Mohamed Kouki
email: mohamed.kouki@ec-nantes.fr

Bogdan Marinescu
email: bogdan.marinescu@ec-nantes.fr

Florent Xavier
email: florent.xavier@rte-france.com

Exhaustive Modal Analysis of Interconnected European Power System

Keywords: coupling/inter-area modes, European power system, dynamic reference model, large-scale systems, modal analysis

The interconnected European power system is evolving due to the new extensions and to the systematic integration of Power Park Modules, i.e., unit or ensemble of units generation electricity (Renewable Energy (RE), Distributed Generation (DG), etc), which is connected to the grid by power electronics and FACTS (STATCOM, HVDC,...). As a consequence, the increase of the size and the emergence of the new oscillatory modes which involve distant dynamic devices, i.e., of coupling modes (modes associated to generator/turbine inertia, and modes associated to Power Park Modules), and of modes related to various regulations. In this paper, a full scan of European power system is given. An exhaustive method that determines all oscillatory modes independently of the system's order, of the topology and without a priori knowledge about the system, is used. The exhaustive method is fully analytic. Indeed, it is based only on the state-space realisation of any given power system and does not require operator manipulation like dynamic simulations. To this end, we quantify and analyse the interaction between the different inputs/outputs of a given system. This leads to the identification of classes of coupled dynamic devices of the power system. Next, starting from a reduced number of classes, a selective modal analysis method is applied to each class to put into evidence a good approximation of all oscillatory modes. Finally, a complete classification of the oscillatory modes found at the previous stage is provided in terms of the types (interarea, electrical, ...) and their characteristics (oscillation frequency, damping, participation, ...). 2900 oscillatory modes were found for the studied model of the European system.

I. INTRODUCTION

During the last years, interconnected European power system has evolved. First, its size has increased, due to the new extensions like, e.g, interconnection with the Turkish power system. Next, its structure changed also due to systematic insertion of Power Park Modules (PPMs) for the renewable sources of for the HVDCs systematically used to reinforce border interconnections (e.g., Spain-France or France-Italy). Therefore, the interconnected dynamic devices are not only the power plants with synchronous machines, but they also contain the PPMs. The eigenanalysis of the resulting mathematical model is becoming very difficult, especially when the oscillatory dynamics are expected in an exhaustive manner, i.e., a full scan of the system has to be done and computation of only some preselected modes it is not enaugh. In fact, an exhaustive analysis should be able to determine all oscillatory modes, not only the inter-area modes, but also the electrical modes (modes related to the electrical parts of distant classic generators and the ones related to the PPMs). For example, In [START_REF] Arioua | Multivariable control with grid objectives of an HVDC link embedded in a large-scale AC grid[END_REF], [START_REF] Arioua | Robust grid-oriented control of high voltage DC links embedded in an AC transmission system[END_REF], [START_REF] Kouki | Eigencalculation of Coupling Modes in Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF] and [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF] coupling modes associated to electrical part of the distant calssic synchrounous generators are highlighted but not of inter-area nature. In [START_REF] Munteanu | Analysis of the interactions between close HVDC links inserted in an AC grid[END_REF], and [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF] oscillatory modes associated to converters of distant HVDCs, PVs and winds were also highlighted and analysed. Therefore, all these oscillatory modes will be called coupling modes in the sequel.

Several approaches of modal analysis that aim to find all coupling modes of large scale power systems have been developed and discussed (see, e.g., [START_REF] Barquin | Generalized selective modal analysis[END_REF]). They are often non exhaustive and provide only a selective modal analysis. This means that, if good initial guesses are given, only few modes can be found in an iterative manner. This supposes that some knowledge about the system is available which is not always the case. To overcome these difficulties, two exhaustive methodologies were discussed and developed in our previous works [START_REF] Kouki | Eigencalculation of Coupling Modes in Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF], [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF]. In fact, they provide a way to systematically excite and put into evidence all the coupling modes of a given power system. In addition, they quantify the interactions between the different dynamic devices of any given power system. Indeed, for each device, one can find the other devices to which it is coupled. These coupled devices should be aggregated into a class, and for all devices a set of classes can be constructed. These latter should be used in computing of the coupling modes based on a specific selective modal analysis method [START_REF] Barquin | Generalized selective modal analysis[END_REF], [START_REF] Rouco | A computer package for analysis of small signal stability in large electric power systems[END_REF].

To put into evidence all the European power system modes, the method proposed in [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF] is used here with some modifications for large-scale use. Importantly, the analysis of the European power system or of a system of comparable size and complexity in exhaustive way is a novelty in modal analysis and has never been addressed before. Indeed, not only the well-known inter-area modes can be found, but all the coupling modes are put into evidence. In addition, determining the low and high frequencies modes of the network becomes feasible.

As well, aggregating the dynamic devices according to the nature of the coupling modes or according to the oscillation frequencies it is also possible. Preliminary modal analysis results of the European power sysem were presented in [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF]. The analysis is carried out on the Dynamic Reference Model (DRM), which was developed for the IPS/UPS EC study and next maintained [START_REF] Luther | Feasibility Aspects of a Synchrounous Coupling of the IPS/UPS with the UCTE[END_REF]. The rest of this paper is organized as follows. Section II provides the mathematical model associated to power system and a brief description of the exhaustive modal analysis methods that were developed and discussed in [START_REF] Kouki | Eigencalculation of Coupling Modes in Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF] and [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF]. Section III details the specific implementation of the analytic method to large-scale systems. The results of the full modal analysis of the European power system are reported in Section IV. Conclusions are given in Section V.

II. BASIC TOOLS

In this section we present the mathematical model used for power system and a brief description of exhaustive modal analysis methodologies developed in [START_REF] Kouki | Eigencalculation of Coupling Modes in Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF] and [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF], which will be used subsequently.

A. Power system description

A power system is often modeled by a set of Differential and Algebraic Equations (DAE) called DAE form

. x= f (x(t), w(t)) 0 = g(x(t), w(t)), (1)
where, x(t) and w(t) are the vectors of differential and algebraic variables, respectively. x represents the state variables (machines speeds, regulators, active power,...), while w mainly contains the bus voltages of the power system. Linearization of (1) around an equilibrium point (x 0 , w 0 ) gives

. x= Ax, (2)
where A ∈ IR n * n . If a vector of input variables u and a vector of output variables y are also considered, (2) leads to an input-output linear model

. x= Ax + Bu y = Cx (3)
where, B ∈ IR n * q is the input matrix, and C ∈ IR l * n is the output matrix.

B. New exhaustive modal analysis methodology

[6] and [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF] proposed an exhaustive modal analysis methods which hold for any type of coupling modes and power systems, in particular those with high power electronics penetration. They are structured into two main parts. First, the identification of classes of the coupled dynamic devices (synchronous generators and converters) is done using the simulation technique [START_REF] Kouki | Eigencalculation of Coupling Modes in Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF] (i.e., exciting, one by one, each dynamic device with well-chosen events (such as shortcircuit)), or by analytic technique [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF] (i.e., using the relative sensitivity technique to quantify the interaction between the inputs/outputs), and by analysing the correlation between the obtained outputs/sensitivities of all dynamic devices. Next, the Selective Modal Analysis (SMA) method [START_REF] Rouco | A computer package for analysis of small signal stability in large electric power systems[END_REF] is applied to each class in order to determine a good approximation of coupling modes. SMA is an efficient approach for simplifying and analyzing a selected part of a large-scale linear system [START_REF] Arioua | Robust grid-oriented control of high voltage DC links embedded in an AC transmission system[END_REF]. It is based on the split of the state vector x into two parts, a relevant part r and a less relevant part z

. x= . r . z = A rr A rz A zr A zz r z . ( 4
)
This partition is illustrated on the Fig. 1.a. Based on this latter, the transfer function H(s) of the less relevant dynamics could be generated.

H(s) = A rz (sI -A zz ) -1 A zr . ( 5
)
The eigencomputation is run only on the relevant dynamics

(a) (b)
Fig. 1: The relevant and less relevant dynamics by taking into account the small effect of the less relevant ones on the eigenvalues of interest. The less relevant dynamics can be exactly introduced in the relevant dynamics by a constant matrix M that satisfies the linear system of [START_REF] Kouki | Eigencalculation of Coupling Modes in Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF].

