M Brienza
email: monica.brienza@unibas.it

A Sauvetre

N Ait-Mouheb

V Bru-Adan

D Coviello

K Lequette

D Patureau

S Chiron

N W Éry

N Wéry

Reclaimed wastewater reuse in irrigation: role of biofilms in the fate of antibiotics and spread of antimicrobial resistance

Keywords: Biofilms, ARGs, antibiotics, reclaimed wastewater reuse, irrigation

Antibiotics degradation by interactions between planktonic and biofilm

• Antibiotic exposure had no impact on the planktonic bacterial communities.

• Resistance selection and propagation in biofilms are generate from bacterial community composition and diversity • Antibiotic exposure does not driven forces for resistance selection and propagation in biofilm

Introduction

The use of reclaimed wastewater for irrigation has become an attractive alternative water resource to alleviate water scarcity. However, there are several potential environmental-and health-related risks associated with this practice. One risk is the introduction of antibiotics and their associated antibiotic resistant bacteria (ARB) and antibiotic resistant genes (ARGs) into the agro(eco)systems [START_REF] Kampouris | Antibiotic resistance gene load and irrigation intensity determine the impact of wastewater irrigation on antimicrobial resistance in the soil microbiome[END_REF][START_REF] Marano | Hidden Resistome: Enrichment Reveals the Presence of Clinically Relevant Antibiotic Resistance Determinants in Treated Wastewater-Irrigated Soils[END_REF][START_REF] Subirats | High-quality treated wastewater causes remarkable changes in natural microbial communities and intI1 gene abundance[END_REF]. Bacterial infections are treated with antibiotics, and, after administration, part of these compounds is excreted unchanged in the urine and feces [START_REF] Junker | Biodegradability of14C-labeled antibiotics in a modified laboratory scale sewage treatment plant at environmentally relevant concentrations[END_REF]. This is a problematic issue because antibiotic residues can promote resistance in bacteria during wastewater treatment process [START_REF] Watkinson | Removal of antibiotics in conventional and advanced wastewater treatment: Implications for environmental discharge and wastewater recycling[END_REF]. Antibiotic resistance can then be acquired by soil-bacteria [START_REF] Gatica | Impact of treated wastewater irrigation on antibiotic resistance in the soil microbiome[END_REF] or even reach the food chain [START_REF] Christou | The potential implications of reclaimed wastewater reuse for irrigation on the agricultural environment: The knowns and unknowns of the fate of antibiotics and antibiotic resistant bacteria and resistance genes -A review[END_REF] when using reclaimed wastewater for crop irrigation. The dissemination of this resistance in the environment might end up with inefficiency of antibiotics use [START_REF] Lucas | Fungal treatment for the removal of antibiotics and antibiotic resistance genes in veterinary hospital wastewater[END_REF]. For these reasons, the development and dissemination of antibiotic resistance are today issues attracting great attention from the general public and has been classified as one of the major challenges for the global health by the WHO [START_REF] Who | Antimicrobial Resistance: global report on surveillance[END_REF].

Conventional wastewater treatment plants (WWTPs) are currently not designed to fully remove antibiotics and ARB&Gs, likely due to the cost and difficulty of treating complex waste mixtures. ARGs have been detected in wastewater treatment plants (WWTPs) [START_REF] Cacace | Antibiotic resistance genes in treated wastewater and in the receiving water bodies: A pan-European survey of urban settings[END_REF][START_REF] Yang | Quantification and characterization of β-lactam resistance genes in 15 sewage treatment plants from East Asia and North America[END_REF] drinking water treatment system [START_REF] Zhang | Antibiotic Resistance Genes in drinking water of China: Occurrence, distribution and influencing factors[END_REF], surface water [START_REF] Tang | Metagenomic analysis of bacterial community composition and antibiotic resistance genes in a wastewater treatment plant and its receiving surface water[END_REF] and groundwater [START_REF] Zainab | Antibiotics and antibiotic resistant genes (ARGs) in groundwater: A global review on dissemination, sources, interactions, environmental and human health risks[END_REF] indicating a broad distribution and contamination of different water environments.

Even if it has been reported that low-energy anaerobic-aerobic treatment reactors, constructed wetlands, and disinfection processes have shown good removal efficiencies [START_REF] Barancheshme | Development of Antibiotic Resistance in Wastewater Treatment Plants[END_REF], ARGs are frequently detected in WWTPs effluents.

Traditionally, it has been suggested that urban WWTPs act as hotspots of ARGs and also mobile genetic elements (MGEs) [START_REF] Guo | Metagenomic analysis reveals wastewater treatment plants as hotspots of antibiotic resistance genes and mobile genetic elements[END_REF][START_REF] Rizzo | Urban wastewater treatment plants as hotspots for antibiotic resistant bacteria and genes spread into the environment: A review[END_REF]. In a recent review including data from WWTPs worldwide, a positive correlation between antibiotics and ARGs commonly detected in WWTPs was observed, except for β-lactam antibiotics and bla genes [START_REF] Wang | Occurrence and fate of antibiotics, antibiotic resistant genes (ARGs) and antibiotic resistant bacteria (ARB) in municipal wastewater treatment plant: An overview[END_REF]. The most frequently detected antibiotics are macrolides (clarithromycin, erythromycin, azithromycin and roxithromycin), quinolones (ofloxacin, ciprofloxacin and norfloxacin), and sulfonamides (sulfamethoxazole), trimethoprim and tetracycline while bla (blaCTXM, blaTEM), sul (sul1, sul2), tet (tetO, tetQ, tetW) and ermB genes are the ARGs commonly reported in WWTPs (Wang et al., 2020a).

Yet, other recent studies have shown controversial results, suggesting that in fact the core resistome of biological wastewater treatment units is different from other parts of urban water ecosystems and not necessarily a hotspot for gene transfer [START_REF] Munck | Limited dissemination of the wastewater treatment plant core resistome[END_REF][START_REF] Quintela-Baluja | Spatial ecology of a wastewater network defines the antibiotic resistance genes in downstream receiving waters[END_REF]. It is therefore important to study the composition and behavior of microbial communities involved in wastewater treatment and transport processes, from the WWTP to the downstream receiving water bodies or in the case of agricultural reuse, to the field and crops.

Biofilms are biological structures made up of bacteria, organic matter and some inorganic matter that together constitute three-dimensional polymer niches [START_REF] Flemming | The biofilm matrix[END_REF]. Biofilms may also be associated with the accumulation of absorbed chemical and pathogenic agents and would thus act as potential reservoirs of materials that would affect the safety of irrigated food produced. However, they lead also to self-depuration due the ability to accumulate organic compounds [START_REF] Huerta | Determination of a broad spectrum of pharmaceuticals and endocrine disruptors in biofilm from a waste water treatment plant-impacted river[END_REF][START_REF] Ruhí | Bioaccumulation and trophic magnification of pharmaceuticals and endocrine disruptors in a Mediterranean river food web[END_REF][START_REF] Wilkinson | Spatial (bio)accumulation of pharmaceuticals, illicit drugs, plasticisers, perfluorinated compounds and metabolites in river sediment, aquatic plants and benthic organisms[END_REF] at concentrations up to few hundred ng g -1 , as well as to degrade them [START_REF] Desiante | Micropollutant biotransformation and bioaccumulation in natural stream biofilms[END_REF][START_REF] Writer | Role of biofilms in sorptive removal of steroidal hormones and 4-nonylphenol compounds from streams[END_REF][START_REF] Sabater | Monitoring the effect of chemicals on biological communities. The biofilm as an interface[END_REF].

It has been suggested that ARB can proliferate within irrigation pipe-based biofilms [START_REF] Blaustein | Irrigation waters and pipe-based biofilms as sources for antibiotic-resistant bacteria[END_REF] and that biofilms in water environments are thought to be hotspots for horizontal gene transfer of ARGs [START_REF] Abe | Biofilms: Hot spots of horizontal gene transfer (HGT) in aquatic environments, with a focus on a new HGT mechanism[END_REF]. In fact, biofilms are formed when bacteria are under stressed conditions. In the distribution system, the presence of antibiotics, residual concentration of chlorine will imply a stress and then will favour biofilm development with growth of bacteria and extracellular polymeric substance (EPS) production. As in activated sludge system, this bacterial growth and activity could favour gene transfer. This is a hypothesis that is discussed in the recent paper of [START_REF] Abe | Biofilms: Hot spots of horizontal gene transfer (HGT) in aquatic environments, with a focus on a new HGT mechanism[END_REF], but was also proposed by [START_REF] Pruden | Balancing Water Sustainability and Public Health Goals in the Face of Growing Concerns about Antibiotic Resistance[END_REF] for the spread of ARG in biofilm grown in drinking water pipe system due to opportunistic pathogens growth. Literature on wastewater transport pipes or drip irrigation systems with reclaimed wastewater is still scarce. A study performed in sewers transporting raw wastewater showed that community composition rather than antibiotic concentration is the main factor driving the diversity of the sewage resistome [START_REF] Auguet | Sewers as potential reservoirs of antibiotic resistance[END_REF]. Lequette and coworkers (2020) have also shown that community composition shifts between different parts of a treated wastewater irrigation system such as pipes and drippers [START_REF] Lequette | Hydrodynamic effect on biofouling of milli-labyrinth channel and bacterial communities in drip irrigation systems fed with reclaimed wastewater[END_REF]. However, no link has been established between antibiotic concentration, community structure, biofilm development and ARGs occurrence.

In this work we address some of these questions in an attempt to get a comprehensive overview of the effect of antibiotics on the dissemination of antibiotic resistance thanks to a deep characterization of biofilms growing in treated wastewater. Therefore, biofouling, microbial activity, community composition and abundances of four ARGs and 2 MGEs were monitored in biofilms exposed to a cocktail of six antibiotics commonly found in wastewater. The role of planktonic bacteria in the spreading of resistance in interaction with biofilms was also investigated by determining antibiotic concentration and microbial activity and composition in the liquid phases during the 14 days of antibiotics exposure.

Material and methods

Reagents and chemicals.

Six antibiotics were selected in this study: amoxicillin (AMX), ofloxacin (OFX), sulfamethoxazole (SMX), trimethoprim (TMP), clarithromycin (CLA) and cephalexin (CEF)) as they are among the most abundant in WWTP effluents and are frequently quantified in the range of ng/L to µg/L and its presence is addressed in many countries. Some of them such as ciprofloxacin can be detected also at concentration around 8 µg/L [START_REF] Anjali | Insights on the current status of occurrence and removal of antibiotics in wastewater by advanced oxidation processes[END_REF]." Moreover, amoxicillin, ciprofloxacin, sulfamethoxazole, and trimethoprim are included in the surface water watch list adopted by the European Commission under the Water Framework Directive (2020/1161). Reference standards for all compounds were obtained from Sigma-Aldrich (St Quentin-Fallavier, France) and were of high purity (> 95%). Individual stock solutions of standards (1000 mg L -1 ) were prepared individually in 100% methanol and stored at -20 °C.

Intermediate solutions (100 to 1 mg L -1 ) were prepared in serial dilution in methanol and stored at -20 °C. Working solutions were prepared in serial dilution in water by mixing all compounds and immediately used for the treatment. All LC-MS grade organic solvents and HPLC water were purchased from Carlo Erba Reagents (Val de Reuil, France) and were of purity higher than 99.9%.

Biofilm growth conditions.

The development of 6-weeks old biofilms was performed in three reactors where 8 high density polyethylene (PE) disks (0.5 cm thick and 5 cm in diameter) were fixed on a grid immerged in the reactor. The choice of PE has been selected to be similar to the material conditions used in drip irrigation.

