Limin Wang

Man-Yuan Guo

Jean-Baptiste Thibaud

Anne-Aliénor Véry
email: very@supagro.inra.fr

Hervé Sentenac

A repertoire of cationic and anionic conductances at the plasma membrane of Medicago truncatula root hairs

Keywords: Medicago truncatula, root hair, K +, cation, Cl -transports, patch-clamp

Root hairs, as lateral extensions of epidermal cells, provide large absorptive surfaces to the root, and are major actors of plant hydromineral nutrition. In contact with the soil, they also constitute a site of interactions between the plant and rhizospheric microorganisms. In legumes, initiation of symbiotic interactions with N 2 -fixing rhizobia is often triggered at the root hair cell membrane in response to nodulation factors secreted by rhizobia, and involves early signaling events with changes in H + , Ca 2+ , K + and Cl -fluxes inducing transient depolarization of the cell membrane. Here, we aimed to build a functional repertoire of the major root hair conductances to cations and anions in the sequenced legume model Medicago truncatula. Five root hair conductances were characterized through patch-clamp experiments on enzymatically recovered root hair protoplasts. These conductances displayed varying properties in terms of voltage dependence, kinetics and ion selectivity. They consisted of hyperpolarization and depolarization-activated conductances for K + , cations or Cl -. Among them, one weakly outwardly-rectifying cationic conductance and one hyperpolarization-activated slowly inactivating anionic conductance were not known as active in root hairs. All five conductances were detected in apical regions of young growing root hairs, using membrane spheroplasts obtained by laser-assisted cell-wall microdissection.

Combined with recent root hair transcriptomes of M. truncatula, this functional repertoire of conductances is expected to help the identification of candidate genes for reverse genetics studies to investigate the possible role of each conductance in root hair growth and interaction with the biotic and abiotic environment.

INTRODUCTION

Root hairs are outgrowths of the apical cells of the epidermis of plant roots, and play important functions in regulating water and nutrient uptake from the soil [START_REF] Peterson | Root hairs: specialized tubular cells extending root surfaces[END_REF][START_REF] Long | Rhizobium symbiosis: Nod factors in perspective[END_REF]. In legumes, they also play a crucial role in the establishment of a symbiotic interaction with nitrogen-fixing rhizobia [START_REF] Fahraeus | The infection of clover root hairs by nodule bacteria studied by a simple glass slide technique[END_REF]. During rhizobial infection, molecular cross-talk occurs between root hairs and rhizobia. Ionic exchanges across the root hair cell membrane, notably involving Ca 2+ , Cl -, K + and H + , are amongst the earliest responses to the nodulation factors (NF) secreted by rhizobia [START_REF] Felle | The role of ion fluxes in Nod-factor signaling in Medicago sativa[END_REF], creating calcium and electrical (by affecting the membrane potential) signals [START_REF] Ehrhardt | Depolarization of alfalfa root hair membrane potential by Rhizobium meliloti Nod factors[END_REF][START_REF] Felle | Nod signal-induced plasma membrane potential changes in alfalfa root hairs are differentially sensitive to structural modifications of the lipochitooligosaccharide[END_REF].

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Despite their importance, the ion conductances from plant root hairs have been poorly investigated, when compared with those from guard cells for example, particularly in situ.

Only a handful of papers have described ion channel currents in root hairs. Such information concerns mainly Arabidopsis. [START_REF] Lew | Substrate regulation of single potassium and chloride ion channels in Arabidopsis plasma membrane[END_REF] recorded, by voltage-clamping Arabidopsis root hair cells using double-barreled microelectrodes, proton pump currents and potassium currents sensitive to the tetraethyl ammonium (TEA) blocker. By using the patch-clamp technique, [START_REF] Gassmann | Inward-rectifying K + channels in root hairs of wheat: A mechanism for aluminium-sensitive low-affinity K + uptake and membrane potential control[END_REF] described an inwardly-rectifying K + conductance in root hairs of wheat. Then, by coupling reverse genetics to patch-clamp studies, Hedrich's group provided evidence that Arabidopsis root hair protoplasts are equipped with two K + inwardly rectifying Shaker channel subunits (AKT1 and AtKC1) and a K + outwardly rectifying one, GORK [START_REF] Ivashikina | K + channel profile and electrical properties of Arabidopsis root hairs[END_REF][START_REF] Reintanz | AtKC1, a silent Arabidopsis potassium channel alpha -subunit modulates root hair K + influx[END_REF]. In addition, hyperpolarization-activated and depolarization-activated calcium channels were reported at the tip of Arabidopsis root hairs (Véry and Davies, 2000;[START_REF] Miedema | Two voltage-dependent calcium channels co-exist in the apical plasma membrane of Arabidopsis thaliana root hairs[END_REF].

In leguminous plants, few analyses have investigated root hair ion channel activity. In Medicago sativa, beside Shaker-like inwardly and outwardly rectifying K + currents, an anionic current was detected by voltage-clamp on intact root hairs (Bouteau et al., 1999;[START_REF] Kurkdjian | Ion currents involved in early Nod factor response in Medicago sativa root hairs: a discontinuous single-electrode voltage-clamp study[END_REF]. The ion channels present in root hairs of Medicago truncatula, whose genome was recently sequenced, were not investigated so far at the functional level.

In this study, we describe five ionic conductances active at the plasma membrane of Medicago truncatula root hairs, their major functional properties being studied by the patch-clamp technique. The term "conductance" in our experiments made in a native context refers to a type of permeation pathway that can be mediated by a single molecular identity or by set of channels of several identities but displaying similar properties. Overall, this study of the ionic conductances of the root hair plasma membrane of M. truncatula is aimed at building a "repertoire", which could help us progress in the identification of the main actors in nutrient uptake and in channel-mediated responses to biotic and abiotic signals in leguminous plants.

RESULTS

With the aim to identify the repertoire of main cationic and anionic conductances present at the plasma membrane of Medicago truncatula root hairs, root hair protoplasts were first recovered by enzymatic digestion of the apical cell wall of root hairs. This enabled to get a

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This article is protected by copyright. All rights reserved. large number of protoplasts, facilitating a detailed characterization by patch-clamp. Several types of conductances were identified, and those which were most often encountered were further characterized. Some were reminiscent of conductances already described in root hair cells from other plants (e.g., Shaker-like conductances), whereas others have not been reported previously in root hair cells. The characterized conductances are described and compared below, sorted based on their sensitivity to voltage and ionic permeability, into three classes: cationic outwardly-rectifying, cationic inwardly-rectifying and anionic conductances.

Cationic outwardly-rectifying conductances

A first set of patch-clamp solutions were designed to investigate K + -permeable conductances in our root hair protoplasts. External and internal (pipette) solutions contained 30 mM and 150 mM K + respectively (Figure 1, Table S1). In order to discard anionic currents, low concentrations of permeant anions were used (2 and 4 mM Cl -in external and internal solutions, respectively), by adding K + as a gluconate salt. The calcium concentration was set to 1 mM externally, and was buffered to a low value internally (free Ca 2+ i < 10 nM). The pH was 5.6 externally and 7.4 internally.

By applying a voltage-clamp protocol with a holding voltage at -75 mV, and 1.5 s-lasting pulses at less negative voltages varying with 15 mV increments up to +75 mV (Figure 1a), outward currents were mainly recorded. Several conductances were distinguished based on the level of rectification and current activation kinetics.

The conductance that was most often observed displayed weak outward rectification (Figure 1b,d). Upon application of depolarizing pulses, its currents comprised an instantaneous component and a time-dependent component, the relative part of the latter increasing with depolarization (Figure 1b). Current activation was smooth with exponential kinetics. The time constant of half activation determined at 45 mV was 67±3 ms (n=3). Small deactivation currents were observed upon return to the holding voltage after the activation pulse (Figure 1b).

Besides this weakly rectifying conductance, a strongly outwardly-rectifying Shaker-like conductance was evidenced in the same experimental conditions (Figure 1c,e). This conductance displayed a typical slow sigmoidal activation of currents and slow deactivation kinetics upon return to the holding voltage (Figure 1c), recalling the kinetics of the Arabidopsis SKOR and GORK outward Shaker channels characterized in Xenopus oocytes and in planta [START_REF] Gaymard | Identification and disruption of a plant Shaker-like outward channel involved in K + release into the xylem sap[END_REF][START_REF] Downey | KDC1, a novel carrot root hair K + channel[END_REF][START_REF] Ivashikina | K + channel profile and electrical properties of Arabidopsis root hairs[END_REF]Hosy et al.,

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2003)

. The time constant of half-activation of Shaker-like currents at 45 mV was 301±58 ms (n=3), thus 4.5 times larger than that of the weakly-rectifying smooth conductance described above which is thereafter called by contrast "fast-activating" outward conductance.

The effect of varying the external K + concentration was examined on both the "fast-activating" and the Shaker-like outward conductances, using a set of external solutions containing K-gluconate concentrations at 3, 10, 30 or 100 mM, in the same background (1 mM CaCl 2 at pH 5.6). Decreasing the external concentration of K + shifted negatively the current-voltage (I-V) relationships of the "fast-activating" conductance, indicating that this conductance is permeable to K + (Figure 1f). This also increased the outward rectification of currents (Figure 1f), which appeared strong with 3 mM external K + , while almost absent with 100 mM external K + . To check whether the rectification involved a regulation by voltage of the corresponding conductance, a combined Goldman-Boltzmann equation was fitted to the mean steady-state I-V relationships (Véry et al., 1995). This revealed that the observed rectification could be mostly explained by the transmembrane K + concentration gradient (Figure 1f). Only a slight voltage gating, not affected by the external K + concentration, had to be included to fully account for the shape of the I-V relationships (Figure 1h): the obtained activation curve upon depolarization displayed an apparent gating charge (z) of 0.17, and a half activation potential (Ea50) of 33 mV. In the Shaker-like conductance, in contrast to the "fast-activating" conductance, the outward rectification was strong whatever the external K + concentration (Figure 1g). Analysis of the Shaker-like gating by depolarization, fitting Boltzmann equations to the deactivation currents, indicated a large apparent gating charge z of 1.3, and Ea50 values varying with the external K + concentration (Figure 1i). The Ea50 variations followed those of the K + equilibrium potential (E K ), however with weaker slope (44 mV per 10 fold activity change; Figure 1i), as already reported for such type of conductance [START_REF] Downey | KDC1, a novel carrot root hair K + channel[END_REF][START_REF] Langer | Poplar potassium transporters capable of controlling K + homeostasis and K + -dependent xylogenesis[END_REF].

