Chang Liu

Wen-Shen Liu

Hermine Huot
email: hermine.huot@univ-lorraine.fr

Yi-Ming Yang

Mei-Na Guo

Jean Louis Morel

Ye-Tao Tang

Rong-Liang Qiu

Responses of ramie (Boehmeria nivea L.) to increasing rare earth element (REE) concentrations in a hydroponic system

Keywords: phytoremediation, ramie, rare earth elements, fractionation, molybdenum, phosphorus

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Rare earth elements (REEs) include lanthanides (from La to Lu) plus yttrium (Y) and scandium (Sc). Lanthanides can be divided into 2 groups: light REEs (LREEs, from La to Eu) and heavy REEs (HREEs, from Gd to Lu). [START_REF] Weber | Rare earth elements: A review of production, processing, recycling, and associated environmental issues[END_REF] Rare earth elements are important strategic resources for modern technologies and clean energy production. [START_REF] Alonso | Evaluating rare earth element availability: A case with revolutionary demand from clean technologies[END_REF] However, the growing demand of these resources has facilitated the exploitation of ion-adsorption REE deposits in southern China, resulting in large areas of REE mine tailings. [START_REF] Liang | State of rare earth elements in different environmental components in mining areas of China[END_REF][START_REF] Liu | Water, sediment and agricultural soil contamination from an ion-adsorption rare earth mining area[END_REF] It was found that the concentrations of REEs in these mine tailings reached 300-1200 mg/kg, which are several times higher than those of the unmined soils nearby. [START_REF] Chao | Structure, variation, and co-occurrence of soil microbial communities in abandoned sites of a rare earth elements mine[END_REF] These abandoned ion-adsorption REE mine tailings are subjected to runoff and erosion, leading to REE contamination of surface water and agricultural land. [START_REF] Liu | Water, sediment and agricultural soil contamination from an ion-adsorption rare earth mining area[END_REF] High concentrations of REEs may induce adverse effects on microorganisms, invertebrates, and plants, [START_REF] Chen | Environmental toxicology of rare earth elements 147 Pm, 141 Ce, 147 Nd[END_REF][START_REF] Hu | Physiological and biochemical effects of rare earth elements on plants and their agricultural significance: A review[END_REF][START_REF] Feng | Study on the acute toxicity of rare earth yttrium to earthworms under the stress of leaching agent ammonium sulfate[END_REF][START_REF] Fujita | Effects of simulated rare earth recycling wastewaters on biological nitrification[END_REF] and pose potential threats to human health. [START_REF] Li | A human health risk assessment of rare earth elements in soil and vegetables from a mining area in Fujian Province, Southeast China[END_REF] Therefore, control and remediation of these REE-contaminated soils have become a serious environmental concern.

Phytoremediation is a low-cost and environmental-friendly method that employs plants to achieve remediation for metal(loid)s polluted soils. [START_REF] Chaney | Phytoremediation of soil metals[END_REF][START_REF] Arthur | Phytoremediation-an overview[END_REF] Among several phytoremediation strategies, phytostabilization uses plants and soil amendments to reduce the availability of metal(loid)s in the soil and limit their migration to the surface and ground waters. [START_REF] Chaney | Phytoremediation of soil metals[END_REF] The plants used for phytostabilization operations can usually survive on soils with high concentrations of certain metal(loid)s and have limited soil-to-shoot bioconcentration factors for metal(loid)s, thereby minimizing the risks of metal(loid)s entering the food chain and transferring to other environmental compartments. [START_REF] Wong | Ecological restoration of mine degraded soils, with emphasis on metal contaminated soils[END_REF][START_REF] Mendez | Phytostabilization potential of quailbush for mine tailings: Growth, metal accumulation, and microbial community changes[END_REF][START_REF] Branzini | Phytostabilization as soil remediation strategy[END_REF] In the past decades, many efforts have been focused on phytostablization of a series of heavy metals, i.e. Cd, Zn, As, Pb, etc. These studies found that a number of plants meet the requirements of phytostablization, e.g. pioneer plant Miscanthus sinensis, commercial plant Vetiveria zizanioides or fiber plant Boehmeria nivea L. [START_REF] Huang | Study on Pb tolerance and remediation potential of lead-contaminated soil of ramie[END_REF][START_REF] Liu | Soil characteristics and heavy metal accumulation by native plants in a Mn mining area of Guangxi, South China[END_REF][START_REF] Chen | Remediation efficiency and pathways of Brassica juncea and Vetiveria zizanioides L. for Pb-contaminated soil[END_REF] These plants improve the soil physico-chemical and biological properties and reduce the availability of heavy metals, thus alleviating the phytotoxicity. [START_REF] Adriano | Role of assisted natural remediation in environmental cleanup[END_REF][START_REF] Badri | Regulation and function of root exudates[END_REF] Those heavy metals are immobilized, or converted to less toxic forms through oxidation-reduction, complexation and precipitation within the roots or the rhizosphere of the plants. [START_REF] Raskin | Phytoremediation of metals: Using plants to remove pollutants from the environment[END_REF][START_REF] Mun | Assessment of Pb uptake, translocation and immobilization in kenaf (Hibiscus cannabinus L.) for phytoremediation of sand tailings[END_REF] Even though there is an increasing interest in the biological responses of plants to REEs, most studies focus on the effects of individual REE on crop plants e.g. rice, soybean and wheat. [START_REF] Hu | Physiological and biochemical effects of rare earth elements on plants and their agricultural significance: A review[END_REF][START_REF] Pagano | Health effects and toxicity mechanisms of rare earth elements-knowledge gaps and research prospects[END_REF][START_REF] Adeel | Cryptic foot prints of rare earth elements on natural resources and living organisms[END_REF] These studies found that REEs can induce a "hormesis effect", that is, REEs can promote crop biomass production at low concentrations, but inhibit crop growth at high concentrations. [START_REF] Redling | Rare earth elements in agriculture with emphasis on animal husbandry[END_REF] For example, La had positive effects on rice growth at 0.36-5.40 μmol/L, but negative effects on rice biomass at concentrations higher than 10.7 μmol/L in solution. [START_REF] Xie | Effect of lanthanum on rice production, nutrient uptake, and distribution[END_REF] In soybean, La promoted plant growth at 5-10 μmol/L, but reduced plant biomass, caused ultrastructural modifications in cell walls and thylakoids at concentrations higher than 20 μmol/L of La in solution. [START_REF] De Oliveira | Bioaccumulation and effects of lanthanum on growth and mitotic index in soybean plants[END_REF] Usually, REEs increase the uptake of nutrients in crop plants under low concentrations, while interfere with nutrient uptake, [START_REF] Sun | Effects and mechanisms of the combined pollution of lanthanum and acid rain on the root phenotype of soybean seedlings[END_REF] destroy cell membrane structure, [START_REF] Wang | Arabinogalactan protein-rare earth element complexes activate plant endocytosis[END_REF] damage chloroplast structure and inhibit photosynthesis under high concentrations. [START_REF] Wen | Combined effects of lanthanum ion and acid rain on growth, photosynthesis and chloroplast ultrastructure in soybean seedlings[END_REF] Nevertheless, little attention has been paid on the response of REE-tolerant species to elevated REE concentrations.

Ramie (Boehmeria nivea L.) is a perennial fiber plant belonging to the Urticaceae family, which colonizes naturally at most of the tropical and subtropical regions. As ramie produces high quality of fibers for agricultural and industrial uses, it has been cultivated in south China for thousands of years (e.g. Jiangxi, Guangdong, Hunan, Hubei provinces). This plant produces high amount of biomass and has a dense root system, thereby acting as a choice of revegetation for reducing runoff and soil erosion. [START_REF] Liu | Effect of growth regulators on yield and fiber quality in ramie (Boemheria nivea L. Gaud.), China grass[END_REF] Moreover, ramie is regarded as a multi-metal tolerant plant with strong tolerance to Pb, Zn, Cd, As, and has strong adaptability to harsh conditions such as drought. [START_REF] Yang | Constitutional tolerance to heavy metals of a fiber crop, ramie (Boehmeria nivea), and its potential usage[END_REF] Studies showed that ramie has very limited soil-to-shoot bioconcentration factors for heavy metals, and most heavy metals are restricted within roots. [START_REF] Huang | Study on Pb tolerance and remediation potential of lead-contaminated soil of ramie[END_REF] For instance, most Cd is bound to the cell walls, or precipitated with phosphate in the roots, while organic acids are not involed in Cd tolerance of ramie. [START_REF] Liu | Cadmium-induced oxidative stress and response of the ascorbate-glutathione cycle in Bechmeria nivea (L.) gaud[END_REF][START_REF] Wang | Subcellular distribution and chemical forms of cadmium in Bechmeria nivea (L.) Gaud[END_REF] However, how this species adapts to high concentrations of REEs remains unknown.

Elucidating the effects of increasing concentrations of REEs on ramie growth and nutrient uptake is key to understanding the mechanisms of tolerance within the aim of using this plant for the phytoremediation at REE-contaminated soils. In this study, a hydroponic experiment was conducted using equimolar solutions of 16 REEs to study the tolerance of ramie to REE stress. More specifically, the influence of REEs on plant biomass, nutrient accumulation, REE accumulation and fractionation, and REE-nutrient interactions in the plant tissues were determined in order to study the mechanisms of tolerance in ramie and evaluate its potential for phytoremediation at REE-contaminated soils.

Materials and methods

Plant material and growth conditions

The seeds of ramie (Zhongzhu No.1) were bought from the Institute of Bast Fiber Crops, Chinese Academy of Agricultural Sciences. Before sowing, approximately 0.5 g of seeds were sterilized in 50 mL of 5% NaClO (AR) solution for 15 min, then rinsed 5 times with deionized water. Subsequently they were sown on quartz sand with high-purity water (18.2 Ω) in a growth chamber at 25°C in the dark for 7 d. After germination, healthy and uniform seedlings with two leaves were transferred to beakers containing 1/2-strength Hoagland nutrient solution (500 μmol/L Ca (NO3)2•4H2O, 500 μmol/L KNO3, 250 μmol/L MgCl2•7H2O, 50 μmol/L KH2PO4, 25 μmol/L KCl, 5 μmol/L H3BO3, 0.5 μmol/L MnCl2, 0.1 μmol/L CuSO4•5H2O, 0.1 μmol/L Na2MoO4•2H2O, 10 μmol/L Fe (III)-EDDHA, 1.0 μmol/L ZnSO4•7H2O, AR). [START_REF] She | The uptake and accumulation characterisics if heavy metals and expression of Cd stress response genes in ramie (Boehmeria nivea (L.) Gaud)[END_REF] The solution pH was adjusted to 5.5 ± 0.1 with 2-morpholinoethanesulphonic acid (MES)-KOH throughout the experimental period. [START_REF] Han | Organic acids promote the uptake of lanthanum by barley roots[END_REF] The nutrient solution was changed every three days. The seedlings were exposed to tungsten lamps under controlled conditions (350 µmol/m 2 /s PAR-light, 14/10 h day/night, 25/20°C day/night and 70% relative humidity). [START_REF] Yuan | The accumulation and fractionation of rare earth elements in hydroponically grown Phytolacca americana[END_REF]

Plant exposure to REEs

After two weeks of nurturing, three healthy and uniform seedlings (each with six leaves) were transferred to a beaker containing 1 L of fresh nutrient solution free of P to avoid precipitation with REEs. Phosphorus was sprayed on the leaves with a KH2PO4 solution (50 μmol/L) once a day. A series of equimolar REE solutions were prepared by mixing chloride salts of 16 REEs (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Y and Sc, Aladdin, America). Eight REE concentrations were tested, i.e. 0, 1.6, 8, 16, 80, 160, 400, 800 μmol/L of REE (each REE with equal molar concentration, i.e. 0, 0.1, 0.5, 1, 5, 10, 25, 50 μmol/L). The nutrient solution with varying REE concentrations was renewed every three days. After four weeks of REE exposure, plants were harvested and separated into leaves, stems and roots. As a result of limited plant biomass, the three ramie seedlings in each beaker were put together as one sample. Each treatment (i.e. each concentration of REEs) was prepared in triplicates.

