Philippe Douchet
email: philippe.douchet@univ-perp.fr

Jérôme Boissier

Stephen Mulero

Hubert Ferté

Margot Doberva

Jean- François Allienne

Eve Toulza

Kévin Bethune

Olivier Rey
email: olivier.rey@univ-perp.fr

Make visible the invisible: Optimized development of an environmental DNA metabarcoding tool for the characterization of trematode parasitic communities

Keywords: Metabarcoding, Parasites, Trematoda, biodiversity assessment, cryptic diversity, freshwater, water-sediment interface

The world's parasitic and mutualistic biodiversity is undergoing major upheavals related to modifications in host community structures, changes in interactions between species, and through coextinction events. Trematodes are an important component of this invisible biodiversity, in terms of species richness, but also because of their role in ecosystems functioning and in the emergence of associated diseases. Together, these elements point to the need for a better assessment and understanding of the structure and dynamics of trematode diversity. In this context, our aim was to develop an optimized eDNA-based metabarcoding approach to detect trematodes and characterize their communities, most of which associated to aquatic environments. The efficiency of this newly developed tool was first assessed by exhaustive in silico and in vitro validation steps. We next assessed the ability of our eDNA-based approach to reconstruct trematode communities compared to a classical trematode monitoring method over four freshwater aquatic ecosystems. Our eDNA-based monitoring tool displayed a high amplification enrichment of trematode DNA, a 100% detection score for tracking back an in vitro mock community composed of 28 trematode species, and high genetic resolution which makes it relevant to discriminate between even phylogenetically close trematode species. Over the four natural ecosystems screened in natura, 33 OTUs were generated from the eDNA-based approach, from which 11 trematode species were identified. In comparison, we identified five trematode species using the classical monitoring method, three of which were also detected by the eDNA-based approach. We believe that this new eDNA-based metabarcoding tool will open new perspectives for fundamental and applied research in community ecology, conservation, and health survey.

Introduction

Global change including climate changes, land-use, fragmentation and human-driven international trades deeply modify the spatial distribution of all living organisms worldwide, and largely participate to the ongoing decline of global biodiversity [START_REF] Ceballos | Accelerated modern human-induced species losses : Entering the sixth mass extinction[END_REF][START_REF] Pereira | Scenarios for Global Biodiversity in the 21st Century[END_REF][START_REF] Sage | Global change biology : A primer[END_REF]. In this regard, mutualistic and parasitic organisms are particularly threatened through coextinction, the most common form of biodiversity loss [START_REF] Carlson | Parasite biodiversity faces extinction and redistribution in a changing climate[END_REF][START_REF] Dunn | The sixth mass coextinction : Are most endangered species parasites and mutualists[END_REF]. In parallel to the massive melting of the overall parasite diversity, we are currently witnessing a significant increase in the spread and abundance of some parasites and consequently the emergence or reemergence of infectious diseases worldwide [START_REF] Gottdenker | Anthropogenic Land Use Change and Infectious Diseases : A Review of the Evidence[END_REF][START_REF] Jones | Global trends in emerging infectious diseases[END_REF][START_REF] White | Emerging zoonotic diseases originating in mammals : A systematic review of effects of anthropogenic land-use change[END_REF]. As for most organisms, there are likely to be few winners and many losers among parasites during this ongoing massive extinction [START_REF] Cable | Global change, parasite transmission and disease control : Lessons from ecology[END_REF][START_REF] Mckinney | Biotic homogenization : A few winners replacing many losers in the next mass extinction[END_REF]. Together, these elements call for a better assessment and a better comprehension of the structure and dynamics of parasite diversity, a still neglected although certainly the richest and most abundant fraction of overall biodiversity [START_REF] Poulin | The diversity of parasites[END_REF].

Digenean trematodes are among the most abundant group of parasitic Platyhelminthes. Estimating the exact number of trematode species is particularly challenging since parasites harbor important cryptic diversity [START_REF] Poulin | Uneven distribution of cryptic diversity among higher taxa of parasitic worms[END_REF]. According to Carlson et al., the number of trematode species could reach up to 44.000 species of which 14% have been described so far [START_REF] Carlson | What would it take to describe the global diversity of parasites[END_REF]. These parasitic worms are ubiquitous over all marine and freshwater ecosystems. They are characterized by a complex life cycle involving some parasitic stages associated to sequential intermediate and definitive obligatory hosts, and often, some free living aquatic stages that allow the parasite to shift from one sequential host to another [START_REF] Esch | The Transmission of Digenetic Trematodes : Style, Elegance, Complexity[END_REF]. Trematodes are well known as pathogenic agents responsible for several infectious diseases including bilharziasis, fasciolosis (liver fluke) and food-borne trematodiases, most of which cause sometimes important health issues for humans, livestock and wild animals [START_REF] Toledo | Digenetic trematodes[END_REF]. However, and despite their sometimes devastating consequences on human and animal health, trematodes are increasingly recognized as key organisms in ecosystem functioning [START_REF] Carlson | A global parasite conservation plan[END_REF][START_REF] Dougherty | Paradigms for parasite conservation : Parasite Conservation[END_REF][START_REF] Marcogliese | Parasites : Small Players with Crucial Roles in the Ecological Theater[END_REF]. However, trematodes are still generally overlooked. This might be partly explained by the fact that they are very difficult to detect in natural environments, their spatial and temporal distribution is influenced by several biotic and abiotic factors, and they also comprise an important fraction of cryptic species hence complexifying their identification [START_REF] Poulin | Uneven distribution of cryptic diversity among higher taxa of parasitic worms[END_REF].

Environmental DNA (eDNA) assays have recently become a valuable tool to detect species from environmental samples such as air, soil, sediments, or water without seeing or catching them [START_REF] Beng | Applications of environmental DNA (eDNA) in ecology and conservation : Opportunities, challenges and prospects[END_REF][START_REF] Taberlet | Environmental DNA[END_REF]. In addition to being less restrictive in terms of sampling effort, eDNA-based methods are generally more efficient than classical monitoring methods to infer species occurrence and thus offer a promising alternative for monitoring cryptic and elusive organisms such as parasites [START_REF] Bass | Diverse Applications of Environmental DNA Methods in Parasitology[END_REF]. So far, targeted eDNA approaches were developed to detect and monitor several trematode species all of which being associated with Human and animal health issues including Schistosoma mansoni [START_REF] Sengupta | Environmental DNA for improved detection and environmental surveillance of schistosomiasis[END_REF], S. haematobium [START_REF] Alzaylaee | Schistosoma species detection by environmental DNA assays in African freshwaters[END_REF], S. japonicum [START_REF] Worrell | Field Detection of Schistosoma japonicum Cercariae in Environmental Water Samples by Quantitative PCR[END_REF], Fasciola hepatica (R. A. [START_REF] Jones | Detection of Galba truncatula, Fasciola hepatica and Calicophoron daubneyi environmental DNA within water sources on pasture land, a future tool for fluke control?[END_REF], Opisthorchis viverrini [START_REF] Hashizume | Application of environmental DNA analysis for the detection of Opisthorchis viverrini DNA in water samples[END_REF], and Ribeiroia ondatrae [START_REF] Huver | Development and application of an eDNA method to detect and quantify a pathogenic parasite in aquatic ecosystems[END_REF]. However, in a more general context of biodiversity assessment, no tool so far exists to detect all trematode species and characterize trematode communities in the environment.

To fill this gap, we here developed an optimized eDNA-based metabarcoding tool for characterizing and monitoring trematode communities in natura. We first describe a new metabarcode specifically developed for trematodes and provide an exhaustive in silico and in vitro validation study regarding its sensibility, specificity, and resolution. We next present a field study to compare the trematode communities characterized by our eDNA-based tool to those identified based on a classical approach combining the shedding of cercaria by mollusks captured in the field and molecular approaches to identify each trematode species. We finally discuss the pros and cons of such eDNAbased approach to characterize trematode communities.

Material and methods

In silico primers' design, optimization, and evaluation

We first built a custom database curated with an exhaustive collection of trematode species' mitogenomes gathered from the GenBank database (i.e., 50 species distributed over 19 phylogenetically distant families [START_REF] Olson | Phylogeny and classification of the Digenea (Platyhelminthes : Trematoda)[END_REF]; Table S1). We next ran ecoPrimers version 0.3 [START_REF] Riaz | ecoPrimers : Inference of new DNA barcode markers from whole genome sequence analysis[END_REF] using the following parameters: we targeted amplicons with a size comprised between 60bp and 300bp; we allowed three mismatches between each primer and the primer-binding sequence, we excluded any mismatches at the two final nucleotides at the 3' end of the primers and we accounted for the circularized nature of the mitogenome. These parameters are those usually recommended for designing primers in an eDNA context [START_REF] Taberlet | Environmental DNA : For Biodiversity Research and Monitoring[END_REF]. At the end of this first step, we next filtered the obtained primers so as to target candidate primers that displayed values greater than 0.96 for both, the coverage index (Bc) (i.e., the ratio between the number of amplified taxa and the total taxa number), and the specificity index (Bs) (i.e., the ratio between the number of taxonomically discriminated taxa and the number of amplified taxa).

We next improved the ability of each retained candidate primer to hybridize with DNA sequences from a wide range of trematode species. To do so, we first performed an in silico hybridization test of each primer using the 50 initial trematode mitogenome database as template (Table S1) and Geneious 4.8.2 software (default settings used). We then edited the primers manually by incorporating adequate degenerated bases at some specific nucleotidic sites. At each degenerated base incorporation step, the melting temperature of the resulting primers and the potential occurrence of secondary structures (e.g., hairpins, self-dimers and hetero dimers) were checked using IDTDNA's Oligo Analyzer (IDT DNA, 2020). We next selected the final primers based on the best compromise between the maximum number of accepted mismatches (up to 2 per pair of degenerated primers), the primer melting temperature (close to 60°C as recommended by Illumina) and the maximum number of nucleotides implicated in secondary structures (<1/3 of the primer length maximum).

The quality of the designed primers was next assessed by quantifying their sensitivity (i.e., the percentage of sequences from our trematode database on which the primers hybridize) and their specificity (i.e., their ability to preferentially hybridize to trematode sequences compared to that of other organisms). These two parameters were computed by performing virtual in silico PCR reactions using ecoPCR version 0.2 [START_REF] Ficetola | An In silico approach for the evaluation of DNA barcodes[END_REF] and using the custom trematode mitogenome database (Table S1) implemented with mitogenomes from 5624 metazoan species distributed among 26 phyla available from the GenBank nucleotide database (Table S2). We set the ecoPCR parameters to authorize up to 3 mismatches per primer and to select only amplicon lengths between 60bp and 400bp. The number of amplified sequences per phylum and per class of Platyhelminthes was retrieved and the number of mismatches between primers and sequence templates was assessed for each taxonomic group present in the database. Finally, we statistically compared the number of amplified sequences obtained per phylum, and per class of Platyhelminthes, and specifically checked for significant enrichment of the Trematoda Class using a series of Fisher exact tests.

In vitro amplification tests

To assess in vitro the efficiency of the designed primers to amplify size-expected amplicons in trematodes, we ran PCRs using 40 DNA extracts from 34 trematode species collected during this study (see section 2.3) and from internal trematode collections (Table S3). PCRs were performed using the GoTaq® G2 Hot Start Polymerase kit of Promega. Each PCR reaction contained Colorless Buffer at 1X, MgCL2 at 1.5mM, dNTPs at 0.2mM, primers at 0.4µM, 1.25 units of GoTaq G2 Hot Start, 2µL of DNA sample and ultrapure water for a total PCR reaction volume of 25µL. The following PCR program was used: An initial denaturation step at 94°C for 3' followed by 40 cycles with a denaturation step at 95°C for 30 sec, a hybridization step at 54°C for 30 sec and an elongation step at 72°C for 15 sec. We finally performed a final elongation step at 72°C for 5'. Ten microliters of the resulting PCR products were migrated on a 2% agarose gel for 30' at 135 V and revealed using a Vilber Infinity 1000 imaging system.