To explore in more details the selectivity of the two outward conductances, reversal potentials of currents were compared in the set of solutions containing varying external K + concentrations. A "tail current" protocol (Figure 2a) was used: in a first step, outwardly rectifying channels were opened by applying a positive voltage, then were progressively closed in less depolarized conditions flanking their reversal potential (Figure 2b,c). The reversal potential of both the Shaker-like and the "fast-activating" outward currents, graphically determined from time decreasing currents at the test voltages (Figure 2d, e), was identical to E K in the presence of 100 mM external K + (Figure 2f,g). When the external K +

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This article is protected by copyright. All rights reserved. concentration was decreased, the reversal potential remained close to E K in the Shaker-like conductance (52 mV shift of Erev per 10-fold change in external K + activity, as compared to 58 mV shift of E K ; Figure 2g), while it clearly deviated from E K in the "fast-activating" conductance (42 mV shift of Erev per 10-fold change in external K + activity; Figure 2f). This indicates that the "fast-activating" conductance is less selective for K + than the Shaker-like outward conductance. The cationic selectivity of both conductances was further addressed by comparing reversal potentials in solutions containing 50 millimole.L -1 of different monovalent cations (K + , Na + , Rb + , Li + or NH 4 + ). The Shaker-like outward conductance showed selectivity for K + and Rb + , compared to the other cations (relative permeability to K + close to 0.3 for Na + , Li + and NH 4 + ; Figure S1b,d). In contrast, the "fast-activating" outward conductance appeared fairly non-selective for the different cations (Figure S1a,c), since permeabilities to K + , Na + and Rb + were almost equal. This conductance displayed however a weak selectivity against Li + (P Li /P K = 0.6 ± 0.01) and, surprisingly, a permeability to NH 4 + larger than that to K + (P NH4 /P K = 1.5 ± 0.2) (Figure S1a,c). At last, the sensitivity to pH of both outward conductances was examined. The outward current was slightly increased in the Shaker-like conductance when the external pH was more alkaline (7.5 versus 5.6) (Figure S1f) similarly as in the outward Shakers GORK and SKOR (Véry et al., 2014). More alkaline pH also increased the outward current in the "fast-activating" conductance (Figure S1e).

Cationic inwardly-rectifying conductances

The presence of inwardly-rectifying K + /cationic conductances in M. truncatula root hair plasma membrane was explored using dedicated protocols starting at a weakly negative holding voltage at which this type of conductance would be closed, then applying hyperpolarizing pulses to activate the conductance (Figure 3a, top).

As previously observed in root hairs from different species, a Shaker-like inwardly rectifying K + conductance [START_REF] Gassmann | Inward-rectifying K + channels in root hairs of wheat: A mechanism for aluminium-sensitive low-affinity K + uptake and membrane potential control[END_REF]Bouteau et al., 1999;[START_REF] Downey | KDC1, a novel carrot root hair K + channel[END_REF][START_REF] Hartje | Functional characterisation of LKT1, a K + -uptake channel from tomato root hairs, and comparison with the closely related potato inwardly rectifying K + -channel SKT1 after expression in Xenopus oocytes[END_REF] was present in the enzymatically-recovered root hair

protoplasts. An example of corresponding current traces and I-V plot obtained with 30 mM K-gluconate and 10 mM CaCl 2 (at pH 5.6) in the external solution and 150 mM K-gluconate (at pH 7.4) in the internal solution is shown in the bottom panel of figure 3a. The inwardly-rectifying conductance was observed upon membrane hyperpolarization from a threshold of about -120 mV (Figure 3b) and displayed the typical slow activation of inward Shaker currents, and fast exponential deactivation when the membrane was weakly polarized

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again (Figure 3a;Zimmermann et al., 1998;[START_REF] Hartje | Functional characterisation of LKT1, a K + -uptake channel from tomato root hairs, and comparison with the closely related potato inwardly rectifying K + -channel SKT1 after expression in Xenopus oocytes[END_REF][START_REF] Reintanz | AtKC1, a silent Arabidopsis potassium channel alpha -subunit modulates root hair K + influx[END_REF]Geiger et al., 2009a). Steady-state currents of the M. truncatula root hair Shaker-like inward conductance were strongly rectifying (Figure 3b), as classically observed for this conductance type. Half saturation (apparent K M ) of this inward conductance upon increasing the external K + concentration occurred at about 34 mM K + (Figure S2).

Since the Shaker-like conductance displayed inward rectification, its selectivity was addressed using "tail current" protocols (Figure 4a), with first activation of the conductance at a quite negative voltage before applying less negative test voltages, for determination of the conductance reversal potential. The analysis of tail currents in the 30 mM K + and 10 mM CaCl 2 -containing external solution gave an Erev value only 3 mV positive to E K (Figure 4a,b). Varying the external K + concentration from 3 to 100 mM (in 1 mM CaCl 2 background to avoid blockage of the conductance at low external K + concentration), Erev values closely followed changes of E K , shifting by 52 mV upon 10-fold change in the external K + activity (Figure 4c). Thus, in the tested solutions both with 1 or 10 mM external Ca 2+ and varying Cl - backgrounds, the inwardly rectifying Shaker-like conductance was confirmed to mainly carry, as expected, K + ions.

Two types of "Slow" anion conductances

For the characterization of the anion conductances at the plasma membrane of M. truncatula root hairs, in order to prevent K + channel activity recording, TEA-Cl and/or CaCl 2 -based patch-clamp solutions without added K + were used. A first set of solutions resulted in almost symmetrical Cl -concentrations on both sides of the membrane: 100 mM at the external face of the membrane and 113 mM at the internal face. The free Ca 2+ concentration of the internal solution was buffered to 1 µM (Table S1). Using these solutions and voltage-clamp protocols composed of a positive holding potential (set at 0 to +20 mV), and 10-s voltage pulses from positive to quite negative values in -15 or -30 mV increments (Figure 5a, b), two types of anion conductances were dominant.

One of these two anion conductances mediated currents that were reminiscent of those elicited by S-type anion conductances described in different plant tissues, including roots and guard cells in different species [START_REF] Schroeder | Two types of anion channel currents in guard cells with distinct voltage regulation[END_REF][START_REF] Frachisse | Characterization of a nitrate-permeable channel able to mediate sustained anion efflux in hypocotyl cells from Arabidopsis thaliana[END_REF][START_REF] Roberts | Plasma membrane anion channels in higher plants and their putative functions in roots[END_REF]Geiger et al., 2009b[START_REF] Geiger | Stomatal closure by fast abscisic acid signaling is mediated by the guard cell anion channel SLAH3 and the receptor RCAR1[END_REF]. This root hair conductance, which was steadily active at the 0 mV holding potential used in our experiments, showed typical slow deactivation of currents upon prolonged application of hyperpolarizing pulses (Figure 5c). Thus, currents

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elicited by this conductance type were maximal at the beginning of the hyperpolarizing pulse and decreased during the pulse, more so and faster when the voltage pulse became more negative (Figure 5c). Current-voltage relationships determined at the end of the 10-s voltage pulses displayed a maximal negative current (anion efflux) at about -175 to -200 mV, in the 100/113 external/internal mM Cl -conditions (Figure 5e). At more negative voltages, the current at the end of the pulse was only slightly reduced. The relative open probability of this conductance (Po/Pomax), whose voltage dependence was analyzed by dividing, for each applied voltage, the steady state current by the driving force, decreased upon hyperpolarization. Using simple Boltzmann functions to describe the regulation by voltage of the S-type conductance open probability, provided good fits to the I-V data (Figure 5e) and allowed to extract an activation curve (Figure 5g) reminiscent of those provided for guard cell S-type channels [START_REF] Geiger | Stomatal closure by fast abscisic acid signaling is mediated by the guard cell anion channel SLAH3 and the receptor RCAR1[END_REF], with a small apparent gating charge (z = 0.2) and a quite negative half activation potential (Ea50 = -158 mV). The reversal potential of this S-type conductance, directly determined from I-V relationships in these ionic conditions with Cl -as dominant anion species was close to E Cl (Figure 5e).

Other root hair protoplasts displayed another kind of anion current pattern with major kinetic differences compared to the above described classical S-type conductance. Actually, upon application of hyperpolarization pulses from a slightly positive holding potential, the initial negative current was not maximal (especially when applying the most negative voltage pulses). Indeed, the current slowly increased at the beginning of the hyperpolarization pulse (Figure 5d). This suggested that hyperpolarization activated this conductance. Additionally to this voltage dependent activation, a voltage-dependent inactivation was observed. This was visible when pulses were applied at voltages more negative than ca -130 mV, in which case a prolonged hyperpolarization led after the initial increase in current to a subsequent current decrease. The reduction of current initiated faster and was stronger when the applied voltage was more negative (Figure 5d). Thus, from the voltage threshold where current inactivation was detected, a peak of current was apparent during the 10-s pulses. I-V relationship at the end of the 10-s pulse displayed a marked maximum of negative current at about -150 mV (Figure 5f). At voltages less negative than -130 mV for which no late current decrease was observed, the slope of the I-V relationship increased with more negative voltages (Figure 5f), in contrast with that of the S-type conductance (Figure 5e). The activation curve of the hyperpolarization-activated conductance extracted from peak currents (Figure 5h) was almost opposite to that of the Medicago root hair S-type conductance, with similar apparent gating charge (z = 0.2) but opposite voltage gating, and half activation potential around -40 mV

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(Figure 5g, h). The steady state current reversed very close to E Cl (+ 3 mV; Figure 5f), suggesting that the slow hyperpolarization-activated conductance mediated anionic currents.

Changing the external Cl -concentration from 100 to 20 mM (which made a 39 mV positive shift in E Cl ) induced a positive shift in the I-V relationships of both S-type and slow hyperpolarization-activated conductances ( Figures 6a, b andS3). The mean shift in Erev was 32 mV in the S-type and 38 mV in the slow hyperpolarization-activated conductance (Figure 6c,d). This confirmed that both conductances were Cl -conductances.

