Imad Aharchaou

Armand Maul

Marie-Noëlle Pons

Danièle Pauly

Hélène Poirot B$

Justine Flayac

François Rodius

Philippe Rousselle

Maximilien Beuret

Eric Battaglia

Davide A L Vignati
email: david-anselmo.vignati@univ-lorraine.fr

Chemosphere

Effects and bioaccumulation of Cr(III), Cr(VI) and their mixture in the freshwater mussel Corbicula fluminea

Keywords: chromium speciation, speciation, mixture exposure, stable isotopes, regression analysis, biomarkers

Chromium has two main oxidation states, Cr(III) and Cr(VI), that can occur simultaneously in natural waters. Current consensus holds that Cr(VI) is of high ecotoxicological concern, but regards Cr(III) as poorly bioavailable and relatively non-toxic. In this work, the effects and bioaccumulation of Cr(III), Cr(VI) and their mixture were studied using the freshwater clam Corbicula fluminea as a model organism. Mixture exposures were carried out using solutions isotopically enriched in 50 Cr(III) or 53 Cr(VI), allowing to quantify the contribution of each redox form to total Cr accumulation in the clams. Following exposure to individual redox forms, Cr(III) accumulated preferentially in the digestive glands and Cr(VI) in the gills of C. fluminea. In mixture exposures, both redox forms accumulated mainly in the gills; the concentration of Cr(III) in the digestive glands being much lowered compared with individual exposures. Both oxidation states affected the expression of biomarkers related to energy reserves, cellular damage and mitochondrial functioning, as well as the expression of mRNA for detoxification genes. The observed effects differed between gills and digestive glands. The present study suggests that Cr(III) is a bioavailable and biologically active elemental species deserving more consideration by the ecotoxicological community.

Introduction

Chromium is a widespread environmental contaminant due to its use in several industrial processes (WHO, 2009;2013). In natural environments, Cr mostly occurs in two oxidation states, Cr(III) and Cr(VI), that have markedly different environmental and biological behaviour.

Chromium(VI) is a recognized human carcinogen and has a high environmental mobility, bioavailability and (eco)toxicity. These biogeochemical properties follow from chromate ions being the most common chemical species of Cr(VI) [START_REF] Gorny | Chromium behavior in aquatic environments: a review[END_REF]. Chromate ions are structural analogues of sulphates and can easily cross biological membranes via facilitated transport [START_REF] Viti | Molecular mechanisms of Cr(VI) resistance in bacteria and fungi[END_REF]. Once inside the cell, Cr(VI) undergoes rapid reduction to Cr(III). The reduction reactions cause oxidative stress damage and produce intracellular Cr(III) that can further react with nucleic acids causing mutagenic and genotoxic effects [START_REF] Kortenkamp | The reduction of chromate is a prerequisite of chromium binding to cell nuclei[END_REF][START_REF] Sawicka | Chromium (III) and chromium (VI) as important players in the induction of genotoxicity -current view[END_REF][START_REF] Zhitkovich | Importance of chromium-DNA adducts in mutagenicity and toxicity of chromium (VI)[END_REF]. Chromate ions also show limited reactivity towards environmental particles, which usually bear a net negative charge at pH values above 4 [START_REF] Warren | Biogeochemical control on metal behaviour in freshwater environments[END_REF]. Chromate ions can therefore undergo long-distance transport and migrate into groundwater [START_REF] Guo | High Hexavalent Chromium Concentration in Groundwater from a Deep Aquifer in the Baiyangdian Basin of the North China Plain[END_REF][START_REF] Izbicki | Cr(VI) occurrence and geochemistry in water from public-supply wells in California[END_REF][START_REF] Oze | Genesis of hexavalent chromium from natural sources in soil and groundwater[END_REF].

Chromium(III) is putatively less bioavailable and (eco)toxic, but increasing evidence suggests that it can also elicit adverse effects in humans as well as in aquatic and terrestrial organisms [START_REF] Aharchaou | Chromium hazard and risk assessment: New insights from a detailed speciation study in a standard test medium[END_REF][START_REF] Park | Contrasting effects of Cr(III) and Cr(VI) on lettuce grown in hydroponics and soil: Chromium and manganese speciation[END_REF][START_REF] Tsuchiyama | Increased levels of renal damage biomarkers caused by excess exposure to trivalent chromium in workers in tanneries[END_REF]. Mechanism of Cr(III) uptake are less studied than those for Cr(VI) and may involve Cr(III)-transferrin complexes, endocytosis of Cr(III)-bearing particles or uptake of hydrophobic Cr(III) complexes [START_REF] Beyersmann | Carcinogenic metal compounds: recent insight into molecular and cellular mechanisms[END_REF][START_REF] Viti | Molecular mechanisms of Cr(VI) resistance in bacteria and fungi[END_REF]. Effects of Cr(III) have also been reported on the structure and permeability of biological membranes [START_REF] Filice | Correlating Live Cell Viability with Membrane Permeability Disruption Induced by Trivalent Chromium[END_REF][START_REF] Suwalsky | Cr(III) exerts stronger structural effects than Cr(VI) on the human erythrocyte membrane and molecular models[END_REF]. Finally, Cr(III) is particle reactive and tends to adsorb onto suspended particulate matter and colloids eventually accumulating in bed sediments [START_REF] Dominik | Speciation and environmental fate of chromium in rivers contaminated with tannery effluents[END_REF][START_REF] Gustafsson | Chromium(III) Complexation to Natural Organic Matter: Mechanisms and Modeling[END_REF][START_REF] Liao | Formation and Transport of Cr(III)-NOM-Fe Colloids upon Reaction of Cr(VI) with NOM-Fe(II) Colloids at Anoxic-Oxic Interfaces[END_REF].

Although ecotoxicological studies are biased toward Cr(VI), risk assessment has been performed for both oxidation states (WHO, 2009;2013). The World Health Organization (WHO) proposes guideline values (Probable No Effect Concentrations -PNEC) of 10 µg L -1 for Cr(III) and 4 µg L -1 for Cr(VI). The Joint Research Centre of the European Commission proposed a PNEC of 1.8 µg L -1 for Cr(III) based on the result of a chronic test with the teleost zebrafish Danio rerio [START_REF] Gomez Cortes | Selection of substances for teh 3rd Watch List under the Water Framework Directive[END_REF]. Several studies also confirmed that both Cr(III) and Cr(VI) can accumulate in marine and freshwater organisms via aqueous and dietary exposure routes [START_REF] Aharchaou | Bioaccumulation and subcellular partitioning of Cr(III) and Cr(VI) in the freshwater green alga Chlamydomonas reinhardtii[END_REF][START_REF] Baumann | Modeling metal bioaccumulation in a deposit-feeding polychaete from labile sediment fractions and from pore water[END_REF][START_REF] Roditi | Rates and routes of trace elements uptake in zebra mussel[END_REF][START_REF] Semeniuk | Chromium uptake and adsorption in marine phytoplankton -Implications for the marine chromium cycle[END_REF][START_REF] Wang | Bioavailability of Cr(III) and Cr(VI) to marine mussels from solute and particulate pathways[END_REF]. The net bioaccumulation and biological effects of Cr in real field conditions therefore likely result from the combined uptake of Cr(III) and Cr(VI). The two oxidation states actually occur simultaneously, albeit in variable proportions, in natural field conditions [START_REF] Ferrari | Chromium bioavailability in aquatic systems impacted by tannery wastewaters. Part 2: New insights from laboratory and in situ testing with Chironomus riparius Meigen (Diptera, Chironomidae)[END_REF][START_REF] Gunkel-Grillon | Toxic chromium release from nickel mining sediments in surface waters, New Caledonia[END_REF][START_REF] Mcclain | Chromium fluxes and speciation in ultramafic catchments and global rivers[END_REF][START_REF] Saputro | Speciation of dissolved chromium and the mechanisms controlling its concentration in natural water[END_REF]. To the best of our knowledge, no existing laboratory study has yet examined the effects and bioavailability of Cr during such conditions of simultaneous exposure to Cr(III) and Cr(VI) that organisms often encounter in the field.

In principle, it is relatively easy to prepare laboratory exposure media containing both Cr redox forms. On the other hand, it remains far from straightforward to understand the relative contribution of Cr(III) and Cr(VI) to the total Cr body burden and biological responses of the exposed organisms. In aqueous matrices, including ecotoxicological test media, the redox speciation of Cr can be determined using ion Chromatography ICP-MS [START_REF] Aharchaou | Chromium hazard and risk assessment: New insights from a detailed speciation study in a standard test medium[END_REF][START_REF] Markiewicz | Chromium and its speciation in water samples by HPLC/ICP-MS -technique establishing metrological traceability: A review since 2000[END_REF][START_REF] Seby | Chromium speciation by hyphenation of high-performance liquid chromatography to inductively coupled plasma-mass spectrometry-study of the influence of interfering ions[END_REF]. However, the redox speciation of Cr in the exposure medium will not be retained in the test organisms, because Cr(VI) is rapidly reduced to Cr(III) in the intracellular environment [START_REF] Beyersmann | Carcinogenic metal compounds: recent insight into molecular and cellular mechanisms[END_REF][START_REF] Sawicka | Chromium (III) and chromium (VI) as important players in the induction of genotoxicity -current view[END_REF][START_REF] Zhitkovich | Importance of chromium-DNA adducts in mutagenicity and toxicity of chromium (VI)[END_REF]. Tracking the fate of Cr(III) and Cr(VI) from exposure solution to biological tissues is however possible using speciated, isotopically enriched solutions (IES). Speciated isotope dilution mass spectrometry has already been used to obtain information on the relative abundance of Cr(III) and Cr(VI) in solid matrices [START_REF] Martone | Determination of Chromium Species in Dietary Supplements Using Speciated Isotope Dilution Mass Spectrometry with Mass Balance[END_REF][START_REF] Séby | Critical assessment of hexavalent chromium species from different solid environmental, industrial and food matrices[END_REF][START_REF] Unceta | Chromium speciation in solid matrices and regulation: a review[END_REF]. Isotopically enriched solutions have also been used successfully to document the importance of dietary uptake of Cu and Zn by freshwater snails [START_REF] Croteau | Novel and Nontraditional Use of Stable Isotope Tracers To Study Metal Bioavailability from Natural Particles[END_REF]2017), but similar studies on Cr are still lacking.

Chromium has four naturally occurring stable isotopes ( 50 Cr = 4.345%, 52 Cr = 83.789%, 53 Cr = 9.501% and 54 Cr = 2.365%) that are widely studied in relation to cosmochemical, geochemical and environmental processes [START_REF] Cole | Encyclopedia of Geochemistry: A Comprehensive Reference Source on the Chemistry of the Earth[END_REF][START_REF] Rodríguez-González | Isotope dilution analysis for elemental speciation: a tutorial review[END_REF][START_REF] Wei | Biogeochemical cycle of chromium isotopes at the modern Earth's surface and its applications as a paleoenvironment proxy[END_REF]. Ecotoxicological studies on Cr ecotoxicity usually employ commercially available Cr(III) and Cr(VI) salts that contain all four isotopes at the corresponding natural abundances. However, materials enriched in one specific isotope and in a specific redox form are also available for purchase [START_REF] Martone | Determination of Chromium Species in Dietary Supplements Using Speciated Isotope Dilution Mass Spectrometry with Mass Balance[END_REF][START_REF] Novotnik | Preparation of Cr(VI) and Cr(III) isotopic spike solutions from 50Cr and 53Cr enriched oxides without the use of oxidizing and/or reducing agents[END_REF]. In the absence of redox interconversions between Cr(III) and Cr(VI) in the exposure medium, the fate of each isotope (i.e., redox form) can be followed from the exposure solution to the biological tissue via a simple ICP-MS analysis at the right m/z value. In the present study, specimen of the freshwater bivalve Corbicula fluminea were exposed to solutions amended with Cr(III) or Cr(VI) salts having natural isotopic abundances, or with a mixture of 50 Cr(III) + 53 Cr(VI). Corbicula fluminea is a widespread filterfeeder that is largely used as a sentinel organism for freshwater biomonitoring of metal accumulation and sublethal effects (de Oliveira and dos Reis Martinez, 2014; [START_REF] Marie | Metallothionein response to cadmium and zinc exposures compared in two freshwater bivalves, Dreissena polymorpha and Corbicula fluminea[END_REF][START_REF] Pereto | Corbicula fluminea: A sentinel species for urban Rare Earth Element origin[END_REF]. The specific objectives of the present study were to determine: (1) the effect of the individual Cr redox forms and of the 50 Cr(III) + 53 Cr(VI) mixture on the expression of several biomarkers and mRNA, and (2) the bioaccumulation of Cr in the bivalves' gills and digestive glands.