MV r = [H(λ 1 )v 1 ...H(λ r )v r ] (6)
with λ r are the interest eigenvalues, and Notice that, in our case, the relevant part is composed by the states of dynamic devices selected in each class. The flowchart (see Fig. 2) summarizes the main steps of the exhaustive approaches [START_REF] Kouki | Eigencalculation of Coupling Modes in Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF], [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF]. This methodology is now improved to apply to large-scale power systems such as the European one.

V r = [v 1 ...v r ]

III. EXHAUSTIVE MODAL ANALYSIS OF MODERN LARGE-SCALE POWER SYSTEMS A. Description of the European case

We use for the European power system, the Dynamic Reference Model (DRM) which was developed for the IPS/UPS EC study and next maintained and currently used for stability studies [START_REF] Luther | Feasibility Aspects of a Synchrounous Coupling of the IPS/UPS with the UCTE[END_REF]. It is a detailed nonlinear model including generators along with their regulations (AVRs, PSS and Governors). It consists of 1074 machines, 7652 buses, 10465 lines and 2550 transformers. The order of the resulting linear model, i.e., n in (2), is 16784. To overcome this problem, an analytic technique, called Relative Sensitivity Technique (RST) [START_REF] Ellis | Selection of control configurations for economic model predictive control systems[END_REF], [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF], [START_REF] Yin | Input-output pairing accounting for both structure and strength in coupling[END_REF] is used here since it is applicable to large-scale systems such as the European one. The RST does not require operator manipulations, such as dynamic simulations, to identify the coupling classes. In this work, the RST provides a matrix called, in the sequel, Relative Sensitivity Matrix (RSM) [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF]. Based on the absolute value of RSM and according to a chosen selection criterion δ, the construction of the classes of coupled dynamic devices of a given power system is done:

• RSM (i, j) ≤ δ⇒ device i and device j are not coupled, • RSM (i, j) > δ ⇒ device i and device j are coupled. This protocol is applied on all the RSM elements. Hence, a quasi-symmetric matrix made up with all coupled devices is obtained. Each row of RSM corresponds to a class. Notice that, the chosen inputs u and outputs are respectively the references of the voltage regulators, and the speeds of synchronous generators.

2) Modes computing: The use of all obtained classes to put into evidence all coupling modes of large-scale power system gives a high redundancy between the computed modes and it is numerically fastidious. Instead, only some classes can be used. To identify them, a selection threshold σ is used as follows:

• if the similarity (i.e., percentage of identic dynamic devices of two classes) between class i and class j ≥ σ and size(class i )>size(class j ) then class i is one of the selected classes. • if the similarity between class i and class j < σ then class i and class j are both selected. Furthemore, in order to improve the computational efficiency of the specific implementation of the exhaustive approach, the stop condition used in [START_REF] Kouki | Eigencalculation of Coupling Modes in Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF], [START_REF] Kouki | Exhaustive Modal Analysis of Large-Scale Interconnected Power Systems with High Power Electronics Penetration[END_REF] is replaced here by a fixed number of iterations k.

IV. ANALYSIS RESULTS FOR THE EUROPEAN POWER SYSTEM

A. Classes and modes

As mentioned above, the number of dynamic devices of European power system is equal to 1074. Therefore, using the RST approximation and according to a selection criterion δ = 0.001, 1074 classes are constructed. As indicated in Table I, according to a selection criterion σ = 50%, the number of final classes is 66 and the size of the largest class is 341 machines. Notice that, the number of coupled machines of the largest class is a little large compared to the total number of machines. This is due to the small value chosen for the selection criterion δ. In general, δ should be small in order to ensure high redundancy in the construction of classes. Therefore, a maximum number of modes will be found for each class and, by the end, the eigencalculation is surely exhaustive in the sense that all the coupling modes of the studied power system will be found. Starting from the final set of classes (66 classes) found for the European power system, SMA is applied on each class in order to put into evidence all coupling modes. The analysis of 66 classes provides real and complex modes. Here, only the complex modes are taken into account. Indeed, only the complex modes are associated to oscillatory phenomena. Furthermore, based on the complex modes, all coupling modes of the European power system can be found. Importantly, this is a challenge and a novelty in modal analysis of large-scale systems.

Table II provides all complex modes (2900 modes) found for the European power system. Their oscillation frequencies are up to 15.64Hz and can be classified as reported in Table III.

In order to prove that the obtained modes are the exact values of the overall system, the classic selective modal analysis method, Modified Arnoldi (MA), was used. The modes generated by the SMA are used as initial guesses of MA iterations which converge to the exact values of the modes of the overall system. Notice that, 80% of 5800 complex modes are very precisely generated at the SMA stage and 20% are approximated with very small differences. This means that the selected classes (66 class) are the significant and efficient ones in the sense that there are no important interaction between the dynamics of each class and the rest of the system.

B. Modes classification

The complex modes found are classified according to their natures, oscillation frequencies, damping ratios and participation factors.

1) Classification according to the nature: The 2900 complex modes indicated in Table II • an EM mode signifies that the most participation factors of its involved dynamic devices are associated to electrical parts (D-axes, or Q-axes, or injectors). • an EMM mode means that the most participation factors of its involved dynamic devices are associated to rotor parts.

• an RM mode means that the most participation factors of its involved dynamic devices are associated to controller parts (Governor, AVR, PSS,...). The first three modes of Table II are electromechanical of interarea nature. They are identified in the low frequency range 0.17Hz to 0.30Hz. The slowest mode of the European power system is the mode number 1 of Table II. The oscillation frequency of this mode is 0.17Hz. As indicated in columns 2 and 5 of Table IV, mode 1 involves 585 geographically distant synchronous generators with the highest participations associated to the rotors. The generators in Spain, Portugal and France swing against the generators in Turkey and Bulgaria. Therefore, this mode is an inter-area one and is one of the well-known modes identified in the studies of European power system performed by TSOs [START_REF] Grebe | Low frequency oscillations in the interconnected system of continental Europe[END_REF]. An example of electrical coupling mode is reported in Table V. This mode involves 11 geographically distant synchronous generators with significant participation factors. The highest participation parts are not associated to the rotors but to Daxes. Therefore, this mode is not an inter-area mode but an electrical coupling mode. Furthermore, Fig. 4 illustrates the responses (ID current and IQ current) of most participating generators (G1, G2 and G11) to a short-circuit at the connection point of generator G1. This short-circuit is applied at t1 = 60s and eliminated at t2 = 60.1s. From this figure, notice an important variation at t = 60 associated to the ID-current wich comes as consequence of electrical coupling. 2) Classification according to the participation factor: Pariticipation factors give an interesting quantification of the spread of the modes, i.e., of how many distant devices they involve. Such classification is given in Fig. 5 for the electromechanical modes. Notice that, participations provided here are relative ones, i.e., purcentages of participation of the dominant device (which has 100%). As a consequence, the group of 197 modes in Fig. 5 are of local nature, i.e., they mainly concern only one dynamic device. The second and third groups in Fig. 5 contain thus the inter-area modes of the system. Notice first that the inter-area modes are majoritary in the EMM class. The inter-area modes usually studied are the ones geographically spread and which involve a large number of machines, as for example, the East-West modes (see also link with the classification according to frequency). Other examples from the analysis done here:

• Mode number 5 in Table II Notice however that, as shown in Fig. 6, most of the interarea modes are less spread modes since they involve a smaller number of machines (less than 18) with significant participation in the mode (more than 10%). This is an important fact for stability analysis. From the control point of view, they can be damped in a more local manner, via classic PSS or POD installed on power converters (as for HVDCs, for example) in a more restricted area.

3) Classification according to the frequency: Classification of the oscillatory modes is done here according to the oscillation range of the frequency as illustrated in Fig. 7. Notice that majority of the electromechanical modes are of high frequencies (982 compared to only 15 modes in the usual These modes are less spread (see also classification 3) and can thus be damped with more local control solutions as, for example, POD control of HVDC links. 4) Classification according to the damping: Classification according to the damping ratio is given in Fig. 8. Based on this classification more than half of the coupling modes are damped at more than 40% and thus do not have significant impact on system dynamics. Most of them are regulator and electrical modes but not electromechanical. Indeed, the critical (with damping less than 5%) ones are of electromechanical nature.

