Alexandra Rahmani

Leila Parizadeh

Marine Baud

Yoannah François

A Bazire

Sophie Rodrigues

Yannick Fleury

Héléna Cuny

Eric Debosse

Joëlle Cabon

ENvironment Leila Parizadeh Littoral

Francois Yoannah

Station LPGP/INRAE Alexis Sysaaf

Bazire

Joelle Cabon

Laure Bellec

Thierry Morin
email: thierry.morin@anses.fr

Potential of Marine Strains of Pseudoalteromonas to Improve Resistance of Juvenile Sea Bass to Pathogens and Limit Biofilm Development

Keywords: Antimicrobial peptides, Pseudoalteromonas, Seabass, Probiotics

published or not. The documents may come

L'archive ouverte

Introduction

The European sea bass (Dicentrarchus labrax) is naturally found in the European waters, from the Northeast Atlantic (off Norway) to the Mediterranean Sea [1]. Sea bass aquaculture represents an important economic market with more than 191 000 tones (t) produced worldwide in 2016, out of which 82 000 t were farmed in Europe [2]. Even though zootechnical improvements have strongly contributed to the optimization of sea bass production over the last decades, the farmed sea bass market is threatened by episodes of epidemics caused by pathogenic agents such as bacteria, viruses or parasites [1], which can result in substantial mortalities and economic losses [3].

Among bacterial diseases affecting sea bass, vibriosis includes infections caused by different Gramnegative Vibrio species such as Vibrio alginolyticus [4], Vibrio parahaemoliticus [START_REF] Khouadja | Characterization of Vibrio parahaemolyticus isolated from farmed sea bass (Dicentrarchus labrax) during disease outbreaks[END_REF], Vibrio anguillarum [6-8] and Vibrio harveyi [START_REF] Breton | Mediterranean n sh pathologies; present status and new developments in prophylactic methods[END_REF][START_REF] Korun | Marine vibrios associated with diseased sea bass (Dicentrarchus labrax) in Turkey[END_REF]. Infection by V. harveyi was rst reported to induce ulcerative lesions in European sea bass in the late 1990s [START_REF] Breton | Mediterranean n sh pathologies; present status and new developments in prophylactic methods[END_REF]. The disease is characterized by lethargy, exophthalmus, ulcers and skin lesions, hemorrhaging in the internal organs, ascites and paleness of the kidney and liver. V. harveyi occurs naturally in marine habitats [START_REF] Zhang | Signi cance of Vibrio species in the marine organic carbon cycle-a review[END_REF] and it has already been described as a pathogenic species for a large range of marine animals, from mollusks to crustaceans and sh [START_REF] Austin | Vibrio harveyi: a signi cant pathogen of marine vertebrates and invertebrates[END_REF]. This pathogen affects several sh species, causing a large range of symptoms [START_REF] Zhang | Vibrio harveyi: a serious pathogen of sh and invertebrates in mariculture[END_REF] such as skin ulcerations in Solea senegalensis [START_REF] Zorrilla | Vibrio species isolated from diseased farmed sole, Solea senegalensis (Kaup), and evaluation of the potential virulence role of their extracellular products[END_REF], tail rot disease in Sparus aurata [START_REF] Haldar | Identi cation of Vibrio harveyi as a causative bacterium for a tail rot disease of sea bream Sparus aurata from research hatchery in Malta[END_REF] or necrotizing enteritis in Paralichthys dentatus [START_REF] Gauger | Vibrio harveyi and other bacterial pathogens in cultured summer ounder, Paralichthys dentatus[END_REF].

During the past decades, several strategies to ght against V. harveyi infection have been developed to limit the use of antibiotics [START_REF] Zhang | Vibrio harveyi: a serious pathogen of sh and invertebrates in mariculture[END_REF], including phage therapy [16], dietary supplements with probiotics [START_REF] Gupta | Growth performance, feed utilization, digestive enzyme activity, innate immunity and protection against Vibrio harveyi of freshwater prawn, Macrobrachium rosenbergii fed diets supplemented with Bacillus coagulans[END_REF] or inhibition of quorum sensing [18]. In 2019, a study investigated the protective e cacy of a Pseudoalteromonas strain in European abalone against infection by V. harveyi ORM4 [START_REF] Offret | Protective E cacy of a Pseudoalteromonas Strain in European Abalone, Haliotis tuberculata, Infected with Vibrio harveyi ORM4[END_REF]. That study demonstrated that abalone immersed prior to infection in a seawater suspension of Pseudoalteromonas hCg-6 -isolated from the hemolymph of healthy marine mollusks [START_REF] Defer | Antimicrobial peptides in oyster hemolymph: The bacterial connection[END_REF] -showed signi cantly improved viability, with a 24% increase in the survival rate [START_REF] Offret | Protective E cacy of a Pseudoalteromonas Strain in European Abalone, Haliotis tuberculata, Infected with Vibrio harveyi ORM4[END_REF]. Pseudoalteromonas, well known to be associated with marine organisms, produce antimicrobial compounds such as alterins, which may contribute to host defense [START_REF] Offret | Spotlight on Antimicrobial Metabolites from the Marine Bacteria Pseudoalteromonas: Chemodiversity and Ecological Signi cance[END_REF][START_REF] Offret | Alterins, a new family of marine antibacterial cyclolipopeptides[END_REF]. Alterins are considered to be one of the largest families of antibacterial cyclolipopeptides from marine bacteria [START_REF] Offret | Alterins, a new family of marine antibacterial cyclolipopeptides[END_REF]. They are especially produced by Pseudoalteromonas strains hCg-6, hCg-42 and hOe-125. Alterocin, a 139 amino acid protein is produced by Pseudoalteromonas sp. 3J6 and exhibits antibio lm effects without bactericidal activities, which is rare and original [START_REF] Jouault | Alterocin, an Antibio lm Protein Secreted by Pseudoalteromonas sp. Strain 3J6[END_REF].

Viral encephalopathy and retinopathy induced by the nervous necrosis virus (NNV), a betanodavirus, is one of the most important diseases for European sea bass. Described in the 1990s, it has been reported to cause mass mortalities in hatcheries and aquaculture farms, as well as in the natural environment [START_REF] Bellance | L'encephalite virale du loup de mer[END_REF][25][26][START_REF] Skliris | Induction of nodavirus disease in seabass, Dicentrarchus labrax, using different infection models[END_REF]. No e cient treatment or vaccine exists to date to ght against this pathogen, which has an extremely wide host spectrum and signi cant recombination capacities due to its genome composed of two RNA fragments [28].

In this study, we investigated the potential of Pseudoalteromonas strains hCg-42, hOe-125 and 3J6 to improve the resistance capacities of European sea bass, after impregnation for several weeks with these probiotic candidates, against V. harveyi and NNV. After characterizing the e ciency of the immersion process used to impregnate the sh with the Pseudoalteromonas strains, sea bass were repeatedly immersed with the probiotic candidates over a period of at least 2 months, and then experimentally infected by V. harveyi or NNV. The presence of probiotic candidates in sh, their impact on bio lm parameters, the kinetics of mortalities associated with the infections and various markers of the immune system were monitored during the impregnation and/or infection phases.