Each individual amplicon from the overall 40 available trematode DNA extracts (Table S3) that displayed a size similar to the theoretical size of the designed metabarcode was sequenced. The forward primer Trem_16S_F1, and the reverse primer Trem_16S_R2 (see results), were used following the same PCR condition as described previously, except that reactions were performed in a final volume of 35 µL. Additionally, and in order to taxonomically assign each trematode, we also sequenced the positive samples using forward C2'b (5'-GAAAAGTACTTTGRARAGAGA-3') and reverse D2 (5'-TCCGTGTTTCAAGACGGG-3') primers [START_REF] Huguenin | MALDI-TOF mass spectrometry : A new tool for rapid identification of cercariae (Trematoda, Digenea)[END_REF], which amplify part of the 28S D2 domain of rDNA. PCRs followed the same conditions as described previously (performed in a final volume of 35µL) except that the hybridization step was set at 50°C for 30 sec and the elongation step at 72°C for 30 sec. All amplicons generated were sequenced on an ABI 3730xl sequencer at the GenoScreen platform (Lille, France). For the taxonomic assignment of each trematode, a MEGABLAST analysis was performed with the recovered 28S D2 sequences. The best hits for each sequence (i.e., hits which presented a similar query cover and a similar identity percentage) were retrieved, and the sequences labelled to the lower taxonomic level were kept.

In natura validation

We assessed our ability to identify the trematode community in natura using our developed eDNA-based tool from water-sediment filtration samples. To this aim we set up a protocol consisting in comparing the trematode community obtained from eDNA samples collected at the water-sediment interfaces to the trematode community characterized based on classical snail emission protocol. The field work was conducted during March and May 2021. We focused on 4 natural sites from Occitanie Region (Southern France) that differ in terms of habitats, and in which the trematode communities were previously at least partially characterized: 1-Salses le château canals (hereafter Salses) consisted in ancient, connected canals used for agriculture yet abandoned since the 1950s'. Water at this site is flowing permanently and three snail species co-occur (Ancylus fluviatilis, Theodoxus fluviatilis, and Potamopyrgus antipodarum). In terms of trematodes, two species were identified including an Opecoelidae species associated with T. fluviatilis and an unidentified trematode producing furcocercariae associated with P. antipodarum. 2-Saint-Génis des Fontaines Canals (hereafter Saint-Génis) consisted in a contemporary urban freshwater canal for domestic water use (i.e., garden irrigation). In this system water flows intermittently and only one snail species was identified (Lymnaea stagnalis). The trematode Australapatemon burti was found to be associated to this snail species locally. 3-Villelongue-del-Mont Lake (hereafter Villelongue) consisted in an old gravels exploitation rehabilitated in water plan for recreational fishing. Species found in this lake are mostly invasive species and two snail species were found (P. acuta and Radix sp.). At this site, the trematodes Posthodiplostomum centrarchi, Telorchis attenuata [START_REF] Moravec | Some endohelminths from the freshwater turtle Trachemys scripta from Yucatan, Mexico[END_REF], and a Diplostomidae that we could not identify at the species level were previously found associated with P. acuta. 4-The Têt River in Néfiach (hereafter Néfiach). This 115-km long river starts its course at an altitude of 2,400 meters and flows to the Mediterranean Sea. At this specific site, 3 snail species and one trematode species were found: Radix sp. (associated with a Plagiorchis species), A. fluviatilis and P. acuta.

At each of these sites, we sampled the water-sediment interface along a 10 m transect against the water flow when discernible (i.e., at Salses, Néfiach and Saint-Génis). The Salses site was sampled twice, in March and May. The water-sediment filtering collection was achieved using disposable water filter capsules with a membrane surface area of 600 cm2 in polyethersulfone and a pore size of 0.45µm (Waterra) and according to recommendations made by Argaly (Argaly; https://www.argaly.com/). Briefly, capsules were connected at the intake end of an electric pump circuit built-up as follow: At the upstream of the capsule, we set up one tube fitted with a check valve (to avoid possible contamination) directly connected to a diaphragm water pump powered by a 12V battery. Downstream the pump an output tube was fitted so as to collect the filtered water into a 10L graduated collection container hence allowing controlling for the total water volume filtered (Figure S1). Filtrations were carried out until the filters membrane clogged, and the volume filtered at each site was recorded. Moreover, at each site, 8L of commercial spring water was filtered following the same protocol as a technical field negative control. Once filtrations completed, capsules were drained from the outlet side, filled with 50mL of Longmire buffer solution [START_REF] Longmire | Use of" lysis buffer" in DNA isolation and its implication for museum collections[END_REF] to preserve the eDNA, closed with end plugs, vigorously agitated, and stored at room temperature until subsequent DNA extraction. During all the sampling process, precautions were taken to avoid any contamination. Operators used disposable sterile gloves and reusable material in close contact with capsules (i.e., tubes fitted with check valves) were decontaminated after each use by successively placing this material in a 10% bleach bath, in a 70% ethanol bath and then in a DNA AWAY bath for 1 minute each.

Once the eDNA sampling achieved, all snails found were systematically collected at each sampling site manually or by scooping the grass on the water bench using a colander. We set the snail collection time at one hour by a unique observer. After their taxonomical identification, snails were individualized on well plates filled with dechlorinated water and left to emit trematodes for 24 hours under a 12h:12h sequential light:dark photoperiod. After measuring trematode prevalence, we then collected and transferred pools of cercariae from the infected snails into 1.5mL tubes and stored at -20°C until DNA extraction for molecular identification analyses.

Molecular approaches

All pre-PCR molecular steps were performed under a sterile PCR hood decontaminated before and after each use as follow: the working surface was successively washed using 10% bleach, 70% ethanol, a DNA AWAY solution, and exposed to UV light for 20 min. The reusable materials were decontaminated following the same protocol.

DNA extractions from pools of cercariae collected from the emitting snails were achieved using the Qiagen DNeasy Blood & Tissue kit. Briefly, we centrifuged the 1.5mL tubes containing the cercarial pools at 14,000 rpm for 10 minutes. After removing the supernatant, we followed the Tissue protocol as recommended by the supplier. All trematodes obtained from our direct approach were sequenced at the two genetic markers including the barcode developed in this study and the 28S D2 rDNA gene domain [START_REF] Huguenin | MALDI-TOF mass spectrometry : A new tool for rapid identification of cercariae (Trematoda, Digenea)[END_REF].

The longmire solution contained in each eDNA field capsule was split in 3 by pouring the contents of each capsule into three 50mL Falcon tubes. For the negative controls, the capsule contents were recovered in one 50mL Falcon tube only. Falcons were centrifugated at 12,000 rpm for 20 minutes and the supernatant was removed. Next, we collected 1g of sediment in the resulting pellet from each falcon tube (triplicates) using a metal spatula cleaned between each subsampling following the decontamination protocol specified above (except for UV exposure). For negative controls, no pellet was observed after centrifugation. In these cases, 500µL of Longmire was retained after discarding the supernatant and processed as the other samples. This pre-extraction step led to the processing of 19 samples (i.e., one negative control per site and 3 extraction replicates for each of the five environmental samples). Total environmental genomic DNA was extracted from each triplicate and negative controls using the Qiagen's Dneasy PowerSoil Pro Kit following supplier recommendation.

A total of 50 individual metabarcoding libraries were prepared. These 50 libraries include the triplicates of each filtering capsule (i.e., 3 x 5 = 15), four negative controls (one per sampling site) and six positive controls, the PCRs of each of these samples being duplicated (i.e., 25 x 2 = 50). The positive controls consisted into two categories of mock communities. The first category of mock community consisted in equimolar pools of 28 DNA extracts (set at a 3.5×10 -3 ng/µL final concentration) from different trematode species from internal collections (Table S3). These positive controls are useful to detect potential competition bias among trematode sequences during the PCR amplification step. The second mock community consisted in equimolar pools of PCR products independently obtained from the same 28 trematodes species. These controls are useful to detect possible biases during the library preparation and sequencing process.

Individual NGS libraries were prepared following the Illumina two-step PCR protocol, using our developed primers set up with Illumina adapters for the first locus-specific PCR. This first PCR was performed using the GoTaq® G2 Hot Start Polymerase kit of Promega. PCR reactions were performed twice into two independent plates, one in a final volume of 15µL used to check the PCR products through electrophoresis, the other in a final volume of 25µl specifically used for NGS sequencing hence limiting manipulation and possible cross-contamination. Except for the final reaction volume, PCR reactions were performed under the same conditions as described in section 2.2.

The resulting PCR products were individually indexed using a second PCR step consisting in eight cycles using the Nextera™ XT Index (Illumina, San Diego, USA) following the manufacturer's instructions. Finally, the 50 libraries were normalized using Sequal-Prep™ plates (Thermo Fischer Scientific, USA) before pooling. The pooled libraries were then purified following the JetSeq™ Clean protocol (Bioline, UK), quality checked on a Bioanalyzer High Sensitivity DNA kit (Agilent, USA) and quantified using a Qubit fluorometric quantification (Thermo Fisher Scientific, USA). Paired-end sequencing (2 x 250 cycles) was performed with a MiSeq Reagent Kit v2 on an Illumina MiSeq™ instrument at the Bio-Environnement platform (University of Perpignan Via Domitia, France).

Data analysis

The resulting amplicon sequence dataset was processed using the Find Rapidly OTUs with Galaxy Solution (FROGS) pipeline implemented in Galaxy [START_REF] Escudié | FROGS : Find, Rapidly, OTUs with Galaxy Solution[END_REF] available from the Genotoul platform (Toulouse, France). (i) The amplicon dataset was first pre-processed by filtering out the sequences so as to keep amplicon sizes from 150 to 400 nucleotides. (ii) The sequences kept were next clustered into operational taxonomic units (OTUs) using the swarm algorithm and using denoising and an aggregation distance of three nucleotides [START_REF] Mahé | Swarm : Robust and fast clustering method for amplicon-based studies[END_REF]. (iii) The dataset was filtered out for chimeras using VSEARCH [START_REF] Rognes | VSEARCH : A versatile open source tool for metagenomics[END_REF]. (iv) Singletons and underrepresented clusters (i.e., clusters whose number of sequences were <0.1% of the total number of sequences) were removed. Each OTU was next assigned to a species through a two-step BLAST affiliation procedure. The first BLAST analysis was computed using the standalone blastn program contained in the BLAST+ package and a custom trematode sequence database containing a total of 88 sequences including the sequences obtained from the amplicons generated by the in silico ecoPCR (i.e., 50 species; see Section 2.1; Table S1), the sequences generated by the in vitro Sanger sequencing (i.e., 26 species over the 34 species sequenced; see Section 2.2; Table S3), and 12 sequences retrieved from the GenBank database (Table S4). The second BLAST analysis was performed using the online MEGABLAST tool without restricting parameters to achieve affiliation of OTUs that could not be assigned in the first BLAST analysis. The obtained OTUs were filtered for presenting minimal blast coverage of 97% and a pairwise identity above 97% with the affiliated sequence. The remaining OTUs were considered as "unassigned". Lastly, we considered that a given OTU was present in a sub sample (i.e., one of the three replicates of a single environmental sample; see section 2.4) if its number of sequences was >0.1% of the total number of sequences in each of the two library assigned to this sub sample and if this OTU was present in both libraries (i.e., the two PCR replicates performed on the single subsample; see section 2.4). This 0.1% threshold was determined as being the most stringent while allowing the retention of the necessary sequences to detect all the 28 species from the control mock communities.

Finally, we assessed the genetic resolution of the developed metabarcode, i.e., to which extent the metabarcode allows us to differentiate between (even close) trematode species. To this aim, we computed pairwise genetic distances between a set of targeted trematode species with various degrees of phylogenetic relatedness. In particular, we combined the 88 sequences contained in our database. All sequences were aligned with T-Coffee on EMBL-EBI [START_REF] Madeira | The EMBL-EBI search and sequence analysis tools APIs in 2019[END_REF]. Together this set of 88 sequences covered 60 genera within the trematode phylogeny. We next built a neighborjoining phenetic tree based on the percentage of nucleotide differences from the obtained alignment using Jalview version 2.11. 1.4 (Waterhouse et al., 2009) for visualization.

Results

In silico primers evaluation and molecular validation

Our stringent computational searching method resulted in a single pair of primers (Trem_16S_F1 and Trem_16S_R2; Table 1) that produces a metabarcode localized at the 3' end of the mitochondrial 16S rDNA gene and which theoretical size is around 220bp depending on the targeted trematode species.