V. CONCLUSION

In this paper, the results of full modal analysis of the European power system are presented. All of the European power system are computed using a fully analytic method. The latter is exhaustive and does not require any a priori knowledge neither about the nature of modes nor about the participating devices. It was extended here to deal with large-scale systems. For a given system, this kind of analysis allows one first to check the model: are any unstable modes? any abnormal frequencies of oscillations? Next, ranges of the dynamics of the system (slowest to fastest modes) are put into evidence. This is useful for control and model reduction actions. Finally, the couplings between the dynamic elements of the system are important to settle protection and control actions. The important novelty is to provide all this information in an exhaustive manner, i.e., to be able to make a general and full scan of the system. For the European system it is provided for the first time such exhaustive information. Dynamics are classified in accordance to their frequencies, spread of the participating devices, damping and nautre of the modes. This improved knowledge of the European power system will be exploited for any further dynamic correction actions: control, tuning of protection, defense plans,...

  is the matrix of the right eigenvectors associated to λ r . The partition of Fig. 1.a could be represented now as shown in Fig. 1.b.

Fig. 2 :

 2 Fig. 2: Flowchart of the exhaustive modal analysis approach

Fig. 4 :

 4 Fig. 4: Terminal current responses to the G1, G2 and G11

Fig. 5 :

 5 Fig. 5: Participation factor classification

Fig. 6 :

 6 Fig. 6: Classification of modes with participation factor ≥ 10%

Fig. 7 :

 7 Fig. 7: Frequency classification

Fig. 8 :

 8 Fig. 8: Damping classification

TABLE I :

 I Coupled Classes of European power system

		Numb.	Numb.
	Sys.	Mac.	Inp./Out.	Tot. Numb. Clas.	Red. Numb. Clas.	Size Larg. Clas.
	DRM	1074	1074/1074	1074	66	341

TABLE II :

 II Comparison between RST method and full model

	Number	Mode of full model	Mode with RST	Freq.(Hz)	Damp.	Type
	1	-0.1247± j*1.1142	-0.2362± j*1.1142	0.1773	11.12	ROT
	2	-0.1980± j*1.5467	-0.2003± j*2.0038	0.2462	12.69	ROT
	3	-0.1489± j*2.0089	-0.1315± j*2.0102	0.3197	7.39	ROT
	. . .	. . .	. . .	. . .	. . .	. . .
	5	-0.2241± j*2.5112	-0.2445± j*2.3463	0.39978	8.89	ROT
	6	-0.9178+j*5.3588	-0.8531+j*5.3899	0.8529	16.88	ROT
	. . .	. . .	. . .	. . .	. . .	. . .
	2198	-3.8818±j*1.2878	-3.8836±j*1.2898	0.2050	94.91	SMQ
	. . .	. . .	. . .	. . .	. . .	. . .
	2308	-0.1943±j*0.2883	-0.1946±j*0.2832	0.0459	55.88	SMD
	. . .	. . .	. . .	. . .	. . .	. . .
	2796	-1.1596±j*1.2160	-1.1167± j*1.2311	0.1935	69.01	GOV
	. . .	. . .	. . .	. . .	. . .	. . .
	2900	-120.4578±j*98.3070	-120.3100± j*98.285	15.6462	77.47	EXC

  are of 3 main types as reported in Table III: Electrical Modes (EM), Electromechanical Modes (EMM) i.e., inter-area and local modes, and Regulation

					41
		40
				37
	Number of modes (%)	30
		22
		EM		EMM	RM
		Fig. 3: Nature classification
	Modes (RM). Their proportions are illustrated in Fig. 3.
	Notice that:
		TABLE III: Base-Type of oscillatory modes
		DRM	EMM	EM	RM
		Numb. Mod.	1091	636	1173
		Freq. Max. (Hz)	3.3985	5.6660	15.6462

  Table VI illustrates the results for mode number 2198 of Table

TABLE IV :

 IV Machines with highest participation in mode 1

	Country	Mac.	Rel. Part. (%)	Phase r. evec. δ (deg)	Rot. part	D-axes part.
	Bulgaria	G1	95.9	-5.4	0.2628	0.0082
	Turkey	G2	100.0	-1.0	01002	0.0025
	Turkey	G3	100.0	-1.0	0.0999	0.0024
	Bulgaria	G4	55.3	-0.8	0.0105	0.0001
	Bulgaria	G5	94.0	4.0	0.0826	0.0611
	. . .	. . .	. . .	. . .	. . .	. . .
	Spain	G581	23.2	164.4	0.0035	0.0002
	Portugal	G582	24.4	165.9	0.0101	0.0019
	France	G583	34.4	175.6	0.0077	0.0001
	Spain	G584	44.2	177.3	0.0127	0.0007
	France	G585	36.8	178.9	0.0648	0.0040

TABLE V :

 V Machines with highest participation in mode 2308 II. This latter is another electrical coupling mode type since it involves 3 distant generators with significant participation parts associated to the Q-axes as given in columns 6 and 8 of Table VI. The last type of oscillatory modes are the RM ones. They are local modes in which state variables of the controllers are dominant.

	Country	Mac.	Rel. Part. (%)	Phase r. evec. δ (deg)	Rot. part	D-axes part.
		G1	100	0.0	0.5251e -2	0.7833
		G2	15.4	69.6	0.6263e -3	0.1239
		G3	3.5	42.5	0.8938e -4	0.2660e -1
		. . .	. . .	. . .	. . .	. . .
	Spain
		G9	1.8	162.5	0.6028e -4	0.1456e -1
		G10	13.4	103.0	0.4318e -3	0.1085
		G11	20.2	103.0	0.6119e -3	0.1582


Thanyakorn Rongsawat

Jean-Benoît Peltier

Jean-Christophe Boyer

Anne-Aliénor Véry

Hervé Sentenac
email: herve.sentenac@inrae.fr

Looking for Root Hairs to Overcome Poor Soils

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U N C O R R E C T E D P R O O F 2Q2 Review
Looking for Root Hairs to Overcome Poor Soils 4Q3 Q4

Thanyakorn Rongsawat, 1,2 Jean-Benoît Peltier, 1 Jean-Christophe Boyer, 1 Anne-Aliénor Véry, 1 and Hervé Sentenac 1, * Breeding new cultivars allowing reduced fertilization and irrigation is a major challenge. International efforts towards this goal focus on noninvasive methodologies, platforms for high-throughput phenotyping of large plant populations, and quantitative description of root traits as predictors of crop performance in environments with limited water and nutrient availability. However, these high-throughput analyses ignore one crucial component of the root system: root hairs (RHs). Here, we review current knowledge on RH functions, mainly in the context of plant hydromineral nutrition, and take stock of quantitative genetics data pointing at correlations between RH traits and plant biomass production and yield components.

Root Hairs Enlarge the Soil-Root Interface

Research efforts aiming at improving understanding of the functioning of root systems are required to better exploit the genetic variation in productivity of crops in poorly fertile soils and to develop new crop cultivars with enhanced capacity for soil resource acquisition [START_REF] Lynch | Root phenotypes for improved nutrient capture: an underexploited opportunity for global agriculture[END_REF][START_REF] Zhu | Evolutionary agroecology: trends in root architecture during wheat breeding[END_REF]. Root system architecture and, at the root-soil interface, production and elongation of root hairs (RHs), are major determinants of the location and volume of exploited soil, which is why RHs take center stage in this review. It has been reported that a single rye plant (Secale cereale) can develop more than 10 10 RHs, representing an underground interface of ~400 m 2 , much larger than that of the aerial parts of the plant [START_REF] Dittmer | A quantitative study of the roots and root hairs of a winter rye plant (Secale cereale)[END_REF]. Figure 1 (Key Figure ) shows dense and long RHs over almost the whole root system in a 2-week old wheat (Triticum turgidum ssp. durum) seedling. The diameter of the RH cylinder around the root in the displayed enlargement is approximately ten times larger than that of the root itself and, thus, the volume of this cylinder would be ca.100 times larger than that of the root. Such a figure indicates that the ability of the root system to take up poorly mobile nutrient ions (e.g., phosphate; see later) can be significantly increased by RH production. Here, we review some major functions of RHs, in the context of plant mineral nutrition, and scrutinize recent attempts to use RH traits in plant breeding programs.