Materials And Methods

Production of the probiotic suspensions

The bacterial strains used in this study are described in Table 1. Pseudoalteromonas strains hCg-42, hOe-125 and 3J6 were selected as probiotic candidates. The Pseudoalteromonas strain RA15, which does not produce any alterin or alterocin compounds, was used as a control strain with no expected antibacterial effect. Pseudoalteromonas strains were grown on Marine Agar (Sigma-Aldrich) or in 5 mL Marine Broth (Sigma-Aldrich) with agitation at 150 rpm for 24 h. A volume of 2.5 mL of this pre-culture was then added to a nal volume of 220 mL of Marine Broth, cultured for 24 h in the same conditions and centrifuged at 3220 × g for 25 min. After removing the culture medium, the cell pellet was washed twice, i.e. centrifuged at 3220 × g for 30 min, and re-suspended in sterile SeaSalts (Sigma-Aldrich) at 3% to obtain the nal suspension for the immersion procedure. The nal concentration of the suspension was controlled by performing serial dilutions in duplicate on Marine Agar, from 10 - 1 to 10 - 8 , for each strain tested. After two days of culture, the concentrations obtained were over 10 9 colony-forming units (CFU)/mL. #32741-2020121509556347 v5). Two batches of European bass from the hatcheries at the Fermes Marine du Soleil (https://fermesmarinesdusoleil.com ) sh farm were used in this study. Batch 1 was composed of 1500 juvenile sea bass with an average weight of 15 g. They were vaccinated against V. anguillarum and Photobacterium damselae subsp. piscicida; this vaccination does not confer any crossprotection against V. harveyi infection. Batch 2 was composed of 1500 juvenile sea bass with an average weight of 5 g, not vaccinated to avoid interference with the NNV challenge. All batches were free of viral septicemia hemorrhagic virus (VSHV), infectious hematopoietic necrosis virus (IPNV), infectious pancreatic infectious virus (IHNV) and NNV. Their health status was checked during the quarantine period following their arrival, with autopsies and general bacteriological and virological analyses.

Fish were kept in ltered seawater (treated on sand and activated carbon lters), sterilized with UV. The water circulation was continuous (with tank water renewal occurring at least once hourly), with a temperature regulated at 20°C ± 2°C and an oxygen saturation greater than or equal to 80%. Throughout the rearing protocols, the sh received a diet appropriate for their weight and adapted to the water temperature (Le Gouessant, pellet diameter ranging from 1.7 to 3.2 mm). If needed, compassionate euthanasia was employed in the event of observation of signs of suffering or at the end of the experiment. Euthanasia involved the addition of a lethal dose of 100 ppm of Eugenol (Fili@Vet Reseau Cristal) to the tank water. Fish were exposed to Eugenol until the complete disappearance of all respiratory activity ceased.

3. Impregnation of sea bass with Pseudoalteromonas

General procedure

The impregnation procedure consisted in immersing the sh for 4 h in a bath of limited volume of hyperoxygenated seawater containing the probiotic strains at a determined concentration. At the end of this session, the tanks were again continuously supplied with seawater. Fish were observed at least every 30 minutes during immersion to visually check for abnormal behavior.

Assessment of the safety of the Pseudoalteromonas strains for European bass

The safety of the tested Pseudoalteromonas strains for the European bass was assessed by immersing three sh from Batch 2 in 5 L of hyperoxygenated seawater at 20°C ± 2 for 4 h at a concentration of 10 7 CFU/mL of Pseudoalteromonas strains, which is tenfold higher than the concentration selected for the impregnation procedure. Experimental treatment groups were composed of Pseudoalteromonas strain hCg-42 combined with strain hOe-125, Pseudoalteromonas 3J6, Pseudoalteromonas RA15 and a control with no bacteria added. The overall condition of the sh (behavior, mobility and respiration) was followed for 1 week after this impregnation.

Evaluation of the persistence of probiotic candidates in sh after one immersion session

Experimental immersions (see Section 3.1.) were performed using 35 sh from Batch 1 per tank in 8 L of seawater following three impregnation treatments: (1) with strain 3J6, in duplicate; (2) with a combination of strains hCg-42 and hOe-125 in duplicate and (3) without bacteria (control). The nal concentration of the probiotic candidates was 10 6 CFU/mL of seawater for each. Right before and after this immersion session, sh were transferred to 50 L tanks, and mucus and gills were sampled on three sh per impregnation treatment to detect probiotic candidates (T0 and T4H). This sampling procedure was repeated at 24 h, 48 h, 72 h and 96 h post-immersion (n = 3 sh per time point and impregnation treatment). Tissues were quick-frozen in liquid nitrogen and then conserved at -80°C until analysis. This procedure is described in Fig. 1.

Long-term impregnation

The experimental procedure is described in Fig. 2. Four experimental treatment groups were formed: (

Control, (2) strain RA15, (3) combined strains hCg-42 + hOe-125 and (4) strain 3J6. For Batches 1 and 2, each group was composed of four biological replicates of 50 L tanks containing 70 sea bass per tank, except for the control group that included ve biological replicates. Immersion was carried out in a volume of 12 L of seawater containing 10 6 CFU/mL of each tested strain. This protocol was repeated every two weeks for 8 weeks for Batch 1 (total of four immersion sessions) and 12 weeks for Batch 2 (total of six immersion sessions). For the control treatment, no bacteria were added to the tanks. Samples of mucus and gills were taken twice, after 4 h of immersion, to detect the probiotic candidates. This sampling was done at T0 (i.e. after the rst immersion session) and T2 (after the second immersion session, 2 weeks after the beginning of the experiment) for Batch 1 and at T0 and T4 (4 weeks after the beginning of the experiment) for Batch 2.

Analysis of bio lm formation

Bio lm formation was monitored on 76 × 26 mm glass slides during the impregnation of sh from Batch 2. To do so, slides were immersed in sh tanks during the rst 8 weeks of impregnation. Thereafter, the slides were collected and stored at + 4°C for 24 h in tank seawater. For analyses, slides were rinsed twice in sterile seawater, dried and stained with 15 µL of 5 µM SYTO-9 (Invitrogen -Life Science Technology) for 45 min. SYTO-9 uorescence was measured with an 488 nm excitation lter and a 500 and 550 nm emission lter by confocal laser scanning microscopy, CLSM, (Zeiss, LSM710) using a 63x oil immersion objective. At least three images were acquired for each slide. 3D images were acquired using ZEN software (Zeiss). ZenBlue software (Zeiss, license-free version) was used to process data from the 3D images. The bio lm stacks were analyzed using COMSTAT software [START_REF] Heydorn | Quanti cation of bio lm structures by the novel computer program COMSTAT[END_REF] to estimate the biovolume (µm 3 /µm 2 ),the average thickness (µm) and the maximum thickness (µm).

5. Effects of the Pseudoalteromonas strains against V. harveyi and NNV

Selection of the sh pathogen isolates

The infectious strain of V. harveyi no. 94473 1811603 AQN553P2 used in this study, kindly provided by Dr. Alain Le Breton (Vet'Eau), was isolated from sea bass during an episode of mortality (Table 1). This strain was cultivated on Tryptic Soy Agar (TSA) medium at 25°C. The red-spotted grouper NNV strain PP160 was produced and titrated on a snakehead sh cell line (SSN-1 cells), as previously described [START_REF] Thiery | Genomic classi cation of new betanodavirus isolates by phylogenetic analysis of the coat protein gene suggests a low host-sh species speci city[END_REF].

The viral suspension was kept at -80°C until used.