The relative permeabilities to nitrate and chloride were compared in the slow hyperpolarization-activated conductance by replacing the external Cl -(20 mM) with the same concentration of NO 3 -(0.5 mM CaCl 2 were however kept; Figure S4a,b). Current-voltage relationships showed a positive shift when NO 3 -replaced Cl -(Figure S4c), the mean reversal potential shifting from +25 ± 2 mV to +47 ± 4 mV (n = 5). This indicated that the slow hyperpolarization-activated conductance was more permeable to Cl -than to NO 3 -, with a relative permeability P NO3 /P Cl of 0.4 Figure S4d). This feature contrasts with reports concerning the S-type anion conductance described in vivo at the guard cell plasma membrane and the cloned S-type guard cell SLAH3 channel [START_REF] Schmidt | Anion selectivity of slow anion channels in the plasma-membrane of guard-cellslarge nitrate permeability[END_REF][START_REF] Frachisse | Characterization of a nitrate-permeable channel able to mediate sustained anion efflux in hypocotyl cells from Arabidopsis thaliana[END_REF][START_REF] Roberts | Plasma membrane anion channels in higher plants and their putative functions in roots[END_REF][START_REF] Geiger | Stomatal closure by fast abscisic acid signaling is mediated by the guard cell anion channel SLAH3 and the receptor RCAR1[END_REF]. This latter conductance was more often observed than the S-type one (in > 70% of ca. 25 protoplasts displaying one of these two conductances).

Effect of the cytosolic Ca 2+ concentration on the activity of the cationic conductances

Ca 2+ represents an important second messenger in root hair cells, in legumes in interaction with rhizobia in particular [START_REF] Long | Rhizobium symbiosis: Nod factors in perspective[END_REF]. Large tip-base variations of cytosolic Ca 2+ in root hairs also support tip growth [START_REF] Felle | The cytosolic Ca 2+ concentration gradient of Sinapis alba root hairs as revealed by Ca 2+ -selective microelectrode tests and Fura-Dextran ratio imaging[END_REF]. We examined whether cytosolic Ca 2+ affected the three cationic conductances observed in M. truncatula root hair protoplasts, i.e., the inward and outward Shaker-like conductances and the "fast-activating" outward cationic one. Using a pipette solution with the free Ca 2+ concentration increased to 1 µM, the three cationic conductances were still observed in the protoplasts (Figure 7a to 7c). However, the mean current was reduced in the two Shaker-like conductances, as compared to low cytosolic Ca 2+ conditions (Figure 7d, e), by ca. 40% for the inward Shaker and 65% for the outward Shaker. The inward Shaker current decrease was accompanied by a slight negative shift (-15 mV) in the threshold of activation. The reversal potentials of currents of both the inward and outward Shaker conductances (-33 ± 3 and -27 ± 2 mV, respectively; n = 3) were similar to

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those recorded at low cytosolic Ca 2+ , remaining close to E K , as expected for these highly K + -selective conductances. In contrast to the Shaker conductances, the fast-activating cationic outward conductance was enhanced by the high cytosolic Ca 2+ concentration, by ca. 50%

(Figure 7f). The apparent outward rectification was reduced, as compared to low cytosolic Ca 2+ conditions (Figures 1b, d and7c, f), and the reversal potential of currents (-18 ± 3.5 mV)

was positively shifted by ca. 15 mV. This shift in Erev may denote a permeability to Ca 2+ in this weakly selective cationic conductance. A Cl -contribution to the current mediated by this conductance could also be possible and contribute to the shift in Erev. Indeed, E Cl was shifted by +28 mV in the experiments with high internal Ca 2+ , and weakly-selective cation conductances described in other plant species were reported to display significant permeability to Cl -in addition to monovalents and Ca 2+ (Wegner and de Boer, 1997;Zhang et al., 2000Zhang et al., & 2002)).

Co-occurence of the different K + conductances in M. truncatula root hair protoplasts

Among the five characterized conductances in our root hair protoplasts, the K + -permeable ones (the "fast-activating" outward conductance and the outward and inward Shaker-like ones) could be evidenced using the same internal solutions, which rendered the analysis of co-occurrence of these three conductances possible with the aim to test the hypothesis of a preferential co-occurrence or of a mutual exclusion. Recordings obtained at low internal Ca 2+

(Figure 8) were more particularly analyzed, since a larger number of recordings (104 protoplasts) were available in this condition.

Regarding the outward K + -permeable conductances, the fast-activating outward type was the predominant one. It was found in 56 of the 104 protoplasts examined (from which 29 displayed the inward Shaker-type conductance and 27 did not display this latter conductance), thus in ca. 55% of the protoplasts. The Shaker-like outward type was on the other hand observed in 24 protoplasts (12 displaying the inward Shaker-type conductance and 12 not displaying this conductance), thus in ca. 30% of the protoplasts. The presence of both types of outward conductances was only occasional, observed in only ~9% of protoplasts (in 8 of the 104 protoplasts). Altogether, these results provide evidence that the fast-activating outward conductance is rather rarely present together with the Shaker-like outward conductance, i.e. that these two conductances tend to be mutually exclusive from each other.

In ca. 15% of the protoplasts (16 protoplasts in total out of the 104), neither of the two outward K + conductances was observed. In that latter case, the protoplasts either displayed an

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instantaneously-activating outwardly-rectifying conductance or a "leak-like" (i.e. "Ohmic") K + one (conductances not further characterized).

The inwardly rectifying Shaker-like K + conductance was found to dominate the membrane conductance to K + in 56 from the 104 protoplasts, i.e., in 54% of the root hair protoplasts (Figure 8). Amongst these 56 protoplasts, the fast-activating outward K + conductance and the Shaker-like outward conductance dominated the membrane outward conductance to K + in 29 and 12 protoplasts, respectively, thus in ca. 52% and 22% of these protoplasts. Similar percentage (56% and 25%, respectively) were found among the 48 protoplasts sorted as displaying no inwardly rectifying Shaker-like K + conductance. Hence, no correlation appeared between the presence of the inward Shaker conductance and that of the outward ones.

The analysis of 16 protoplasts patch-clamped at high cytosolic Ca 2+ (Figure S5) suggested that the same conclusions could be drawn: the fast-activating cationic conductance and the outward Shaker-type one dominated the outward conductances, the fast-activating cationic one being the most often encountered. Co-occurrence of the two outward conductances was rare (1 out of 16 protoplasts). The inward Shaker type dominated the inward conductances and was observed jointly with either the outward Shaker type (in 75% of protoplasts displaying this outward Shaker conductance: 3 out of 4 protoplasts) or the fast outward conductance (in ca. 60% of protoplasts displaying this fast conductance: 5 out of 8), confirming with high cytosolic Ca 2+ , the absence of correlation between the presence of inward Shaker conductance and that of the cationic outward ones.

Conductances active in apical membrane of young root hairs

With the aim to investigate whether the characterized root hair conductances were active at the tip of young growing root hairs, the laser cell-wall ablation technique was used (Véry and Davies, 2000). The cell-wall of selected growing root hairs was burnt at the apex by the laser after a short plasmolysis of the cells, allowing the recovery of spheroplasts of apical membrane extruding from the cut wall upon deplasmolysis (Figure 9A). These spheroplasts were analyzed in the whole-cell configuration in the same ionic conditions and with the same voltage-clamp protocols as those used in the previous patch-clamp experiments on enzymatically-recovered protoplasts. The main conductances characterized in enzymatically-recovered protoplasts, i.e., the outwardly and inwardly rectifying Shaker-like ones, the fast-activating outward cationic one and the two slow anionic ones (Figures 1b,c,

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3a and 5c, d), were identified from their typical kinetics and I-V relationships in laser-obtained root hair spheroplasts (Figure 9b to 9f). This provided evidence that all the conductances characterized in enzymatically-recovered protoplasts were present in the apical membrane of growing root hairs.

DISCUSSION

The identified "repertoire" of M. truncatula root hair conductances compared to that of other plant cell types

An important number of plasma membrane plant conductances have already been characterized in varying cell types [START_REF] Hedrich | Ion channels in plants[END_REF]. However, integrative descriptions of the plasma membrane conductance equipment are still lacking for most cell types, except for guard cells, the main model of membrane transport biology in plants so far, in which a number of native plasma membrane conductances have been characterized since the 1980's, first in Vicia faba then in Arabidopsis (Ward et al., 2009), and to a lesser extent for xylem parenchyma cells in barley [START_REF] Köhler | Step by step: Deciphering ion transport in the root xylem parenchyma[END_REF]. Here, our aim was to build a repertoire of the major K + and Cl -conductances active at the plasma membrane of root hairs in the sequenced model legume Medicago truncatula.

Five root hair conductances from M. truncatula were characterized in this study. Among them, three are reminiscent of conductances described at the root hair membrane of other species. These conductances are the common K + Shaker-like inwardly and outwardly rectifying ones (Fig. 1c and3a; Véry et al., 2014), already described in root hair cells from, e.g., Arabidopsis or Medicago sativa (Bouteau et al., 1999;[START_REF] Ivashikina | K + channel profile and electrical properties of Arabidopsis root hairs[END_REF][START_REF] Reintanz | AtKC1, a silent Arabidopsis potassium channel alpha -subunit modulates root hair K + influx[END_REF], and the S-type anion conductance observed for instance in root hairs from bean (Figure 5c; [START_REF] Dauphin | Effect of desiccation on potassium and anion currents from young root hairs: Implication on tip growth[END_REF].

The kinetics, voltage threshold of activation, sensitivity to external K + and affinity for K + of the M. truncatula root hair inwardly rectifying Shaker-type conductance (Figure 3, Figure S2) are similar to those of counterparts described in root hairs from other plant species [START_REF] Gassmann | Inward-rectifying K + channels in root hairs of wheat: A mechanism for aluminium-sensitive low-affinity K + uptake and membrane potential control[END_REF]Bouteau et al., 1999;[START_REF] Reintanz | AtKC1, a silent Arabidopsis potassium channel alpha -subunit modulates root hair K + influx[END_REF]. Similarly, the [START_REF] Frachisse | Sulphate is both a substrate and an activator of the voltage-dependent anion channel of Arabidopsis hypocotyl cells[END_REF][START_REF] Downey | KDC1, a novel carrot root hair K + channel[END_REF][START_REF] Ivashikina | K + channel profile and electrical properties of Arabidopsis root hairs[END_REF]. On the other hand, none of these studies provided a detailed analysis of the cationic selectivity of any root hair Shaker-type outward conductance. The present analysis indicates that the ionic selectivity of the M. truncatula root hair Shaker-type conductance (Figure 2g, Figure S1b,d) is quite similar to that determined, by using the Xenopus oocyte expression system, for the Arabidopsis outward Shaker SKOR, which is specifically expressed in the root stele [START_REF] Gaymard | Identification and disruption of a plant Shaker-like outward channel involved in K + release into the xylem sap[END_REF][START_REF] Johansson | External K + modulates the activity of the Arabidopsis potassium channel SKOR via an unusual mechanism[END_REF].