Adhesion to Soil Particles, Soil Penetration, and Rhizosheath Formation

RHs enhance seedling survival upon soil disruption by favoring root anchoring [START_REF] Choi | Root hairs enhance Arabidopsis seedling survival upon soil disruption[END_REF]. They also provide grip for root tip penetration in soil. For instance, during germination of maize on a soil displaying a moderate penetrometer resistance, wild-type seedlings took ~16 h to anchor themselves to the soil, compared with >30 h for hairless mutant seedlings, most of which did not become anchored securely [START_REF] Bengough | Root hairs aid soil penetration by anchoring the root surface to pore walls[END_REF]. The strength of the grip can be increased by root exudation of adhering molecules [START_REF] Galloway | Cereal root exudates contain highly structurally complex polysaccharides with soil-binding properties[END_REF], as also shown in clinging-climber species, such as English ivy, and their specialized RHs [START_REF] Huang | Nanospherical arabinogalactan proteins are a key component of the high-strength adhesive secreted by English ivy[END_REF]. However, when the resistance to vertical uprooting forces is compared between arabidopsis (Arabidopsis thaliana) wild-type plants and mutant plants impaired in RH development or lateral root production, the conclusion was that RHs do not contribute to whole-plant anchoring in this operational definition [START_REF] Bailey | The role of root system architecture and root hairs in promoting anchorage against uprooting forces in Allium cepa and root mutants of Arabidopsis thaliana[END_REF].

At a later stage of root system development, RHs and root exudation of adhesive molecules are involved in rhizosheath (see Glossary) formation [START_REF] Galloway | Cereal root exudates contain highly structurally complex polysaccharides with soil-binding properties[END_REF], which contributes to plant adaptation to abiotic and biotic conditions, as prevention of water loss, nutrient and water acquisition, and

Highlights

Plant breeding for improved belowground traits, allowing reduced fertilization and irrigation inputs, can contribute to the development of sustainable agriculture practices.

Root hairs (RHs) increase the volume of exploited soil, and have major roles in nutrient and water uptake as well as in beneficial interactions with soil microorganisms.

Plant engagement in mycorrhizal symbiosis also increases the volume of exploited soil, but appears less efficient than RH development in terms of biomass production in some soil conditions.

Evidence that plant biomass production can be positively correlated to RH length is available.

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Glossary

Rhizosheath: root-soil sheath resulting from agglutination and adhesion to the root of soil particles. Operational descriptions are proposed from the weight of soil that adheres to roots that have been freshly excavated and submitted to a soil removal treatment (e.g., hand shaking or sonication in water) of standardized vigor [START_REF] Pang | Unwrapping the rhizosheath[END_REF][START_REF] George | Understanding the genetic control and physiological traits associated with rhizosheath production by barley (Hordeum vulgare)[END_REF][START_REF] Brown | The rhizosheatha potential trait for future agricultural sustainability occurs in orders throughout the angiosperms[END_REF]. Shaker channels: highly K + -selective voltage-gated channels. The plant Shaker family comprises both hyperpolarization-activated and depolarization-activated channels, involved in K + uptake or K + secretion, respectively. The name 'Shaker' comes from the phenotype of a fly (Drosophila melanogaster) mutant characterized by rapid abnormal movements, in which the first channel of this family was cloned a few years before the cloning of channels of the same type in plants.

Transceptor: acronym of 'transporter' and 'receptor'. A transceptor is endowed with the capacity to mediate membrane transport and the ability to sense and signal the availability of a given solute.

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mechanical defense against herbivorous and plant parasites [START_REF] Pang | Unwrapping the rhizosheath[END_REF]. A strong correlation has been found between RH length and rhizosheath weight in wheat [START_REF] Delhaize | The genetics of rhizosheath size in a multiparent mapping population of wheat[END_REF]. The correlation found in barley (Hordeum vulgare) is weaker [START_REF] George | Understanding the genetic control and physiological traits associated with rhizosheath production by barley (Hordeum vulgare)[END_REF], and no significant correlation has been observed in 58 other species except for those with quite short RHs [START_REF] Brown | The rhizosheatha potential trait for future agricultural sustainability occurs in orders throughout the angiosperms[END_REF]. It has been proposed that, when RH length exceeds ~300 μm, other factors have increasing importance in rhizosheath size and stability, which might include RH density and RH morphology (e.g., bent or hooked forms that would trap more soil, and root and microbial mucilage) [START_REF] Pang | Unwrapping the rhizosheath[END_REF][START_REF] Brown | The rhizosheatha potential trait for future agricultural sustainability occurs in orders throughout the angiosperms[END_REF].

Nutrient and Water Acquisition

Major lines of evidence that RHs contribute significantly to nutrient ion acquisition from the soil can be sorted as follows: (i) nutrient starvation results in increased RH density and length [START_REF] Bates | Stimulation of root hair elongation in Arabidopsis thaliana by low phosphorus availability[END_REF][START_REF] Janes | Cellular patterning of Arabidopsis roots under low phosphate conditions[END_REF]; (ii) mutant plants displaying impaired RH growth show poor nutrient ion uptake and biomass production; furthermore, nutrient accumulation is positively correlated with RH length under nutrient-deficient conditions [START_REF] Ahn | Expression of KT/KUP genes in Arabidopsis and the role of root hairs in K + uptake[END_REF][START_REF] Canales | Nitrate induction of root hair density is mediated by TGA1/TGA4 and CPC transcription factors in Arabidopsis thaliana[END_REF][START_REF] Haling | Root hairs improve root penetration, root-soil contact, and phosphorus acquisition in soils of different strength[END_REF][START_REF] Tanaka | Characteristics of a root hair-less line of Arabidopsis thaliana under physiological stresses[END_REF]; (iii) genotypes with longer RHs have been shown in barley and wheat to be better adapted to low nutrient soil [START_REF] Gahoonia | Barley genotypes with long root hairs sustain high grain yields in low-P field[END_REF][START_REF] Klinsawang | Effects of root hair length on potassium acquisition in rice (Oryza sativa l.)[END_REF]; and (iv) evidence that RHs contribute directly to nutrient uptake has been obtained by various electrophysiological approaches [START_REF] Babourina | K + transport by Arabidopsis root 374 hairs at low pH[END_REF][START_REF] Lew | Immediate and steady state extracellular 376 ionic fluxes of growing Arabidopsis thaliana root hairs under 377 hyperosmotic and hypoosmotic conditions[END_REF][START_REF] Meharg | NO3transport across the 380 plasma membrane of Arabidopsis thaliana root hairs: kinetic 381 control by pH and membrane voltage[END_REF] or by using dedicated growth devices ensuring that only RHs had access to the nutrient source [START_REF] Gahoonia | Direct evidence on 384 participation of root hairs in phosphorus ( 32 P) uptake from 385 soil[END_REF].

Evidence has also been obtained that RHs can facilitate water uptake [START_REF] Tanaka | Characteristics of a root hair-less line of Arabidopsis thaliana under physiological stresses[END_REF][START_REF] Ahmed | Hydraulic processes in roots and the 387 rhizosphere pertinent to increasing yield of water-limited grain 388 crops: a critical review[END_REF][START_REF] Carminati | Root hairs enable high transpiration 390 rates in drying soils[END_REF]. For instance, the absence of RHs affects water absorption and drought tolerance in arabidopsis [START_REF] Tanaka | Characteristics of a root hair-less line of Arabidopsis thaliana under physiological stresses[END_REF]. In barley, analyses of the relationship between transpiration rate and xylem suction in wild type and hairless mutant plants provided direct evidence that RHs contribute to water uptake in drying soils in rapidly transpiring plants by increasing the soil-root interface [START_REF] Carminati | Root hairs enable high transpiration 390 rates in drying soils[END_REF]. RHs are also involved in the formation of rhizosheaths, which are more developed in mesophytic grasses in drier conditions [START_REF] Pang | Unwrapping the rhizosheath[END_REF], which also supports the hypothesis that the control of RH development has a role in plant adaptation to drought conditions.