Materials and methods

Organism collection and acclimation

Individuals of Corbicula fluminea (size 2-4 cm) were collected in the Moselle River (France) at GPS coordinates 49°07'06.5'' N, 6°09'39.0'' E and kept until the end of sampling in unfiltered site water (pH = 8.0, conductivity = 1823 µS cm -1 , oxygen saturation = 103%). Upon return to the laboratory, organisms were kept in aerated water at 15 ± 0.1 °C and acclimated to the test medium (Cristaline® mineral water, spring Aurèle, see composition in Table S1) by replacing 1/3 site water with spring water at 24 h intervals over four days (Potet et al., 2016). The viability of organisms was checked at each medium renewal by testing their ability to close the valves after a gentle pressure on the ligament.

Exposure set-up

A preliminary test was performed to establish suitable exposure conditions for the measurement of sublethal effects and bioaccumulation of Cr. After acclimation (section 2.1), twelve individuals of C. fluminea were transferred to 3 L aquaria containing 1.5 L of Cristaline® mineral water amended with [START_REF] Barrera-Escorcia | Lipid peroxidation and metallothionein induction by chromium and cadmium in Oyster Crassostrea virginica (Gmelin) from Mandinga Lagoon, Veracruz[END_REF][START_REF]VI) 3.1 ± 0.1 1.8 ± 0.4 50 Cr(III) + 53 Cr(VI) 13.7 ± 1.6 3.8 ± 2.6 Cr(III) 2.8 ± 0.2 1.7 ± 0.2 Cr(III) 19.8 ± 5.8 5.5 ± 3.6 Cr(VI) 2.9 ± 0.3 1.6 ± 0.3 Cr(VI) 4.3 ± 1.7 5.8 ± 4.7 LOOH µmol/g prot Control 64.3 ± 9[END_REF]500,1000,2000 and 5000 µg L -1 of Cr(III) or Cr(VI). Trivalent Cr was added as CrCl3 • 6H2O (Alfa Aesar, purity 99.5 % min, crystalline, CAS 10060-12-5) and hexavalent Cr as K2Cr2O7 (VWR, purity 99.8 % min, CAS 7778-50-9). Both Cr salts had natural isotopic abundances (NIA). Exposure medium was continuously aerated by bubbling ambient air in each aquarium. Organisms were kept at 15 °C under artificial daylight (800-1000 Lux) and a 12 h light -12 h dark photoperiod. Organisms were not fed during the experiments. No mortality was observed across the entire concentration range for neither Cr(III) nor Cr(VI) (data not shown) and exposure concentrations for subsequent experiments were selected as described in the next paragraph.

In a first experiment (exp #1) for biomarker and mRNA measurements in the gills and digestive glands, organisms were exposed to 5, 50 and 500 µg L -1 of either Cr(III) or Cr(VI) using salts with NIA. Exposures lasted for 24 h or 96 h with daily renewal of exposure medium.

Chromium concentrations were selected according to three criteria: absence of mortality, potential for sublethal effects and prospective environmental relevance. Twelve acclimated organisms were used for each concentration and exposure time. Each Cr redox form and exposure time were matched to a batch of control organisms not exposed to Cr. Other experimental conditions were the same as previously described.

A second experiment (exp #2) examined the relationships between biomarker expression and Cr bioaccumulation in the gills and digestive glands of C. fluminea. Organisms were exposed to: 50 µg L -1 of Cr(III) with NIA, 50 µg L -1 of Cr(VI) with NIA, or a mixture of 50 µg L -1 of Cr(VI) + 50 µg L -1 of Cr(III) using IES of 53 Cr(VI) and 50 Cr(III). Both IES were obtained from Applied Isotope Technologies -AIT, Pittsburg, PA, USA: batch # SH1332, Lot # CR12222014JC; PN-20529, isotopic enrichment 97.7 % for 53 Cr(VI) and batch # 030129-04-A, Lot # Cr04272010A; PN-2; isotopic enrichment 97.3 % for 50 Cr(III). All tests lasted for 96 h and were carried out under the same conditions as described above with 12 individuals for each scenario plus two control batches of 12 individuals each. The selected concentrations were considered a good compromise between practical constraints (notably the availability of IES and analytical detection in biological tissues) and environmental relevance for contaminated environments [START_REF] Gunkel-Grillon | Toxic chromium release from nickel mining sediments in surface waters, New Caledonia[END_REF][START_REF] Hayzoun | Organic carbon, and major and trace element dynamic and fate in a large river subjected to poorly-regulated urban and industrial pressures (Sebou River, Morocco)[END_REF].

Chromium measurements in aqueous and biological matrices

In exp #1, total and total filterable Cr concentrations were measured at t = 0, 2, 4, 6 and 24 h (n = 1 to 3). Filtration was carried out using 0.45 µm membrane filters (Merck Millipore, Millex 33mm, PVDF, reference # SLHV033NB). For experiments lasting 96 h, measurements were carried out over a single renewal period of 24 h and considered representative for all exposure periods. Samples were acidified at 1 % v/v with concentrated nitric acid (65%, VWR, Suprapur) and analysed by Flame Atomic Absorption Spectroscopy (FAAS; Perkin-Elmer AAnalyst 100) or Graphite Furnace Flame Atomic Absorption (GFAAS; Varian spectra 880Z+). Measurement accuracy was verified using certified reference materials SPS-SW1 (2.00 ± 0.02 µg L -1 ) and WW1 (200 ± 1 µg L -1 ), both from Spectrapure Standards (Oslo, Norway).

In exp #2, Cr concentrations were measured by ICP-MS (iCAPQ, Thermo-Scientific) at t = 0 h and t = 24 h. Measurements were performed at m/z 52 for solutions amended with Cr salts having NIA and at m/z 50 and 53 for the 50 Cr(III) + 53 Cr(VI) mixture. All analyses were performed using Kinetic Energy Discrimination (KED) mode. Because the ICP-MS calibration was performed using solutions with NIA, concentration results for 50 Cr and 53 Cr were corrected to account for the use of IES. A correction for signal drift in the internal standard was also applied using the following formulas: 'baseline drift' is the ratio between raw counts of the internal standard ( 45 Sc) for a given sample divided by the average raw counts for the calibration standards (n = 7) at the beginning of the ICP-MS run. All Cr measurements in exposure solutions were performed on aliquots sampled from aquaria containing living organisms.

Chromium levels in gills and digestive glands of C. fluminea were assayed by ICP-MS at m/z 50, 52 and 53 using KED mode after mineralization with 0.5 mL of concentrated nitric acid (65% v/v, Suprapur, VWR) and 0.1 mL of H2O2 (30% v/v, Suprapur, VWR). Mineralization was carried out in acid-washed polystyrene tubes placed in a stove at 65 °C overnight. Raw analytical results were corrected as necessary using equations 1 to 3. Gills and digestive glands were recovered by immediately dissecting exposed organisms at the end of the exposure period.

Tissues were frozen at -20 °C and then freeze-dried over 24 h (CRIOS-80, Cryotech). The accuracy of the mineralization procedure was checked using the reference material TORT-2 (lobster hepatopancreas) from the National Research Council of Canada (NRC-CNRC). The certified value for total Cr was 0.77 ± 0.15 mg kg -1 (uncertainty expressed as 95% confidence limits). The ICP-MS detection and quantification limits were calculated as three and ten times the standard deviation of replicate analytical blanks (n = 14).

Biomarker expression

After each exposure period, gills and digestive glands were pooled from two individuals of the same batch to obtain sufficient material for biomarker measurement and stored at -80 °C pending analysis. A total of four samples per exposure conditions was therefore available for biomarker analysis. Immediately before analysis, samples were thawed on ice and their wet weight determined. Tissues were then transferred to microtubes (2 mL) and manually crushed in 50mM phosphate buffer (98% v/v, pH = 7.6) supplemented with 1% v/v of phenylmethylsulfonyl fluoride (PMSF) and 1% v/v of a 1 mM mixture of L-serine and borate to inhibit proteases [START_REF] Garaud | Multibiomarker assessment of cerium dioxide nanoparticle (nCeO2) sublethal effects on two freshwater invertebrates, Dreissena polymorpha and Gammarus roeseli[END_REF][START_REF] Sroda | Seasonal variability of antioxidant biomarkers and energy reserves in the freshwater gammarid Gammarus roeseli[END_REF]. The quantity of added buffer was adjusted to the mass of tissue based on a ratio of 4 mL of buffer to 500 mg of tissue (wet weight). Samples were then centrifuged for 5 min (250 g) at 4 °C. Part of the resulting tissue homogenate was used as such for biomarker analysis. A second aliquot of homogenate was again centrifuged for 20 min (1000 g) at 4 °C. The resulting supernatant was subject to a final centrifugation (50 min, 20000 g, 4 °C) to recover the cytosolic fraction for the measurements of enzymatic activities.

Tissue homogenates were used to assay markers of cellular damage (LOOH, Lipid Hydroperoxide; ACP, Acid phosphatase), the status of energy reserves (Prot1, proteins; TG, triglycerides), Cholesterol (Chol.), and the activity of the mitochondrial Electron Transport Systems (ETS). Other markers of oxidative stress (TAC, Total antioxidant capacity; GPx-tot, Glutathione peroxidase; GST, Glutathione S-transferase; CAT, Catalase), cellular damage (CSP-3, Caspase-3) and energy reserves (Prot2, proteins; LDH, Lactate dehydrogenase) were measured in the cytosolic fraction. All biomarkers were analysed using an automated spectrophotometer (Konelab 20 XTi, Thermo Scientific) according to the methods described in [START_REF] Garaud | Multibiomarker assessment of cerium dioxide nanoparticle (nCeO2) sublethal effects on two freshwater invertebrates, Dreissena polymorpha and Gammarus roeseli[END_REF], [START_REF] Bertrand | The influence of salinity on the fate and behavior of silver standardized nanomaterial and toxicity effects in the estuarine bivalve Scrobicularia plana[END_REF] and references therein.

RNA extraction and RT-qPCR analysis

Gills and digestive glands were excised from 3 individuals per condition after 96 h exposure (exp #1). Non-digestive tissues were carefully removed from digestive glands. Samples were then transferred separately into 1 mL of 4M guanidine thiocyanate and stored at -80° C. Total RNAs were extracted using a RNeasy mini kit (Qiagen, Hilden, Germany) following manufacturer's instructions. Genomic DNA was digested after extraction with DNAse I (Qiagen) and a new total RNA purification was performed with RNeasy mini kit. Total RNA purity and quantity were assessed by optical density measurements on a Nanodrop spectrophotometer (A260 nm and absorbance ratio 260/280 and 260/230). RNA integrity was evaluated by capillary electrophoresis using a Bioanalyzer 2100 (Agilent, CA, USA).

RT-qPCR was performed on 8 detoxification genes (GST-pi, CAT, Mn SOD, CuZn SOD, TR, MT, HSP70 and Se-GPX) and PS3 as reference gene using the methodologies described in Koehle-Divo et al. ( 2019) and the sequences reported in Table S2. cDNA was reverse-transcribed from total RNA (900 ng) using the Superscript IV Reverse Transcriptase with oligo(dT)18

according to the manufacturer recommendations (Invitrogen, Carlsbad, CA, USA). Real-time PCR was performed on a StepOnePlus PCR system using the Fast SYBR® green master mix (Applied Biosystem®, CA, USA). Amplifications were performed with 50 ng of template, or water (no template control), or 50 ng total RNA (no RT control) using 200 to 300 nM of primers in a final volume of 20 µL [START_REF] Koehle-Divo | A sub-individual multilevel approach for an integrative assessment of CuO nanoparticle effects on Corbicula fluminea[END_REF]. The cycling conditions were 20 s at 95°C, followed by 40 cycles of 10 s at 95°C (except for CuZn SOD, HSP70, PS3 and GST-pi: 3 s at 95°C) and 30 s at 60° C. A melting curve analysis (65° C-95° C) was performed to ensure the absence of artefacts and the expression of the genes of interest was normalized to that of PS3 using the 2 -ΔΔCt method.

Statistical methods and data treatment

Correlation circle graphs among biomarkers (exp #1 and 2) and detoxification genes (exp #1)

were obtained by Principal Component Analysis (PCA) using the R-package 'FactoMineR'.

Biomarkers response was modelled for both exp #1 and 2. For exp #2, accumulation of 'total chromium' was added to the list of modelled variables. The values, levels and coding of the factors considered in the models for the two experiments are presented in Table S3. The levels mentioned in Table S3 were then arranged according to a 2 2 x 4 (exp #1) and a 2 3 (exp #2) randomized factorial design with 4 and 3 replicates per combination of the factors in exp #1 and exp #2, respectively. The expression of the 8 mRNAs was also modelled for exp #1, but with a single exposure duration (96h) and three replicates for each factor combination. Parametric statistical methods and regression analysis were performed for biomarkers, mRNAs and Cr accumulation as described below. The homogeneity of variances, associated with all the factor combinations within the replicate groups, was preliminary verified using the modified Levene's test (i.e., Brown-Forsythe's test for homogeneity of variances) after log transformation of the explained variables.