Each reactor was filled with 4 L of treated wastewater, and water level was kept constant by means of constant water addition using a peristaltic pump ensuring a hydraulic retention time equal to 24 h and no external nutrients were supplied for biofilm growth. Treated wastewater was collected from the wastewater treatment plant operated on the basis of stabilization ponds with three successive lagoons (13,680, 4784 and 2700 m 3 , respectively), without addition of disinfection processes and had a nominal capacity of 1500 Inhabitant Equivalent [START_REF] Manasfi | Impact of long-term irrigation with municipal reclaimed wastewater on the uptake and degradation of organic contaminants in lettuce and leek[END_REF]). The reactors were operated in dark in order to reproduce the irrigation pipe conditions in anaerobic conditions (O2< 0.1 mg L -1 ) by continuously feeding with treated wastewater stored in 200 L tanks in the laboratory (T = 20 ± 2 °C). The mix in the reactor was maintained under magnetic stirring. Fresh treated wastewater was changed once per week in the storage tank.

Experimental set-up, design and operation.

The experimental set-up was divided into two groups: treated wastewater (W) and sterilized wastewater (SW) . W-group gathered set-up made with treated wastewater coming from a secondary biological treatment, applied after a lagoon treatment (Montpellier, France). The SW-group gathered set-up made with the same treated wastewater that was sterilized (120°C, 30 min). After 6 weeks, two PE disks colonized by biofilm were taken out from the reactor and were placed in each set-up made of glass container filled with 650 mL of the treated (W) or sterilized treated (SW) wastewater spiked with selected antibiotics at a final concentration of 10 µg L -1 (each). Control containers were made using wastewater (C_BW) and sterilized wastewater (C_BSW) without antibiotics. The experiments were made in triplicate.

The experiments were conducted using identical glass containers. Each container was operated in the dark at room temperature, under anoxic conditions, and under slight shaking using an orbital shaker at 30 rpm. Anoxic conditions were created by purging the air trapped in closed containers with nitrogen.

Anoxic conditions were chosen to mimic conditions found for biofilms inside drip-irrigation pipes. The experimental set-up (Table 1SM) consisted in 14 containers divided as follows: (1) one control with biofilm in wastewater (C_BW); (2) one control with biofilm in sterilized wastewater (C_BSW); (3) 3 containers with biofilm exposed to spiked wastewater (BW); (4) 3 containers with biofilms exposed to spiked sterilized wastewater (BSW), (5) 3 containers with spiked wastewater, and, (6) 3 containers with spiked and sterilized wastewater. All containers were incubated during two weeks and water was changed after 7 days, with fresh antibiotic solution for the spiked containers. Analysis were performed at the start and end of experiment.

Analytical procedures

Physicochemical characterization of the treated wastewater.

Treated wastewater was collected from WWTP located at Murvièl -Les-Montpellier (Hérault, France). pH, dissolved oxygen, conductivity, nitrate, nitrite, ammonia, total nitrogen, phosphate, suspended solid (TSS) and chemical oxygen demand (COD) concentrations were monitored to follow the wastewater treatment performances. COD, ammonia, nitrate, nitrite, total nitrogen and phosphorus concentrations were measured with a spectrophotometer (DR1900, Hach Company, Loveland, CO, USA) using Hach reagents® after filtration using glass filters according to the Standard Methods (APHA, 2005).

Conductivity, dissolved oxygen and pH were measured with probes (TetraCon® 925 and pH-Electrode Sentix® 940, WTW, Wilhelm, Germany). The physicochemical characteristics of the treated wastewater are listed in Table 2SM.

Antibiotic analysis.

Liquid samples (100 mL) were collected from each glass container. Collected samples were filtered through 0.45 µm syringe filters, frozen and stored at -20°C until analysis. For the analysis of antibiotics, extraction and purification were implemented through HLB OASIS cartridges (200 mg sorbent / 6cc) from Waters. The conditioning phase was performed with 4 mL MeOH followed by 4 mL UPW, 100 mL of filtered sample was passed through the cartridge followed by 4 mL of UPW for washing.

Cartridges were dried under vacuum until total dryness and finally retained analytes were eluted with 4 mL of MeOH. Eluates were totally evaporated under a gentle nitrogen flow at 55°C and residues were reconstituted in 200 µL of LC-MS mobile phase (ACN/water, 5/95%, v/v) and filtered with 0.22 µm PTFE syringe filters before injection.

Antibiotics were quantified using LC-HRMS in positive mode of ionization using an Exactive Orbitrap mass spectrometer (Thermo Fisher Scientific, USA). Specifically, 10 µL of each sample was injected on a Zorbax Eclipse XDB-C-18 column (3 i.d. × 150 mm length × 5 µm particle size) at a flow rate of 0.150 mL min -1 . Samples were eluted using a water (A): acetonitrile (B) mobile phase as follows: start at 5% B, increase to 100% B over 15 min, held for 5 min and back to 5% B in 1 min. Full scan MS mode with a scan range from 90 to 900 m/z, at a resolution of 50,000 FWHM, Automatic Gain Control (AGC) target of 1E6 ions at injection time of 50 ms provided the optimum parameters for high sensitivity together with sufficient data point per peak.

Biofilm characterization.

Biofilm samples were collected by rubbing a sterile swab over the PE disk with the help of a cylindrical hollow punch (inner surface of 0.8 cm 2 ). For each container, 2 biofilms samples were collected, one on each PE-disk. Samples were stored at -20°C until DNA extraction.

Determination of biofilm thickness.

At the beginning and end of the spiked studies, the biofilm thickness and quantity was compared in the same measurement area. Three-dimensional Optical Coherence Tomography (OCT), Thorlabs GANYMEDE II OCT (LSM03 lens, lateral resolution: 8 μm; Thorlabs GmbH, Lübeck, Germany), was used to investigate the biofilm evolution as non-invasive imaging. The axial voxel size in water (n = 1.333) of GANYMEDE II was 2.1 μm and center wavelength was 930 nm.

Firstly, 3-D OCT datasets were processed in Fiji (running on ImageJ version 1.50b [START_REF] Schindelin | Fiji -an open platform for biological image analysis[END_REF]). Secondly, an in-house code was used to detect the pixels associated to the biofilm using MATLAB R2018r (MathWorks, Natick, MA, USA, version 2018b). A threshold (adapted to each dataset) was applied to binarize the dataset and the region above the threshold intensity losses during simple penetration. The intensity contrast between the fouling pixels and the remaining parts (PE plates or water) was significant to allocate these last areas to the back-ground (black). In addition, suspended solids and outliers were removed applying the 'clean and majority functions' of Matlab. For each position (x and y coordinates), the pixels associated with the fixed biofilm (up to the threshold) were summed (on the z coordinate) to obtain the biofilm volume.

In order to compare the mean thickness of biofilm, the biofilm quantity was normalized by surface of the corresponding measurement zones. Taking into account the pixel and mm scale ratios, the mean biofilm thickness (mm) was calculated for each measuring volume after data analysis according to Equation 1:

Mean biofilm thickness =V fouling / S measurement 1)
where Vfouling is the volume of biofilm measured from OCT acquisitions (mm 3 ) and Smeasurement is the surface of OCT acquisitions of the sample determined by the taken image (mm 2 ).

Quantification of microbial activity.

Microbial activity in biofilm and wastewater samples was determined by adenosine triphosphate (ATP) measurement. The ATP concentration was determined in water and biofilm samples by measuring the amount of light produced in a luciferin-luciferase assay. The generated light signal is measured as Relative Light Units (RLU) by using the luminometric assay DENDRIDIAG kit (GLBiocontrol, France). The ATP data from water were expressed in ng L -1 and from biofilm in pg cm -2 .

Microbial community analysis.

DNA isolation.

Water (25 mL) and biofilm (0.8 cm 2 ) samples were collected. Water samples was vacuum filtered through polycarbonate track-etch membrane filters with a diameter of 50 mm and a pore size of 0.22 µm (Millipore, FRANCE). Filters were then transferred to a bead tube of the FastDNA TM Spin Kit for soil (MP biomedicals, Solon, USA) following the manufacturer's instructions. DNA purity and concentrations were estimated by spectrophotometry (Infinite NanoQuant M200, Tecan, Austria). The extracted DNA was stored at -20 °C.

Quantification of total bacteria and ARGs by qPCR.

The amplification reactions were performed in triplicates, and for two dilutions to check for the absence of PCR reaction inhibition. Reaction mixtures (12.5 µL) contained 6.5 µL of SsoADV Univer Probes or Sybr Super Mix (Biorad) and primers at concentrations indicated in Table 2SM. Total bacterial quantification was performed by qPCR targeting the V9 region from 16S rDNA. 100 nM forward primer BAC338 (5'-ACTCCTACGGGAGGCAG-3'), 250 nM of reverse primer BAC805 (5'-GACTACCAGGGTATCTAAT CC-3') and 50 nM of probe BAC516 (Yakima Yellow - TGCCA GCAGC CGCGG TAATA C - TAMRA) (Yu et al., 2005). The cycling parameters were 2 min at 95 °C for pre-incubation of the DNA template, followed by 40 cycles at 95 °C for 15 s for denaturation and 60 °C for 60 s for annealing and amplification.

Four ARGs and two genes linked to dissemination of antibiotic resistance were quantified by qPCR using Sybr Green chemistry: qnrS (reduced susceptibility to fluoroquinolone); sul1 (resistance to sulphonamides); blaTEM (resistance to β-lactams), ermB (resistance to macrolide), intl1 encoding the integrase of Classe I integrons used as a proxy for the potential capacity of the bacterial community to disseminate resistance [START_REF] Ma | The prevalence of integrons as the carrier of antibiotic resistance genes in natural and man-made environments[END_REF] and IS613 (transposon). The selected genes were quantified in wastewater and biofilm DNA using qPCR systems and PCR programs described in Table 3SM. Specificity of amplification was validated both by melting curve analyses and by checking the size of the amplicons on 2100 Bioanalyzer (Agilent, Waldbronn, Germany). All qPCR assays were performed with high R 2 values (0.9999 in average), high efficiency (90-100%) validating the gene quantification.

Non-parametric statistical tests (Kruskal-Wallis and Conover tests) were performed to compare concentrations of ARGs.

Analysis of bacterial communities by Illumina sequencing.

The PCR amplified the V4-V5 region of 16S rRNA gene with 30 cycles (annealing temperature 65°C) using the primers 515U (5′-GTGYCAGCMGCCGCGGTA-3′) and 928U (5′-CCCCGYCAATTCMTTTRAGT-3′) [START_REF] Wang | Conservative fragments in bacterial 1S rRNA genes and primer design for 16S ribosomal DNA amplicons in metagenomic studies[END_REF]. Adapters were added for multiplexing samples during the second sequencing amplification step. The resulting products were purified and loaded onto the Illumina MiSeq cartridge for sequencing of paired 300 bp reads according to the manufacturer's instructions (v3 chemistry). Sequencing and library preparation tasks were performed at the Genotoul Lifescience Network Genome and Transcriptome Core Facility in Toulouse, France (get.genotoul.fr). Mothur (version 1.39.5) [START_REF] Schloss | Introducing mothur: open-source, platformindependent, communitysupported software for describing and comparing microbial communities[END_REF] was used to associate forward and reverse sequences and clustering at four different nucleotides over the length of the amplicon. UCHIME [START_REF] Edgar | UCHIME improves sensitivity and speed of chimera detection[END_REF] was used to identify and remove chimera. Sequences that appeared fewer than three times in the entire data set were removed. Remaining sequences were aligned using SILVA SSURef NR99 version 128 [START_REF] Schloss | Introducing mothur: open-source, platformindependent, communitysupported software for describing and comparing microbial communities[END_REF]. Taxonomic assignment was done using Blast (http://www.ncbi.nlm.nih.gov/BLAST/) to locate publicly available sequences closely related to the sequences obtained from the samples. Sequences with 97% similarity were sorted into operational taxonomic units (OTUs) (Nguyen et al., 2016). The chloroplast sequences from 16S rRNA sequences were removed from the raw data and represented less than 1%. Reads were grouped at the 97% similarity level.