The in silico PCR performed based on our virtual community of 5674 mitochondrial genomes indicated that these primers were able to amplify an amplicon for 94.3% of the Platyhelminthe species (i.e., 100 species over the 106 represented species) while respectively amplifying only 18.3%, 14.3%, 5.5%, 1.5% and 0.1% of the Mollusca, Acanthocephala, Porifera, Arthropoda and Chordata (Figure 1a). Among Platyhelminthes, 100% of Trematoda, Cestoda and Monogenea were successfully amplified (Figure 1b). The Fisher's exact tests indicated that the Platyhelminthes are significantly enriched compared to Mollusca (p-value=3 e-44 ). Moreover, our set of primers showed no mismatches for 92% of the trematode species tested, and only 1 mismatch for the remaining 8% species (Figure S2).

Concerning the in vitro validation step, we obtained a single amplicon of ~220 bp as expected from the in silico analysis for all of the 40 DNA extracts from the 34 targeted trematodes species (Table S3).

Field trematodes survey

Five trematode species were identified overall from the four sampled sites based on our snail cercarial emission approach (Table 2). Among these 5 species of trematodes, the occurrence of 3 species (i.e., Echinostoma revolutum, Hypoderaeum sp. and Trichobilharzia physellae) was newly recorded at these sites (Table 2). At Salses in March, 382 P. antipodarum were collected but none emitted cercariae. At the same site, in May, 292 P. antipodarum and 56 T. fluviatilis were collected. Among the P. antipodarum, none emitted cercariae, while 8.9% of the collected T. fluviatilis emitted cotylicercariae of an Opecoelidae species. At Saint-Génis, 191 L. stagnalis were collected of which 2.1% emitted cercariae of two different species including E. revolutum and Hypoderaeum sp. At Villelongue, 131 P. acuta were collected, two of which emitted furcocercariae that were taxonomically assigned to T. physellae and P. centrarchi based on the obtained DNA sequences. Finally, at the Néfiach site, 222 Radix sp. were collected, and none emitted cercariae.

eDNA survey

A total of 5 capsules were obtained from the four sampled sites. The volumes filtered were 50L for each of the two capsules used at Salses in March and May, 20L at Saint-Génis, 18L at Villelongue and 25L at Néfiach.

Overall, a total of 6,426,325 sequences were obtained from the MiSeq sequencing corresponding to 50 (25 x 2) libraries. After pre-processing, 84% of these sequences were kept (i.e., 5,394,194). At this stage, the number of sequences per library ranged from 5 (negative control) to 211,107 with an average of 107,884 sequences per library (Figure S3). After all filtering steps, 4,693,991 sequences, grouped into 53 OTUs, were kept for subsequent analyses.

Regarding the positive controls, 1,885,902 sequences, grouped into 28 OTUs, were retained at the end of the bioinformatic analyses. The number of sequences per library ranged from 90,666 to 199,089 with an average of 157,159 sequences per library (Figure S3). After affiliation the OTUs were consistently assigned to the 28 species initially assembled in the mock communities (Figure 2). In the mock communities consisting in equimolar pools of PCR products, the sequencing resulted in an almost equally distributed number of reads per species as expected (Figure 2a). Conversely, we did observe amplification biases in the mock communities consisting in PCRs from equimolar pools of DNA (Figure 2b). Importantly, irrespective of the nature of the mock communities, the results obtained across each replicate and each PCR replicate indicated a strong technical replicability (Figure 2).

Regarding the libraries from the field eDNA samples and negatives, 2,808,089 sequences were retained at the end of the bioinformatic analyses. The number of reads obtained ranged from 0 (Negative control) to 186,198. One negative control displayed a non-negligeable number of sequences (Salses negative control PCR Duplicate 1: N= 133,203; PCR Duplicate 2: N = 3) (Figure S3). These sequences were affiliated to S. mansoni, one of the Schistosoma species reared and which DNA is frequently manipulated in our laboratory for about 40 years. Sequences of S. mansoni and S. bovis were also unexpectedly detected in two field subsamples from a single site (i.e., Saint-Génis subsample 1: N1= 87,614; N2= 64,125 and subsample 2: N1=83,071; N2=71,881) where no Schistosoma species may occur. This clearly indicates that despite all precautions taken from eDNA sampling to library preparation, contamination may not be totally prevented.

Excluding sequences of the two OTUs assigned as contamination in field samples and negatives, 2,368,192 sequences were retained at the end of the bioinformatic analyses. Among field samples, the number of sequences per library ranged from 4,411 to 182,311 with an average of 78,940 sequences per library (Figure S3). These sequences were grouped into 33 OTUs of which 11 were assigned to trematode species (accounting for 74.9% of the sequences obtained from field sampling), 1 OTU (1.5% of the sequences) corresponded to a cestode species and 21 OTUs (23.6% of the sequences) were unaffiliated according to our filtering parameters. Among the 11 trematode species, the occurrence of 4 species (i.e., Notocotylus sp., Petasiger phalacrocoracis, Plagiorchis maculosus and Plagiorchis koreanus) was newly recorded (i.e., not detected by the field trematodes survey method used in this study and not previously recorded on the study site) overall the four sampled sites (Table 2)

Comparison between eDNA monitoring and field trematodes survey

Overall, 13 trematode species were identified among the four sampling sites when combining the results from our two approaches (i.e., eDNA and field trematodes survey). Eleven trematode species (i.e., 84.6 %; 11/13) were identified based on the eDNA-based method, while five trematode species (38.5%; 5/13) were identified according to the classical survey (Figure 3). Only three trematode species were congruently detected by both methods at the same site (Figure 3). In this regard, eDNA monitoring confirmed the detection obtained by field trematode survey for 60% (3/5) of the detection. The two species P. centrarchi and T. physellae detected by the direct approach at the Villelongue site were not detected by eDNA monitoring. Conversely, field trematode survey confirmed the detection obtained by eDNA monitoring for only 27.3% (3/11) of the detected species (Figure 3).

At the Salses site, two sampling campaigns were achieved in March and May of the same year. At this site, the eDNA method allowed the detection of the Opecoelidae gen. sp. and the Notocotylus sp. only in March. No trematode was detected by this method in May. Conversely, the conventional method only detected the Opecoelidae gen. sp. in May. No trematodes were detected by this method in March.

Genetic resolution of the developed metabarcode

In order to assess the resolution of the metabarcode (i.e., the ability of the metabarcode to discriminate trematodes at the species level), the percentage of nucleotide difference in sequence was quantified between each pair of the 88 trematode sequences. The inter-specific variability exceeded 2.81%, value observed between two closely related species (i.e., Diplostomum spathaceum and D. pseudospathaceum) (Figure 4). Furthermore, the phenetic construction from the 88 samples allowed a clustering of the sequences into 9 superfamilies (Figure 4) among the 25 existing superfamilies of digenetic trematodes [START_REF] Olson | Phylogeny and classification of the Digenea (Platyhelminthes : Trematoda)[END_REF][START_REF] Toledo | Digenetic Trematodes[END_REF].

Discussion

Marker efficiency and reproducible protocol

Our results clearly indicate that the mitochondrial 16S rDNA gene metabarcode developed in this study is powerful to detect and identify trematode species from complex environmental DNA samples.

First, the obtained barcode has a strong specificity to the Trematoda class (i.e., ability of primers to preferentially hybridize to trematodes compared to other organisms). Indeed, our in silico validation tests revealed a huge enrichment of trematode (and to a lower extent cestodes and monogenes) DNA sequences when computing virtual PCR reaction and using a highly diverse DNA sequence database as template. Such specificity was also experimentally confirmed since only 33 OTUs were identified from environmental samples that are likely to contain complex mixtures of DNA from many different organisms [START_REF] Taberlet | Towards nextgeneration biodiversity assessment using DNA metabarcoding[END_REF]. Importantly however, only 11 OTUs (i.e., 74.9% of the overall sequences after Bio-informatic filtering steps) were affiliated to trematodes, and 1 OTU to a cestode species while the remaining 21 OTUs could not be affiliated to any organisms according to our filtering parameters.

Second, combined with this high specificity, the developed 16S rDNA metabarcode is also sensitive (i.e., ability of primers to amplify sequences for as much trematodes as possible) enough to detect all trematode species used in our in silico and in vitro validation steps. In this regard, the developed primers showed no mismatch with 92% of the trematode DNA sequences available in our database and only 1 mismatch for the remaining 8% species. We are thus confident that DNA from almost all trematode species could be amplified and detected using this metabarcode, although probably at some varying efficient degrees depending on the number of mismatches with the targeted sequences and the complexity of the DNA sequence community. In fact, our results from the mock community indicate that some species are likely to be more easily detectable than others especially when DNA from different trematode species is present in the same eDNA sample.

Our metabarcode also displays strong genetic resolution (i.e., level of intrinsic genetic diversity assessed over amplicons), which is crucial for identifying organisms to the lowest taxonomic levels. In this respect, the lowest phenetic distance computed in this study is 2.81 % between two closely related Diplostomum species. This is in line with a recent study that confirms that the mitochondrial 16s rDNA gene is particularly suitable for trematode species identification [START_REF] Chan | Assessing the suitability of mitochondrial and nuclear DNA genetic markers for molecular systematics and species identification of helminths[END_REF]. Finally, although we did not specifically test for the resolution of our metabarcode to discriminate Cestoda and Monogenea, we have clear in silico evidence that this marker could also be useful to characterize communities of Cestoda, Monogenea and probably most Platyhelminthes. Further in silico and in vitro studies are required to assess the performance of this marker (i.e., sensibility and genetic resolution) for other Platyhelminthe species.

In natura proof of concept

According to our sampling design, the eDNA approach appeared more powerful than classical trematode surveys to detect trematodes established locally. This result echoes with those generally obtained in empirical studies that compare classical prospections and eDNA approaches [START_REF] Deiner | Environmental DNA metabarcoding : Transforming how we survey animal and plant communities[END_REF][START_REF] Hänfling | Environmental DNA metabarcoding of lake fish communities reflects long-term data from established survey methods[END_REF][START_REF] Mulero | Malacological survey in a bottle of water : A comparative study between manual sampling and environmental DNA metabarcoding approaches[END_REF][START_REF] Valentini | Next-generation monitoring of aquatic biodiversity using environmental DNA metabarcoding[END_REF]. However, we believe that this particularly holds for trematodes given their complex life cycles and the difficulty to detect them in natura. In fact, trematodes spend more of their life within their definitive hosts, and to a lower extent, within their intermediate hosts. At these parasitic stages, trematodes are thus invisible [START_REF] Toledo | Digenetic Trematodes[END_REF]. Trematode free living stages, including miracidia and cercaria are generally microscopic and particularly difficult to collect and handle from the field for direct identification. Classical approaches to study trematodes in natural environments thus generally rely on the inspection of trematode eggs in vertebrate hosts feces, on the cercarial emissions from snails collected from the field and through inspection of developing trematode stages in dissected snail hosts. Although commonly used to target specific species of trematodes in epidemiological contexts [START_REF] Akindele | Epidemiology and Burden of Schistosoma Haematobium Infection Among School Children in Osun State, Nigeria[END_REF][START_REF] Bärenbold | Estimating sensitivity of the Kato-Katz technique for the diagnosis of Schistosoma mansoni and hookworm in relation to infection intensity[END_REF], the former approach is tedious, time-consuming and inconceivable to trace the entire trematode community circulating at a given site. The cercarial emission approach is often jointly used for such prospects but may also be difficult to set up for several reasons. First, prevalence of trematodes in mollusks is generally low in natural populations [START_REF] Rachprakhon | Very low prevalence of Opisthorchis viverrini s.l. Cercariae in Bithynia siamensis siamensis snails from the canal network system in the Bangkok Metropolitan Region, Thailand[END_REF] which implies the sampling of a huge number of snails to detect only few positive infection. Second, even when snails are infected by trematodes, emission of cercaria generally occurs after a maturing period within the snail (i.e., prepatent period) meaning that not all the infected snails collected during field prospection will emit cercariae. Additionally to this prepatent period, some trematode species are characterized by specific circadian rhythms and will thus be emitted at specific times of the day or night [START_REF] Théron | Chronobiology of Trematode Cercarial Emergence : From Data Recovery to Epidemiological, Ecological and Evolutionary Implications[END_REF]. If snails are not monitored for a period long enough, some trematodes can thus be missed. This is how snail dissection is used in addition to cercarial emission to better assess trematode communities and prevalence estimates. Snail dissection is however laborious and require specific knowledge in terms of dissection practices. In this study, we based our classical sampling method on the detection and identification of free-living stages of trematodes, which are more likely to be detected and identified by our eDNA-based method as well. Despite our sampling effort and a 24h cercarial emission period we only found 5 trematode species using this direct approach while we found more than twice (i.e., N = 11) using our eDNA based protocol. This huge difference might be explained by our strategy to collect eDNA samples from the water-sediment interface. This sampling scheme allowed collecting not only trematode DNA in its free form or associated to free-living trematode stage swimming in the water column, but also trematode DNA from the sediment either free but also and more likely associated to dead trematode miracidia and cercaria. In fact, most free-living trematodes die before encountering a permissive (intermediate or definitive) host [START_REF] Morley | Cercariae (Platyhelminthes : Trematoda) as neglected components of zooplankton communities in freshwater habitats[END_REF] and we might thus expect that most of these free-living dead stages sediment. This strategy also allowed us to obtain true sampling replicates by splitting the sediment obtained from each single capsule into three. In this regard, the results obtained from the triplicates are very similar and this indicates that our approach is highly repeatable. Another non-exclusive hypothesis to explain the difference of communities retrieved from the two approaches, is that since no snail dissection was performed, we could have missed some trematode species and thus underestimate the trematode community richness locally based on our direct approach.

