Aurore Dubuffet
email: aurore.dubuffet@uca.fr

Marina Chauvet

Anne Moné

Didier Debroas

Cécile Lepère

Aurore Dubuffet°

A phylogenetic framework to investigate the microsporidian communities through metabarcoding and its application to lake ecosystems

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INTRODUCTION

For long underestimated, the diversity of parasites and their central role in the functioning of ecosystems are nowadays recognized [START_REF] Lafferty | Parasites dominate food web links[END_REF][START_REF] Dobson | Colloquium paper: homage to Linnaeus: how many parasites? How many hosts?[END_REF][START_REF] Lafferty | Parasites in food webs: the ultimate missing links[END_REF]. For example, it is estimated that about 40% of species, all ecosystems combined, are parasites [START_REF] Dobson | Colloquium paper: homage to Linnaeus: how many parasites? How many hosts?[END_REF] and that more than 75% of the trophic links include parasites [START_REF] Lafferty | Parasites dominate food web links[END_REF]. However, these parasites only represent 5% of the described species [START_REF] Costello | Parasite rates of discovery, global species richness and host specificity[END_REF] and further studies are needed to reveal the extent of their diversity and to clarify their role in ecosystems. The advent of molecular approaches, in particular metabarcoding, has revealed a considerable diversity of microeukaryotes (<200 μm) in different types of ecosystems, including parasite groups which can represent up to 50% of the total diversity [START_REF] Cleary | Unexpected prevalence of parasite 18S rDNA sequences in winter among Antarctic marine protists[END_REF][START_REF] Lepère | Diversity and biogeography of picoeukaryotes: new insights into the rare biosphere[END_REF][START_REF] Mahé | Parasites dominate hyperdiverse soil protist communities in Neotropical rainforests[END_REF]. In lacustrine ecosystems these studies highlighted the presence of several group of putative parasites, such as chytrids (Fungi), Apicomplexa and Perkinsea (Alveolates) [START_REF] Debroas | Overview of freshwater microbial eukaryotes diversity: a first analysis of publicly available metabarcoding data[END_REF]. Some groups of parasites, however, are largely absent from these studies (see Bass et al., 2015 for review). This is the case of Microsporidia, which are obligate unicellular intracellular parasites related to Fungi [START_REF] Vavra | Microsporidia and 'the art of living together[END_REF]. Because of their rapid evolution and significant reduction of their genomes [START_REF] Keeling | MICROSPORIDIA : Biology and Evolution of Highly Reduced Intracellular Parasites[END_REF], they are genetically divergent from other eukaryotes, making the primers conventionally targeting the conserved regions of the rRNA 18S gene ineffective [START_REF] Marande | Eukaryotic diversity and phylogeny using small-and large-subunit ribosomal RNA genes from environmental samples[END_REF][START_REF] Hadziavdic | Characterization of the 18S rRNA Gene for Designing Universal Eukaryote Specific Primers[END_REF][START_REF] Hugerth | Systematic Design of 18S rRNA Gene Primers for Determining Eukaryotic Diversity in Microbial Consortia[END_REF]. Some studies attempted to characterize the diversity of this phylum using Microsporidia-specific primers, mainly in particular hosts such as daphnids, amphipods or mosquitoes [START_REF] Weigl | Natural distribution and coinfection patterns of microsporidia parasites in the Daphnia longispina complex[END_REF][START_REF] Quiles | Microsporidian infections in the species complex Gammarus roeselii (Amphipoda) over its geographical range: evidence for both host-parasite co-diversification and recent host shifts[END_REF][START_REF] Trzebny | A new method of metabarcoding Microsporidia and their hosts reveals high levels of microsporidian infections in mosquitoes (Culicidae)[END_REF] but more rarely in environmental samples [START_REF] Ardila-Garcia | Microsporidian diversity in soil, sand and compost of the pacific Northwest[END_REF][START_REF] Williams | Group-specific environmental sequencing reveals high levels of ecological heterogeneity across the microsporidian radiation[END_REF].

Based on ultrastructural, ecological (e.g. host species, habitat, transmission route) and tissue tropism criteria, more than 1300 Microsporidia species have been described so far [START_REF] Vossbrinck | Molecular phylogeny and evolution of mosquito parasitic Microsporidia (Microsporidia: Amblyosporidae)[END_REF], most infecting one or several closely related hosts. This may nevertheless represent only a minor fraction of their real diversity, as the number of microsporidian species is estimated to approach the number of all animal species [START_REF] Keeling | MICROSPORIDIA : Biology and Evolution of Highly Reduced Intracellular Parasites[END_REF][START_REF] Larsen | Inordinate Fondness Multiplied and Redistributed: the Number of Species on Earth and the New Pie of Life[END_REF]. Microsporidia are found in all ecosystems, and particularly in aquatic ecosystems which harbour about 50% of the described genera (Stentiford et al., 2013a). Vertebrates, crustaceans and insects alone account for nearly 80% of known microsporidian hosts (Stentiford et al., 2013a;[START_REF] Weiss | Microsporidia: Pathogens of Opportunity[END_REF][START_REF] Stentiford | Microsporidia -Emergent Pathogens in the Global Food Chain[END_REF]. Such proportion however is likely due to biases towards the search of infectious agents in hosts of sanitary (human, disease vector insects such as mosquitoes), veterinarian (domestic and farm animals), agronomic (auxiliary insects and pests) or ecological importance (daphnids, amphipods). As a result, the global diversity of Microsporidia is largely unknown, particularly in small sized hosts, such as protists which represent only 10 of the 7366 microsporidian SSU sequences listed in the databases (NCBI consultation 18/11/2020 with the key words 'small subunit ribosomal'+ microsporidia).

In this work we used a targeted metabarcoding approach to specifically study the microsporidian diversity in lake ecosystems, focusing on planktonic fractions containing small sized hosts. Highthroughput sequencing classically produces thousands of sequences, which need to be assigned taxonomically in order to perform comparisons between samples. Such assignations are based on databases that link sequences to taxonomy. We first checked the consistency between available databases and phylogenetic data, and found that they are either weakly detailed or contain several errors.

We thus built a comprehensive reference database which positions microsporidian SSU rRNA gene sequences within a coherent ranked phylogenetic framework covering the microsporidian diversity.

Moreover, we addressed the question of OTU (Operational Taxonomic Unit) definition in Microsporidia. Indeed, sequences of taxonomic marker genes with high similarity scores are classically clustered into OTUs, themselves considered as proxies for species. We therefore determined the clustering threshold that maximizes the recovery of distinct species and/or genera for microsporidian SSU sequences. Using this framework, we highlighted a considerable uncovered microsporidian diversity in our samples, and evidenced spatio-temporal variations in the abundance of many Microsporidia species.

MATERIALS AND METHODS

Microsporidia database

An exhaustive search of all Microsporidia species for which a SSU sequence is available was conducted (last update: march 2020), using hundreds of publications and the NCBI database (nucleotides and WGS sequences). We focused on the canonical Microsporidia, characterized by a specific invasion apparatus known as the polar tube, an extreme genome compaction, a lack of true mitochondria, and a rRNA operon of prokaryote-like size and structure [START_REF] Wadi | Evolution of microsporidia: An extremely successful group of eukaryotic intracellular parasites[END_REF]. We also included sequences from Chytridiopsida and Metchnikovellida, considered as unusual Microsporidia as they lack some of the typical features of Microsporidia [START_REF] Mikhailov | Genomic survey of a hyperparasitic microsporidian Amphiamblys sp. (Metchnikovellidae)[END_REF][START_REF] Corsaro | Filling gaps in the microsporidian tree: rDNA phylogeny of Chytridiopsis typographi (Microsporidia: Chytridiopsida)[END_REF]. When several sequences were found for a species, we kept the longest or the most consensual one. We found 337 microsporidian species with a SSU sequence. We then selected the species for which the full length of the SSU rRNA gene was available (at least 90% of the V1-V9 region, 265 sequences, Supplementary Information 1).

Their taxonomic assignations were searched in public databases (NCBI, SILVA, PR², PHYMYCO-DB) and compared with several microsporidian phylogenies [START_REF] Vossbrinck | Molecular phylogeny of the Microsporidia: ecological, ultrastructural and taxonomic considerations[END_REF][START_REF] Smith | The ecology and evolution of microsporidian parasites[END_REF][START_REF] Vossbrinck | Molecular phylogeny and evolution of mosquito parasitic Microsporidia (Microsporidia: Amblyosporidae)[END_REF][START_REF] Corsaro | Filling gaps in the microsporidian tree: rDNA phylogeny of Chytridiopsis typographi (Microsporidia: Chytridiopsida)[END_REF] and a recent taxonomic review [START_REF] Wijayawardene | Notes for genera: basal clades of Fungi (including Aphelidiomycota, Basidiobolomycota, Blastocladiomycota, Calcarisporiellomycota, Caulochytriomycota, Chytridiomycota, Entomophthoromycota, Glomeromycota, Kickxellomycota, Monoblepharomycota, Mortierellomycota, Mucoromycota, Neocallimastigomycota, Olpidiomycota, Rozellomycota and Zoopagomycota)[END_REF].