Ion Transport Systems at the RH Plasma Membrane

Molecular analyses, including reverse genetics approaches, most often carried out in arabidopsis, have provided information on the transporters and channels involved in nutrient ion acquisition by roots. Here, we summarize our current understanding regarding potassium (K), nitrogen (N), phosphorus (P), and sulfur (S) acquisition (Figure 2). K + uptake from the soil by arabidopsis roots is essentially mediated by the high-affinity K + transporter AtHAK5 and the Shaker channels AtAKT1 and AtKC1 [START_REF] Véry | Molecular biology of K + transport across 392 the plant cell membrane: what do we learn from comparison 393 be6 ween plant species[END_REF]. Evidence is available that these three K + transport systems are expressed in RHs [START_REF] Ahn | Expression of KT/KUP genes in Arabidopsis and the role of root hairs in K + uptake[END_REF][START_REF] Reintanz | AtKC1, a silent Arabidopsis potas-395 sium channel alpha -subunit modulates root hair K + influx. 396[END_REF].

NO 3

-acquisition by roots involves transporters belonging to three different families: nitrate transporter 1/peptide transporter family (NPF), NRT2, and NRT3 (also named NAR2 for 'nitrate assimilation related family') [START_REF] Wang | Nitrate transport, signaling, and use 398 efficiency[END_REF]. The NPF family comprises the extensively studied AtNPF6.3 'transceptor' (endowed with a dual NO 3 -transport/signaling function [START_REF] Gojon | Nitrate transceptor(s) in plants[END_REF]), initially named CHL1 or AtNRT1.1. This membrane protein behaves both as a dual-affinity NO 3 -transporter and as a NO 3 -sensor mediating NO 3 -regulated auxin transport, thereby having an important role in root development [START_REF] Wang | Nitrate transport, signaling, and use 398 efficiency[END_REF][START_REF] Krouk | Nitrate-regulated auxin transport by 402 NRT1.1 defines a mechanism for nutrient sensing in plants. 403[END_REF]. Transcriptome data provide evidence that AtNPF6.3/AtNRT1.1 transcripts are present in arabidopsis RHs [START_REF] Huang | Diversification of root hair development 405 genes in vascular plants[END_REF]. The NRT2 family comprises AtNRT2.1 and AtNRT2.2, which physically interact with a member of the NRT3 family, AtNRT3.1 (also named NAR2.1) to form

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heteromeric structures that contribute significantly to high-affinity NO 3 -uptake from the soil solution [START_REF] Wang | Nitrate transport, signaling, and use 398 efficiency[END_REF][START_REF] Yong | Characterization of an intact two-407 component high-affinity nitrate transporter from Arabidopsis 408 roots[END_REF]. AtNRT2.1 and AtNRT3.1 display expression in RHs as shown by reporter gene experiments and/or transcriptome analyses [START_REF] Huang | Diversification of root hair development 405 genes in vascular plants[END_REF][START_REF] Nazoa | Regulation of the nitrate transporter 410 gene AtNRT2.1 in Arabidopsis thaliana: responses to nitrate, 411 amino acids and developmental stage[END_REF]. RHs also express members of the AMT, PHT1, and Sultr families, involved in ammonium [START_REF] Yuan | The organization of high-affinity ammonium 414 uptake in Arabidopsis roots depends on the spatial arrangement 415 and biochemical properties of AMT1-type transporters[END_REF], phosphate [START_REF] Młodzińska | Phosphate uptake and 418 allocation -a closer look at Arabidopsis thaliana L. and Oryza 419 sativa L. Front[END_REF][START_REF] Nussaume | Phosphate import in plants: focus 421 on the PHT1 transporters[END_REF][START_REF] Wang | Molecular mechanisms of phosphate 423 transport and signaling in higher plants[END_REF], and sulfate [START_REF] Takahashi | Sulfate transport systems in plants: 426 functional diversity and molecular mechanisms underlying reg-427 ulatory coordination[END_REF] uptake in arabidopsis, respectively [START_REF] Huang | Diversification of root hair development 405 genes in vascular plants[END_REF].

Close homologs of all the above-cited channels and transporters, (AtAKT1, AtKC1, AtHAK5, NPF6.3, AtNRT2.1, and AtNRT3.1) as well as ammonium, phosphate, and sulfate transporters from the AMT, PHT1, and Sultr families, have been identified in Medicago truncatula RH transcriptome data by analyses focused on membrane transport systems [START_REF] Damiani | Nod factor effects on root hair-specific 429 transcriptome of Medicago truncatula: focus on plasma Front[END_REF], and can be found in other RH transcriptomes from both dicots and monocots [START_REF] Huang | Diversification of root hair development 405 genes in vascular plants[END_REF]. Shaker channels and members of the HAK, NRT2, NRT3, AMT, PHT, and Sultr transporter families are also present in the moss Physcomitrella patens [41-44] i . Altogether, these observations suggest that major components of the RH equipment involved in plant mineral nutrition were acquired very early during plant evolution.

The sensitivity of RH length and density to nutrient and water availability in the soil (see earlier) may involve a role of RHs as sensors of soil conditions. Such a hypothesis of early sensing of water deficit conditions has received support from studies in barley wild type and hairless mutant plants.

Analysis of transcriptomes from roots that were sampled at the onset of a water stress revealed that more genes were induced in the roots of the wild-type plants, including, for example, genes involved in abscisic acid biosynthesis [START_REF] Kwasniewski | Transcriptome analysis reveals 445 the role of the root hairs as environmental sensors to maintain 446 plant functions under water-deficiency conditions[END_REF]. In terms of nutrient sensing, the arabidopsis NO 3 transporter AtNPF6.3/AtNRT1.1 and the Shaker K + channel AtAKT1, which have both been proposed to behave as transceptors, able to sense and signal the availability of their substrates, NO 3 -and K + , respectively [START_REF] Gojon | Nitrate transceptor(s) in plants[END_REF][START_REF] Shahzad | Food for thought: how 449 nutrients regulate root system architecture[END_REF], are both expressed in RHs (see earlier). [START_REF] Huang | Diversification of root hair development 405 genes in vascular plants[END_REF][START_REF] Damiani | Nod factor effects on root hair-specific 429 transcriptome of Medicago truncatula: focus on plasma Front[END_REF]. The expression of genes encoding high-affinity uptake systems can be repressed in rich media, which might be why expression in root hairs of genes such as AtHAK5 and AtNRT2;1 is not always revealed by transcriptome analyses of plants grown on rich media while evidenced by other studies using more diluted media (e.g., [START_REF] Ahn | Expression of KT/KUP genes in Arabidopsis and the role of root hairs in K + uptake[END_REF][START_REF] Nazoa | Regulation of the nitrate transporter 410 gene AtNRT2.1 in Arabidopsis thaliana: responses to nitrate, 411 amino acids and developmental stage[END_REF]). See [START_REF] Ahn | Expression of KT/KUP genes in Arabidopsis and the role of root hairs in K + uptake[END_REF][START_REF] Véry | Molecular biology of K + transport across 392 the plant cell membrane: what do we learn from comparison 393 be6 ween plant species[END_REF][START_REF] Reintanz | AtKC1, a silent Arabidopsis potas-395 sium channel alpha -subunit modulates root hair K + influx. 396[END_REF][START_REF] Wang | Nitrate transport, signaling, and use 398 efficiency[END_REF][START_REF] Huang | Diversification of root hair development 405 genes in vascular plants[END_REF][START_REF] Yong | Characterization of an intact two-407 component high-affinity nitrate transporter from Arabidopsis 408 roots[END_REF][START_REF] Nazoa | Regulation of the nitrate transporter 410 gene AtNRT2.1 in Arabidopsis thaliana: responses to nitrate, 411 amino acids and developmental stage[END_REF][START_REF] Yuan | The organization of high-affinity ammonium 414 uptake in Arabidopsis roots depends on the spatial arrangement 415 and biochemical properties of AMT1-type transporters[END_REF][START_REF] Nussaume | Phosphate import in plants: focus 421 on the PHT1 transporters[END_REF][START_REF] Takahashi | Sulfate transport systems in plants: 426 functional diversity and molecular mechanisms underlying reg-427 ulatory coordination[END_REF].

