0.087
	Mg 2+		0.375	0.134	2.868
	Mn 2+	mol m -3	0.029	< 7.1 10 -6	< 7.1 10 -6
	Na +		0.399	0.023	0.636
	NH 4 +		0.071	< 1.9 10 -3	0.020
	Zn 2+		0.411	< 7.8 10 -6	< 7.8 10 -6
	Cl -		0.480	0.436	1.242
	NO 3 -P ions	mol m -3	3.842 < 2.5 10 -4	3.219 0.174	11.907 < 2.5 10 -4
	SO 4 2-		0.653	0.416	0.793
	Charge balance mol c m -3	0.112	1.102	0.387

Table 5 .

 5 Comparison of measured and simulated pH in the rhizosphere of three plant species grown in three soils. The simulation conditions are given in Table3.

	pH measurement conditions	Soil A		Soil B		Soil C
		Measured Simulated Measured Simulated Measured Simulated
				Ryegrass
	Average pH along the surface of a single young root			7.8***	7.6	6.5*	6.9
	Average pH in root network			7.9***	8.0	7.7***	7.6
	Bulk soil (start of simulation)			7.4	7.4	6.2	6.2
				Alpine pennycress
	Average pH along the surface of a single young root	6.8***	6.4	8.4***	7.6	7.8***	6.9
	Average pH in root network	7.0***	7.7	7.9***	8.0	n.d. ‡	7.7
	Bulk soil (start of simulation)	5.6	5.6	7.5	7.5	6.1	6.1
				Maize
	Average pH along the surface of a single young root	4.9***	6.4	7.1***	7.6	6.0	6.9
	Average pH along the surface of a single older root	5.5***	6.5	7.5***	7.6	6.5*	6.9
	Bulk soil (start of simulation)	5.6	5.6	7.4	7.4	6.2	6.2
	‡ n.d.: not determined.
	* Significant at the 0.05 probability level
	*** Significant at the 0.001 probability level

Table 6 .

 6 Global sensitivity analysis. Mean percentage of each ion in the total absolute charge uptake.

		Part in the total
	Ion	absolute charge uptake (%)
		1 week	4 weeks
	Ca 2+	0.83	1.36
	K +	16.69	21.10
	Mg 2+	3.65	6.01
	NH 4 +	6.79	5.91
	NO 3 -	63.69	54.88
	P ions -	7.51	9.32
	SO 4 2-	0.85	1.40

Table 7 .

 7 Simulation of the rhizosphere pH using the composition of 37 soils sampled worldwide. Mean ion concentrations at the rhizoplane after 0, 1, 4, 8 and 12 weeks of root uptake.

	Ion			Initial	Concentration at the rhizoplane, mol m -3 1 week 4 weeks 8 weeks	12 weeks
	Ca 2+	4.81	5.10	5.16	5.16	5.16
	Mg 2+	1.02	1.09	1.10	1.10	1.09
	K +			0.72	0.68	0.66	0.63	0.61
	NO 3	-	7.83	6.69	4.39	2.48	1.26
	SO 4	2-	1.57	1.70	1.70	1.70	1.70
	NH 4	+	0.28	0.24	0.20	0.18	0.16
	P ions	4.70E-03	3.20E-03	2.74E-03	2.51E-03	2.36E-03
	Na +			1.05	1.11	1.11	1.11	1.11
	Cl -			1.54	1.60	1.60	1.60	1.60
	Charge balance	+1.20	+2.74	+5.08	+6.95	+8.12
	ΔpH	0	0.9	1.1	1.1	1.1

Table 8 .

 8 Simulation of the rhizosphere pH using the composition of 37 soils sampled worldwide. Cumulated root uptake of the different ions and part in the total absolute charge uptake after 1, 4, 8 and 12 weeks of root uptake.

			Ion	Cumulated root uptake, mol m -1 1 week 4 weeks 8 weeks 12 weeks	Part in the total absolute charge uptake, % 1 week 4 weeks 8 weeks 12 weeks
	Ca 2+	3.85E-07 1.54E-06 3.08E-06 4.62E-06	1.0	1.4	1.8	2.1
	Mg 2+	1.26E-06 5.07E-06 1.01E-05 1.52E-05	2.8	3.5	4.3	4.9
	K +			1.55E-05 6.10E-05 1.20E-04 1.76E-04	20.3	26.8	32.0	35.2
	NO 3	-	1.20E-04 3.95E-04 6.28E-04 7.79E-04	69.8	61.8	55.6	51.7
	SO 4	2-	2.95E-07 1.18E-06 2.36E-06 3.54E-06	0.9	1.2	1.5	1.7
	NH 4	+	7.45E-06 2.54E-05 3.96E-05 4.87E-05	4.8	4.7	4.1	3.7
	P ions	3.20E-07 1.27E-06 2.54E-06 3.80E-06	0.4	0.5	0.7	0.8
	Na +			0	0	0	0	0	0	0	0
	Cl -			0	0	0	0	0	0	0	0
	Cations					29.0	36.5	42.2	45.8
	Anions					70.6	63.0	57.1	53.3
	Charge balance -9.47E-05 -2.99E-04 -4.51E-04 -5.27E-04

ACKNOWLEDGMENTS

Jean-Marc Custos had a PhD grant from Région Lorraine (Décision 613-2008) and from Département Environnement et Agronomie of INRA. The authors are grateful to Romain Goudon for carrying out the analyses of the soil solutions.

Michaelis constant for P ions 5.8 10 -3 mol m -3 𝑆 NO 3 -Source of NO 3 -from mineralization 1.70 10 -7 mol m -3 s -1 # † Source: [START_REF] Nye | The measurement and mechanism of ion diffusion in soils. VIII -A theory for the propagation of changes of pH in soils[END_REF]. ‡ Source: [START_REF] Li | Diffusion of ions in sea water and in deep-sea sediments[END_REF]. § Source: [START_REF] Blossfeld | Rhizosphere pH dynamics in trace-metal-contaminated soils, monitored with planar pH optodes[END_REF]; ¶ Source: [START_REF] Sterckeman | Applying a mechanistic model to cadmium uptake by Zea mays and Thlaspi caerulescens: Consequences for the assessment of the soil quantity and capacity factors[END_REF].

# Source: for root absorption parameters and source of nitrate, references are given in Supplementary Material.