In vitro characterization of the effect of Pseudoalteromonas strains against V. harveyi

The antimicrobial activity of Pseudoalteromonas strains against V. harveyi was assessed using the welldiffusion method. Brie y, bacterial strains were grown in Marine Broth at 18°C under shaking (100 rpm) for 24 h for V. harveyi or 72 h for Pseudoalteromonas strains (Pseudoalteromonas hCg-42, Pseudoalteromonas hOe-125, Pseudoalteromonas 3J6 as a positive control strain and Pseudoalteromonas RA15 as a negative control strain). Then, 1 mL of each Pseudoalteromonas culture was centrifuged (6000 × g for 5 min at 4°C) and the supernatant was inactivated 5 min at 100°C before being tested against target bacteria. To test the activity of these supernatants, Marine Agar plates were inoculated on the surface with 5 mL of an exponential growth culture of V. harveyi adjusted to 1 × 10 6

CFU mL - 1 . When dry, wells (4 mm in diameter) were punched into the Marine Agar medium and supernatants (20 µL) or controls (Marine Broth for negative control and polymyxin B at 1 µg mL - 1 as positive control against Gram-negative target bacteria) were dropped into agar punch holes of 0.4 cm.

After overnight incubation at the optimal growth temperature of the strain, antibacterial activity was detected based on the appearance of an inhibition halo around wells.

Experimental infection of impregnated sea bass with V. harveyi

After two steps of cultivation on TSA medium at 25°C for 24 h, 5 mL of TSB was added to the culture plate to resuspend the bacteria. Optical density (OD) was measured at 600 nm wavelength and the suspension was diluted in TSB to calibrate the concentration for sh injection.

To de ne the 50% lethal dose (LD50), four different concentrations of V. harveyi were tested in 10 sh per concentration (from Batch 1), by intraperitoneal injection (IP) (100 µL per sh). The LD 50 was de ned as the concentration inducing the mortality closest to 50% one week after infection at a water temperature of 20°C ± 2. For the infection, European sea bass resulting from long-term impregnation (50 sh from Batch 1 with an average weight of 32.6 g per tank; 4 replicates per treatment), were anesthetized with Eugenol

(1:500 nal concentration) and individually IP injected with a volume of 100 µL containing 4.6 × 10 8 CFU of V. harveyi (Fig. 2). One tank of the control group was treated with sterile TSB as a negative control of the infection (i.e. non-infected control group). Mortalities were followed daily for 2 weeks at 20°C ± 2 and all survivors were euthanized with Eugenol.

Experimental infection with NNV

Following Castric et al. [START_REF] Castric | Sea bream Sparus aurata, an asymptomatic contagious sh host for nodavirus[END_REF], a stock of betanodavirus, PP160 strain (red-spotted grouper nervous necrosis virus [RGNNV/RGNNV] genotype), isolated from diseased D. labrax displaying typical signs, was produced at 24°C on the SSN-1 cell line (L15 medium, 10% FBS, pH7.6) and frozen 10 days postinoculation. Cell debris were removed by centrifugation for 15 min at 2000 × g; the virus was then aliquoted and stored at - 80°C. Before the challenge, viral titration was carried out on one of the aliquots after a single freeze-thaw cycle based on the tissue culture infectious dose technique (TCID50) described in Castric et al. [START_REF] Castric | Sea bream Sparus aurata, an asymptomatic contagious sh host for nodavirus[END_REF]. The infectious titer of the viral production was calculated according to the Kärber method [START_REF] Kärber | Tabellen zur näherungsweisen Bestimmung von Titern[END_REF], and was found to be 1 × 10 8 TCID50/mL. Fish from Batch 2 (17 tanks with approximately 54 sh per tank) were immersed for 4 h in a static bath of 6 L of aerated seawater at 26°C ±2 containing 1 × 10 5 TCID50/mL of the PP160 strain. Negative control sh (with no probiotic supplementation) underwent the same protocol, but using non-infected SSN-1 cell supernatant. Mortalities were followed daily during 6 weeks at 26°C ± 2 and all survivors were euthanized with Eugenol.

6. DNA extraction procedure DNA of a 2 mL pure culture of each Pseudoalteromonas strain was extracted using the Monarch Genomic DNA Puri cation Kit (Biolabs, protocol without lysozyme); and DNA of mucus and gills was extracted using the DNeasy PowerSoil Pro Kit (Qiagen). For gills, 250 mg was placed in 2 mL Lysing Matrix D tubes (MPbio) containing 1.4 mm ceramic beads. Sterile PBS was added to obtain a volume-toweight ratio of 1.6:1 and tissues were homogenized using the Precellys Evolution tissue homogenizer equipped with the Cryolys Evolution cooling system, at 4°C and 10 000 rpm, 4 × 30 s. After homogenization, 400 µL was used for DNA extraction using the DNeasy PowerSoil Pro Kit (Qiagen) homogenization tubes with 800 µL of solution CD1 and again homogenized at 4°C and 4500 rpm, 2x 30 s. Mucus extraction was initiated directly at this stage. After this step, DNA was extracted according to the manufacturer's recommendations for both gills and mucus. DNA was eluted in 100 µL of elution buffer and concentration was measured using the NanoDrop 1000 spectrometer (ThermoScienti c). DNA was diluted to 5 ng/µL for use in qPCR analyses.

7. qPCR analyses for quanti cation of Pseudoalteromonas strains qPCRs were performed using the CFX Touch96 thermocycler (Bio-Rad) and the qPCR kit Fast start essential DNA green master (Roche LifeScience). The primers OB-PVP 3 (5'-CTTGCTTGGAAATGGGCTGA-3') and OB-PVP 4 (5'-TGGCCCGTAAGCATTGTATAAA-3') were designed to target a 51 bp CDS, speci c to Pseudoalteromonas strains hCg-6 (CDS HCG6B_v1_a1650, accession number CP102371-CP102372), hCg-42 (CDS HCG42B_v1_3844, accession number CP118496-CP118498) or hOe-125 (CDS HOE125B_v1_a0382, accession number CP118914-CP118915), without discrimination (same CDS sequence with different names). As these CDSs are common to the Pseudoalteromonas antimicrobial strains (hCg or hOe), this sequence will be called AntiMicCDS in the manuscript. Used on a mix of two strains, this qPCR did not allow discrimination, but gave a copy number of the AntiMicCDS as a proxy of the quantity of the two strains. The primers OB-PVP 5 (5'-CTTTCAGCAAACACAATGGCA-3') and OB-PVP 6 (5'-GCCTTGTCGCTCTTCCACAT-3') were designed to target the gene coding for the alterocin that allow detection of the Pseudoalteromonas strain 3J6. Each qPCR reaction (20 µL nal volume) was composed of 2 µL of a mix of primers 3 + 4 or 5 + 6, diluted to 3.75 µM in sterile milliQ water, 10 µL of the SYBR Green qPCR mix solution and 8 µL of DNA diluted at 5 ng/µL.

Cycling conditions were as follows: 95°C for 10 min and 40 cycles at 95°C for 15 s and 60°C for 10 s. A melting curve analysis was performed at the end of the qPCR at 95°C for 15 s, 60°C for 10s, and 95°C for 15 s. Each run included a blank control (H 2 O control) and appropriate positive DNA controls used to perform a standard curve from 10 5 copies/µl to 10 1 copies/µl. The LD PCR (limit of detection of the PCR)

was determined according to the standard NF U47-600-2. Brie y, dilutions of bacterial DNA were tested three times in qPCR in 8 replicates for each strain. LD PCR was de ned as the last number of copies at which ampli cation (level de ned as the threshold cycle, Ct) can be determined on at least 23 of the 24 replicates of a dilution. In this study, the LD PCR corresponded to 5 copies for both AntiMicCDS and alterocin gene ampli cation. The qPCR results were analyzed using CFX MAESTRO software (Bio-Rad).