Plasma membrane "slowly-inactivating" anion conductances, although reported in various cell types including root hairs [START_REF] Frachisse | Characterization of a nitrate-permeable channel able to mediate sustained anion efflux in hypocotyl cells from Arabidopsis thaliana[END_REF][START_REF] Köhler | The delivery of salts to the xylem: three types of anion conductance in the plasmalemma of the xylem parenchyma of roots of barley[END_REF][START_REF] Dauphin | Effect of desiccation on potassium and anion currents from young root hairs: Implication on tip growth[END_REF]Zhang et al., 2004a;[START_REF] Hosy | A procedure for localisation and electrophysiological characterisation of ion channels heterologously expressed in a plant context[END_REF][START_REF] Mori | CDPKs CPK6 and CPK3 function in ABA regulation of guard cell S-type anion-and Ca 2+ -permeable channels and stomatal closure[END_REF], have so far mainly been analyzed (by patch-clamp or in situ voltage-clamp) at the membrane of guard cells [START_REF] Schroeder | Two types of anion channel currents in guard cells with distinct voltage regulation[END_REF][START_REF] Forestier | Pharmacological properties of slow anion currents in intact guard cells of Arabidopsis. Application of the discontinuous single-electrode voltage-clamp to different species[END_REF]Vahisalu et al., 2008, Geiger et al., 2009b;[START_REF] Guzel Deger | Guard cell SLAC1-type anion channels mediate flagellin-induced stomatal closure[END_REF]. Here, two slow anion conductances manifested themselves at the plasma membrane of M. truncatula root hairs. From current kinetics and shape of I-V relationships (Figure 5c to 5f), one of the two anion conductances of M. truncatula resembled S-type conductances described in the literature [START_REF] Köhler | The delivery of salts to the xylem: three types of anion conductance in the plasmalemma of the xylem parenchyma of roots of barley[END_REF][START_REF] Mori | CDPKs CPK6 and CPK3 function in ABA regulation of guard cell S-type anion-and Ca 2+ -permeable channels and stomatal closure[END_REF], and in particular the one previously identified in root hairs of bean (Phaseolus vulgaris) subjected to hydric stress [START_REF] Dauphin | Effect of desiccation on potassium and anion currents from young root hairs: Implication on tip growth[END_REF].

The second slowly inactivating type of anion conductances, which was the most frequently observed anionic conductance in M. truncatula root hair protoplasts in our experimental conditions, is reminiscent (to our knowledge) of only one Cl -conductance described so far, that in pea mesophyll cells (Elgenza and Volkenburgh, 1997;[START_REF] Roberts | Plasma membrane anion channels in higher plants and their putative functions in roots[END_REF]. From its slow kinetics of current inactivation along 10-s hyperpolarization pulses (Figure 5d), it may be classified in the slow-type anion conductances. However, the general shape of its I-V relationships at steady-state was quite different from that of S-type channels, more closely resembling that of rapid-type conductances (Figures 5f, 6b, S3 and S4;[START_REF] Frachisse | Sulphate is both a substrate and an activator of the voltage-dependent anion channel of Arabidopsis hypocotyl cells[END_REF]Vahisalu et al., 2008;[START_REF] Diatloff | R type anion channel -A multifunctional channel seeking its molecular identity[END_REF][START_REF] Roelfsema | Anion channels: master switches of stress responses[END_REF]. Furthermore, a distinctive feature of this M. truncatula root hair anion conductance, displayed also by the pea mesophyll conductance described by Elgenza and Volkenburgh (1997), is that, upon hyperpolarization, the current first displays a slow activation before inactivating (Figures 5d andS3). Thus, this conductance seems better described as a hyperpolarization-activated slowly inactivating one, conversely to S-type (depolarization-activated) conductances.

Hyperpolarization-activated anion conductances displaying no or slow inactivation of currents have been reported [START_REF] Kurkdjian | Ion currents involved in early Nod factor response in Medicago sativa root hairs: a discontinuous single-electrode voltage-clamp study[END_REF]Zhang et al., 2004a;[START_REF] Roberts | Plasma membrane anion channels in higher plants and their putative functions in roots[END_REF].

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They however all display instantaneous current activation upon membrane hyperpolarization, with the exception of the pea mesophyll conductance (Elgenza and Volkenburgh, 1997), which shares with the M. truncatula conductance slow initial activation of currents upon hyperpolarization. Thus, from kinetics of current activation/inactivation, the hyperpolarization-activated M. truncatula root hair and pea mesophyll anion conductances could be considered as a particular type of hyperpolarization-activated anionic conductances which might be legume-specific. Finally, it should be noted that the M. truncatula hyperpolarization-activated conductance, besides its original kinetic properties, also displayed an unusual selectivity, showing higher permeability to Cl -than to NO 3 -(Figure S3).

This contrasts with the commonly reported higher NO 3 -to Cl -permeability of anion conductances (S-type, rapidly inactivating and hyperpolarization-activated ones) in many cell types and with the first cloned anion channels characterized in heterologous system [START_REF] Hedrich | Malate-induced feedback regulation of plasma membrane anion channels could provide a CO 2 sensor to guard cells[END_REF][START_REF] Schmidt | Anion selectivity of slow anion channels in the plasma-membrane of guard-cellslarge nitrate permeability[END_REF][START_REF] Roberts | Plasma membrane anion channels in higher plants and their putative functions in roots[END_REF]Geiger et al., 2009b[START_REF] Geiger | Stomatal closure by fast abscisic acid signaling is mediated by the guard cell anion channel SLAH3 and the receptor RCAR1[END_REF][START_REF] Gutermuth | Pollen tube growth regulation by free anions depends on the interaction between the anion channel SLAH3 and calcium-dependent protein kinases CPK2 and CPK20[END_REF][START_REF] Maierhofer | A single-pore residue renders the Arabidopsis root anion channel SLAH2 highly nitrate selective[END_REF]. Thus, the characterization of this M.

truncatula anion conductance represents a particularly interesting contribution to our knowledge of plant anion channel diversity.

The last conductance of the present M. truncatula root hair "repertoire", the fast-activating outwardly-rectifying cation conductance, weakly selective among cations, had not been previously reported in root hair cells, but is reminiscent of conductances already described in other cell types. A variety of depolarization-activated non-selective cation channels have been reported in the literature: e.g., in xylem parenchyma (Wegner and Raschke, 1994;

Wegner and de Boer, 1997), Arabidopsis root / root epidermis (Demidchik and Tester, 2002;[START_REF] Shabala | Extracellular Ca 2+ ameliorates NaCl-induced K + loss from Arabidopsis root and leaf cells by controlling plasma membrane K + -permeable channels[END_REF], Arabidopsis and Thlaspi mesophyll [START_REF] Piñeros | Differences in whole-cell and single channel ion currents across the plasma membrane of mesophyll cells from two closely related Thlaspi species[END_REF][START_REF] Shabala | Extracellular Ca 2+ ameliorates NaCl-induced K + loss from Arabidopsis root and leaf cells by controlling plasma membrane K + -permeable channels[END_REF], bean seed coat and cotyledon dermal cells (Zhang et al., 2000(Zhang et al., , 2002(Zhang et al., & 2004b)).

In all these studies, a fast or 'instantaneous' kinetics of current activation was observed. The reported conductances were mostly depicted as only weakly more permeable to cations than to anions, with P cation /P Cl in the range of 2 to 4 (Wegner and de Boer, 1997;Zhang et al., 2000Zhang et al., & 2002)), and weakly discriminating among monovalent cations. The M. truncatula fast-activating outwardly-rectifying conductance displayed strong similarity, with respect to cation selectivity and voltage dependence, with a weakly voltage-dependent non selective cation channel described in wheat roots [START_REF] Davenport | A weakly voltage-dependent, nonselective cation channel mediates toxic sodium influx in wheat[END_REF]. Both conductances poorly discriminated among monovalent cations and displayed the highest permeability to NH 4 + and the lowest one to Li + (Figure S1). The M. truncatula fast-activating outward conductance is also reminiscent of the weakly cationic-selective NORC conductance of

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barley root xylem parenchyma permeable to both monovalents and Ca 2+ (Wegner and Raschke, 1994; Wegner and de Boer, 1997). Analysis of the regulation by voltage of NORC in xylem parenchyma cells indicated in contrast to the Shaker-like conductance, a non-K + -dependent gating, and a smaller gating charge than that in Shakers (~2.5 fold smaller;

Wegner and de Boer, 1997). The voltage-gating of the M. truncatula fast-activating conductance was also non-K + -dependent, and the gating charge was very low (Figure 1h), the observed rectification being mostly explained by transmembrane concentration gradients (Figure 1f).

Gene candidates for the characterized ionic conductances from M. truncatula root hairs

Based on RNA-seq analysis of the M. truncatula root hair transcriptome from plants grown in the same experimental conditions as those used for the present electrophysiological analyses (Damiani et al., 2016), molecular correspondence for some of the macroscopic conductances evidenced in root hair protoplasts can be proposed (Figure 10). Since Medtr5g077770 is the unique outward Shaker gene present in the genome of M. truncatula and is expressed in root hairs (Damiani et al., 2016), it certainly encodes the Shaker outwardly rectifying K + conductance of the root hair membrane. Regarding the inwardly-rectifying Shaker K + conductance, the candidate genes are Medtr3g108320 and Medtr4g113530, which are homologous to AtKC1 and AKT1 genes in Arabidopsis, and are both expressed in M.

truncatula root hairs (Damiani et al., 2016). In Arabidopsis, both AKT1 and AtKC1 Shaker subunits contribute to the root hair inward K + conductance [START_REF] Reintanz | AtKC1, a silent Arabidopsis potassium channel alpha -subunit modulates root hair K + influx[END_REF]Wang et al., 2010). AtKC1 is unable to form by itself functional inward channels at the plasma membrane but associates with AKT1 in heteromeric channels [START_REF] Reintanz | AtKC1, a silent Arabidopsis potassium channel alpha -subunit modulates root hair K + influx[END_REF][START_REF] Duby | AtKC1, a conditionally targeted Shaker-type subunit, regulates the activity of plant K + channels[END_REF]. Thus, as in Arabidopsis, the inward Shaker-type conductance in Medicago is likely composed of both MtAKT1 homomeric channels and MtAKT1/MtKC1 heteromers.