Regarding the 13 trematodes species detected by both approaches in this study, all except T. physellae, were already reported in France or in Europe [START_REF] Aksenova | First molecular identification of Australapatemon burti (Miller, 1923) (Trematoda : Digenea: Strigeidae) from an intermediate host Radix labiata (Rossmaessler) (Gastropoda: Lymnaeidae) in Europe[END_REF][START_REF] Bock | The life-cycle of Plagiorchis maculosus (Rudolphi, 1802) Braun, 1902 (Trematoda : Plagiorchiidae), a parasite of swallows (Hirundinidae)[END_REF][START_REF] Demkowska-Kutrzepa | A review of the helminths co-introduced with Trachemys scripta elegans -a threat to European native turtle health[END_REF][START_REF] Faltýnková | A revision of Petasiger Dietz, 1909 (Digenea : Echinostomatidae) and a key to its species[END_REF][START_REF] Kvach | Aquaculture as a possible vector in the spread of Posthodiplostomum centrarchi (Hoffman, 1958) (Digenea : Diplostomidae) in Europe[END_REF][START_REF] Muñoz-Antolí | The life cycle and transmission dynamics of the larval stages of Hypoderaeum conoideum[END_REF][START_REF] Tkach | Molecular and morphological differentiation between species of the Plagiorchis vespertilionis group (Digenea, Plagiorchiidae) occurring in European bats, with a re-description of P. vespertilionis (Müller, 1780)[END_REF][START_REF] Toledo | An update on human echinostomiasis[END_REF]. To our knowledge, this is the first record of the avian schistosome T. physellae in Europe. From an ecological point of view, all trematode species detected by both approaches, except one (i.e., Notocotylus sp.), are known to use at least one co-occurring host species at the target sampling sites. In fact, Notocotylus sp. is the only trematode detected only through eDNA (i.e., not detected by classical method in this study and not previously recorded on the study site) that could not have been identified to the species level. Notocotylus is a genus composed of more than 40 species [START_REF] Chaisiri | Notocotylus loeiensis n. sp. (Trematoda : Notocotylidae) from Rattus losea (Rodentia: Muridae) in Thailand[END_REF] using molluscan hosts of the superorders Caenogastropoda and Heterobranchia [START_REF] Gonchar | Transatlantic discovery of Notocotylus atlanticus (Digenea : Notocotylidae) based on life cycle data[END_REF] and definitive hosts which range from waterfowl birds and small mammals [START_REF] Boyce | Identification of a new species of digenean Notocotylus malhamensis n. sp. (Digenea : Notocotylidae) from the bank vole (Myodes glareolus) and the field vole (Microtus agrestis)[END_REF]. Hypotheses about potential host species of this trematode occurring at the sampling site would therefore be premature without further investigation. Three other trematode species were detected only by eDNA (i.e., P. phalacrocoracis, P. maculosus and P. koreanus). These trematodes, all detected at the Néfiach site, use molluscan hosts of the genus Radix [START_REF] Bock | The life-cycle of Plagiorchis maculosus (Rudolphi, 1802) Braun, 1902 (Trematoda : Plagiorchiidae), a parasite of swallows (Hirundinidae)[END_REF][START_REF] Huguenin | MALDI-TOF mass spectrometry : A new tool for rapid identification of cercariae (Trematoda, Digenea)[END_REF][START_REF] Kudlai | Diversity of Plagiorchis (Trematoda : Digenea) in high latitudes: Species composition and snail host spectrum revealed by integrative taxonomy[END_REF] which also occur at this site. Furthermore, the presence of P. maculosus is consistent with the presence of swallows (i.e., Hirundinidae; [START_REF] Bock | The life-cycle of Plagiorchis maculosus (Rudolphi, 1802) Braun, 1902 (Trematoda : Plagiorchiidae), a parasite of swallows (Hirundinidae)[END_REF], observed in high abundance at this site and that act as definitive host for this parasite. In the same vein, cormorants (Phalacrocorax sp.) or bats of the family Vespertilionidae, respectively definitive hosts of P. phalacrocoracis and P. koreanus, although these two vertebrate host species were not observed during sampling, are both present at high abundance in the Occitanie Region. Finally, two trematode species (i.e., P. centrarchi and T. attenuata), detected at Villelongue, are considered as invasive species in Europe [START_REF] Demkowska-Kutrzepa | A review of the helminths co-introduced with Trachemys scripta elegans -a threat to European native turtle health[END_REF][START_REF] Stoyanov | Morphology and molecules reveal the alien Posthodiplostomum centrarchi Hoffman, 1958 as the third species of Posthodiplostomum Dubois, 1936 (Digenea : Diplostomidae) in Europe[END_REF]. As most recreational lakes, the Villelongue site harbor several exotic introduced vertebrates (e.g., Lepomis gibbosus) and invertebrates (Crayfish) including the mollusk P. acuta. The four trematode species identified in this lake are all associated with P. acuta might have been introduced together with their intermediate and definitive hosts. The detection of trematode species associated with these exotic vertebrate and invertebrate species could have been hence expected. Together the detections of all trematode species in this study are thus ecologically congruent.

Among the 13 species of trematodes that were identified using both approaches, two species (i.e., T. physellae and P. centrarchi) were however missed by our eDNA based approach at the Villelongue site. Several non-exclusive hypotheses can be proposed to explain such unexpected results. First, these parasites were present at low abundance and below the detection threshold of our eDNA sampling strategy. In accordance with this hypothesis, we observed low prevalence of these two species in mollusks (0.8% for each of them) at the time of the study. Second, additionally to low abundance among snails, free living stages of these two species might be present at low abundance in the environment due to suboptimal environmental conditions. Indeed, the release of cercariae by mollusks largely depends on abiotic factors such as temperature and light and of the trematode species [START_REF] Al-Jubury | Temperature and light effects on Trichobilharzia szidati cercariae with implications for a risk analysis[END_REF][START_REF] Koprivnikar | Interspecific and Intraspecific Variation in Cercariae Release[END_REF]. Third, amplification biases during PCR of the eDNA samples might disfavor the amplification of these two species compared to that of other trematode species. In this respect, one of the two species (i.e., P. centrarchi) was part of the positive mock communities and is among the species for which we obtained the fewest number of sequencing reads. We believe that temporal eDNA sampling replicates at the same targeted sites could help disentangling these three hypotheses. More generally, we argue that temporal studies are also required to better assess the detection dynamics of trematodes from aquatic eDNA samples.

Limitations

We noticed few drawbacks that require specific attention regarding the e-DNA based tool developed in this study. First, although our approach is promising to qualitatively characterize trematode communities, our mock communities indicate that this approach cannot be used to precisely quantify the relative abundance of each trematode species. Indeed amplification biases exist and may result from (i) differences in the number of mismatches between primers and trematode DNA sequences, highlighted for four species in our in silico analysis, (ii) different genome sizes or copy numbers of targeted loci between species, as well as differences in organism biomass and (iii) factors such as the proportion of GCs in amplicons or the length of amplicons [START_REF] Krehenwinkel | Estimating and mitigating amplification bias in qualitative and quantitative arthropod metabarcoding[END_REF]. Second, it is important to note that our proof of concept was conducted in a temperate environment where ecosystems display generally poor biodiversity. The efficiency of this metabarcoding tool to characterize trematode communities in tropical aquatic ecosystems characterized with more diverse organisms' communities remains to be assessed. Importantly however, given the efficiency of the tool to reconstruct our mock community of 28 species, some of which being tropical, we are confident that it could be applied to tropical environments successfully. Third, we warn on the importance of avoiding any handling of environmental samples in laboratories working on trematodes to prevent contamination. In fact, despite our stringent decontamination protocols and particular care during manipulations, we detected some Schistosoma species in our sequencing libraries which clearly result a contamination process. These two schistosome species have been manipulated (including living forms, DNA, RNA, cells) in the laboratory for over 40 years. However, because none of the other trematode species detected by eDNA were handled at our laboratory before, their detection is unlikely to result from laboratory contamination. Finally, and probably most importantly, an important current limitation of our tool is the poor number of trematode mitochondrial 16S rDNA gene sequences available in the existing databases. In the future, we will improve this aspect by systematically sequencing trematode species for this marker.

Applications

Despite its limitations, the present eDNA-based tool has several applications both for fundamental and practical studies in ecology and for epidemiological surveillance. Indeed, this eDNA approach could be particularly useful to draw general patterns of the spatial and temporal dynamics of trematode communities in aquatic environments. Moreover, although not specifically assessed in this study, this tool could also be useful to document the lifecycle of some still understudied trematode species. For instance, combined with eDNA approaches on definitive hosts feces or urine [START_REF] Duval | Urine DNA (uDNA) as a non-lethal method for endoparasite biomonitoring : Development and validation[END_REF][START_REF] Huggins | A novel coprodiagnostic molecular method for qualitative detection and identification of parasitic nematodes in amphibians and reptiles[END_REF], on bulk mollusks or on bulk invertebrates [START_REF] Ruppert | Past, present, and future perspectives of environmental DNA (eDNA) metabarcoding : A systematic review in methods, monitoring, and applications of global eDNA[END_REF], it might be possible to identify new intermediate or definitive hosts for some trematode species. In an ecological prospective, this metabarcode also constitutes a new tool to detect and monitor invasive species. For instance, we here detected T. attenuata which presence is potentially problematic because it can cause a possible threat on the native amphibian and reptile species [START_REF] Cardells | Helminths from the red-eared slider Trachemys scripta elegans (Chelonia : Emydidae) in marshes from the eastern Iberian Peninsula : First report of Telorchis attenuata (Digenea : Telorchiidae)[END_REF][START_REF] Demkowska-Kutrzepa | A review of the helminths co-introduced with Trachemys scripta elegans -a threat to European native turtle health[END_REF]. Trematodes also display several characteristics that make them promising for developing biodiversity indexes [START_REF] Huspeni | Trematode Parasites as Estuarine Indicators : Opportunities, Applications, and Comparisons with Conventional Community Approaches[END_REF]. For instance, in Californian estuaries, [START_REF] Hechinger | Host diversity begets parasite diversity : Bird final hosts and trematodes in snail intermediate hosts[END_REF] found that the diversity of birds is highly correlated with the diversity of trematodes characterized from their snail hosts. In addition, these biodiversity indices could provide additional information to existing ones. Indeed, whereas a traditional bird survey provides a snapshot of bird presence, trematode communities characterization provides a record of the bird community that has visited a site during the lifetime of the infested mollusks, and thus provides an integration through time of bird presence [START_REF] Hudson | Is a healthy ecosystem one that is rich in parasites?[END_REF]. Besides these applications in the field of ecology and biological conservation, this eDNA approach also constitute a promising tool to perform health surveys. Indeed, the detection of some specific trematode species that are pathogenic for humans, livestock, or wildlife in the environment is primordial to identify active transmission sites. Although several specific tools were recently developed to target some of the most pathogenic trematodes (at least for humans) (e.g., Fasciola hepatica; R. A. Jones et al., 2018, or Schistosoma mansoni;[START_REF] Sengupta | Environmental DNA for improved detection and environmental surveillance of schistosomiasis[END_REF], no tools exist to detect all trematode species. In this regard, two species of trematodes causing human and animal echinostomiasis (i.e., E. revolutum and Hypoderaeum sp.) were detected in the urban irrigation canals at Saint-Génis.