The Shannon diversity index, the Simpson index, and the nonparametric richness estimator Chao1 were calculated [START_REF] Chao | Non-parametric estimation of the classes in a population[END_REF]. To compare bacterial community structures, a dissimilarity matrix (Bray-Curtis) was performed and visualized using principal coordinate analysis (PCoA). The sequencing data analysis was processed under R v3.4 (www.r-project.org) through R-Studio (http://www.rstudio.com/) using the phyloseq package [START_REF] Mcmurdie | Phyloseq: A Bioconductor Package for Handling and Analysis of High-Throughput Phylogenetic Sequence Data[END_REF]. Relative abundancies of putative antibiotic resistant bacteria were determined by looking for taxonomic groups of interest based on scientific literature (Table 3SM). Concentrations were obtained by multiplying relative abundancies by bacterial concentration given by the measurement of total 16S rDNA by qPCR.

Results and discussion

Degradation of antibiotics by planktonic and biofilm bacteria.

Belonging to the most frequently detected antibiotics in wastewater (Wang et al., 2020a), CLA, OFX, SMX and TMP were included in this study together with two beta-lactam antibiotics, CEF and AMX.

Among these six antibiotics, AMX, CEF and OFX have bactericidal activity, whereas CLA, SMX and TMP are mainly bacteriostatic. However, SMX and TMP can be bactericidal when used together and CLA may also be bactericidal depending on the concentration and targeted organism.

Minimum inhibitory concentration (MIC) is defined as the lowest concentration of an antimicrobial that inhibits the visible growth of a microorganism after overnight incubation. By taking advantage of the fact that an antibiotic concentration that kills or inhibits growth of at least some bacteria will, by consequence, be selective at the community level. However, the concentrations of antibiotics are dynamic in time (degradation and sorption) and space (microhabitats in biofilms). The mechanisms of resistance selection could be thus interrupted depending of these concentration variations. Antibiotic concentrations were thus monitored in the medium after the 14 d of incubation and removals were calculated as shown in Table 1. The removal was reported as variation of the concentrations during the experiments. It was not possible to separate adsorption process from degradation due to limited quantity of biofilm mass. Whatever the conditions and molecules, the calculated removals were all below 50% underlying low degrading activity of the different ecosystems but in counterpart it enables favorable conditions for resistance selection. Moreover, removals were observed even in the sterilized wastewater set-up underlying a residual bacterial activity of these systems. This residual activity can be explained by the results on microbial community characterization (3.3) that showed a loss of diversity (due to sterilization) but a highest absolute abundance of 16SrDNA (growth of opportunistic bacteria) in planktonic bacteria under this condition. In addition, among the selected antibiotics, the natural fate of them is well studied. Due their characteristic physic-chemical, the only antibiotics that can readily undergo hydrolysis in aquatic environments due to their unstable beta-lactam ring are the beta-lactams (cephalexin and amoxicillin) while no hydrolysis for sulfamethoxazole was detected at several pH studied by Trovô et al; (2009 ); Similar was observed for

Trimethoprim by Lam and co-workers (2004), where dark hydrolysis and biodegradation in the microcosms over the course of the study from 1.5 to 82 d demonstrated any important in the loss of the compounds. For this reason, we are strongly convince that the biodegradation is mainly driver from specific bacteria communities.

CEF and AMX were totally removed in all reactors and therefore no differential behavior between the applied conditions could be inferred. In fact, beta-lactams are antibiotics that can readily undergo hydrolysis in aquatic environments due to their unstable beta-lactam ring.

TMP removal was very low (<25%) and observed only in reactors without biofilm, suggesting a main role of planktonic bacteria in its degradation. Addition of biofilm may modify the interactions and metabolism of both communities preventing the TMP removal. In the literature, degradation of TMP has been mainly attributed to nitrifying conditions which are far from our conditions [START_REF] Batt | Enhanced biodegradation of lopromide and trimethoprim in nitrifying activated sludge[END_REF], so it is mean that nitrifying bacteria play an important role on its degradation. So, the low removal of TMP is in agreement with [START_REF] Luo | Trimethoprim: kinetic and mechanistic considerations in photochemical environmental fate and AOP treatment[END_REF] and Perez et al. ( 2005) also reported that no biodegradation was observed in conventional activated sludge treatment. In addition, it was demonstrated by [START_REF] Kong | The performance and evolution of bacterial community of activated sludge exposed to trimethoprim in a sequencing batch reactor[END_REF] that the presence of TMP, the protein and polysaccharide contents increased to protect the cells from unfavourable conditions. This can justify why the low decrease of its concentration was observed in our experiments.

For SMX, the removal (ranging between 6 and 38.5%) increased in wastewater with biofilm by comparison to wastewater without biofilm. This removal could be assigned in part to biodegradation by biofilm communities and in part to sorption processes. Antibiotic sorption to biofilm is strongly dependent on physicochemical properties of biofilm and also on antibiotic structure. Wunder and coworkers [START_REF] Wunder | Sorption of antibiotics to biofilm[END_REF] demonstrated that speciation and size of the antibiotic are important factors affecting the interactions between antibiotics and biofilm. The degradation of SMX is also well documented in the literature [START_REF] Chen | Overview of sulfonamide biodegradation and the relevant pathways and microorganisms[END_REF][START_REF] Larcher | Biodegradation of sulfamethoxazole: Current knowledge and perspectives[END_REF]. Several studies have focused on biological degradation of SMX under aerobic conditions but few have been investigated

under anaerobic and anoxic conditions. Baumgarten and co-workers [START_REF] Baumgarten | Long term laboratory column experiments to simulate bank filtration: factors controlling removal of sulfamethoxazole[END_REF] reported anaerobic and anoxic column tests of contaminated water at 0.25 and 4.5 µg L -1 of SMX and observed 27% removal under anoxic condition (49-day half-life) and 51% under anaerobic conditions (16-day half-life).

OFX removals were observed in all reactors between 30 and 49%, excepted when biofilm was present in wastewater; it supposed a main contribution of the planktonic bacteria and potential competition for co-substrates when biofilm is added in presence of a more diverse community. Fluoroquinolone compounds have a fluorine atom that contributes to their recalcitrance in the environment [START_REF] Natarajan | Microbial cleavage of CF bond[END_REF] and only few studies reported their biodegradation. Few bacteria and fungi species were shown to be able to transform fluoroquinolones [START_REF] Kim | Modification of norfloxacin by a Microbacterium sp. strain isolated from a wastewater treatment plant[END_REF][START_REF] Manasfi | Biodegradation of fluoroquinolone antibiotics and the climbazole fungicide by Trichoderma species[END_REF], also under anaerobic conditions [START_REF] Shu | Anaerobic biodegradation of levofloxacin by enriched microbial consortia: Effect of electron acceptors and carbon source[END_REF].

Clarithromycin showed opposite trend to OFX. Its removal, ranging between 11 and 47%, increased in presence of biofilm in wastewater, as observed for SMX, and decreased in presence of biofilms in sterilized wastewater.

To summarize, planktonic bacteria were able to degrade the six antibiotics in sterilized wastewater when bacterial community is less diverse but more active, suggesting specific metabolism towards these compounds. Interestingly, SMX was less degraded by planktonic bacteria when wastewater was not sterilized. A plausible explanation could be that low abundant degrading bacteria, generally outcompeted by dominant species in wastewater planktonic communities, might become dominant after sterilization. The presence of biofilm had a positive effect on removal of SMX and CLA in wastewater and only in SMX in sterilized wastewater. Of special interest is the observation that biofilm impacted negatively the removal of OFX and TMP in wastewater and of TMP and CLA in sterilized wastewater revealing interactions between biofilm and planktonic bacteria. Such inter-species interactions have been shown to contribute to antibiotic tolerance in complex bacterial communities (Aranda-Díaz et al., 2020).

Impact of antibiotics on biofilm thickness and activity

Biofilm thickness was determined by three-dimensional optical coherence tomography (OCT) imaging.

Before exposure to antibiotics, the mean initial thickness averaged 1.76 mm. After 14 days, the OCT measurements show that the mean biofilm thickness (which could give biofilm biomass) was not affected by the presence of antibiotics and the sterilization of wastewater (Figure 1aSM). Therefore, the 6 added antibiotics at a concentration of 10 µg L -1 each did not promote the detachment of biofilms from the PE support. However, OCT results are not representative of changes induced at the bacterial community or activity level.

Often, bacterial respiration measured as total ATP produced is used as a parameter for microbial activity measurement (van der Wielen & van der Kooij, 2010). ATP concentration was monitored both in planktonic communities (liquid phase) and in biofilms before and after the antibiotic exposure (Figure 1b,cSM). The average ATP concentrations differed in each experimental group (W and SW). When biofilm was put in presence of treated wastewater, microbial activity was in general lower than in t0.

Exposure to antibiotics induced a slight increase of microbial activity. The opposite effect was observed when biofilms were put in presence of sterilized wastewater where the presence of antibiotics induced a reduction in microbial activity in the biofilm (from 7.37 to 6.96 Log eq.bact./ml). Besides, the activity was in any case higher in biofilms put in presence of SW than in W.

Concerning planktonic bacteria, their activities showed different results. Exposure to antibiotics induced an increase on microbial activity in wastewater (5.85 to 6.26 Log eq.bact./ml). This effect disappeared in samples containing biofilms (5.71 to 5.51 Log eq.bact./ml). When using SW, a reduction in microbial activity was observed at t0 due to sterilization (2.97 Log eq.bact./ml) but recovered after 14 days to reach 5.72 Log eq.bact./ml which was similar to the initial ATP value in wastewater. In SW samples containing biofilm, the effect was similar but to a lower extent (5.19 to 6.02 Log eq.bact./ml).

Bacterial cellular respiration measured as production of ATP is today considered a good indicator of microbiological activity and since antibiotics are designed to kill or inhibit the growth of bacteria, it is expected that bacterial cellular respiration is reduced after exposure to antibiotics. However, Lobritz and coworkers identified two different mechanisms depending on antibiotic bacteriostatic or bactericidal effect. Growth inhibition from bacteriostatic antibiotics was associated with suppressed cellular respiration whereas cell death from most bactericidal antibiotics was associated with accelerated respiration. However, in combination, authors observed that suppression of cellular respiration by the bacteriostatic antibiotic was the dominant effect, blocking bactericidal killing [START_REF] Lobritz | Antibiotic efficacy is linked to bacterial cellular respiration[END_REF].

Following this observation, we can conclude that the increase observed on ATP production after 14 days in planktonic bacteria in W and SW or in biofilms in presence of wastewater corresponds to cell proliferation. This bacterial growth in presence of antibiotics could be explained by resistance, degradation or both mechanisms. TMP could have a predominant role since it is not degraded and it is known to inhibit bacterial DNA synthesis.

Anyway, even if ATP assay detects the activity of various types of microbial species, the methodology also has its limitations, as some pathogenic microbes may be present without eliciting a response in ATP; since they are present in relatively low numbers compared to the total number of bacteria [START_REF] Vang | Evaluation of ATP measurements to detect microbial ingress by wastewater and surface water in drinking water[END_REF]. To overcome this limitation, characterization of bacterial communities was carried out.