The molecular identity is much more speculative for the outwardly rectifying weakly selective conductance found in M. truncatula root hairs. Several gene families have been proposed to encode weakly selective plasma membrane cation channels, the most frequently proposed ones being the CNGC, GLR, annexin, MCA and OSCA families (Demidchik et al., 2002;White et al., 2002;[START_REF] Jammes | Calcium-permeable channels in plant cells[END_REF]Yuan et al., 2014). Transporters from the KUP/HAK/KT family also are known as weakly selective cationic transport systems (Véry et al., 2014) and KUP/HAK/KT members have been proposed to mediate cation effluxes in vivo (Osakabe et al., 2013). Their selectivity between the monovalents K + , Na + and Li + is however

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not well determined and transporters are believed to give rise to low current levels when compared to channels. Channel members from the families proposed to encode weakly selective plasma membrane cation channels have been suggested through in planta genetic analyses to be involved in different cation transports including that of Ca 2+ (Ali et al., 2007; Laohavisit et al., 2012; Wang et al., 2013; Kong et al., 2016), but their electrophysiological properties have not been extensively characterized. Medicago annexin 1 (Medtr8g038210), coding for one of the six annexins present in root hairs (de Carvalho-Niebel et al., 1998; Figure 10), has been shown to conduct cations across planar bilayer membrane [START_REF] Kodavali | Structural and functional characterization of annexin 1 from Medicago truncatula[END_REF]. However, lack of information on MtAnn1 regulation by voltage prevents comparison with the cationic conductances characterized in the present study. M. truncatula RNA-seq data indicate that root hairs expressed a large number of genes from CNGC, GLR and annexin families (Figure 10). Genes from these families could be proposed as molecular candidates of the weakly outwardly rectifying cationic conductance of M. truncatula root hairs.

The characteristic S-type anion conductance found in M. truncatula root hair protoplasts, based on its current kinetics and voltage gating (Figure 5c, e, g and 9e), is likely encoded by one or several of the 3 SLAC genes (Geiger et al., 2009b[START_REF] Geiger | Stomatal closure by fast abscisic acid signaling is mediated by the guard cell anion channel SLAH3 and the receptor RCAR1[END_REF] expressed in Medicago root hairs (Damiani et al., 2016; Figure 10). On the other hand, the molecular identity of the hyperpolarization-activated slowly inactivating Cl -conductance found in M. truncatula root hair protoplasts is more difficult to predict. The shape of its I-V relationship is reminiscent of that of R-type conductances, among which the Arabidopsis guard cell one has been attributed

to the malate transporter AtALMT12 [START_REF] Meyer | AtALMT12 represents an R-type anion channel required for stomatal movement in Arabidopsis guard cells[END_REF] [START_REF] Felle | Nod signal-induced plasma membrane potential changes in alfalfa root hairs are differentially sensitive to structural modifications of the lipochitooligosaccharide[END_REF][START_REF] Meyer | AtALMT12 represents an R-type anion channel required for stomatal movement in Arabidopsis guard cells[END_REF][START_REF] Mumm | C-terminus-mediated voltage gating of Arabidopsis guard cell anion channel QUAC1[END_REF] in contrast to the M. truncatula hyperpolarization-activated anion conductance (Figure 5d, f andh). This M. truncatula anion conductance also contrasts with cloned plasma membrane anion channels from SLAC and ALMT families by a higher permeability to Cl -than to NO 3 - [START_REF] Piñeros | Not all ALMT1-type transporters mediate aluminum activated organic acid responses: the case of ZmALMT1an anion-selective transporter[END_REF]Zhang et al., 2008;Geiger et al., 2009b;[START_REF] Geiger | Stomatal closure by fast abscisic acid signaling is mediated by the guard cell anion channel SLAH3 and the receptor RCAR1[END_REF][START_REF] Ligaba | Maize ZmALMT2 is a root anion transporter that mediates constitutive root malate efflux[END_REF]

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Conclusion

Five M. truncatula root hair conductances were identified in enzymatically obtained protoplasts, two of which having no reported counterparts in root hairs from other plant species. The fact that all these conductances could be also recorded in root hair spheroplasts obtained using a laser assisted procedure provides evidence that their presence in enzymatically obtained protoplasts does not result from artefactual gene expression during the cell wall digestion but that they are actually active at the root hair cell membrane.

Detection of their presence in our root hair spheroplasts also indicates that they can be active at least in the root hair apical region, involved in tip growth as well as in biotic interactions (e.g., with nitrogen-fixing rhizobia). The three cationic conductances were observed at both low and high cytosolic Ca 2+ conditions and their activity was regulated by the internal Ca 2+ concentration, indicating that they could play a role in varying physiological conditions. The fact that the two outwardly rectifying K + conductances characterized in the enzymatically obtained protoplasts are mutually exclusive from each other is compatible with the hypothesis that the enzymatic procedure provides a mix of protoplasts from root hairs at different developmental stages, displaying distinctive conductance equipment and/or distinctive cytosolic conditions, which would differentially modulate ion channel activities and would not be reversed during the patch-clamp experiments. The fact that the Shaker-type inwardly rectifying K + conductance can be either present in, or absent from, the protoplasts is also compatible with the above hypotheses.

The present report provides the most documented conductance "repertoire" for root hair cells. Members from this repertoire can be expected to play a role in plant mineral nutrition and/or mechanisms involved in root hair tip growth (e.g., via turgor regulation) or in various signaling events. For instance, they might be involved in the ionic responses to Nod factors secreted by rhizobia, known to rapidly trigger (within 1 or 2 min) changes in Ca 2+ , Cl -and K + fluxes leading, along with changes in H + fluxes, to electrical and calcium signaling events that constitute the earliest responses identified so far to Nod factors [START_REF] Felle | The role of ion fluxes in Nod-factor signaling in Medicago sativa[END_REF]. In the sequenced legume M. truncatula, the molecular identity of several members of the characterized repertoire could be hypothesized, which can pave the way to future analysis through reverse genetics of the role of these conductances in the multiple physiological functions of root hairs.

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EXPERIMENTAL PROCEDURES

Plant cultivation

M. truncatula A17 (cv. Jemalong A17) was used in this study. Seeds were placed in a small container and chemical scarification was performed by soaking them in pure sulfuric acid for around 8 minutes. All H 2 SO 4 liquid was then carefully removed. Seeds were washed several times with chilled water to avoid heating damage. Finally, the seeds were immersed in clean water for 1.5-2h. Then, the scarified seeds were sterilized by immersion for 3 minutes in a bleach 6% solution and were rinsed with sterile water several times. Sterilized seeds were transferred onto an agar Petri-dish (0.5 % (w/v) in deionized water) and kept in the dark at 4°C in order to break dormancy and get a synchronization of germination. Two days later, the plate was move to a growth chamber at 21°C for germination for 15 to 24 h. Seedlings (~9)

were transferred to a square Petri dish of solid Fahraeus medium, overlaid by an autoclaved filter paper. The medium (modified from Vincent, 1970) contained 10 g/L of purified agar (Euromedex) and 0.5 mM MgSO 4 , 0.7 mM KH 2 PO 4 , 0.8 mM Na 2 HPO 4 , 1 mM CaCl 2 , 20 µM Fe-citrate, and 0.1 mg/L of MnSO 4 , CuSO 4 , ZnSO 4 , H 3 BO 3 and Na 2 MoO 4 . The pH was adjusted with KOH to 7.5. The lower part of the plate was wrapped in aluminum foil. The plate was placed at 25°C in a growth chamber.

Enzymatic isolation of root hair protoplasts

Three root apical portions of 3-4 cm were cut from 7 to 12 day-old seedlings grown on Fahraeus solid medium, and were incubated in an enzyme solution containing 1% (w/v) cellulase RS (Onozuka, Japan), 0.5% (w/v) macerozyme R-10 (Onozuka, Japan), 0.1% (w/v) pectolyase Y23 (Seishin Pharmaceutical, Japan), 1 mM CaSO 4 , 25 mM MES and 250 mM D-sorbitol, in which the pH was adjusted to 5.5 by addition of KOH. During the digestion, the cell wall at the tip of the root hairs was first digested and protoplasts appeared at the tip of root hair. The follow-up of the cell-wall digestion and protoplast release under the microscope enabled to confirm the root hair origin of recovered protoplasts. After 1h and 20 min of digestion at 28°C with shaking at 60 rpm, the released protoplasts were filtered through a 40 µm nylon mesh and washed twice with the enzyme free medium after slow-speed centrifugation at 4℃. Finally, the protoplasts were re-suspended in the enzyme free medium and stored in the dark at 4℃ until patch-clamp measurements were made.

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Isolation of spheroplasts of apical root hair membrane by laser microdissection

Two day-old seedlings were used. Plasmolysis of root hairs was necessary prior to laser cell-wall cutting in order to prevent cell membrane damage by the laser energy (Figure 9a;

Véry and Davies, 2000). Root hair plasmolysis was achieved within 3 min by using a solution containing 350 mM sorbitol and 5 mM CaCl 2 . 0.2% Fluostain I (Sigma), a cell-wall binding dye absorbing UV light, was added to the plasmolysis solution in order to potentiate the action of the UV laser pulses on the cell wall. After cutting the apical cell-wall of a few root hairs with the laser (up to 7 min), another solution with lower osmolarity (200 mM sorbitol)

was added to the chamber for root hair deplasmolysis (achieved within 1-3 min). During deplasmolysis, a rapid extrusion of spheroplasts of apical membrane out of holes made in root hair cell-wall was observed (Figure 9a). Root hair spheroplasts were excised by shaking the chamber. Batches of root hair spheroplasts (total number < 10) for patch-clamp experiments were renewed every hour. S1. Details on bath solutions are given in the result section.

Whole

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Shaker-like outward conductance, from one representative protoplast. Initial deactivation currents (Itail) recorded upon return to the holding potential after the activation steps were plotted against the voltage of the activation steps. Boltzmann equations were adjusted to the latter relationships, with the same z for all K + concentrations: z = 1.27, Ea50 = -25, -6, 13 and 40 mV in 3, 10, 30 and 100 mM K + , respectively. P o /P o max vs voltage relationships were obtained by dividing I tail data by the maximal value of I tail extrapolated from the Boltzmann fits. Solid lines were drawn with Boltzmann parameters obtained from the fits.

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(b) Difference between initial and final tail currents (Ii-If tail) plotted against the voltage, for determination of Erev. E K was -37 mV.