Cytoplasm

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RH Production versus Engagement in Mycorrhizal Symbiosis

Both RH production and engagement in mycorrhizal symbiosis result in increased soil exploration and exploitation. Arbuscular endomycorrhizal colonization (AM) is associated with either a decrease [START_REF] Orfanoudakis | Both the arbuscular mycorrhizal 452 fungus Gigaspora rosea and Frankia increase root system 453 branching and reduce root hair frequency in Alnus glutinosa[END_REF][START_REF] Sun | Effect of arbuscular mycorrhi-456 zal fungi inoculation on root traits and root volatile organic com-457 pound emissions of Sorghum bicolor[END_REF] or an increase [START_REF] Wu | Mycorrhiza alters the profile of root hairs 460 in trifoliate orange[END_REF] in RH density and length, depending in some species on the root type (lateral root order) [START_REF] Wu | Mycorrhiza alters the profile of root hairs 460 in trifoliate orange[END_REF]. The decrease has been proposed to result from changes in the root metabolic status and competition for available photosynthates between RH production and the fungus [START_REF] Orfanoudakis | Both the arbuscular mycorrhizal 452 fungus Gigaspora rosea and Frankia increase root system 453 branching and reduce root hair frequency in Alnus glutinosa[END_REF]. The increase has been associated with changes in the expression of auxin metabolism and transport genes, which are likely to impact RH development [START_REF] Liu | Mycorrhiza stimulates root-hair growth 462 and IAA synthesis and transport in trifoliate orange under 463 drought stress[END_REF] (see later). In ectomycorrhizal symbiosis, evidence is available that fungal secretion of the auxin antagonist hypaphorine can inhibit RH development [START_REF] Ditengou | Root hair elongation is inhibited by 465 hypaphorine, the indole alkaloid from the ectomycorrhizal fun-466 gus Pisolithus tinctorius, and restored by indole-3-acetic acid[END_REF][START_REF] Rigas | Root gravitropism and root hair develop-469 ment constitute coupled developmental responses regulated 470 by auxin homeostasis in the Arabidopsis root apex[END_REF].

Mycorrhizal fungal hyphae can have a smaller diameter (ca. 4 μm for Glomus intraradices and 5 μm for Glomus mosseae [START_REF] Drew | Beyond the rhizosphere: growth and 473 function of arbuscular mycorrhizal external hyphae in sands of 474 varying pore sizes[END_REF]) compared with RHs (ca. 10 μm in arabidopsis [START_REF] Grierson | Root hairs[END_REF]), which allows exploration of smaller soil pores. Furthermore, they extend far beyond the limits of the RH cylinder. Thus, mycorrhizal symbiosis may be hypothesized to be more efficient for exploiting the soil compared with the promotion of RH elongation and density. This question has been investigated in barley by comparing wild type and hairless mutant (brb) plants inoculated or not by different endomycorrhizal fungi. Mycorrhizas were found to substitute for RHs in P uptake, but the additional P was most often used less efficiently, in terms of plant growth, compared with P provided by RHs [START_REF] Jakobsen | Contrasting phosphate acquisition of 478 mycorrhizal fungi with that of root hairs using the root hairless 479 barley mutant[END_REF]. A similar study used several barley lines that were differentially affected in RH development (hairless, short or intermediate, or wild-type RH length phenotypes). The plants were naturally colonized by a live community present in the soil. Endomycorrhizal symbiosis did not fully compensate for the absence of RHs with regard to both P acquisition and biomass production [START_REF] Brown | Interactions between root hair length 481 and arbuscular mycorrhizal colonisation in phosphorus defi-482 cient barley (Hordeum vulgare)[END_REF]. A third series of similar experiments, using the hairless barley mutant brb and the corresponding wild-type genotype grown under well-watered or drought conditions showed that, with respect to biomass production, endomycorrhizal symbiosis compensated for the absence of RHs in the latter condition but not in the former [START_REF] Li | Relative importance of an arbuscular mycor-484 rhizal fungus (Rhizophagus intraradices) and root hairs in plant 485 drought tolerance[END_REF]. Altogether, these reports indicate that AM associations can be less efficient in P-acquisition and biomass production compared with RHs in some environmental conditions.

Organic Compound Exudation, Plant Growth-Promoting Rhizobacteria, and Stimulation of RH Development

Plants exude large amounts of organic compounds into the soil, rendering the rhizosphere a rich niche for the development of microbial communities. The actual amount of carbon (C) invested in root exudation, which can vary from 5% to 50% of the net photosynthesized C [START_REF] Haichar | Stable isotope probing of carbon 487 flow in the plant holobiont[END_REF][START_REF] Venturi | Signaling in the rhizosphere[END_REF], depends on the health of the plant, its rate of growth, its nutrient and water status, and its microbiota. Root exudation has been compared in wild type and hairless mutant barley plants, revealing that the amount of exuded C was three times higher in wild-type plants compared with the hairless mutant [START_REF] Holz | Root hairs increase rhizosphere extension 491 and carbon input to soil[END_REF]. Furthermore, experiments carried out with wild type and hairless mutant barley plants showed that an absence of RHs significantly reduced the diversity of the bacterial community [START_REF] Robertson-Albertyn | Root hair mutations 493 displace the barley rhizosphere microbiota[END_REF]. Bacterial attraction by root exudates probably involves selective chemotaxis processes [START_REF] Bulgarelli | Structure and functions of the bacte-496 rial microbiota of plants[END_REF].

The microbial community thriving in the rhizosphere can include up to 10 9 bacteria per gram of soil, belonging to diverse taxa [START_REF] Robertson-Albertyn | Root hair mutations 493 displace the barley rhizosphere microbiota[END_REF][START_REF] Bulgarelli | Structure and functions of the bacte-496 rial microbiota of plants[END_REF]. Within this population, bacteria generically named plant growth promoting rhizobacteria (PGPR) can be recruited by roots to engage in beneficial interactions. PGPR promote plant growth via very diverse mechanisms, such as improved plant mineral nutrition resulting from solubilization of poorly soluble nutrient sources, production of phytohormones that affect root development, and protection against phytoparasites [START_REF] Vacheron | Plant growth-promoting rhizobacteria 498 and root system functioning[END_REF].

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Promotion of RH development, likely to result in improved mineral nutrition, has been reported in response to diverse PGPR and in various plant species [START_REF] Contesto | Effects of rhizobacterial ACC deam-500 inase activity on Arabidopsis indicate that ethylene mediates 501 local root responses to plant growth-promoting rhizobacteria[END_REF][START_REF] Galland | The ethylene pathway contributes 504 to root hair elongation induced by the beneficial bacteria Phyllobacterium brassicacearum STM196[END_REF][START_REF] Jain | Root hair deformation, bacterial attachment, and plant growth in wheat-Azospirillum associations[END_REF][START_REF] López-Bucio | Bacillus megaterium rhizobacteria promote growth and alter root-system architecture through an auxin-and ethylene-independent signaling mechanism in Arabidopsis thaliana[END_REF][START_REF] Poitout | Local signalling pathways regulate the Arabidopsis root developmental response to Mesorhizobium loti inoculation[END_REF][START_REF] Ribaudo | Azospirillum sp. promotes root hair development in tomato plants through a mechanism that involves ethylene[END_REF][START_REF] Vacheron | Differential contribution of plantbeneficial functions from Pseudomonas kilonensis F113 to root system architecture alterations in Arabidopsis thaliana and Zea mays[END_REF][START_REF] Zamioudis | Unraveling root developmental programs initiated by beneficial Pseudomonas spp. bacteria[END_REF]. The increase in RH Q5 length can be important, by more than 100% [START_REF] Contesto | Effects of rhizobacterial ACC deam-500 inase activity on Arabidopsis indicate that ethylene mediates 501 local root responses to plant growth-promoting rhizobacteria[END_REF][START_REF] Poitout | Local signalling pathways regulate the Arabidopsis root developmental response to Mesorhizobium loti inoculation[END_REF][START_REF] Zamioudis | Unraveling root developmental programs initiated by beneficial Pseudomonas spp. bacteria[END_REF], making this response to the bacterial inoculation the easiest to detect and, thus, the most straightforward way in laboratory experiments to check whether a given plant species can interact with a given PGPR.

Large differences in the capacity to promote RH elongation have been observed between PGPR strains [START_REF] Contesto | Effects of rhizobacterial ACC deam-500 inase activity on Arabidopsis indicate that ethylene mediates 501 local root responses to plant growth-promoting rhizobacteria[END_REF]. This raises the question of whether such differences are correlated with PGPR capacity to promote plant growth and, thus, are indicative of symbiosis effectiveness. However, pathogenic strains of the bacterium Pseudomonas syringae have been shown to promote RH elongation in arabidopsis, similar to that seen with beneficial Pseudomonas spp. bacteria Q6 [START_REF] Pecenková | Early Arabidopsis root hair growth stimulation by pathogenic strains of Pseudomonas syringae[END_REF].