Roots were washed with Milli-Q H2O and desorbed with 0.1 M EDTA-Na2 (Aladdin, America) solution for 30 min, and then washed 3 times with Milli-Q H2O. Leaves and stems were washed only with Milli-Q H2O. Plant tissues were dried at 105°C for 30 min and then at 60°C for 48 h. Dry biomass of plant tissues was weighed, then ground to fine powder (< 0.15 mm). The fresh roots were scanned and the determination of root characteristics (length, surface area, volume) was conducted using WinRhizo software.

Chemical analysis

Aliquots of plant material (100 ± 1 mg) were introduced into Teflon crucibles with covers, and wetted with 1-2 drops of Mill-Q water before digestion in a mixture of concentrated HNO3 (3 mL, GR), HCl (1 mL, GR) and HClO4 (1 ml, GR) on a hot plate at 200 ± 5°C for 12 h. The mineralization solution was then evaporated at 180°C until almost complete dehydration (<1 mL). The residues were dissolved in 2 mL of concentrated HNO3, dried again, and re-dissolved in 15 mL of 2% HNO3.

Concentrations of REEs were determined using an inductively coupled plasma-mass spectrometer (ICP-MS, Agilent 7700x, USA). The internal standard used during analysis was 115 In at 1.5 mg L -1 in 2% HNO3, and 0.1 mL of indium solution was added to the sample solutions. Macro-and micro-nutrients were analyzed using an inductively coupled plasma optical emission spectrometer (ICP-OES, Agilent VISTA, USA). All the analyses were conducted at ALS Chemex (Guangzhou) Go., LTD. The total REE concentration (∑REEs) was calculated as the sum of the concentrations of the 16 REEs.

REE fractionation analysis

Rare earth elements are a group of elements with similar chemical properties but they can be fractionated in soils and plants due to small but detectable variations in their behavior derived mainly from differences in ionic radii with increasing atomic number. [START_REF] Laveuf | A review on the potentiality of rare earth elements to trace pedogenetic processes[END_REF] The ratio between LREEs and HREEs is a common index used in REE fractionation analysis. Moreover, REEs are trivalent elements, except for Ce and Eu that have variable valences (Ce 4+ and Ce 3+ ; Eu 2+ and Eu 3+ ) in the environment, which can induce positive or negative anomalies in the REE distribution patterns (i.e. enrichment or depletion in Ce and Eu compared to REEs of close atomic number). [START_REF] Redling | Rare earth elements in agriculture with emphasis on animal husbandry[END_REF] In hydroponic experiments, only Ce may be subjected to redox reactions, thus only Ce anomaly was considered. Rare earth element fractionation patterns and anomalies are reliable tools to trace mechanisms associated with REE accumulation and tolerance in plants under REE stress. [START_REF] Wei | Rare earth elements in naturally grown fern Dicranopteris linearis in relation to their variation in soils in South-Jiangxi region (southern China)[END_REF][START_REF] Xu | Distribution of rare earths and heavy metals in field-grown maize after application of rare earth containing fertilizer[END_REF][START_REF] Ding | Fractionation mechanisms of rare earth elements (REEs) in hydroponic wheat: An application for metal accumulation by plants[END_REF][START_REF] Liang | A review of fractionations of rare earth elements in plants[END_REF] For instance, a HREE enrichment in the leaves of wheat may be attributed to the facilitated transport of HREEs by small organic acids complexation. [START_REF] Ding | Fractionation mechanisms of rare earth elements (REEs) in hydroponic wheat: An application for metal accumulation by plants[END_REF] The ratio of LREEs to HREEs in our hydroponic solution was calculated as: ∑(LREE molar concentration in solution × LREE molar mass) / ∑(HREE molar concentration in solution × HREE molar mass, (mg/kg)/(mg/kg)), and amounted to 0.65. Therefore, ΣLREE/ΣHREE >0.65 in the plant tissues means a LREE enrichment, while ΣLREE/ΣHREE <0.65 means a HREE enrichment. Cerium anomaly ( * Ce) was calculated as {2Ce/[La + Pr ]} ((mg/kg)/(mg/kg)); * Ce >1 refers to a positive anomaly, and * Ce <1 refers to a negative anomaly. The transfer factors, which are defined as the REE concentrations in one compartment divided by the concentrations in another compartment, were also calculated through the REE transfer route (e.g. from solution to roots, from roots to stems, from stems to leaves and from roots to shoots). [START_REF] Tyler | Rare earth elements in soil and plant systemsa review[END_REF]

Tolerance index analysis

Tolerance index (TI) is the ratio of the dry weight of ramie tissue for a given treatment to the dry weight of ramie tissue for the control. Tolerance index values lower than 1 indicate a decrease in biomass and suggest that plants are stressed, whereas TI values equal to 1 indicate no effect. Tolerance index values greater than 1 indicate a net increase in biomass and suggest a growth promoting effect. [START_REF] Audet | Heavy metal phytoremediation from a meta-analytical perspective[END_REF] A value of 0.5 is considered to be the acceptable minimum of biomass production for plants growing in a metal-contaminated site. [START_REF] Deivanai | Phytostabilization potential of yard long bean in removing cadmium from soil[END_REF]

Data analysis

The SPSS 20.0 statistical analysis software was used for data analysis. All results were presented as arithmetic means with standard deviation calculated on three replicates (beakers). The differences among the treatments were tested using an ANOVA followed by Duncan test at a 0.05 significance level. For one ramie organ, values with different letters refer to significant differences between the different treatments (p <0.05). For some elements, the data were not normally distributed and were transformed to follow a normal distribution before being tested. Matrices of correlation were performed on the elemental concentrations (REEs and nutrients) obtained form all the treatments for ramie roots and leaves separately, using a Pearson correlation two-sided test.

Results

Plant growth

A significant effect of REE concentrations in solution to the growth of ramie is recorded (Figs.

1; S1). Compared to the control (0 μmol/L REE treatment), the root biomass is higher under 1.6 μmol/L REE treatment, while both the shoot and root biomasses show no obvious variations in the range of 8-80 μmol/L REE treatments. However, a significant decrease in the production of both the shoot and root biomasses occurs at 160 μmol/L and higher REE concentration treatments. The root total length, surface area and volume are the highest under the 1.6 μmol/L REE treatment (Table S1), while no significant decrease of the total length and surface area is shown even at 800 μmol/L REE treatment, compared with the control. Under lower REE concentrations (1.6-80 μmol/L), the tolerance index (TI) of ramie varies between 0.70 and 2.45 for the roots, stems and leaves (Table S2). At higher REE concentrations (160-800 μmol/L), it decreases below 0.7 for roots and below 0.5 for leaves, and toxicity symptoms are recorded, i.e. yellow leaves and dead leaves with strong necrosis areas (Fig. S1).

The TI of ramie stems are greater than one under all the REE treatments. Though TI of ramie leaves and roots shows no significant differences among treatments of 0-80 μmol/L REEs, the highest TI of ramie roots and leaves is recorded at 1.6 μmol/L REEs treatment. Hence, it can be assumed that REEs induce an hormesis effect, and toxicity occurs between 80 and 160 μmol/L REEs in the solution.

REE accumulation in ramie organs

The total REE concentrations in different organs of ramie are shown in Fig. 2. In general, the roots have much higher REE concentrations than the stems and leaves. At lower REE concentrations in solution (1.6-8 μmol/L), the REE concentrations in ramie organs are in the order of roots > leaves > stems (Fig. 2). This order is slightly different for higher REE concentrations treatments (>8 μmol/L), namely roots > stems > leaves. Root REE concentrations increase sharply with increasing REE concentrations in solution. At 8 μmol/L REEs in solution, REE concentration in the roots is 3.6 times higher than at 1.6 μmol/L REEs in solution. In stems and leaves, REE concentrations show a slight decrease with the increasing REE concentrations in solution up to 80 μmol/L, then increase markedly at higher REE concentration treatments (160-800 μmol/L). Within the tolerance range determined in this

REE fractionation in ramie organs

The ratio of the sum of LREE concentrations to the sum of HREE concentrations (∑LREE/∑HREE) in the different organs ranges from 0.23 to 0.68 under all the REE treatments, except for the stems at 1.6 μmol/L REEs (∑LREE/∑HREE=1.21) (Figs. 3a; S2). In the leaves, the ∑LREE/∑HREE ratios are lower than 0.65 (HREE enrichment) under all the treatments, and they are lower under 1.6-80 μmol/L than under 160-800 μmol/L REE treatments. In the stems, the ∑LREE/∑HREE ratios are lower at 8-80 µM REE treatments (0.38-0.51) and higher at 160-800 μmol/L REE treatments (0.58-0.61). In both the roots and the whole plant, the ∑LREE/∑HREE ratios are lower under the 1.6-16 μmol/L REE treatments (<0.65) than the 80-800 μmol/L REE treatments (around 0.65) (no significant fractionation). Under the same REE concentration treatment, the ∑LREE/∑HREE ratios in the leaves are lower than those in the stems. Different ramie organs and the whole plant show positive Ce anomalies (1.06-1.48) under all the REE treatments (Figs. 3b,S2), that is, ramie organs accumulate more Ce than its neighbours (La and Pr). Cerium shows positive anomalies during the uptake (solution-to-plant) process, but no significant anomalies during the transport (roots-to-leaves) process (Fig. S3).

Concentrations of scandium are lower than those of the lanthanides (La to Lu) under all the REE treatments in the leaves, whereas in the roots they are higher compared to those of the lanthanides from 80 to 800 μmol/L REE treatments. Concentrations of yttrium in ramie leaves and stems are similar to those of the lanthanides, while in the roots they are lower under all the REE treatments (Tables S3-S5).

Concentrations and relationships of nutrients in ramie organs

The addition of REEs in the solution significantly decreases the concentrations of Ca and Mn in ramie roots, stems and shoots, especially under 80-800 μmol/L treatments (Fig. 4). The addition of REEs in the solution also decreases K, Mg, Cu and Zn concentrations in ramie roots, stems and leaves (Tables S6-S8). On the contrary, the concentrations of P in all the plant organs and Mo in the roots increase with the increasing REE concentrations in the solution. The addition of REEs in the solution has no obvious influence on the Fe concentrations in the roots, stems and leaves. The values in the matrix are the Pearson correlation coefficient (R) between element concentrations. The higher the R 2 value is, the higher degree of correlation (positive or negative) between the two element concentrations is.