Immune system analyses

Immune parameters were monitored based on ow cytometry analyses using a protocol adapted from Danion et al. [START_REF] Danion | Effects of in vivo chronic hydrocarbons pollution on sanitary status and immune system in sea bass (Dicentrarchus labrax L.)[END_REF]. For each sh sampled, an approximate volume of 300 µL of blood was withdrawn from the caudal vein using a lithium heparin vacutainer (BD Vacutainer LH 85 U.I.). These whole blood samples were diluted with 7 mL of Leibovitz 15 medium (L15 Eurobio) containing 20 U heparin lithium (Sigma). Then, an aliquot of 7 mL was loaded on a Ficoll gradient (Histopaque® 1077, Eurobio) to a density of 1.077 g cm - 3 . After centrifugation (400 × g, 30 min, 15°C), leucocytes at the interface were collected and washed twice (1200 × g, 5 min, 4°C) with L15 medium. Phagocytosis activity and leucocyte mortality were analyzed on an Attune NxT Acoustic Focusing Flow Cytometer (Invitrogen Molecular Probes). For each sample, 15,000 events were counted and subpopulations (i.e. lymphocytes, monocytes and granulocytes) were subsequently determined using gates based on size and complexity in the general acquisition data. Phagocytosis activity was analyzed on 300 µL of leucocyte samples and 15 µL of a 1:10 dilution of uorescence beads (2.7 × 10 10 particles mL - 1 , Fluorosphere® carboxylate-modi ed microspheres, diameter 1 µm, MolecularProbes) after incubation for 30 min at room temperature. Using green uorescence, phagocytosis activity was determined on the monocyte/granulocyte population. Fluorescence beads were rst analyzed alone to remove the area that corresponds only to these beads, not to the mix of beads and leucocytes.

Cell mortality was analyzed on 300 µL of leucocyte suspensions after 30 min incubation at 4°C in the dark with 15 µL of propidium iodide (PI, 1.0 g L - 1 , MolecularProbes). This procedure was carried out on whole immune cells without distinguishing subpopulations during all data acquisition. Dead cells were counted in the red uorescence gate, because only dead cells are able to internalize the red uorescent PI.

For Batch 1 (V. harveyi challenge), immune parameters were analyzed at 15 days post-infection (dpi) (i.e. at the end of the infection challenge) on 6 sh per treatment (n = 4 for the negative control). For Batch 2 (NNV challenge), immune parameters were analyzed at the end of the impregnation phase (12 weeks of impregnation) on 8 sh per treatment and during the infection phase at 96 hours and 7 dpi on 6 sh per treatment per sampling time.

Statistical analyses

All statistical analyses were performed using multiscale comparison tests. Speci cally, we performed an ANOVA test followed by the Tukey HSD post-hoc analysis or a Kruskal-Wallis test followed by the Dunn test for post-hoc analysis depending on our data.

Results

Safety of Pseudoalteromonas strains for sea bass

Immersion of European bass during 4 h with 10 7 CFU/mL of probiotic candidates, a quantity 10 times greater than that used subsequently, did not induce any sign of animal suffering or any behavior change in the week following immersion. No mortality was observed.

Persistence of the probiotic candidates in sh tissues after one immersion session

In the gills, an average of 1.20 × 10 4 copies of the AntiMicCDS and alterocin genes were detected at the end of the immersion step (T4H) but no persistence was observed after this time (T24H, T48H, T72H and T96H post-immersion; Fig. 3).

In the mucus, the AntiMicCDS and alterocin genes were detected at 4.95 × 10 4 and 2.49 × 10 4 copies on average, respectively, after the 4 h immersion session (Fig. 4A). Signi cant levels of the alterocin gene were still found after 3 to 4 days post-immersion (Fig. 4B). Results of the qPCR on the control sh showed that some marine bacteria present naturally on the sh carried the alterocin gene, but at a level much lower than that detected in impregnated sh.

Detection of the Pseudoalteromonas strains during long-term impregnation

Throughout the long-term impregnation, the average copy number of the AntiMicCDS gene was estimated at 9.3 × 10 3 in the gills and 4.8 × 10 5 in the mucus of the sh from Batch 1, and at 2.13 × 10 3 in the gills and 1.44 × 10 5 in the mucus of the sh from Batch 2. For the 3J6 group, the average number of alterocin gene copies was estimated at 5 × 10 3 in the gills and 3.5 × 10 5 in the mucus for the sh from Batch 1 and at 3.3 × 10 3 in the gills and 4.61 × 10 4 in the mucus for the sh from Batch 2.

Bio lm development during impregnation with Pseudoalteromonas strains

Mean thickness (µm), biovolume (µm 3 /µm 2 ) and maximal thickness (µm) of the bio lms that formed on the surface of glass slides immersed 8 weeks in water with impregnated or control sh are shown in Figure 6A, 6B and 6C, respectively. For these three parameters, the control slides were always statistically different from those immersed with at least one of the other experimental groups. The maximal thickness of the bio lm on the control slides (16.47 µm) was signi cantly higher than those immersed with the experimental groups (10.22µm; 11.77µm; 11.62µm for RA15, Combined strain and 3J6 groups respectively). Also, the biovolume of the bio lm on the control slides was 10.54 µm 3 /µm 2 in average against 6.60 µm 3 /µm 2 in RA15 group; 8.83 µm 3 /µm 2 in the combined strains group; 9.84 µm 3 /µm 2 in 3J6 group. As we can observe in Figure 6E to 6G, the bio lm was also composed of different type of organisms, such as microalgae and bacteria and was not homogenously distributed on the slide. Indeed, a reduction of maximal thickness and of biovolume in the impregnated-groups suggest modi cation of bio lm architecture in these experimental groups.

5. Impact of the probiotics on V. harveyi

In vitro effects

The culture supernatant of Pseudoalteromonas hCg-42 inhibited V. harveyi growth. However, the culture supernatant of Pseudoalteromonas hOe-125, 3J6 and RA15 did not show antibacterial activity against the V. harveyi strain used in this study (Fig. S1).

Monitoring of cumulative mortalities during the V. harveyi challenge

Cumulative mortalities after IP injection of V. harveyi are presented for each group in Figure 7. The percentages of survival within a group were rst compared using an ANOVA test to analyze the inter-tank variability. One replicate from the infected-control group and one replicate from the hCg-42+hOe-125 group differed statistically from the three other tanks (infected-control group: p values ranged from < 0.05 to < 0.001; hCg-42+hOe-125 group: p values ranged from 10 -3 to 10 -5 ) and were removed from all further analyses. The percentage of survival in the non-infected control group was 100% after IP injection with sterile TSB medium. In the infected control and 3J6 groups, the survival rates were similar, with 37% and 31.3%, respectively (Kruskal-Wallis test, p = 1). In the hCg-42+hOe-125 group, survival was 61.5% and was statistically different from the RA15 group (p-value: 9.71 x 10 -11 ). Further, no statistical difference was observed between the hCg-42+hOe-125 and the non-infected control groups (p = 0.18). For the negative probiotic control strain RA15, the percentage of survival, 48.6%, was signi cantly different from the infected control group (p = 4.96 x 10 -4 ).

Impact of probiotics on mortality induced by NNV infection

Cumulative mortalities after NNV infection are presented for each group in Figure 8. The percentage of survival in the negative control group was 88%. In the infected control, the hCg-42+hOe-125 and the RA15 groups, the survival rates were 54.72%, 56.44% and 55.01%, respectively. No statistical differences were observed among these three treatment groups, which were all statistically different from the negative control. For the 3J6 group, survival was 45.19%, statistically lower than all the other probioticsupplemented groups and to the infected-control group.