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Percentage of sequences amplified by in silico PCR with primers developed in this study (a)

Figure 4 :

 4 Figure 4: Phenogram illustrating the relationships between trematode species based on the

Figure S1 :

 S1 Figure S1 : Sampling device used in this study. Reusable material decontaminated is the part of the reusable material that undergoes cleaning after each use by successively placing this material in a 10% bleach bath, in a 70% ethanol bath and then in a DNA AWAY bath for 1 minute each.

Figure S2 :

 S2 Figure S2: (a) Number of mismatches between primers and matrices sequences according to taxonomic selection and (b) sequence representation showing the most conserved bases on primer hybridization sites in 50 trematode species. The total height of the letters depicts the information content of the position, in bits.

Figure

  Figure S3: Number of sequences per library after pre-process (i.e., without N), after FROGS analysis (i.e., Clustered, chimera removed and OUT filtered), and after all bioinformatic analysis

Table 1 :

 1 Primers designed in this study and their characteristics as determined in IDTDNA's Oligo Analyzer (parameters were set with 0.4 µM of primers, 1.5 mM of Mg 2+ and 0.2mM of dNTPs)

	Primer name	Sequence (5'-3')	Average Tm (°C)	GC content (%)	Length (bp)
	Trem_16S_F1	GACGGAAAGACCCCRAGA	61.8	58.3	18
	Trem-16S_R2	CRCCGGTYTTAACTCARYTCAT	62.8	45.5	22

Table 2 :

 2 List of the 13 trematode species found with eDNA & classical sampling with their status detected or not according to the sampling method. Historical record and molluscan hosts observed at the sampling sites as well as other intermediate or definitive hosts observed are mentioned.

	Sites	Species	Observed before this study (see section 2.2)	Detected by classical sampling (This study)	Detected by eDNA (This study)	Mollusk hosts observed at sampling sites	Intermediate/definitive hosts observed at sampling sites
		Opecoelidae gen. sp.	Yes	Yes	Yes	T. fluviatilis	Gammarus sp.
	Salses
		Notocotylus sp.	No	No	Yes	N.I	N.I
		Australapatemon burti	Yes	No	Yes	L. stagnalis	N.I
	Saint-Génis	Echinostoma revolutum	No	Yes	Yes	L. stagnalis	N.I
		Hypoderaeum sp.	No	Yes	Yes	L. stagnalis	N.I
		Diplostomidae gen.2 sp.	Yes	No	Yes	P.acuta	N.I
	Villelongue	Posthodiplostomum Centrarchi	Yes	Yes	No	P.acuta	Lepomis gibbosus
		Trichobilharzia physellae	No	Yes	No	P.acuta	N.I
		Telorchis attenuata	Yes	No	Yes	P.acuta	N.I
		Plagiorchis sp.	Yes	No	Yes	Radix sp.	N.I
		Plagiorchis maculosus	No	No	Yes	Radix sp.	Hirundinidae
	Néfiach	Plagiorchis koreanus	No	No	Yes	Radix sp.	N.I
		Petasiger phalacrocoracis	No	No	Yes	Radix sp.	N.I
		N.I : Life cycle not investigated at the study site

  S3: Number of sequences per library after pre-process (i.e., without N), after FROGS analysis (i.e., Clustered, chimera removed and OUT filtered), and after all bioinformatic analysis

	230000 220000 210000 200000 190000 180000 170000 160000 150000 140000 130000 120000 110000 100000 90000 80000 70000 60000 50000 40000 30000 20000 10000 0	Mo111	Mo112	Mo121	Mo122	Mo131	Mo132	Mo211	Mo212	Mo221	Mo222	Mo231	Mo232	NS1	NS2	NG1	NG2	NV1	NV2	NN1	NN2	S11	S12	S21	S22	S31	S32	TS11	TS12	TS21	TS22	TS31	TS32	G11	G12	G21	G22	G31	G32	V11	V12	V21	V22	V31	V32	N11	N12	N21	N22	N31	N32
													without N		Clustered/Chimera-removed/Filtered			Post-analysis

Table S3 :

 S3 DNA sample collection of trematodes species used in this study Present in the 16S trematode database in addition to sequences of the tables S1 and S4, in order to perform the first BLAST analysis to assign each OTU to a species

	Species	Origin	Species	Origin
	Posthodiplostomum cuticola a b	IHPE collection	Diplodiscus subclavatus a b	IHPE collection
	Diplostomidae gen.1 sp. a b	IHPE collection	Notocotylus sp. a b	IHPE collection
	Sphincterodiplostomum sp. b	IHPE collection	Schistosoma mansoni a	IHPE collection
	Diplostomum pseudospathaceum a b IHPE collection	Schistosoma haematobium (Egypte) a	IHPE collection
	Parastrigea robusta a b	IHPE collection	Schistosoma haematobium (Sénégal)	IHPE collection
	Cotylurus marcogliesei a b	IHPE collection	Schistosoma bovis b	IHPE collection
	Australapatemon burti a b	IHPE collection	Schistosoma hybride	IHPE collection
			haematobium/bovis haplotype bovis a
	Apharyngostrigea cornu b	IHPE collection	Trichobilharzia physellae b	IHPE collection
	Leptophallus nigrovenosus a b	IHPE collection	Trichobilharzia physellae	Field trematode survey
	Paralepoderma cloaciola a b	IHPE collection	Fasciola gigantica a	IHPE collection
	Macrodera longicollis a b	IHPE collection	Fasciola hepatica a	IHPE collection
	Plagiorchis sp. a b	IHPE collection	Dicrocoelium dendriticum a	IHPE collection
	Plagiorchis koreanus a b	IHPE collection	Paramphistomidae gen. sp. a b	IHPE collection
	Skrjabinoeces similis a b	IHPE collection	Opecoelidae gen. sp. a b	IHPE collection
	Echinostoma miyagawai	IHPE collection	Opecoelidae gen. sp.	Field trematode survey
	Echinostoma revolutum a	IHPE collection	Trematoda sp. a b	IHPE collection
	Echinostoma revolutum	Field trematode survey	Posthodiplostomum centrarchi a b	IHPE collection
	Echinoparyphium_recurvatum a b	IHPE collection	Posthodiplostomum centrarchi	Field trematode survey
	Hypoderaeum sp.	Field trematode survey	Telorchis attenuata a b	IHPE collection
	Petasiger phalacrocoracis a b	IHPE collection	Diplostomidae gen.2 sp. b	IHPE collection

a : Present in the two mock communities b :

Table S4 :

 S4 Additional 16S sequences from the GenBank Database used in our 16S trematode database, in order to perform the first BLAST analysis to assign each OTU to a species

	Accession number	Species
	MH536507.1	Alaria americana
	MH536508.1	Cardiocephaloides medioconiger
	KR269763.1	Diplostomum spathaceum
	AP017706.1	Echinostoma caproni
	KT008005.1	Echinostoma paraensei
	MH536511.1	Hysteromorpha triloba
	MK033132.1	Opisthorchis sudarikovi
	MH322000.1	Paragonimus kellicoti
	AY382770.1	Paragyliauchen fractoporus
	AY197350.1	Schistosoma edwardiense
	AY197349.1	Schistosoma hippopotami
	MH536513.1	Tylodelphys immer

This study was funded by the French Agency for Food, Environmental and Occupational Health & Safety (PNREST 2019/1/059 Molrisk). SM thanks the Occitania region (MOLRISK) and the EU (FEDER funds). This study is set within the framework of the "Laboratoires d'Excellences (LABEX)" TULIP (ANR-10-LABX-41). Finally, the authors warmly thank the BioEnvironnement platform (University of Perpignan) for sequencing and bioinformatics services.

The OTUs table underlying the main results of the study is archived on Dryad https://datadryad.org/stash. All sequences generated in this study are incorporated into the NCBI GenBank database.

Supporting Informations (Tables)


Emmanuel Gnandi1

Michel Boyom

Stéphane Puechmorel

CANONICAL FOLIATIONS OF STATISTICAL MANIFOLDS WITH STATISTICAL MODELS

The sheaf of solutions J ∇ of the Hessian equation on a gauge structure ( , ∇) is a key ingredient for understanding important properties from the cohomological point of view. In this work, a canonical representation of the group associated by Lie third's theorem to the Lie algebra formed by the sections of J ∇ is introduced. On the foliation it defines, a characterization of compact hyperbolic leaves is then obtained. We conclude that these leaves are equipped with a statistical model structure.

I

Hyperbolicity is quite an important notion in geometry and can be tackled using different approaches. In its original work [START_REF] Koszul | Domaines bornés homogenes et orbites de groupes de transformations affines[END_REF], Koszul introduces the development map Q to define hyperbolic manifold as those for which the image by Q of their universal coverings is an open convex domain without straight line. In [START_REF] Koszul | Déformations de connexions localement plates[END_REF], a cohomological characterization is given, stating that a necessary condition for a gauge structure ( , ∇) to be hyperbolic is the existence of a closed 1-form admitting a positive definite covariant derivative. Finally, if ( , , ∇) is a gauge structure on an Hessian manifold, then ( , ∇) is hyperbolic iff [ ] = 0 in Koszul-Vinberg cohomology [START_REF] Boyom | Foliations-webs-hessian geometry-information geometry-entropy and cohomology[END_REF].In [START_REF] Shima | The geometry of Hessian structures[END_REF], Shima observes that Fisher information of all classical used statistical models are Hessian metrics. As Amari did, he also proved that any locally flat statistical manifold is a Hessian manifold. These structures arised from works of many mathematicians (e.g. J.-L. Koszul, Y. Matsushima, A. Nijenhuis, E. B. Vinberg) in their attempt to solve the Gerstenhaber conjecture in the category of locally flat manifold.The geometry of locally flat hyperbolic manifold is also named Koszul geometry (see [START_REF] Gerstenhaber | On the deformation of rings and algebras[END_REF]),the conjecture of Gerstenhaber say that every restricted theory of deformation generates its proper theory of cohomology.This conjecture has been solved in 2006 (see [START_REF] Nguiffo Boyom | The cohomology of Koszul-Vinberg algebras[END_REF]) by Michel Boyom. In [START_REF] Boyom | Foliations-webs-hessian geometry-information geometry-entropy and cohomology[END_REF][START_REF] Nguiffo Boyom | The last formula of jean-louis koszul[END_REF] he also realize the homological version of the Hessian geometry.The aim of this paper is to study hyperbolicity of the leaves of canonical foliations in statistical manifolds using representations in the affine group of a finite dimensional vector space and equipped these leaves with statistical model.The paper is organized as follows:in section (2), we summarize the main results,section 3 is devoted to basics notions which are used,in section [START_REF] Boyom | Numerical properties of koszul connections[END_REF], is devoted to useful notions in the category of locally flat manifolds and their KV-cohomology. In section [START_REF] Boyom | Sur les structures affines homotopes à zéro des groupes de lie[END_REF] we introduce the notion of hyperbolicity in sense of Koszul, Vey.In section [START_REF] Cartan | Homological algebra[END_REF], the Hessian equation on a gauge structure is introduced along with some basic facts. In section [START_REF] Chevalley | Cohomology theory of lie groups and lie algebras[END_REF], the canonical group representation is defined, from which a vanishing condition in cohomology is given in Theorem 7.2. Finally, the case of statistical manifolds is treated in section (8) and a characterization of hyperbolic compact leaves is given.In section [START_REF] Kobayashi | Intrinsic distances associated with flat affine or projective structures[END_REF] we prove that the universal covering of these leaves are convex cones.In the last section [START_REF] Kobayashi | Projectively invariant distances for affine and projective structures[END_REF] we show that leaves are bearing a statistical model structure pertaining to the exponential family. In the final section [START_REF] Koszul | Domaines bornés homogenes et orbites de groupes de transformations affines[END_REF], a explicit construction is given and conclusion drawn. (2) The cohomology class [ ∇ ] vanishes;

(3) The affine representation

∇ → ( ∇ ( ), ∇ ( ) ∈ Aff( ∇ )
is conjugated to the linear representation

∇ → ∇ ( ) ∈ ( ∇ ).
Theorem 2.2. Let ( , , ∇, ∇ ★ ) be a compact statistical manifold. Then the foliation F ∇ (resp. F ∇ * ) is a Hessian foliation in ( , , ∇) (resp. ( , , ∇ * )).