In exp #1, the biomarkers' responses were modelled separately for gills and digestive glands using the following log-linear regression model: S3).

For modelling mRNA responses, Eq. 4a simplified to Eq. 4b because the model did not include exposure duration:

Log10 (mRNA) = β0 + β1Cr + β2Co + β12Cr*Co (Eq. 4b)
In exp #2, a log-linear regression model was used to describe the response of each biomarker (Eq. 5a) and of total Cr bioaccumulation (Eq. 5b) as a function of the tissue (T) and the presence/absence of trivalent (Cr3) and hexavalent (Cr6) chromium.

Log10 (biomarker) = β0 + β1T + β2Cr3 + β3Cr6 + β12T*Cr3 + β13T*Cr6 + β23Cr3*Cr6 + β123T*Cr3*Cr6 (Eq.5a) Log10 (total Cr) = β0 + β1T + β2Cr3 + β3Cr6 + β12T*Cr3 + β13T*Cr6 + β23Cr3*Cr6 + β123TxCr3*Cr6 (Eq.5b)
The testing of the regression coefficients (i.e., the βs') and the subsequent construction of reduced regression models was performed using a stepwise backward procedure with a significance threshold set at α = 0.01 for the statistical tests. All statistical analyses were performed using R software (version 3.0.0).

The Cr bioconcentration factors (BCF, in L g -1 ) were calculated as:

𝐵𝐶𝐹 = 𝐶 𝑏𝑖𝑜𝑡𝑎 𝐶 𝑤𝑎𝑡𝑒𝑟 (Eq. 6)
where Cbiota and Cwater are the concentrations measured in biological material (in µg g -1 ) and in water (in µg L -1 ), respectively.

Chromium speciation calculations were performed using VisualMinteq version 3.1. No changes were made to the default database and the precipitation of amorphous Cr(OH)3 was allowed as possible solid species. The average composition of Cristaline® mineral water (Table S1) was used to describe the characteristics of the aqueous matrix and the Debye-Hückel equation was used for activity correction. Total and filterable Cr concentrations at t = 0 were slightly lower than expected for solutions amended with 50 or 500 µg L -1 in exp #1 and within 15% of the expected values in the other cases (Table S4). Over the 24 h water renewal periods, total and filterable Cr levels varied by less than 15-20 % in the case of Cr(VI), but decreased by 70-95 % in the case of Cr(III) (Figure 1, Table S4). Measurements taken at t = 0.5, 2, 4 and 6 h during exp #1 suggest that the concentration decrease began early after Cr(III) addition into the exposure medium (Table S4).

Results and discussion

Chromium concentrations and behaviour in test solutions

The concentrations of the two Cr redox form showed a similar behaviour when added alone or in combination to the exposure medium (Figure 1; Table S4). A dilution error during the use of IES (see section 2.2) resulted in an initial total filterable Cr(III) concentration of 104 ± 1.9 µg L -1 instead of the targeted 50 µg L -1 value (Figure 1B). Once analytically identified, the higher initial S4 for tabulated data.

The decrease in total and filterable chromium concentrations in all solutions amended Cr(III)

agrees with the known tendency of Cr(III) to form insoluble Cr(OH)3 species at circumneutral pH [START_REF] Rai | Environmental chemistry of chromium[END_REF]. Speciation calculation predicted that approx. 81 % and 98 % of the added Cr(III) would precipitate for initial concentrations of 50 and 500 µg L -1 . According to VisualMinteq calculations, the theoretical maximum soluble Cr(III) concentration in Cristaline® water is about 10 µg L -1 at a pH of 7.4 (measured at the source); in reasonable agreement with experimentally measured Cr concentrations at 24 hours (Fig. 1 and Table S4). On the other hand, the model does not predict the formation of any precipitate for added Cr(III) concentrations of 5 µg L -1 , in disagreement with the experimental results (Table S4).

Because VisualMinteq is a thermodynamic model, the agreement between theoretical solubility and experimentally measured concentrations at t = 24 h suggests that the exposure system reached quasi equilibrium within 24 hours. The decrease in filterable Cr concentrations agrees with the hypothesis of a rapid formation of Cr(III) bearing particles in the test medium [START_REF] Aharchaou | Chromium hazard and risk assessment: New insights from a detailed speciation study in a standard test medium[END_REF]. On the other hand, the decrease in total Cr concentrations suggests that other phenomena also occurred during the experiments (e.g., adsorption onto the vessels' walls and on the organisms' shells). These phenomena may also explain the observed concentration decrease in solutions initially amended with 5 µg L -1 of Cr(III); i.e. a concentration below the theoretical solubility limit.

Chromium bioaccumulation and bioavailability

Chromium levels in control organisms were above the analytical quantification limits (see section 3.1) only for 52 Cr: 0.74 ± 0.17 µg g -1 for gills (G) and 0.32 ± 0.09 µg g -1 for digestive glands (DG). Results for Cr accumulation were therefore corrected for the corresponding blank values in specimen exposed to solutions prepared using Cr salts with NIA (for which measurements were performed at m/z 52). No blank corrections were necessary for specimen exposed to 50 Cr(III) and 53 Cr(VI) using IES. Duplicate analysis of the reference material TORT2

gave blank corrected Cr concentrations (measured at m/z = 52) of 0.643 and 0.740 µg g -1 corresponding to percentage recoveries of 83 and 96 %, respectively.

For specimen exposed to individual Cr redox forms, Cr(III) accumulation was higher in DG than in G (10.2 ± 5.5 vs. 2.98 ± 1.05 µg g -1 ), while the opposite situation was observed for Cr(VI)

(2.64 ± 1.11 vs. 12.9 ± 0.19 µg g -1 ) (Fig. 2). In organisms exposed to the Cr(III) and Cr(VI) mixture, Cr accumulation was higher in G than in DG for both redox forms (Mann-Whitney-Wilcoxon test; P=0.029). with a high explanatory power and predictive capacity (F test, P < 10 -4 ; r 2 = 0.954; coefficient of variation of the residuals = 34.11%). Except for the three-way interaction T*Cr3*Cr6 (P=0.03), all factors had a statistically significant effect (P<0.01) on the bioaccumulated total Cr concentration and could be ranked as follows: Cr6 ≥ Cr3*Cr6 ≥ Cr3 > T*Cr6 ≥ T > T*Cr3. The results of the fitted model agreed well with the experimentally measured total Cr concentrations (Table 1).

The fitted bioaccumulation model (Eq. 7) suggests that both direct and interaction effects control Cr(III) and Cr(VI) bioaccumulation by C. fluminea. The interaction effects between the Cr redox forms and the tissues (e.g., T*Cr3 and T*Cr6) agree with the experimental observation that exposure to Cr(III) or Cr(VI) led to a preferential accumulation of Cr in DG or G, respectively. Equation 7 also suggests the presence of an antagonistic effect between Cr(III) and Cr(VI) (e.g., Cr3*Cr6) with a magnitude (i.e., the value of the coefficient in Eq. 7) being comparable to the direct effects of both redox forms. This antagonistic effect is confirmed by the experimental results (Table 1). Specifically, the accumulation of Cr Although the cellular uptake mechanisms for Cr(VI) and Cr(III) are known to differ [START_REF] Beyersmann | Carcinogenic metal compounds: recent insight into molecular and cellular mechanisms[END_REF][START_REF] Viti | Molecular mechanisms of Cr(VI) resistance in bacteria and fungi[END_REF] [START_REF] Hu | Occurrence and speciation of polymeric chromium(III), monomeric chromium(III) and chromium(VI) in environmental samples[END_REF]. Previous studies on Cr bioaccumulation by other bivalves confirm the bioavailability of Cr(VI) and Cr(III) from dissolved or particulate pathways [START_REF] Parlak | Accumulation and Loss of Chromium by Mussels (M. galloprovincialis)[END_REF][START_REF] Roditi | Rates and routes of trace elements uptake in zebra mussel[END_REF][START_REF] Wang | Bioavailability of Cr(III) and Cr(VI) to marine mussels from solute and particulate pathways[END_REF]. Extending the isotopic approach to simultaneous exposure to dissolved Cr(VI) and particulate-bound Cr(III) would allow to better understand Cr bioaccumulation in real field conditions where simultaneous exposure to Cr(III)

and Cr(VI) can occur.

Table 1. Total chromium concentrations (95% confidence intervals are given in brackets) accumulated in gills and digestive glands for each combination of factors in experiment 2 as estimated by the fitted reduced model described in Eq. 7. The corresponding arithmetic means of measured concentrations (in italics) are reported for comparison purposes (see Table S5).

Nominal exposure concentrations were 0 or 50 µg L -1 except for '$' that was 100 µg L -1 (see Organisms exposed to Cr(III) showed a higher BCF in DG than in G (0.5 vs. 0.15 L g -1 ), while the opposite was true for Cr(VI) (0.05 vs. 0.22 L g -1 ). In exposures to the Cr(III) + Cr(VI) mixture, both redox forms showed higher BCFs in G than in DG: 0.08 vs. 0.02 for Cr(III) and 0.14 vs. 0.03 for Cr(VI) (all values in L g -1 ). Calculations of BCF for Cr(III) were done using the geometric means of the measured filterable concentrations at t = 0 and t = 24 h to account for the temporal decrease in Cr concentration (Fig. 1, Table S4). Existing studies on Cr accumulation in bivalves are biased toward Cr(VI) or do not specify Cr speciation in the exposure matrices (Table S6). Potet et al. (2016) exposed two species of dreissenids to Cr(VI) using experimental conditions similar to those adopted in the present study. Despite the differences in the measured Cr burdens, the BCF factors for the two dreissenids (Table S6) are similar to those observed in the present study for C. fluminea exposed to Cr(VI). de Oliveira and dos Reis Martinez (2014) exposed specimen of C. fluminea along a contamination gradient in a stream impacted by landfill leachate and observed BCF much higher than those reported in the present study (Table S6). We surmise that, during such in situ exposures, the bulk of accumulated Cr originated from particle ingestion rather than uptake of aqueous Cr species. Uptake of particle-bound Cr(III) has been confirmed using radiolabelled material in laboratory studies [START_REF] Roditi | Rates and routes of trace elements uptake in zebra mussel[END_REF][START_REF] Wang | Bioavailability of Cr(III) and Cr(VI) to marine mussels from solute and particulate pathways[END_REF]. Comparison with other published results is less straightforward because of interspecies differences and variability of exposure conditions (Table S6). Nonetheless, our results clearly highlight the need for more research on the bioavailability of Cr(III) or Cr(III) + Cr(VI) mixtures.

Biomarker responses

Biomarkers expression could be properly measured except for TAC in gills (both experiments)

and CAT in gills and digestive glands (exp #2 only). The expression of energy related biomarkers (Prot 1, Prot 2, TG, LDH), Chol and some markers of oxidative stress (TAC, GST and CAT) was higher in digestive glands (DG) than in gills (G) for both experiments and regardless of exposure conditions (Tables S7 andS8). Among the markers of cellular damage, ACP activities were similar in the two tissues, while CSP-3 levels were higher in G than in DG for both experiences.

Lipid hydroperoxide (LOOH) behaved differently in exp #1 (DG > G) and exp #2 (G > DG); most likely because of the changes in LOOH levels among the specimen of C. fluminea not expose to Cr between the two experiments (Tables S7 andS8). In some cases, the experimental values were too low for quantification (blank cells in Tables S7 andS8) and the corresponding results were excluded from further statistical analysis.

For exp #1, the first two PCA factors explained 58.2% and 66.3% of the inertia for G and DG, respectively. Apart from Chol, Prot1, Prot2, TG and CAT, LDH, the associations between the biomarkers differed between tissues (Fig. 3). Note however that CAT and LDH were negatively correlated in G and positively correlated in DG. ACP and CSP3 were associated with Chol, Prot1, Prot2, TG in G and with CAT and LDH in DG. LOOH was linked to either GST (G) or ETS (DG) and seemed to respond differently to Cr exposure compared with other biomarkers. In DG, the first PCA axis opposed some energy-related biomarkers (Prot, TG) and Chol to those indicative of oxidative or cellular damage (Fig. 3B).