(c) Variation of Erev with the external K + activity. Currents were successively recorded in external solutions containing 3, 10, 30 or 100 mM K-gluconate, and an ionic background of 1 mM CaCl 2 and 10 mM Mes/Tris (pH 5.6). The osmolarity of the bath was adjusted to 280 mosmol/L with D-sorbitol. The pipette solution was the same as in Figure 3c. Data are mean ± SE (n = 3 to 8). The solid line shows a logarithmic fit to Erev data, while the dashed line indicates the K + equilibrium potential.

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This article is protected by copyright. All rights reserved. Figure 7. Effect of the cytosolic Ca 2+ concentration on the inward and outward K + conductances of enzymatically-isolated root hair protoplasts.

Currents were recorded 10 to 20 min after whole cell access in the presence of 30 mM K + in the bath solution (same solution as in Figure 1b,c). (a, d) Shaker-like inwardly rectifying conductance (b, e) Shaker-like outwardly rectifying conductance (c, f) Cationic fast activating outward conductance (a-c) Representative recordings of the three cationic conductances at high cytosolic Ca 2+ concentration. The pipette solution, which contained 150 mM K + was identical to that described in Figure 1 except that 4.65 mM CaCl 2 were added to increase the free Ca 2+ concentration in the cytosol to 1 µM (Table S1). (b-f) Similar ionic conductances to those described in enzymatically-isolated protoplasts, observed in young root hair spheroplasts: example of current traces (top) and corresponding I-V relationships (bottom). The Shaker-like inward K + conductance (b), fast-activating outward cationic conductance (c) and the Shaker-like outward K + conductance (d) were recorded in the presence of 30 mM K-gluconate (pH 5.6) externally, and 150 mM K-gluconate (pH 7.4) internally (same solutions as in Figure 1). S-type anion currents (e) and hyperpolarization-activated anion currents (f) were recorded in the presence of 100 mM external and 113 mM internal Cl -(same bath and pipette solutions as in Figure 5).

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This article is protected by copyright. All rights reserved. Proposed candidate genes are based on M. truncatula root hair RNA-seq analysis (Damiani et al., 2016) and functional information available on characterized members from the corresponding gene families in other plant species (Geiger et al., 2009;[START_REF] Jammes | Calcium-permeable channels in plant cells[END_REF]

  kinetics, voltage gating, sensitivity to external K + of the M. truncatula Shaker-like outward conductance (Figure1 c, g, i) are fully consistent with the results from in situ analyses on similar root hair conductances in other species or with data obtained in Xenopus oocytes for the Arabidopsis GORK Shaker channel, which displays expression in root hairs (Bouteau etAccepted ArticleThis article is protected by copyright. All rights reserved.

  -cell patch clamp Patch-clamp experiments were performed in the whole-cell configuration. Patch-clamp pipettes were pulled using a DMZ-Universal Puller (Zeitz-Instruments GmbH, Germany) from borosilicate capillaries (GC150F-7.5; Phymep, France) and fire polished (by the DMZ-Universal Puller). Microelectrode resistance was about 10 MOhms in experimental solutions. A reference Ag/AgCl half-cell completed the circuit. The patch clamp amplifier was an Axopatch 200B (Axon Instruments Inc., USA). Currents passing through the protoplast/spheroplast membrane were measured at least 5 min after seal formation and obtaining the whole-cell configuration, in order to allow the pipette solution to diffuse and control the protoplast cytosol ionic composition. Data were sampled at 1 kHz. The Clampex module of the pClamp9 software (Axon Instruments Inc., USA) was used for design of voltage-clamp pulse protocols and data (applied voltages and resulting currents) acquisition. Analysis was performed using the Clampfit module of pClamp10 and SigmaPlot 11 (Systat Software Inc., USA). Applied voltages were corrected for liquid junction potentials. Main pipette solutions designed for the characterization of the different ionic conductances are given in Table

FIGURE LEGENDS Figure 1 .

 LEGENDS1 FIGURE LEGENDS

Figure 2 .

 2 Figure 2. Reversal potential of two M. truncatula root hair outwardly rectifying conductances in K + -containing solutions. (a) Example of voltage-clamp "tail current" protocol used for the determination of the reversal potential of currents. (b, d, f) Fast-activating outwardly rectifying conductance.(c, e, g) Shaker-like outwardly rectifying conductance. (b, c) Representative traces of currents recorded using the "tail current" protocol in the presence of 30 mM external K + (same bath and pipette solutions as in Figure1b, c). A zoom on tail currents (dashed circle) is displayed on the right of each panel. The arrows mark the position of zero current through the outwardly rectifying conductances.(d, e) Difference between initial and final tail currents (Ii-If tail) plotted against the voltage, for determination of the reversal potential of currents(Erev). The K + equilibrium potential (E K ) was -37 mV.(f, g) Variation of Erev with the external K + activity. Data are means ± SE (n ≥4). The solid lines show a logarithmic fit to Erev data, while the dashed lines indicate the K + equilibrium potential.

Figure 3 .

 3 Figure 3. Activation kinetics and voltage dependence of a root hair inwardly rectifying conductance in a K + -containing solution.(a) Voltage-clamp "activation" protocols (top) and kinetics of Shaker-like inwardly rectifying current activation (bottom). Whole-cell patch-clamp experiments were performed on root hair protoplasts isolated by enzymatic treatment. The osmolarity of the bath and pipette solutions was adjusted to 280 and 300 mosmol/L, respectively, with D-sorbitol. Shaker-like currents were recorded using a bath solution containing 30 mM K-gluconate, 10 mM CaCl 2 and 10 mM Mes/Tris (pH 5.6), and a pipette solution identical to that in Figure1. (b) I-V relationships drawn from steady state currents shown in (a).

Figure 4 .

 4 Figure 4. Selectivity analyses of the Shaker-like inward conductance.(a) Example of "tail current" protocol used for determination of the reversal potential of Shaker-like inward currents (top) and representative current traces recorded using the "tail current" protocol (bottom). Bath and pipette solutions used for the Shaker-like inward conductance were the same as in Figure3c. A zoom on tail currents (dashed circle) is displayed on the right of bottom panel. The arrows mark the position of zero current through the analyzed conductance.

Figure 5 .

 5 Figure 5. Activation kinetics and voltage dependence of two M. truncatula root hair anion conductances. (a, b) Voltage-clamp protocols used for the analysis of S-type (c, e, g) and, respectively, hyperpolarization-activated (d, f h) anion conductances. Whole-cell patch-clamp experiments were performed on root hair protoplasts isolated by enzymatic treatment. The external solution contained 10 mM CaCl 2 , 80 mM TEA-Cl, 10 mM Hepes/Tris (pH 7.5), and 100 mM D-sorbitol. The pipette solution was composed of 100 mM TEA-Cl, 2 mM MgCl 2 , 2 mM Mg-ATP, 5 mM EGTA, 4.65 mM CaCl 2 (free Ca 2+ 1 µM), 10 mM Hepes/Tris (pH 7.4), and 90 mM D-Sorbitol. (c, d) Representative current traces. Anion currents were recorded 7 to 10 minutes after whole-cell access. (e, f) Current-voltage relationships, at the end of the 10 s voltage pulses. The position of the equilibrium potential for Cl -(E Cl ) is indicated by the arrow. In (e), the solid line shows a fit to currents by the function I= G*(V-Erev)*P o/ P o max, where G is the macroscopic conductance when the channels are open, and P o/ P o max follows a Boltzmann distribution (P o/ P o max = 1/(1+exp(z*F*(-V+Ea50)/R/T)); z=0.21, Ea 50 =-158 mV). (g, h) Voltage gating. (g) The relative open probabilities were determined from the fit to I-V data shown in (e). The solid line shows the Boltzmann fit. (h) The maximal hyperpolarization-activated currents recorded between +45 and -150 mV (insert) were fitted (solid line) with the same functions as in (e), however using a Boltzmann distribution in opposite direction (I=G*(V-Erev)/(1+exp(z*F*(V-Ea50)/R/T)); z=0.20, Ea50=-41 mV).

Figure 6 .

 6 Figure 6. Sensitivity of S-type (a, c) and hyperpolarization-activated (b, d) anion conductances to external Cl -. The voltage-clamp protocols were similar to those shown in Figure 5a, b. Whole cell currents were measured at the end of the 10 s pulses.

  (c, d) Variation of the reversal potential of currents with the logarithm of the external Cl - activity. Erev values were obtained from I-V relationships. Data are means ± SE (n = 3 in (c), and n = 4 in (d)). The solid lines show logarithmic fits to experimental data, while the dashed lines indicate E Cl .

  Figure7. Effect of the cytosolic Ca 2+ concentration on the inward and outward K + conductances of enzymatically-isolated root hair protoplasts. Currents were recorded 10 to 20 min after whole cell access in the presence of 30 mM K + in the bath solution (same solution as in Figure1b, c). (a, d) Shaker-like inwardly rectifying conductance (b, e) Shaker-like outwardly rectifying conductance (c, f) Cationic fast activating outward conductance (a-c) Representative recordings of the three cationic conductances at high cytosolic Ca 2+ concentration. The pipette solution, which contained 150 mM K + was identical to that described in Figure1except that 4.65 mM CaCl 2 were added to increase the free Ca 2+ concentration in the cytosol to 1 µM (TableS1). (d-f) Comparison of I-V relationships of the three cationic conductances at low and high cytosolic Ca 2+ concentrations. Currents were measured at steady state. The pipette solution with low Ca 2+ concentration (<10 nM) was that described in Figure1. Data are means ± SE (n = 10 in (d), n = 5 and 4 at low and high Ca 2+ , respectively in (e), and n = 5 in (f)).

Figure 8 .

 8 Figure 8. Frequency of the different K + conductances recorded at low cytosolic Ca 2+ concentration.Protoplasts were isolated enzymatically. The external medium contained 30 mM K + . Bath and pipette solutions were those described in the legend to Figure1b, c. The surface of each rectangle is proportional to the number of protoplasts in which the corresponding current type was observed.

Figure 9 .