As none of the public databases was satisfying (see results), we built an extensive phylogeny of Microsporidia, based on their full-length SSU gene sequences. These sequences, as well as 6 outgroup sequences (from Rozella allomycis, Paramicrosporidium sp., Nucleophaga sp. and Mitosporidium daphnia), were aligned using the online version of MAFFT (ver. 7; Computational Biology Research Consortium [http://mafft.cbrc.jp/alignment/server/index.html]; [START_REF] Katoh | MAFFT version 5: improvement in accuracy of multiple sequence alignment[END_REF] with the E-INS-i multiple alignment method, leaving gappy regions, a gap opening penalty of 1.0 and gap extension penalty extension of 0.0. The alignment was checked using MEGA 7 [START_REF] Kumar | MEGA7: Molecular Evolutionary Genetics Analysis Version 7.0 for Bigger Datasets[END_REF] and both ends were trimmed. The resultant alignment was analysed by Maximum Likelihood (ML) and Neighbor joining (NJ) using MEGA 7.0 [START_REF] Kumar | MEGA7: Molecular Evolutionary Genetics Analysis Version 7.0 for Bigger Datasets[END_REF]. A General Time-Reversible (GTR) model of evolution with gamma-distributed rate heterogeneity (G) and a proportion of invariable sites (I) (GTR + G + I) was used to create the ML tree, with 500 bootstrap replications and partial deletion (85%). The NJ tree was obtained using the Kimura-2-parameter model with a gamma distribution of 4, partial deletion (85%) and 1000 bootstraps. Metadata (habitat and hosts) were reported on the ML tree using iTOL [START_REF] Letunic | Interactive tree of life (iTOL) v3: an online tool for the display and annotation of phylogenetic and other trees[END_REF]. Similar phylogenetic analyses were done using the region of the SSU gene which was chose for the metabarcoding analysis described below (V1-V3, corresponding to the first three variables regions of the SSU gene). These trees were used to define three phylogenetic ranks inside the Microsporidia phylum: clade, group and genus. This classification (see results and discussion section) was subsequently used to assign the Operational Taxonomic Units (OTUs) obtained from environmental samples.

OTU definition

We tested different clustering thresholds on selected SSU sequences (species with a clear genus affiliation, and for which the full length of the SSU gene was available, 255 sequences) in order to determine which one could maximize the recovery of distinct genera and species. Full-length and V1-V3 sequences were aligned with the Needleman-alignment algorithm and a distance matrix was obtained with the pairwise.seqs command in Mothur [START_REF] Schloss | Introducing mothur: open-source, platform-independent, community-supported software for describing and comparing microbial communities[END_REF]. Then, the average neighbor algorithm was used with clustering thresholds ranging from 100% (unique sequences) to 90% in steps of 1%. The number of obtained OTUs was then compared to the number of species (255) and genera (104) in the dataset (% recovery).

Study sites and sampling procedure

The study was conducted in two lakes (Massif Central, France). Lake Pavin (45 • 55 N and 2 • 54 E) is an oligo-mesotrophic lake with an area of 0.44 km 2 and a maximum depth of 92 m, at an altitude of 1197 m above sea level. It is a meromictic lake characterized by the presence of two permanent stratified layers. Lake Aydat (45°39′ 50″ N, 2° 59′ 04″ E) is a dimictic and eutrophic lake with a maximum depth of 15 m, situated at an altitude of 825 m. In April, May and June 2016, water samples were collected in the euphotic zone of both lakes at a permanent station located at the deepest zone of the water column. 6-10 L of water were filtered successively through 150 μm, 50 μm, and 10 μm pore-size filters in order to concentrate Microsporidia spores (<10 µm fraction) and planktonic species susceptible to host Microsporidia (10-50 and 50-150 µm fractions). The microbial biomass of each size-fraction was collected on 0.8µm pore-size (pressure, <10 kPa) polycarbonate filters (Millipore, Molsheim, France) and stored at -80°C until further analysis.

Nucleic acids extraction, amplification and sequencing

Filters were covered with the extraction buffer (50 mM glucose, 10 mM EDTA, 25 mM Tris). 0.1g of 0.1 mm glass beads were added and samples were homogenized in a bead beater (3 pulses for 30s at 30 Hz). Tubes were then centrifuged during 1 min at 600 g. EDTA pH8 and a lysozyme solution (final concentrations 0.1 M and 1.5 mg.ml -1 respectively) were added to the supernatant and incubated at 37°C for 30 min. Then sodium dodecyl sulfate (1.8%, final), proteinase K (100 μg.ml -1 , final) and RNaseA (50 μg.ml -1 , final) were added, and samples were incubated at 37°C for at least 60 min. A cetyltrimethylammonium bromide (CTAB) solution (final concentration 1% in a 0.7 M NaCl solution) was added, and samples were incubated at 65°C for 10 min. Nucleic acids were extracted with phenolchloroform-isoamyl alcohol (25:24:1); the aqueous phase containing nucleic acids was kept and purified by adding chloroform-isoamyl alcohol (24:1). After isopropanol (0.6 vol) addition, the nucleic acids were precipitated at -20°C for 12 h. After centrifugation, the DNA pellet was ethanol rinsed and resuspended in 20 μL of water. Amplification of the V1-V3 region of the Microsporidia small subunit rDNA was performed using the universal microsporidian V1F (CACCAGGTTGATTCTGCCTGAC) and 530R

(CCGCGGCTGCTGGCAC) primers set [START_REF] Baker | Phylogenetic Relationships among Vairimorpha and Nosema Species (Microspora) Based on Ribosomal RNA Sequence Data[END_REF][START_REF] Weiss | Utility of microsporidian rRNA in diagnosis and phylogeny: a review[END_REF] modified by adding barcodes. Each PCR was performed in a total volume of 30 μL containing 3 μL of 10× NH4 reaction buffer, 1.2 μL of 50 mM MgCl2, 0.15 μL of EurobioTaqII (Eurobio, Les Ulis, France, 5 U/μL), 0.6 μL of 10 mM each dNTP, 0.3 μL of 50 mg/mL BSA and 1.2 μL of each 5 μM primer. The amplification conditions consisted of initial denaturation at 94 °C for 5 min followed by 35 cycles for 1 min at 94 °C, 1 min 10 s at 61 °C and 1min at 72 °C, and a final elongation of 7 min at 72 °C.

The PCR products were run on a 1.2 % agarose gel electrophoresis. The amplicons were purified and concentrated using the MinElute gel extraction kit (Qiagen®, Courtaboeuf, France), and quantified using the Agilent 2200 TapeStation system and the D1000 ScreenTape kit (Agilent Technologies, Les Ulis, France). Finally, the purified amplicons for each sample were pooled at equimolar concentrations and sent to GATC Biotech (Konstanz, Germany) for library preparation and paired-end (2x300 bp)

sequencing on a MiSeq Illumina instrument (San Diego, CA).

Amplicons analysis & phylogenetic affiliation

The MiSEQ data were assembled with the VSEARCH tool [START_REF] Rognes | VSEARCH: a versatile open source tool for metagenomics[END_REF]. These sequences were then subjected to four cleaning procedures: (i) no undefined bases (Ns), (ii) a minimum sequence length of 300 bp, (iii) no sequencing error in the forward and reverse primers and (iv) no chimeras [START_REF] Edgar | UCHIME improves sensitivity and speed of chimera detection[END_REF]. After this cleaning step and demultiplexing process, the remaining DNA sequences were clustered at 99%. The representative OTUs were then affiliated by similarity and phylogeny using a custom curated reference database (Freshwater-database: https://github.com/panammeb/Freshwaterdatabase). This database contains eukaryote references extracted from the SSURef SILVA database [START_REF] Pruesse | SILVA: a comprehensive online resource for quality checked and aligned ribosomal RNA sequence data compatible with ARB[END_REF], typical freshwater lineages identified in previous studies (e.g. [START_REF] Debroas | Evidence for an active rare biosphere within freshwater protists community[END_REF], and 528 sequences from the 265 microsporidian species (107 genera) we analysed in the microsporidian phylogenetic tree (full length SSU sequences). After a comparison of the OTUs with the Freshwater-database by a similarity approach (VSEARCH tool), trees of OTUs with their closest reference sequences were built in FastTree [START_REF] Price | FastTree 2 -approximately maximum-likelihood trees for large alignments[END_REF] (see the detailed pipeline in [START_REF] Debroas | Overview of freshwater microbial eukaryotes diversity: a first analysis of publicly available metabarcoding data[END_REF]. Taxonomic assignment was conducted according to two methods: nearest neighbour (NN)

and last common ancestor (LCA) affiliations. This process was implemented in the pipeline PANAM (Phylogenetic Analysis of Next-generation Amplicons; https://github.com/panammeb/) and is described in more detail in [START_REF] Taib | Phylogenetic affiliation of SSU rRNA genes generated by massively parallel sequencing: new insights into the freshwater protist diversity[END_REF] Affiliations to groups and genera were considered as valid when identity scores were ≥85% and ≥94.5%, respectively (empirical thresholds based on comparisons between affiliations and phylogenetic analyses, not shown).