Regression analysis did not yield statistically significant models for Chol, LDH, TG and, on a tissue-specific basis Prot 1, Prot 2, GST and LOOH (Table S9). These biomarkers appear unsuitable to detect adverse effects of Cr in G and/or DG of C. fluminea. Regression analysis was not performed for CAT and TAC because the missing values prevented comparisons between tissues and experiments. Statistically significant direct effects of Cr redox form and/or Cr concentration were observable for ACP and CSP3 in gills, while exposure duration had a direct effect on ACP in gills and ETS in both gills and digestive glands (Table S7). The responses of other biomarkers in exp #1 were mostly driven by interaction effects among exposure duration, Cr redox state and Cr exposure concentration (Table S9). In particular, interaction effects involving Cr(VI) or Cr(III) concentration were identified for Prot1, LOOH, and CSP3 in DG and Prot2, ACP, CSP3 and ETS in G (Table S9). In the other cases, interaction effects involved only exposure duration and/or Cr oxidation state. Interpreting interaction effects is a complex task and is better illustrated graphically for some biomarkers with the highest r 2 ; e.g. ETS and CSP3 in G and ETS and LOOH in DG (Table S9 and Fig. 4). The activities of ETS in G at 96 h decreased with increasing Cr(III) concentrations and increased with increasing Cr(VI) concentrations, but showed no clear trends at 24 h. The response of CSP3 to Cr(III) showed opposite trends at 24 h and 96 h, while CSP3 levels in organisms exposed to Cr(VI) increased with concentration regardless of exposure duration. In DG, ETS levels peaked in specimen exposed to 50µg L -1 Cr(III) for 96 h, but no clear trends [START_REF] Barmo | Pleiotropic effects of hexavalent chromium (CrVI) in Mytilus galloprovincialis digestive gland[END_REF][START_REF] Ciacci | Immunomodulation in Mytilus galloprovincialis by non-toxic doses of hexavalent Chromium[END_REF][START_REF] Ciacci | Effects of sublethal, environmentally relevant concentrations of hexavalent chromium in the gills of Mytilus galloprovincialis[END_REF]Potet et al., 2016). The present study suggests that Cr(III) can also affect biomarker expression in bivalves and should be considered as a biologically active chromium form deserving more attention in future studies.

For exp #2, neither Cr(III) nor Cr(VI) showed any direct effect on biomarker responses, but differences in biomarker expression existed between tissues. Expression levels of Chol, TG, Prot1, Prot2 and GST were significantly higher (P < 10 -4 ) in DG than in G, while the opposite (P < 0.01) was observed for LOOH, CSP3 and ETS (Table S9). The simultaneous exposure to Cr(III) and Cr(VI) did not seem to modify biomarkers responses compared with exposure to individual redox forms of Cr; at least for the selected aqueous exposure levels and the corresponding bioaccumulated concentrations (see section 3.5).

Gene expression

The expression levels of all mRNAs could be properly quantified except for Se-GPX in G.

The first two PCA factors explained 75% and 72% of the inertia for the G and DG, respectively (Fig. 5). The expression patterns of mRNAs differed between the two tissues. For example, the correlations between Mn SOD and CAT as well as TR and CuZn SOD were observed only in G, while Mn SOD and MT were correlated only in DG.

Regression models (Table S10) also pointed to tissue-dependent relationships between mRNA expression and Cr exposure. In the gills, expression of mRNAs for GST-pi, CAT, CuZn SOD, Mn SOD, MT and HSP70 increased with Cr(III) and slightly decreased with Cr(VI) concentrations. In digestive glands, significant regression models were obtained only for TR, MT and HSP70. The expression of TR and HSP70 increased with Cr concentration for both redox states, although the response was higher in the presence of Cr(VI). At the opposite, MT expression decreased with Cr concentration regardless of the element oxidation state (Table S10). Changes in the expression levels of GST-pi, CAT, MnSOD, CuZnSOD, MT, TR and HSP70

were driven by both Cr redox state and concentration in at least one tissue, while MT expression in digestive glands was driven only by Cr concentration (Table S10). The effects on mRNA expression are mostly marked in the gills compared to the digestive glands (Table S10).

Modifications of GST-pi, CAT, MT and HSP70 mRNA levels have been reported in the gills and digestive glands of Mytilus galloprovincialis exposed to Cr(VI) [START_REF] Ciacci | Effects of sublethal, environmentally relevant concentrations of hexavalent chromium in the gills of Mytilus galloprovincialis[END_REF]Franzellitti et al., 2012), while we are unaware of previous report for Cr(III) effects on mRNA expression in C.

fluminea or other bivalves. [START_REF] Ciacci | Effects of sublethal, environmentally relevant concentrations of hexavalent chromium in the gills of Mytilus galloprovincialis[END_REF] also observed that the effects of Cr(VI) on mRNA GST-pi, CAT, MT isoforms and HSP70 differed between male (decreased expression) and female (increased expression); a distinction that was not taken into account in the present study.

Conclusions

Both Cr(III) and Cr(VI) can accumulate in gills and digestive glands of the freshwater mussel

Corbicula fluminea and affect biomarkers' and mRNA expression. Future research on Cr in bivalves and, likely, other aquatic organisms should therefore consider both redox forms and not focus exclusively on Cr(VI). Exposure to Cr(III)+Cr(VI) mixtures does not markedly modify the accumulation of Cr(III) and Cr(VI) in gills compared with exposures to individual redox forms.

On the other hand, the presence of Cr(VI) seems to decrease Cr(III) accumulation in digestive glands. Typical laboratory-based scenarios may therefore miss possible interactions between Cr(VI) and Cr(III) when exposing organisms to individual Cr redox forms. This caveat applies to the study of Cr uptake, bioaccumulation and ecotoxicity in both laboratory and real field conditions. Studies under more diversified exposure scenarios, including exposure via dietary pathways, are required to confirm the general validity of this hypothesis.

The regression approach suggests that biomarkers for the activity of mitochondrial electron transport systems (ETS) and cellular damage (ACP and CSP-3) are the most suitable to study the effect of Cr on C. fluminea. However, the biomarker response varies depending on the selected tissue and must also consider the complex interactions between Cr oxidation state, Cr concentration and exposure duration. In the case of antioxidant genes, regression analysis suggests that CAT and MnSOD mRNA levels are the most relevant to study in the gills. For digestive glands, regression for mRNA levels of TR and HSP70 were the most significant. In all Table S8. Arithmetic means ± 1 standard deviation (n = 4; n = 3 for $) for the values of cellular biomarkers measured in gills and digestive glands during exp # 2.

Table S9. Regression analysis of the biomarkers' expression Table S10. Regression analysis of the mRNA expression 15), 15174-15186. 3 Chen H, Zha J, Yuan L, Wang Z. 2015. Effects of fluoxetine on behavior, antioxidant enzyme systems, and multixenobiotic resistance in the Asian clam Corbicula fluminea. Chemosphere. 119, 856-62. 4 Unpublished probes 5 Navarro A, Faria M, Barata C, Piña B. 2011. Transcriptional response of stress genes to metal exposure in zebra mussel larvae and adults. Environ Pollut. 159, 100-7.

Table S3. Factors, levels and coding values used for regression modelling in the two experiments. Tissue was not considered as a factor in experiment 1 because data for gills and digestive glands were treated separately, while only one exposure duration (96 h) was used in experiment 2 (see text, section 2.2, for details). In the column 'Factor', Cr3 and Cr6 are the chosen variable names for the corresponding Cr redox species. Table S9. Regression analysis of the biomarkers' expression (after Log10 transformation) as a function of the different explanatory variables: exposure duration (D), Cr redox form (Cr), exposure concentration (Co), tissue (T) and their interactions (*). A probability threshold of 0.01 was used for the construction of all fitted models. r 2 : coefficient of determination.

  Cr concentrations corrected for isotopic abundances and for the ICP-MS baseline drift (see below); xx Cr ICP-MSsoftware are the Cr concentrations calculated by the ICP-MS software based on normal isotopic abundances;

  Log10 (biomarker) = β0 + β1D + β2Cr + β3Co + β12D*Cr + β13D*Co + β23Cr*Co + β123D*Cr*Co (Eq. 4a) where, (D) is the exposure duration, (Cr) indicates the chromium redox form, (Co) the exposure concentration and the '*' sign indicates the interactions between factors (Table

Figure

  Figure of merits for FAAS and GFAAS were the same as previously reported(Aharchaou et

  Cr(III) concentration did not prevent the study of the interactions between Cr(III) and Cr(VI) on biomarker expression nor of the simultaneous bioaccumulation of Cr(III) and Cr(VI) by C. fluminea.

Figure 1 .

 1 Figure 1. Panel A: temporal evolution of total (NF) and filterable (F, < 0.45 µm) Cr concentrations in solutions amended with 50 µg L -1 of either Cr(III) or Cr(VI) with natural

Figure 2 .

 2 Figure 2. Arithmetic means of Cr concentrations accumulated in gills (hatched bars) and

  (VI) decreased by about 50% in both G and DG of organisms exposed to the Cr(III)+Cr(VI) mixture compared with the Cr(VI)-only exposure. The presence of Cr(VI) did not affect Cr(III) levels in G, but lowered Cr(III) accumulation in DG by about 10-fold. This antagonistic effect is unlikely to arise from differences in the uptake of the three Cr isotopes considered in the present study. Indeed, in specimen exposed to Cr(VI) or Cr(III) solutions prepared using salts with NIA, the coefficient of variation of the measured Cr concentrations obtained at m/z 50, 52 and 53 was between 8 and 12%; i.e., not large enough to account for the observed decreases in Cr(III) and Cr(VI) bioaccumulation during mixture exposure.

  Cr =50 Cr(III) + 53 Cr(VI); B, 50 Cr(III); C, 53 Cr(VI) ‡ Example of calculation using Eq. 7. Remembering that the actual numerical values of the factors Cr3, Cr6 and T in Eq. 7 correspond to the codes in TableS3, total Cr accumulation in gills (code = -1) for organisms exposed to 50 µg L -1 of Cr(III) (code = +1) and 0 µg L -1 of Cr(VI) (code = -1) leads to the following development for Eq. 7: Log10(total Cr)=0.405 -[0; or 3.51 µg g -1 on a linear scale.

Figure 3 .

 3 Figure 3. Correlation circle graphs of the PCA (A: gills; B: digestive glands) for the biomarker

  [START_REF] Wang | Cr accumulation and its effects on nutrient elements uptake and oxidative response in Corbicula fluminea exposed to hexavalent chromium[END_REF] observed that exposure to Cr(VI) increased lipid peroxidation and decreased catalase activity in C. fluminea. Effects of Cr(VI) exposure on various biomarkers have been previously reported in Mytilus galloprovincialis and Dreissena spp.

Figure 5 .

 5 Figure 5. Correlation circle graphs of the PCA (A: gills; B: digestive glands) for the mRNA

3 1

 3 Arini A, Daffe G, Gonzalez P, Feurtet-Mazel A, Baudrimont M. 2014. Detoxification and recovery capacities of Corbicula fluminea after an industrial metal contamination (Cd and Zn): a one-year depuration experiment. Environ Pollut. 192, 74-82. 2 Koehle-Divo V, Pain-Devin S, Bertrand C, Devin S, Mouneyrac C, Giamberini L, Sohm B. 2019. Corbicula fluminea gene expression modulated by CeO2 nanomaterials and salinity. ESPR. 26(

  TableS4, exp #2, for actual exposure concentrations in each scenario). Errors are expressed as ± 1 standard deviation. Analytical results for individual samples are given in TableS5. NIA, solutions with Natural Isotopic Abundances; IES, Isotopically Enriched Solutions.

	The fitted bioaccumulation model (Eq. 5b) obtained from experimental data was:
	Log10(total Cr) = 0.405 -0.164 T + 0.249 Cr3 + 0.286 Cr6 + 0.092 T*Cr3 -0.166 T*Cr6 -0.271
	Cr3*Cr6	(Eq. 7)

  , further research is required to understand the mechanisms behind the observed antagonist effect. Future studies should also clarify if the accumulation of Cr(III) solely reflects absorption of soluble elemental species inside the tissues or if adsorption/absorption of Cr-bearing nanoparticles (see section 3.1) also contributes to the Cr

body burdens of C. fluminea. The role of Cr-bearing nanoparticles in Cr(III) bioaccumulation may have a real environmental relevance because polymeric Cr(III) species can occur in environmental samples

  section 2.2). Concentrations in organisms exposed to either Cr(III) or Cr(VI) have been corrected for the corresponding blank values of control organisms, while no blank corrections were necessary for organisms exposed to the Cr(III)+Cr(VI) mixture (see section 3.2).

				Total accumulated Cr concentration (µg g -1 )
	Cr(III) Cr(VI)		Gills	Digestive glands
			Mean	95% CI	Mean	95% CI
		Estimated	0.488	[0.335; 0.710]	0.323	[0.222; 0.471]
	no	no
		Measured	0.737	---	0.319	---
		Estimated	13.6	[9.36; 19.8]	1.96	[1.35; 2.85]
	no	yes
		Measured	12.9	---	2.64	---
		Estimated	3.51 ‡	[2.41; 5.1]	5.42	[3.72; 7.88]
	yes	no
		Measured	2.98	---	10.2	---
		Estimated	8.06	[5.54; 11.7]	2.70	[1.85; 3.93]
		Measured	8.80 A
	yes $	yes
			2.49 B

Table S1 :

 S1 Composition of Cristaline® mineral water (spring Aurèle) used as a test medium for experiments on biomarker activities in and Cr accumulation by specimen of Corbicula fluminea. All values are in mg L -1 .