Plasticity of RH Development and Adaptation to External Conditions

RH development is strongly responsive to environmental factors, nutrient availability, and rhizosphere microbial communities (see earlier), as well as soil porosity, strength, and water content [START_REF] Haling | Root hair length and rhizosheath mass depend on soil porosity, strength and water content in barley genotypes[END_REF]. Different patterns of RH distribution within the root epidermis can be identified among plant species (Box 1) but the question of whether a given pattern has specific advantages and in what environmental conditions is poorly documented.

Studies in arabidopsis (RH distribution pattern 3; Box 1) to investigate how external biotic or abiotic conditions can impact epidermal cell differentiation and RH morphogenesis, have brought to light hormone-driven processes [START_REF] Vissenberg | Hormonal regulation of root hair growth and responses to the environment in Arabidopsis[END_REF]. Evidence has been obtained that auxin and ethylene have major roles in the control of epidermal cell differentiation and RH development and that the pathways allowing these two hormones to affect RH formation are significantly congruent [START_REF] Grierson | Root hairs[END_REF][START_REF] Vissenberg | Hormonal regulation of root hair growth and responses to the environment in Arabidopsis[END_REF][START_REF] Cui | Regulation and functional diversification of root hairs[END_REF][START_REF] Bruex | A gene regulatory network for root epidermis cell differentiation in Arabidopsis[END_REF][START_REF] Zhang | Multiple phytohormones promote root hair elongation by regulating a similar set of genes in the root epidermis in Arabidopsis[END_REF]. Proper auxin distribution is required for correct cell fate assignment and RH formation (both initiation site selection and tip growth). In arabidopsis, auxin regulates RH formation by acting downstream of RHD6, and probably primarily via RSL4 [START_REF] Masucci | Hormones act downstream of TTG and GL2 to promote root hair outgrowth during epidermis development in the Arabidopsis root[END_REF][START_REF] Yi | A basic helix-loop-helix transcription factor controls cell growth and size in root hairs[END_REF], two transcription factors with central roles in RH development (Box 2). Ethylene also acts on RH formation downstream of RHD6 [START_REF] Masucci | Hormones act downstream of TTG and GL2 to promote root hair outgrowth during epidermis development in the Arabidopsis root[END_REF]. Evidence is also available that jasmonic acid, strigolactones, and cytokinins are positive regulators of RH growth, whereas brassinosteroids and abscisic acid are negative regulators [START_REF] Grierson | Root hairs[END_REF][START_REF] Vissenberg | Hormonal regulation of root hair growth and responses to the environment in Arabidopsis[END_REF][START_REF] Cui | Regulation and functional diversification of root hairs[END_REF][START_REF] Han | Arabidopsis ZINC FINGER PROTEIN1 acts downstream of GL2 to repress root hair initiation and elongation by directly suppressing bHLH genes[END_REF].

The widely reported increase in RH density and length in response to low P availability involves induction of RH-expressed auxin-inducible transcription factors, including RSL2 and RSL4 [START_REF] Bhosale | A mechanistic framework for auxin dependent Arabidopsis root hair elongation to low external phosphate[END_REF] and ethylene-mediated events with a role in the control of RH gene expression [START_REF] Song | The molecular mechanism of ethylenemediated root hair development induced by phosphate starvation[END_REF]. Promotion of RH development by PGPR species has also been shown to involve auxin [START_REF] Zamioudis | Unraveling root developmental programs initiated by beneficial Pseudomonas spp. bacteria[END_REF], ethylene [START_REF] Galland | The ethylene pathway contributes 504 to root hair elongation induced by the beneficial bacteria Phyllobacterium brassicacearum STM196[END_REF], and a complex interplay of auxin and ethylene signaling pathways [START_REF] Poupin | A complex molecular interplay of auxin and ethylene signaling pathways Is involved in Arabidopsis growth promotion by Burkholderia phytofirmans PsJN[END_REF]. Nevertheless, a case of PGPR-induced promotion of RH elongation poorly dependent on auxin and ethylene signaling mechanisms has been reported in arabidopsis [START_REF] López-Bucio | Bacillus megaterium rhizobacteria promote growth and alter root-system architecture through an auxin-and ethylene-independent signaling mechanism in Arabidopsis thaliana[END_REF].

Finally, although control of RH longevity is likely to be of major importance in root system adaptation to soil abiotic and biotic factors, RH longevity is still poorly documented. Reported values in barley vary from a few days to 2-3 weeks [START_REF] Henry | Natural (non-pathogenic) death of the cortex of wheat and barley seminal roots, as evidenced by nuclear staining with acridine orange[END_REF][START_REF] Holden | Use of nuclear staining to assess rates of cell death in cortices of cereal roots[END_REF][START_REF] Mcelgunn | Formation, elongation, and longevity of barley root hairs[END_REF]. Evidence has been obtained that apoptosis-like programmed cell death (AL-PCD), characterized by protoplast retraction, nuclear DNA fragmentation, and sensitivity to inhibitors of caspase-3-like activity, occurs in arabidopsis RHs in response to heat shock, salt stress, and reactive oxygen species (ROS; H 2 O 2 ) treatment [START_REF] Hogg | An in vivo root hair assay for determining rates of apoptotic-like programmed cell death in plants[END_REF][START_REF] Tan | Nuclear dynamics and programmed cell death in Arabidopsis root hairs[END_REF]. Basal AL-PCD rates ranging from ca. 5% to 15% have been monitored in arabidopsis seedlings classically grown on agar plates [START_REF] Tan | Nuclear dynamics and programmed cell death in Arabidopsis root hairs[END_REF]. Thus, it is tempting to hypothesize that the extension of functional RH zones might be regulated by AL-PCD in response to local environmental conditions. 236 maize [START_REF] Zhu | Mapping of QTL controlling root hair length 585 in maize (Zea mays L.) under phosphorus deficiency[END_REF], and wheat [START_REF] Delhaize | The genetics of rhizosheath size in a multiparent mapping population of wheat[END_REF][START_REF] Horn | Mapping of quantitative trait loci for root 588 hair length in wheat identifies loci that co-locate with loci for 589 yield components[END_REF][START_REF] James | Rhizosheaths on wheat grown in acid 591 soils: phosphorus acquisition efficiency and genetic control[END_REF][START_REF] Liu | Analysis of aneuploid lines of bread wheat 594 to map chromosomal locations of genes controlling root hair 595 length[END_REF]). A seminal study of the correlation between RH length and 237 biomass production and yield, reported by Gahoonia and Nielsen [START_REF] Gahoonia | Barley genotypes with long root hairs sustain high grain yields in low-P field[END_REF], explored the biological 238 diversity within 38 barley cultivars. Large variations in RH length, from ~0.4 mm to >1.3 mm, 239 were observed in hydroponically grown plants, consistent with variations thereafter observed in 240 field conditions. Then, a set of ten representative cultivars was tested in field experiments, with 241 different levels of soil P availability. The complete set of results indicated that barley genotypes 242 with long RHs displayed higher tolerance to low P conditions, and expressed higher yield poten-243 tials both in low and high P soils [START_REF] Gahoonia | Barley genotypes with long root hairs sustain high grain yields in low-P field[END_REF]. In a similar experiment, characterization of barley mutant 244 lines with various RH phenotypes, screened from a mutagenized population, showed that RH 245 length was important for shoot P accumulation and biomass production, especially under com-246 bined water and phosphorus stress [START_REF] Brown | What are the implications of variation 597 in root hair length on tolerance to phosphorus deficiency in 598 combination with water stress in barley (Hordeum vulgare)[END_REF]. However, for grain yield, only the presence of RHs, and 247 not RH length, was critical. The difference in RH length between the genotypes classified as 248 'Short RH' and 'Long RH' (0.54 mm vs 0.69 mm) in this report was small compared with the 249 differences observed within the set of barley cultivars previously used by Gahoonia and Nielsen

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250 [START_REF] Gahoonia | Barley genotypes with long root hairs sustain high grain yields in low-P field[END_REF]. Evidence that plant biomass production can be positively correlated to RH length in barley 251 has also been provided by phenotyping 175 lines from a doubled-haploid (DH) population, 252 using rhizosheath size as a surrogate for RH length and the normalized difference vegetation 253 index (NDVI) as a surrogate for crop biomass. These analyses revealed colocation between 254 QTLs for root rhizosheath size and NDVI-estimated biomass [START_REF] Gong | QTL mapping of root traits 581 in phosphorus-deficient soils reveals important genomic re-582 gions for improving NDVI and grain yield in barley[END_REF]. QTLs for RH length have

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also been found to colocate with QTL for yield components in wheat [START_REF] Horn | Mapping of quantitative trait loci for root 588 hair length in wheat identifies loci that co-locate with loci for 589 yield components[END_REF]. However, further work is still required to investigate whether, and under what environmental conditions, longer RHs benefit grain yield.