7. Monitoring of the immune system parameters 7.1. Immune parameters at 15 days post-infection with V. harveyi

Flow cytometry analyses at 15 dpi showed that the percentage of dead cells only differed for lymphocyte populations (Figure S2). The group impregnated with strain 3J6 had a statistically lower percentage of dead lymphocytes compared with the infected-control group and the hCg-42+hOe-125 group (p-value <0.05).

Phagocytosis was measured on both the monocyte and the granulocyte populations (Fig. S3). In the group impregnated with strain 3J6, phagocytosis was lower than in the non-infected control (p = 0.009).

No other statistical difference was observed for phagocytosis between the different groups.

Immune parameters before and after infection with NNV

Flow cytometry analyses were at 12 weeks of impregnation (i.e. just before the infection), at 96 h post-infection and at 7 dpi (Figs. S4-S6). The percentage of dead cells only differed for the lymphocyte population at 7 dpi (Fig. S4). The group impregnated with strain 3J6 had a statistically greater percentage of dead lymphocytes compared with the infected control group and non-infected control group (p < 0.05).

Phagocytosis was measured on monocyte and granulocyte populations (Figure S6). In the hCg-42+hOe-125-impregnated group, a decreased phagocytosis rate was observed compared with the infected control (p < 0.05) at 7 dpi. No other statistical difference in phagocytosis was observed between groups.

Discussion

Among the large diversity of pathogenic Vibrio species found in the aquatic environment, V. harveyi is currently the most problematic species for European bass, repeatedly responsible for signi cant economic losses in intensive farms [7][34]. The main treatments used to limit the losses associated with vibriosis include antibiotic therapies and vaccination, but they have limited e cacy and can lead to antibiotic resistance [START_REF] Breton | Mediterranean n sh pathologies; present status and new developments in prophylactic methods[END_REF][START_REF] Zhang | Vibrio harveyi: a serious pathogen of sh and invertebrates in mariculture[END_REF][START_REF] Bourouni | Antimicrobial resistance and potential probiotic application of Enterococcus spp. in sea bass and sea bream aquaculture[END_REF]36]. Over the past years, probiotics have emerged as alternatives to limit the spread of antibiotic resistance and promote the prevention of these opportunistic diseases [START_REF] Wang | Effects of compound probiotics on growth performance, rumen fermentation, blood parameters, and health status of neonatal Holstein calves[END_REF].

Several feed suppliers now supplement their sh feed with probiotics (mostly lactic acid-producing bacteria or Bacillus). However, these probiotics have often been developed for terrestrial animals and are therefore not necessarily suitable for sh, which live in a different environment and have different physiological needs [38]. In our study, we focused on the genus Pseudoalteromonas, which was rst described in 1995 [START_REF] Gauthier | Phylogenetic analysis of the genera Alteromonas, Shewanella, and Moritella using genes coding for small-subunit rRNA sequences and division of the genus Alteromonas into two genera, Alteromonas (emended) and Pseudoalteromonas gen. nov., and proposal of twelve new species combinations[END_REF]. Pseudoalteromonas are marine bacteria found worldwide. This genus contains 41 species and over 3772 Pseudoalteromonas strains have been isolated to date [START_REF] Offret | Spotlight on Antimicrobial Metabolites from the Marine Bacteria Pseudoalteromonas: Chemodiversity and Ecological Signi cance[END_REF]. These species and strains are of ecological signi cance due to the production of natural anti-fouling substances [START_REF] Bowman | Bioactive compound synthetic capacity and ecological signi cance of marine bacterial genus Pseudoalteromonas[END_REF][START_REF] Dobretsov | Inhibition of biofouling by marine microorganisms and their metabolites[END_REF], as well as metabolites that possess antimicrobial or antibio lm activity [START_REF] Offret | Spotlight on Antimicrobial Metabolites from the Marine Bacteria Pseudoalteromonas: Chemodiversity and Ecological Signi cance[END_REF][START_REF] Defer | Screening for antibacterial and antiviral activities in three bivalve and two gastropod marine molluscs[END_REF][START_REF] Desriac | Antimicrobial Peptides from Marine Proteobacteria[END_REF][START_REF] Dheilly | Antibio lm activity of the marine bacterium Pseudoalteromonas sp. strain 3J6[END_REF][START_REF] Rodrigues | Antibio lm activity of the marine bacterium Pseudoalteromonas sp. 3J6 against Vibrio tapetis, the causative agent of brown ring disease[END_REF]. This antimicrobial or antibio lm potential was investigated in this study for sh application by testing the ability of three potentially probiotic Pseudoalteromonas strains to limit or even prevent mortalities induced by a pathogenic strain of V. harveyi in bacteria-impregnated European sea bass.

Although the oral route, administered through feed, is generally the most frequently used route to supplement animals with probiotics, we chose to deliver our probiotic candidates to the sh using repeated sessions of bath immersion, each lasting several hours. One possible eld application of this administration route is a permanent drip of a stabilized probiotic solution of probiotics in sh farm tanks.

We used a bath impregnation method to promote this impregnation through the sh environment, so as to limit the variability that can be associated with individual feed intake. We studied the fate of our probiotic strains in two major sh tissues in direct contact with the aquatic environment: the gills and the mucus. We found signi cant quantities of the probiotic strains in these tissues in the hours following immersion, but were not able to measure persistence over time. Screening for probiotics is frequent in assays of feed supplementation, particularly in the intestine [46], where probiotics administered via feed coatings can be more easily detected. The lack of apparent persistence of our probiotics in the tissues analyzed is not necessarily problematic if the impregnation process nevertheless has bene cial effects.

We also investigated bio lm formation and structure during impregnation. Our results indicate that the maximal bio lm thickness in the non-impregnated control was signi cantly higher than in the repeated impregnation with probiotic candidates. Furthermore, considering the other two parameters of bio lm development (maximum thickness and biovolume), the non-impregnated control always differed from at least one other experimental treatment group. This difference suggests that the impregnation of sea bass with probiotic candidates tends to modify these bio lm parameters after 8 weeks of repeated immersion sessions. For the probiotic hCg-42 + hOe-125 and 3J6 strains, these results may be partly associated with the antimicrobial or antibio lm properties of alterins or alterocin, respectively, although for the 3J6 group, there was no major reduction in the bio lm parameters measured after 8 weeks in seawater. For the RA15 strain, for which no antibio lm or antimicrobial activity has been described to date, the bio lm also appeared to have been modi ed compared with the non-impregnated control, potentially highlighting an interesting property of this strain. However, we must note that the evolution of these parameters does not allow us to obtain information on the composition of the bio lm, especially in terms of pathogens.

The main result of this work is the drastic reduction in the cumulative mortalities in sh experimentally infected with V. harveyi impregnated with the hCg-42 + hOe-125 strains (25% survival gain) or the RA15 strain (10% survival gain). Because the sh were IP injected, it appears more likely that the Pseudoalteromonas strains led to an inherent improvement in their ability to ght the infection rather than an enhancement of the skin barrier through an antibacterial action. Although qPCR did not persistently detect the Pseudoalteromonas strains in the gills or mucus on the days following immersion, the metabolite compounds potentially produced by the strains, which had colonized the sh and/or which were present in their environment, seem to have been signi cantly effective in helping ght the infection.

Indeed, strains hCg-42 and hOe-125 have been reported to produce alterins, which are cyclolipopeptides with demonstrated antimicrobial activity against several pathogens [START_REF] Offret | Alterins, a new family of marine antibacterial cyclolipopeptides[END_REF]. An improvement in survival rates had already been reported for the European abalone Haliotis tuberculata impregnated 4 h with the Pseudoalteromonas strain hCg-6 and then injected with V. harveyi ORM4 [START_REF] Offret | Protective E cacy of a Pseudoalteromonas Strain in European Abalone, Haliotis tuberculata, Infected with Vibrio harveyi ORM4[END_REF]. In this case, abalone were exposed to the probiotic strain right before the infection. Here, we demonstrated that the repeated administration of probiotics had a signi cant effect on the survival rates after an infection challenge, which can occur several days after the last immersion bath.