The color of the box refers to the significance level of the correlation (p-value).

Discussion

Response of ramie to increasing REE concentrations

Investigations on plant tolerance to REEs have been often conducted on crop plants. These studies showed that crop plants always have relatively low tolerance to REEs. [START_REF] Adeel | Cryptic foot prints of rare earth elements on natural resources and living organisms[END_REF] However, in the REE hyperaccumulator plant Phytolacca americana, a decrease of plant biomass was recorded at mixed REE concentrations up to 50-100 μmol/L in the hydroponic solution (similar pH and nutrient composition with this study). [START_REF] Yuan | The accumulation and fractionation of rare earth elements in hydroponically grown Phytolacca americana[END_REF] In this study, increasing REE concentrations in solution have no adverse effect on ramie growth from 1.6 to 80 μmol/L, suggesting that ramie has a tolerance capacity similar to the REE hyperaccumulator plant P. americana, and much stronger than common crop plants.

The Ca, Mn and K concentrations in ramie roots and shoots are significantly decreased by the addition of REEs in the solution. It has been proposed that REEs might be taken up through K + , Ca 2+ ion channels, hence the addition of REEs might reduce the uptake of K and Ca. [START_REF] Han | Organic acids promote the uptake of lanthanum by barley roots[END_REF][START_REF] Yuan | The accumulation and fractionation of rare earth elements in hydroponically grown Phytolacca americana[END_REF] Studies also found that Mn and REEs were co-localized in the REE hyperaccumulator Dicranopteris linearis, [START_REF] Liu | Spatially-resolved localization of lanthanum and cerium in the rare earth element hyperaccumulator fern Dicranopteris linearis from China[END_REF] and the addition of LaCl3 could inhibit the uptake of Mn by P. americana, [START_REF] Xu | Preliminary analysis of manganese uptake mechanism in the hyperaccumulator Phytolacca americana L[END_REF] indicative of a competitive uptake of REEs and Mn in plants. The decrease of the nutrients (e.g. K, Ca, Mn, Mg, Zn) in ramie organs may be also ascribed to the damage of cells (e.g. cell members and chloroplast) under high REE concentrations. Some studies highlighted that the most significant toxic symptom of REEs is the inhibition of P uptake. [START_REF] Saatz | The influence of gadolinium and yttrium on biomass production and nutrient balance of maize plants[END_REF] However, an increase of P concentration with the addition of REEs is found in both ramie roots and shoots. Based on previous studies on wheat, REE fractionation (e.g. ∑LREE/∑HREE) between roots and solution was mainly controlled by cell wall absorption and phosphate precipitation. [START_REF] Ding | Fractionation mechanisms of rare earth elements (REEs) in hydroponic wheat: An application for metal accumulation by plants[END_REF] Moreover, electron microscopy and X-ray microanalysis have also shown deposits containing REEs and P in the extracellular space of root tissue of Cucumis sativus. [START_REF] Zhang | Comparative toxicity of nanoparticulate/bulk Yb2O3 and YbCl3 to cucumber (Cucumis sativus)[END_REF] The positive correlation between REEs and P is coherent with the assumption that phosphate precipitation might contribute to the detoxification and accumulation of REEs in both ramie roots and shoots. The positive relationship between Mo and REE concentrations in the roots is suggestive of Mo involving in dealing with REE stress in ramie, e.g. the potential formation of REE molybdate complexes. A similar increase of Mo concentration was also observed in the roots of maize under 10 μmol/L Y treatment. [START_REF] Saatz | The influence of gadolinium and yttrium on biomass production and nutrient balance of maize plants[END_REF] Moreover, Mo was considered to have a synergistic effect with P, [START_REF] Basak | Interaction of phosphorus and molybdenum in relation to uptake and utilization of molybdenum, phosphorus, zinc, copper and manganese by rice[END_REF] which might enhance the precipitation of REEs with P and/or Mo in plants. Molybdenum was also found to increase the expression level of genes related to heavy metal (Cd) uptake, transport and sequestration. [START_REF] Ali | CBF/DREB transcription factor genes play role in cadmium tolerance and phytoaccumulation in Ricinus communis under molybdenum treatments[END_REF][START_REF] Marwa | Can selenium and molybdenum restrain cadmium toxicity to pollen grains in Brassica napus?[END_REF] Though Mo concentrations in ramie shoots are not influenced by REE addition in the hydroponic experiment, high concentrations of REE exposure may lead to Mo and P deficiency within plants in the field. [START_REF] Hu | Combined effects of simulated acid rain and lanthanum chloride on chloroplast structure and functional elements in rice[END_REF] Nevertheless, the potential role of P and Mo in the uptake and accumulation of REEs in ramie remains to be proved.

REE accumulation and fractionation in ramie with increasing REE concentrations

Most studies focused on the effects of REEs individually on plants, however, the soils always contain varying concentrations of the 16 mixed REEs. [START_REF] Liang | A review of fractionations of rare earth elements in plants[END_REF][START_REF] Zhao | Ecological risk assessment of neodymium and yttrium on rare earth element mine sites in Ganzhou, China[END_REF] In ramie, the whole plant and roots show HREE enrichments under 1.6-16 μmol/L treatments or no significant fractionation under 80-800 μmol/L treatments, suggestive of a preferential uptake of HREEs, or an unselective uptake of REEs by ramie roots depending on the range of REE concentrations in solution. Lower ∑LREE/∑HREE ratios are recorded in the leaves than the stems. This preferential accumulation of HREE in ramie leaves under 1.6-80 μmol/L treatments could be ascribed to the preferential root-to-leaf transport of HREEs compared with LREEs. Previous studies have demonstrated that small organic acids in the xylem sap may play a key role in the HREE enrichment of wheat leaves. [START_REF] Ding | Fractionation mechanisms of rare earth elements (REEs) in hydroponic wheat: An application for metal accumulation by plants[END_REF] In the xylem saps of the REE hyperaccumulator plant P. americana, citric acid facilitated the transport of HREEs rather than LREEs, leading to a HREE enrichment in the leaves. [START_REF] Yuan | The accumulation and fractionation of rare earth elements in hydroponically grown Phytolacca americana[END_REF] Therefore, the significant HREE enrichment in ramie leaves could indicate that selective complexation by organic chelators contributes to the long distance transfer of HREEs in xylem. The decrease of the HREE enrichment pattern in the leaves under higher REE concentrations (160-800 μmol/L) could be a result of toxicity. The lower root-toshoot translocation factor of Sc compared to those of the lanthanides suggests that Sc tends to be restricted within the roots. Alternatively, Y behaves rather similarly with the lanthanides during the uptake and transport processes.

Brioschi et al. [START_REF] Brioschi | Transfer of rare earth elements (REE) from natural soil to plant systems: Implications for the environmental availability of anthropogenic REE[END_REF] speculated that the positive Ce anomaly in plant xylem sap could be associated to the formation of REE-oxyhydroxide particles during the transport process. However, no significant Ce anomaly is found during the transport process in our study. This means that the Ce positive anomalies in both the ramie roots and leaves are owing to the uptake processes.

One possible reason is that Ce (Ⅲ) may be oxided to CeO2 (Ⅳ) particles on the roots surface under aeration condition. These particles may be solubilized by organic acids secreted by ramie roots, or probably activate endocytosis of the root cells membrane, [START_REF] Wang | Arabinogalactan protein-rare earth element complexes activate plant endocytosis[END_REF] thus promoting the preferential adsorption and uptake of Ce. Though a large part of Ce exists as CeO2 in the soils and is thus less available than other REEs, the higher concentration of Ce and the preferential absorption of Ce on the roots may result in a positive Ce anomaly in the leaves of a series of plant species. [START_REF] Miao | Environmental biogeochemical characteristics of rare earth elements in soil and soil-grown plants of the Hetai goldfield, Guangdong Province, China[END_REF] Under acid soil condition, Thomas et al. [START_REF] Thomas | Rare earth elements (REEs): Effects on germination and growth of selected crop and native plant species[END_REF] found that Ce was the most toxic REE for the local plant species. Under hydroponic condition, Rodrigues et al. [START_REF] Rodrigues | Foliar application of rare earth elements on soybean (Glycine max (L)): Effects on biometrics and characterization of phytotoxicity[END_REF] also found that foliar application of Ce was more harmful than that of La.

Implications for phytoremediation

A suitable plant species for phytostabilization should exhibit high tolerance to pollutants, high biomass production and retain the pollutants in roots or in the rhizosphere. The REE concentrations in the soil solution (simulated by 0.001 mol/L CaCl2 extraction) of ionadsorption REE mine tailings in southern China ranged from 1.41 to 66 μmol/L, [START_REF] Liu | Reclamation with organic amendments and plants remodels the diversity and structure of bacterial community in ionadsorption rare earth element mine tailings[END_REF] which are generally within the REE tolerance range of ramie. Therefore, ramie might be a good candidate for the phytomanagement on these REE-rich mine soils. Rare earth elements easily form precipitates with P in the soil and soil solution. However, phosphate might also play an indispensable role in REE tolerance and detoxification within both ramie roots and shoots. Similarly, Mo may also involve in dealing with the high concentration of REEs within plants. The addition of P and Mo fertilizers could be a good management strategy for the phytostabilization on REE-contaminated soils using ramie. Cerium is though to be less soluble in the soils in many former studies. However, future phytoremediation practices should pay more attention to Ce, as the plants could take up more this REE than its neighbours. Ramie has the advantage of producing high amounts of biomass and fibers and developing an extensive root system. As a consequence, promoting the use of ramie in mine tailings may fulfill a dual purpose of stabilizing the soils and contaminants and providing a potential economic value. Field experiments are underway to confirm to use of ramie for phytostabilization of ionadsorption REE mine tailings.

Conclusions

The current hydroponic experiment revealed that ramie could adapt to REE concentrations up to 80 μmol/L, which is comparable to the REE hyperaccumulator plant P. americana. Moreover, REEs in ramie are mainly retained within the roots and the upward transfer is limited. Under high REE concentrations (160-800 μmol/L), the uptake of nutrients especially Ca and Mn by ramie is largely inhibited. On the contrary, P and Mo concentrations in ramie roots increase with the increase of REE concentration in the solution. Precipitation of P and Mo with REEs could be involved in the mechanisms developed by the plant to deal with the high concentrations of REE within ramie roots and shoots. Moreover, significant REE fractionations are recorded in ramie tissues during the uptake, transport and accumulation processes. The preferential adsorption and uptake of Ce by the roots and the preferential transport of HREEs by the stems could explain the positive Ce anomaly and HREE enrichment in ramie leaves, respectively. Our study supports the idea that ramie could be a good candidate for phytoremediation at REE-contaminated soils, e.g. ion-adsorption REE mine tailings with high concentrations of REEs in southern China.