	Element Concentration
	Ca	106
	Mg	4.2
	Na	3.5
	K	1.5
	HCO3	272
	Cl	3.8
	SO4	50
	NO3	1
	Si	8.3
	F -	0.9
	pH	7.2

Table S2 .

 S2 Primer sequences used in RT-qPCR analysis.

			Accession			Amplicon	Concentration
			Number			length	used for qPCR
	Gene Name	(GenBank)	Forward (5'-3')	Reverse (5'-3')	(bp)	(nM)	GO biological process Ref.
								Response	to
	Catalase	EF446609	CCTTCAGTGCGTAGTCTGG	CCTGTTCTCTGACCGTGG	196	200	oxidative stress	1
	Cu-Zn Superoxide						Response	to
	dismutase	KF218347.1 GAATGTTGTGGCTGATGAGG	GGATCGTTCTCCTGTCAATG	76	250	oxidative stress	2
								Response	to
	Se-GPx		EF547366.1 TATGCCCTGTTCCCGTGTTC	CTGCCATCTGGACCAATCAG	148	200	oxidative stress	3
			AN					Response to reactive
	GST pi		AY885667.1 CTGGGCGACCGAATACAAATC	GATGGCATTGGACTGGGCTA	90	250	oxygen species	3
								Cellular response to
								stress, heat, protein
	HSP70		KJ461738.1 AAGAAGGTTGGTGGTGAACG	TCCCAAATGTGTGTCTCCTG	121	250	refolding	4
								Cellular metal ion
								homeostasis, cellular
	Metallothionein						response to oxidative
	(isoform 1)	EF185126	CGGCTATCTCCCGCGA	AGCTTTTACCAGAACCAAACAGT	210	200	stress	1
	Mn	Superoxide						Response	to
	dismutase	EF446611	CTGTAGGTAGTAAGCGTGCTCC	GGATCATTTGACGAGATGAAGGC	200	200	oxidative stress	1
	Ribosomal protein						Translation (Used as
	S3		AJ517687	CAGTGTGAGTCCCTGAGATACAAG AAACGGTTGATAGGACGCAAG	154	250	housekeeping gene)	5
	Thioredoxin						Cell	redox
	reductase	KF275126	AATGCGGTGGCTTCCTACA	TCTGCTGGTCAAATCCTCGC	89	200	homeostasis

Table S4 .

 S4 Total and total filterable (< 0.45 µm) Cr concentrations in aliquots of Cristaline® water amended with different levels of either Cr(III) or Cr(VI) at different incubation times (in hours).

	Experiment 1. All values are in µg L -1 . Concentrations at t=0, 2, 6 and 24 hours are arithmetic means ± one standard deviations of triplicate
	measurements (duplicate measurements for t = 2). Concentrations at t = 0.5, 1 and 4 hours were measured only in one replicate. Bdl, below
	detection limit.
		Expected concentrations			Measured concentrations
			t = 0	t = 0.5 t = 1	t = 2	t = 4	t = 6	t = 24
	Cr(III) control unfiltered	0	0.19 ± 0.12	0.07	bdl	bdl	bdl 0.19 ± 0.27	1.0 ± 1.4
	Cr(III) control <0.45 µm	0	0.07 ± 0.11	bdl	bdl 0.35 ± 0.49 bdl 0.17 ± 0.28 0.97 ± 1.27
	Cr(VI) control unfiltered	0	0.07 ± 0.06	bdl	bdl	bdl	bdl 0.08 ± 0.10 0.42 ± 0.40
	Cr(VI) control <0.45 µm	0	0.10 ± 0.17	bdl	bdl	bdl	bdl 0.07 ± 0.06 0.24 ± 0.33
	Cr(III) unfiltered	5	4.59 ± 0.27	3.85	3.03 2.26 ± 0.23 1.41 1.64 ± 0.46 1.18 ± 0.29
	Cr(III) filtered 0.45 µm	5	4.44 ± 0.41	3.73	4.09 2.33 ± 0.18 1.37 1.54 ± 0.45 0.68 ± 0.46
	Cr(VI) unfiltered	5	4.32 ± 0.36	3.99	3.93 4.22 ± 0.67 3.97 4.32 ± 0.48 3.64 ± 0.44
	Cr(VI) filtered 0.45 µm	5	4.93 ± 0.35	4.24	4.32 4.48 ± 0.44 4.24 4.50 ± 0.42 4.13 ± 0.40
	Cr(III) unfiltered	50	31.3 ± 4.04	26	24	22.0 ± 2.8	15	17.3 ± 2.9 8.33 ± 2.08
	Cr(III) filtered 0.45 µm	50	30.7 ± 1.53	29	24	22.0 ± 1.4	13	14.7 ± 2.5 6.00 ± 1.00
	Cr(VI) unfiltered	50	36.0 ± 5.00	31	32	33.5 ± 4.9	30	32.7 ± 3.2	29.3 ± 3.1
	Cr(VI) filtered 0.45 µm	50	34.3 ± 2.52	31	33	31.5 ± 4.9	31	33.7 ± 3.2	30.3 ± 2.5
	Cr(III) unfiltered	500	434 ± 86.0	309	281	228 ± 2	136	124 ± 24	246 ± 210
	Cr(III) filtered 0.45 µm	500	362 ± 25.4	304	274	148 ± 16	83	55.7 ± 9.7	18 ± 5.6
	Cr(VI) unfiltered	500	363 ± 26.8	354	351	363 ± 30	327	361 ± 26	312 ± 32
	Cr(VI) filtered 0.45 µm	500	378 ± 20.8	361	351	364 ± 28	336	360 ± 28	318 ± 28

Table S5 .

 S5 Chromium concentrations accumulated in the gills (G) and digestive glands (DG) of individual specimen of Corbicula fluminea exposed for 96 hours to solutions amended with either Cr(III) or Cr(VI) salts having natural isotopic abundances (NIA) or to a solution simultaneously amended with a mixture of 50 Cr(III) and 53 Cr(VI) using isotopically enriched solutions (IES). n.m., isotope not measured.

	Cr species/isotopes	Exposure Concentration (µg L -1 )	Tissue	52 Cr (µg g -1 )	50 Cr(III) (µg g -1 )	53 Cr(VI) (µg g -1 )
				2.034	3.192	2.611
			Gills	4.381 2.332	5.069 3.105	4.615 2.727
	Cr(III) -NIA	50		3.161 4.998	3.469 5.097	3.222 4.257
			DG	6.191 13.250	7.455 13.984	6.398 11.140
				16.293	19.656	16.797
			Gills	13.058 12.783	12.445 11.099	11.856 11.057
	Cr(VI) -NIA	50		2.490	2.318	2.234
			DG	3.812	3.533	3.399
				1.613	1.656	1.508
				n.m.	2.528	8.038
			Gills	n.m. n.m.	2.923 1.891	7.692 5.582
	50 Cr(III) + 53 Cr(VI) -IES	100 + 50		n.m. n.m.	2.616 0.969	3.911 1.134
			DG	n.m. n.m.	0.232 1.394	1.437 2.002
				n.m.	0.354	0.539

Table S6 .

 S6 Comparison of Cr bioaccumulation (µg g -1 dry weight) in bivalves under different exposure conditions from published literature studies. $, exposure temperature = 12°C ; #, exposure temperature = 17°C ; & = µg g -1 wet weight ; BCF = Bioconcentration factor.

		Table S6 (continued)
		Organism	Acclimation (d)	Cr form	Cr (µg L -1 )	Duration (d)	Test medium	Tissue	Cr (µg g -1 )	BCF (L g -1 )
						1			3400 &	3400
		Organism Caelatura companyoi 4 Mytilus galloprovincialis 1 Crassostrea virginica 5 Dreissena Crassostrea virginica 6 Mytilus edulis 6	Acclimation (d) 15 3 23 1 month	Exposure conditions Cr form Cr (µg L -1 ) Unspecified 20000 Cr(VI) 0.001 0.01 0.05 unspecified 88 144 0.001 0.01 0.05 5 Cr(VI) 5 10 5 10 15	Duration (d) 2 5 10 15 20 4 7 12 weeks 4 $ 1	Unspecified Test medium Filtered Artificial seawater seawater Artificial seawater	Whole organism (no shell) Tissue DG G DG G DG Soft tissue (no DG) Whole organism (no shell)	Bioaccumulation 8400 16800 28200 42400 Cr (µg g -1 ) 65000 1.5 2.5 2.7 27.47 ± 18.65 13.89 ± 5.69 19.41 ± 2.87 28.19 ± 12.85 1 1.2 2.2 0.5 ± 0.2 3.12 ± 0.45 5.63 ± 1.15 4.83 ± 1.32 9.41 ± 3.37 1.9 ± 0.7 8.24 -22.59	4200 3360 2820 2827 BCF (L g -1 ) 3250 1500 250 54 0.31 0.16 0.13 0.2 1000 120 44 0.1 0.62 0.56 0.97 0.941 0.13 2.3 -21 A
		polymorpha 2 Dreissena r. Corbicula fluminea 7	4 0	Cr(VI) Cr total	5 15 0.70 -3.58 5 15	4 # 5 15 4 $ 30 1	Cristaline® mineral water In situ (river)	Whole G organism (no shell)	0.7 ± 0.1 1.3 ± 0.7 17.51 -31.44 29.83 -34.32 1 ± 0.2 0.9 ± 0.4 22.75 4.05 -9. 05	0.14 0.09 4.9 -45 A 8.3 -49 A 0.2 0.06 6.3 -32 A 1.1 -13 A
		bugensis 2			5 15 0.70 -3.58	4 # 5 15		DG	1.2 ± 0.3 1.2 ± 0.3 3.28 -14.81 4.80 -17.31	0.24 0.08 0.9 -21 A 1.3 -25 A
	945 946	Mytilus galloprovincialis 3 A, the BCF range covers the ratios (minimum accumulated concentration / maximum exposure concentration) and (maximum accumulated 7 Cr(VI) 0.5 28 Whole 1.4 ± 0.2 2.8 30 3.51 -18.22 1.0 -26 A Artificial organism 5 2.5 ± 0.4 0.5 seawater (no shell) 50 15.4 ± 6.1 0.31 concentration / minimum exposure concentration).
	947

Table S7 .

 S7 Arithmetic means ± 1 standard deviation (n = 4; n = 3 for $) for the values of cellular biomarkers measured in gills and digestive 964 glands of the freshwater bivalve Corbicula fluminea exposed to 0, 5, 50 and 500 µg L -1 of Cr(III) or Cr(VI) during 24 or 96 h (exp #1). w.w.