Regarding the NNV challenge, we did not show any increase in the survival rate after impregnation with probiotic candidates. This result is not surprising because the candidates have been reported to in uence bacterial development, not viral multiplication. This result suggests that repeated probiotic impregnation does not improve the capacity of sea bass to ght viral infections.

In the particular case of strain 3J6, the mortalities occurred more quickly than in the other treatments, especially in the non-impregnated and infected control treatment groups, and this strain did not appear to confer any protective effect. The mortalities reported in the viral infection were even signi cantly higher than in other conditions. Because this strain has a potential to limit bacterial bio lm formation, it may nevertheless be worthwhile to determine if the bacterial composition of the tank water (particularly in bio lms), may have increased the mortality rate by modifying the potentially protective microbiota and/or by inducing stress to the sh.

A surprising result in the case of V. harveyi infection was the increase in the survival rate obtained with strain RA15, which has never been reported to produce any antibio lm or antibacterial compounds. In the case of NNV infection, the results reported suggest that this strain showed the same pattern as in the combined hCg-42 + hOe-125 treatment. In our study, it appears that (i) the RA15 strain was able to produce one or more compounds different from the ones reported for strains hCg-42, hOe-125 and 3J6

with an impact on V. harveyi or (ii) RA15 may have outcompeted V. harveyi, thereby limiting its spread. This result highlights the importance of including this strain in future investigations.

Due to the very sudden nature of the death caused by V. harveyi on sea bass, it was not possible to harvest moribund animals; we therefore analyzed live sh at a de ned time post-infection. At 15 dpi, ow cytometry analyses did not show differences among conditions, although the 3J6 strain showed a signi cant reduction in phagocytosis activity compared with the non-infected control group. Furthermore, during the viral challenge, we also observed slight modi cations in immune parameters, without a clear, de ned pattern. Further investigations are in progress to determine the effect of the impregnation on the microbiome of different sea bass tissues before and after a pathogen infection, but also to study other immune parameters such as gene expression or speci c antibody response.

Finally, our results suggest that a higher dose of Pseudoalteromonas strains could further improve the survival rates after a bacterial infection. In the hCg-42 + hOe-125-impregnated sh, we doubled the survival rates compared with the RA15-impregnated sh. However, the number of bacteria in the dual strain impregnation was double that of the RA15 impregnation, because each strain was added at a concentration of 10 6 CFU/mL during the immersion sessions. Higher impregnation doses should therefore be tested, particularly given that a concentration of 10 7 CFU/mL did not reveal any toxicity for sea bass. Testing higher doses can help design an industrialization process, because the probiotics must be produced at higher levels. Research efforts are also needed to investigate if the protective effect observed can be reproduced in real farm conditions and with bacterial pathogens other than Vibrio.

Conclusion

This study highlighted the protective effect of Pseudoalteromonas strains (hCg-42 combined with hOe-125 and strain RA15), signi cantly improving the survival rates of sea bass after experimental infection with V. harveyi. Based on a worldwide production of sea bass estimated at approximately 200 000 t in 2016, a 10 to 25% gain in survival following a bacterial infectious episode is economically signi cant for marine sh farmers. This improved survival could be further enhanced by a positive effect of these strains on the survival and the growth of the larval stages as well as on potential bio lm control. This proof of concept is a rst step in the development of a solution applicable to sea bass farms, based on the potential of these marine bacteria in a circular aquaculture approach.

Assessment

Supplementary Files

This is a list of supplementary les associated with this preprint. Click to download.

2 Figure 3 4 Figure 4 3 Figure 5

 234435 Figure 2

Figure 7 Survival 4 Figure 8

 748 Figure 7

FigS1Rahmanietal

  FigS1Rahmanietal.2023.tif FigS2Rahmanietal.2023.tif FigS3Rahmanietal.2023.tif

Table 1

 1 Fish experimentation was carried out in strict accordance with European guidelines and recommendations on animal experimentation and welfare (European Directive 2010/63/EU).Experimental procedures were validated by the ANSES animal ethics committee (ANSES/ENVA/UPC No. 16) and authorized by the French Ministry of National Education, Higher Education and Research (APAFIS

		Bacterial strains used in this study
	Strain	Description	Production	Reference
			of alterins
			ou
			alterocin
	Pseudoalteromonas	Type strain with antibacterial activity isolated	Alterins	[20, 22,
	hCg-6	from hemolymph of Crassostrea gigas		47]
	Pseudoalteromonas	Strain with antibacterial activity isolated from	Alterins	[20, 22,
	hCg-42	hemolymph of Crassostrea gigas		47]
	Pseudoalteromonas	Strain with antibacterial activity isolated from	Alterins	[22, 48]
	hOe-125	hemolymph of Oestrea edulis
	Pseudoalteromonas	Strain with antibio lm activity isolated from an	Alterocins	[23, 44,
	sp. 3J6	underwater substratum		49]
	Pseudoalteromonas	Strain isolated from Arthrocnemum	None	[50]
	rhizospherae RA15	macrostachyum, plant growth promoter, without
		any antibacterial or antibio lm activity. Used as a
		negative control
	Vibrio harveyi no.	Pathogenic strain provided by Dr. Alain Le Breton	None	[7]
	94473 1811603	-VetEau
	AQN553P2

ACKNOWLEDGEMENTS

We warmly thank Dr. Alain Le Breton (VetEau, Grenade-sur-Garonne, France) for providing the pathogenic strain of Vibrio harveyi used in this study.

FUNDING

This work was funded by the European Maritime, Fisheries, and Aquaculture Fund (EMFAF), measure 47, under the name "Projet PaqMan : Développement de probiotiques en Aquaculture Marine", OSIRIS number PFEA470019FA1000012.

ETHICS COMMITTEE APPROVAL

All animal experiments were approved by the French Ministry of National Education, Higher Education and Research under the authorization number APAFIS #32741-2020121509556347 v5. AUTHOR CONTRIBUTIONS TM, and MD acquired funds and coordinated the study. AR performed all experimental assays, the production of protective strains and the analyses of the samples harvested with advice of TM, MD, MB, YFr, ED and LP and LB. LP and YFl performed in vitro inhibition tests on V. harveyi. HC performed primers design for qPCR. AB, SR and AR performed bio lm measurement by microscopy. JC and LL produced the viral suspension for the NNV challenge. AR, TM and MD wrote the manuscript ( rst draft was written by AR). The manuscript was carefully reviewed by other co-authors, who all approved the nal version.