Figure 1

 1 Figure 1 Variations in ramie root and shoot biomass in response to increasing REE concentrations in solution.

  study (1.6-80 μmol/L REEs), REE concentrations in the leaves range from 53 to 249 mg/kg. The translocation factors (REE concentrations in leaves / REE concentrations in roots) are lower for the treatments in the range of 1.6-80 μmol/L REEs (0.01-0.09) than for those in the range of 160-800 μmol/L REEs (0.08-0.16). The lowest translocation factor (0.01) is recorded at 80 μmol/L REE concentration in solution (TablesS3-S5).

Figure 2

 2 Figure 2 Rare earth element concentrations in ramie roots, stems and leaves with increasing REE concentrations in solution.

Figure 3

 3 Figure 3 Rare earth element fractionation in ramie tissues with increasing REE concentrations in solution: (a) the ratio of LREEs to HREEs and (b) the Ce anomaly (b). The ΣLREE/ΣHREE ratio in the hydroponic solution is 0.65 and is indicated by the dashed line, ΣLREE/ΣHREE >0.65 and <0.65 refers to a LREE and HREE enrichment in the plant, respectively, and *Ce >1 refers to positive Ce anomaly.

Figure 4

 4 Figure 4 Concentrations of nutrients in ramie leaves, stems and roots with increasing REE concentration in solution: (a) Ca concentrations, (b) Mn concentrations, (c) P concentrations and (d) Mo concentrations.

Figure 5

 5 Figure 5 Correlation matrices performed on elemental concentrations obtained under 0-800 μmol/L REE treatments in ramie roots (upper triangular matrix) and in leaves (lower triangular matrix).

Fig. S1

 S1 Fig. S1 Ramie growth in response to increasing REE concentrations in solution after 4 weeks treatment. The solution contains 0, 1.6, 8, 16, 80, 160, 400, 800 μmol/L REEs (each REE with equal molar concentration, i.e. 0, 0.1, 0.5, 1, 5, 10, 25, 50 μmol/L).

Fig. S2

 S2 Fig. S2 Rare earth element concentrations (REE distribution patterns) in (a) leaves, (b) stems and (c) roots with increasing REE concentrations in solution.

Fig. S3

 S3 Fig. S3 Rare earth element fractionation patterns during the uptake (a) and transport (b) processes.

Table S1 .

 S1 Root characteristic parameters of ramie in response to increasing REE concentrations in solution

	Treatment	Total Root Length	Root Surface Area	Root Volume
	(µmol/L of REEs)	(cm)	(cm 2 )	(cm 3 )
	0	1170±350 b	212±31.2 c	3.15±0.05 b
	1.6	1610±174 a	284±9.37 a	4.00±0.17 a
	8	1610±123 a	249±5.76 b	3.08±0.09 bc
	16	1290±33.9 b	252±3.36 b	3.94±0.21 a
	80	1100±71.4 b	199±19.3 c	2.85±0.37 bcd
	160	1050±34.3 b	193±4.72 c	2.82±0.05 bcd
	400	1270±14.2 b	202±16.1 c	2.57±0.38 cd
	800	1190±58.7 b	201±11.1 c	2.71±0.17 d

Table S2 .

 S2 Ramie tolerance index (TI) with increasing REE concentrations in solution

	Treatment (µmol/L of REEs)	Leaves	Stems	Roots
	0	1.00 a	1.00 b	1.00 abc
	1.6	1.02±0.28 a	1.13±0.31 b	1.50±0.49 a
	8	0.70±0.31 ab	1.76±1.00 ab	1.00±0.39 abc
	16	0.91±0.18 a	2.45±0.86 a	1.02±0.16 abc
	80	0.71±0.26 ab	1.97±0.89 ab	1.08±0.54 ab
	160	0.39±0.15 b	1.63±0.48 ab	0.65±0.19 b
	400	0.44±0.17 b	1.16±0.17 b	0.62±0.29 b
	800	0.38±0.05 b	1.27±0.63 ab	0.49±0.12 c

Table S3 .

 S3 Rare earth element concentrations in ramie leaves (mg/kg) with increasing REE concentrations in solution

	Treatment (µmol/L of REEs)	1.6	8	16	80	160	400	800
	La	13.0±2.00 bc	3.80±1.16 cd	2.58±0.46 de	2.04±1.00 e	68.2±7.70 b	77.2±33.4 b	178±103 a
	Ce	11.2±1.59 bc	4.08±1.15 cd	3.11±0.48 d	2.56±1.14 d	71.0±7.97 b	79.3±33.7 b	183±105 a
	Pr	7.16±1.18 bc	2.61±0.78 cd	1.95±0.34 de	1.52±0.78 e	45.7±5.18 b	51.0±21.3 b	118±68.0 a
	Nd	8.50±1.48 bc	3.13±0.99 cd	2.19±0.38 de	1.71±0.86 e	47.9±5.56 b	53.6±22.9 b	125±71.5 a
	Sm	7.26±1.20 bc	3.08±0.93 cd	2.24±0.46 de	1.75±0.86 e	49.2±5.77 b	55.2±23.2 b	128±73.8 a
	Eu	11.6±1.79 bc	4.89±1.50 cd	3.45±0.67 de	2.66±1.35 e	73.9±7.41 b	83.3±35.5 b	190±105 a
	Gd	15.8±2.44 bc	6.07±2.11 cd	3.65±0.64 de	2.75±1.28 e	77.2±8.22 b	86.5±37.7 b	203±117 a
	Tb	15.8±2.53 bc	6.36±2.24 cd	3.84±0.73 de	3.08±1.43 e	81.5±8.79 b	90.2±38.1 b	209±119 a
	Dy	12.1±1.81 bc	5.12±1.90 cd	2.92±0.54 de	2.42±1.07 e	58.0±7.36 b	64.3±27.9 b	152±86.5 a
	Ho	20.3±3.43 bc	9.06±3.64 cd	4.67±0.79 d	4.08±1.85 d	85.3±11.4 b	95.4±41.5 b	224±124 a
	Er	20.0±3.44 bc	9.87±4.05 cd	4.97±0.85 d	4.65±2.06 d	83.1±11.1 b	93.0±40.8 b	222±123 a
	Tm	19.0±3.19 bc	10.6±4.33 cd	5.31±0.87 d	5.50±2.35 d	85.9±10.6 b	97.4±42.7 b	226±124 a
	Yb	16.7±2.63 bc	10.5±4.25 cd	5.65±0.96 d	5.93±2.57 d	88.2±12.2 b	100±44.3 b	232±124 a
	Lu	17.1±2.88 bc	11.8±5.03 cd	5.99±1.04 d	6.64±2.99 d	89.1±12.9 b	101±45.1 ab	237±127 a
	Sc	0.72±0.39 f	1.43±0.11 e	1.99±0.78 de	3.36±1.25 cd	28.5±7.18 b	19.9±7.01 bc	56.3±17.4 a
	Y	13.9±2.17 bc	5.69±2.38 c	2.39±0.39 d	2.28±0.93 d	44.6±5.24 b	47.5±22.1 b	110±58.5 a
	∑REEs	210±33.5 bc	98.0±36.3 cd	56.9±10.0 d	52.9±23.3 d	1080±126 b	1200±516 b	2790±1550 a

Table S4 .

 S4 Rare earth element concentrations in ramie stems (mg/kg) with increasing REE concentrations in solution

	Treatment (µmol/L of REEs)	1.6	8	16	80	160	400	800
	La	27.6±4.62 bc	5.13±0.59 d	8.14±3.23 cd	9.59±9.81 d	362±286 ab	332±144 ab	1180±658 a
	Ce	20.4±3.07 bc	5.08±0.24 cd	7.58±2.72 cd	8.64±8.02 d	349±279 bc	330±141 ab	1180±640 a
	Pr	10.8±1.88 bc	2.82±0.16 c	4.26±1.77 c	4.84±4.88 c	218±176 ab	211±91.4 ab	758±406 a
	Nd	11.0±1.69 bc	3.31±0.27 c	4.44±1.65 c	5.00±4.95 c	226±190 ab	215±96.0 ab	778±419 a
	Sm	8.11±1.62 bc	2.90±0.20 c	4.04±1.58 c	4.61±4.49 c	219±182 ab	219±97.6 ab	819±441 a
	Eu	12.3±1.97 bc	4.49±0.17 c	6.44±2.28 c	7.22±7.16 c	329±267 ab	333±143 ab	1220±654 a
	Gd	14.1±2.11 bc	5.43±0.15 c	7.35±2.55 c	8.82±7.91 c	341±283 ab	335±150 ab	1260±703 a
	Tb	12.3±1.97 bc	5.43±0.15 c	7.75±2.61 c	9.66±8.16 c	356±288 ab	353±152 ab	1310±743 a
	Dy	7.96±0.95 bc	4.22±0.23 c	5.93±1.85 c	7.75±6.28 c	262±214 ab	252±112 ab	969±558 a
	Ho	10.8±1.51 bc	7.09±0.40 c	9.35±2.65 c	13.1±9.26 c	381±300 ab	367±158 ab	1390±813 a
	Er	9.41±1.09 b	7.09±0.40 b	9.49±2.77 b	13.4±9.12 b	376±301 a	360±155 a	1390±838 a
	Tm	7.62±0.97 c	6.93±0.24 c	9.04±2.46 c	13.3±8.27 bc	365±291 ab	355±151 ab	1360±824 a
	Yb	6.17±0.73 d	6.63±0.58 cd	8.73±2.44 cd	12.6±7.88 bc	359±291 ab	349±147 ab	1380±836 a
	Lu	6.00±0.67 d	7.39±0.69 cd	9.06±2.41 c	13.3±7.62 bc	357±284 b	362±153 b	1400±856 a
	Sc	1.32±1.51 d	4.21±5.62 cd	1.49±0.95 d	2.34±0.71 bcd	14.8±9.99 bc	24.2±17.3 ab	1500±64.4 a
	Y	8.50±1.39 bc	5.32±0.30 d	5.99±1.40 cd	9.23±5.90 cd	219±178 ab	202±88.0 ab	7660±454 a
	∑REEs	174±23.6 bc	83.5±6.95 c	109±34.8 c	143±110 c	4730±3820 ab	4600±1990 ab	17300±9880 a

Table S5 .

 S5 Rare earth element concentrations in ramie roots (mg/kg) with increasing REE concentrations in solution

	Treatment (µmol/L of REEs)	1.6	8	16	80	160	400	800
	La	160±31.7 d	434±67.4 abc	313±45.6 cd	463±141 abc	407±116 bc	562±173 ab	899±429 a
	Ce	169±32.8 d	546±83.0 bc	422±57.1 cd	618±174 ab	544±135 bc	719±217 ab	1130±468 a
	Pr	108±20.2 c	386±55.5 ab	308±39.4 bc	441±117 ab	370±94.1 ab	471±143 ab	696±302 a
	Nd	114±21.6 c	425±60.6 ab	341±42.3 bc	475±121 ab	402±103 ab	507±149 ab	743±316 a
	Sm	119±22.6 c	496±66.8 ab	416±42.8 bc	568±137 ab	480±112 ab	585±169 ab	833±345 a
	Eu	177±33.2 c	702±92.5 ab	583±59.3 bc	786±189 ab	653±147 ab	819±235 ab	1160±486 a
	Gd	180±33.2 c	644±72.3 ab	499±56.0 bc	672±185 ab	571±141 ab	739±211 ab	1090±494 a
	Tb	185±34.3 c	668±80.9 ab	513±56.8 bc	685±185 ab	592±142 ab	767±223 ab	1130±500 a
	Dy	131±26.4 c	466±56.4 ab	356±43.6 bc	465±131 ab	402±99.0 ab	525±154 ab	777±344 a
	Ho	184±38.5 c	626±73.9 ab	475±54.7 bc	613±174 ab	537±137 b	713±216 ab	1060±469 a
	Er	181±38.1 c	616±79.5 ab	465±47.3 bc	588±171 ab	511±131 ab	687±205 ab	1000±434 a
	Tm	183±40.4 c	649±83.2 ab	502±46.9 bc	612±173 ab	532±129 abc	707±209 ab	1020±436 a
	Yb	192±42.4 b	712±83.2 a	570±49.8 ab	674±179 a	582±136 ab	762±216 a	1070±459 a
	Lu	191±41.6 b	706±78.6 a	557±51.9 ab	655±162 a	569±128 ab	745±206 a	1060±463 a
	Sc	36.5±7.38 f	249±30.8 e	409±43.6 de	2420±156 cd	3820±725 bc	5330±1400 b	8430±2280 a
	Y	81.9±17.5 d	244±40.5 abc	176±29.7 cd	240±70.4 abc	219±59.9 bc	297±90.1 ab	496±235 a
	∑REEs	2390±474 e	8570±1090 cd	6900±760 d	11000±2340 bc	11200±2400 bc	15000±4210 ab	22600±7750 a

Table S6 .

 S6 Nutrient concentrations in ramie leaves (mg/kg) with increasing REE concentrations in solution

	Treatment (µmol/L of REEs)	0	1.6	8	16	80	160	400	800
	Na	266±48.8 bc	162±68.4 d	167±23.5 d	248±26.4 c	650±69.5 a	332±126 bc	254±16.3 c	449±202 ab
	Mg	8320±1050 a	8550±246 a	6630±855 ab	5190±261 b	6330±2250 ab	6240±151 ab	6850±1380 ab 6420±972 ab
	P	1710±217 ab	1390±340 b	2110±499 ab	1710±389 ab	1970±949 ab	2680±908 a	2920±358 a	2820±386 a
	S	3281±421 abc	2610±599 bcd	3480±686 abc	1900±474 d	2540±566 cd	3780±238 ab	3880±360 a	3780±794 ab
	K	26000±1570 ab 31400±5680 a	30100±2430 ab 18800±354 c	23800±4060 b	24600±1100 b	25000±3200 b 28300±4100 ab
	Ca	51300±5060 a	43300±2100 ab 26700±3200 bc 23200±1460 c	28700±9490 bc	24000±3970 c	26600±6930 bc 25100±4950 bc
	Mn	59.6±8.99 a	45.6±6.30 ab	16.9±3.77 c	15.9±1.04 c	19.4±9.46 c	19.0±1.99 bc	20.6±4.14 bc	20.6±3.76 bc
	Fe	160±8.91 a	137±12.7 a	151±23.3 a	160±20.7 a	260±229 a	179±9.99 a	171±26.9 a	171±15.8 a
	Cu	15.9±2.00 a	9.28±0.31 ab	6.63±1.64 bc	5.12±1.01 c	6.00±2.56 c	6.25±1.22 bc	7.09±1.56 bc	7.05±1.74 bc
	Zn	124±8.41 a	93.5±21.0 ab	54.4±11.6 bc	46.7±2.59 c	54.8±30.0 c	57.0±5.56 bc	64.9±19.1 bc	61.3±10.0 bc
	Mo	14.7±1.37 a	15.9±7.61 a	18.6±5.15 a	14.2±4.29 a	12.4±2.83 a	16.8±1.19 a	18.3±2.12 a	14.6±4.33 a

Table S7 .

 S7 Nutrient concentrations in ramie stems (mg/kg) with increasing REE concentrations in solution

	Treatment (µmol/L of REEs)	0	1.6	8	16	80	160	400	800
	Na	1000±443 a	738±599 ab	1210±323 a	198±76.4 b	388±68.9 ab	414±195 ab	500±71.9 ab	975±547 a
	Mg	4580±1330 a	4070±241 ab	2430±626 cd	1800±132 d	2990±727 abc	1850±174 d	2530±563 bcd	2490±616 cd
	P	1190±331 c	1080±121 c	1750±602 abc	1330±179 bc	1740±808 bc	2530±615 ab	2670±399 ab	3090±793 a
	S	1170±758 c	1300±82.0 bc	990±104 c	1440±286 abc	2370±349 a	2000±452 ab	1630±530 abc	2180±1020 ab
	K	51500±7960 a	44700±8060 ab	22500±6120 cd 13100±575 e	17700±3580 de 16700±7270 de 20600±3630 cde	32800±4800 bc
	Ca	32500±790 a	28000±2230 ab	10700±2090 c	7280±1320 d	11200±2820 bc 8560±2920 cd	11500±2370 bc	11800±2530 bc
	Mn	70.4±22.2 a	57.8±15.3 ab	16.8±6.76 cd	10.5±2.08 d	18.9±6.81 bc	14.1±3.16 cd	14.4±1.69 cd	16.5±2.23 bc
	Fe	204±27.1 a	195±81.5 ab	168±156 ab	111±8.27 ab	140±96.9 ab	127±29.6 ab	240±84.0 a	89.1±52.9 b
	Cu	13.8±2.08 a	7.00±1.96 ab	5.18±1.76 bc	4.09±0.15 c	5.79±2.06 bc	7.28±1.34 ab	7.76±2.16 ab	14.9±8.38 a
	Zn	253±10.9 a	245±65.0 a	87.4±26.8 b	50.7±5.46 c	75.5±28.3 bc	91.3±31.1 b	90.9±10.3 b	208±144 a
	Mo	7.05±0.88 a	6.68±2.35 a	7.30±3.27 a	5.25±0.43 a	5.21±3.68 a	7.44±2.31 a	8.44±1.21 a	9.52±2.75 a

Table S8 .

 S8 Nutrient concentrations in ramie roots (mg/kg) with increasing REE concentrations in solution

	Treatment (µmol/L of REEs)	0	1.6	8	16	80	160	400	800
	Na	318±17.8 ab	353±134 a	229±3.52 abcd	158±14.1 cd	149±61.4 d	204±29.0 bcd	261±75.9 abc	330±130 ab
	Mg	1940±517 ab	2020±471 a	1270±288 c	1220±67.4 c	1400±64.3 abc 1280±301 bc	1180±172 c	1120±185 c
	P	1640±117 cd	1530±267 d	2200±462 abc	2070±219 bc	2620±1160 ab	2800±449 ab	3080±757 ab	3440±724 a
	S	3360±322 a	1910±335 b	1780±276 b	1970±93.9 b	1990±144 b	2100±224 ab	1920±167 b	1870±237 b
	K	36500±2340 a 25800±2870 ab 14500±1110 bc 13100±950 cd 11800±2090 cd 13400±1380 cd 13000±460 cd 11500±2900 d
	Ca	6030±337 a	5900±302 a	3690±320 ab	3070±248 bc	3310±155 bc	2650±735 c	3350±266 bc	3420±721 bc
	Mn	70.2±25.1 a	40.5±1.97 ab	17.1±3.98 bc	12.1±1.16 c	12.9±2.42 c	12.7±2.05 c	13.6±3.75 c	16.0±3.73 bc
	Fe	1540±158 a	1110±265 ab	1370±157 ab	467±615 b	1320±242 ab	1430±251 a	1300±234 ab	1490±315 a
	Cu	26.7±1.00 a	12.7±1.13 cd	13.3±1.32 bcd	5.20±6.78 d	13.0±2.03 cd	14.1±1.20 bc	14.9±2.69 bc	17.8±2.67 ab
	Zn	512±140 a	246±24.8 ab	97.1±24.4 bc	81.0±17.3 bc	70.0±30.5 c	78.3±13.1 c	70.6±7.97 c	74.9±14.5 c
	Mo	57.7±3.93 de	36.8±9.50 f	47.9±5.10 ef	59.4±6.62 de	137±4.55 cd	197±33.2 bc	229±60.5 ab	295±70.9 a

Acknowledgements

The authors are indebted to the LabEx Ressources 21 for its financial support. The "International Joint Lab -ECOLAND" is acknowledged for supporting this work. The author C. Liu would like to thank the China Scholarship Council for PhD scholarship.

Foundation item: Project supported by the Key R&D Program of Jiangxi province (20192ACB70016), the National Natural Science Foundation of China (41771343), the 111 project (B18060) and the Sino-French Cai Yuanpei Programme (38896SC).

Author contributions

Chang Liu and Ye-Tao Tang conceived the study; Chang Liu, Mei-Na Guo and Yi-Ming Yang performed the experiments; Chang Liu, Yi-Ming Yang and Wen-Shen Liu analyzed the data, prepared tables and figures; Chang Liu, Wen-Shen Liu, Hermine Huot, Jean Louis Morel and Ye-Tao Tang wrote and reviewed the manuscript; Rong-Liang Qiu, Jean Louis Morel and Ye-Tao funded and supervised all aspects of the project.

Supplementary material

Responses of ramie (Boehmeria nivea L.) to increasing rare earth element (REE) concentrations in a hydroponic system Chang Liu a,b , Wenshen Liu a,c,d , Hermine Huot a,* , Yiming Yang a , Meina Guo a,b , Jean Louis Morel b , Yetao Tang a,c,d , Rongliang Qiu


Etienne Gourc
email: etienne.gourc@gmail.com

Christophe Vergez
email: vergez@lma.cnrs-mrs.fr

Pierre-Olivier Mattei
email: mattei@lma.cnrs-mrs.fr

Samy Missoum
email: smissoum@arizona.edu

Nonlinear dynamics of the wolf tone production

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Introduction

Players of bowed string instruments might experience difficulties to produce a steady tone of good quality for certain ranges of playing parameters, regardless of the quality of the instrument. This phenomenon, mainly occurring with cellos is generally referred to as the wolf tone in the musical community and is characterized by a distinct warbling sound. There have been many attempts to explain the phenomenon. Among the first, White [START_REF] White | The origin of the wolf-note in bowed string instruments[END_REF] showed that the wolf tone occurs when the played note coincides with a strong resonance of the body of the instrument. Almost at the same time, Raman [START_REF] Raman | Xliii. on the "wolf-note" in bowed stringed instruments[END_REF] recorded the motion of the string during the wolf tone using optical lever, and suggested that the sympathetic resonance of the body of the cello takes energy from the string, causing cyclic alternance of bowed string motion and was the first to introduce the notion of minimum bow force. This notion was later revisited by Schelleng [START_REF] Schelleng | The bowed string and the player[END_REF] who formulated maximum and minimum bow force limits as a function of the playing parameters for which Helmholtz motion can take place. In order to account for the dynamic of the body, Schelleng introduced a resistance parameter and showed that the minimum bow force is affected by this resistance. It is however quite difficult to give a physical interpretation of this resistance parameter and Woodhouse revisited the formula to incorporate the measured body behavior [START_REF] Woodhouse | On the playability of violins. part ii: Minimum bow force and transients[END_REF].

Since then, considerable efforts have been made to derive accurate models of bowed string instruments. Some of the necessary ingredients, such as the string bending stiffness, finite width of the bow or the frictional behavior of the rosin are presented in [START_REF] Woodhouse | The bowed string as we know it today[END_REF]. Inacio [START_REF] Inácio | Computational modelling of string-body interaction for the violin family and simulation of wolf notes[END_REF] developed a modal model of bowed string instrument able to incorporate measured or simulated body behavior. They were able to reproduce the behavior of the cello at the wolf region, and showed the dependence of the beating frequency on the bowing parameters. A linear stability analysis of this model is presented in [START_REF] Debut | What can we learn about the wolf phenomenon from a linearized analysis?[END_REF]. A similar model was used to investigate the effect of wolf killing device in [START_REF] Debut | Modelling and experiments on string/body coupling and the effectiveness of a cello wolf-killing device[END_REF]. Mansour used a digital wave-guide modelling approach to analyze the behavior of plucked and bowed string instruments [START_REF] Mansour | Enhanced wave-based modelling of musical strings. part 2: Bowed strings[END_REF]. They also revised the minimum bow force [START_REF] Mansour | On minimum bow force for bowed strings[END_REF] formula accounting for the fact that close to the wolf region, the force transmitted at the bridge departs from the assumed perfect sawtooth considered in earlier minimum bow force calculation. They revealed that when the played note is close to the body resonance frequency, the dependence of the minimum bow force on the playing parameters is complex and deviates from the original Schelleng's diagram.

The previous mentioned studies involve either strong hypothesis on the behavior of the system allowing to make analytical predictions or complex model needing intensive numerical calculation. Although bowed string instruments exhibit strong nonlinear behavior due to the essential nonlinear nature of friction, very few studies analyzed the bifurcation behavior of these systems. Friedlander studied a model of non-dissipative string with rigid terminations, bowed at single point at an integer fraction of the string length and showed that for this simple model, all periodic solutions are unstable [START_REF] Friedlander | On the oscillations of a bowed string[END_REF]. It has been shown later that this instability was due to the absence of dissipation on the model [START_REF] Woodhouse | On the stability of bowed string motion[END_REF]. In musical acoustics, methods dedicated to the analysis of nonlinear dynamical systems such as continuation methods have been mostly used to investigate the behavior of wind instruments. The software DDE-biftool [START_REF] Sieber | Dde-biftool manualbifurcation analysis of delay differential equations[END_REF] has been used to investigate the periodic solution and bifurcations of flute-like instruments in [START_REF] Terrien | Flute-like musical instruments: a toy model investigated through numerical continuation[END_REF]. The oscillation threshold of a clarinet as a function of model parameters has been investigated in [START_REF] Karkar | Oscillation threshold of a clarinet model: A numerical continuation approach[END_REF]. Properties of saxophone bifurcation diagrams have been investigated in [START_REF] Colinot | Multiple two-step oscillation regimes produced by the alto saxophone[END_REF][START_REF] Gilbert | Minimal blowing pressure allowing periodic oscillations in a simplified reed musical instrument model: Bouasse-benade prescription assessed through numerical continuation[END_REF]. A bifurcation diagram based classification of different models of trumpets has been proposed in [START_REF] Fréour | Numerical continuation of a physical model of brass instruments: Application to trumpet comparisons[END_REF]. To the authors' knowledge, the only study investigating the bifurcation behavior of bowed string instruments is presented in [START_REF] Vigué | Investigation of periodic solutions of a bowed string toy model[END_REF]. They used the continuation software Manlab [START_REF] Guillot | A generic and efficient taylor series-based continuation method using a quadratic recast of smooth nonlinear systems[END_REF] and a regularized friction model to compute the bifurcation diagram of a toy model of bowed string instrument consisting of two string modes. They showed the existence of period doubling bifurcation yielding to a first register Helmholtz motion. Although these methods are mostly suited to analyze low dimensional, simplified models, they allow one to extract essential features of the dynamics of the system. A key feature which certainly explains the lack of nonlinear analysis of bowed string instruments is the discontinuous nature of friction which is generally not compatible with available software.

The present paper is devoted to the analysis of string body interaction in the wolf tone region, using a simplified model of bowed string instrument. In the second section, the toy model of cello is presented. The conservative partial differential equation is projected, by using a standard Galerkin method, on a real modal basis, prior to adding modal damping. In the third section, numerical simulations are used to demonstrate that this simple model is able to reproduce warbling oscillations, characteristic of the wolf tone. In the fourth section, we present the mapping procedure to compute the periodic solutions and their stability. The special case of grazing bifurcation, encountered for non-smooth dynamical systems is also addressed. In section five, the bifurcation diagrams obtained for bowing parameters in the vicinity of 1 : 1 resonance between the string and the body are presented. It is shown that bifurcations of the periodic stick-slip motion are responsible for beating oscillations. Finally, we show that the bifurcations of periodic solutions may also be interpreted in term of minimum bow force.

Description of the model

The simplified cello model is depicted in Fig. 1. It consists of a linear string connected at the origin to a linear oscillator of mass m and stiffness k, representing the dynamic of the body. The string has a variable effective length x L ∈ [0, L], where L is the distance from the bridge to nut. It is subjected to a pre-stress tensile force N and its mass per unit length is denoted by ρ. The bow with constant velocity V b exerts a friction force at a distance x B from the origin.

Equation of motion and boundary conditions

Denoting by y(x, t) the transverse displacement of the string, the linearized equations of motion are given by N y (x, t) -ρÿ(x, t) + F N µ( ẏ(x, t))δ(x -x B ) = 0

(1)

The dots and the primes denote differentiation with respect to time and position, respectively. δ(x) is the Dirac delta, F N the normal bow force and µ( ẏ) the Coulomb's friction coefficient defined by

µ( ẏ)    = µ d if ẏ(x B , t) < V b ∈] -µ s , µ s [ if ẏ(x B , t) = V b = -µ d if ẏ(x B , t) > V b (2)
where µ d and µ s refer to the dynamic and static friction coefficient, respectively. The geometric boundary condition at x = x L (nut) is defined by y(x L , t) = 0. The mechanical boundary condition at x = 0 (bridge), describing the connection between the string and the body, is given by N y (0, t) = -ky(0, t) -mÿ(0, t)

Non-dimensional quantities are introduced as

x = x L , xL = x L L , xB = x B L , ỹ = y L , t = ω = 1 L N ρ t m = m ρL , k = kL N , F = F N N , V = V b ρ N (4)
Using Eq. ( 4), the non-dimensional equation of motion is given by (the tilde have been omitted for brevity)

y (x, t) -ÿ(x, t) + F µ( ẏ(x, t))δ(x -x B ) = 0 (5)
Now the dots and the primes denote differentiation with respect to the non-dimensional time and position, respectively. The non-dimensional mechanical boundary condition is given by y (0, t) = -ky(0, t) -mÿ(0, t)

Modal discretization

The partial differential equation of motion Eq. ( 5) is discretized using modal decomposition. Neglecting the contribution of the friction force, the transverse displacement of the string is obtained through the separation of variables

y(x, t) = Nm n=1 Y n (x)v n (t) (7)
where N m is the number of modes retained in the modal decomposition, v n (t) the modal displacement of the n th mode and Y n (x) the associated mode shape. The mode shapes are normalized using the following orthogonality condition

mY n (0)Y s (0) + x L 0 Y n (x)Y s (x)dx = δ ns (8)
where δ ns is the Kroenecker delta. The unknowns of the linear eigenproblem are found by imposing y(x, t) = Y n (x)e jωnt where j = √ -1. The mode shapes are given by

Y n (x) = C n sin(ω n x + φ n ), φ n = nπ -ω n x L (9)
The natural frequencies ω n are obtained as the solution of the following transcendental equation

(k -mω 2 n ) sin(ω n x L ) + ω n cos(ω n x L ) = 0 ( 10
)
For the sake of comparison, neglecting the effect of the bridge (i.e. considering a hinge-hinged boundary conditions), the natural frequencies and mode shapes satisfies

ω n = nπ x L , φ n = 0, C n = 2 x L (11)
Substituting Eq. ( 7) into Eq. ( 5), performing Galerkin decomposition and introducing modal damping, we obtain the following modal model of the cello

v + 2ζω v + ω 2 v = F µ( ẏ(x b , t))Y (x b ) (12)
where

v(t) = [v 1 (t), . . . , v Nm (t)] T , Y (x) = [Y 1 (x), . . . , Y Nm (x)] T and ω = diag([ω 1 , . . . , ω Nm ]), ζ = diag([ζ 1 , . . . , ζ Nm ]
) gather the natural frequencies and modal damping, respectively.

Slip and stick dynamics

Before describing the dynamics of the system during sliding and sticking phases of the motion, it is convenient to rewrite Eq. ( 12) in state space form. Introducing the state vector y = [v, v] T , Eq. ( 12) is expressed by

ẏ = Ay + bµ(y) (13)
where

A = 0 Nm×Nm I Nm×Nm -ω 2 -2ζω , b = 0 Nm×1 F Y (x B ) (14)
The friction coefficient is now expressed by

µ(y)    = µ d if h T y < V ∈] -µ s , µ s [ if h T y = V = -µ d if h T y > V , h = 0 Nm×1 Y (x B ) (15)

Stick motion

During stick motion, the following condition holds

h T y = V (16)
such that the string and the bow velocity are equal at the bowing location. As indicated by Eq. ( 15), the friction coefficient does not take a steady value during sticking phases of the motion. Instead, the friction coefficient µ(y) varies to counteract the non-friction forces acting on the string. Multiplying Eq. ( 13) by h T and substituting h T ẏ = 0 from Eq. ( 16), we have

µ(y) = Ly, L = - 1 h T b h T A (17)
The equation governing the dynamics of the system during the sticking phase of motion is obtained by substituting Eq. ( 17) into Eq. ( 13) as

ẏ = Ry, R = A + bL (18)
Equation ( 18) is linear and its solution can be expressed using matrix exponential as follows

y(t 1 ) = S(t 1 -t 0 )y(t 0 ), S(t 1 -t 0 ) = e (t 1 -t 0 )R (19)
Stick-slip transition occurs at t = t i if the friction coefficient expressed by Eq. ( 17) exceeds the static friction coefficient µ s giving

Ly(t i ) = ±µ s (20)

Slip motion

We denote by slip motion, motion occurring for h T y = V . Eq. ( 13) becomes

ẏ = Ay ± bµ d (21)
Equation ( 21) is linear and its solution can be expressed using matrix exponential as follows Bridge-nut string length

L = 0.
y(t 1 ) = M (t 1 -t 0 ) (y(t 0 ) ∓ p) ± p, M (t) = e tA , p = ∓A -1 bµ d (22)
Let us assume that at time t = t i we have h T y(t i ) = V such that the string velocity at the bowing point is equal to the bow velocity. Two scenarios are possible depending on the non friction forces acting on the string [START_REF] Kuznetsov | One-parameter bifurcations in planar filippov systems[END_REF]. If the non friction forces acting on the string exceed the dynamic friction force, another slip motion with opposite dynamic friction coefficient takes place. On the contrary, if the non friction forces acting on the string do not exceed the dynamic friction force, stick motion is triggered. These conditions are expressed by Slip-slip transition if

Ly(t i ) > µ d Slip-stick transition if Ly(t i ) ≤ µ d (23)

Numerical evidence

The physical parameters of the system, considering the C2 string of the cello extracted from [START_REF] Inácio | Computational modelling of string-body interaction for the violin family and simulation of wolf notes[END_REF] are gathered in Table 1. The relation between parameters given in Table 1 and the physical parameters in Eqs. (1-3) is described in Appendix A. For all the calculation, a modal damping ζ = 0.1% has been used for each string-body coupled modes.

The evolution of the natural frequencies of the coupled system for the parameters given in Table [START_REF] White | The origin of the wolf-note in bowed string instruments[END_REF] is depicted in Fig. 2(a). Solid lines (orange) correspond to the natural frequencies of the coupled system obtained by solving Eq. ( 10) while dashed lines (blue) correspond to the natural frequencies of the hinged-hinged string given by Eq. ( 11), and adimensional frequency of the body ω = k/m. It is observed that frequency coalescence is avoided by the occurrence of a frequency veering phenomenon [START_REF] Perkins | Comments on curve veering in eigenvalue problems[END_REF]. As depicted in Fig. 2(b,c), where the mode-shapes of the first two modes are plotted for x L = 0.33 and x L = 0.4, the veering phenomenon gives rise to modal hybridation. Close to frequency coalescence, for x L = 0.33, the first two modes have comparable string amplitude and out of phase motion at the origin (i.e. bridge location) having also comparable amplitude, suggesting modal hybridation. For x L = 0.4, away from frequency coalescence, the first mode shows large string displacement with small displacement at the origin, while the second mode shows comparatively small string amplitude with large bridge motion. This suggests localization of mode, the first one corresponding to string motion and the second one to body motion.

In order to perform numerical simulation of the equation of motion Eq. ( 12), the truncation order of the modal basis needs to be determined. It has been shown in [START_REF] Inácio | Computational modelling of string-body interaction for the violin family and simulation of wolf notes[END_REF] that a high number of mode (80) is needed to reach computational convergence. It seems difficult if not impossible to perform numerical continuation and bifurcation analysis on such a system as suggested by the intricate bifurcation diagram of a modal model of a bowed hinged-hinged string comprising only two modes [START_REF] Vigué | Investigation of periodic solutions of a bowed string toy model[END_REF]. In order to investigate the effect of modal hybridation, only the first two modes have been kept in the modal discretization aiming to show the main phenomena involved in the dynamics of the system. Using the same range of playing parameters as in [START_REF] Inácio | Computational modelling of string-body interaction for the violin family and simulation of wolf notes[END_REF], a numerical bifurcation diagram computed for a normal bow force of F N = 2N and a bow velocity of V b = 0.1ms -1 is depicted in Fig. 3 for increasing and decreasing x L around the 1 : 1 string-body resonance. On the vertical axis, the points correspond to the trajectory of the bridge (i.e. v(0, t) = n Y n (0)v n (t)) intersecting the Poincaré section defined by

S = {v ∈ R : v(0, t) = 0, v(0, t) > 0} (24)
Two branches of periodic solutions are observed. When x L is increasing (blue dots in Fig. 3), the solution jumps from one stable branch to the other stable branch close to x L = 0.349. The time history at the jump location is depicted in Fig. 4(a) and it is observed that the jump between the two stable periodic solutions causes deep amplitude modulation due to the low damping of the system. When x L is decreasing (orange dots in Fig. 3), the periodic solution loses its stability giving rise to complex aperiodic motion as depicted in Fig. 4(b). In this case, the motion is weakly chaotic suggesting the presence of a strange attractor. As x L is further decreasing, aperiodic motion is annihilated and the motion jumps back to the stable periodic branch close to x L = 0.334. The above observations yield the following remarks: (i) even considering only two modes, string body interaction may give rise to stable aperiodic motion, (ii) upward or downward glissando gives rise to different behavior, (iii) even if no stable aperiodic solution is observed during the forward glissando, the jump between two stable branches yields deeply modulated motion. Note that the remarks (ii,iii) are consistent with the results of numerical simulations presented in [START_REF] Inácio | Computational modelling of string-body interaction for the violin family and simulation of wolf notes[END_REF] who mentions that 1), F N = 2N and V b = 0.1ms -1 . Forward points (blue) are obtained when x L is increased from 0.32 to 0.36. Downward points (orange) are obtained when x L is decreased from 0.36 to 0.32 the wolf tone does not arise at exactly the same value for the upward and downward glissando.

Computation of periodic solutions

The knowledge of the periodic solution and associated bifurcations gives a deep insight into the dynamics of the system. Due to the presence of Coulomb friction, the system is piecewise smooth, which prevents the use of classical methods. This problem can be addressed by using a mapping procedure [START_REF] Lenci | Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation[END_REF].

Mapping

We look for periodic solutions of Eq. ( 12) with one slip and one stick motion per period denoted as single slip solution for brevity. It is described by slip motion from t = 0 to t = t 1 given by ( 21):

y I = M (t 1 )(y 0 -p) + p stick motion from t = t 1 to t = t 2 given by ( 18):

y II = S(t 2 )y I ( 25
)
where y 0 is the unknown initial condition vector and the unknown time instants t 1 and t 2 are be computed by taking into account slip-stick and stick-slip transition conditions such that slip stick transition given by ( 22): h 1 (y 0 , t 1 ) ≡ h T y I -V = 0 stick slip transition given by ( 19): h 2 (y 0 , t 1 , t 2 ) ≡ Ly II -µ s = 0 (26)

Note that the solution is valid if Ly I ≤ µ d and of course The first condition coming from Eq. ( 23) ensures that the event is a slip-stick transition and not a slip-slip transition. The second and third conditions ensure that no secondary events are present during slipping and sticking phases of the motion. Finally, periodic solutions satisfy the following condition g(y 0 , t 1 , t 2 ) ≡ y IIy 0 = 0 (28) System (26,28) is solved numerically using a Newton procedure for the triplet (y 0 , t 1 , t 2 ).

Ly(t 1 ) ≤ µ d h T y(t) < V for t ∈ [0, t 1 [ |Ly(t)| < µ s for t ∈ [t 1 , t 2 [ (27)

Regular bifurcation

In this section, we look for the stability of the single slip periodic solutions. This section is devoted to the analysis of classical codimension 1 bifurcations. The stability of the periodic solutions is determined by looking at the Floquet multipliers of the Poincaré return map, computed using a condensation procedure. Let us denote by ( ŷ0 , t1 , t2 ) a set of periodic solution. Adding perturbations yields to

y 0 = ŷ0 + δy 0 , t 1 = t1 + δt 1 , t 2 = t2 + δt 2 (29)
Note that the perturbed solution must also satisfy the event conditions h 1 and h 2 in Eq. ( 26), such that

h 1 ( ŷ0 + δy 0 , t1 + δt 1 ) = 0 h 2 ( ŷ0 + δy 0 , t1 + δt 1 , t2 + δt 2 ) = 0 (30)
Substituting perturbations into Eq. ( 28), taking into account the previous relation and expanding the result into first order Taylor series gives

  0 0 g( ŷ0 + δy 0 , t1 + δt 1 , t2 + δt 2 )   = J   δy 0 δt 1 δt 2   ( 31
)
where J is the matrix of partial derivatives evaluated at the periodic solution and is given by

J =        ∂h 1 ∂y 0 ∂h 1 ∂t 1 0 ∂h 2 ∂y 0 ∂h 2 ∂t 1 ∂h 2 ∂t 2 ∂g ∂y 0 ∂g ∂t 1 ∂g ∂t 2        (32)
Removing dependencies upon δt 1 and δt 2 (condensation procedure) gives

g( ŷ0 + δy 0 , t1 + δt 1 , t2 + δt 2 ) = ∂g ∂y 0 δy 0 - ∂g ∂t 1 ∂g ∂t 2    ∂h 1 ∂t 1 0 ∂h 2 ∂t 1 ∂h 2 ∂t 2    -1    ∂h 1 ∂y 0 ∂h 2 ∂y 0    δy 0 ( 33
)
Substituting g( ŷ0 + δy 0 , t1 + δt 1 , t2 + δt 2 ) = ỹIIy 0 -δy 0 , where ỹII is the perturbed return point of the map, into Eq. ( 33) and taking the derivative with respect to the perturbation δy 0 finally gives

d ỹII dδy 0 = I + ∂g ∂y 0 - ∂g ∂t 1 ∂g ∂t 2    ∂h 1 ∂t 1 0 ∂h 2 ∂t 1 ∂h 2 ∂t 2    -1    ∂h 1 ∂y 0 ∂h 2 ∂y 0    (34)
Eq. ( 34) represents the variation of return point subjected to small perturbations, in other words, the Jacobian matrix of the Poincaré return map, whose eigenvalues are the Floquet multipliers. The periodic solution is stable when all Floquet multipliers are located inside the unit circle and unstable otherwise. When a periodic solution loses its stability, the resulting solution depends on the manner in which the Floquet multiplier leaves the unit circle [START_REF] Nayfeh | Applied nonlinear dynamics: analytical, computational, and experimental methods[END_REF]. A Floquet multiplier leaving the unit circle through +1 corresponds to a fold bifurcation, a Floquet multiplier leaving the unit circle through -1 corresponds to a flip (or period doubling) bifurcation and if a pair of complex conjugate Floquet multipliers leave the unit circle it corresponds to a Neimark-Sacker bifurcation.

Grazing bifurcation

In addition to classical bifurcations, system (6) may also encounter grazing bifurcation associated with non-smooth dynamical systems. A grazing bifurcation occurs when the flow is tangent to the transition plane. We can distinguish two types of grazing bifurcation. The first one denoted as velocity grazing bifurcation arise when slip stick transition occurs tangentially or

h T y = V h T ẏ = 0 (35)
The second type of grazing bifurcation, denoted as force grazing bifurcation and occurs when stick slip transition is tangent as 1), F N = 2N and V b = 0.1ms -1 . Triangles and squares correspond to Neimark-Sacker and Grazing bifurcations, respectively. Solid and dashed lines correspond to stable and unstable solutions, respectively.

Ly = ±µ S L ẏ = 0 (36)
The computation of periodic stick slip regimes as well as the stability analysis have been embedded in a pseudo-arclength continuation scheme [START_REF] Kuznetsov | Elements of applied bifurcation theory[END_REF] to track branches of periodic solutions. The continuation of bifucation points has also been implemented using the minimally extended system approach described in [START_REF] Kuznetsov | Elements of applied bifurcation theory[END_REF].

Bifurcation behavior near the wolf note

The periodic solutions and associated bifurcations near 1 : 1 string body resonance are now analyzed. The bifurcation diagram of periodic stick slip regimes obtained using the same bowing parameters as those used for the numerical bifurcation diagram in Fig. 3 is depicted in Fig. 5. Solid and dashed lines correspond to stable and unstable periodic solution, respectively, and triangles and squares indicate Neimark-Sacker and velocity grazing bifurcation, respectively. Figures 5(a) and (b) are two different representations of the same result. Figure 5(a) shows the evolution of the pulsation of the stick-slip oscillation (ω osc = 2π/(t 1 + t 2 )) while Fig. 5(b) shows the evolution of the maximum amplitude of the motion at the bridge. Comparison between Fig. 5(a) and Fig. 2(a) shows that the frequency of the stick slip oscillations follows the natural frequency of the system. Figure 5(b) can be directly compared to Fig. 3. As a first general remark, the periodic solutions are divided in two distinct branches. A first branch is entirely stable and stops at x L = 0.35 due to a grazing bifurcation. The second presents a pair of Neimark-Sacker bifurcations at x L = (0.22, 0.344) between which the periodic solutions are unstable. The branch vanishes at x L = 0.21 due to the presence of a grazing bifurcation. The presence of these bifurcations explains the behavior observed with numerical simulations in Fig. 3. A wider picture of the system dynamics is depicted in Fig. 6 where the locus of the bifurcation points is depicted in the (x L , F N ) plane for the parameters given in Table [START_REF] White | The origin of the wolf-note in bowed string instruments[END_REF] and V b = 0.1ms -1 . Orange lines correspond to velocity grazing bifurcation, blue lines (bold and thin) to Neimark-Sacker bifurcation, purple lines to flip bifurcation and yellow lines to force grazing bifurcation. This bifurcation diagram is more easily understood by making the parallel with classical bifurcation diagram depicted in Fig. 7 where the same system parameters are used and only the bow normal force F N is changed, as indicated by the horizontal dashed lines in Fig. 6. The italic numbers indicate the bifurcation points to facilitate the comparison between Fig. 6 and Fig. 7.

(
The same notation as in Fig. 5 has been used and cross and circles corresponds to flip and force grazing bifurcations, respectively. When comparing Fig. 5(b) and Fig. 7(a) (F N has increased from 2N to 5N ), we observe that the small portion of stable branch for x L ∈ [0.21, 0.22] does not exist any more, and a Neimark-Sacker bifurcation point (labeled (2)) has emerged on the right stable branch corresponding to large amplitude motion of the bridge. Increasing the bow force to F N = 7N (Fig. 7(b)), a new pair of Neimark-Sacker bifurcations appears on the lower branch (labeled (4) and ( 5)) yielding to a stable portion of periodic stick-slip regime. Increasing the bow force to F N = 7.2N , a pair of Neimark-Sacker bifurcation is destroyed. The lower branch is now almost entirely stable and two stable periodic solutions coexist over a large range of x L . Again, increasing the normal bow force to F N = 8N , the upper branch splits into two branches due to the appearance of a force grazing bifurcation (labeled (14) and (15)). It is also observed that large bow force also gives rise to flip bifurcations (labeled (16)).

From the above results, it is tempting to make a parallel between the minimum bow force that limits the existence of stick-slip periodic oscillations, generally used to explain the occurrence of the wolf tone, and the bifurcations of periodic solutions. Note that if the dynamic of the body is ignored by considering one string mode with hinged-hinged boundary conditions, the only bifurcation that limits the existence of single slip motion is the grazing bifurcation depicted by the orange dash-dotted line in Fig. 6. It is observed that the locus of velocity grazing bifurcation in the coupled and uncoupled case is similar except close to the frequency coalescence between the string and the body resonance. The situation changes if we consider the dynamics of the coupled system. Let us consider that the player is performing a downward glissando, decreasing x L from 0.4. According to Fig. 7, single slip motion will lose its stability subsequent to the Neimark-Sacker bifurcation indicated by the bold blue line in Fig. 6. However, due to the S-shaped look of the Neimark-Sacker bifurcation curve, a small increase of the bow force from F N = 7 (Fig. 7(b)) to F N = 7.2 (Fig. 7(c)) greatly extends the stability range of single slip motion. The bow force at which this sudden increase of the playability domain occurs can be viewed as the minimum bow force under which the wolf tone phenomenon takes place. The classical Schelleng's minimum bow force diagram, shows that the minimum bow force increases when the bridge-bow distance (denoted by x B in the present model) decreases [START_REF] Schelleng | The bowed string and the player[END_REF]. Figure 8, depicting the locus of the bifurcation points (similar as in Fig. 6) for different values of x B , shows a similar trends. Although the diagrams presented in Fig. 6 and in Fig. 8 clearly present similarities, interesting changes are noticed. The stable portion of branch observed in Fig. 7(b), which is due to the horizontal S-shaped Neimark-Sacker bifurcation curve, exists for a larger range of normal bow force when x B diminishes but vanishes when x B = 0.7, showing a more complex dependence upon the bow bridge distance than suggested by Schelleng's diagram, as already mentioned by Mansour [START_REF] Mansour | On minimum bow force for bowed strings[END_REF].

Conclusion

The bifurcation behavior of a simple model of cello played in the wolf region has been investigated. The model consists of a two mode modal decomposition of a linear string, subjected to a coulomb friction force, terminated at its end with a mass spring oscillator representing the dynamic of the body. It is shown that exact frequency coalescence between the frequency of the string and the body is avoided by a veering phenomenon. The mode shape at the veering region shows hybrid mode with comparable string amplitude and out of phase motion of the body. Numerical simulations show the existence of jumps phenomenon close to 1 : 1 string body resonance as well as the existence of stable quasi-periodic responses. A mapping procedure that allows to compute the periodic solution and the stability of single slip motion hes been presented. This mapping procedure has been embedded in a continuation procedure allowing us to track branch of periodic solutions as well as branch of bifurcation points. Two scenario giving rise to warbling oscillation are explained by bifurcation of periodic solution. (i) a periodic solution will lose its stability subsequent to a grazing bifurcation and jump to the competing stable branch, causing transient modulation. (ii) the periodic solution loses its stability due to the presence of a Neimark-Sacker bifurcation that can give rise to stable quasi-periodic response. Finally, it is shown that the presence of these bifurcation limits the range of possible single slip motion giving a novel interpretation of minimum bow force criterion. Although the model used in this paper is greatly simplified, the use of dedicated nonlinear tools to analyze the motion of bowed string instruments can provide interesting information about the behavior of these systems.

Figure 1 :

 1 Figure 1: Simplified model of cello

Figure 2 :

 2 Figure 2: (a): Evolution of the natural frequencies as a function of x L . Solid lines (orange) refer to the coupled system and dashed lines (blue) to the uncoupled system. Vertical dot-dashed lines indicate two particular values x L = 0.33 and x L = 0.4. (b,c): Corresponding mode shapes for x L = 0.33 and x L = 0.4, respectively.

Figure 3 :

 3 Figure 3: Numerical bifurcation diagram for parameters given in Table (1), F N = 2N and V b = 0.1ms -1 . Forward points (blue) are obtained when x L is increased from 0.32 to 0.36. Downward points (orange) are obtained when x L is decreased from 0.36 to 0.32

Figure 4 :

 4 Figure 4: Time history of the bridge motion corresponding to particular points of Fig. 3. (a) x L = 0.349 for increasing x L . (b) x L = 0.34 for decreasing x L .

Figure 5 :

 5 Figure 5: Bifurcation diagram of periodic stick-slip regimes for parameters given in Table (1), F N = 2N and V b = 0.1ms -1 . Triangles and squares correspond to Neimark-Sacker and Grazing bifurcations, respectively. Solid and dashed lines correspond to stable and unstable solutions, respectively.

Figure 7 :

 7 Figure 7: Bifurcation diagram of periodic stick slip regimes for parameters given in Table (1), V b = 0.1ms -1 . (a): F N = 5, (b): F N = 7, (c): F N = 7.2 and (d): F N = 8. Triangles and squares correspond to Neimark-Sacker and velocity grazing bifurcations, respectively, cross and circles to flip and force grazing bifurcations, respectively. Solid and dashed lines corresponds to stable and unstable solutions, respectively.

Figure 8 :

 8 Figure 8: Locus of the bifurcation points in the (x L , F N ) plane for parameters given in Table (1) and V b = 0.1ms -1 . (a): x B = 0.5 and (b): x B = 0.7. Orange lines correspond to velocity grazing bifurcation, blue lines to Neimark-Sacker bifurcation and yellow lines to force grazing bifurcation.

Table 1 :

 1 7m Open string natural frequency ω string = 2π × 64.5Hz String mass per unit length ρ = 14 × 10 -3 kgm -1 Main body resonance frequency ω body = 2π × 196Hz Main body resonance damping ratio ζ body = 0.7% Main body resonance mobility M body = 0.163ms -1 N -1 Physical parameters extracted from Inacio, et al.[6]

	Bridge-bow distance	0.04m
	Static friction coefficient	0.4
	Dynamic friction coefficient	0.2

  Figure 6: Locus of the bifurcation points in the (x L , F N ) plane for parameters given in Table (1) and V b = 0.1ms -1 . Orange lines correspond to velocity grazing bifurcation, blue lines to Neimark-Sacker bifurcation, purple lines to flip bifurcation and yellow lines to force grazing bifurcation.

	(12)	(13) (14)	(15)	(16)		(17) (18)
	(9)					(10) (11)
		(4)		(5)	(6)	(7) (8)
						1) (2) (3)

Acknowledgements

This work has been carried out in the framework of the Labex MEC (Contract No. ANR-10-LABX-0092) and of the A*MIDEX project (Contract No. ANR-11-IDEX-0001-02), funded by the Investissements d'Avenir French Government program managed by the French National Research Agency (ANR).

Appendix A. Expression of physical parameters from parameter given in Table 1 The physical parameters of system Eqs. [START_REF] White | The origin of the wolf-note in bowed string instruments[END_REF][START_REF] Raman | Xliii. on the "wolf-note" in bowed stringed instruments[END_REF][START_REF] Schelleng | The bowed string and the player[END_REF] are not directly obtained from the identified parameters of the C2 string of a cello given in Table 1 and extracted from [START_REF] Inácio | Computational modelling of string-body interaction for the violin family and simulation of wolf notes[END_REF]. The pre-stress tensile force N is related to the natural pulsation ω string and full length L of the string as follows

In order to obtain meaningful values for the mass m and stiffness k of the body, used in the expression of the mechanical boundary conditions (3), the main body resonance mobility peak has been fitted by a damped spring-mass oscillator. The natural pulsation ω body , damping ratio ζ body and amplitude of the mobility peak M body are related to the mass m, stiffness k and damping c of the equivalent damped harmonic oscillator as follows The non-dimensional parameters used for computations are then obtained by using Eq. ( 4).