Since the clustering threshold we used (99%) may generate OTUs resulting from sequencing errors, we only kept for the final analysis OTUs representing more than 0.05% of the total reads in at least one sample. Two samples were removed from the analysis since they contained less than 20 000 reads. Data were finally normalized using rarefaction with the phyloseq package on R (R Development Core Team, 2011;[START_REF] Mcmurdie | Phyloseq: an R package for reproducible interactive analysis and graphics of microbiome census data[END_REF].

Diversity indices

Community richness and diversity indices (Richness observed, Chao1, Shannon and Simpson) were calculated using vegan package on R (R Development Core Team, 2011; [START_REF] Oksanen | Vegan: Community Ecology Package. R package Version 2[END_REF]. Nonmetric multidimensional scaling (NMDS) ordinations, based on a Bray-Curtis dissimilarity matrix [START_REF] Bray | An ordination of upland forest communities of southern Wisconsin[END_REF][START_REF] Paliy | Application of multivariate statistical techniques in microbial ecology[END_REF], were performed to depict the difference in Microsporidia community compositions among the samples using the R vegan package (R Development Core Team, 2011; [START_REF] Oksanen | Vegan: Community Ecology Package. R package Version 2[END_REF].

Phylogenetic analysis of OTUs

All OTUs obtained in this analysis were assigned at least to a clade, but 24% of them were not assigned to a group. We therefore performed phylogenetic analyses (for each clade) on OTUs not assigned to a group in order to determine whether new groups could be highlighted. Besides, only 48% of the OTUs were assigned to a genus. We checked the accuracy of affiliations to genera and highlighted potential new genera through phylogenetic analyses in each identified group. We focused our analyses on abundant OTUs (>1% of reads in at least one sample, 95 OTUs), as they represent between 53 and 92% of the total reads in each sample. We included in these analyses several sequences from undescribed microsporidian species (found through BLAST search and/or bibliographic review), in addition to species from our database. After alignment using the MAFFT algorithm [START_REF] Katoh | MAFFT version 5: improvement in accuracy of multiple sequence alignment[END_REF], ML trees were obtained using a General Time-Reversible (GTR) model of evolution with gamma-distributed rate heterogeneity (G) and a proportion of invariable sites (I) (GTR + G + I), with 200 bootstrap replications and partial deletion (85%), using MEGA 7 [START_REF] Kumar | MEGA7: Molecular Evolutionary Genetics Analysis Version 7.0 for Bigger Datasets[END_REF]. Relative abundances were plotted on these trees using iTOL [START_REF] Letunic | Interactive tree of life (iTOL) v3: an online tool for the display and annotation of phylogenetic and other trees[END_REF].

RESULTS AND DISCUSSION

OTU definition for Microsporidia

Metabarcoding analyses usually produce thousands of sequences, which are then clustered into OTUs to produce 'species equivalents' [START_REF] Eren | A framework for analysis of metagenomic sequencing data[END_REF][START_REF] Alberdi | Scrutinizing key steps for reliable metabarcoding of environmental samples[END_REF]. For eukaryotic SSU sequences, a similarity threshold of 95% [START_REF] Caron | Defining DNA-Based Operational Taxonomic Units for Microbial-Eukaryote Ecology[END_REF][START_REF] Debroas | Evidence for an active rare biosphere within freshwater protists community[END_REF] or alternatively 97% [START_REF] Cleary | Unexpected prevalence of parasite 18S rDNA sequences in winter among Antarctic marine protists[END_REF][START_REF] Bista | Annual timeseries analysis of aqueous eDNA reveals ecologically relevant dynamics of lake ecosystem biodiversity[END_REF]) is usually employed. However, an incorrect clustering threshold could cause oversplitting or undersplitting of sequences into species equivalents, resulting in either inflated or underestimated species counts [START_REF] Alberdi | Scrutinizing key steps for reliable metabarcoding of environmental samples[END_REF]. We therefore addressed this issue by applying different clustering thresholds on the full-length and V1-V3 sequences of the Microsporidia species we selected. Such analysis indicated that an important proportion of microsporidian species has highly similar SSU sequences (about 13% of species clustered together with a 99% clustering threshold) or even identical sequences (0.4% (full-length) and 2.7% (V1-V3) of species clustered together with a 100% clustering threshold) (Supplementary Information 2). None of the clustering thresholds may in this way generate OTUs that could be identified with high accuracy at the species level, thus highlighting the limits of this gene marker for species discrimination. Besides, thresholds for species delineation may vary according to the lineages, since some species belonging to the same genera have sometimes less than 90% identity (Zheng et al 2016), while some share more than of 99.5% identity in other genera [START_REF] Tokarev | A formal redefinition of the genera Nosema and Vairimorpha (Microsporidia: Nosematidae) and reassignment of species based on molecular phylogenetics[END_REF]. For the later genera, combination of different molecular markers is then sometimes required to improve phylogenetic resolution (Bacela-Spychalska et al 2018; [START_REF] Tokarev | A formal redefinition of the genera Nosema and Vairimorpha (Microsporidia: Nosematidae) and reassignment of species based on molecular phylogenetics[END_REF]. As 100% of genera could nevertheless be distinguished using a 99% threshold on V1-V3 sequences, we chose to keep this threshold and only consider affiliations to genera as valid in further analyses.

Available databases are not reliable for microsporidian classification

OTUs assignment to phylogenetic clades requires the use of a reliable reference database, in which species are classified into several taxonomic ranks. As assignment lies on the comparison between OTUs and reference sequences, the efficiency of OTU classification depends on the degree to which taxonomic classification of reference sequences fits their evolutionary relationships. However, is it largely recognized that taxonomic designation of Microsporidia, based primarily on morphological characters, is inconsistent with phylogenetic analyses [START_REF] Vossbrinck | Molecular phylogeny of the Microsporidia: ecological, ultrastructural and taxonomic considerations[END_REF]Stentiford et al., 2013b;[START_REF] Tokarev | A formal redefinition of the genera Nosema and Vairimorpha (Microsporidia: Nosematidae) and reassignment of species based on molecular phylogenetics[END_REF]. We thus primarily checked whether taxonomic information contained in various public databases was consistent with microsporidian phylogenetic data, focusing on the species for which the full length of the SSU gene was available (265 sequences). We found that taxonomic data found in GenBank [START_REF] Federhen | The NCBI Taxonomy database[END_REF] are largely in disagreement with the phylogenetic clusters identified in the literature [START_REF] Vossbrinck | Molecular phylogeny of the Microsporidia: ecological, ultrastructural and taxonomic considerations[END_REF][START_REF] Smith | The ecology and evolution of microsporidian parasites[END_REF][START_REF] Vossbrinck | Molecular phylogeny and evolution of mosquito parasitic Microsporidia (Microsporidia: Amblyosporidae)[END_REF][START_REF] Corsaro | Filling gaps in the microsporidian tree: rDNA phylogeny of Chytridiopsis typographi (Microsporidia: Chytridiopsida)[END_REF]; Supplementary Information 1). In SILVA [START_REF] Quast | The SILVA ribosomal RNA gene database project: improved data processing and web-based tools[END_REF], many of these microsporidian species are lacking (12%, Supplementary Information 1) or when present, often assigned incorrectly to phyla which are not Microsporidia (21%, Supplementary Information 1). Besides, SILVA, as well as PR² (a database dedicated to protists; [START_REF] Guillou | The Protist Ribosomal Reference database (PR²): a catalog of unicellular eukaryote small sub-unit rRNA sequences with curated taxonomy[END_REF], do not include any taxonomic rank between phyla and genera (see Supplementary Information 1).