Theorem 2.3. In a statistical manifold ( , , ∇, ∇ ★ ) the following assertions are equivalent:

(1)

[ ∇ ] = 0 ∈ 2 (F ∇ ) (2) [ ∇ ★ ] = 0 ∈ 1 ( ∇ ★ , ∇ ★ ) (3) ( ∇ ★ , ∇ ★ ) has a fixed point. ( 4
) ( ∇ ★ , ∇ ★ ) is affinely conjugated to its linear component ∇ ★ .
Theorem 2.4. Let ( , , ∇) being a compact Hessian manifold. If the characteristic obstruction satisfies:

{[ ∇ ] ∪ [ ∇ ★ ]} = {0} ∪ {0} . Then ( , ∇) is an hyperbolic manifold. Theorem 2.5. Let ( , ∇) be a closed hyperbolic manifold.Then [ ∇ ★ ] = 0 ∈ 1 ( ∇ ★ , ∇ ★ ) and [ ∇ ] = 0 ∈ 1 ( ∇ , ∇ )
Theorem 2.6. Let ( , ∇) be a closed hyperbolic manifold.Then ( , ∇ ★ ) is also hyperbolic manifold.

Theorem 2.7. Let ( , , ∇) be a statistical manifold and let F ∇ be its canonical Hessian foliation.

If the characteristic obstruction vanishes:

{[ ∇ ] ∪ [ ∇ ★ ]} = {0} ∪ {0} .
Then up to a affine diffeomorphism every compact leaf of F ∇ is the quotient of a sharp open convex cone by a discrete group of (R ) Theorem 2.8. Let ( , , ∇) be a statistical manifold and let F ∇ be its canonical Hessian foliation. If the characteristic obstruction vanishes:

{[ ∇ ] ∪ [ ∇ ★ ]} = {0} ∪ {0} .
Then up to affine diffeomorphism, every compact leaf of F ∇ is a parametric space of statistical models for a measurable set (Ω ★ , B (Ω ★ ))

< ∇ , > ,where is defined by

(Θ, ) = ∫ ( ∇) -< Θ, > ∫ Ω ★ -< Θ, > ( ∇) with (∇) ⊂ (R )

B

Most of the material presented here can be found in greater detail in [START_REF] Boyom | Foliations-webs-hessian geometry-information geometry-entropy and cohomology[END_REF]. Only the required notions will be introduce in this section. In the sequel, ( ) stands for the real Lie algebra of smooth vector fields on . The convention of summation on repeated indices will be used thorough the document.

Definition 3.1. A Koszul connection is a first order differential operator ∇ of ★⊗2 to . It is usually denoted by ( , ) → ∇ and has the following properties:

(1) ∇ = ∇ (2) ∇ = ( ) + ∇ ∈ ∞ ( ), ∀ , ∈ X( )
The torsion tensor ∇ and the curvature tensor ∇ are defined by:

(1)

∇ ( , ) = ∇ -∇ -[ , ] (2)
∇ ( , ) = ∇ ∇ -∇ ∇ -∇ [ , ]
Definition 3.2. A gauge structure is a pair ( , ∇) where ∇ is a Koszul connection in M.

Definition 4.2.

A locally flat manifold is a pair ( , ∇) where ∇ is a linear Koszul connection whose curvature ∇ and torsion ∇ both vanish identically:

(1) ∇ -∇ -[ , ] = 0 (2) ∇ ∇ -∇ ∇ -∇ [ , ] = 0 4.1.
KV-cohomology of locally flat manifolds. Let ( , ∇) be a locally flat manifold.Let A = (X( ), ∇) be the Koszul-Vinberg algebra associated to ( , ∇).For a locally flat manifold the Koszul-Vinberg complex is defined as follows:

Definition 4.3. Let (∇), ≥ 0 be the vector spaces:

0 (∇) = { ∈ ∞ ( ), ∇ 2 = 0} (∇) = Hom R (⊗ ( ), ∞ ( ))
and let : ( ) → +1 ( ) be the coboundary operator:

         = ∀ ∈ 0 (∇) ( 1 ⊗ ... ⊗ +1 ) =Σ 1 (-1) [ ( (.. ⊗ ˆ ⊗ .. ⊗ +1 )) ( ) -Σ ≠ (.. ⊗ ˆ ⊗ .. ⊗ ∇ ⊗ ..]
The complex ( (∇), ) is the Koszul-Vinberg complex of ( , ∇), with cohomology groups denoted by (∇). The following proposition gives a cohomological definition of a Hessian manifold .

Proposition 4.2 ( [14]

). A locally flat manifold ( , , ∇) is Hessian manifold in the sense of [START_REF] Shima | Geometry of hessian manifolds[END_REF] if vanishes identically .

H

The opposite of geodesic completeness is hyperbolicity in the sense of Vey [ [20]].Hyperbolic affine manifolds are closely related to Hessian manifold.This notion of hyperbolicity has been studied by many authors Koszul[ [13], [START_REF] Koszul | Domaines bornés homogenes et orbites de groupes de transformations affines[END_REF], [START_REF] Koszul | Variétés localement plates et convexité[END_REF]], Vey[ [20], [START_REF] Vinberg | Theory of homogeneous convex cones[END_REF]],Kobayashi[ [9], [START_REF] Kobayashi | Projectively invariant distances for affine and projective structures[END_REF]].The notion of hyperbolicity is among the fundamental notions of the geometry of Koszul.It may be addressed and studied form many perspectives. We highlight here the algebraic topology point of view, which is richer than the Riemannian geometry one. As an example, the theory of affine representations of Lie groups plays a key role in Lie group theory of Heat after Jean-Marie Souriau.

5.1. Developing map of locally flat manifold. Given a locally flat structure ( , ∇), we fix a point 0 ∈ and we consider the pairs {0, [0, 1]} and { 0 , M}. We go to deal with the space of differentiable paths

: {0, [0, 1]} → { 0 , M} .
The notation means that (0) = 0 .

The quotient modulo the fixed ends homotopy is denoted by M. The application

M [ ] → ( [ ]) = (1) ∈ M
is is a universal covering of M.The universal covering of ( , ∇) is denoted by ( , ∇). Consider a path c, ∈ [0, 1],let be the parallel transport

: T 0 M → T ( ) M.
The developing map D is defined as it follows:

: M [ ] → ( [ ]) = ∫ 1 0 -1 ( ( ) ∈ T 0 M Definition 5.1. A locally flat manifold ( , ∇) is called hyperbolic if is a diffeomorphism
of ˜ onto a convex domain not containing any straight line (Koszul [13]).

Remark 5.2. This definition is equivalent to the fact that is diffeomorphic to the orbit space

=

where is an open convex domain not containing any straight line in R and is a discrete subgroup of the Lie group ( ) (see J. L. Koszul [START_REF] Koszul | Déformations de connexions localement plates[END_REF][START_REF] Vey | Sur une notion d'hyperbolicité des variétés localement plates[END_REF]).

Theorem 5.3. (see [START_REF] Koszul | Déformations de connexions localement plates[END_REF]) For a locally flat manifold ( , ∇) being hyperbolic, it is necessary that it exists a de Rham closed differential 1-form on whose covariant derivate ∇ is positive definite. If is compact then this condition is also sufficient.

Remark 5.4. The existence of the nonvanishing Koszul 1-form proves that M is fibered over S 1 [START_REF] Tischler | On fibering certain foliated manifolds overs1[END_REF].Topological consequences follow,the Euler characteristic X( ) = 0 and the first Betti number 1 ( ) ≥ 1.

In [START_REF] Boyom | Numerical properties of koszul connections[END_REF] Michel Boyom has given a cohomological analogue of the theorem above and his result states as follows:

Theorem 5.5. Let ( , , ∇) be a compact Hessian manifold then the following assertions are equivalent:

• [ ] = 0 ∈ 2 (∇).
• ( , ∇) is hyperbolic 6. H ∇ Definition 6.1. The Hessian operator ∇ 2 : ( ) → 1 2 ( ) is, for a fixed ∈ ( ), the covariant derivative of the 1 1 ( )-tensor ∇ . For any triple ( , , ) of vector fields, its expression is given by:

∇ 2 , = ∇ (∇ ) -∇ ∇ . Proposition 6.1. The product ( , ) ∈ ( ) ↦ → ∇ has associator ∇ 2
Proof. This a direct consequence of def. 6.1, since: ., ) be a system of local coordinate functions of and let ∈ X( ). We set

= , = ∇ = Σ Γ .
The principal of symbol of the Hessian differential operator can be expressed as (6.1)

(∇ 2 ) ( , ) = Σ Ω , ( )
where

Ω ( ) = 2 + Γ + Γ -Γ + Γ + Γ Γ -Γ Γ (6.2) The geometric symbol ∇ of ∇ 2 is ∇ ( ) ( , ) = Σ 2
he Hessian differential operator ∇ 2 is involutive (The Koszul-Spencer complex is acyclic).

Definition 6.3 ( [4]

). Let ( , ∇) be a gauge structure. The sheaf of solutions of its Hessian equation, denoted by J ∇ ( ), is the sheaf of associative algebras:

↦ → { ∈ ( ), ∇ 2 = 0}
with product defined in prop. 6.1.

The space of sections of J ∇ ( ) will be denoted in the sequel by ∇ .

Proposition 6.2. The pair ( ∇ , ∇) is an associative algebra with commutator Lie algebra ( ∇ , [-, -] ∇ ) where the bracket [ , ] ∇ is:

[ , ] ∇ = ∇ -∇ .
When ∇ has vanishing torsion, ( , ∇) is said to be a symmetric gauge structure and the Lie algebra ( ∇ , [-, -] ∇ ) is obviously a Lie subalgebra of the Lie algebra of vector fields ∇) is symmetric gauge structure, then the Lie subalgebra ∇ ⊂ X( ) is finite-dimensional over the field of real numbers.

(X( ), [-, -]); Proposition 6.3 ( [4]). If ( ,
The next propositions shows that ∇ may be trivial.

Proposition 6.4. Let (M,g) be a Riemmian manifold and let Ric be its Ricci curvature tensor.It comes:

∇ ⊂ ( )
Proof. Let (X,Y) be a couple of vector fields on M and let ∈ ∇ .We have:

∇ ( , ) = (∇ 2 ) ( , ) -(∇ 2 ) ( , )
Taking the trace we deduce that:

( , ) = 0 Then ∇ ⊂ ( ) Corollary
∇ = ∇ + ∇ 2
we deduce that ∇ is subset of killing vector field on M.By assumption < 0, From the theorem of Bockner we deduce that

∇ = {0}
As an easy consequence of Lie's third theorem, it comes: Up to a Lie group isomorphism, it exists a unique simply connected Lie group ∇ whose Lie algebra is isomorphic to the Lie algebra ∇ . Definition 6.4. [START_REF] Palais | A global formulation of the lie theory of transformtion groups[END_REF] ∇ is called completely integrable if its the linear counterpart of a locally effective differentiable acion:

∇ × M ( , ) → . ∈ M.
In a compact manifold M every infinitesimal action of a finite dimensional Lie group is integrable. That is due to the fact every vector field X is complete in the meaning that X is a generator of a one parameter subgroup of the group of diffeomorphisms be a finite dimensional real vector space. Its group of affine isomorphisms, denoted by Aff( ), is defined as the semi-direct product:

Aff( ) = GL( )

where is by abuse of notation the group of translations of .

There is a natural affine representation of the Lie algebra ∇ in itself as a vector space:

∇ → ( ) = (∇ , ) ∈ ( ∇ ) × ∇ = ( ∇ ).
with affine action given by:

( ). = ∇ + , ∀ ∈ ∇ .
By virtue of the universal property of simply connected finite dimensional Lie groups, there exist a unique continuous affine representation

∈ ∇ → ( ∇ ( ), ∇ ( )) ∈ ( ∇ ).
Proposition 7.1. With the above notations, ∇ is a linear representation of ∇ in ∇ and ∇ is a ∇ valued 1-cocycle of Proof. The couple (f,q) is a continuous homomorphism of the Lie group ∇ on the Lie group ( ∇ ), thus, for any 1 , 2 ∈ ∇ :

( ( 1 ). ( 2 ), ( 1 ) ( 2 ) + ( 1 )) = ( ( 1 . 2 ), ( 1 . 2 )) .
The cohomology of the Lie group ∇ value in his Lie algebra ∇ is defined by the complex [START_REF] Boyom | Sur les structures affines homotopes à zéro des groupes de lie[END_REF][START_REF] Cartan | Homological algebra[END_REF][START_REF] Chevalley | Cohomology theory of lie groups and lie algebras[END_REF] : The condition : (2) The cohomology class [ ∇ ] vanishes;

... → ( ∇ , ∇ ) → +1 ( ∇ , ∇ ) → +2 ( ∇ , ∇ ) →
∇ ( 1 . 2 ) = ∇ ( 1 ) ∇ ( 2 ) + ∇ ( 1 ) ∀ 1 , 2 ∈ ∇ is equivalent to ∇ ∈ 1 ( ∇ , ∇ ).
(3) The affine representation

∇ → ( ∇ ( ), ∇ ( ) ∈ Aff( ∇ )
is conjugated to the linear representation

∇ → ∇ ( ) ∈ ( ∇ ).
Proof. Let us first show that (1) implies (2). Let -0 be a fixed point of the affine action ( ∇ , ∇ ), then ∇ ( ) (-0 ) + ∇ ( ) = -0 , ∀ ∈ ∇ . Therefore one has:

∇ ( ) = ∇ ( ) ( 0 ) -0 , ∀ ∈ ∇ .
So the cocycle is exact. To prove that (2) implies (3), consider the affine isomorphism ( , 0 ). It is nothing but the translation by 0 → + 0 ;

We calculate ( , 0 ) ( ∇ ( ), ∇ ( )) ( , 0 ) -1 = ( ∇ ( ), 0 ∇ ) Where ∇ stands for the zero element of the vector space ∇ . Finally, (3) implies (1). This assertion means that there exists an affine isomorphism ∇ → ( ) + 0 ∈ ∇ such that ( , 0 ) ( ∇ ( ), ∇ ( )) ( , 0 ) -1 = ( ∇ ( ), 0 ∇ ), ∀ ∈ ∇ The calculation of the left member yields the following identities:

( ) : ∇ ( ) = ∇ ( ) (7.1) ( ) : ( ∇ ( )) + 0 -[ ( ∇ ( ) -1 ] ( 0 ) = 0 ∇ . (7.2)
The identity ( ) yields:

∇ ( ) = ∇ ( ) ( -1 ( 0 )) --1 ( 0 ), ∀ ∈ ∇ .
Taking into account identity ( ), we obtain:

∇ ( ) = ∇ ( ) ( 0 ) -0 , ∀ ∈ ∇ .
So the vector -0 is a fixed point of the affine representaion ( ∇ , ∇ ).

Definition 7.3. The affine representation ( ∇ , ∇ ) is called the canonical affine representation of the gauge structure ( , ∇)

When the infinitesimal action ∇ is integrable, the proposition above is a key tool to relate the canonical affine representaion of ( , ∇) and the hyperbolicity problem for the orbits of ∇ .

8. A 8.1. Statistical manifolds. In this section, We restrict our attention to Riemannian statistical manifolds (excluding the pseudo-Riemannian case) whose particular cases are non-degenerate Fisher information metrics of statistical models and their family ofconnections, ∈ R. We recall that a statistical manifold can be viewed as a triple ( , , ∇) formed of a Riemannian manifold ( , ) and a symmetric gauge structure ( , ∇) which are linked by the following identity:

(∇ ) ( , ) -(∇ ) ( , ) = 0, ∀( , , ) ⊂ X( )
The -dual, ∇ ★ is defined as:

(∇ ★ , ) = ( , ) -( , ∇ ), ∀( , , ).
Depending on needs, we will use the alternative definition: Definition 8.1. A statistical manifold is a quadruple ( , , ∇, ∇ ★ ) formed a positive Riemannian manifold ( , ) and pair of symmetric gauge structure ( , ∇, ∇ ★ ) which are linked by the following identity

( , ) -(∇ , ) -( , ∇ ★ ) = 0, ∀( , , ).
Example 1: Any Riemannian manifold ( , , ∇ ) where ∇ is the Levi-Civita connection of is a statistical manifold as ∇ is self-dual. We say in such a case that is a trivial statistical manifold.

Example 2: Let ( , , ∇, ∇ * ) any statistical manifold, the family ( , , ∇ ( ) , ∇ (-) ) ∈R where ∇ ( ) = 1+ 2 ∇ + 1- 2 ∇ * is a statistical manifold for any ∈ R. 8.2. Hyperbolic leaves in statistical manifold. Let ( , , ∇, ∇ * ) be a compact statistical manifold and ∇ be its Levi-Civita connection.We consider three gauge structures ( , ∇), ( , ∇ ★ ) and ( , ∇ ).By assumption, M is compact,therefore ∇ , ∇ ★ and ∇ are the infinitesimal counterpart of the following locally effective differentiable dynamical systems:

∇ × → (8.1) ∇ ★ × → (8.2) ∇ × → (8.3)
Remark 8.2. In the Riemannian geometry viewpoint, the orbits of ∇ are flat Riemannian manifolds. Up to an isometry and a finite covering, each such an n-dimensional orbit is a flat cylinder of over an Euclidean torus [START_REF] Wolf | Space of constant curvature[END_REF], namely

( T Γ × R -, 0 )
with the flat metric 0 induced from the ambient Euclidean metric of R . The integer k is the first Betti number of the orbit and Γ is finite group of isometry. Proof.

(1) By assumption is compact then ∇ is integrable,we conclude that ∇ is an infinitesimal action of ∇ on M.We have ∇ ( , ) ∇ = 0 then F ∇ is locally flat foliation in ( , ∇). The ∇ -orbits are ∇-auto-parallel then ∇ = 0. The leaves of F ∇ are Hessian submanifolds of ( , , ∇).

(2) By using the same arguments for ∇ ★ , the theorem is proved. Theorem 8.5. In a compact statistical manifold ( , , ∇, ∇ ★ ) the following assertions are equivalent:

(1) [ ∇ ] = 0 ∈ 2 (F ∇ ) (2) [ ∇ ★ ] = 0 ∈ 1 ( ∇ ★ , ∇ ★ ) (3) ( ∇ ★ , ∇ ★ ) has a fixed point. (4) ( ∇ ★ , ∇ ★ ) is affinely conjugated to its linear component ∇ ★ .
Proof. According to Theorem 7.2, assertions (2), ( 3) and (4) are equivalent. Therefore, it is sufficient to prove that assertions (1) and (2) are equivalent. Let us demonstrate first that (1) implies [START_REF] Barbaresco | Koszul information geometry and souriau lie group thermodynamics[END_REF] Since the class [ ∇ ] vanishes, it exist a de Rham closed differential 1-form such that:

∇ ( , ) = -( ) + (∇ ), ∀( , ).
By the defining property of statistical manifolds, it comes:

∇ ( , ) = ∇ (∇ ★ , ) + ∇ ( , ∇ ), ∀( , , ).
Let be the unique vector field satisfying:

( ) = ∇ ( , ), ∀ .
Using once again the defining property of statistical manifolds, we get:

∇ ( , ) -∇ (∇ ★ , ) -∇ ( , ∇ ) = 0.
Since left hand member is ∞ ( )-multilinear we can assume that ∈ ∇ and we get the following identity

∇ (∇ ★ , ) = ∇ ( , ) -∇ ( , ∇ ) = ∇ ( , ), ∀( , ) ⊂ X( ). Thus one has: ∇ ★ (-) -= 0 ∇ , ∀ ∈ X( ).
So theis a fixed point of ( ∇ ★ , ∇ ★ ).

Let us demonstrate now that (2) implies (1).

Let us assume that ( ∇ ★ , ∇ ★ ) has a fixed point 0 ∈ ∇ ★ . Then:

∇ ★ ( ) ( 0 ) + ∇ ★ ( ) = 0 , ∀ ∈ ∇ ★ .
To every ∈ ∇ ★ , we assign the one parameter subgroup

{ ( ), ∈ R} ⊂ ∇ ★ .
We have:

∇ ★ ( ( )) ( 0 ) + ∇ ★ ( ( )) = 0 , ∀ ∈ R.
Calculating the derivative at = 0, one obtains:

∇ ★ 0 + = 0. Finally: ∇ ( 0 , ) = ∇ (∇ ★ 0 , ) + ∇ ( 0 , ∇ ) Using ∇ ★ 0 = -, ∀
, one obtains the following identity:

∇ ( 0 , ) = -∇ ( , ) + ∇ ( 0 , ∇ ). By putting ( ) = -∇ ( 0 , ), it comes: ∇ ( , ) = ( ) -(∇ ), ∀( , )
Then one has [ ∇ ] = 0 ∈ 2 (∇), concluding the proof.

Corollary 8.1. Let ( , , ∇, ∇ ★ ) such that both ∇ and ∇ ★ be integrable. We assume that leaves of both F ∇ and F ∇ ★ satisfy one among the assertions of Theorem 8.5. Then every compact leaf of F ∇ (resp. F ∇ ★ ) is hyperbolic. [START_REF] Vey | Sur une notion d'hyperbolicité des variétés localement plates[END_REF], [START_REF] Kobayashi | Projectively invariant distances for affine and projective structures[END_REF], [START_REF] Wu | Some theorems on projective hyperbolicity[END_REF]). Let M be a closed hyperbolic manifold.Then M is a quotient of a properly convex cone.

Theorem 8.8. Let ( , ∇) be a closed hyperbolic manifold.Then

[ ∇ ★ ] = 0 ∈ 1 ( ∇ ★ , ∇ ★ ) and [ ∇ ] = 0 ∈ 1 ( ∇ , ∇ )
Proof.

Step 1: Koszul [START_REF] Koszul | Déformations de connexions localement plates[END_REF] From theorem 9.7, the universal covering ˜ is diffeomorphic to a convex cone not containing any straight line of R ,let's call it by Ω and a affine map from Ω to define by = • D -1 ,where : ˜ -→ and the developing map

D : ˜ -→ Ω.Let ∈ X(R ) define by ( ) = =1
is homotheties fields (ie this field is generated by a 1-parameter group of homotheties of center 0 and ratio ,then for all ∈ X(R ),we have ∇ 0 = where ∇ 0 is canonical flat connection on R .The vector field is (R )-invariant.Since Ω is salient then any affine automorphism of Ω is restriction of an element of (R ).The restriction of to Ω is invariant by the affine automorphisms of Ω. Therefore, there exists one and only one vector field on , such that = .Since is locally an isomorphism of locally flat manifolds ie( (∇ 0 ) = ∇):

∇ = ∇ 0 ∇ = then [ ∇ ] = 0 ∈ 1 ( ∇ , ∇ ) step 2
By assumption ( , ∇) is compact hyperbolic, so it exists a closed differentiable one form such that ∇ is positive define. Then = ∇ is Hessian metric and locally =

2

.We thus conclude that ( , , ∇) is a Hessian manifold. Let ∇ ★ be the dual of ∇, defined by:

(∇ ★ , ) = . ( , ) -( , ∇ )
∇ ★ is a locally flat connection on M.