 9 Figure 9. Ionic conductances in spheroplasts of young root hair apical membrane, isolated by laser cell-wall ablation. (a) Recovery of M. truncatula root hair spheroplasts by laser microdissection. (Left) Root hair plasmolysis after 3 min in 350 mosmol.L -1 solution. The arrow points to the plasma membrane retracted from the cell wall. (Middle) The apical cell-wall of a plasmolyzed root hair was cut with the laser. (Right) Spheroplast of apical root hair membrane extruded upon deplasmolysis in 275 mosmol.L -1 solution).(b-f) Similar ionic conductances to those described in enzymatically-isolated protoplasts, observed in young root hair spheroplasts: example of current traces (top) and corresponding I-V relationships (bottom). The Shaker-like inward K + conductance (b), fast-activating outward cationic conductance (c) and the Shaker-like outward K + conductance (d) were recorded in the presence of 30 mM K-gluconate (pH 5.6) externally, and 150 mM K-gluconate (pH 7.4) internally (same solutions as in Figure1). S-type anion currents (e) and hyperpolarization-activated anion currents (f) were recorded in the presence of 100 mM external and 113 mM internal Cl -(same bath and pipette solutions as in Figure5).

Figure 10 .

 10 Figure 10. Candidate genes coding for the identified M. truncatula root hair conductances.

ACKNOWLEDGEMENTS

We are grateful to Julie Cullimore and Clare Gough for critical reading of the manuscript. This work was supported in part by an ANR grant (ANR-11-BSV7-010-02) (to H.S.), and a doctoral grant of the China Scholarship Council to M.-Y.G.

Accepted Article

This article is protected by copyright. All rights reserved.

Voltage gating, affinity and selectivity analysis

Voltage gating parameters (z: apparent gating charge, Ea50: half activation potential) were obtained through Boltzmann fits to initial deactivation currents (Itail) upon return to a voltage

where the channel open probability was nil (Itail = Itailmax/(1+exp(z*F(V-Ea50)/R/T)) in inwardly-rectifying conductances or Itailmax/(1+exp(z*F(-V+Ea50)/R/T)) in outwardly rectifying conductances, and F, R and T had their usual meaning), or through fits to steady-state currents using Boltzmann combined to Goldman equations, the Goldman et al., 1995), where Gl was the limit conductance at voltages infinitely negative (in inwardly rectifying conductances) or infinitely positive (in outwardly rectifying conductances)). When the concentration of permeant ions was almost the same on both sides of the membrane, I = G*(V-Erev) replaced the Goldman equation for modelling currents through open channels.

For determination of K + transport affinity, a Michaelis-Menten equation was fitted to Gl values (obtained by the Goldman combined to Boltzmann fits to steady state currents) plotted

The permeability ratios of different monovalent ions in K + and anionic conductances were

CONFLICT OF INTEREST

The authors declare that there is no conflict of interest. Table S1. Pipette solutions used to study the different ionic conductances.

Accepted Article

This article is protected by copyright. All rights reserved.


Pierre Gehl -Brgm

Bayesian updating for rapid earthquake loss assessment of road network systems

Keywords: Bayesian inference, critical infrastructure, seismic risk, loss updating, road network

Within moments following an earthquake event, observations collected from the affected area can be used to define a picture of expected losses and to provide emergency services with accurate information. A Bayesian Network framework could be used to update the prior loss estimates based on ground-motion prediction equations and fragility curves, considering various field observations (i.e., evidence). The present study explores the applicability of approximate Bayesian inference, based on Monte-Carlo Markov-Chain sampling algorithms, to a real-world network of roads where expected loss metrics pertain to the accessibility between damaged areas and hospitals in the region. Observations are gathered either from free-field stations (for updating the ground-motion field) or from structure-mounted stations (for the updating of the damage states of infrastructure components). It is found that the proposed Bayesian approach is able to process a system comprising hundreds of components with reasonable accuracy, time and computation cost. Emergency managers may readily use the updated loss distributions to make informed decisions.

Introduction

Several rapid response systems have been developed worldwide, as detailed in the review by [START_REF] Guérin-Marthe | Rapid earthquake response: the state-of-the art and recommendations with a focus on European systems[END_REF]. While such systems are mostly applied to common buildings and the estimation of casualties, the treatment of the performance loss of critical infrastructure and its consequences remains mostly overlooked. A rigorous rapid response system should ensure the propagation of the uncertainties due to the estimation of ground shaking up to loss predictions. Therefore, this paper investigates a proof-of-concept rapid response procedure that would integrate the following features, in answer to the aforementioned gaps: (1) loss estimation for built areas and infrastructure systems, (2) integration of various types of observations to constrain the predictions, (3) propagation of all sources of uncertainty, from hazard to losses, and (4) ability to treat real-world systems over large areas.

Bayesian Networks (BNs) have emerged as a very promising mathematical tool, well adapted to seismic risk analyses that mobilize a probabilistic framework and dependencies between many variables [START_REF] Bensi | Efficient Bayesian network modeling of systems[END_REF]. The inference operations on a BN enable the combination of the initial estimates (i.e., prior distribution provided by predictive models) and of field observations (i.e., providing evidence at some nodes) in order to generate updated posterior distributions of the variables of interest. Therefore, thanks to their probabilistic inference capabilities, BNs constitute an adequate solution for the updating of damage and loss estimates in the crisis phase [START_REF] Bensi | Framework for post-earthquake risk assessment and decision making for infrastructure systems[END_REF]. While many previous studies have investigated the scalability issues of BNs [START_REF] Tien | Algorithms for Bayesian network modeling and reliability assessment of infrastructure systems[END_REF][START_REF] Gehl | Approximate Bayesian network formulation for the rapid loss assessment of real-world infrastructure systems[END_REF][START_REF] Byun | Matrix-based Bayesian Network for efficient memory storage and flexible inference[END_REF], their actual application to large real-world systems remains a challenge when considering the spatial distribution of the ground-motion field.

Pending further developments of BN algorithms that are able to address some of the scalability issues, it is proposed here to adopt a more pragmatic approach based on a sampling inference algorithm (i.e., Monte-Carlo Markov-Chain sampling). The objective is to exploit state-of-the-art techniques in order to demonstrate the use of BNs in an operational capacity during the rapid response phase (see Figure 1).

Bayesian model

Relying on previous developments regarding the post-earthquake loss assessment of infrastructure systems [START_REF] Bensi | Efficient Bayesian network modeling of systems[END_REF][START_REF] Bensi | Framework for post-earthquake risk assessment and decision making for infrastructure systems[END_REF][START_REF] Cavalieri | Bayesian networks and infrastructure systems: Computational and methodological challenges[END_REF][START_REF] Gehl | Approximate Bayesian network formulation for the rapid loss assessment of real-world infrastructure systems[END_REF], a tool using Bayesian updating is proposed for rapidly estimating losses to road networks. A BN is designed with the OpenBUGS tool [START_REF] Lunn | The BUGS project: Evolution, critique and future directions (with discussion)[END_REF], which enables the modelling of continuous and discrete variables in the same BN. The OpenBUGS library is freely available, integrated in the R environment (www.r-project.org), and it uses approximate inference via Monte-Carlo Markov-Chain (MCMC) sampling. The developed BN relies on five main types of variables:

•

Spatially distributed intensity measure (IM) at the locations of infrastructure components (e.g., road bridges), which represents the distribution of the logarithm of the ground-motion parameter of interest (e.g., peak ground acceleration, PGA). A correlation structure is modelled to represent the contribution of intra-and inter-event error terms of the related ground-motion model (GMM) to the spatial distribution of the IMs at the various sites.

• Seismic capacity (C) of the infrastructure components, which represents the distribution of the logarithm of the seismic response components (e.g., fragility parameters expressed in PGA). A correlation structure may be introduced in order to represent structural similarities between components of the same typology, for instance.

•

Damage state (DS) of the components, depending on the IM level at the components' sites and on their seismic capacity C. If only two damage states are considered (i.e., bridge is functional or non-functional), the following convention is adopted: DS = 1 if the bridge is intact, DS = 0 if non-functional. The assumption of binary damage states is followed for the rest of the study. Therefore, for component i, the damage state DS i is determined as follows:

(1)

•

Accessibility of a minimum link set (MLS) between two locations A and B of the road network. A MLS is defined as a minimum set of components whose joint survival ensures survival of the system (here, the connectivity between A and B). In the case of complex networks containing intersections or alternative routes, multiple MLSs exist between A and B, representing the number of different possible routes to reach the destination. By definition, a MLS represents a sub-system of components in series. The following convention is adopted: MLS = 1 if the MLS is accessible and MLS = 0 if not. Therefore, for a given MLS k containing p components, the associated variable MLS k is defined as follows: MLS k = DS 1 x … x DS p (in the case of binary damage states).

•

Connectivity between the locations A and B, representing the system performance (S) for this specific objective. In a connectivity analysis, S is then determined by a system of MLSs in parallel (i.e., one accessible MLS is enough to ensure the connectivity between A and B):

S = MLS 1 + … + MLS q .
The decomposition of a road network into MLSs is illustrated in Figure 2, where 4 MLSs are identified, each containing a different subset of components. Then, the corresponding BN may be built as shown in Figure 3. Two types of observations may be entered as evidence in the BN:

• Strong-motion recordings by seismic stations, which corresponds to evidence entered at the level of the IM variables;

• Identification of the damage states of some components, e.g. with near-real time structural monitoring [START_REF] Tubaldi | Examining the contribution of near real-time data for rapid seismic loss assessment of structures[END_REF], which corresponds to evidence entered at the level of the DS variables.

With the BN implemented in the OpenBUGS tool, the evidence is then propagated through the related variables. In the inference algorithm, several MCMC chains are initiated: each chain is built with a Gibbs sampling scheme, where variables are successively sampled from the posterior distribution of previous variables. As a result, the BN generates thousands of samples for all variables, which are then assembled to estimate their posterior distributions given the evidence. Examples of these realisations are provided in Table 1 and Table 2, where for a given earthquake event and related field observations, posterior statistics of variables of interest may be extracted from the BN inference. Table 1. Illustrative example of n MCMC samples for the damage states of the 10 components in Figure 2, assuming DS 7 = 0 (failure of component #7) as an evidence.

Sample # DS 1 DS 2 DS 3 DS 4 DS 5 DS1 6 DS 7 DS 8 DS 9 DS 10 1 1 1 0 1 1 1 0 0 1 1 2 0 1 0 0 1 1 0 0 0 1 … 0 0 0 1 1 0 0 1 1 0 n 1 1 1 1 1 1 0 1 0 0 Table 2.
Illustrative example of n MCMC samples for the accessibility of the 4 MLSs in Figure 2, assuming DS7 = 0 (failure of component #7) as an evidence.

Sample # MLS 1 MLS 2 MLS 3 MLS 4 1 0 1 0 0 2 0 1 0 0 … 0 0 0 0 n 0 1 1 0
From Table 1 andTable 2, it is possible to extract damage and loss estimates in the form of probabilities, such as P(DS i =0) the probability of failure of component i, or P(MLS j =0) the probability of MLS j being inaccessible.

Definition of prior distributions

The prior distributions of the variables that are probabilistically defined (IM, C) are assumed to follow normal/lognormal distributions, whose parameters are obtained from predictive uncertain models.

In the case of IM, the mean of the logarithm of the ground-motion parameter distribution (µ logIM ) is given by a ground-motion model (GMM), based on the earthquake characteristics that are assumed to be known with confidence shortly after the event. The covariance matrix ∑ IM is assembled as follows:

(2)

where σ η and σ ξ respectively represent the standard deviations of the inter-and intra-event error terms, which are given by the GMM. The term ρ ij represents the spatial correlation of the intra-event error between sites i and j, and it may be defined by available models in the literature (e.g., [START_REF] Jayaram | Correlation model for spatially distributed ground-motion intensities[END_REF].

The seismic response C of components is provided by fragility curves, where the elements of µ logC correspond to the median fragility, and the standard deviations of C correspond to the fragility dispersion β. The dispersion term β may be further decomposed into β R and β M , which respectively represent the uncertainty due to record-to-record variability and the uncertainty due to imperfect knowledge or modelling of the component [START_REF] Crowley | Probabilistic damage assessment of buildings due to induced seismicity[END_REF]. A third type of uncertainty, related to the definition of the damage state threshold, is neglected here for simplification purposes. Therefore, the covariance matrix ∑ C is expressed as follows:

(3)

The term ρ R ij , representing the correlation of the response due to record-to-record variability between components i and j, is very difficult to quantify without the knowledge of the seismic records used in the derivation of the fragility curves. A qualitative rationale may postulate that components i and jif they are spatially very close to each othermay experience ground motion inputs with similar characteristics in terms of duration, frequency content, etc. and therefore their record-to-record variability should be fully correlated. On the other hand, spatially-distant components are likely to be subjected to different ground-motion (e.g., different spectral shapes), and therefore their record-torecord variability should be uncorrelated. Such considerations are discussed in [START_REF] Silva | Uncertainty and correlation in seismic vulnerability functions of building classes[END_REF], who advocates the use of a correlation structure similar to the one that models the spatial correlation of the intra-event error. Although deserving further investigation, this assumption is also used here for the characterization of ρ R ij . The correlation of the component-to-component variability within the same class/typology, represented by ρ M ij , also requires more knowledge of how the corresponding fragility curves are derived. Therefore, it is proposed to consider the extreme cases in the application (see Section 4), namely fully correlated or uncorrelated variability, in order to investigate the impact of these assumptions on the posterior distributions. The decomposition of the dispersion into β R and β M is not always detailed in available fragility models (i.e., only the global dispersion β is specified). Various assumptions regarding this decomposition are also tested in the application example.

Application

Description of the case-study area

The proposed BN approach for rapid response is applied to a road network composed of 118 bridges (i.e., vulnerable components), which connects 53 municipalities (i.e., built areas) located in a valley around Bagnères-de-Luchon (Pyrenees, France). The case-study area is detailed in Figure 4. The typologies of the 118 bridges are identified based on photographs and aerial pictures, and their conditional probability of failure is defined by fragility functions, some of which are taken from the SYNER-G database [START_REF] Crowley | Probabilistic damage assessment of buildings due to induced seismicity[END_REF]. In total, 18 different fragility curves have been assigned (3 models for 83 single-span bridges, 3 for 7 continuous multi-span bridges, and 12 for 28 arch bridges). The fragility curve corresponding to the first limit state is considered as the threshold of the loss of functionality of the bridge (i.e., failure of the component), assuming that even small structural damage might be enough to prevent safe passage.

The studied area is surrounded by several seismic stations (see Figure 4), which are used as sources of observations to constrain estimates of the strong-motion field. All selected fragility curves use PGA as IM; therefore, the regional GMM by [START_REF] Tapia | Desarrollo Y Aplicación de Métodos Avanzados Para La Caracterización de La Respuesta S smica Del Suelo a Escala Regional Local[END_REF] is applied here for the estimation of the prior distribution of IM. For PGA, the spatial correlation model by [START_REF] Jayaram | Correlation model for spatially distributed ground-motion intensities[END_REF] is used with a correlation distance of 8.5 km. Finally, site effects are modelled via soil amplification factors, which were estimated from local investigations and soil measurements [START_REF] Roullé | Performing seismic scenario in the Luchon -Val d'Aran area, Central Pyrenees[END_REF].

The road network is simplified by considering only paths that need to go through bridges and by building abstract layers of Super-Nodes and Super-Edges [START_REF] Gehl | Rapid earthquake loss updating of spatially distributed systems via sampling-based Bayesian inference[END_REF], as shown in Figure 5. From this conceptualization step, the MLS decomposition is performed via a recursive algorithm [START_REF] Cavalieri | Bayesian networks and infrastructure systems: Computational and methodological challenges[END_REF]. The connectivity between the town of Bagnères-de-Luchon (point A) and the Northern part of the network (point B) is investigated here, which leads to 60 MLSs (i.e., 60 different possible routes between A and B).

Loss updating results

The updating capabilities of the BN are tested with a hypothetical Mw 6.3 earthquake scenario, located south of the road network (0.60° lon, 42.65° lat). Hypothetical observations from 7 seismic stations and damage measures on 5 bridges are set as evidence in the BN (2 survivals and 3 failures). It is assumed that one of the monitored bridges belongs to the identified MLSs, and it has been observed as intact (see Figure 5).

Following the discussion of Section 3 on the assumptions to be made for the covariance models, three different correlation hypotheses are tested for the response C of bridges:

• Corr1: no correlation is introduced, so that ∑ C is simply a diagonal matrix.

• Corr2: only the correlation of the record-to-record variability is introduced, with a correlation model decreasing with inter-bridge distance (i.e., Eq. 3 with ρ M ij = 0).

•

Corr3: in addition to the correlation of the record-to-record variability, a full correlation between bridges of the same type (i.e., using the same fragility model) is assumed, i.e. ρ M ij = 1 if bridges i and j are in the same typology. Global results for the system connectivity are detailed in Table 3, where it is shown that the proposed approach is also able to identify which MLS is the most likely to remain accessible (i.e., "best" MLS). A significant difference is observed between the prior and posterior probabilities of disconnection, due to the assumption that the damage states of 5 bridges were entered as evidence (see Figure 5). The evidence of a bridge's state has an impact on the system at various levels:

•

The observation of a bridge failure directly modifies the accessibility of the MLS(s) to which it belongs, and in turn system connectivity.

•

If the seismic response C is modelled with a constrained correlation model (i.e., Corr2 or Corr3), then the observation of a bridge's state may modify the seismic response of other bridges, in turn modifying their probability of failure and ultimately the accessibility of the MLSs to which they belong.

•

Finally, the observation of a bridge's state may also modify the distribution of the IM at the base of the bridge, in turn modifying the ground-shaking field in the vicinity.

The differences between the three correlation models are noticeable: the extreme configurations Corr1 and Corr3 should be used as upper and lower bounds of the model outcome, pending an in-depth investigation of appropriate correlation models for the seismic response. Furthermore, each assumption leads to the identification of a different MLS as the least affected route, which could have a large impact on emergency operations. An example of the changes in the rapid estimate of the post-earthquake condition of the network is provided in Figure 6, both in terms of failure probability of bridges and identifying the most accessible MLS.

Conclusions

This study has investigated the implementation of a Bayesian Network-based framework for improving situational awareness during the rapid response phase following an earthquake event. It has been demonstrated that a BN built in OpenBUGS environment is able to provide updated losses for a real-world road network and built areas based on available observations. The BN is solved with a MCMC sampling scheme, which delivers approximate posterior distributions: this approximate solution, as opposed to exact inference algorithms, is a necessary trade-off in order to treat systems of hundreds of components exposed to spatially distributed hazards. Moreover, the sampling inference scheme used in OpenBUGS can combine continuous (e.g., intensity measures, seismic capacities) and discrete variables (e.g., damage states), which has the benefit of introducing exact distributions instead of discretizing continuous variables. Regarding road networks, when connectivity loss is used as a simple system indicator, the decomposition of the system into MLSs is also essential in reducing the complexity of the BN. As a result, in the studied example, posterior distributions are generated within 20 or 30 minutes. Moreover, the decomposition into MLSs has the benefit of identifying specific routes associated with probabilities of accessibility: this information has the potential to be used by emergency managers to set up dedicated evacuation routes or safe itineraries to hospitals. Each MLS can also be associated with a travel distance or travel duration, in order to develop more elaborate performance indicators than connectivity loss. Finally, the results of such a BN application can constitute the starting point of rapid repair strategies for bridges, in order to improve the seismic resilience of transportation systems [START_REF] Sun | Agent-based model on resilience-oriented rapid responses of road networks under seismic hazard[END_REF].

Fig. 1 -

 1 Fig.1-Proof-of-concept of the procedure for the rapid earthquake loss assessment of infrastructure systems.

Fig. 2 -

 2 Fig. 2 -Decomposition of the A-B routes into four MLSs, for an illustrative network. Black dots represent intersections in the network, and grey rectangles represent components. The connectivity between A and B constitutes the system performance S.

Fig. 3 -

 3 Fig. 3 -BN corresponding to the illustrative example of Figure 2. The node IM is in bold because it represents the vector of spatially correlated IMs at the sites.

Fig. 4 -

 4 Fig. 4 -Situation map of the Luchon case-study area.

Fig. 5 -

 5 Fig. 5 -Left: abstraction of the network topology for connectivity analysis; Right: MLS decomposition between points A and B of the network.

Fig. 6 -

 6 Fig. 6 -Left: prior distribution using only the characteristics of the earthquake event; Right: posterior distribution using field observations (with Corr3 model).

Table 3 .

 3 Results of the Bayesian updating, in terms of probability of disconnection between points A and B(i.e., S = 0) and of identification of the "best" MLS, for the various correlation assumptions.

		Prior		Posterior
			Corr1	Corr2	Corr3
	Pr(Disconnection)	0.095	0.342	0.353	0.435
	"Best" MLS	#45	#49	#21	#3

Acknowledgements

This research was funded by the European Union's Horizon 2020 research and innovation program, grant number 821046, project TURNkey.