Concluding Remarks and Future Perspectives

RHs have major roles in plant autotrophy and fitness by contributing to the anchorage of growing root tips into the soil, to soil mining for water and nutrient acquisition, and to interactions with soil microorganisms. The RH cell model is extensively used to decipher processes of cell fate and celltype patterning in plants as well as the mechanisms involved in tip growth. However, it is clear that many questions regarding RH development and functions at the soil-root interface deserve to be further investigated, such as the mechanisms involved in epidermal cell fate in species displaying a RH patterning different from that in arabidopsis (Box 1), or the environmental and internal determinants of RH longevity and apoptosis-like programmed cell death.

In terms of crop breeding, evidence is available that the genetic variation in root system and RH traits can be used to reduce fertilization and irrigation inputs and increase crop yield in poor soil conditions [START_REF] Burridge | A case study on the efficacy of root 601 phenotypic selection for edaphic stress tolerance in low-input 602 agriculture: common bean breeding in Mozambique[END_REF][START_REF] Strock | Seedling root architecture and its 605 relationship with seed yield across diverse environments in 606 Phaseolus vulgaris[END_REF]. With the objective of contributing to a vital new green revolution, strong efforts have been aimed at developing methodologies and imaging platforms for high-throughput phenotyping (HTP) of root trait variation in large genotype collections to identify promising germplasm and markers for selection. So far, the root traits that have been analyzed by such HTP approaches and shown to be positively associated with yield in field experiments, at least in some soil and environmental conditions, most often correspond to macroscopic features, such as 'primary root length' in oilseed rape [START_REF] Thomas | High-throughput phenotyping (HTP) 608 identifies seedling root traits linked to variation in seed yield and 609 nutrient capture in field-grown oilseed rape (Brassica napus L.)[END_REF], or 'seminal root number', 'total root length' [START_REF] Xie | Identifying seedling root architectural traits associated with yield and yield components in wheat[END_REF], 'narrow root angle' (thought to result in a steep root phenotype) [START_REF] Khokhar | Juvenile root traits show limited correlation with grain yield, yield components and grain mineral composition traits in Indian wheat under hostile soils[END_REF], or 'root diameter' [100] in wheat. None of the HTP methodologies used in these studies were reported to have included RH development in the targeted root traits, probably because the phenotyping procedure was not aimed at obtaining and analyzing high-resolution images. RH traits that have to be taken into account are length and density as well as the sensitivity of these parameters to abiotic and biotic conditions, such as reduced nutrient and water availability or presence of PGPR. It is also likely that the location and relative extension of the root zones bearing live RHs are important traits. HTP methodologies could also be used to screen, for instance, collections of crop cultivars together with collections of beneficial soil bacteria by phenotyping the RH responses to the inoculated bacteria before assessing the effects of selected bacterial partners on plant growth in soil conditions. Various low-cost HTP methodologies can be used/adapted for such research objectives, including the so-called 'paper-roll' and 'pouch and wick' setups and other 2D phenotyping methodologies [101]. Our group is currently developing a similar 2D HTP approach using rhizobox-like devices in which root growth occurs (Figure 1) on a piece of non-putrescible tissue under automatically controlled watering. More complex 3D phenotyping methodologies could also be developed using heterogeneous transparent substrates that mimic soil features [102,103].

RH phenotyping in plants grown in field conditions can provide valuable criteria for plant breeding programs. Gentle and careful washing of excavated root systems can allow quantification of RH traits, namely length and density [START_REF] Dittmer | A quantitative study of the roots and root hairs of a winter rye plant (Secale cereale)[END_REF]104], but this is poorly amenable to high-throughput procedures. By contrast, such analyses of RH genotypic variation in the field have provided information in line with the results of HTP analyses (paper-roll type) and have been found to be effective in breeding programs for edaphic stress tolerance in low-input agriculture [START_REF] Burridge | A case study on the efficacy of root 601 phenotypic selection for edaphic stress tolerance in low-input 602 agriculture: common bean breeding in Mozambique[END_REF]104]. Thus, evidence is already available that a combination of HTP approaches with assessment of the selected germplasm in field conditions can increase the efficiency and speed up plant breeding for lowinput agriculture [START_REF] Burridge | A case study on the efficacy of root 601 phenotypic selection for edaphic stress tolerance in low-input 602 agriculture: common bean breeding in Mozambique[END_REF][START_REF] Strock | Seedling root architecture and its 605 relationship with seed yield across diverse environments in 606 Phaseolus vulgaris[END_REF]105]. This suggests that further development/adaptation of HTP Outstanding Questions

Figure 1 .

 1 Figure 1. Main photo: 2-week-old seedling grown in a rhizobox. Inset: part of the root system of a plant grown for 2 months in soil in a pot, showing root hairs in old parts of the root system. Wheat cultivar: Oued Zenati.

234

  Quantitative Trait Loci of RH Production and Plant Yield Q7 235 Quantitative trait loci (QTLs) of RH length have been identified in various crops (e.g., barley [89],

b0: 2 Box 1 .

 21 Root Hair Distribution Patterns b1:3 RH cell distribution within the root epidermis varies among angiosperms. The distribution patterns have been sorted into b1:4 three basic types: random, alternating, and position-dependent, named types 1, 2, and 3, respectively [106,107] (Figure I).

b1: 5

 5 Figure I. Root Hair Distribution Patterns.

b1: 20

 20 Trends in Plant ScienceTrends in Plant Science, Month 2020, Vol. xx, No. xx 7

b0: 2 Box 2 . 3 Figure

 223 Figure I describes the core of the current model of the regulation of RH development by the intrinsic developmental

Figure I .

 I Figure I. Arabidopsis Root Hair Development. Arrows indicate positive control. Blunted lines indicate negative

U

Q8

  What are the sensing and signaling mechanisms allowing RH production and elongation to respond to soil features (nutrient ion availability, soil moisture, texture, porosity…)? How long do RHs stay alive and functional in the soil and what are the internal and external determinants of RH lifespan? Do the different RH distribution patterns have specific physiological advantages and under what environmental conditions? What are the comparative costbenefit ratios of RH production and mycorrhization, in terms of biomass production and grain yield? What is the physiological meaning of the stimulation of RH elongation by rhizobacteria, and can this stimulation be operationally considered, in HTP methodologies, as an indication of an engagement in beneficial interactions? Q9 What kinds of high-throughput root trait phenotyping methodologies could take into RHs consideration? What have been the consequences of generations of selective breeding for increased yields in artificial soil conditions on the ability of plants to invest photosynthates in root development and functions? Trends in Plant Science Trends in Plant Science, Month 2020, Vol. xx, No. xx 9 phenotyping of RH traits and of their responses to abiotic and biotic conditions would contribute significantly to such programs. and reactive oxygen species 431 networks. Front. Plant Sci. 7, 794 432 41. De Michele, R. et al. (2012) Ammonium and urea transporter 433 inventory of the Selaginella and Physcomitrella genomes.

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i https://phytozome.jgi.doe.gov/pz/portal.html#!info?alias=Org_Ppatens

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Acknowledgments

We thank Martin Boeglin for help in preparing Figure 1. This work was supported by the Q10

French National Research Agency (ANR) [ANR-11-BSV7-010-02 (to H.S. and A-A.V.) and ANR ANR-19-CE32-0011 (to J-B.P.)], an EU Erasmus Mundus grant, Alfabet project (to T.R.) and an ERANET EU Arimnet2 grant (no. 618127) (to H.S.).

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