Copy numbers of AntiMicCDS (representing strains hCg-42 and hOe-125) or alterocin (representing strain 3J6) genes in gills and mucus after 4 h of immersion at T0 and T2 (Batch 1) and at T0 and T4 (Batch 2) for control and probiotic treatments. The AntiMicCDS gene was ampli ed for the control, RA15 and hCg-42+hOe-125 groups. The alterocin gene was ampli ed for the control, RA15 and 3J6 groups. ****: p ≤ 10 -


Pierre Develter
email: pierre.develter@onera.fr

Jonathan Bosse
email: jonathan.bosse@onera.fr

Olivier Rabaste
email: olivier.rabaste@onera.fr

Philippe Forster
email: pforster@parisnanterre.fr

Jean-Philippe Ovarlez
email: jean-philippe.ovarlez@onera.fr

Sur

Sur la probabilité de détection du filtre adapté pour la détection distance-Doppler de cibles hors-grille

come

Introduction

Détecter des signaux bruités est un problème rencontré dans de nombreux domaines [START_REF] Kay | Fundamentals of statistical processing[END_REF]. Classiquement, quand les signaux dépendent de paramètres inconnus, le détecteur du Test du Rapport de Vraisemblance Généralisé (GLRT) est utilisé : il consiste à remplacer les paramètres inconnus dans le test du rapport de vraisemblance par leur Estimateur du Maximum de Vraisemblance (MLE). Cependant, certains paramètres non-linéaires n'admettent pas de MLE sous forme analytique. En pratique, pour palier ce problème, l'espace de paramètre est discrétisé en une grille fixe, et des tests sont réalisés pour les valeurs de paramètres sur la grille, de manière à obtenir des tests indépendants. Cependant, en réalité les vrais paramètres de cibles ne correspondront pas exactement aux paramètres sur grille, créant là une désadaptation entre les signaux reçus et les signaux de référence. Cette désadaptation peut dégrader les performances de détection des tests. Quand le GLRT correspond au filtre adapté, cette perte peut atteindre 3dB par paramètre inconnu.

La détection en présence d'erreurs de modèle sur le signal a été considérée dans plusieurs contextes généraux, comme dans le cas d'une désadaptation dans un cône au-tour du signal de référence [START_REF] Besson | Detection of a signal in linear subspace with bounded mismatch[END_REF] ou d'une désadaptation contrainte quadratiquement [START_REF] De Maio | Fractional QCQP with applications in ML steering direction estimation for radar detection[END_REF], parmi d'autres contextes [START_REF] Bandiera | Advanced Radar Detection Schemes Under Mismatched Signal Models[END_REF][START_REF] Liu | Robust detection in mimo radar with steering vector mismatches[END_REF][START_REF] Besson | Adaptive detection of a signal known only to lie on a line in a known subspace, when primary and secondary data are partially homogeneous[END_REF]. Alors que le problème de désadaptation lié à une grille est couramment examiné en reconstruction parcimonieuse [START_REF] Tang | Compressed sensing off the grid[END_REF][START_REF] Lasserre | Bayesian sparse Fourier representation of off-grid targets with application to experimental radar data[END_REF] ou en estimation (en radar, estimation de la direction d'arrivée [START_REF] Dai | Root sparse Bayesian learning for off-grid DOA estimation[END_REF]), il reste relativement peu étudié en détection [START_REF] Bosse | Adaptive subspace detectors for off-grid mismatched targets[END_REF].

La façon la plus directe de lutter contre cette désadaptation est de se passer de la grille en utilisant le "vrai" GLRT qui consiste à calculer le maximum du rapport de vraisemblance sur tout l'espace continu où les paramètres peuvent se situer. Cependant, ses propriétés statistiques dépendent du maximum d'un continuum de variables non-indépendantes et leur étude peut être délicate, et sa relation P F A -seuil n'est pas connue. Dans [START_REF] Develter | Sur la probabilité de fausse alarme du filtre adapté normalisé pour la détection de cibles hors-grille[END_REF], nous avons calculé la relation P F A -seuil du Filtre Adapté Normalisé (NMF). De manière analogue, dans cet article nous nous proposons de calculer une nouvelle relation P F A -seuil asymptotique pour le filtre adapté (MF) avec deux (distance et Doppler) paramètres inconnus, en présence de bruit blanc gaussien complexe. Pour cela, nous allons utiliser l'espérance de caractéristiques d'Euler [START_REF] Adler | On excursion sets, tube formulas and maxima of random fields[END_REF].

La section 2 formalise le problème ainsi que le détecteur GLRT hors-grille étudié dans cet article. Dans la section 3, nous calculons la relation P F A -seuil asymptotique de ce détecteur, qui est validée numériquement en section 4.

Notations : Les matrices sont en gras et en majuscules, les vecteurs en gras. Pour toute matrice A ou vecteur, A T est la transposée de A et A H est la transposée Hermitienne de A. I est la matrice identité et CN (µ, Γ) est la distribution normale complexe circulaire de moyenne µ et de matrice de covariance Γ. ⊗ est le produit de Kronecker. a r et a i sont les parties complexes et imaginaires du vecteur complexe a.

Formulation du problème

Notre problème consiste à détecter un signal connu s(ξ) ∈ C N M d'amplitude inconnue, dépendant d'un paramètre inconnu ξ et plongé dans du bruit. Formellement, cela correspond au test d'hypothèses binaire suivant :

H 0 : r = n , H 1 : r = a s(ξ) + n , (1)
où n ∈ C N M est un vecteur de bruit distribué selon une loi gaussienne complexe circulaire n ∼ CN (0, Γ) de matrice de covariance connue Γ et a ∈ C est l'amplitude complexe de la cible. Dans la suite de cet article, le nombre de paramètres inconnus est fixé à deux : la distance de la cible et son décalage Doppler. La distance de la cible au radar se déduit du délai τ avant le retour du signal émis. Dans le reste de cet article, nous considérons que le signal reçu échantillonné sur une impulsion pour un certain retard τ est modélisé par le chirp linéaire suivant :

b j (τ ) = 1 √ K Π T P (t j -τ ) e iπk(tj -τ ) 2 , 1 ≤ j ≤ M
où les t j = jT s sont les instants d'échantillonnage avec

T s = 1 B où B est la bande du chirp, relié avec le scalaire k ∈ R par k = B T p . K
d(θ) = 1 √ N 1, e 2iπθ , .
. . , e 2iπ(N -1)θ T , avec ξ = {τ, θ}. Le modèle de signal distance-Doppler que nous utilisons est défini de manière classique comme le produit de Kronecker des composantes Doppler et distance :

s(ξ) = b(τ ) ⊗ d(θ). (2)
Quand ξ est connu et l'amplitude complexe a est inconnue, le GLRT de ce problème, appelé GLRT "oracle", devient le test du Filtre Adapté [START_REF] Scharf | Signal detection in Gaussian noise of unknown level: an invariance application[END_REF][START_REF] Scharf | Matched subspace detectors[END_REF] :

s(ξ) H Γ -1 r 2 s(ξ) H Γ -1 s(ξ) H1 ≷ H0 w 2 .
( Dans ce cas, la relation P F A -seuil est donnée par l'équation suivante :

P F A = e -w 2 . ( 4
)
Puisque les vrais paramètres de cible sont inconnus, les tests sont généralement évalués pour des valeurs de paramètres ξ sur une grille discrète. Cependant, dans la réalité les paramètres de cible ne correspondent jamais exactement aux paramètres de la grille. Pour une résolution de Fourier usuelle, la perte en SNR du filtre adapté peut atteindre 3 dB par paramètre inconnu en bord de case de résolution (donc 6 dB en détection distance-Doppler), ce qui dégrade les performances de détection, comme illustré en Figure 1.

Le "vrai" GLRT, nommé GLRT hors-grille ou MF horsgrille, consiste à évaluer (3) pour tous les ξ de manière continue dans le domaine de recherche D, que l'on appellera une cellule (par exemple, un intervalle

h N , h + 1 N × h ′ B , h ′ + 1 B
) dans la suite :

max ξ∈D s(ξ) H Γ -1 r 2 s(ξ) H Γ -1 s(ξ) H1 ≷ H0 w 2 . ( 5
)
Les performances de détection empiriques de ce détecteur sont montrées en Figure 1. Elles sont proches du GLRT oracle (qui connaît parfaitement la position des cibles), alors que le Filtre Adapté montre une perte en performance de détection liée à la présence de cibles horsgrille. Pour garantir une P F A donnée, on doit fixer un seuil adéquat w 2 dans (5). Pour cela, on doit évaluer la statistique du maximum d'un continuum de variables nonindépendantes. A notre connaissance, il n'y a pas de solution connue à ce problème dans notre contexte. La section suivante comble ce manque dans le cas bruit blanc (Γ = I).