Characterization of bacterial communities

Richness and diversity indexes in planktonic bacterial communities were determined (Table 2). As expected, sterilization had a relevant impact in richness and diversity, being SW the group with lower richness and diversity indexes. The presence of biofilm increased the richness of planktonic communities in both BW and BSW but only the diversity of BSW communities. Overall, exposure to antibiotics had almost no impact on richness. Antibiotics impacted differentially the diversity of planktonic communities. Whereas antibiotic exposure increased the diversity of planktonic communities in BSW, it reduced it in BW.

Sterilization and presence of biofilms both had an impact on the structure of planktonic bacterial communities (Figure 1). Samples corresponding to the four tested conditions BSW, BW, SW and W formed four independent clusters. SW cluster was very distant from the other three groups, showing that SW community was the most different from the others. Inter-cluster distances between SW and W was bigger than between SW and BSW and also bigger than between W and BW, showing that sterilization was the most dominant factor in deciphering planktonic community structure. The biggest intercluster distance was found between SW and BW, revealing that in addition to sterilization, the presence of biofilm introduced also differences in the community structure. Interestingly, inter-cluster distance between BSW and W and between BSW and BW were similar indicating that when biofilms were present, the disruption caused by sterilization was in part restored. Intra-cluster distances were similar in all groups. Control samples (non-exposed to antibiotics) C_BSW and C_BW, clustered together in their respective group BSW and BW, revealing that antibiotic exposure had no impact on the structure of the planktonic bacterial communities.

In order to go deeper in the bacterial structure, taxonomic classification in biofilm and planktonic communities was performed at genera level (Figure 2 and 2SM) and phyla level (figure 3SM). Pedobacter, and finally a group gathering BW and BSW. Interestingly, the planctonic community in C_BSW was more similar to the one of SW than the one of BSW, with a remaining dominance of Flavobacterium. To compare, the planktonic community in C_BW was very similar to the one of BW.

It seems thus that an impact of the presence of antibiotic is only observable when sterilized wastewater was used inducing loss of diversity and richness.

Taxonomic annotation was employed to also estimate the relative bacteria abundance at the phylum level (Fig 3SM). Whereas Proteobacteria and Bacteroidetes were the predominant phyla in planktonic (average of 55 and 30% respectively) and biofilm (average of 55 and 20% respectively) bacterial communities, the abundance of other minor phyla was different among them. In biofilm samples, the other detected phyla were Chloroflexi, Firmicutes, Planctomycetes and Acidobacteria in similar relative abundances between control and exposed samples and Actinobacteria, Gemmatimonadetes and Latescibacteria at lower abundances (Fig 3SM b). In wastewater, only Actinobacteria and Firmicutes were also present and when biofilm was present other phyla as Chloroflexi, Planctomycetes and TM6_Dependentiae were present (Fig 3SM a). Previous studies reported that these phylogenetic groups were ubiquitous in both laboratory and pilot-scale anaerobic bioreactors [START_REF] Calderón | Analysis of microbial communities developed on the fouling layers of a membrane-coupled anaerobic bioreactor applied to wastewater treatment[END_REF][START_REF] Gao | Membrane fouling in an anaerobic membrane bioreactor: Differences in relative abundance of bacterial species in the membrane foulant layer and in suspension[END_REF][START_REF] Khuntia | Accumulation of antibiotic-resistant genes in anaerobic biofilm reactor fed with household chemical products[END_REF][START_REF] Lu | Pyrosequencing reveals highly diverse microbial communities in microbial electrolysis cells involved in enhanced H2 production from waste activated sludge[END_REF] and fit with the regular structure pattern of bacterial communities in WWTPs effluents [START_REF] Do | 16S rRNA gene based bacterial community structure of wastewater treatment plant effluents[END_REF]. Moreover, Proteobacteria, followed by Firmicutes, Bacteroidetes, and Actinobacteria were found to be the most mobile phyla associated with the transfer of ARGs and MGEs [START_REF] Hu | The bacterial mobile resistome transfer network connecting the animal and human microbiomes[END_REF]. Some phyla like Firmicutes, Bacteroidetes, and Beta-Proteobacteria were also depicted to be highly correlated with some antibiotic resistant strains such as ciprofloxacin resistant heterotrophs, ciprofloxacin-resistant enterococci, and sulfamethoxazole-resistant enterobacteria [START_REF] Novo | Antibiotic resistance, antimicrobial residues and bacterial community composition in urban wastewater[END_REF].

Experiments with sterilized wastewater were performed with the aim of reducing the number of planktonic bacteria (only 3 to 4 phyla were present), thus limiting the competition for nutrients in the reactors in favor of biofilm-forming bacteria. In sterilized wastewater conditions, by comparison to nonsterilized one, relative abundances of Actinobacteria, Planctomycetes, and TM6_Dependentiae were drastically reduced almost to extinction whereas Firmicutes increased. In reactors containing biofilms (BW), relative abundances of planktonic Firmicutes, Planctomycetes, Chloroflexi and TM6_Dependentiae phyla increased. TM6_Dependentiae and Planctomycetes phyla increased also in reactors containing biofilms and sterilized wastewater (BSW) but not Firmicutes, and this increase was higher in presence of antibiotic. Following these observations, TM6_Dependentiae and Planctomycetes can be phyla with a potentially relevant role in antibiotic resistance or degradation among planktonic bacteria. Furthermore, biofilm-forming bacteria seem to be involved in this mechanism since TM6_Dependentiae and Planctomycetes were not present in reactors containing only SW but their abundance increased in presence of antibiotics only when biofilms are present in the reactors.

Wastewater shaped the biofilm communities and conversely biofilm communities influenced the planctonic one. Exchange of taxons may occur on both sides. However, less changes are observed when diversity and richness are higher in both media.

Abundance of ARGs and putative ARBs in biofilms and wastewater

Numerous studies have investigated the occurrence and prevalence of ARGs through wastewater treated by different technologies [START_REF] Conte | Characterization of CTX-M enzymes, quinolone resistance determinants, and antimicrobial residues from hospital sewage, wastewater treatment plant, and river water[END_REF][START_REF] Lee | Quantitative and qualitative changes in antibiotic resistance genes after passing through treatment processes in municipal wastewater treatment plants[END_REF]. Biofilm grown under certain levels of antibiotic concentration are expected to develop a bacterial community more tolerant to the direct stress that is made up by antibiotics. Nevertheless, to our knowledge, the present study is the first one investigating the abundance of ARGs and their variation in biofilm and wastewater after long exposure with the most common detected antibiotics in wastewater.

The total and relative abundances of four ARGs (sul1,blaTEM,qnrS,ermB), and two MGEs (intl1, IS613) were estimated in biofilms and wastewaters collected from all the reactors. The abundance of 16S rDNA gene was also determined to quantify the total bacterial population.

As shown on the figure 4SM, copy numbers of bacterial 16S rDNA genes were 3*10 7 copies/ml in the treated wastewater and was statistically lower than the one in the sterilized water (6.8*10 7 copies/ml); this higher abundance for this less diverse planktonic community is probably due to the emergence and growth of opportunistic bacteria in absence of bacterial competition (Fahrenfeld et al., 2013). The presence of biofilms lowered these 16S abundances in both wastewaters whereas it increased diversity and change the microbial structure. This growth reduction in presence of biofilm may be due to the competition for substrates and/or higher growth into the biofilm by comparison to the planktonic communities as also shown with ATP measurements (fig 1SM b,c). The abundance of bacterial 16S

rDNA genes was also determined in biofilm and it was in a range from 1.3 to 3.3*10 9 gene/cm 2 (figure 4SM). No statistical differences were observed between the conditions, even with the abundance at the beginning of the experiment, but the lowest abundance was observed for the controls (C_BW, C_BSW).

Abundance of ARGs in wastewater

On the 6 targeted ARGs, qnrS, blaTEM and IS613 were below the quantification limits at the start or at the end of the experiment and were thus not considered in the data analysis (Table 5SM).

In the wastewater, the ARG concentration was globally 1.5log lower that the one measured in biofilm (fig 5SM andfig 3). Moreover, the absolute abundances of sul1 and intl1 were 2log higher than the one of ermB. The absolute abundance of ermB was not statistically different from one condition to another but tendencies were similar to the two other ARGs. For sul1 and intl1, their absolute abundances were similar in sterilized and non-sterilized wastewater; as the 16S increased in SW, the relative abundance was thus lower in SW which may suggest that the opportunistic bacteria did not carry these ARGs. Their absolute abundances were however significantly reduced (as similarly observed for 16S rDNA) by the presence of biofilm only in presence of non-sterilized wastewater. In the presence of sterilized wastewater, even if biofilm reduced the 16S planktonic abundance, the ARG absolute abundance remained similar, which ended up with a highest relative abundance. It underlines that when activity, competition and diversity are reduced between planktonic and biofilm communities, the dissemination of ARGs is favored.

Abundance of ARGs in biofilm

Similarly, to the wastewater samples, two ARGs (sul1 and ermB) and one MGE (intl1) were detected in all biofilm samples during the experiment, ermB being always the less concentrated one (2log lower).

The three other genes were not quantified as their values were always below detection limits.

Comparison between the different conditions was thus performed and checked statistically based on Kruskall Wallis-Conover tests (p-value<0.05) excepted for the controls because of low number of data (Fig 3).

The absolute abundance of sul1 was similar whatever the conditions and no increase was observed during the 14 days even if SMX concentration remained at a high level (Table 1). The spread of sul genes by horizontal gene transfer usually occurs rapidly in natural communities thanks to their linkage to class 1 integrons [START_REF] Gillings | Using the class 1 integron-integrase gene as a proxy for anthropogenic pollution[END_REF][START_REF] Sköld | Sulfonamide resistance: mechanisms and trends[END_REF] that was not observed in our experiment.

However, its relative abundance significantly increased during the 14 days and was higher in presence of sterilized WW. For the two other genes, a significant increase of the absolute and relative abundances was observed during the 14 d in presence of sterilized wastewater and for ermB these abundances were higher than the one observed in presence of non-sterilized wastewater. It seems that this condition, biofilm in presence of SW, is favorable to the gene dissemination in the biofilm compartment. It reveals that the differences in structure and diversity between planktonic and biofilm communities play a key role in the development and dissemination of resistance.

Considering all ARG results on wastewater and biofilm samples, we can first say that exposure to antibiotics is not the main parameter driving resistance selection and propagation which is in agreement with previously published data [START_REF] Auguet | Sewers as potential reservoirs of antibiotic resistance[END_REF]. Indeed, community structure, diversity and activity are the driven force. The condition that favors dissemination of ARG either in planktonic or biofilm communities is when wastewater is sterilized and put in presence of biofilm, underlying the important role of the biofilm. In such condition, the presence of biofilm allowed recovering a certain diversity of the planktonic community with a specific structure, dominated by Planctomycetaceae, . When using non-sterilized WW, the planktonic community also evolved in presence of biofilm but also the biofilm community, meaning that both communities influence each other. But these interactions seem to be different when diversity is previously modified and decreased as if the ecosystem was less efficient or resistant allowing more gene dissemination. It means that it is not only the genes themselves but which bacteria harboring these genes that must be considered to understand the dynamic of antibiotic resistance in wastewater and biofilms in contact with wastewater.

Putative ARB in biofilm

To further investigate the effect of the different conditions on less-dominant bacteria, genera known to harbor antibiotic-resistant-genes ('putative ARBs') based on the literature (Table 3SM) were looked for in sequencing data. Their absolute abundancies in wastewater are gathered in Figure 4.

Among genera known to harbor ARG, some were not detected in control wastewater but were abundant in wastewater spiked with antibiotics. Streptococcus, Exiguobacterium, Acholeplasma.

Methylophylaceae and Porphyromonadaceae were identified in control but their absolute concentrations were higher in wastewater spiked with antibiotics. Similar observation was reported by [START_REF] Shen | Pharmaceutical exposure changed antibiotic resistance genes and bacterial communities in soilsurface-and overhead-irrigated greenhouse lettuce[END_REF], where the abundance of Methylophilaceace increased after exposure. One possible explanation may be related to biodegradation properties of bacteria in the order of Methylophilales [START_REF] Liu | Metaproteomics reveals major microbial players and their biodegradation functions in a large-scale aerobic composting plant[END_REF].

Additionally, the genus Exiguobacterium has been recently characterized as a carrier of some ARGs encoding resistance to beta-lactam and sulfonamides [START_REF] Carneiro | Genome sequence of Exiguobacterium antarcticum B7, isolated from a biofilm in Ginger Lake, King George Island, Antarctica[END_REF]. The presence of antibiotics may then have favored these 5 groups of putative ARBs. Others, such as Mycobacterium, were found in controls and wastewater but not in sterilized wastewater. The effect of the presence of biofilms on the concentration of Mycobacterium, Clostridium sensus stricto 1 and Porphyromonadaceae is noticeable: absent in SW, they were found at the same concentration as in W in the presence of biofilms (BSW),

showing that the biofilm is a reservoir of putative ARBs that can colonize the planktonic phase. Finally, the presence of biofilm decreased the concentration of some putative ARBs, as it was also seen at a global level for total 16S rDNA. Concentration of most putative ARBs (Erysipelotrichaceae_UCG-004, Geobacter, Porphyromonadaceae, Methylophylaceae, Streptococcus, Exiguobacterium, Acholeplasma)

were lower in BW than in W (Fig 4).

Biofilms can then act in two opposite ways depending on the wastewater microbiome composition and diversity: on one side, biofilms can lower the level of ARB in wastewater but on the other side, it can also serve as a reservoir of ARB and re-contaminate low diversity wastewater exposed to these biofilms.

This means also that dissemination is a matter of diversity in between ecosystems and in our case of interaction between planktonic and biofilm microbial communities.

In this sense, a strategy to limit the proliferation of the ARGs in biofilms are needed. methods used to reduce clogging include precipitation, filtration, chlorination, acidification and pressure flushing.

The chlorination procedure has been recognised as the least expensive method to treat biological-growth related clogging but it has several disadvantages such as the formation of by-products. Recently, biological method based on the role of quorum sensing molecules such as N-acyl-homoserine lactones on biofilm formation in wastewater has been r studied by several groups [START_REF] Maddela | Roles of quorum sensing in biological wastewater treatment: A critical review[END_REF]. Finding a way to interrupt quorum sensing (a process called quorum quenching) would be a tool to counteract proliferation of ARGs in biofilms growing in wastewater pipes. Lately, several approaches have been studied to regulate biofilm formation, including the use of exogenous molecules such as quorum sensing inhibitors or enzymes degrading quorum sensing molecules [START_REF] Sun | Quorum sensing regulation methods and their effects on biofilm in biological waste treatment systems: A review[END_REF] or strategies using or inducing specific bacterial species with this ability [START_REF] Noori | Quorum quenching bacteria isolated from industrial wastewater sludge to control membrane biofouling[END_REF]. All these methods are well studied in membranes but application in irrigation pipes is still poorly studied, and the latest method can be an interesting future perspective.

Conclusions

• Antibiotics were partially degraded and remained above lowest MIC for environmental samples maintaining conditions for resistance selection. Degradation driven by interactions between planktonic and biofilm bacteria is hypothesized.

• Biofilm thickness was not affected by the 6 added antibiotics at a concentration of 10 µg L -1 each.

• Based on ATP measurements, the presence of antibiotics by comparison to the C_BSW and C_BW did not affect the microbial growth of planktonic bacteria after 14 days in W, SW and BSW but reduced microbial activity in biofilms exposed to SW.

• Antibiotic exposure had no impact in the structure of planktonic bacterial communities.

Sterilization was the most dominant factor explaining the planktonic community structure.

• Bacterial community composition and diversity are the driven forces for resistance selection and propagation in biofilms, rather than exposure to antibiotics.

• The abundance of some genera known to harbor ARG like Streptococcus, Exiguobacterium, Acholeplasma, Methylophylaceae and Porphyromonadaceae increased in reclaimed wastewater containing antibiotics.

• Interactions between planktonic, biofilm microbial communities and antibiotic resistance dissemination should be further investigated.

Strategies to combat the proliferation of ARGs in biofilms are needed. These could be developed to module biofilm bacterial diversity, based on quorum quenching, or using quorum sensing

  Bengtsson-Palme and Larsson determined the upper boundaries for minimal selective concentrations and suggested individual safety margins for antibiotics based on the extent of available MIC data (Bengtsson-Palme & Larsson, 2016). Predicted no-effect concentrations (PNECs) were calculated from available MIC values in the public EUCAST database and were used as an indicator for resistance selection. For the six antibiotics used here, the PNECs calculated were (in µg L -1 ): AMX (0.25), CEF (4), CLA (0.25), OFX (0.5), TMP-SMX (0.5). The concentration used in this study (10 µgL -1 ) is in any case above the estimated PNECs values, setting favorable conditions for resistance selection in our experimental conditions.

  Fig 2presents the 30 most abundant genera in all biofilm samples. The type of wastewater (sterilized or not) had an impact on the bacterial communities of biofilms. Indeed, BW and C_BW samples were dominated by several genera like Comamonadaceae, Anaerolineaceae (including some but not all BW and C_BW samples), Arenimonas, Flavobacterium, Thiocapsa, Run-SP154 and an unculture bacteria as dominant taxons. When wastewater was sterilized, the biofilms were exclusively dominated by the Comamonadaceae family. The predominance of Comamonadaceae family is common in biofilm evolved in chlorinated drinking water distribution system and the family was the main colonizers of biofilms from chlorinated drippers[START_REF] Lequette | Effects of the chlorination and pressure flushing of drippers fed by reclaimed wastewater on biofouling[END_REF].Investigation of the most dominant genera in wastewater (Figure2SM) showed that initial sterilization reduced the richness and shifted communities from dominance of taxon '12up' (Rhodocyclaceae) and Chitinophacaeae to dominance of Flavobacterium (Flavobacteriaceae), Fluviicola (Cryomorphaceae) and Pedobacter (Sphingobacteriaceae). Relative abundancy of Pirellula, Algoriphagus, Stenotrophomonas and Rickettsiales also increased in sterilized wastewater samples. The presence of biofilms impacted the planktonic community underlying interactions between both media. It reduced the dominant taxons and improved the diversity in both W and SW samples, with increased abundancy of Arenimonas, Planctomycetaceae, Cloacibacterium, Romboutsia, TM6-Dependentiae, Methyloversatilis, Legionella, Thermomonas, and Zoogloea, probably due to detachment and/or growth events. The dominant genera in biofilm (Comamonadaceae) did not colonized the planktonic community. Four types of bacterial communities were then identified in wastewater: W first with dominance of Rhodocyclaceae, SW with low diversity and dominance of Flavobacterium or

Flavobacteriaceae

  and Xanthomonadaceae (Fig 2SM), that may influence the biofilm community dominated by Comamonadaceae (Fig 2)

  financially supported by MUSE (Montpellier University of Excellence) Also part of this work supported by the European Union in the framework of the Project AIM (Attraction and Mobility of Researchers), under the responsibility of the Italian Ministry of University and Research. Authors would like to acknowledge Geoffrey Froment, Marine Muffat-Jandet, and François Liron from INRAE Montpellier for their technical assistance and good advices.

Figure 2 .

 2 Figure 2. Heatmap showing the relative abundance of the 30 most dominant bacterial genera in biofilms after 2 weeks of antibiotics exposure (excepted control).

Figure 3 :

 3 Figure3: Absolute (cm 2 ) and relative ([16S rDNA]) abundance of sul1, intl1 and ermB in biofilms before and after 2 weeks of antibiotics exposure (excepted control). Average and standard deviation values were obtained from 6 values (2 duplicated biofilms in 3 containers), apart for control biofilms (average of 2 duplicated biofilms). *: significance differences between conditions (Kruskal-test, p<0.05).

Figure 4 .

 4 Figure 4. Concentration in wastewater of genera known to harbor ARB ('putative ARB') after two weeks of antibiotics exposure (excepted control).

Table 1 :

 1 Antibiotic removal (in %) in containers with and without biofilm in treated and sterilized wastewater. Values are means of the three experimental replicates (n=3). Cephalexin and Amoxicillin were completely removed whatever the conditions. None of the antibiotics were quantified in the controlled container, BCWW and BCSWW.

		OFX	SMX	TRI	CLA
			% removal
	W	48.9 ± 0.6 6.2 ± 0.1 18.7 ± 0.4	31.6 ± 1.1
	BW	0.0	38.5 ± 0.3	0.0	46.8 ± 0.4
	SW	30.5 ± 0.9 15.3 ± 0.4 24.2 ± 0.8	35.6 ± 0.6
		30.9 ±
	BSW	0.12	20.5± 0.34	0±0.01	11.3 ± 0.11

Table 2 :

 2 Richness and diversity indices in wastewater samples after two weeks of experiment (average and standard deviation for the three containers per condition, one value per control containerin bold) Figure1. Principal coordinate analysis (PCoA) of the bacterial communities structure in wastewater samples after 2 weeks of antibiotics exposure (excepted control).

		Richness indices	Diversity indices
		Observed	Chao1	Simpson	Shannon
	C_BW	552	577	0.98	4.89
	BW	550±33	593±50	0.95±0.04	4.36±0.41
	W	462±51	496±44	0.95±0.02	4.22±0.41
	C_BSW	345	374	0.88	3.18
	BSW	349±46	371±48	0.95±0.01	3.83±0.23
	SW	134±7	142±6	0.88±0.02	2.93±0.21

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


El-Mehdi El Arar
email: el-mehdi.el-arar@uvsq.fr

Devan Sohier
email: devan.sohier@uvsq.fr

Pablo De Oliveira Castro

Eric Petit
email: eric.petit@intel.com

The Positive Effects of Stochastic Rounding in Numerical Algorithms

Keywords: Stochastic rounding, Floating-point arithmetic, Polynomial evaluation, Horner's algorithm, Numerical integration

Recently, stochastic rounding (SR) has been implemented in specialized hardware but most current computing nodes do not yet support this rounding mode. Several works empirically illustrate the benefit of stochastic rounding in various fields such as neural networks and ordinary differential equations. For some algorithms, such as summation, inner product or matrixvector multiplication, it has been proved that SR provides probabilistic error bounds better than the traditional deterministic bounds.

In this paper, we extend this theoretical ground for a wider adoption of SR in computer architecture. First, we analyze the biases of the two SR modes: SR-nearness and SR-up-or-down. We demonstrate on a case-study of Euler's forward method that IEEE-754 default rounding modes and SR-up-or-down accumulate rounding errors across iterations and that SR-nearness, being unbiased, does not. Second, we prove a O( √ n) probabilistic bound on the forward error of Horner's polynomial evaluation method with SR, improving on the known deterministic O(n) bound.

I INTRODUCTION

In floating-point arithmetic, rounding errors occur because of the finite representation of floating-point numbers in computers.

The accumulation of rounding errors can significantly reduce the accuracy of a computation [START_REF] Parker | Monte Carlo Arithmetic: exploiting randomness in floating-point arithmetic[END_REF], p. 8.

Stochastic arithmetic proposed in the 1950s by von Neumann and Goldstine [START_REF] Neumann | Numerical inverting of matrices of high order[END_REF] is a computing paradigm developed as a model for exact computation with imprecise data. However, hardware units proposing stochastic rounding (SR) are still unavailable in most computer machines. Various specialized processors have introduced it such as: Graphcore IPUs which supports SR for binary32 and binary16 arithmetic [START_REF]Graphcore Limited[END_REF]- [START_REF]c AI-Float™-Mixed Precision Arithmetic for AI: A Hardware Perspective[END_REF], or Intel neuromorphic chip Loihi [START_REF] Davies | Loihi: A neuromorphic manycore processor with on-chip learning[END_REF] to improve the accuracy of biological neuron and synapse models. Moreover, several related patents are owned by major chip designers such as AMD [START_REF] Loh | Stochastic rounding logic[END_REF], NVIDIA [START_REF] Alben | Stochastic rounding of numerical values[END_REF], IBM [START_REF] Bradbury | Reproducible stochastic rounding for out of order processors[END_REF], [START_REF] Bradbury | Stochastic rounding floating-point multiply instruction using entropy from a register[END_REF], and other computing companies [START_REF] Henry | Processor with memory array operable as either cache memory or neural network unit memory[END_REF]- [START_REF] Lifsches | In-memory stochastic rounder[END_REF]. These developments support the idea of hardware implementations using stochastic rounding becoming more available in the future.

Stochastic arithmetic has two main applications [START_REF] Croci | Stochastic Rounding: Implementation, Error Analysis, and Applications[END_REF]. First, it can be used to estimate empirically the numerical error of complex programs. In this context, stochastic arithmetic introduces a random noise in each floating-point operation to simulate the effect of rounding errors. To make this error analysis automatic, various tools such as Verificarlo [START_REF] Denis | Verificarlo: Checking Floating Point Accuracy through Monte Carlo Arithmetic[END_REF], Cadna [START_REF] Jézéquel | CADNA: a library for estimating round-off error propagation[END_REF] and Verrou [START_REF] Févotte | VERROU: a CESTAC evaluation without recompilation[END_REF] have been developed. Second, stochastic arithmetic is sometimes used as a replacement for the default deterministic rounding mode in numerical simulations. Indeed, it has been demonstrated that in multiple domains, SR provides positive effects compared to the deterministic IEEE-754 [START_REF]IEEE Standard for Floating-Point Arithmetic[END_REF] default rounding mode.

In particular, in the neural networks field, using SR instead of deterministic rounding [START_REF] Gupta | Deep Learning with Limited Numerical Precision[END_REF] enables training with smaller data types and better accuracy. Indeed, the noise during training has been observed to help regularization and avoid model over-fitting [START_REF] Höhfeld | Probabilistic rounding in neural network learning with limited precision[END_REF], [START_REF] Hoehfeld | Learning with limited numerical precision using the cascade-correlation algorithm[END_REF]. The positive effect of SR extends also to the calculation of the solution of ordinary differential equations (ODEs) in low precision [START_REF] Hopkins | Stochastic rounding and reduced-precision fixedpoint arithmetic for solving neural ordinary differential equations[END_REF], [START_REF] Fasi | Algorithms for Stochastically Rounded Elementary Arithmetic Operations in IEEE 754 Floating-Point Arithmetic[END_REF] where SR reduces the accumulation of rounding errors by avoiding stagnation phenomenon when the step decreases. Various other applications such as PDEs, Quantum mechanics, Quantum computing use SR to improve their results [START_REF] Croci | Stochastic Rounding: Implementation, Error Analysis, and Applications[END_REF].

The IEEE-754 norm defines five rounding modes for floating-point arithmetic which are all deterministic: round to nearest ties to even (default), round to nearest ties away, round to zero, round to +∞, and round to -∞. In section II, we present a floating-point arithmetic background, and, we describe two stochastic rounding modes defined in [START_REF] Parker | Monte Carlo Arithmetic: exploiting randomness in floating-point arithmetic[END_REF], p. 34: SR-nearness and SR-up-or-down. Section III presents our first contribution: we study the bias and we compare the two stochastic rounding modes above and the default rounding mode in the IEEE-754 norm (RN-nearest32) on rectangular integration, which is at the basis of Euler's Method for ODE. We show that, contrarily to SR-up-or-down, SR-nearness is unbiased. An exact expression and an estimation of the bias are given for SR-up-or-down. We show how the accumulation of errors with both SR-up-or-down and IEEE-754 modes leads to results significantly less accurate than with SR-nearness.

At the beginning of section IV, a probabilistic background, as well as some properties of SR-nearness rounding are presented. We recall some techniques that have been used in order to obtain a probabilistic bound for the inner product on √ n instead of the deterministic bound on n. We explain two approaches proposed by Higham and Mary [START_REF] Higham | A New Approach to Probabilistic Rounding Error Analysis[END_REF] and Ilse, Ipsen and Zhou [START_REF] Ipsen | Probabilistic Error Analysis for Inner Products[END_REF]. We extend these techniques for our second contribution and showing that they can still be used in situations where one multiplication operand is affected by an error. In particular, using SR-nearness and without additional assumption, we provide a probabilistic bound in O( √ n) rather than the deterministic bound which is in O(n). We conclude this section with numerical experiments comparing the bounds above for the Chebyshev polynomial.

II Background

A. Floating-point arithmetic

A normal floating-point number in such a format is a number x for which there exists a triple (s, m, e) such that x = ±m × β e-p , where β is the basis, e is the exponent, p is the working precision, and m is an integer (the significand) such that β p-1 ≤ m < β p . This triple is unique. We only consider normal floating-point numbers; detailed information on the floating-point format most generally in use in current computer systems is defined in the IEEE-754 norm [START_REF]IEEE Standard for Floating-Point Arithmetic[END_REF].

Let us denote F ⊂ R the set of normal floating-point numbers and x ∈ R. Upward rounding x and downward rounding x are defined by:

x = min{y ∈ F : y ≥ x}, x = max{y ∈ F : y ≤ x},
clearly, x ≤ x ≤ x , with equalities if and only if x ∈ F . The floating-point approximation of a real number x = 0 is one of x or x .

fl(x) = x(1 + δ ), (1)
where δ = fl(x)-x

x is the relative error: |δ | ≤ β 1-p . In the following, we note u = β 1-p . IEEE-754 mode RN (round to nearest, ties to even) has the stronger property 1 that |δ | ≤ 1 2 β 1-p = 1 2 u. For x, y ∈ F , the considered rounding modes verify fl(x op y) ∈ { x op y , x op y } for op ∈ {+, -, * , /}. Moreover, for IEEE-754 RN [START_REF]IEEE Standard for Floating-Point Arithmetic[END_REF] and stochastic rounding [START_REF] Connolly | Stochastic rounding and its probabilistic backward error analysis[END_REF] the error in one operation is bounded:

fl(x op y) = (x op y)(1 + δ ), |δ | ≤ u, (2)
specifically for RN we have |δ | ≤ 1 2 u. Assume that x is a real that is not representable: x ∈ R \ F . The machine-epsilon or the distance between the two floatingpoint numbers enclosing x is ε(x) = xx = β e-p . The fraction of ε(x) rounded away, as shown in figure 1, is We note x the integer part of x. The following lemma gives an important property of downward rounding.

θ (x) = x-x x -x x x x 1 2 ε(x) θ (x)ε(x)
Lemma II.1. Let x ∈ R \ F . β p-e x = β p-e x .
Proof. We know that β p-e x , β p-e x ∈ Z, and x < x < x , then β p-e x < β p-e x < β p-e x . We thus have β p-e x ≤ β p-e x < β p-e x . 1 In many works focusing on IEEE-754 RN, u is chosen to be 1 2 β 1-p .

Since xx = β e-p , then β p-e xβ p-e x = 1 and β p-e x ≤ β p-e x < β p-e x + 1.

B. Stochastic arithmetic

In order to define the two principal stochastic rounding modes, consider x the random variable of the distribution of results after random rounding of x. Then

x = random_round(x) = round(x + β e-p ξ ),
where ξ is a random variable that can be discrete or continuous and round is the default IEEE-754 rounding mode to the nearest.

• SR-nearness [START_REF] Parker | Monte Carlo Arithmetic: exploiting randomness in floating-point arithmetic[END_REF], p. 34 is defined for an ξ uniform random variable on ] -1 2 ; 1 2 [, hence ξ has mean 0 and standard deviation 1/ √ 12. In other words, SR-nearness consists in rounding up

x x x 1 -θ (x) θ (x)
x ∈ R \ F with probability θ (x) = (x -x )/( x -x ),
we round down with probability 1θ (x), proportional to the distances between x and the closest representable numbers.

Stott Parker shows that SR-nearness is unbiased [START_REF] Parker | Monte Carlo Arithmetic: exploiting randomness in floating-point arithmetic[END_REF], p. 34 so E( x) = x. • SR-up-or-down is defined for a ξ uniform random variable on ]θ (x); 1θ (x)[, ξ is biased with mean In other words, SR-up-or-down consists in rounding x up or down with probability 1 2 . SR-up-or-down mode can be expressed [START_REF] Parker | Monte Carlo Arithmetic: exploiting randomness in floating-point arithmetic[END_REF], p. 34 in terms of θ (x): since the two outcomes of SR-up-or-down mode are equiprobable, we have E( x) = x + x 2 , which allow us to write the bias as

E( x -x) = x + x 2 -x, because θ (x) = x-x x -x E( x -x) = ( x -x )( 1 2 -θ (x)) = ε(x)( 1 2 -θ (x)).
Thus, we conclude that SR-up-or-down is biased and the expected value depends on θ (x) and ε(x).

III Integrating a constant function

Rectangular integration rule is a classic approximation for performing numerical integration: the area under a curve is approximated by a sum of N rectangle areas:

b a f (t)dt ≈ N-1 ∑ k=0 h f (a + kh)
where h = b-a N . In particular, rectangular rule is one of the resolution techniques for ODE using Euler's forward method.

Verrou's tutorial [START_REF] Févotte | Etudier la qualité numérique d'un code avec Verrou[END_REF] integrates the cosinus function with the rectangular rule; with deterministic round to nearest or SRup-or-down modes, the solution is biased. When the number of integration steps grows, this bias can become high and degrade the quality of the solution. In this section, we show why deterministic and SR-up-or-down modes are sometimes biased with rectangular rule.

We perform the analysis on a constant function f (t) = 1 for all t ∈ [0; 1]. With f constant, the evaluation error is zero, making it clear how the numerical error accumulates on the summation.

Denote x = 1 = ∑ N-1 k=0 h, where h = 1/N. The distribution x is produced by summing N times the integration step h. We note s k the random variable for the partial sum at step 0 ≤ k ≤ N -1 and s k the exact expected result, with s N-1 = x. SR-up-or-down : As shown before, for each s k we introduce a bias corresponding to

E( s k -s k ) = ε(s k )( 1 2 -θ (s k )), from the definition of θ (s k ), we have 0 < θ (s k ) < 1, then -1 2 < 1 2 -θ (s k ) < 1 2 and |E( s k -s k )| < 1 2 ε(s k ).
Table I shows these different values for N = 20. Interestingly, in this table, we note that θ (s k ) is constant between two powers-of-the-base except for the first value. For example for 9 < k < 20, s k stays within [2 -1 ; 2 0 ) and both θ (s k ) and E(S ks k ) are constant. Let us show why that is always the case.

Suppose s k ∈ [β e ; β e+1 ). Then ε(s k ) = β e-p . At each step the next partial sum is computed as, s k+1 = fl(s k ) + h, in that case, using the lemma II.1, we have

θ (s k+1 ) = β p-e h -β p-e h = θ (h).
Thus θ (s k+1 ) depends only on h and e. Recursively for all l > 0 satisfying

s k+l ∈ [β e ; β e+1 ), θ (s k+l ) = β p-e h - β p-e h = θ (h) is constant. The bias E( s k -s k+l ) = ε(s k )( 1 2 -θ (s k+l )) = β e-p ( 1 2 -β p-e h -β p-e h ),
is also constant in this interval.

Between two powers of the base, θ remain constant as well as the bias. Because the bias is constant (and, consequently, its sign too), it accumulates across iterations.

The total bias can be written as,

E( x -x) = E( s N-1 -s N-1 ) = ε(s N-1 )( 1 2 -θ (s N-1 )), and
|E( x -x)| < 1 2 ε(s N-1 ).
We can neglect the effect of the first partial sum in each power-of-the-base interval. Since s k increases arithmetically, and the size of the power of the base interval grows geometrically, the bias in the last interval usually dominates since it contains more summation terms and has a larger ε.

Numerical experiment. We have verified numerically that the above expression for the bias closely predicts the bias measured with SR-up-or-down stochastic rounding.

We consider a fixed number of iterations N. We ran one time the C program in listing 1 with each of the two previously defined stochastic rounding modes as well as round to nearest.

Listing 1: Fixed-step rectangle integration of a constant float h = 1/N; float s = 0.0; for (int i=0 ; i < N ; i++) { s += h*1; } return s; Figure 4 shows that SR-nearness mode samples is unbiased regardless of the number N of rectangles. The unbiased nature is unsurprising, since SR-nearness mode can be seen as a sub case of Monte Carlo Arithmetic (MCA). Stott Parker proves [START_REF] Parker | Monte Carlo Arithmetic: exploiting randomness in floating-point arithmetic[END_REF], p. 46 that the expectation of a sum of terms with MCA is the exact mathematical result.

On the other hand, SR-up-or-down mode and RN-binary-32 samples have a bias, which confirms the previous results for SR-up-or-down mode. The maximal amplitude of the bias for both SR-up-or-down and increases with N because of errors accumulation. The bias is reproducible and constant across different runs.

In conclusion, this example illustrates that SR-nearness is unbiased not only for one elementary operation, but, even in other numerical methods such as rectangular integration, it is much closer to the expected value than SR-up-or-down or RN-binary32, in particular for N large. In the remainder of this paper, we focus on SR-nearness.

IV Probabilistic bound

The aim of this section is to introduce a probabilistic bound in O( √ nu) on the forward error of Horner algorithm, based on the Azuma-Hoeffding inequality and martingale properties. We first recall some probabilistic properties, then provide a review of the results made on the forward error in the numerically computed inner product, and we conclude with numerical experiments illustrating the previous results.

A. Probabilistic Background

A random variable Y is said to be mean independent of random variable X if and only if its conditional mean E[Y /X = x] equals its unconditional mean E(Y ) for all reals x such that the probability that X = x is not zero and we write

E[Y /X] = E(Y ). The random sequence (X 1 , X 2 , . . .) is mean independent if E[X k /X 1 , ..., X k-1 ] = E(X k ) for all k.
Proposition IV.1. Let X and Y two real random variables: X and Y are independents =⇒ X is mean independent from Y =⇒ X and Y are uncorrelated. The reciprocals of these two implications are false.

Under stochastic rounding, the elementary operations are stochastically rounded, hence, for x ∈ R, if fl(x) = x(1 + δ ) is obtained by SR-nearness then δ is a random variable such that E(δ ) = 0. For x 1 , x 2 , x 3 ∈ R, such that c = x 1 op x 2 op x 3 where op ∈ {+, -, * , /}, and

fl(c) = (x 1 op x 2 )(1 + δ 1 ) op x 3 (1 + δ 2 )
obtained from SR-nearness. δ 1 , δ 2 are random variables such that E(δ 1 ) = E(δ 2 ) = 0.

The following lemma has been proven in [START_REF] Connolly | Stochastic rounding and its probabilistic backward error analysis[END_REF], Lem 5.2 and shows that SR-nearness satisfies the property of mean independence.

Lemma IV.1. For some δ 1 , δ 2 , ..., in that order obtained from SR-nearness, the δ k are random variables with mean zero such that

E[δ k /δ 1 , ..., δ k-1 ] = E(δ k ) = 0.
Finally, we need to recall the concept of a martingale and the Azuma-Hoeffding inequality for a martingale [START_REF] Mitzenmacher | Probability and Computing: Randomized Algorithms and Probabilistic Analysis[END_REF].

Definition IV.1. A sequence of random variables M 1 , ..., M n is a martingale with respect to the sequence X 1 , ..., X n if, for all k,

• M k is a function of X 1 , ..., X k ,

• E(|M k |) < ∞, and • E[M k /X 1 , ..., X k-1 ] = M k-1 .
Lemma IV.2. (Azuma-Hoeffding inequality). Let M 0 , ..., M n be a martingale with respect to a sequence X 1 , ..., X n . We assume that there exist a k < b k such that a k ≤ M k -M k-1 ≤ b k for k = 1 : n. Then for any A > 0

P(|M n -M 0 | ≥ A) ≤ 2 exp - 2A 2 ∑ n k=1 (b k -a k ) 2 .
In the particular case a k = -b k we have

P |M n -M 0 | ≥ n ∑ k=1 b 2 k 2 ln(2/λ ) ≤ λ ,
where 0 < λ < 1.

B. Inner product bound

Under SR-nearness, the deterministic bound on the error of an inner product y = a b, where a, b ∈ R n , is proportional to nu, where u is the unit roundoff of the floating-point arithmetic in use. Wilkinson [START_REF] Wilkinson | Rounding errors in algebraic processes[END_REF], sec 1.33 had the intuition that the roundoff error accumulated in n operations is proportional to √ nu rather than nu. Based on the mean independence of errors established in Lemma IV.1, Higham and Mary [START_REF] Higham | A New Approach to Probabilistic Rounding Error Analysis[END_REF] and Ilse, Ipsen and Zhou [START_REF] Ipsen | Probabilistic Error Analysis for Inner Products[END_REF] have proved this result for SR. Both works build on the mean independence property of SR. This allows them to form a martingale, and then to apply the Azuma-Hoeffding concentration inequality. The difference between these two works is in the way they form the martingale. In [START_REF] Higham | A New Approach to Probabilistic Rounding Error Analysis[END_REF], the martingale is built using the errors accumulated in the whole process, while [START_REF] Ipsen | Probabilistic Error Analysis for Inner Products[END_REF] forms it by following step-by-step how the error accumulates. In particular, they distinguish between the multiplications and additions computed in the inner product, and carefully monitor their mean independences. In the following, we adapt this construction to Horner's algorithm.

C. Horner algorithm bound

Horner algorithm is an efficient way of evaluating polynomials. When performed in floating-point arithmetic this algorithm may suffer from catastrophic cancellations and yield a computed value with less accurate than expected.

In the following, we derive a probabilistic bound for the computed fl(P(x)) based on the previous methods applied for the inner product.

Model IV.1. Let P(x) = ∑ n i=0 a i x i , Horner's rule consists in writing this polynomial as

P(x) = ((a n x + a n-1 )x + a n-2 )x... + a 1 x + a 0 .
We define by induction the following sequence

r 0 = a n , r 2k-1 = r 2k-2 x, r 2k = r 2k-1 + a n-k , for all 1 ≤ k ≤ n. Likewise, We define r 0 = a n , r 2k-1 = r 2k-2 x(1 + δ 2k-1 ), r 2k = ( r 2k-1 + a n-k )(1 + δ 2k ),
for all 1 ≤ k ≤ n, with δ 2k-1 and δ 2k represent the rounding errors from the products and the additions, respectively.

Let us define Z i := r ir i for all 1 ≤ i ≤ 2n. The total forward error is

|Z 2n | = | r 2n -r 2n | = |fl(P(x)) -P(x)|. The computation of Z 2n introduces δ 1 , ..., δ 2n such that |δ k | ≤ u for all 1 ≤ k ≤ 2n.
We prove by induction that the accumulation of the (1 + δ k ) errors on the a i x i term for 0

≤ i ≤ n -1 reaches ∏ 2n k=2n-2i (1 + δ k ) := ϕ i , and for i = n, it reaches ∏ 2n k=1 (1 + δ k ) := ϕ n . Hence |fl(P(x)) -P(x)| = n ∑ i=0 a i x i (ϕ i -1) ≤ n ∑ i=0 |a i x i ||ϕ i -1| ≤ n ∑ i=0 |a i x i |γ 2n ,
where

γ 2n = (1 + u) 2n -1 = 2nu + O(u 2 ) (we recall that, ∀k, |δ k | ≤ u). Finally |fl(P(x)) -P(x)| |P(x)| ≤ cond 1 (P, x)γ 2n , (3)
where cond 1 (P, x) :=

∑ n i=1 |a i x i | |P(x)|
is the condition number of the evaluation of P in x using the 1-norm. The deterministic bound is proportional to nu. In the following, we prove a probabilistic bound in O( √ nu). The partial sums forward errors satisfy

Z 2k-1 = r 2k-1 -r 2k-1 = r 2k-2 x(1 + δ 2k-1 ) -r 2k-2 x = xZ 2k-2 + r 2k-2 xδ 2k-1 , Z 2k = r 2k -r 2k = ( r 2k-1 + a n-k )(1 + δ 2k ) -r 2k-1 -a n-k = r 2k-1 + ( r 2k-1 + a n-k )δ 2k -r 2k-1 = Z 2k-1 + ( r 2k-1 + a n-k )δ 2k ,
for all 1 ≤ k ≤ n. The sequence Z 1 , ..., Z 2n does not form a martingale with respect to δ 1 , ..., δ 2n because, due to the multiplication in odd steps,

E[Z 2k-1 /δ 1 , ..., δ 2k-2 ] = xZ 2k-2 .
In order to form a martingale and use the Azuma-Hoeffding inequality, we define the following variable change

Y i = Z i x (i+1)/2 ,
where (i + 1)/2 is the integer part of (i + 1)/2, we thus have

Y 2k-1 = Y 2k-2 + 1 x k-1 r 2k-2 δ 2k-1 , (4)
Y 2k = Y 2k-1 + 1 x k ( r 2k-1 + a n-k )δ 2k , (5)
for all 1 ≤ k ≤ n.

Theorem IV.1. The sequence of random variables Y 1 , ...,Y 2n is a martingale with respect to δ 1 , ..., δ 2n .

Proof. We check that the three conditions of definition IV.1 are satisfied. Throughout the proof, we note the set F k = {δ 1 , ..., δ k }.

• The recursion in Model IV.1 shows that Y i is a function of δ 1 , ..., δ i for all 1 ≤ i ≤ 2n.

• E(|Y i |) is finite because x and a k are finite for all ni ≤ k ≤ n and |δ j | ≤ u for all 1 ≤ j ≤ i.

• We prove that E[Y i /F i-1 ] = Y i-1
by distinguishing the even and odd case. Firstly, using the mean independence of δ 1 , ...δ 2k-1 and equation ( 4) we obtain

E[Y 2k-1 /F 2k-2 ] = E[Y 2k-2 /F 2k-2 ] +E[ 1 x k-1 r 2k-2 δ 2k-1 /F 2k-2 ] = Y 2k-2 + 1 x k-1 r 2k-2 E[δ 2k-1 /F 2k-2 ] = Y 2k-2 .
Secondly, using the mean independence of δ 1 , ...δ 2k and equation [START_REF]c AI-Float™-Mixed Precision Arithmetic for AI: A Hardware Perspective[END_REF] we obtain

E[Y 2k /F 2k-1 ] = E[Y 2k-1 /F 2k-1 ] +E[ 1 x k ( r 2k-1 + a n-k )δ 2k /F 2k-1 ] = Y 2k-1 + 1 x k ( r 2k-1 + a n-k )E[δ 2k /F 2k-1 ] = Y 2k-1 .
Lemma IV.3. The above martingale Y 1 , ...,Y 2n satisfies

|Y i -Y i-1 | ≤ C i u, 1 ≤ i ≤ 2n, where C 2k-1 = |a n |(1 + u) 2k-2 + k-1 ∑ j=1 |a n-j ||x| -j (1 + u) 2(k-j)-1 , C 2k = |a n |(1 + u) 2k-1 + k ∑ j=1 |a n-j ||x| -j (1 + u) 2(k-j) , for all 1 ≤ k ≤ n. Proof. Note that Y 0 = 0, then |Y 1 -Y 0 | = |Y 1 | = |a n |
and the equality holds for C 1 . Using equation ( 4)

|Y 2k-1 -Y 2k-2 | ≤ 1 |x| k-1 | r 2k-2 |u. However | r 2k-2 | ≤ | r 2k-3 |(1 + u) + |a n-k+1 |(1 + u) ≤ | r 2k-4 ||x|(1 + u) 2 + |a n-k+1 |(1 + u),
by induction we obtain

| r 2k-2 | ≤ |a n ||x| k (1 + u) 2k-2 + k-1 ∑ j=1 |a n-j ||x| k-j (1 + u) 2(k-j)-1 .
This completes the proof for C 2k-1 for all 1 ≤ k ≤ n. A similar approach can be applied to proving the same result for C 2k for all 1 ≤ k ≤ n.

We now have all the tools to state and then demonstrate the main result of this section: Theorem IV.2. Under SR-nearness, for all 0 < λ < 1 and with probability at least 1λ

|fl(P(x)) -P(x)| |P(x)| ≤ cond 1 (P, x) √ uγ 4n ln(2/λ ), (6)
where cond 1 (P, x) =

∑ n i=1 |a i x i | |P(x)|
is the condition number of the polynomial evaluation and γ 4n = (1 + u) 4n -1.

P |Y 2n | ≤ u 2n ∑ i=1 C 2 i 2 ln(2/λ ) ≥ 1 -λ , it follows that |Z 2n | ≤ u 2n ∑ i=1 (|x| n C i ) 2 2 ln(2/λ ),
with probability at least 1λ , where

|x| n C 2k = |a n ||x| n (1 + u) 2k-1 + k ∑ j=1 |a n-j x n-j |(1 + u) 2(k-j) ≤ (1 + u) 2k-1 k ∑ j=0 |a n-j x n-j | ≤ (1 + u) 2k-1 n ∑ j=0 |a j x j |, for all 1 ≤ k ≤ n. Hence, (|x| n C 2k ) 2 ≤ (1 + u) 2(2k-1) n ∑ j=0 |a j x j | 2 .
In a similar way

(|x| n C 2k-1 ) 2 ≤ (1 + u) 2(2k-2) n ∑ j=0 |a j x j | 2 . Thus 2n ∑ i=1 (|x| n C i ) 2 ≤ n ∑ j=0 |a j x j | 2 2n-1 ∑ i=0 ((1 + u) 2 ) i = n ∑ j=0 |a j x j | 2 ((1 + u) 2 ) 2n -1 (1 + u) 2 -1 = n ∑ j=0 |a j x j | 2 γ 4n u 2 + 2u . As a result |fl(P(x)) -P(x)| = |Z 2n | ≤ n ∑ j=0 |a j x j | uγ 4n 2 + u 2 ln(2/λ ),
with probability at least 1λ . Finally

|fl(P(x)) -P(x)| |P(x)| ≤ cond 1 (P, x) √ uγ 4n ln(2/λ ),
with probability at least 1λ .

Remark IV.1. The bounds (3) and ( 6) have the same condition number, but differ in another factor: γ 2n for (3) against √ uγ 4n ln(2/λ ) for [START_REF] Davies | Loihi: A neuromorphic manycore processor with on-chip learning[END_REF].

For n such that 2nu < 1, [START_REF] Higham | Accuracy and stability of numerical algorithms[END_REF], Lemma 3.1 implies

γ 2n ≤ 2nu 1-2nu , it follows that for 4nu < 1 √ uγ 4n ≤ 4nu 2 1 -4nu = u √ n 2 √ 1 -4nu .
For n large, Taylor's formula implies γ n = nu + O(u 2 ) and γ 2n ≈ 2nu. This approach can't be used for √ γ 4n because it's indeterminate in 0. However,

lim u→0 √ uγ 4n √ nu = 2 ⇐⇒ √ uγ 4n ≈ 2 √ nu.
Eventually, the probabilistic bound for the forward error of Horner's algorithm is in O( √ nu).

D. Numerical experiments.

In this section, we illustrate that the probabilistic bound is tighter than the deterministic bound for SR-nearness forward error on a numerical application: the evaluation of the Chebyshev polynomial. We use Horner's method to evaluate the polynomial P(x) = T N (x) = ∑ N 2 i=0 a i (x 2 ) i where T N is the Chebyshev polynomial of degree N. Consider an even N = 2n. We use single-precision (binary32) for both SR-nearness and round to nearest ties to even. All SR computations are repeated 30 times with verificarlo [START_REF] Denis | Verificarlo: Checking Floating Point Accuracy through Monte Carlo Arithmetic[END_REF]; we plot all samples and the forward error of the average of the 30 SR instances. The following error bounds and evaluations apply: Chebyshev polynomial is ill-conditioned near 1: in figure 5, we evaluate T 20 (x) for x ∈ [ 8 64 ; 1]. As expected and due to catastrophic cancellations among the coefficients, the condition number increases from 10 0 to 10 7 which explains the increase of numerical error from 10 -7 to 10 -1 . The probabilistic bound is closer to the forward error points than the deterministic bound even for a small n = 10. The average of SR-nearness stays below RN-binary32 for almost all points.

In Figure 6, the two previous bounds and the forward error are divided by the condition number cond(P, x), and the evaluation in x = 24/26 ≈ 0.923 is plotted for various polynomial degrees N. The comparison between the two bounds is fairly visible in figure 6. By increasing N, the deterministic bound draws away from the forward error faster than the probabilistic bound and for N large, the gap becomes more interesting. SRnearness points are between 10 -8 and 10 -10 , versus 10 -7 for the probabilistic bound and 10 -6 for the deterministic bound.

V Conclusion.

Stochastic rounding has drawn a lot of attention in various domains [START_REF] Croci | Stochastic Rounding: Implementation, Error Analysis, and Applications[END_REF], [START_REF] Gupta | Deep Learning with Limited Numerical Precision[END_REF], [START_REF] Hopkins | Stochastic rounding and reduced-precision fixedpoint arithmetic for solving neural ordinary differential equations[END_REF], [START_REF] Fasi | Algorithms for Stochastically Rounded Elementary Arithmetic Operations in IEEE 754 Floating-Point Arithmetic[END_REF] due to its efficiency compared to the default rounding mode. The fact that SR-nearness satisfies mean independence (a weaker property than independence) leads to an expected value that coincides with the exact value. We have shown that the bias in SR-up-or-down can significantly reduce the precision of the computation, even on simple algorithms such as rectangular integration section III, and that SR-nearness can remain unbiased and provide the full expected precision on them. We also discussed that using SR-nearness leads to having a probabilistic bound in O( √ nu), compared to the O(nu) deterministic bound for the inner product forward error. We have shown this property for Horner's method using Azuma-Hoeffding inequality and martingale properties. As opposed to the study made for the inner product, the issue of this algorithm is that the martingale does not appear explicitly; nevertheless, a change of variable shows that it is present, allowing the use of concentration inequalities. As future work we will investigate more complex algorithms with non-explicit martingales.

Fig. 1 .

 1 Fig. 1. θ (x) is the fraction of ε(x) to be rounded away.

Fig. 2 .

 2 Fig. 2. SR-nearness.

1 2 -Fig. 3 .

 23 Fig. 3. SR-up-or-down.

θ

  (s k+1 ) = β p-e (s k+1s k+1 ) = β p-e s k+1β p-e s k+1 = β p-e fl(s k ) + β p-e hβ p-e fl(s k ) + β p-e h . Since fl(s k ) ∈ F , we have β p-e fl(s k ) ∈ Z and β p-e fl(s k ) + β p-e h = β p-e fl(s k ) + β p-e h .

Figure 4 Fig. 4 .

 44 Figure 4 plots the three distributions over N.

Proof.

  Recall that | r 2nr 2n | = |Z 2n | = |x n ||Y 2n |. Therefore, Y 1 , ...,Y 2n is a martingale with respect to δ 1 , ..., δ 2n and Lemma IV.3 implies |Y i -Y i-1 | ≤ C i u for all 1 ≤ i ≤ 2n. Using the Azuma-Hoeffding inequality yields

  Probabilistic bound = cond 1 (P, x) √ uγ 4n ln(2/λ ), Deterministic bound = cond 1 (P, x)γ 2n , SR-nearness = |fl(P(x)) -P(x)| |P(x)| .

Fig. 5 .

 5 Fig. 5. Probabilistic bound with λ = 0.5 vs deterministic bound of the computed forward errors of Horner's rule for Chebyshev polynomial T 20 (x).

Fig. 6 .

 6 Fig. 6. Forward errors/cond(P,x) of Horner's rule for Chebyshev polynomial T N (24/26).

TABLE I s

 I k , θ , bias and ε for N = 20.

	k	s k	θ (s k )	E(S k -s k )	ε(s k )
	2 0.150... 0.7500 -3.725290e-09 1.490116e-08
	3 0.200... 0.2500	3.725290e-09 1.490116e-08
	4 0.250... 0.6250 -3.725290e-09 2.980232e-08
	5 0.300... 0.6250 -3.725290e-09 2.980232e-08
	6 0.350... 0.6250 -3.725290e-09 2.980232e-08
	7 0.400... 0.6250 -3.725290e-09 2.980232e-08
	8 0.450... 0.6250 -3.725290e-09 2.980232e-08
	9 0.500... 0.3125	1.117587e-08 5.960464e-08
	10 0.550... 0.8125 -1.862645e-08 5.960464e-08
	11 0.600... 0.8125 -1.862645e-08 5.960464e-08
	12 0.650... 0.8125 -1.862645e-08 5.960464e-08
	13 0.700... 0.8125 -1.862645e-08 5.960464e-08
	14 0.749... 0.8125 -1.862645e-08 5.960464e-08
	15 0.799... 0.8125 -1.862645e-08 5.960464e-08
	16 0.849... 0.8125 -1.862645e-08 5.960464e-08
	17 0.899... 0.8125 -1.862645e-08 5.960464e-08
	18 0.949... 0.8125 -1.862645e-08 5.960464e-08
	19 0.999... 0.8125 -1.862645e-08 5.960464e-08
	Finally

Acknowledgment

This research was supported by the French National Agency for Research (ANR) via the InterFLOP project (No. ANR-20-CE46-0009).