We have:

( , ) = . ( ) -(∇ )
thus, it exists ★ ∈ X( ) such that ( ) = ( ★ , ),then we conclude that

( , ) = (∇ ★ ★ , )
for all ∈ X( ). We conclude that:

∇ ★ ★ = then [ ∇ ★ ] = 0 ∈ 1 ( ∇ ★ , ∇ ★ )
Corollary 8.2. Let ( , ∇) be a closed hyperbolic manifold.Then there exists closed one form Θ such that = ∇ ★ Θ. Theorem 8.9. Let ( , ∇) be a closed hyperbolic manifold.Then ( , ∇ ★ )is also hyperbolic manifold.

W .

Convex cones are examples of bounded domains. The studies of convex cones have been and continue to be among high standing subjects in geometry and in analysis. The pioneering works are those of Elie Cartan, but there is a wealth of subsequent works, see cartan1935domaines, koszul1961domaines, vinberg1967theory and many others. Nowadays the analysis in convex cones plays interesting role in the information geometry, bar-baresco2013information. Theorem 3.3 provides conditions under which compact orbits of ∇ are hyperbolic. We are going to demonstrate that up to difeomorphisms, compact leaves of F ∇ are quotient homogeneous convex cones. According to our previous notation, if is a compact leaf of F ∇ then ( ˜ ) is a homogeneous convex cone. For convenience, we introduce the following notion:

Let ( , , ∇, ∇ ★ ) be an integrable statistical manifold whose Hessian foliations are denoted by {F ∇ , ∇ , ∇} and

F ∇ ★ , ∇ ★ , ∇ ★ .
Henceforth, by abuse of notation, a leaf of

{F ∇ , ∇ , ∇}
is an Hessian manifold that we denote by

( , ∇ , ∇).
Its -dual is: ( , ∇ , ∇ ★ ). Theorem 9.5 links the Hessian class ∇ , namely:

[ ∇ ] ∈ 2 (∇)
and the radiant class of ( , , ∇ ★ ), namely:

[ ∇ ★ ] ∈ 1 ( ∇ ★ , ∇ ★ ).
We deduce that if is compact then:

[ ∇ ] ∪ [ ∇ ★ ]
is a characteristic obstruction of ( , ∇) being hyperbolic. Henceforth we assume these obstructions vanish.

We have already pointed out that if ˜ the universal covering of , it admits a unique locally flat structure ( ˜ , ∇) such that the covering map : ˜ → is a gauge morphism between ∇ and ∇. Theorem 9.1. [START_REF] Koszul | Domaines bornés homogenes et orbites de groupes de transformations affines[END_REF]

Let {( , , ∇), [ ∇ ] ∪ [ ∇ ★ ]} being a compact Hessian manifold. If: {[ ∇ ] ∪ [ ∇ ★ ]} = {0} ∪ {0} .
Then for ( ˜ ) being a cone, it is sufficient that ˜ being homogeneous under the group of transformation of ( ˜ , ∇)♣ Theorem 9.2. Let ( , , ∇) be a statistical manifold and let F ∇ be its canonical Hessian foliation. If the characteristic obstruction vanishes:

{[ ∇ ] ∪ [ ∇ ★ ]} = {0} ∪ {0} .
Then, up to a affine diffeomorphism, every compact leaf of F ∇ is the quotient of a sharp open convex cone by a discrete group of (R )

♣

Proof. The action

∇ × →
gives rise to the action

∇ × ˜ → ˜ .
( , ∇) is homogeneous under the action of the Lie group ∇ and ( , ∇) is the quotient of ( ˜ , ∇) under the action of the fundamental group 1 ( ).

Therefore both 1 ( ) and ∇ are subgroups of the group

( ˜ , ∇).
It is easy to check that ∇ is included in the normalizer of 1 ( ). Since is transitively acted on by ∇ , every orbit of ∇ in ˜ is an open submanifold of ˜ . Since ˜ is connected the action of ∇ on ˜ is transitive.From the theorem 10.2 we deduce that the compact leaves of F ∇ are diffeomorphic to the quotient of a sharp open convex cone by a discrete group of (R ). where is constant.

As ★ ★ = then ( 1 , ) is hyperbolic.Now let : R → 1 the covering map, and consider the map (R, ˜ ) → ( 1 , )

.The compacts orbits is ∇ ( , ) is quotient R >0 Γ , here Γ is discrete subgroup of (R)

C

The characterization of hyperbolicity given in theorem 8.5 has an important application in elucidating the structure of some statistical manifolds as statistical models. In a future work, an explicit construction of an exponential family on leaves of a statistical manifold will be given along with an application to the classification of compact exponential models in dimension 4. R

Theorem 2 . 1 .( 1 )

 211 proved in this paper are summarized below. The following statements are equivalent : The affine action ∇ ∇ ( , ) → ∇ ( ). + ∇ ( ) ∈ ∇ has a fixed point;

Definition 4 . 1 .

 41 An affinely flat structure in a m-dimensional manifold is defined by a complete atlas:A = {( , Φ )} whose local chart changes Φ -1 • Φ are restrictions of affine transformations to the mdimensional affine space R .

∇ 2 ,

 2 = .( . ) -( . ). Definition 6.2. The Hessian equation ∇ is defined by: HE(∇) : ∇ 2 = 0 6.1. Local expression of Hessian equation of ∇. Let ( 1 , .

6 . 1 .Proposition 6 . 5 .

 6165 If ∇ is the Levi-Civita connection of an Einstein Riemannian manifold,then: ∇ = {0} Let ( , ) be a compact Riemanian manifold such that the < 0 then: ∇ = {0} Proof. By using the formula

with differential operator : ( 1

 1 , ......, +1 ) = ∇ ( 1 ). ( 2 , ......., +1 ) + (-1) (.., +1 , ...) + (-1) ( 1 , ...., )

Theorem 7 . 2 .( 1 )

 721 The next definition thus makes sens: The cohomology class [ ∇ ] ∈ 1 ( ∇ , ∇ ) is called the radiant class of the affine representation ( , ). We are now in position to state one of the main results of the article: The following statements are equivalent : The affine action ∇ ∇ ( , ) → ∇ ( ). + ∇ ( ) ∈ ∇ has a fixed point;

Example 3 :

 3 Let ( , ) some Riemannian manifold with ∇ the Levi-Civita connection with respect to g, and let ∈ \ {0}. The triplet ( , , ∇, ∇ * ) is a statistical manifold where ∇ = ∇ + ( , ) ( , ) and ∇ * = ∇ -( , ) ( , ) .

Notation 8 . 3 .Theorem 8 . 4 .

 8384 F ∇ is the foliation whose leaves are orbits of the Lie group ∇ and ∇ the restriction of to F ∇ . Let ( , , ∇, ∇ ★ ) be a compact statistical manifold. Then the foliation F ∇ (resp. F ∇ * ) is a Hessian foliation in ( , , ∇) (resp. ( , , ∇ * )).

8. 3 .Theorem 8 . 6 .Theorem 8 . 7

 38687 Cohomological characteristical of hyperbolicity of Hessian manifolds. The following nullity of Hessian class,nullity of radiant class affine representation and hyperbolicity of locally flat manifold the following theorem is another formulation of the hyperbolicity of a locally flat manifold in the sense of Koszul. It proves that the hyperbolicity can be described by the nullity of the radiant class of the affine dynamics of the dual connection. Let ( , , ∇) being a compact Hessian manifold.If the characteristic obstruction {[ ∇ ] ∪ [ ∇ ★ ]} = {0} ∪ {0} Then ( , ∇) is an hyperbolic manifold . ( [

1 .

 1 Characteristic function and Formalism of Barbaresco. The locally flat structure of ( , ∇) rises up on its universal covering ˜ as a locally flat structure ( ˜ , ∇) for which the projection : ( ˜ , ∇) -→ ( , ∇) is affine map.The locally flat structure on ˜ is defined by the local affine diffeomorphism D : ( ˜ , ∇) -→ (R , ∇ 0 ) where ∇ 0 is the standard locally flat connection define by ∇ 0 = 0 ∀ = 1, 2, ..., . The development map gives rise to an representation (∇) of the fundamental group 1 ( ) of in (R ), called the holonomy representation of . It is defined by the following commutative diagram: ∇ 0 ) ( ∇) / / (R , ∇ 0 ) D • = (∇) • D < ˜ , ∇, 1 ( ) >-→ (Ω, ∇ 0 , (∇)) ⊂ (R , ∇ 0 )

By the theorem 53,Ω is sharp open convex cone of R and (∇) ⊂ (R ), thus it exists a local diffeomorphism ˜ fitting in the next diagram:

(Ω, ∇ 0 , (∇))

Let Ω sharp open convex cone of R and let Ω the closure of Ω. The set:

is called the dual cone of Ω.The holonomy (∇) act on Ω ★ by dual representation

where ∈ (∇) J-L Koszul and E.B. Vinberg have introduced the following integral of a sharp convex cone Ω:

where is the Lebesgue measure of R ★ . is an analytic function on Ω,with Ω ( ) ∈]0, +∞[,called the Koszul-Vinberg characteristic function of cone Ω. It satisfies [START_REF] Vey | Sur les automorphismes affines des ouverts convexes saillants[END_REF]:

(1) Ω is logarithmically strictly convex and Φ Ω ( ) = ( Ω ) is strictly convex.

(2) If ∈ (∇) then Ω ( ) = ( ) -1 Ω ( ).

(

In [START_REF] Barbaresco | Koszul information geometry and souriau lie group thermodynamics[END_REF],Frédéric Barbaresco introduced a probability density called Koszul density and defined by: For ∈ Ω and ∈ Ω ★ ,we define

Then { ( , ), ∈ Ω ★ } is an exponential family of probability distributions on Ω ★ parameterized by ∈ Ω.

10.2. Construction of Probability density on leaves of F ∇ . Our aim is to provide the leaves of F ∇ with a statistical model structure.

(∇) ⊂ ( , R) is an amenable group, thus it exists a mean ∈ ( ∞ ( (∇)), R) defined by:

and satisfying:

(1) If > 0, ( ) > 0.

(

Let:

, :

Then:

, ∈ ∞ ( (∇))

Proposition 10.1.

Proposition 10.2. Let the map

By the Theorem of Fubini-Tonelli,we have

Theorem 10.1. Let ( , , ∇) be an integrable statistical manifold and let F ∇ be its canonical Hessian foliation. If the characteristic obstruction vanishes:

Then, up to an affine diffeomorphism, every compact leaf of F ∇ is a parametric space of statistical models for a measurable set (Ω ★ , B (Ω ★ ))

< , > Where is define by:

We shall give a example of compact hessian manifold whose orbits are quotient of statistical models by a discrete subgroup.Let R 2 be a 2-dimensional real affine space with the natural flat affine connection and let { , } be an affine coordinate system of R 2 .

Let Ω be a domain define by > 0 and > 0.Consider a Riemannian metric on Ω given by = 1 2 2 + 1 2 2 . Then ( , ) is Hessian structure on Ω. Let Σ and be linear transformations on Ω defined by :

Then <Σ, > leave the Hessian structure ( , ) invariant. We denote Γ the group generate by {Σ, }, we can also write Γ as < 2 0 0 1 ; 2 0 0 3 >. Γ acts properly discontinuously on Ω and Ω Γ is is compact Hessian manifold which is diffeomorphic to a Torus. Let us denote by the projection from Ω to Ω Γ and by ( , ) the Hessian structure on Ω Γ .Since the space of all Γ-invariant parallel 1-forms on Ωis spanned by and .The space of all -parallel 1-forms on Ω Γ is spanned by and Φ where = ★ and = ★ Φ.Let and ˜ be a vector field on Ω Γ defined by ( ) = ( , ) and Φ( ) = ( ˜ , )for each vector field on Ω Γ .Then = ★ ( ) and ˜ = ★ ( ) and the vector space ∇ of all Hessian vector fields on Ω Γ is spanned by < , ˜ >.Since 2 is compact the ∇ is integrable.Let ( ) and ( ) is a 1-parameter group of transformations generated by and by .Let ∇ =< ( ), ( ) >. Consider > 0 and > 0, the compact homogeneous the orbit ∇ ( , ) = { . ( , ), ∈ ∇ } is a circle. Then we conclude that ( 1 , , 1 ) is Hessian manifold and satisfy = 0 and 1 = 1