Une relation P F A -seuil asymptotique pour le filtre adapté horsgrille distance-Doppler

Dans cette section, on démontre une nouvelle relation P F A -seuil asymptotique pour le filtre adapté hors-grille sous bruit blanc en utilisant les caractéristiques d'Euler moyennes. L'article [START_REF] Adler | On excursion sets, tube formulas and maxima of random fields[END_REF] propose un cadre théorique général pour résoudre ce type de problèmes.

La réponse du filtre adapté peut être considérée comme le champ aléatoire Y (ξ) suivant :

Y (ξ) = s(ξ) H r 2 . ( 6
)
Pour utiliser les résultats de [START_REF] Adler | On excursion sets, tube formulas and maxima of random fields[END_REF], il est nécessaire d'introduire le champ aléatoire gaussien : X(α, ξ) = Y (ξ). Ainsi la P F A à caractériser peut s'écrire :

X(α, ξ) = (γ 1 (ξ) cos α + γ 2 (ξ) sin α ) T r, (7)
où γ 1 (ξ) = s r (ξ) s i (ξ) , γ 2 (ξ) = -s i (ξ) s r (ξ) , r =
P F A = P max ξ∈D Y (ξ) > w 2 = P max α,ξ∈[0,2π]×D X(α, ξ) > w .
(8) On peut aisément montrer que X est stationnaire et on note Λ la matrice de covariance de son gradient ∂X ∂α , ∂X ∂ξ .

L'ensemble d'excursion A w (X) associé à X pour un seuil w est défini comme l'ensemble des paramètres tels que X(α, ξ) excède w [START_REF] Adler | On excursion sets, tube formulas and maxima of random fields[END_REF] :

A w (X) = {(α, ξ), X(α, ξ) > w} . (9)
Dans [START_REF] Adler | Random fields and geometry[END_REF], il est montré que l'espérance de la caractéristique d'Euler de A w (X), E(φ(A w (X))), est une estimation précise de la probabilité que X(α, ξ) excède w, et donc de la P F A voulue :

|P F A -E (φ (A w (X)))| < O e -c w 2 , ( 10
)
pour un certain c > 1. La caractéristique d'Euler φ(.) est une caractérisation topologique d'un ensemble : les lecteurs intéressés peuvent étudier les chapitres correspondants de [START_REF] Adler | Random fields and geometry[END_REF]. Notre résultat est basé sur le théorème fondamental pour les champs aléatoires définis sur R 3 suivant :

Théorème 3.1 [START_REF] Adler | On excursion sets, tube formulas and maxima of random fields[END_REF] Soit X un champ aléatoire centré, isotrope et stationnaire sur l'espace de paramètres T ⊂ R 3 avec pour matrice de covariance du gradient Λ = λ I. Supposons de plus que ∂T , définie comme la frontière de T , est C 2 sauf sur les arêtes, les creux de longueurs finies et les sommets. Alors

E(φ(A w (X))) = |T | ρ 3 (w) + |∂T | 2 ρ 2 (w) + H(∂T ) π ρ 1 (w) + φ(T )Ψ w σ , (11)
où H(.) est une fonction de la courbure détaillée dans [START_REF] Worsley | Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images[END_REF],

σ 2 est la variance de X (constante puisque X est sta- tionnaire), ρ k (w) = e -w 2 /2σ 2 λ k/2 (2π) (k+1)/2 σ k H k-1 w σ avec H k le k-ième polynôme de Hermite et Ψ(t) = +∞ t e -u 2
P F A = πB 6 ∆τ ∆θ (N 2 -1) K + 1 K -1 2w 2 -1 + π 3 B∆τ K + 1 K -1 + ∆θ N 2 -1 w + 1 e -w 2 +O e -c w 2 pour un c > 1. (12)
Trois termes apparaissent dans la formule : le premier rend compte de l'intérieur de la cellule. Les deuxième et troisième termes représentent respectivement les arêtes et les coins de la cellule : leur somme est égale à la somme des P F A du filtre adapté hors-grille dans les cas où un seul paramètre est inconnu (distance ou Doppler). Donnons une idée de la preuve de ce corollaire. On obtient le résultat en calculant la caractéristique d'Euler de l'ensemble d'excursion de X définie dans (9) grâce au Théorème 3.1, puis en appliquant le résultat [START_REF] Bosse | Adaptive subspace detectors for off-grid mismatched targets[END_REF] pour approximer la P F A . Tout d'abord, notons que dans notre cas σ 2 = 1/2. Puisque pour tout ξ ∈ D, X(0, ξ) = X(2π, ξ), X est périodique selon α et l'espace de paramètres T peut être vu comme un pavé droit de dimensions 2π, ∆τ, ∆θ ν : c'est ce qui compte dans l'hypothèse d'isotropie. Un des arguments qui nous a permis de trouver cette relation est que X est stationnaire sous bruit blanc. Malheureusement, la méthode employée ne peut pas être appliquée sous bruit colorée (Γ ̸ = I).

Résultats numériques

Pour vérifier [START_REF] Adler | On excursion sets, tube formulas and maxima of random fields[END_REF], nous avons évalué empiriquement la relation P F A -seuil du MF hors-grille via 5 × 10 7 tirages de Monte Carlo dans une cellule de résolution [0, 1/N ] × [0, 1/B], avec des vecteurs b et d de taille fixée à 10. Les résultats sont montrées dans la figure 3. La relation observée correspond très bien à la relation asymptotique calculée dans la section précédente, sauf pour des P F A proches de 1, de manière analogue au cas du NMF [START_REF] Develter | Sur la probabilité de fausse alarme du filtre adapté normalisé pour la détection de cibles hors-grille[END_REF].

Conclusion

Dans cet article, nous avons analysé la statistique du GLRT hors-grille, qui est plus robuste en présence de cibles hors-grille que le filtre adapté. Nous avons calculé et évalué numériquement une relation P F A -seuil asymptotique pour une utilisation en détection distance-Doppler, qui sera utile pour la communauté de détection radar. La prochaine étape serait d'obtenir une relation similaire avec une matrice de covariance quelconque (Γ ̸ = I).

Figure 1 -

 1 Figure 1 -Performance en détection distance-Doppler avec des cibles hors-grille tirées aléatoirement et uniformément dans une cellule pour le détecteur oracle, le GLRT (5) et le Filtre Adapté (3). P F A = 10 -6 , N = 10, K = 40.

Figure 2 -

 2 Figure 2 -Espace de paramètre phase-distance-Doppler T . Il s'agit d'un pavé droit avec deux faces collées ensemble.

/ 2 √Corollaire 3 . 1 . 1

 2311 2π du désigne la queue d'une loi gaussienne centrée réduite. |T | désigne le volume de T et |∂T | la surface de ∂T . En appliquant ce résultat général à notre problème, on peut montrer le corollaire suivant : La relation P F A -seuil du GLRT horsgrille testé sur une cellule D = [τ 1 , τ 2 ] × [θ 1 , θ 2 ] est :

Figure 3 -

 3 Figure 3 -Relations théorique (12) et empirique obtenues sous bruit blanc en utilisant 5 10 7 tirages de Monte Carlo avec K = 40, et N = 10. La relation sur-grille (4) est montrée à titre de comparaison.