							Gills			Digestive glands
	965 966	weight. Biomarker	Exposure (Type)	Exposure (Duration)	0 µg L -1	5 µg L -1	50 µg L -1	500 µg L -1	0 µg L -1	5 µg L -1	50 µg L -1	, wet 500 µg L -1
	967	Biomarker LOOH µmol/g prot	Exposure (Type) Cr(III) Cr(VI) Cr(III) Cr(III) Cr(VI)	h Exposure (Duration) 24 h h	41.3 ± 2.5 34.4 ± 3.3 0 µg L -1 19.1 ± 3.5 33.1 ± 2.1 31.2 ± 2.9	37.7 ± 3.3 35.5 ± 2.5 5 µg L -1 18.7 ± 3.5 38.7 ± 2.0 40.7 ± 3.6	Gills 42.0 ± 10.6 33.4 ± 4.5 37.1 ± 4.4 50 µg L -1 17.3 ± 4.9 38.0 ± 2.8	30.4 ± 4.9 36.3 ± 2.9 500 µg L -1 18 ± 4.6 37.0 ± 3.1 39.4 ± 2.2	27.0 ± 2.5 24.9 ± 3.0 0 µg L -1 38.7 ± 1.5 21.0 ± 1.8 26.5 ± 2.7	Digestive glands 30.2 ± 2.0 24.9 ± 4.4 20.9 ± 1.5 23.8 ± 2.1 5 µg L -1 50 µg L -1 36.0 ± 6.4 37.1 ± 6.3 21.4 ± 1.6 24.0 ± 2.3 24.8 ± 3.6 26.7 ± 2.1	22.9 ± 1.1 24.3 ± 4.8 500 µg L -1 41.4 ± 4.8 27.2 ± 5.7 21.0 ± 1.0
		Protein content 1 (homogenate) mg/g w.w. ACP Protein content 2 (cytosolic fraction) mg/g w.w. µmol/g prot/h LDH Cholesterol mg/g w.w. µM NADPH/g prot/h Triglycerides mg/g w.w. GST µmol CDNB/ g prot/min CSP3 ETS mmol O2/g prot/h µmol pNA/g prot/h Catalase (CAT) mmole H2O2/g prot/min	Cr(VI) Cr(III) Cr(III) Cr(VI) Cr(VI) Cr(III) Cr(III) Cr(VI) Cr(VI) Cr(III) Cr(III) Cr(VI) Cr(VI) Cr(III) Cr(III) Cr(VI) Cr(VI) Cr(III) Cr(VI) Cr(III) Cr(III) Cr(VI) Cr(III) Cr(VI) Cr(VI) Cr(III) Cr(III) Cr(VI) Cr(VI) Cr(III) Cr(III) Cr(VI) Cr(VI) Cr(III) Cr(VI) Cr(III) Cr(VI) Cr(III) Cr(VI)	96 h h h 24 h 96 h h h 24 h 96 h h h 24 h 96 h h 24 h h 96 h h h	16.1 ± 4.8 20.4 ± 1.3 107 ± 17 117 ± 7 14.9 ± 2.0 9.7 ± 1.8 115 ± 5 95 ± 7 12.6 ± 1.8 15.1 ± 0.6 75 ± 13 114 ± 13 10 ± 0.7 1.0 ± 0.3 96 ± 10 179 ± 32 0.6 ± 0.2 0.7 ± 0.1 0.6 ± 0.1 1.3 ± 0.4 391 ± 46 440 ± 44 249 ± 11 348 ± 44 1 ± 0.5 1.2 ± 0.1 6.1 ± 2.0 4.0 ± 0.9 0.9 ± 0.1 2.3 ± 0.1 7.2 ± 1.7 3.1 ± 0.6 2.4 ± 0.1 2.1 ± 0.1 1.4 ± 0.1 51 ± 5.8 41 ± 1.3 41 ± 4.2 36 ± 13	18.9 ± 2.8 18.1 ± 1.2 116 ± 9 113 ± 8 15.7 ± 1.1 11.9 ± 0.8 106 ± 7 102 ± 8 12.9 ± 1.4 12.4 ± 1 100 ± 12 108 ± 17 10.9 ± 0.8 0.7 ± 0.2 90 ± 12 85 ± 3 0.7 ± 0.1 0.7 ± 0.1 0.6 ± 0.01 1.2 ± 0.2 397 ± 67 366 ± 25 277 ± 75 597 ± 87 1.2 ± 0.2 1.2 ± 0.1 10.2 ± 5 6.3 ± 0.5 0.9 ± 0.1 2.4 ± 0.3 6.3 ± 2.7 4.7 ± 1.1 2.2 ± 0.1 1.9 ± 0.2 2.3 ± 0.1 47 ± 2.8 40 ± 6.9 43 ± 2.7 49 ± 8.9	18.7 ± 1.3 16.9 ± 4.5 107 ± 16 113 ± 8 16.6 ± 1.2 12.3 ± 1.4 106 ± 7 105 ± 1 12.9 ± 1.2 10.7 ± 2.3 106 ± 32 98 ± 18 11.1 ± 0.8 0.6 ± 0.2 111 ± 58 75 ± 16 0.7 ± 0.1 0.6 ± 0.2 0.6 ± 0.01 1.1 ± 0.4 384 ± 170 301 ± 47 387 ± 106 443 ± 57 1 ± 0.1 1 ± 0.3 8.0 ± 5.5 5.9 ± 1.4 0.8 ± 0.1 2.3 ± 0.3 4.1 ± 0.8 5.3 ± 1.8 2.3 ± 0.1 1.8 ± 0.4 2.2 ± 0.2 42 ± 7.7 37 ± 4.2 42 ± 17 45 ± 4.3	18.1 ± 1 13.4 ± 2.4 108 ± 10 104 ± 6 16.2 ± 2.6 11.8 ± 0.3 92 ± 9 95 ± 6 12.6 ± 1.1 8.7 ± 1.4 108 ± 33 103 ± 18 11.3 ± 1.6 0.6 ± 0.1 183 ± 32 120 ± 16 0.7 ± 0.1 0.5 ±0.1 0.6 ± 0.1 1.2 ± 0.5 367 ± 114 303 ± 35 329 ± 65 441 ± 165 0.9 ± 0.2 0.8 ± 0.2 18.7 ± 6.5 10.5 ± 5.2 0.9 ± 0.2 2.3 ± 0.2 3.0 ± 1.2 9.5 ± 3.5 2.2 ± 0.1 1.4 ± 0.3 2.2 ± 0.3 43 ± 5.5 39 ± 3.5 34 ± 11 46 ± 9.3	35.9 ± 7.2 39.8 ± 2.5 81 ± 37 132 ± 47 36.8 ± 5.8 27.8 ± 2.5 103 ± 31 84 ± 24 28.2 ± 5.8 29.3 ± 2.4 129 ± 27 145 ± 39 25.6 ± 2.2 1.5 ± 0.2 129 ± 71 90 ± 27 1.5 ± 0.3 1.6 ± 0.2 1.6 ± 0.3 4.5 ± 0.6 708 ± 190 710 ± 221 607 ± 173 558 ± 275 3.8 ± 0.8 5.5 ± 0.2 4.9 ± 3.3 2.9 ± 1.6 5.0 ± 1.1 1.1 ± 0.2 9.4 ± 7.9 1.2 ± 2.7 1.8 ± 0.4 1.6 ± 0.1 1.8 ± 0.3 72 ± 15 61 ± 46 63 ± 22 39 ± 33	44.1 ± 7 37.2 ± 5.2 89 ± 28 124 ± 15 34.8 ± 4.2 25.5 ± 3.5 111 ± 33 87 ± 16 31.5 ± 5.9 26.3 ± 2.8 116 ± 30 133 ± 32 24.7 ± 2.4 1.4 ± 0.3 94 ± 73 138 ± 50 1.7 ± 0.2 1.5 ± 0.3 1.6 ± 0.2 5.2 ± 1.8 887 ± 208 632 ± 144 643 ± 205 701 ± 173 6.1 ± 0.8 5.2 ± 1.2 6.3 ± 4.2 10 ± 5.2 4.8 ± 0.7 1.1 ± 0.1 6.8 ± 7.7 2.6 ± 2.6 1.5 ± 0.2 1.8 ± 0.2 1.7 ± 0.2 89 ± 25 82 ± 16 52 ± 22 67 ± 37	37.9 ± 4.8 34.8 ± 7.6 120 ± 41 132 ± 4 34.7 ± 5.8 26.9 ± 4.7 104 ± 38 122 ± 37 25.7 ± 2.6 25.4 ± 6.4 157 ± 78 107 ± 48 24.3 ± 4 1.4 ± 0.4 130 ± 77 156 ± 79 1.5 ± 0.1 1.5 ± 0.4 1.5 ± 0.2 4.1 ± 0.9 955 ± 611 746.7 ± 43 612 ± 260 841 ± 337 6.1 ± 1 4.8 ± 0.9 7.6 ± 7.5 8.5 ± 6.2 4.7 ± 0.7 1.3 ± 0.2 4.8 ± 7.1 5.8 ± 6.4 1.7 ± 0.2 3.6 ± 1.4 1.9 ± 0.3 115 ± 39 $ 78 ± 43 50 ± 27 $ 67 ± 21	31.0 ± 5.3 99 ± 27 125 ± 23 33.3 ± 10 39.8 ± 3.5 113 ± 29 88 ± 17 30.4 ± 1.9 23.6 ± 5.1 121 ± 50 175 ± 40 24.9 ± 6.8 25.8 ± 2.3 93 ± 34 112 ± 35 1.6 ± 0.3 1.5 ± 0.2 1.5 ± 0.3 1.6 ± 0.2 583 ± 146 1068 ± 205 605 ± 80 695 ± 174 4.7 ± 0.6 4.5 ± 1.1 7.1 ± 5.7 19 ± 8.4 4.9 ± 1.5 4.0 ± 0.4 1.7 ± 1.3 1.2 ± 1 1.4 ± 0.2 1.8 ± 0.4 2.2 ± 0.4 1.9 ± 0.3 105 ± 63 99 ± 39 62 ± 36 73 ± 28

Table S10 .

 S10 Regression analysis of the mRNA expression (after Log10 transformation) as a function of the Cr redox form (Cr) and the exposure concentration (Co) after 96 h of exposure (experiment 1). A probability threshold of 0.01 was used for the construction of all fitted models. r 2 : coefficient of determination. G, gills; DG, digestive glands. See also tableS2for the coding of Cr redox forms. The coding for Cr(III) is '-1', meaning that mRNA expression increases with concentration because the minus sign of the coefficient for 'Cr*Co' and the minus sign for Cr(III) factor give a + sign The opposite situation occurs for Cr(VI).

			Experiment 1				Experiment 2
	Biomarker mRNA	Tissue Tissue Fitted model Fitted model Log10 (biomarker)	F-test	F-test P-value	r 2	Fitted model Log10 (biomarker)	F-test P-value	r 2
	Chol	G DG	Log10 (mRNA)	P-value	0.209 0.881 r 2		-0.104 + 0.171 T	< 10 -4	0.772
	LDH TG GST pi	G DG G G DG DG	0.207 -2.671 10 -4 Cr*Co (a)	0.012 0.706	0.010 0.255 0.559 0.107 0.510		0.118 + 0.459 T	0.866 < 10 -4	0.927
	Prot1 Prot2 GPx	G DG G G DG	1.563 + 1.096•10 -4 D*Cr*Co Not performed (b) 1.061 -0.035 D*Cr -1.008•10 -4 D*Co + + 1.195•10 -4 D*Cr*Co	0.405	0.059 0.003 < 10 -4	0.129 0.302	1.287 + 0.173 T 1.148 + 0.173 T	< 10 -4 < 10 -4	0.800 0.924
	GST CAT LOOH ACP CuZn SOD	DG G G DG G DG DG G G DG G DG	2.558 + 0.049 D*Cr -1.482 -6.294 10 -4 Cr*Co 1.384 + 0.038 D*Cr -1.416•10 -4 D*Cr*Co 2.036 -0.018 D -0.753•10 -4 Co -0.516 -10.148 10 -4 Cr*Co 2.009 -0.046 D*Cr 0.725 -0.069 Cr + 3.724•10 -4 Co -4.168•10 -4 D*Co + < 10 -3 0.183 < 0.01 0.090	0.180 0.001 0.475 0.055 0.114 < 10 -4 < 10 -4 0.276 0.006 < 10 -4	0.161 0.320 0.290 0.118 0.460	2.786 + 0.191 T 1.570 -0.205 T	< 10 -4 < 10 -4 0.962	0.703 0.968
	CSP3 ETS Mn SOD	DG G G DG	+ 3.911•10 -4 D*Cr*Co 0.550 -9.121•10 -4 D*Co -2.146 -4.477 10 -4 Cr*Co 0.316 -0.042 D + 0.974•10 -4 Cr*Co + 1.094•10 -4 < 10 -3 D*Cr*Co 0.201	< 10 -3 0.472 < 10 -4	0.175 0.439	0.848 -0.180T 0.347 -0.115 T	0.005 < 10 -4	0.310 0.844
		DG G	0.224 + 0.072 D -0.051 D*Cr	0.116	< 10 -4	0.440
	TR
		DG	-1.407 + 0.074 Cr + 3.966 10 -4 Co	0.002	0.439
		G	0.423 -3.556 10 -4 Cr*Co	0.016	0.237
	MT
		DG	0.801 -2.862 10 -4 Co	0.011	0.259
		G	0.458 -4.447 10 -4 Cr*Co	<0.01	0.268
	HSP70
		DG	0.120 + 0.091 Cr + 5.077 10 -4 Co	<10 -4	0.668
	(a)

(b) Insufficient GPx level.

Acknowledgements

This work has been performed as a part of the PhD thesis of Imad Aharchaou who received financial support from the French Research Ministry. The research was partially supported by the French National Research Agency through the national program "Investissements d'avenir" with the reference "ANR-10-LABX-0021/ LABEX RESSOURCES21", excellence action 86. CRediT author statement IA: Conceptualization, Investigation, Validation, Writing (Original Draft) AM: Formal analysis, Visualization, Writing (Original Draft) M-N P: Resources, Validation, Writing (Review & Editing) DP: Investigation HP: Investigation JF: Investigation, Methodology, Writing (Review & Editing) FR: Investigation, Methodology, Writing (Original Draft) PR: Investigation MB: Investigation EB: Conceptualization, Supervision, Validation, Funding acquisition, Writing (Original Draft) DAL V: Conceptualization, Supervision, Project Administration, Funding acquisition, Validation, Visualization, Writing (Original Draft)

Highlights

cases, regression analysis suggest that mRNA levels respond to both Cr(III) and Cr(VI), further supporting the recommendation for more comparative studies on both Cr oxidation states. S2: Primer sequences used in RT-qPCR analysis.