PHYMYCO-DB [START_REF] Mahé | PHYMYCO-DB: A Curated Database for Analyses of Fungal Diversity and Evolution[END_REF], a database dedicated to Fungi, does not contain microsporidian sequences. The existing databases being inappropriate, another one based on the phylogeny rather than on morphological characters, was thus required. Although the microsporidian taxonomy should probably be revised (as discussed in Vossbrinck and[START_REF] Vossbrinck | Molecular phylogeny of the Microsporidia: ecological, ultrastructural and taxonomic considerations[END_REF]Stentiford et al., 2013b), our objective was simply here to provide a practical classification of microsporidian species which could be used to analyse metabarcoding data.

Extensive phylogeny of Microsporidia

We built a phylogenetic tree which is twice more exhaustive than the previously published reference trees [START_REF] Vossbrinck | Molecular phylogeny of the Microsporidia: ecological, ultrastructural and taxonomic considerations[END_REF][START_REF] Vossbrinck | Molecular phylogeny and evolution of mosquito parasitic Microsporidia (Microsporidia: Amblyosporidae)[END_REF]. NJ and ML phylogenetic trees obtained on the full-length sequences placed canonical Microsporidia in a monophyletic group, with Metchnikovellids and Chytridiopsidae at the basal position of the microsporidian phylogenetic tree (Figure 1). A total of 10 clades were identified (9 for canonical Microsporidia numbered with Latin names, and one for Metchnikovellidae), all being supported by high bootstrap values (>85%).

Former reference trees identified five major clades in canonical Microsporidia [START_REF] Vossbrinck | Molecular phylogeny of the Microsporidia: ecological, ultrastructural and taxonomic considerations[END_REF][START_REF] Vossbrinck | Molecular phylogeny and evolution of mosquito parasitic Microsporidia (Microsporidia: Amblyosporidae)[END_REF]. It should however be noted that clade numbering between these two reference publications differs, as "clade 5" in [START_REF] Vossbrinck | Molecular phylogeny and evolution of mosquito parasitic Microsporidia (Microsporidia: Amblyosporidae)[END_REF] [START_REF] Vossbrinck | Molecular phylogeny of the Microsporidia: ecological, ultrastructural and taxonomic considerations[END_REF][START_REF] Vossbrinck | Molecular phylogeny and evolution of mosquito parasitic Microsporidia (Microsporidia: Amblyosporidae)[END_REF][START_REF] Trzebny | A new method of metabarcoding Microsporidia and their hosts reveals high levels of microsporidian infections in mosquitoes (Culicidae)[END_REF]. However, we found only a weak overlap for the clade II, which species composition already differed markedly between the former reference phylogenies (Supplementary Information 4). As we included several new species in the current phylogenies, it appears that former clade II could be split into three different clades (here II, VIII and IX). Eventually, we introduce two distinct novel clades (VI and VII), containing species which were either unclassified [START_REF] Vossbrinck | Molecular phylogeny of the Microsporidia: ecological, ultrastructural and taxonomic considerations[END_REF][START_REF] Smith | The ecology and evolution of microsporidian parasites[END_REF] or included in clade 5 [START_REF] Vossbrinck | Molecular phylogeny and evolution of mosquito parasitic Microsporidia (Microsporidia: Amblyosporidae)[END_REF] probably due to long branch attraction.

Within these clades, we furthermore identified 22 'groups' of species, also supported by high bootstrap values (≥95%) and previous phylogenetic studies (Figure 1). We named these groups according to the names given in these studies when available, or according to a representative genus (i.e. genera namelike group) for the others. Again, a strong discrepancy between these phylogenetic data and taxonomy provided in NCBI was found (Supplementary Information 5). However, our data were mostly in agreement with the microsporidian taxonomy, since most of the phylogenetic 'groups' we identified corresponded to families reviewed in Wijayawardene et al., 2018 (Supplementary Information 6). Other groups included species classified in various families (such as the 'Aquatic group' [START_REF] Vossbrinck | Molecular phylogeny and evolution of mosquito parasitic Microsporidia (Microsporidia: Amblyosporidae)[END_REF], which contains representatives of the Caudosporidae, Gurleyidae and Agglomeratidae families [START_REF] Wijayawardene | Notes for genera: basal clades of Fungi (including Aphelidiomycota, Basidiobolomycota, Blastocladiomycota, Calcarisporiellomycota, Caulochytriomycota, Chytridiomycota, Entomophthoromycota, Glomeromycota, Kickxellomycota, Monoblepharomycota, Mortierellomycota, Mucoromycota, Neocallimastigomycota, Olpidiomycota, Rozellomycota and Zoopagomycota)[END_REF]), or species not assigned yet to any family (such as representatives of the Orthosomella-like or Nematocida-like groups). All these groups were recovered in the phylogeny based on the V1-V3 region, highlighting the relevance of this region (Supplementary Information 7).

Some trends in terms of habitats and to a lesser extent of hosts are characterizing these clades or groups (Figure 1). Clades I, VI, VII, VIII, as well as Neopereziidae (clade V) mostly contain species infecting freshwater hosts, such as Crustacea, aquatic Diptera or Bryozoa, while clades II, IV1, IX and Tubulinosematidoidea (clade V) contain mostly species infecting terrestrial hosts. Clade III contain aquatic (freshwater and marine) species. One group within clade IV2 however, contains microsporidian species infecting very different hosts (ciliates, insects, crustacean, human) found in diverse habitats (terrestrial, freshwater, brackish). We named this group 'Arlequin group' to reflect this variability.

Clades and groups highlighted by this phylogenetic analysis and literature review were thereafter used to classify OTUs in our metabarcoding analysis. Sequences included in the phylogeny, as well as other full-length SSU sequences of the same species (total of 528 sequences), were included in a database used to study eukaryotic communities in freshwater environments [START_REF] Debroas | Overview of freshwater microbial eukaryotes diversity: a first analysis of publicly available metabarcoding data[END_REF].

Application of the proposed methodology to study the lacustrine microsporidian diversity

So far, only two studies attempted to characterize the diversity of Microsporidia in environmental samples, focusing mostly on terrestrial and marine habitats [START_REF] Ardila-Garcia | Microsporidian diversity in soil, sand and compost of the pacific Northwest[END_REF][START_REF] Williams | Group-specific environmental sequencing reveals high levels of ecological heterogeneity across the microsporidian radiation[END_REF]. Both studies highlighted new diversity across all areas of the microsporidian tree. However, due to the low number of sequences obtained in these studies (25 and 745 respectively, with a maximum of 20 OTUs per sample), comparisons of microsporidian communities between habitats were either not done or could only uncover a general variability between samples. In this study, we use our expertise database combined with the use of specific primers to characterize the diversity of Microsporidia in two lake ecosystems (planktonic samples) as well as its spatio-temporal variations through metabarcoding.

Diversity

After quality filtering and clustering at 99%, 1 135 287 reads were obtained, and 91% of them were assigned to Microsporidia. After removal of the two samples which contained < 20 000 reads (Pavin, April, fraction 50-150 µm and Pavin, June, fraction 10-50µm) and normalisation, 323 904 reads were assigned to the microsporidian classification we defined above. This analysis revealed a considerable diversity, since 1531 OTUs were obtained in total. However, the Microsporidia community was dominated by few highly abundant OTUs, since 95 "abundant" OTUs (>1% in at least one sample) covered more than 75% of the total reads. These results highlight, together with those of [START_REF] Ardila-Garcia | Microsporidian diversity in soil, sand and compost of the pacific Northwest[END_REF][START_REF] Williams | Group-specific environmental sequencing reveals high levels of ecological heterogeneity across the microsporidian radiation[END_REF], the necessity to sample more environments in order to reveal the extent of the environmental microsporidian diversity.

All OTUs obtained in this study belonged to the canonical Microsporidia, and were distributed within 7 clades (over the 10 clades identified for Microsporidia). The microsporidian community was clearly dominated by clades IV2 and I, which represent respectively 76% and 20% of the OTUs (Figure 2). The remaining OTUs were classified into clades IV1 (<2%), V (<1.5%), II (<0,2%), VIII (<0.2%) and IX (<0.2%).

Nine groups (over the 21 belonging to the canonical Microsporidia) were represented. Three of them accounted for more than 74% of the OTUs: Aquatic outgroup (clade I, 19%), Arlequin and Orthosomella-like groups (clade IV2, 45% and 10% respectively) (Figure 2). 47% of OTUs were assigned to a genus, of which Crispospora, Pseudoberwaldia and Conglomerata were the most represented (35%, 5% and 4%, respectively). Phylogenetic analyses confirmed these affiliations (Figure 3 and Supplementary Information 8), which supports the accuracy of the developed methodology. As expected, most of these genera are known to infect hosts having at least one aquatic stage across their full development (crustaceans: daphnids or cyclops, insects: chironomids, Figure 2).

Besides the identified genera, many OTUs were not assigned to a genus (53%) nor sometimes even to a group (23%). OTUs not assigned to any genus within the Arlequin group (Figure 3), Aquatic outgroup (Supplementary Information 8) and Orthosomella-like group (Supplementary Information 9) were often related to undescribed Microsporidia found in freshwater hosts, such as amphipods, cladocerans, copepods (crustacean), caddisflies, mayflies and mosquitoes (insects). OTUs not assigned to any group mostly belonged to clade IV2 (21%), and few to clades IV1 and V (1% each) (Figure 2). Phylogenetic analysis of these OTUs confirmed that they do not belong to any of the groups we formerly identified (clade IV2: Figure 4; clades IV1 and V: Supplementary Information 10-11). Their closest relatives were often Microsporidia amplified from environmental samples [START_REF] Ardila-Garcia | Microsporidian diversity in soil, sand and compost of the pacific Northwest[END_REF][START_REF] Williams | Group-specific environmental sequencing reveals high levels of ecological heterogeneity across the microsporidian radiation[END_REF] or from amphipods sampled in the lake Baikal [START_REF] Ironside | Accumulation and exchange of parasites during adaptive radiation in an ancient lake[END_REF]. In clade IV2, they notably form 5 new groups (Van9, BFAS6, Van10, BPAR5 and Van8 groups, Figure 4), each having less than 85% similarity with groups identified in this clade through phylogenetic analysis of known species (Supplementary Information 12). These results suggest that for freshwater habitats, a large proportion of genera and/or families remains to be described. As our analysis focused on small sized fractions (< 150 µm), this novel diversity may correspond at least partly to not yet described microsporidian species infecting small hosts. Some studies report infections of ciliates and rotifers by Microsporidia in freshwater environments [START_REF] Fokin | Euplotespora binucleata n. gen., n. sp. (Protozoa: Microsporidia), a parasite infecting the hypotrichous ciliate Euplotes woodruffi, with observations on microsporidian infections in ciliophora[END_REF][START_REF] Wolska | Parasites of zooplankton and periphyton assemblages in the littoral zone of lakes in Drawa National Park, Poland[END_REF]Stentiford et al., 2013a;[START_REF] Yakovleva | The first case of microsporidiosis in Paramecium[END_REF], but only two sequences from ciliates, and none from rotifers are currently available in the databases. Further sampling efforts aiming to identify hostmicrosporidian interactions in these small size organisms will be necessary in order to confirm this hypothesis.

Spatio-temporal variations

α-diversity of microsporidian community varied among the different samples (Supplementary Information 13). When focusing on community richness (Richness observed, Chao1 index, number of OTUs), the two lakes seem to present an opposite pattern, as richness increased from April to June in lake Aydat while it decreased in Lake Pavin. Similar patterns were observed for Shannon and Simpson indexes (Supplementary Information 13 B).

While no trend for the different size fractions was detected, variations in the relative abundances of some taxa were observed between sampling sites, and to a lesser extent across time (in lake Aydat notably), as suggested by the non-metric multidimensional scaling (Figure 5A). For example, while

OTUs identified as Crispospora (IV2-Arlequin group), Conglomerata and Pseudoberwaldia (I-Aquatic outgroup) sp. are very abundant in lake Aydat (up to 91, 17 and 5%, respectively, shown in black, lightblue and deep blue in Figure 5B), their relative abundances were in comparison low in lake Pavin (maximum 11, 0.2 and 0.3%, respectively, except in one sample where they reached 19, 2 and 3%).

OTUs belonging to clade IV1 from unknown group(s) had similar pattern, reaching up to 13% in Aydat, and being nearly absent in Pavin (<0.4%) (striped green on Figure 5B). These trends were also observed at the OTU level, some being clearly more abundant in one lake than in the other (Figure 3 and Supplementary Information 8 and 10). At the opposite, lake Pavin was clearly dominated by Microsporidia belonging to unknown group(s) in the clade IV2 or to unknown genera in the Aquatic outgroup (clade I) (striped orange and dotted blue respectively on Figure 5B). Phylogenetic analysis of these unassigned OTUs within clade IV2 revealed that they belong to 3 de novo groups (Van 9, BFAS6 and Van8) (Figure 4). In particular, a cluster of closely related OTUs in group Van8 was far more abundant in lake Pavin (some OTU reaching up to 54% of the total reads) than in lake Aydat (<1.3%) (Figure 4). Similarly, phylogenetic analysis of abundant OTUs belonging to the Aquatic outgroup revealed that several OTUs clustering with Octosporea muscadomesticae and unknown Microsporidia found in freshwater hosts were far more abundant in lake Pavin (up to 21%) than in lake Aydat (<0.5%) (Supplementary Information 8).

Relative abundances of some taxa can also significantly vary across the sampling period (Figure 5B).

For example, abundances of Pseudoberwaldia sp. (Aquatic outgroup, clade I) increased over time in lake Aydat (deep blue in Figure 5B, Supplementary Information 8), as did Microsporidia in the Van8 group (clade IV2) in lake Pavin, which clearly dominates in June (>84% of the reads) (Figure 4).

Besides, Microsporidia belonging to the Orthosomella group were much more abundant in April compared to May and June (dotted green in Figure 5B, Supplementary Information 9). Although these general trends can be observed at the group or genus level, striking differences can also be observed at the OTU level: in OTUs assigned to the genus Crispospora for example, OTU3989 and OTU7498 were more abundant in April (up to 85 and 29%), while OTU19444 was more abundant in May (up to 45%), and OTU24464 in June (up to 21%), suggesting these OTUs might belong to different species (Figure 3 and Supplementary Information 14A). As genetic distances between these OTUs can be lower than 2% (Supplementary Information 14B), such data support the relevance of a clustering threshold fixed at 99%.

These spatio-temporal variations might be related to the abundance of the hosts present in these lakes at the time of sampling, as Microsporidia have generally a narrow host specificity. Indeed, these two lakes differ in their zooplankton species compositions [START_REF] Lair | Répartition spatio-temporelle, biomasse et production des populations zooplanctoniques du lac d'Aydat en période estivale[END_REF][START_REF] Devaux | Contribution à l'étude limnologique du lac Pavin (France). II. Relations entre paramètres abiotiques -Phyto et zooplancton dans la zone 0-20 mètres[END_REF][START_REF] Lair | Effects of invertebrate predation on the seasonal succession of a zooplankton community: a two year study in Lake Aydat, France[END_REF][START_REF] Quiblier-Lloberas | Impact of grazing on phytoplankton in Lake Pavin (France): Contribution of different zooplankton groups[END_REF], which could themselves be influenced by differences in the nutrients content, altitude and mixing type recorded in these two lakes [START_REF] Lepère | Vertical structure of small eukaryotes in three lakes that differ by their trophic status: a quantitative approach[END_REF]. Considerable temporal variations in the abundances of these species have also been recorded, which could in part explain the temporal variations observed in some Microsporidia lineages. For example, increases of the Cladoceran population are repeatedly recorded in lake Aydat in spring, which could explain the increase of Agglomeratidae (Pseudoberwaldia and Conglomerata species) across our sampling period [START_REF] Lair | Effects of invertebrate predation on the seasonal succession of a zooplankton community: a two year study in Lake Aydat, France[END_REF].

Monitoring the presence and abundances of both the microsporidian and other eukaryotic species across a significant period of time could help, for example through co-occurrence network analyses, to highlight potential new host-microsporidian interactions. This could be of particular interest for the Van8 group, which clearly dominates in lake Pavin, and which hosts are totally unknown. This group, as well as the other dominant lineages (Crispospora sp. and Agglomeratidae species) might play a significant role in the regulation of hosts populations, that further work may help to determine.

Conclusion

In this work, we propose a methodology which aims at studying the microsporidian diversity through metabarcoding. We notably provide a robust database, based on an extensive phylogenetic and bibliographic analysis, which classify microsporidian species in a coherent framework. Using this database, we explored the microsporidian diversity in lacustrine samples through a metabarcoding approach, and highlighted a substantial amount of Microsporidia phylogenetically distant from any described microsporidian species, which might represent undescribed genera and families. Further studies, including larger datasets (in terms of geography and/or ecology) will help to reveal the extent of Microsporidia diversity, which is obviously far from being fully uncovered. Besides, further analyses will be necessary to determine which hosts these Microsporidia are able to infect, a necessary step to understand the role of these parasites in the ecosystem functioning. Within these clades, 1 to 5 groups are identified (highlighted with branches and species names colours), for a total of 22 groups. References to other studies that support these groups are provided, and listed in Bibliography for Supplementary Information. Colour strips indicate hosts (different colours can be given for a single species when different hosts are known) and habitat (different colours can be given for a single species when different habitats are known). Tree annotated using the interactive Tree of Life (iTOL) server [START_REF] Letunic | Interactive tree of life (iTOL) v3: an online tool for the display and annotation of phylogenetic and other trees[END_REF]. The alignment used to build the phylogenetic tree is provided in Supplementary Information 3.

Figure 1 :

 1 Figure 1: Microsporidian phylogenetic tree based on full length SSU rDNA of 265 microsporidian species and six outgroup taxa (1127 positions). Values above branches indicate the ML and Neighbour

Figure 2 :

 2 Figure 2: Krona chart representing the affiliation of the 1531 OTUs found in the analysis. Inner

Figure 3 :

 3 Figure 3: Maximum Likelihood phylogeny of abundant OTUs (>1% of reads in at least one sample) affiliated to the Arlequin group (clade IV2) with other sequences belonging to this group.

Figure 4 :

 4 Figure 4: Maximum Likelihood phylogeny of abundant OTUs (>1% of reads in at last one sample) not assigned to any group, within clade IV2, with other sequences belonging to this clade. Numbers

Figure 5 :

 5 Figure 5: A. Non-metric multidimensional scaling (NMDS) ordination of the microsporidian OTU relative abundances across samples based on Bray-Curtis distances (stress: 0.09). Ellipses in the

ACKNOWLEDGMENTS

The authors are very grateful to Laetitia Juillard for her priceless tenacity to devise the right PCR conditions during her Master traineeship, and for her help with sampling and DNA extractions. We also gratefully acknowledge constructive comments by two anonymous referees. Authors acknowledge the financial support from the "Fédération des Recherches en Environnement" through the CPER "Environment" founded by the Region Auvergne, the French government and the FEDER (European Community).

CONFLICT OF INTEREST

The authors declare they have no competing interests


Arnaud Badel
email: arnaud.badel@unige.ch

Louis Antognazza

Mathieu Therasse

Markus Abplanalp

C Schacherer

Michel Decroux

Hybrid model of quench propagation in coated conductors for fault current limiters

1

Introduction ReBCO Coated Conductors (the so-called 2 nd generation HTS) are promising for resistive Fault Current Limiters (FCL) due to their high normal-state resistivity and current densities, which should lead to very compact FCL designs. Moreover, their price is decreasing and CC-based FCL devices will be significantly cheaper than bulk BSCCO FCL when the industrial stage will be reached. One of the major issues for an efficient use of CC in FCL, along with the difficulty to obtain homogeneous characteristics over long lengths, is the low normal zone propagation velocities (NZPV) that they display. These velocities are 2 orders of magnitude lower than what is observed with ReBCO deposited on sapphire [START_REF] Heinrich | Quenching of superconductivity and propagation of the resulting normal phase in YBCO films[END_REF], in the range of 10 -2 to 10 -1 m/s for current densities (J) slightly above the critical value Jc. In current limiting applications, low NZPV leads to slow increase of the total resistance. Finally, this slower establishment of the current limiting regime leads to an increased temperature excursion of the hot spot, which may be damaging for the conductor.

Slow NZPV in CC come from their poor thermal characteristics, with high thermal capacity and low thermal conductivity. However if this is established qualitatively, an accurate propagation model is needed to determine quantitatively the respective influences of the different thermal parameters on the velocity: geometry of the layers, materials and interfaces characteristics. This information could be used to design FCL architectures making an optimal use of the conductors and / or optimize the conductor for this specific application.

Modelling CC conductor presents two challenges. From the electrical point of view, the strong nonlinearity of the relation between electric field and current density in the superconducting dissipative state must be introduced. From the thermal point of view, the low thermal conductivity and the high thermal capacity of the layers make it impossible to consider the cross section as thermally uniform, especially considering the very high power density dissipated in the conducting layers. The very high aspect ratio of the layers also makes the modelling difficult using numerical methods such as FEM. One dimensional transition propagation numerical models have been proposed, solving the heat diffusion equation along the CC length. They allow fast calculations but make assumptions about the thermal propagation in the CC cross section [START_REF] Daibo | Evaluation of the normal-zone propagation characteristics of REBCO coated conductors with laminated Cu tape[END_REF][START_REF] Pelegrin | Numerical and experimental analysis of normal zone propagation on 2G HTS wires[END_REF][START_REF] Celentano | Hot spot stimulated transition in YBCO coated conductors: experiments and simulations[END_REF]. It is difficult in this case to discriminate the influence of the various layer parameters. On the contrary, 2D and 3D multi-physic models using FEM or FVM implement the electric and thermal equations at the microscopic scale [START_REF] Chan | Three-dimensional micrometer-scale modeling of quenching in high-aspect-ratio YBa2Cu3O7-delta coated conductor tapes -part I: model development and validation[END_REF]. However, the convergence issues coming from the power law implementation and the rapidly growing size of the problems make it difficult to extend such methods to long lengths of conductor and all the more to apply it to large scale application simulations.

2

Novel Hybrid model In order to assert the influence of the layers characteristics on the ReBCO CC thermal behaviour in fault current limiter applications, we developed a new hybrid model for 2D and simple 3D simulations of CC in self-field. It combines analytical calculations to render the highly non-linear behaviour of the superconducting material, with numerical Finite Element calculations for a detailed evaluation of the thermal propagation in the CC layers. The possibility to simulate 3D geometries is presently limited to simple yet common cases, such as the study of transition propagation between adjacent conductors. For now, only the influence of the temperature on the current density repartition is considered, which is a reasonable assumption if the current is close-to or above the critical current. The principle of the model is to use a thermal-only FEM model of the conductor, and calculate analytically the local heat dissipation, similarly to [START_REF] Roy | Quench propagation in coated conductors for fault current limiters[END_REF]. The particularity of our approach is that the evaluation of the heat dissipation is independent of the FEM problem, and may thus be conducted on a different scale from the FEM meshing. We demonstrate that this scale can be significantly higher while maintaining the accuracy, which allows faster and larger simulations.

Thermal model of the conductor

The thermal modelling of the conductor is made using COMSOL ® software. A generic CC architecture is considered. It includes a substrate, an insulating buffer layer, the ReBCO layer and a silver shunt. The thermal conductivities and heat capacities of the layers and their temperature dependence are considered. From the thermal point of view, the silver shunt and the ReBCO layer can be merged into one (later called ReBCO/Ag layer) due to their thinness and the very good quality of the interface between. They will however be considered separately from the electrical point of view to evaluate the current sharing (cf. § 2.3). We suspect that the thermal properties of the interfaces between ReBCO and Buffer, and between Buffer and Substrate, may have a strong impact on the thermal behaviour of the CC. In order to account for these interfaces, an equivalent interface is added in the model between the substrate and the buffer layer. Finally, the thermal model makes the assumption that there is no thermal exchange between the modelled conductor and the outside (cooling bath, thermal link, etc.). This assumption is supported by preliminary results obtained taking the exchange with the boiling nitrogen into consideration, without significant effect.

Equivalent interface evaluation. We reported recently a method [START_REF] Antognazza | Measurement of the DyBCO / substrate thermal conductance in coated conductors[END_REF] to measure an overall conductance regrouping the conductance of the ReBCO / Buffer interface and the Buffer / Substrate interface, but also the equivalent thermal conductance of the Buffer layer itself. As the buffer layer is considered separately in the proposed model, its influence must be deducted from the values obtained through this measurement technique in order to obtain the thermal conductance of the modelled equivalent interface.

ReBCO/Ag layer. In order to account for the localization of the heat dissipation along the conductor length, the ReBCO/Ag layer is split in blocks, each of them considered as an independent heat source. A schematic view of the resulting model geometry is presented Figure 1, for a 2D longitudinal cross section of a conductor. At each step of solving, the average temperatures on these blocks are calculated. The amount of heat produced in each block is then evaluated analytically in a function outside of the FEM problem, using the average temperature obtained. For 3D problems, the ReBCO/Ag layer is also split in along its width. For processing reasons, the number of blocks in the width of the tape must be constant along the conductor.

Evaluation of the heat dissipation

In the model, the substrate is electrically insulated from the ReBCO and Silver layers due to the buffer layer. In consequence, evaluating the heat dissipation in each block requires only the current densities in the silver and ReBCO layers and the resistivity of these layers. For ReBCO, two cases are considered: Above Tc it is in the normal state where it has an actual resistivity. In the superconducting state above Jc but below Tc, a power law is used to evaluate the flux flow resistivity varying with the temperature and the current density. Below Jc(T) the superconductor is in the non-dissipative state and the resistivity is zero.

Power law. A modified formulation [START_REF] Decroux | Properties of YBCO films at high current densities: fault current limiter implications[END_REF] is used to calculate the flux flow resistivitysuper(J,T) for the superconductor in the dissipative state. With this formulation, the calculated resistivity truly reaches zero at Jc(T).

    0 ,1 n super c J JT JT         (1)
where Tc is the critical temperature, Jc(T) is the critical current at the temperature T. n and 0

(respectively the order of the power law and the equivalent resistivity at 2.Jc) are the two parameters that must be obtain through the characterization of the conductor.

The Jc(T) curve can be derived from the measured Jc at one specific temperature, typically 77 K, through [START_REF] Daibo | Evaluation of the normal-zone propagation characteristics of REBCO coated conductors with laminated Cu tape[END_REF]. This equation comes from the Ginzburg-Landau formulation [START_REF] Poole | Superconductivity[END_REF] and was used successfully to predict the critical current of HTS wires in a wide range of temperature [START_REF] Curtz | Patterning of ultrathin YBCO nanowires using a new focused-ion-beam process[END_REF].

  1 24 2 0 11 cc cc TT J T J TT                                  ( 2
)
where Jc0 is the critical current expected at 0 K in self field.

The interest of this power law formulation is the reduced number of parameters that must be obtained through characterization (Jc(77), n, ) and the invariance of these parameters in the studied range of temperature and current densities (see below Figure 3).

Current sharing. The current densities in the two conducting layers of each block are calculated within a Matlab ® function, in which the power law formulation is implemented. This function contains the characteristic parameters Jc(77), n and The temperature on the blocks is an input of this function, obtained from the thermal FEM calculation. As the current densities considered are high (close to or abvoe Jc) we consider that the current repartition in the conductor width is only influenced by the temperature.

For 2D problem considering the conductor along its length (Figure 1), the total current density passing through one block is constant. An iterative method is being used to evaluate the current sharing between the two conducting layers. It starts with 100 % of the current flowing through the ReBCO layer. It then evaluates the ReBCO resistivity and calculates the resulting current sharing. This new current distribution is used in the next step to evaluate again the resistivity of the ReBCO layer, and so on until the current distribution variation between two steps is lower than a given criterion. In order to insure the convergence of this iterative method, the current distribution at a given step is not obtained directly from the resistivity obtained in the previous step but from a pondered average obtained from the resistivities calculated in the two previous steps (Figure 2). For 3D problems, we developed a modified evaluation function that calculate the current sharing between superconductor and resistive layers similarly as in 2D, but within another iterative loop that calculate the distribution of current in the width of the conductor. This function takes the average temperatures of all the blocks forming the ReBCO/Ag layer at a given cross-section, and evaluates the equivalent resistance of each of them considering a homogenous current density. The actual current densities in the different blocks are then obtained by iteration in a similar way as explained above for the 2D problem. This calculation of the current repartition along the width of the conductor only considers the influence of the temperature profile in the cross-section. Other influences of the 3D geometry, conductor path (turns, constrictions) or inhomogeneous magnetic flux density are not considered, which explains why the code is presently limited to simple 3D cases.

3

Results of Theva CC simulation This model (2D version) has been used to simulate Theva Coated Conductor [11]. It consists of a hastelloy substrate on which an MgO insulating layer is deposited using ISD (inclined substrate deposition). On this MgO layer, DyBCO is grown by co-evaporation technique. A thin protective layer of silver is added on top.

Modelling data

The simulated 2D geometry represent a 20 mm long CC. As the problem is extended by symmetry with regards to its side on the left, the total simulated length is actually 40 mm. The simulation depth of the 2D problem, which represents the width of the modelled CC, is 3 mm. In the model, the characteristics of each block forming the DyBCO/Ag layer can be defined independently, including the current density flowing through it. In order to maintain the continuity, this current density is the same in every block, except for the initial dissipating zone. This initial dissipating zone is a 400 m-long length of conductor, at the left extremity of the modelled ReBCO/Ag layer, where the current density value is 1.5 times larger than elsewhere (Figure 3). This increased current density reproduces the effect of a constriction in which the conductor width would be reduced to 2 mm. It initiates the transition of the CC. As the problem is symmetrized, this initial dissipating zone is actually 800 m long and it is at the middle of the total 40 mm simulated length. Electrical characterization was conducted on this conductor to obtain the power law parameters required by the model, following the methodology presented in [START_REF] Therasse | Electrical characteristics of DyBCO coated conductors at high current densities for fault current limiter application[END_REF]. A critical current density of 9 10 9 A/m 2 has been measured, with a power law order n of 1.6 and an initial resistivity 0 of 0.32 nΩ.m (Figure 4). The thermal and electrical characteristics of the modelled layers are interpolated from bibliography data in the range of 77 to 600 K. The extreme values are regrouped in Table 1. The equivalent interface conductance was derived from measurement using the new characterization method reported in [START_REF] Antognazza | Measurement of the DyBCO / substrate thermal conductance in coated conductors[END_REF]. The obtained value was 300 W/K/cm 2 (Table 1). Even if this value is higher than expected, it is considerably lower than the equivalent conductance of the buffer layer (11.10 3 W/K/cm 2 at 77 K). It will therefore have a stronger influence on the thermal behaviour. In the simulations the current density was considered to stay constant in time. The simulations were stopped when the initial dissipating zone (hot spot) reached 600 K. This temperature is considered as an acceptable limit for the tape operation [START_REF] Schwarz | Thermodynamic behaviour of a coated conductor for currents above Ic[END_REF].

Simulated propagation velocity and heat dissipation

The length where the temperature exceeds 90 K is obtained from the FEM model for each time step, and used to derive the NZPV. Figure 5 shows the evolution of the propagation velocities with time, for a current density of 1.1 Jc and 1.78 Jc. For 1.1 Jc the obtained propagation velocity is constant, as it is assumed classically. However, for higher current densities such as 1.78 Jc, the propagation velocity tends to increase. This is probably due to the pre-heating of the whole length of the sample, which at such high current densities is in the dissipating state since the beginning of the simulation. Apart from the temperature gradient, the localization of the heat dissipation can also be obtained from the model. Figure 6 displays the heat dissipation gradient in the first 3 mm of the simulated sample under 1.1 Jc when the hot spot reaches 600 K. The heat dissipation is expressed in power per surface of tape, taking into account the thickness of the dissipating layer (DyBCO/Ag). The maximum value reaches 1 kW/cm 2 .

Influence of the modelling block size

In the proposed model, the heat dissipation in the DyBCO/Ag layer is evaluated on blocks having an adjustable length, larger than the FEM mesh. The dependence of the simulated propagation velocity with these block lengths is presented Figure 7, for a fixed FEM meshing of 10 m along the modelled length (20 mm). These simulations were conducted for a current density of 1.1 Jc. It can be observed that the calculated value converge rapidly when the block length is reduced. For blocks shorter than 200 m, the error is below 5 %, which validates the idea of evaluating the heat dissipation on a larger scale than the FEM meshing of the thermal problem. It is also observed that the value does not tend toward the limit monotonously but rather oscillates around it. This tends to prove that no systematic error is introduced by this macroscopic evaluation.

4

Agreement with experimental results Propagation velocities on Theva CC were measured using small samples obtained by photo-lithography process. The transition is initiated by a constriction at the middle of the sample. This reduction of the width of the superconducting line leads to an increase of the current density by a factor 1.5, as it was considered in the simulations. The progression of the transition is deduced from voltage measurement along the sample length (Figure 8). The results of the 4 experiments that were conducted show a significant dispersion, with measured NZPV ranging from 12.5 to 13.5 cm/s for a current density of 1.1 Jc (see the boundaries Figure 7). The mean value is around 13 cm/s, very close from the simulated values which converge to 12.8 cm/s. It demonstrates that it is possible to obtain an accurate propagation velocity using only standard characterization of the conductor and no free adjustment parameter. The NZPV were measured for increasing current densities during the 4 experiments. The results are compared Figure 9 with the simulated values. The data range of the measurements is limited because the temperature increase for current densities in the range of 1.6 Jc is so high that the samples would be destroyed before measurement can be made. However, there is a fairly good agreement between simulation and experiments in the measured range. The results are also in good agreement with older measurement conducted on similar conductors, reported in [START_REF] Antognazza | Comparison between the behavior of HTS thin film grown on sapphire and coated conductors for fault current limiter applications[END_REF]. Finally, we applied the model to YBCO deposited on sapphire, with electrical characteristics extracted from [START_REF] Antognazza | Comparison between the behavior of HTS thin film grown on sapphire and coated conductors for fault current limiter applications[END_REF]. The simulated velocities range from 20 to 60 m/s under current densities varying from 1.5 to 2 Jc, with Jc being around 3 10 10 A/m 2 . These results are in good agreement with other available experiment data [START_REF] Antognazza | Comparison between the behavior of HTS thin film grown on sapphire and coated conductors for fault current limiter applications[END_REF][START_REF] Antognazza | Simulation of the behavior of superconducting YBCO lines at high current densities[END_REF]. The good match between simulation results and experiments for both low and high thermal conductivity substrates proves that the proposed model is flexible enough to accurately render the behaviour of conductors with a wide range of characteristics.

5

Parametric CC simulations Considering the good agreement between the existing Coated Conductors behaviour and the simulation results, this model can be used to extrapolate with a certain degree of confidence the velocities that would be obtained with other coated conductor architectures. Moreover, simulations require short processing times: it takes less than 5 minutes to determine the behaviour (NZPV, temperature rise) of a given CC architecture in given conditions. The model was used to conduct parametric studies on two parameters that we believed to be critical regarding the thermal behaviour of the CC: Firstly the equivalent thermal conductance of the interfaces between the layers, secondly the thickness of the substrate itself. In order to have simulation results that can be compared from the application point of view, the electrical characteristics of the conductor remain must identical throughout the parametric studies. The properties and thicknesses of the DyBCO and Silver layers remain therefore unchanged.

Influence of substratebuffer interface

As stated above in § 3.1, we recently reported measurements for the equivalent thermal conductance of the layer interfaces [START_REF] Antognazza | Measurement of the DyBCO / substrate thermal conductance in coated conductors[END_REF], at 300 W/K/cm². In order to determine the influence of this conductance value on the propagation velocities, we carried out parametric calculations, for values ranging from 1 to 10 000 W/K/cm². The results are presented Figure 10, for a current density of 1.1 Jc. There is only slight variation of the NZPV when the interface conductance is above 100 W/K/cm 2 . It proves that for the existing samples the interfaces could be considered perfect without introducing significant errors. A drastic increase of the propagation velocity can be observed when the equivalent conductance of the interfaces becomes lower than 40 W/K/cm², with velocities up to 7 m/s when the DyBCO layer is almost thermally decoupled from the other layers. However, the variation of the interface thermal conductance has also an effect on the temperature rise of the hot spot. This temperature rise will be faster if the interface conductance decrease, due to the weak thermal capacity of the superconducting line. There is therefore a trade-off between quench propagation length and survivability of the CC: For low interface conductance the NZPV is higher and propagate the transition on longer length but the survival time is shorter due to the faster hot spot temperature rise. For high interface conductance the temperature rise of the hot spot is slower and the survival time is longer, but the NZPV is lower which leads to smaller transited length. In order to determine if there is an optimal between these two extremes, Figure 11 shows the temperature profiles along the CC for different interface conductances. These temperature profiles are not taken at the same time, but for each of them at the moment where the hot spot reaches 600 K. An optimal zone can indeed be observed between 4 and 20 W/K/cm², where the transited length is longer. Further simulations will be necessary to determine whether this optimal zone exists in every configuration and experiments will be needed to determine if such optimal behaviour can indeed be obtained practically.

Influence of substrate thickness

As it was stated above, existing samples shows interface thermal conductances high enough to consider the YBCO layer perfectly coupled with the substrate. Figure 12 shows the influence of the substrate thickness on the propagation velocity, for a current density of 1.1 Jc. There is a continuous increase of the propagation speed for thinner substrate, as it was already observed by other groups [START_REF] Roy | Quench propagation in coated conductors for fault current limiters[END_REF]. At the same time the temperature profile tends to become stiffer, as shown Figure 13. In conclusion, reducing the thickness of the hastelloy substrate is not favourable for the survivability of the CC in FCL applications, as it leads to a fast rise of the hot spot temperature and an increased non-uniformity of the heat dissipation.

Effect of the substrate width

The 2D simulations makes the assumption that the superconducting layer has the same width than the substrate width. It is not the case in the experiment where the superconducting line has a width of 1 mm on a substrate that is 10 mm large. The agreement between the simulations and the experiments can therefore only be explained if the heat diffusion along the substrate width is small compared to the heat diffusion induced by the quench propagation along its length. In order to verify this assumption, simulations using the 3D version of the model were conducted for geometry closer to the one actually used for the measurements, consisting of a superconducting line (20 mm long, 1 mm wide) on a 20 mm per 10 mm substrate (Figure 14). The simulation was conducted with input data similar to the one used in 2D simulations, the transition being initiated in the middle of the central line. Figure 15 present the simulated temperature distribution (z axis) on the surface of the sample (x,y plane) at the moment where the hotspot reaches 600 K, with the simulated geometry superimposed. There is significant heat diffusion along the width of the sample, with the temperature reaching 90 K at up to 1 mm of the line on each side. This transverse diffusion of the heat softens slightly the temperature profile along the superconducting line. However, the NZPV is not affected by this different temperature distribution. It has a value of 12.7 cm/s for 1.1 Jc, almost equal to the 12.8 cm/s obtained in 2D.

6

First consideration for FCL design optimization CC FCL takes usually the form of low inductance coils, in which the quench propagates only along the length of the conductor. It is commonly accepted that the worst operating condition for a FCL is an impedance fault leading to fault currents not so high above the critical current [START_REF] Tixador | Impact of conductor inhomogeneity on FCL transient performance[END_REF], in which case the conductor quenches only locally, which leads to little to no limitation of the current rise, and highly inhomogeneous heat dissipation. In order to limit the temperature rise in such condition, FCL designs usually include external shunt or extra normal conducting layers on the conductor in order to reduce the resistance per unit length. The maximal electric field usually accepted is around 0.5 V/cm, instead of up to 10 V/cm if the CC was used without additional shunt and up to 40 V/cm for FCL using YBCO deposited on sapphire wafer [START_REF] Antognazza | Test of YBCO thin films based fault current limiters with a newly designed meander[END_REF]. This low equivalent resistivity per unit length increases the amount of conductor necessary to reach the desired limiting factor. A solution to accept higher electric field with a given CC architecture could be the design of FCL geometries increasing the transition propagation velocity and therefore the homogeneity of the heat dissipation. We already proposed meander geometries with thermal links between the legs in order to initiate the transition on adjacent legs [START_REF] Antognazza | Heat propagation velocities in coated conductors for fault current limiter applications[END_REF]. The results obtained from the 3D simulations of a superconducting line demonstrate that heat propagates significantly along the direction transverse to the superconducting line even if this phenomenon does not have an influence on the propagation velocity along the line. In consequence initiating the transition of adjacent lines, such as the legs of a meander, is possible if the distance between them is sufficiently small.

Conclusion

We developed a fast-processing model that is able to reproduce quantitatively the behaviour of CC conductor at the transition and under self-field, using standard conductor characterization and without the need of free adjustment parameters. It was used to determine the influence of both substrate thickness and interfaces conductance value on the transition propagation velocity and temperature uniformity. The results show that in general, a substantial increase of the transition propagation velocity cannot be achieved without at the same time lowering the heat dissipation uniformity, which is not favourable for FCL applications. It illustrates the need of developing new substrates with higher thermal characteristics in order to make compact and reliable CC-based FCL. The proposed model is also able to simulate simple 3D geometries. The preliminary results show that there is non-negligible heat diffusion perpendicularly to the superconducting line, which may be useful for FCL made of CC plates, using meander geometries. Future works will focus on adapting the model to different kind of conductors, such as architectures with electrical connections between substrate and shunt, multi-layer buffers, and so on. It will also aim at using the model to optimize meander designs taking advantage of this transverse heat propagation to obtain substantially higher equivalent propagation velocities.

Figures Direction of current flow

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 12345 Figure 1: Schematic view of Model geometry in 2D

Figure 6 :

 6 Figure 6: Heat dissipation per surface unit in the first 10 mm of the simulated length (Current density 1.1 Jc)

Figure 7 :

 7 Figure 7: Dependence of the simulated NZPV with the length of the blocks used for the evaluation of the heat dissipation

Figure 8 :

 8 Figure 8: Theva sample etching for velocity measurement. The transition starts in the constriction at the middle of the sample, and its propagation is measured electrically.

Figure 9 :

 9 Figure 9: Comparison of simulated NZPV and experimental measurement for increasing current densities.

Figure 10 :

 10 Figure 10: Simulated NZPV evolution with the equivalent interface thermal conductance

Figure 12 :

 12 Figure 12: NZPV evolution with the substrate thickness

Figure 13 :

 13 Figure 13: Temperature profile of the Ag/DyBCO layer when the hot spot reaches 600 K, for increasing values of substrate thickness

Figure 14 :Figure 15 :

 1415 Figure 14: Simulated 3D geometry consisting of a 20 mm long 1 mm wide superconducting line on a 20 x 10 mm substrate + buffer.

Present address: European Organization for Nuclear Research / BE -RF, CERN CH-1211, Geneva 23, Switzerland

Acknowledgment

This work was supported by the Swiss National Science Foundation through the National Center of Competence in Research MaNEP (Materials with Novel Electronic Properties).