Supporting information for

Table S3: Factors, levels and coding values used for regression modelling in the two experiments.

Table S4. Total and total filterable (< 0.45 µm) Cr concentrations in aliquots of Cristaline® water Table S5. Chromium concentrations accumulated in the gills (G) and digestive glands (DG) Table S6. Comparison of Cr bioaccumulation (µg g -1 dry weight) in bivalves under different exposure conditions from published literature studies Table S7. Arithmetic means ± 1 standard deviation (n = 4; n = 3 for $) for the values of cellular biomarkers measured in gills and digestive glands during exp # 1.

Experiment 2. NIA, solutions with natural Cr isotopic abundances; IES, isotopically enriched solutions. Concentration in control exposures are given for 52 Cr only considering that only the concentration of this isotope was above the ICP-MS detection limit (see main text, section 3.1 for details). All values are arithmetic means ± 1 standard deviations, n = 3, t is in hours.


Jean-Éric Pin

Pedro V Silva

A Mahler's theorem for functions from words to integers *

In this paper, we prove an extension of Mahler's theorem, a celebrated result of p-adic analysis. Mahler's original result states that a function from N to Z is uniformly continuous for the p-adic metric dp if and only if it can be uniformly approximated by polynomial functions. We prove the same result for functions from A * to Z, where dp is now the profinite metric defined by p-groups (pro-p metric).

This paper was originally motivated by two main research lines of automata theory, but resulted into an approximation theorem that goes far beyond our original project. We first present our original motivations and then describe our main result. Recall that a variety of languages is a class of regular languages closed under Boolean operations, left and right quotients and inverse morphisms.

Motivations

Our first motivation was the study of regularity-preserving functions f from A * to B * , in the following sense: if X is a regular language of B * , then f -1 (X) is a regular language of A * . More generally, we were interested in functions preserving a given variety of languages V: if X is a language of V, then f -1 (X) is also a language of V. There is an important literature on the regular case [START_REF] Stearns | Regularity preserving modifications of regular expressions[END_REF][START_REF] Kosaraju | Regularity preserving functions[END_REF][START_REF] Seiferas | Regularity-preserving relations[END_REF][START_REF] Pin | Operations and transductions that preserve rationality[END_REF][START_REF] Pin | Une application de la représentation matricielle des transductions[END_REF][START_REF] Berstel | Operations preserving recognizable languages[END_REF], including the authors recent paper [START_REF] Pin | A topological approach to transductions[END_REF]. A similar problem was also recently considered for formal power series [START_REF] Droste | On transformations of formal power series[END_REF]. A remarkable contribution to the second problem can be found in [START_REF] Reutenauer | Variétés et fonctions rationnelles[END_REF], where a characterization of sequential functions preserving aperiodic languages (respectively group-languages) is given.

Our second motivation was the study of certain reductions. A fundamental idea of descriptive set theory is to use continuous reductions to classify topological spaces: given two sets X and Y , Y reduces to X if there exists a continuous function f such that X = f -1 (Y ). For Polish spaces, this gives rise to the Wadge hierarchy [START_REF] Wadge | Reducibility and determinateness in the Baire space[END_REF], nowadays entirely described and very well understood. Wagner [START_REF] Wagner | On ω-regular sets[END_REF] was the first to consider the restriction of the Wadge hierarchy to ω-rational languages. He proved in particular that the hierarchy doesn't change if continuous functions are replaced by sequential functions, a much more restricted class of functions (see [START_REF] Perrin | Infinite Words[END_REF]Chapter 5] for more details).

Our idea was to consider similar reductions for regular languages (of finite words this time). The first obstacle was to find an appropriate topology, but there is a natural candidate: the profinite topologies, a notion first introduced in [START_REF] Hall | A topology for free groups and related groups[END_REF]. Indeed, by Eilenberg's theorem, to each variety of languages V corresponds a unique variety of finite monoids V, which in turn defines the pro-V topology [START_REF] Pin | Uniformities on free semigroups[END_REF]. We shall not give here the precise definition, but it suffices to know that, in the most interesting cases, this topology can be defined by a metric d V . Let us call V-reduction a uniformly continuous function between the metric spaces (A * , d V ) and (B * , d V ). These V-reductions define a hierarchy similar to the Wadge hierarchy among regular languages, which we would like to explore. Note that a different notion of reduction for regular languages was recently considered in [START_REF] Selivanov | A reducibility for the dot-depth hierarchy[END_REF]. The first author had very instructive discussions with Selivanov and Kunc in June 2006 on the comparison between these two reductions and this paper is partly motivated by this conversation.

Regularity-preserving functions and V-reductions are actually strongly related. Indeed, one can show that a function from (A * , d V ) to (B * , d V ) is uniformly continuous if and only if, for every language L in V(B * ), f -1 (L) belongs to V(A * ). This encouraging fact lead us to search for a more precise description of V-reductions. However, apart from general results, not so much is known on pro-V topologies, except when V is a variety of finite groups. Among groups, the variety G p of p-groups, where p is a given prime, is of special interest for two reasons. First, Eilenberg and Schützenberger gave a very nice description of the languages recognized by a p-group (see Proposition 2.2 below). Second, a special case of the metric d p has been widely studied in mathematics: indeed, the free monoid over a one-letter alphabet is isomorphic to N, and the metric d p is known as the p-adic metric. The completion of the metric space (N, d p ) is the space of p-adic numbers and p-adic analysis is the branch of number theory that deals with functions of p-adic numbers [START_REF] Amice | Interpolation p-adique[END_REF][START_REF] Robert | A course in p-adic analysis[END_REF][START_REF] Murty | Introduction to p-adic analytic number theory[END_REF].

Our main result takes advantage of this powerful mathematical framework to provide a characterization of the G p -reductions from A * to N, that is, the uniformly continuous functions from (A * , d p ) to (N, d p ). It turns out that this characterization extends a celebrated result of number theory, Mahler's theorem (see http://en.wikipedia.org/ wiki/Mahler's_theorem), giving our result a mathematical interest on its own. Our result states that a function from A * to N is uniformly continuous for d p if and only if it can be uniformly approximated by a sequence of polynomial functions. Before stating this result in a precise form, we need a few formal definitions.

The p-adic and pro-p topologies

In the sequel, A denotes a finite alphabet, A * is the free monoid on A and 1 denotes the empty word.

Let p be a prime number. Recall that a p-group is a finite group whose order is a power of p. Let u and v be two words of A * . A p-group G separates u and v if there is a monoid morphism from A * onto G such that ϕ(u) = ϕ(v). One can show that any pair of distinct words can be separated by a p-group. We now set r p (u, v) = min { n | there is a p-group of order p n separating u and v }

d p (u, v) = 2 -rp(u,v)
with the usual convention min ∅ = -∞ and 2 -∞ = 0. One can show that d p is an ultrametric, that is, satisfies the following properties, for all u, v, w ∈ A * :

(1)

d p (u, v) = 0 if and only if u = v, (2) d p (u, v) = d p (v, u), (3)
d p (u, v) max(d p (u, w), d p (w, v)) One can also show that the concatenation product on A * is uniformly continuous for this metric. It follows that the completion of the metric space (A * , d p ) is naturally equipped with a structure of monoid, which is in fact a compact group, called the free pro-p group. The topology defined by the metric d p is usually called the pro-p topology in the literature.

There is a nice connection [START_REF] Pin | Topologie p-adique sur les mots[END_REF] between this topology and a generalization of the binomial coefficients. Let u and v be two words of

A * . Let u = a 1 • • • a n , with a 1 , . . . , a n ∈ A. Then u is a subword of v if there exist v 0 , . . . , v n ∈ A * such that v = v 0 a 1 v 1 . . . a n v n .
Following [START_REF] Eilenberg | Automata, Languages and Machines[END_REF][START_REF] Lothaire | Combinatorics on words[END_REF], we define the binomial coefficient of u and v by setting

v u = |{(v 0 , . . . , v n ) | v = v 0 a 1 v 1 . . . a n v n }| .
Observe that if a is a letter, then v a is simply the number of occurrences of a in v, also denoted by |v| a . Also note that if

A = {a}, u = a n and v = a m , then v u = m n
and hence these numbers constitute a generalization of the classical binomial coefficients. The next proposition, whose proof can be found in [START_REF] Lothaire | Combinatorics on words[END_REF]Chapter 6], summarizes the basic properties of the generalized binomial coefficients and can serve as an alternative definition.

Lemma 2.1 Let u, v ∈ A * and a, b ∈ A. Then (1) u 1 = 1, (2) u v = 0 if |u| |v| and u = v, ( 3
) ua vb = u vb if a = b u vb + u v if a = b
A third way to define the binomial coefficients is to use the Magnus automorphism of the algebra Z A of polynomials in noncommutative indeterminates in A defined by µ(a) = 1 + a for all a ∈ A. One can show that, for all u ∈ A * ,

µ(u) = x∈A * u x x (2.1)
which leads to the formula

u 1 u 2 x = x1x2=x u 1 x 1 u 2 x 2 (2.2)
The connection between the pro-p topology and the binomial coefficients comes from the characterization of the languages recognized by a p-group given by Eilenberg and Schützenberger (see [4, Theorem 10.1, p. 239]). Let us call a p-group language a language recognized by a p-group. Note that such a language is necessarily regular.

Proposition 2.2 A language of A * is a p-group language if and only if it is a Boolean combination of the languages

L(x, r, p) = {u ∈ A * | u x ≡ r mod p},
for 0 r < p and x ∈ A * .

Let us set now

r ′ p (u, v) = min |x| x ∈ A * and u x ≡ v x (mod p) d ′ p (u, v) = p -r ′ p (u,v) .
It is proved in [START_REF] Pin | Topologie p-adique sur les mots[END_REF]Theorem 4.4] that d ′ p is an ultrametric uniformly equivalent to d p . We shall use this result under a slightly different form.

Theorem 2.3 A function f : A * → B * is uniformly continuous for d p if and only if, for every regular language L of A * recognized by a p-group, the language f -1 (L) is also recognized by a p-group.

Proof. Let L be a language recognized by a p-group G of order p k . Then there exists a monoid morphism ϕ :

A * → G such that L = ϕ -1 (ϕ(L)). If f is uniformly continuous for d p , there exists n > 0 such that if r p (u, v) n, then r p (f (u), f (v)) k. It follows in particular that f (u) and f (v) cannot be separated by G and hence ϕ(f (u)) = ϕ(f (v)).
Let (ψ i ) i∈I be the family of all monoid morphisms from A * onto a p-group H i of order p n . Let ψ : A * → i∈I H i be the morphism defined by ψ(x) = (ψ i (x)) i∈I and let H be the range of ψ. Then H is a p-group and if ψ(u) = ψ(v), then r p (u, v) n and thus ϕ(f (u)) = ϕ(f (v)). We claim that

ψ -1 (ψ(f -1 (L)) = f -1 (L) First, f -1 (L) is clearly a subset of ψ -1 (ψ(f -1 (L)). To prove the opposite inclusion, let u ∈ ψ -1 (ψ(f -1 (L)). Then ψ(u) ∈ ψ(f -1 (L)), that is, ψ(u) = ψ(v) for some v ∈ f -1 (L). It follows that ϕ(f (u)) = ϕ(f (v)) and since f (v) ∈ L, f (u) ∈ ϕ -1 (ϕ(L)) and finally f (u) ∈ L since L = ϕ -1 (ϕ(L))
. This proves the claim and shows that f -1 (L) is a p-group language.

Suppose now that if L is a p-group language, then f -1 (L) is also a p-group language. Let ϕ be a morphism from A * onto a p-group G. For each g ∈ G, ϕ -1 (g) is a p-group language and hence f -1 (ϕ -1 (g)) is recognized by a morphism ψ g : A * → H g onto a p-group. Let ψ : A * → g∈G H g be the mapping defined by ψ(x) = (ψ g (x)) g∈G and let H = ψ(A * ). Then H is also a p-group and if

ψ(u) = ψ(v), then ψ g (u) = ψ g (v) for all g ∈ G. Since ψ g recognizes f -1 (ϕ -1 (g)), it follows that u ∈ f -1 (ϕ -1 (g)) if and only if v ∈ f -1 (ϕ -1 (g)) and hence ϕ(f (u)) = ϕ(f (v)). Now let k ∈ N.
If we consider all the morphisms ϕ from A * onto a p-group of order p k , and take n ∈ N large enough so that every group H corresponding to ϕ has order p n , it follows that

r p (u, v) > n ⇒ r p (f (u), f (v)) > k
holds for all u, v ∈ A * . This shows that f is uniformly continuous for d p .

In the case of a one-letter alphabet, A * is isomorphic to the additive monoid N and the definition of d p can be further simplified. If n is a non-zero integer, recall that the p-adic valuation of n is the integer

ν p (n) = max k ∈ N | p k divides n
By convention, ν p (0) = +∞. The p-adic norm of n is the real number

|n| p = p -νp(n) .
The p-adic norm satisfies the following axioms, for all n, m ∈ N:

(1) |n| p 0,

(2) |n| p = 0 if and only if n = 0, (3) |mn| p = |m| p |n| p , (4) |m + n| p max{|m| p , |n| p }. Finally, the metric d p can be defined by

d p (u, v) = |u -v| p .
and is known as the p-adic metric. Since Z is naturally embedded in the p-adic completion of N, the definitions above can be readily extended to Z. In the sequel, it will be more convenient to use the metric space (Z, d p ) in place of (N, d p ).

Mahler's expansions

The classical Stone-Weierstrass approximation theorem states that a continuous function defined on a closed interval can be uniformly approximated by a polynomial function. In particular, if a function f is infinitely differentiable in the neighbourhood of 0, it can be approximated, under some convergence conditions, by its Taylor polynomials

k n=0 f (n) (0) n! x n
The p-adic analogue of these results is Mahler's theorem [START_REF] Mahler | An interpolation series for continuous functions of a p-adic variable[END_REF]. For a fixed k ∈ N, the binomial polynomial function u → u k defines a uniformly continuous function from (N, d p ) to (Z, d p ). The Mahler's expansion of a function f from N to Z is defined as the series

∞ k=0 (∆ k f )(0) u k
where ∆ is the difference operator, defined by

(∆f )(u) = f (u + 1) -f (u)
Mahler's theorem states that f is uniformly continuous for d p if and only if its Mahler's expansion converges uniformly to f . Of course, the most remarkable part of the theorem is the fact that any uniformly continuous function can be approximated by polynomial functions, in contrast to Stone-Weierstrass approximation theorem, which requires much stronger conditions. For instance, if f is the Fibonacci sequence defined by f (0

) = f (1) = 1 and f (n) = f (n -1) + f (n -2) for n 2, then f (n) = ∞ k=0 (-1) k+1 f (k) n k
This function is not uniformly continuous for d p for any choice of p. If f (n) = r n , then

f (n) = ∞ k=0 (r -1) k n k
and f is uniformly continuous for d p if and only if p divides r -1.

The first step to extend Mahler's theorem to functions from words to integers is to define a suitable notion of Mahler's expansion for these functions.

Let f : A * → Z be a function. For each letter a, we define the difference operator ∆ a by (∆ a f )(u) = f (ua)f (u)

One can now define inductively an operator ∆ w for each word w ∈ A * by setting (∆ 1 f )(u) = f (u), and for each letter a ∈ A,

(∆ aw f )(u) = (∆ a (∆ w f ))(u).
It is easy to see that these operators can also be defined directly by setting

∆ w f (u) = 0 |x| |w| (-1) |w|+|x| w x f (ux) (3.3) For instance, ∆ aab f (u) = -f (u) + 2f (ua) + f (ub) -f (uaa) -2f (uab) + f (uaab).
For a fixed v ∈ A * , the function

u → u v
from A * to Z which maps a word u to the binomial coefficient u v is uniformly continuous for d p . This family of functions, for v ranging over A * , is locally finite in the sense that, for each u ∈ A * , the binomial coefficient u v is null for all but finitely many words v. In particular, if (m v ) v∈A * is a family of integers, there is a well-defined function from A * to Z defined by the formula

f (u) = v∈A * m v u v
We can now state our first result, which doesn't require any assumption on f . Theorem 3.1 Let f : A * → Z be an arbitrary function. Then there exists a unique family f, v v∈A * of integers such that, for all u ∈ A * ,

f (u) = v∈A * f, v u v (3.4)
This family is given by

f, v = (∆ v f )(1) = 0 |x| |v| (-1) |v|+|x| v x f (x) (3.5)
Proof. First observe that, according to (3.3)

(∆ v f )(1) = 0 |x| |v| (-1) |v|+|x| v x f (x) (3.6) Thus v∈A * (∆ v f )(1) u v = v∈A * |x| |v| (-1) |v|+|x| v x u v f (x) = x∈A * (-1) |u|+|x|   0 |v| |u| (-1) |v|+|u| u v v x   f (x) = f (u)
Intuitively, the coefficient u ↑ v, x is the number of pairs of subsequences of x which are respectively equal to u and v and whose union gives the whole sequence x. For instance, ab ↑ ab = ab + 2aab + 2abb + 4aabb + 2abab ab ↑ ba = aba + bab + abab + 2abba + 2baab + baba Also note that u ↑ v, u = u v . We shall need the following relation (see [7, p.131]). For all

v 1 , v 2 ∈ A * , u v 1 u v 2 = x∈A * v 1 ↑ v 2 , x u x (4.11)
Formula 4.11 leads to an explicit computation of the Mahler's expansion of the product of two functions.

Proposition 4.1 Let f and g be two functions from A * to N. Then the coefficients of the Mahler's expansion of f g are given by the formula:

f g, x = v1,v2∈A * f, v 1 g, v 2 v 1 ↑ v 2 , x Proof. Indeed, if f (u) = v∈A * f, v u v and g(u) = v∈A * g, v u v , then f g(u) = v1,v2∈A * f, v 1 g, v 2 u v 1 u v 2
and the result follows by Formula (4.11).

It is now easy to prove the result announced at the beginning of this section.

Proposition 4.2 Mahler polynomials form a subring of the ring of all functions from A * to Z for addition and multiplication.

Proof. It is clear that the difference of two Mahler polynomials is a Mahler polynomial. Further Proposition 4.1 shows that Mahler polynomials are closed under product.

Mahler's theorem

We are now ready to state our main result, which extends Mahler's theorem. In this section, uniform continuity always refers to the metric d p .

Theorem 5.1 Let f (u) = v∈A * f, v u v be the Mahler's expansion of a function from A * to Z. Then f is uniformly continuous if and only if

lim |v|→∞ | f, v | p = 0. Proof. Suppose that lim |v|→∞ | f, v | p = 0. Then there exists s ∈ N such that, if |v| s, ν p ( f, v ) r. Setting g(u) = |v|<s f, v u v and h(u) = |v| s f, v u v
we get f = g + h. Further p r divides f, v for |v| s. Since g is a Mahler polynomial, it is uniformly continuous and there exists t ∈ N such that d p (u, u ′ ) p -t implies g(u) ≡ g(u ′ ) (mod p r ) and hence f (u) ≡ f (u ′ ) (mod p r ). Thus f is uniformly continuous. This proves the easy direction of the theorem. The key argument for the opposite direction is the following approximation result. Theorem 5.2 Let f : A * → N be a uniformly continuous function. Then there exists a Mahler polynomial P such that, for all u ∈ A * , f (u) ≡ P (u) (mod p).

Proof. We first prove the theorem for some characteristic functions related to the binomial coefficients. The precise role of these functions will appear in the course of the main proof.

Let x ∈ A * and let s be an integer such that 0 s < p. Let χ s,x : A * → N be the function defined by χ s,x (u) = 1 if u x ≡ s (mod p) 0 otherwise Lemma 5.3 There is a Mahler polynomial P s,x such that, for all u ∈ A * , χ s,x (u) ≡ P s,x (u) (mod p).

Proof. Let

P s,x (u) = - u x u x -1 • • • u x -(p -
then f (u) ≡ f (v) (mod p)
It follows that the value of f (u) modulo p depends only on the residues modulo p of the family u x 0 |x| n . Let C be the set of all families r = {r x } 0 |x| n such that 0 r x < p. For 0 i < p, let C i be the set of all families r of C satisfying the following condition: if, for 0 |x| n, u x ≡ r x (mod p), then f (u) ≡ i (mod p) (5.12)

The sets (C i ) 0 i<p are pairwise disjoint and their union is C. We claim that, for all

u ∈ A * , f (u) ≡ 0 i<p iP i (u) (mod p) (5.13)
where P i is the Mahler polynomial

P i = r∈Ci 0 |x| n P rx,x (5.14)
First consider, for r ∈ C, the characteristic function

χ r (u) = 0 |x| n χ rx,x (u)
By construction, χ r is defined by χ r (u) = 1 if, for 0 |x| n, u x ≡ r x (mod p) 0 otherwise and it follows from (5.12) and from the definition of C i that

f (u) ≡ 0 i<p i r∈Ci χ r (u) (mod p) ( 5
P = 0 i<p iP i (u)
is a Mahler polynomial.

Theorem 5.2 can be extended as follows.

Corollary 5.4 Let f : A * → N be a uniformly continuous function. Then, for each positive integer r, there exists a Mahler polynomial P r such that, for all u ∈ A * , f (u) ≡ P r (u) (mod p r ).

Proof. We prove the result by induction on r. For r = 1, the result follows from Theorem 5.2. If the result holds for r, there exists a Mahler polynomial P r such that, for all u ∈ A * , f (u) -P r (u) ≡ 0 (mod p r ). Let g = f -P r . Since g is uniformly continuous, there exists a positive integer n such that if u x ≡ v x (mod p) for |x| n, then g(u) ≡ g(v) (mod p 2r ). It follows that 1 p r g(u) ≡ 1 p r g(v) (mod p r ), and thus 1 p r g is uniformly continuous.

Applying Theorem 5.2 to 1 p r g, we get a Mahler polynomial P such that, for all u ∈ A * , 1 p r g(u) ≡ P (u) (mod p) Setting P r+1 = P r + p r P , we obtain finally f (u) ≡ P r+1 (u) (mod p r+1 ) which concludes the proof.

We now conclude the proof of Theorem 5.1. For each positive integer r, there exists a Mahler polynomial P r such that, for all u ∈ A * , f (u) ≡ P r (u) (mod p r ). Using (3.5) to compute explicitly the coefficients f -P r , v , we obtain f -P r , v ≡ 0 (mod p r ) Since P r is a polynomial, there exists an integer n r such that for all v ∈ A * such that Mahler's theorem is often presented as an interpolation result (see for instance [9, p. 57]). This can also be extended to functions from words to integers. Given a family of integers (c v ) v∈A * , one can ask whether there is a (uniformly) continuous function f from the free pro-p group to Z such that f (v) = c v . Then answer is yes if and only if lim |v|→∞ |m v | p = 0, where m v = 0 |x| |v| (-1) |v|+|x| v x c x .

Conclusion

We proved an extension of Mahler's theorem for functions from words to integers. It would be interesting to find a suitable extension for functions from words to words. It would also be interesting to see whether other results from p-adic analysis can be extended to the word case.

  |v| n, P r , v = 0. It follows | f, v | p < p -r and thus lim |v|→∞ | f, v | p = 0.

  Then P s,x is a Mahler polynomial by Proposition 4.2. If u x ≡ s (mod p), then P s,x (u) ≡ 0 (mod p). If u x ≡ s (mod p), then by Al-Haytham's theorem,

	u x	≡	v x	(mod p)

1) u xs P s,x (u) ≡ -(p -1)! ≡ 1 (mod p) It follows that P s,x (u) ≡ χ s,x (u) (mod p) in all cases.

We now prove Theorem 5.2. Since f is uniformly continuous, there exists a positive integer n such that if, for 0 |x| n,

Acknowledgement

The authors would like to thank Daniel Barsky and Gilles Christol for pointing out several references.

* The authors acknowledge support from the AutoMathA programme of the European Science Foundation. The second author acknowledges support from Project ASA (PTDC/MAT/65481/2006) and C.M.U.P., financed by F.C.T. (Portugal) through the programmes POCTI and POSI, with national and European Community structural funds.

in view of the following relation from [START_REF] Lothaire | Combinatorics on words[END_REF]Corollary 6.3.8]:

Uniqueness of the coefficients f, v follows inductively from the formula

a straightforward consequence of (3.4).

The series defined by (3.4) is called the Mahler's expansion of f . For instance, let f : {0, 1} * → N be the function mapping a binary word onto its value as a binary number. Thus f (010111) = f (10111) = 23. Then one has

Mahler polynomials

A function f : A * → Z is a Mahler polynomial if its Mahler's expansion has finite support, that is, if the number of nonzero coefficients f, v is finite. In this section, we prove in particular that Mahler polynomials are closed under addition and product. We first introduce a convenient combinatorial operation, the infiltration product. We follow the presentation of [START_REF] Lothaire | Combinatorics on words[END_REF]. Let Z A be the ring of formal power series in noncommutative indeterminates in A. Any series s is written as a formal sum s = u∈A * s, u u, a notation not to be confused with our notation for the Mahler's expansion. The infiltration product is the binary operation on Z A , denoted by ↑ and defined inductively as follows: