Pierre-Guy Sauriau
email: pierre-guy.sauriau@univ-lr.fr

Manon Dartois

Vanessa Becquet

Fabien Aubert

Valérie Huet

Martine Bréret

Amélia Viricel

Eric Pante

Multiple genetic marker analysis challenges the introduction history of Ulva australis (Ulvales, Chlorophyta) on French coasts

Keywords: atpI-H, green macroalgae, introduced species, ITS2, rbcL, trnA-N, tufA

The green seaweeds Ulva australis and U. pertusa were described from southern Australia and Japan, respectively. They are conspecific and U. australis, the currently accepted taxon, is native to temperate marine waters in northeastern Asia, and known to be introduced overseas into Australasia, the Americas and Europe. Although the genetics of U. australis have been investigated elsewhere, along French coasts the origins and history of the introduction of this species need to be clarified. We used mitochondrial, plastid and nuclear markers to differentiate introduced populations of U. australis along the French Atlantic coasts. The plastid tufA gene used as a barcoding marker revealed a well-defined species with a higher haplotype diversity in native vs. introduced areas. The ITS2 region (nuclear) and rbcL (plastid) were used to compare French specimens with the lectotype of U. australis. Putative geographic origins of the genetically determined U. australis were examined using genetic markers with better resolution, the plastid atpI-H combined with the mitochondrial trnA-N.

Origin(s) and introduction history of French specimens were inferred from the comparison between their haplotypes and those previously described in native and non-native temperate areas worldwide. Our results indicate that the presence of U. australis along the French Atlantic and Mediterranean coasts is the result of multiple introductions and independent pathways, and suggest that historical oyster transfers from Japan and British Columbia can only partially explain the observed patterns in genetic markers. Alternative hypotheses for the timing and pathways of introductions are proposed in the light of the historic background of maritime transport networks and trade between north-eastern Asia and Europe.

Introduction

The green macroalgal genus Ulva Linnaeus, which includes species previously named under the genus Enteromorpha Link [START_REF] Hayden | Linnaeus was right all along: Ulva and Enteromorpha are not distinct genera[END_REF][START_REF] Shimada | Molecular phylogenetic analyses of the Japanese Ulva and Enteromorpha (Ulvales, Ulvophyceae), with special reference to the free-floating Ulva[END_REF], is distributed worldwide in different marine, brackish and freshwater ecosystems. According to AlgaeBase, amongst the 407 historical names presently reported as Ulva sp. in the database, only 86 have been flagged as taxonomically accepted species [START_REF] Guiry | World-wide electronic publication[END_REF]. This spectacular number of synonyms reflects, to some extent, the notorious problems in distinguishing Ulva species on the basis of their habitat, morphology, anatomy, cytology and physiology [START_REF] Bliding | A critical survey of European taxa in Ulvales, Part II. Ulva, Ulvaria, Monostroma, Kornmannia[END_REF][START_REF] Hoeksema | The taxonomy of Ulva (Chlorophyceae) from the coastal region of Roscoff (Brittany, France)[END_REF][START_REF] Phillips | Field, anatomical and development studies on southern Australian species of Ulva (Ulvaceae, Chlorophyta)[END_REF][START_REF] Fort | Extensive variations in diurnal growth patterns and metabolism among Ulva spp. strains[END_REF]2020a).

Difficulties in the delineation of Ulva species are linked to their high degree of phenotypic plasticity, but have also been enhanced by human-mediated biological invasions in coastal waters. Ulva species are common biofoulers on ship hulls [START_REF] Mineur | Hull fouling on commercial ships as a vector of macroalgal introduction[END_REF], mariculture structures and products [START_REF] Verlaque | Morphology and reproduction of Asiatic Ulva pertusa (Ulvales, Chlorophyta) in Thau Lagoon (France, Mediterranean Sea)[END_REF][START_REF] Boudouresque | Biological Invasion: the Thau lagoon, a Japanese biological island in the Mediterranean Sea. In Global change: mankind marine environmental interactions[END_REF], and hitchhikers in ballast waters [START_REF] Flagella | Identification of cryptic Ulva species (Chlorophyta, Ulvales) transported by ballast water[END_REF]. International trans-oceanic shipping and marine aquaculture are recognized as major vectors of introductions of non-indigenous species (NIS) in coastal waters [START_REF] Carlton | Patterns of transoceanic marine biological invasions in the Pacific Ocean[END_REF][START_REF] Katsanevakis | Invading European Seas: assessing pathways of introduction of marine aliens[END_REF]. Distinguishing between native species and NIS, which often remained overlooked [START_REF] Haydar | Predicting invasion patterns in coastal ecosystems: relationship between vector strength and vector tempo[END_REF], is still a major challenge in the context of marine Ulva species [START_REF] Baamonde López | Is the cryptic alien seaweed Ulva pertusa (Ulvales, Chlorophyta) widely distributed along European Atlantic coasts?[END_REF][START_REF] Hofmann | Cryptic diversity of Ulva (Ulvales, Chlorophyta) in the Great Bay Estuarine System (Atlantic USA): introduced and indigenous distromatic species[END_REF][START_REF] Kraft | Investigations into southern Australian Ulva (Ulvophyceae, Chlorophyta) taxonomy and molecular phylogeny indicate both cosmopolitanism and endemic cryptic species[END_REF][START_REF] Wolf | Ulva (Chlorophyta, Ulvales) biodiversity in the North Adriatic Sea (Mediterranean, Italy): cryptic species and new introductions[END_REF][START_REF] Steinhagen | Cryptic, alien and lost species: molecular diversity of Ulva sensu lato along the German coasts of the North and Baltic Seas[END_REF].

Molecular methods have been employed for a few decades to overcome these difficulties and new light has been shed on Ulva spp. taxonomy, to identify and delineate species [START_REF] Coat | Ulva armoricana (Ulvales, Chlorophyta) from the coasts of Brittany (France). II. Nuclear rDNA ITS sequence analysis[END_REF][START_REF] Baamonde López | Is the cryptic alien seaweed Ulva pertusa (Ulvales, Chlorophyta) widely distributed along European Atlantic coasts?[END_REF][START_REF] Flagella | Identification of cryptic Ulva species (Chlorophyta, Ulvales) transported by ballast water[END_REF], critically reexamine inventory records [START_REF] Wolf | Ulva (Chlorophyta, Ulvales) biodiversity in the North Adriatic Sea (Mediterranean, Italy): cryptic species and new introductions[END_REF][START_REF] Steinhagen | Cryptic, alien and lost species: molecular diversity of Ulva sensu lato along the German coasts of the North and Baltic Seas[END_REF], infer phylogenetic relationships among taxa [START_REF] Hayden | Linnaeus was right all along: Ulva and Enteromorpha are not distinct genera[END_REF][START_REF] Shimada | Molecular phylogenetic analyses of the Japanese Ulva and Enteromorpha (Ulvales, Ulvophyceae), with special reference to the free-floating Ulva[END_REF], elucidate introduction history [START_REF] Couceiro | Evidence for multiple introductions of the Pacific green alga Ulva australis Areschoug (Ulvales, Chlorophyta) to the Iberian Peninsula[END_REF] and characterize type materials to clarify species taxonomy and biogeography (Hanyuda & Kawai, 2018;[START_REF] Hughey | Genetic analysis of the Linnaean Ulva lactuca (Ulvales, Chlorophyta) holotype and related type specimens reveals name misapplications, unexpected origins, and new synonymies[END_REF]. Both the nuclear ribosomal internal transcribed spacer DNA (ITS nDNA) and the chloroplast-encoded rbcL gene have been tested alone [START_REF] Coat | Ulva armoricana (Ulvales, Chlorophyta) from the coasts of Brittany (France). II. Nuclear rDNA ITS sequence analysis[END_REF][START_REF] Flagella | Identification of cryptic Ulva species (Chlorophyta, Ulvales) transported by ballast water[END_REF] or in tandem (e.g. [START_REF] Shimada | Molecular phylogenetic analyses of the Japanese Ulva and Enteromorpha (Ulvales, Ulvophyceae), with special reference to the free-floating Ulva[END_REF][START_REF] Couceiro | Evidence for multiple introductions of the Pacific green alga Ulva australis Areschoug (Ulvales, Chlorophyta) to the Iberian Peninsula[END_REF]. However, compared with these two molecular markers, the plastid elongation factor tufA was recommended by [START_REF] Saunders | An evaluation of rbcL, tufA, UPA, LSU and ITS as DNA barcode markers for the marine green macroalgae[END_REF] as a more suitable barcode marker for most marine green macroalgae including the genus Ulva. This was further confirmed by [START_REF] Kirkendale | A molecular survey of Ulva (Chlorophyta) in temperate Australia reveals enhanced levels of cosmopolitanism[END_REF], [START_REF] Lee | Taxonomy of Ulva causing blooms from Jeju Island, Korea with new species, U. pseudo-ohnoi sp. nov. (Ulvales, Chlorophyta)[END_REF] and [START_REF] Fort | Foliose Ulva species show considerable inter-specific genetic diversity, low intraspecific genetic variation, and the rare occurrence of inter-specific hybrids in the wild[END_REF] for Australian, Korean and west European specimens, respectively, as they provided estimates of intra-and inter-specific divergence in Ulva species. Our current sequencing survey of hundreds of Ulva specimens along the French Atlantic coast found that many tufA sequences fully matched GenBank sequences of U. australis Areschoug and U. pertusa Kjellman, a junior synonym (see details in [START_REF] Couceiro | Evidence for multiple introductions of the Pacific green alga Ulva australis Areschoug (Ulvales, Chlorophyta) to the Iberian Peninsula[END_REF], from Korea in the native area (Kang et al., 2019), and disjunct areas where the species has been introduced e.g. Australasia [START_REF] Heesch | Ulva, Umbraulva and Gemina: genetic survey of New Zealand taxa reveals diversity and introduced species[END_REF][START_REF] Kirkendale | A molecular survey of Ulva (Chlorophyta) in temperate Australia reveals enhanced levels of cosmopolitanism[END_REF], the Italian North Adriatic [START_REF] Wolf | Ulva (Chlorophyta, Ulvales) biodiversity in the North Adriatic Sea (Mediterranean, Italy): cryptic species and new introductions[END_REF] and German North Sea [START_REF] Steinhagen | Cryptic, alien and lost species: molecular diversity of Ulva sensu lato along the German coasts of the North and Baltic Seas[END_REF]. U. australis, recognized to be of north-eastern Asian origin [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF]Hanyuda & Kawai, 2018) was thought to have been introduced into the Mediterranean Sea and along North European coasts through multiple introduction events and different vectors including maritime transport and mariculture [START_REF] Baamonde López | Is the cryptic alien seaweed Ulva pertusa (Ulvales, Chlorophyta) widely distributed along European Atlantic coasts?[END_REF][START_REF] Couceiro | Evidence for multiple introductions of the Pacific green alga Ulva australis Areschoug (Ulvales, Chlorophyta) to the Iberian Peninsula[END_REF][START_REF] Verlaque | Biological invasion: long term monitoring of the macroalgal flora of a major European harbor complex[END_REF].

To compare our specimens unequivocally to the type material of U. australis, recently characterized with molecular markers (Hanyuda & Kawai, 2018), and reanalyse earlier records of the species in Brittany [START_REF] Coat | Ulva armoricana (Ulvales, Chlorophyta) from the coasts of Brittany (France). II. Nuclear rDNA ITS sequence analysis[END_REF], we conducted molecular analyses based on ITS2 and rbcL sequences. We also used both plastid and mitochondrial non-coding sequences, atpI-H and trnA-N, as suggested by [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF]. This latter combination of markers has allowed significant differences in genetic diversity to be resolved among native (Japan and Korea) and non-native populations of U. australis [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF]2018) and contributed to clarifying the introduction history of U. australis in Australia in terms of pathways and timing (Hanyuda & Kawai, 2018). This approach is used here to provide biogeographic information about the origin of our U. australis specimens identified based on ITS2, rbcL and tufA and to re-evaluate commonly accepted hypotheses concerning the vectors and timing of introductions of U. australis along the French coasts in view of the history of maritime transport between Europe and Northeast Asia.

Materials and methods

Sample collection

Our sampling strategy was based on collecting large sample sizes (> 100 specimens) from three sites separated by over 460 km. Green algae with large and flat rigid thalli, pale to dark green in colour and with a stiff base, were sampled from rocky shores at three sites during low spring tides from 22 January to 21 February 2019. The three sites are in Brittany at Roscoff (48.7327'N, 3.9868'W, Parmentier dock, Estacade, Ile Verte) and Concarneau (47.8601'N,3.9148'W,pointe du Cabellou) and in Vendée at La 1.4189'W,La Grière). This last site is a sand-influenced rocky shore and sampling was restricted to midshore pools. At each site, c. 120 living specimens (blade up to 7 cm 2 ) growing attached to rocks were collected by hand and placed individually in small plastic bags. Samples were separated by at least several metres. In the lab, each specimen was rinsed with filtered seawater to remove epiphytes and debris. Following individual photography (Olympus Stylus TG-830), specimens were preserved at -80°C in individual plastic bags numbered with letters (sampling sites) and serial numbers.

DNA extraction, amplification and sequencing

Frozen tissue from each thallus was ground to a powder in liquid nitrogen. Whole genomic DNA was extracted from 0.3 mg of powder using the NucleoSpin Tissue Kit (Macherey-Nagel). The manufacturer's standard protocol for tissues was followed, except for the following steps: (1) an overnight tissue digestion in proteinase K and (2) DNA was eluted in two steps, each consisting of a 3 min incubation with 25 μl of dH 2 O pre-heated at 70°C, for a final volume of 50 μl. DNA quality and quantity were assessed using a Nanodrop ND-2000 spectrophotometer (Thermo Scientific), a Qubit 1.0 (Thermo Scientific) fluorometer (dsDNA HS Assay Kit), and 1x agarose gel electrophoresis. Primers for tufA were designed based on [START_REF] Saunders | An evaluation of rbcL, tufA, UPA, LSU and ITS as DNA barcode markers for the marine green macroalgae[END_REF] but modified to reduce the number of degenerate bases using the complete Ulva chloroplast genomes available on Genbank on 16 March 2019 (Supplementary table S1). The nuclear internal transcribed spacer 2 (ITS2) together with the plastid ribulose 1, 5-bisphosphate carboxylase large chain (rbcL) gene were used to compare our sequences with the type material of U. australis housed in the Swedish Museum of Natural History (http://herbarium.nrm.se/specimens/A2025). The lectotype was characterized by Hanyuda & Kawai (2018) based on partial sequences of ITS2 (LC331301, 242 bp) and rbcL (LC331300, 197 bp). Following ITS2 and rbcL validations, further analyses of genetic variability used the intergenic regions of chloroplast atpI and atpH (atpI-H) and mitochondrial trnA and trnN (trnA-N) genes amplified with primers from [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF]. Primer sequences and PCR amplification cycles are presented in Supplementary table S1 and fig. S1. PCRs were carried out with a Sensoquest labcycler using the TaKara ExTaq reaction kit (Takara Bio).

PCR amplicons were checked on a 1x agarose gel electrophoresis prior to purification and Sanger sequencing in both forward and reverse directions by Eurofins Genomics (Ebersberg-Germany: www.eurofinsgenomics.eu/).

Data analysis

For each DNA sequence, chromatograms were cleaned manually with Geneious Prime 2019 (www.geneious.com); primer sequences were trimmed and sequences were checked for the presence of ambiguities and stop codons. Forward and reverse chromatograms were then assembled and consensus sequences were submitted to Genbank (Accession Numbers in Supplementary table S2). Newly generated consensus sequences were compared with sequences available on Genbank and aligned using Muscle 3.8.425 [START_REF] Edgar | MUSCLE: multiple sequence alignment with high accuracy and high throughput[END_REF]. Our last search on GenBank (17 October 2020) returned 181 tufA sequences identified as U. australis (174 sequences) or U. pertusa (7 sequences) including the two recent releases of complete chloroplast genomes of U. pertusa LC507117 [START_REF] Mitsuhashi | Construction of genomic marker sets based on the chloroplast genome of a green alga, Ulva pertusa (syn. Ulva australis), leads to simple detection of Ulva species[END_REF] and MN853875 (Han, direct submission 9 December 2019). Fifty-five tufA sequences not yet available on Genbank were also utilized including our sequences (MT078952 from Roscoff and MT078953 from Concarneau), and 10 sequences manually retrieved from the doctoral dissertation of Melton (2017). Our final data set contained 219 sequences since 2 and 15 sequences were excluded due to inconsistencies and mismatches in sequence length, respectively (see details in Supplementary table S3). DnaSP v6 [START_REF] Rozas | DnaSP v6: DNA Sequence Polymorphism analysis of large datasets[END_REF] was used to identify haplotypes from the 659 bp-long tufA alignment performed on our data set. Uncorrected p distances were calculated from the tufA alignment matrix using PAUP* v.4.0 [START_REF] Swofford | PAUP*. Phylogenetic Analysis Using Parsimony (* and Other Methods)[END_REF] and are presented as percentages. A median-joining network [START_REF] Bandelt | Median-joining networks for inferring intraspecific phylogenies[END_REF] was constructed to visualize relationships among tufA haplotypes using PopART with default settings (http://popart.otago.ac.nz). Two tufA sequences of U. fenestrata were used as an outgroup as there is consistent support for a close taxon relationship of tufA between U. australis and U. fenestrata [START_REF] Kirkendale | A molecular survey of Ulva (Chlorophyta) in temperate Australia reveals enhanced levels of cosmopolitanism[END_REF][START_REF] Hughey | Genetic analysis of the Linnaean Ulva lactuca (Ulvales, Chlorophyta) holotype and related type specimens reveals name misapplications, unexpected origins, and new synonymies[END_REF][START_REF] Kang | Species composition, diversity, and distribution of the genus Ulva along the coast of Jeju Island, Korea based on molecular phylogenetic analysis[END_REF]: HQ610325 from [START_REF] Saunders | An evaluation of rbcL, tufA, UPA, LSU and ITS as DNA barcode markers for the marine green macroalgae[END_REF] and MK456404 as the holotype of U. fenestrata [START_REF] Hughey | Genetic analysis of the Linnaean Ulva lactuca (Ulvales, Chlorophyta) holotype and related type specimens reveals name misapplications, unexpected origins, and new synonymies[END_REF]. Sequence similarity between the type material and our specimens for ITS2 and rbcL was calculated using BLASTn [START_REF] Zhang | A greedy algorithm for aligning DNA sequences[END_REF] with default parameters (http://blast.ncbi.nlm.nih.gov/Blast.cgi). BLASTn was also used for [START_REF] Coat | Ulva armoricana (Ulvales, Chlorophyta) from the coasts of Brittany (France). II. Nuclear rDNA ITS sequence analysis[END_REF]'s molecular data for testing the two ITS sequences U.rot.4 (MT078968) and U.rot.5 (MT078969) erroneously labelled as U. rotundata. These two sequences with corrected labels were submitted to Genbank on behalf of B. de Reviers (Supplementary table S2).

Correspondence between our specimens and the type material of U. australis using ITS2 and rbcL allowed the atpI-H and trnA-N intergenic sequences of our specimens to be aligned with, respectively, the 15 (haplotypes cH1 to cH15) and 27 (haplotypes mH1 to mH27) sequences defined by [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF] as descriptive of the genetic diversity of native U. australis populations. All sequences were concatenated and analysed following [START_REF] Hudson | A statistical test for detecting geographic subdivision[END_REF] using DnaSP v6 [START_REF] Rozas | DnaSP v6: DNA Sequence Polymorphism analysis of large datasets[END_REF] in order to detect haplotypes identical to any of the 50 combined haplotypes H1-H48 and H49-H50 described by [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF] and Hanyuda et al. (2018), respectively. The metadata associated with Ulva sequences deposited in GenBank, including geographic location, were retrieved from flatfiles. Geographic information, when absent, was manually compiled from published references (Supplementary table S3). Maps were edited with ArcGIS ArcMap 10.6 (ESRI).

Results

Sequencing success across markers

A total of 339 Ulva samples were sequenced for tufA. Forty-one samples matched the U. australis (or U. pertusa) tufA sequences, and 298 represented other Ulva species (Table 1). Amplification and/or sequencing of ITS2 and rbcL were less successful. Of the 41 samples putatively assigned to U. australis, only 32 and 28 were successfully sequenced at ITS2 and rbcL, respectively. However, sequencing of the atpH-I and trnA-N intergenic regions was successful for all of the same 41 samples (Table 1). This last dataset was used to identify which combined haplotypes among those previously described from Japan were also present in France.

tufA sequences

The 41 specimens identified as U. australis were unevenly distributed among localities: 31 samples were from Roscoff and 10 from Concarneau; no U. australis specimens were detected from La Tranche in Vendée (Table 1). All 41 tufA sequences (659 bp) are identical to 106 Genbank sequences, and differ by 1, 2 and 3 substitutions from 105, 6 and 1 sequences, respectively. Uncorrected p distances calculated between our sequences (from Roscoff and Concarneau) and U. australis sequences produced by others ranged from 0-0.46%.

The haplotype network revealed two well-separated groups, the U. fenestrata outgroup, and the other incorporating all U. australis haplotypes. Thirty-eight substitutions separated the U. fenestrata outgroup from its closest relative U. australis. Within the alignments for U. australis, there were 8 variable sites (Supplementary fig. S2) but only up to 3 substitutions per sequence (Fig. 1). The network of U. australis is star-like and included two very common haplogroups here named tuH0 and tuH1, which are associated with six rare haplotypes (Fig. 1). Our tufA sequences from Brittany matched tuH0, the commonest (163 samples), recorded from Northern Asia (27%), Australasia (18%), North and South America (9%) and Western Europe (46%) including the Mediterranean Sea (Figs 1,2). TufA sequences differing from tuH0 by one transition constitute the second haplogroup tuH1 (103 samples), from Northern Asia (91%), British Columbia (BC), California (5%) and Western Europe (4%). All other tufA sequences (7 samples) with at least 2 mutations from tuH0 were from north-eastern Asia (Fig. 2) within the Yellow Sea from Donggang, China [START_REF] Du | DNA barcoding assessment of green macroalgae in coastal zone around Qingdao, China[END_REF] to Jeju Island, Korea [START_REF] Kang | Species composition, diversity, and distribution of the genus Ulva along the coast of Jeju Island, Korea based on molecular phylogenetic analysis[END_REF][START_REF] Lee | Taxonomy of Ulva causing blooms from Jeju Island, Korea with new species, U. pseudo-ohnoi sp. nov. (Ulvales, Chlorophyta)[END_REF]. For simplicity, these tufA sequences were labelled tuH2 and tuH3, as they had 2 and 3 substitutions, respectively. Substitutions in tuH2a and tuH2b haplotypes were at the 5' end of the tufA gene and tuH2c, tuH2d, tuH2e and tuH3 haplotypes substitutions were at the 3' end (see Supplementary fig. S2). The two commonest haplogroups and five of the six rare haplotypes were all recorded in Asia (Fig. 1), from Jeju Island (Fig. 2, Yellow Sea map).

Visual inspection of 14 aligned sequences shorter than 659 bp (excluded from our data set) and two sequences from BC positioned in 349-1122 bp of the complete tufA sequence (see GenBank numbers in Supplementary table S3) concurred with the assumptions that (1) both tuH0 and tuH1 haplogroups are present within the Bohai Sea, (2) tuH0 is recorded in Chile and (3) tuH1 occurred along both sides of the North Atlantic (Germany and Virginia, USA). However, one sequence (MH538644) collected in Schleswig-Holstein, Germany, near Cuxhaven at the mouth of the Elbe River [START_REF] Steinhagen | Cryptic, alien and lost species: molecular diversity of Ulva sensu lato along the German coasts of the North and Baltic Seas[END_REF] and two identical sequences (HQ610379 and HQ610380) from BC [START_REF] Saunders | An evaluation of rbcL, tufA, UPA, LSU and ITS as DNA barcode markers for the marine green macroalgae[END_REF] differed from tuH0 by one and two transitions, respectively. They would represent two new additional haplotypes not shown here (Fig. 1) as the nucleotide substitutions are out of the 659 bp range of our analysis.

ITS2 and rbcL regions

Thirty-one of the 35 ITS2 sequences, 25 from Roscoff (MT078941) and 6 from Concarneau (MT078940), were identical to the ITS2 sequence from the lectotype of U. australis (Supplementary table S4). Two other sequences from Roscoff and one from Concarneau were poorer quality (50-92% of untrimmed based with phred score > 40). Two were < 184 bp (134 and 164 bp, respectively) and one Roscoff sequence had three ambiguities (S instead of C or G). Another sequence, from Concarneau, had one substitution (T instead of G at position 84).

These variants could be due to the co-amplification of multiple alleles/copies of ITS2. The ITS1-5.8S-ITS2 sequences MT078968 (U.rot.4) and MT078969 (U.rot.5) originated from [START_REF] Coat | Ulva armoricana (Ulvales, Chlorophyta) from the coasts of Brittany (France). II. Nuclear rDNA ITS sequence analysis[END_REF] and incorporated ITS2 sequences identical to that of the lectotype U.

australis.

Among all the ITS2 sequences related to Ulva australis and/or U. pertusa available in GenBank (Supplementary table S4), 172 sequences from the literature (80 U. australis and 92 U. pertusa) fully matched the ITS2 of lectotype material. Only six additional ITS2 sequences revealed intra-and inter-individual variation with one (5 specimens) to three (1 specimen) substitutions or indels. AJ234321, initially labelled U. pertusa but misidentified as U. fasciata [START_REF] Hayden | Linnaeus was right all along: Ulva and Enteromorpha are not distinct genera[END_REF], differed from the lectotype of U. australis at 25 positions, and has the lowest percentage identity with the lectotype ITS2.

In contrast to ITS2, the 28 rbcL sequences from France (195 bp) displayed no variation, were identical between Roscoff (MT078958) and Concarneau (MT078959), and showed 100% similarity with the lectotype rbcL gene sequence of U. australis.

atpI-H and trnA-N marker regions

For each marker, three haplotypes were detected, resulting in four combined haplotypes, three identified as H5, H42, H44 by [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF], and a new combination H51 (Table 1).

Two combined haplotypes, H42 and H44, were recorded at Roscoff, the former with 90% relative abundance. At Concarneau, H5, H42 and H51 were recorded. Both H42 and H51 are rare (10% relative abundance; Table 1, Fig. 3). H42, shared by Roscoff and Concarneau, was previously reported from the Thau Lagoon (Mediterranean Sea) by [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF]; haplotype H5 was previously observed in the Netherlands (Fig. 3).

Discussion

Conspecificity of Ulva australis and Ulva pertusa

Since its first morphological description by Kjellman from Japan in 1897, and until the recent work of [START_REF] Couceiro | Evidence for multiple introductions of the Pacific green alga Ulva australis Areschoug (Ulvales, Chlorophyta) to the Iberian Peninsula[END_REF], the specific status of Ulva pertusa has not been seriously challenged (see discussion in [START_REF] Tanner | The taxonomy and morphological variation of distromatic ulvaceous algae (Chlorophyta) from the Northeast Pacific[END_REF]. Very common in Japan, U. pertusa is polymorphic [START_REF] Okamura | Icones of Japanese Algae[END_REF] even including 'stalked-Ulva' forms [START_REF] Hiraoka | Two different genetic strains of stalked-Ulva (Ulvales, Chlorophyta) grow on intertidal rocky shores in Ebisujima, central Japan[END_REF] and floating green tides [START_REF] Yoshida | Persistent occurrence of floating Ulva green tide in Hiroshima Bay, Japan: seasonal succession and growth patterns of Ulva pertusa and Ulva spp. (Chlorophyta, Ulvales)[END_REF]. In contrast, the status of the South Australian U. australis, which was described before U. pertusa, has long been controversial. Several nomenclatural synonyms have been proposed, including U. rigida C. Agardh [START_REF] Agardh | Till algernes systematik[END_REF] and U. lactuca Linnaeus by Womersley (1956, p. 354) following the view that U. lactuca is a cosmopolitan species with multiple varieties. Contradictory opinions on U. australis were finally clarified using ITS1 and rbcL analyses on Spanish, Japanese and Australian specimens [START_REF] Couceiro | Evidence for multiple introductions of the Pacific green alga Ulva australis Areschoug (Ulvales, Chlorophyta) to the Iberian Peninsula[END_REF]. Molecular evidence of the identity of U. australis and U. pertusa presented by [START_REF] Couceiro | Evidence for multiple introductions of the Pacific green alga Ulva australis Areschoug (Ulvales, Chlorophyta) to the Iberian Peninsula[END_REF] corroborates previous results of [START_REF] Kraft | Investigations into southern Australian Ulva (Ulvophyceae, Chlorophyta) taxonomy and molecular phylogeny indicate both cosmopolitanism and endemic cryptic species[END_REF], who distinguished a clade of U. australis and U. pertusa from a more speciose clade with U. laetevirens, U. rigida and U. scandinavica using ITS and rbcL markers. They reinstated U. australis as a valid species and suggested U. pertusa as its junior synonym (see details in the Appendix S1 of [START_REF] Kraft | Investigations into southern Australian Ulva (Ulvophyceae, Chlorophyta) taxonomy and molecular phylogeny indicate both cosmopolitanism and endemic cryptic species[END_REF]. [START_REF] Kirkendale | A molecular survey of Ulva (Chlorophyta) in temperate Australia reveals enhanced levels of cosmopolitanism[END_REF] confirmed this view using rbcL and tufA markers.

Despite known limitations associated with the use of ITS [START_REF] Saunders | An evaluation of rbcL, tufA, UPA, LSU and ITS as DNA barcode markers for the marine green macroalgae[END_REF], more recent advances in secondary structural analysis of ITS2 have advocated the use of ITS2 as a potential DNA barcode for the class Chlorophyceae [START_REF] Buchheim | Internal transcribed spacer 2 (nu ITS2 rRNA) sequence-structure phylogenetics: towards an automated reconstruction of the green algal tree of life[END_REF]. Species identification of Ulva specimens based on ITS2 [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF]) also supports the conspecificity of U. australis and U. pertusa. All available ITS2 sequences labelled as U. australis or U. pertusa in GenBank (Supplementary table S4) are identical or nearly identical to the historic lectotype material of U. australis characterized with ITS2 and rbcL by Hanyuda & Kawai (2018). The inter-specimen variability i.e. 1 to 6 substitution(s) here reported from this large data set is consistent with previous analyses made on a more restricted data set (Melton, 2017, fig. 3.11, p. 98). Our synopsis of ITS2 data suggests misidentification of two specimens previously labelled U. fasciata (AF099728) and U. lactuca (AF099729), and their molecular identification as U. australis (100% match with the lectotype ITS2 sequence). Both were collected in Victoria, Australia [START_REF] Woolcott | Ulva and Enteromorpha (Ulvales, Ulvophyceae, Chlorophyta) in eastern Australia: comparison of morphological features and analyses of nuclear rDNA sequence data[END_REF]. Until the recent study of [START_REF] Hughey | Taxonomic revisions based on genetic analysis of type specimens of Ulva conglobata, U. laetevirens, U. pertusa and U. spathulata (Ulvales, Chlorophyta)[END_REF], the type material of U. pertusa was thought to be unavailable (Tanner, 1979, p. 219). However, both ITS and rbcL sequences are available from specimens collected at Yokohama [START_REF] Shimada | Molecular phylogenetic analyses of the Japanese Ulva and Enteromorpha (Ulvales, Ulvophyceae), with special reference to the free-floating Ulva[END_REF], one of three syntype Japanese localities, together with Hakodate and Yenoshima (Kjellman, 1897): ITS2 (AB097657) and rbcL (AB097627) sequences from the specimen voucher SAP:095069 were identical to ITS2 (LC331301) and rbcL (LC331300) sequences from the lectotype of U. australis, respectively. Most recently, [START_REF] Hughey | Taxonomic revisions based on genetic analysis of type specimens of Ulva conglobata, U. laetevirens, U. pertusa and U. spathulata (Ulvales, Chlorophyta)[END_REF] retrieved the type material of U. pertusa from the Museum of Evolution (UPS) at Uppsala University, and confirmed that U. pertusa is conspecific with U. australis on the basis of rbcL gene sequences from one newly designated lectotype (MT815855) from Yenoshima and two syntype specimens (MT815856, MT815857).

A large body of literature shows the high level of genetic similarity in rbcL gene sequences between populations of U. australis and U. pertusa from Australia [START_REF] Kraft | Investigations into southern Australian Ulva (Ulvophyceae, Chlorophyta) taxonomy and molecular phylogeny indicate both cosmopolitanism and endemic cryptic species[END_REF][START_REF] Kirkendale | A molecular survey of Ulva (Chlorophyta) in temperate Australia reveals enhanced levels of cosmopolitanism[END_REF], BC-Canada [START_REF] Saunders | An evaluation of rbcL, tufA, UPA, LSU and ITS as DNA barcode markers for the marine green macroalgae[END_REF], California [START_REF] Hayden | Linnaeus was right all along: Ulva and Enteromorpha are not distinct genera[END_REF][START_REF] Hayden | A molecular systematic study of Ulva (Ulvaceae, Ulvales) from the northeast Pacific[END_REF], Chile [START_REF] Oróstica | Nuevo registro de Ulva australis (Ulvaceae, Chlorophyta) en el norte de Chile[END_REF], China [START_REF] Du | DNA barcoding assessment of green macroalgae in coastal zone around Qingdao, China[END_REF][START_REF] Kang | Re-evaluation of green tide-forming species in the Yellow Sea[END_REF], Japan [START_REF] Matsumoto | Systematics of green algae resembling Ulva conglobata, with a description of Ulva adhaerens sp. nov. (Ulvales, Ulvophyceae)[END_REF], Korea [START_REF] Kang | Re-evaluation of green tide-forming species in the Yellow Sea[END_REF], Italy [START_REF] Wolf | Ulva (Chlorophyta, Ulvales) biodiversity in the North Adriatic Sea (Mediterranean, Italy): cryptic species and new introductions[END_REF] and Spain [START_REF] Baamonde López | Is the cryptic alien seaweed Ulva pertusa (Ulvales, Chlorophyta) widely distributed along European Atlantic coasts?[END_REF][START_REF] Couceiro | Evidence for multiple introductions of the Pacific green alga Ulva australis Areschoug (Ulvales, Chlorophyta) to the Iberian Peninsula[END_REF].

Synopsis of French records

Ulva australis is now widespread within the Thau lagoon (Mediterranean Sea), particularly within oyster farming zones [START_REF] Verlaque | Morphology and reproduction of Asiatic Ulva pertusa (Ulvales, Chlorophyta) in Thau Lagoon (France, Mediterranean Sea)[END_REF], and is one of the commonest macroalgal NIS species, as U. pertusa, in the lagoon [START_REF] Boudouresque | Biological Invasion: the Thau lagoon, a Japanese biological island in the Mediterranean Sea. In Global change: mankind marine environmental interactions[END_REF]. The first record is thought to be the herbarium specimen labelled Ulva sp. from Thau in 1984 collected by Ben Maiz (see details and references in [START_REF] Verlaque | Morphology and reproduction of Asiatic Ulva pertusa (Ulvales, Chlorophyta) in Thau Lagoon (France, Mediterranean Sea)[END_REF]. Molecular identification of U. australis at Thau was recently confirmed [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF]. Along the French Atlantic coast, U. australis was first reported as U. pertusa from Arcachon Bay in [START_REF] Wolff | Non-indigenous marine and estuarine species in The Netherlands[END_REF]-2006[START_REF] Verlaque | Etude de la flore introduite dans le Bassin d'Arcachon[END_REF], and as U. australis from Port of Le Havre in 2012 [START_REF] Verlaque | Biological invasion: long term monitoring of the macroalgal flora of a major European harbor complex[END_REF]. All these records were based on morphological diagnoses [START_REF] Okamura | Icones of Japanese Algae[END_REF][START_REF] Verlaque | Morphology and reproduction of Asiatic Ulva pertusa (Ulvales, Chlorophyta) in Thau Lagoon (France, Mediterranean Sea)[END_REF], and as pointed out by [START_REF] Verlaque | Biological invasion: long term monitoring of the macroalgal flora of a major European harbor complex[END_REF], still required molecular confirmation. To the best of our knowledge, the first molecular sequences came from Brittany [START_REF] Coat | Ulva armoricana (Ulvales, Chlorophyta) from the coasts of Brittany (France). II. Nuclear rDNA ITS sequence analysis[END_REF], including two ITS1-5.8S-ITS2 sequences misidentified as U. rotundata Bliding. They are representative of at least six specimens collected from Roscoff (attached thalli from Île Verte, Embarcadère and Pointe du Bloscon) plus one specimen from Lannion Bay (floating thalli during a green tide event at Plestin-les-Grèves) in 1994-1995.

Woolcott & King (1999) clearly suspected the U. rotundata specimens to be misidentified as they wrote 'There are again strong molecular similarities between Australian isolates here' i.e. [START_REF] Coat | Ulva armoricana (Ulvales, Chlorophyta) from the coasts of Brittany (France). II. Nuclear rDNA ITS sequence analysis[END_REF]'. This taxonomic inconsistency was also highlighted by [START_REF] Malta | Free-floating Ulva in the southwest Netherlands: species or morphotypes? A morphological, molecular and ecological comparison[END_REF] following a comparison of ITS between U. rotundata from [START_REF] Coat | Ulva armoricana (Ulvales, Chlorophyta) from the coasts of Brittany (France). II. Nuclear rDNA ITS sequence analysis[END_REF] and specimens labelled U. rigida from Flinders, Australia (UrigAus: AF153494, collected by van Oppen) and Oosterschelde, The Netherlands (UrigOS: AF153490). Blastn comparisons among these sequences confirmed their similarity (Supplementary table S4). On the other hand, [START_REF] Shimada | Molecular phylogenetic analyses of the Japanese Ulva and Enteromorpha (Ulvales, Ulvophyceae), with special reference to the free-floating Ulva[END_REF] noted that U. rotundata from Brittany clustered with U. pertusa from Japan in phylogenetic ITS1-5.8S-ITS2 trees. [START_REF] Hayden | A molecular systematic study of Ulva (Ulvaceae, Ulvales) from the northeast Pacific[END_REF] highlighted a similar grouping in ITS and rbcL trees of U. rotundata from Britanny and U. pertusa specimens from Japan and California. Baamonde [START_REF] Baamonde López | Is the cryptic alien seaweed Ulva pertusa (Ulvales, Chlorophyta) widely distributed along European Atlantic coasts?[END_REF] first recognized the presence of U. pertusa in Europe based on the molecular data of [START_REF] Coat | Ulva armoricana (Ulvales, Chlorophyta) from the coasts of Brittany (France). II. Nuclear rDNA ITS sequence analysis[END_REF] following [START_REF] Shimada | Molecular phylogenetic analyses of the Japanese Ulva and Enteromorpha (Ulvales, Ulvophyceae), with special reference to the free-floating Ulva[END_REF] and [START_REF] Hayden | A molecular systematic study of Ulva (Ulvaceae, Ulvales) from the northeast Pacific[END_REF]. This conclusion was also accepted by [START_REF] Flagella | Identification of cryptic Ulva species (Chlorophyta, Ulvales) transported by ballast water[END_REF] before the final suggestion that U. pertusa is a synonym of U. australis [START_REF] Couceiro | Evidence for multiple introductions of the Pacific green alga Ulva australis Areschoug (Ulvales, Chlorophyta) to the Iberian Peninsula[END_REF]. Molecular identification of U. australis at Roscoff was also confirmed by [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF].

U. australis 'and European Ulva species in

Our observations of U. australis at Roscoff from the same sampling sites as those selected by [START_REF] Coat | Ulva armoricana (Ulvales, Chlorophyta) from the coasts of Brittany (France). II. Nuclear rDNA ITS sequence analysis[END_REF] suggest that the species has been present at Roscoff over the past 25 years. Concarneau (Pointe du Cabellou) is a new locality for U. australis in Southern Brittany (Fig. 3, Table 1). Observations here supplement previous reports of U. australis in Brittany (Fort et al., 2020a[START_REF] Fort | Foliose Ulva species show considerable inter-specific genetic diversity, low intraspecific genetic variation, and the rare occurrence of inter-specific hybrids in the wild[END_REF] during spring and summer 2018 green tides at BegMeil (near Concarneau), Bay of Brest (Moulin Blanc), Roscoff and Lannion Bay (Plestin les Grèves) (Fig. 3), demonstrating the relative dominance of U. australis in green tides in Brittany. These observations are scattered both temporally and spatially (Fig. 3) suggesting that the distribution of U. australis along the French Channel, Bay of Biscay (from Brest to Biarritz) and Mediterranean coasts might be more continuous than presently reported.

Current distribution of Ulva australis in Europe

Ulva australis has been reported from Germany, the Netherlands, Ireland, Spain and Italy incorporating the oldest records and corrected misidentifications as U. pertusa [START_REF] Baamonde López | Is the cryptic alien seaweed Ulva pertusa (Ulvales, Chlorophyta) widely distributed along European Atlantic coasts?[END_REF][START_REF] Couceiro | Evidence for multiple introductions of the Pacific green alga Ulva australis Areschoug (Ulvales, Chlorophyta) to the Iberian Peninsula[END_REF]. The first records in the German Wadden Sea were in Helgoland and Schleswig-Holstein in 2014 [START_REF] Steinhagen | Cryptic, alien and lost species: molecular diversity of Ulva sensu lato along the German coasts of the North and Baltic Seas[END_REF]. Records from the Netherlands came from the Oosterschelde at least from 1995 [START_REF] Stegenga | Zeewieren van de voormalige oesterputten bij Yerseke[END_REF], the Dutch Delta area (Fort et al., 2018[START_REF] Fort | Extensive variations in diurnal growth patterns and metabolism among Ulva spp. strains[END_REF](Fort et al., , 2020a)), where U. pertusa was recognized in herbarium specimens collected in the area as early as 1993 [START_REF] Wolff | Non-indigenous marine and estuarine species in The Netherlands[END_REF] and references therein), the entrance of Rotterdam port [START_REF] Gittenberger | Port of Rotterdam survey and monitoring non-native species conform HELCOM/OSPAR protocol[END_REF] and the Dutch Wadden Sea including major ports [START_REF] Gittenberger | Native and non-native species of the Dutch Wadden Sea in 2014[END_REF]. Stegenga et al. (1997, p. 42 note n° 8) mentioned the occurrence of an aberrant Ulva species with a thicker thallus than U. rigida in Zeeland, referencing a 1984 collection, but it may not be represented in the material housed at the Leiden Herbarium (Stegenga, pers. comm. 19 Feb. 2020). Ulva australis has also been reported from Galway, Ireland since 2018 (Fort et al., 2020a). In Spain, the oldest record was morphologically recognized from a herbarium specimen collected from La Coruña (Galicia) in 1990 [START_REF] Baamonde López | Is the cryptic alien seaweed Ulva pertusa (Ulvales, Chlorophyta) widely distributed along European Atlantic coasts?[END_REF]. Spanish records of live specimens came from both the Atlantic [START_REF] Fort | Foliose Ulva species show considerable inter-specific genetic diversity, low intraspecific genetic variation, and the rare occurrence of inter-specific hybrids in the wild[END_REF] and Mediterranean coasts [START_REF] Couceiro | Evidence for multiple introductions of the Pacific green alga Ulva australis Areschoug (Ulvales, Chlorophyta) to the Iberian Peninsula[END_REF]. The first Italian records were from Venice in 2010 [START_REF] Manghisi | A cryptic alien seaweed spreading in Mediterranean coastal lagoons[END_REF] and U. australis was widespread within the lagoon in 2011 [START_REF] Wolf | Ulva (Chlorophyta, Ulvales) biodiversity in the North Adriatic Sea (Mediterranean, Italy): cryptic species and new introductions[END_REF]. Ulva pertusa at Naples was mentioned by [START_REF] Wolf | Ulva (Chlorophyta, Ulvales) biodiversity in the North Adriatic Sea (Mediterranean, Italy): cryptic species and new introductions[END_REF] and [START_REF] Cormaci | Flora marina bentonica del Mediterraneo: Chlorophyta[END_REF] according to [START_REF] Flagella | Identification of cryptic Ulva species (Chlorophyta, Ulvales) transported by ballast water[END_REF].

A closer examination of the [START_REF] Flagella | Identification of cryptic Ulva species (Chlorophyta, Ulvales) transported by ballast water[END_REF] data suggests an erroneous interpretation of their ITS1 phylogenetic tree. Only U. ohnoi Hiraoka and Shimada and U. fasciata Delile, today known as U. lactuca Linnaeus [START_REF] Hughey | Genetic analysis of the Linnaean Ulva lactuca (Ulvales, Chlorophyta) holotype and related type specimens reveals name misapplications, unexpected origins, and new synonymies[END_REF], were reported from ballast waters in the Neapolitan harbour.

Worldwide genetic variability in U. australis tufA

Among the eight tufA haplotypes (659 bp) found in this study (Fig. 1), seven matched those reported from Jeju Island [START_REF] Kang | Species composition, diversity, and distribution of the genus Ulva along the coast of Jeju Island, Korea based on molecular phylogenetic analysis[END_REF][START_REF] Lee | Taxonomy of Ulva causing blooms from Jeju Island, Korea with new species, U. pseudo-ohnoi sp. nov. (Ulvales, Chlorophyta)[END_REF]. The eighth was found only at Donggang Rizhao, China, some 700 km northeast in the Yellow Sea [START_REF] Du | DNA barcoding assessment of green macroalgae in coastal zone around Qingdao, China[END_REF]. At Jeju Island, [START_REF] Lee | Taxonomy of Ulva causing blooms from Jeju Island, Korea with new species, U. pseudo-ohnoi sp. nov. (Ulvales, Chlorophyta)[END_REF] reported seven haplotypes for tufA including one haplotype with three substitutions from tuH0 not included in our analysis. All these results support a scenario of a drastic population bottleneck following the introduction of U. australis outside its native range: (1) a high level of genetic diversity for tufA within the native area [START_REF] Kang | Species composition, diversity, and distribution of the genus Ulva along the coast of Jeju Island, Korea based on molecular phylogenetic analysis[END_REF][START_REF] Lee | Taxonomy of Ulva causing blooms from Jeju Island, Korea with new species, U. pseudo-ohnoi sp. nov. (Ulvales, Chlorophyta)[END_REF], (2) the predominance of a unique haplogroup (tuH0) worldwide and (3) the occurrence of a second one (tuH1) in a few European and North American countries (Figs 1, 2). According to [START_REF] Rius | Marine invasion genetics: from spatio-temporal patterns to evolutionary outcomes[END_REF], a founder effect is not the most common scenario in marine invasions. Introduced populations exhibiting similar or even higher genetic diversity than native populations have been described, as multiple introductions from large native regions with highly genetically structured populations can lead to increased diversity at introduction sites (e.g. [START_REF] Simon-Bouhet | Multiple introductions promote range expansion of the mollusc Cyclope neritea (Nassariidae) in France: evidence from mitochondrial sequence data[END_REF]. Nonetheless, all results for tufA from Australia [START_REF] Kirkendale | A molecular survey of Ulva (Chlorophyta) in temperate Australia reveals enhanced levels of cosmopolitanism[END_REF] and New Zealand [START_REF] Lawton | Algal bioremediation of waste waters from land-based aquaculture using Ulva: selecting target species and strains[END_REF][START_REF] Lawton | Correction: Algal bioremediation of waste waters from land-based aquaculture using Ulva: Selecting target species and strains[END_REF] showed a lack of genetic diversity (only tuH0, Figs 1,2). This supports previous findings that U. australis, despite its abundance and spread along these temperate coasts, is a NIS in Australia [START_REF] Kirkendale | A molecular survey of Ulva (Chlorophyta) in temperate Australia reveals enhanced levels of cosmopolitanism[END_REF] and New Zealand [START_REF] Heesch | Ulva, Umbraulva and Gemina: genetic survey of New Zealand taxa reveals diversity and introduced species[END_REF]. A similar conclusion was reported for combined genetic markers atpI-H and trnA-N in U. australis from Australasia compared with Japan and Korea [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF](Hanyuda et al., , 2018) ) : 1 and 5 combinations in Australia and New Zealand, respectively vs. 48 in both Japan and Korea.

The first records of U. australis as U. pertusa in North America are recent and were validated by molecular analyses in BC in [START_REF] Wolff | Non-indigenous marine and estuarine species in The Netherlands[END_REF]-2006[START_REF] Saunders | An evaluation of rbcL, tufA, UPA, LSU and ITS as DNA barcode markers for the marine green macroalgae[END_REF] and California in 1999 [START_REF] Hayden | A molecular systematic study of Ulva (Ulvaceae, Ulvales) from the northeast Pacific[END_REF]. [START_REF] Aguilar-Rosas | First record of Ulva pertusa Kjellman (Ulvales, Chlorophyta) in the Pacific coast of Mexico[END_REF] Canada [START_REF] Saunders | An evaluation of rbcL, tufA, UPA, LSU and ITS as DNA barcode markers for the marine green macroalgae[END_REF] and Chile [START_REF] Melton | Molecular systematics and organellar genomics of the green macroalgal genus Ulva (Ulvophyceae, Chlorophyta) with a phylogenomic analysis of the "core chlorophyta[END_REF][START_REF] Oróstica | Nuevo registro de Ulva australis (Ulvaceae, Chlorophyta) en el norte de Chile[END_REF], but only tuH1 in California [START_REF] Saunders | Long distance kelp rafting impacts seaweed biogeography in the Northeast Pacific: the kelp conveyor hypothesis[END_REF]. However, visual checks of [START_REF] Saunders | An evaluation of rbcL, tufA, UPA, LSU and ITS as DNA barcode markers for the marine green macroalgae[END_REF] tufA sequences (774 bp) suggested the occurrence of a third haplotype exclusive to BC, Canada, and not recorded from Korea [START_REF] Kang | Species composition, diversity, and distribution of the genus Ulva along the coast of Jeju Island, Korea based on molecular phylogenetic analysis[END_REF][START_REF] Lee | Taxonomy of Ulva causing blooms from Jeju Island, Korea with new species, U. pseudo-ohnoi sp. nov. (Ulvales, Chlorophyta)[END_REF]. This may support the view of repeated introductions into BC) waters from different Asian populations. Scarcity of full-length tufA data from Japan [START_REF] Mitsuhashi | Construction of genomic marker sets based on the chloroplast genome of a green alga, Ulva pertusa (syn. Ulva australis), leads to simple detection of Ulva species[END_REF] hinders the discussion of this hypothesis for now. The status of U. australis along the North-eastern Pacific coast as a NIS or a native (Amphi-Pacific with disjunct temperate areas) was already questioned by [START_REF] Miller | A review of non-native seaweeds from California, USA and Baja California, Mexico[END_REF]. Among vectors involving transoceanic transport of NIS from the western to the eastern North Pacific ocean, the oyster trade from Japan and international shipping are major invasion vectors [START_REF] Carlton | Patterns of transoceanic marine biological invasions in the Pacific Ocean[END_REF][START_REF] Wonham | Trends in marine biological invasions at local and regional scales: the Northeast Pacific Ocean as a model system[END_REF], and are well documented in terms of biogeography and timing. Ulva australis is a common fouling alga on the oyster Magallana gigas (Thunberg) cultivated on rafts in Japan [START_REF] Miyazaki | On fouling organisms in the oyster farm[END_REF]. Successful introductions of oyster seed from Japan to BC and Puget Sound started at the beginning of the 19 th century and lasted to the 1970s [START_REF] Ruesink | Introduction of non-native oysters: ecosystem effects and restoration implications[END_REF]. Oysters originated from the main Japanese farming centres i.e. Miyagi-Sendai (East Japan), Nagasaki-Kumamoto and Hiroshima (SW Japan) [START_REF] Fujiya | Oyster farming in Japan[END_REF]. Maritime transport between Japan and the USA is well-documented in terms of historical changes in port hierarchy and geographic specializations [START_REF] Maurette | Progrès des ports de Yokohama et de Kobé[END_REF][START_REF] Ducruet | Inside the pond : an analysis of Northeast Asia's longterm maritime dynamics[END_REF]. Yokohama and Osaka-Kobe have been the main Japanese ports involved in the trade with Western and Eastern overseas countries since the mid-19 th century [START_REF] Maurette | Progrès des ports de Yokohama et de Kobé[END_REF]. However, natural dispersal of U. australis may also have contributed to its wide amphi-Pacific distribution as tsunamis are known to occur on the scale of geological times [START_REF] Dawson | Tsunami deposits in the geological record[END_REF], particularly in seismically active coasts like those of Japan [START_REF] Garrett | A systematic review of geological evidence for Holocene earthquakes and tsunamis along the Nankai-Suruga Trough, Japan[END_REF]. The 2011 Great East Japan Earthquake and Tsunami was responsible for the arrival of huge amounts of marine debris on North-eastern Pacific shores, including derelict boats fouled with U. australis. Use of combined haplotypes atpI-H and trnA-N [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF] elucidated the biogeographic origin of these U. australis stranded on the Oregon shores since their combined haplotypes matched those found in Iwate Pref.-Northern Japan (Hanyuda et al., 2018). Finally, the amphi-Pacific map of these combined haplotypes in U. australis highlighted the wide distribution of the unique haplotype H5 (Hanyuda et al., 2016, Fig. 1). This surprisingly fully matches the aquaculture history of oyster transfers from the East coast of Japan (Miyagi Pref.) to Australasia and Americas [START_REF] Ruesink | Introduction of non-native oysters: ecosystem effects and restoration implications[END_REF] and references therein for Australia, New Zealand and Americas).

Known secondary transfers of oysters within American countries, e.g. USA to Chile [START_REF] Ruesink | Introduction of non-native oysters: ecosystem effects and restoration implications[END_REF] do not contradict the view of contemporary oyster transfers as a major vector for U. australis within the Pacific region.

Challenging the introduction history of Ulva australis in France

In Europe (France and the Netherlands), two combined haplotypes H5 and H42 (Fig. 3) were previously found, but their distinct geographic distribution suggested repeated introduction events [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF]. Haplotype H5 was found in the Netherlands (fig. 1 in [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF] in U. australis collected in 2005 within the Oosterschelde in the vicinity of Yerseke, which has been the main Magallana gigas culture centre since the mid 1960s in the Netherlands [START_REF] Haydar | Predicting invasion patterns in coastal ecosystems: relationship between vector strength and vector tempo[END_REF]. Haplotype H5 was found along the Miyagi (Japan), BC and west USA coasts (fig. 1 in [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF], regions known to be the biogeographic primary sources of imported M. gigas into the Oosterschelde [START_REF] Ruesink | Introduction of non-native oysters: ecosystem effects and restoration implications[END_REF]. At Concarneau, 80% of U. australis specimens harboured the same H5 (Table 1, Fig. 3), in line with this interpretation. Indeed, the rebound of the French oyster industry was linked to the national RESUR operation based on massive and repeated imports of M. gigas adults from BC and spat from Miyagi in the 1970s [START_REF] Grizel | Introduction into France of the Japanese oyster (Crassostrea gigas)[END_REF].

However, all other combined haplotypes we found at Concarneau (H42 and H51) and Roscoff (H42 and H44) (Table 1) conflict with the above view of the oyster trade as a unique vector of introduction of U. australis into Brittany. Previous results obtained from Roscoff and Thau by [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF]Hanyuda et al. ( ) (only H42 found in specimens collected in 2005Hanyuda et al. ( and 2002 at Roscoff and Thau, respectively, Fig. 3) also strengthen this conflict and furthermore operated several ship lines among Yokohama, Kobe, Shanghai, Hong-Kong, Singapore, Calcutta, Mumbai and Marseille (Berneron-Couvenhes, 2007;[START_REF] Metzger | Relations between Marseille and East and Southeast Asia[END_REF]. Historical analysis of global maritime transports since the late 19th century de facto highlighted the leading role of Yokohama and Kobe ports together with the emerging role of Chinese ports in the early 20th century [START_REF] Ducruet | Asian cities in the global maritime network since the late nineteenth century[END_REF], and the leading role of Marseille and Le Havre, as the main French port at that time [START_REF] Ducruet | Analyser les trafics portuaires mondiaux en 1890 et 1925 à partir des registres du Lloyd's[END_REF]. Long-term port activities are under the control of regional conflicts and world wars [START_REF] Ducruet | Analyser les trafics portuaires mondiaux en 1890 et 1925 à partir des registres du Lloyd's[END_REF]. Numerous wrecks from trading ships from Japan from WWI are known from the Mediterranean and the Atlantic (https://uboat.net/wwi/ships_hit/; see also [START_REF] Hiroaki | Les naufragés du Nagata Maru[END_REF]. Changes in the northeastern Asian port hierarchy between Japan, Korea and China have also been noticeable since the 1980s [START_REF] Ducruet | Port competition and network polarization in the East Asian maritime corridor[END_REF][START_REF] Ducruet | Asian cities in the global maritime network since the late nineteenth century[END_REF], so contemporary European records of U. australis may be linked to vessel traffic originating from ports within the Bohai and Yellow Seas. More phylogeographic studies along Korean and Chinese coasts within the native area of U. australis are required to test these hypotheses.

The patchy records of U. australis along the French coasts suggest that its presence is underestimated, mostly due to historical misidentifications as Ulva species show great morphological diversity and plasticity. The use of molecular markers for population genetics and biogeography is a key tool contributing to a better interpretation of the introduction history of U. australis. Using markers alongside systematic field sampling using large sample sizes and analysing herbarium specimens will lead to a much deeper understanding of the biogeographic origin of introduced populations, vector hierarchy and timing of introductions.

Including an interdisciplinary exploration of human maritime activities is crucial since coastal macroalgal biodiversity has been heterogeneously reshaped over centuries by transport and aquaculture activities. [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF]. *: new combination neither described by [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF] nor Hanyuda et al. (2018), -: no record. S3.

  also recorded U. australis (determined by ITS2 sequences) from Baja California, Mexico in 2006-2007 and suggested the species to be a NIS in range expansion via transoceanic shipping. Along the Eastern Pacific coast, two tufA haplotypes (tuH0 and tuH1) were observed (Figs 1, 2), in

Fig. 1 :

 1 Fig. 1: Median-joining network of tufA haplotype (659 bp) in Ulva australis. Grey: U.

Fig. 2 :

 2 Fig. 2: Map of sampling sites showing the relative occurrence of the tufA haplotypes (659 bp)

Fig. 3 :

 3 Fig. 3: Map of combined atpI-H and trnA-N haplotypes in western Europe. Square: this

Table 1 :

 1 Combined haplotypes (H) of chloroplast (cH) and mitochondrial (mH) haplotypes found in Ulva australis following numbering by

Acknowledgements

We are grateful to Drs Bruno de Reviers (Museum National d'Histoire Naturelle, Paris), Takeaki Hanyuda (Kobe University), Antoine Fort and Ronan Sulpice (National University of Ireland, Galway), Stacy Krueger-Hadfield (University of Alabama at Birmingham), Hyung Woo Lee (Department of Biology, Jeju), Herre Stegenga (Naturalis Biodiversity Centre, Leiden), Sophie Steinhagen (University of Gothenburg, Strömstad),Wei Song (Key Laboratory of Science and Engineering for Marine Ecology and Environment, Qingdao), Marc Verlaque and Fred Mineur (MIO, Marseille) for their kind advice, personal data communication and comments on Ulva species. Nadia Améziane, director of the Marine Station at Concarneau, kindly provided access to the Marine Station facilities during our field trips in Brittany. We also thank Michel Prineau, Laureen Beaugeard, Aya Ghedmasi and Tony Robinet for help in the field and in the laboratory. The authors thank the molecular core facility of the LIENSs laboratory. We also gratefully acknowledge comments from Pr. Christine Maggs and anonymous reviewers that allowed us to greatly improve the manuscript.

Funding details 23

This research was supported by the CNRS, La Rochelle Université, Parc naturel régional du Marais poitevin (contract 180166-04/04/2018), the project DEVOTES (DEVelopment Of innovative Tools for understanding marine biodiversity and assessing good Environmental Status, FP7, grant no. 308392), and the project ECONAT funded by the Contrat de Plan Etat-Région, CNRS and the European Regional Development Fund (FEDER).

challenge the initial suggestion that U. australis could have been introduced into the Thau lagoon as early as the 1970s via oyster M. gigas transfers in [1971][1972][1973][1974][1975][1976] [START_REF] Verlaque | Morphology and reproduction of Asiatic Ulva pertusa (Ulvales, Chlorophyta) in Thau Lagoon (France, Mediterranean Sea)[END_REF]. Based on the distribution map of the combined haplotypes H42 and H44 in Japan and Korea (fig. 1 in Hanyuda et al., 2016), it is suggested that U. australis found at Concarneau, Roscoff and Thau originated from Kobe (Japan) and/or Southeastern Korea. A closer examination of microsatellite genotypes (fig. 2 in [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF] identifies Kobe Port as the most likely origin of U. australis populations at Roscoff, although the genotype found at Thau was more geographically spread (from Kobe to Kumamoto in Japan and in Southeastern Korea). The finding of a new H51 haplotype at Concarneau does not contradict this view. According to [START_REF] Hanyuda | Genetic diversity and biogeography of native and introduced populations of Ulva pertusa (Ulvales, Chlorophyta)[END_REF] this may reflect a mutation that arose since the first introductions or, more likely, insufficient sampling in the native range as further exemplified by the study of tsunami marine debris (Hanyuda et al., 2018). These results indicate maritime transport between Japan (at least) and France as another vector for the introduction of U. australis into Mediterranean and Atlantic waters. Aguilar-Rosas et al. We thus speculate that these repeated and independent events have been linked to the maritime trade between France and Southeast Asia in the late 19 th or early 20 th centuries. The maritime history of Japan was profoundly reorganized during the Meiji era and a new foreign trade policy was initiated for 1868 [START_REF] Horie | Foreign trade policy in the early Meiji era[END_REF] that actively promoted cooperation and business with the USA and Colonial Countries. The opening of the Suez Canal on November 1869 also offered new shipping routes between Europe and Asia [START_REF] Fletcher | The Suez Canal and World Shipping, 1869-1914[END_REF][START_REF] Berneron-Couvenhes | Les Messageries Maritimes. L'essor d'une grande compagnie de navigation française, 1851-1894[END_REF], and favoured maritime companies from leading European countries i.e. Great Britain, France, The Netherlands, Italy and Germany [START_REF] Miller | Europe and the maritime world: A twentieth century history[END_REF]. For instance, the 'Compagnie des Messageries Maritimes' set up in 1851 at Marseille

Supplementary information

Supplementary information can be found online at http://dx.doi.org Supplementary table S1

Author contributions

P.-G. Sauriau: original concept, fieldwork, analysis of molecular data, drafting and editing manuscript; M. Dartois: original concept, fieldwork, molecular analysis, analysis of molecular data, drafting and editing manuscript; V. Becquet: molecular analysis, drafting and editing manuscript; F. Aubert: fieldwork, laboratory analysis; V. Huet: fieldwork, laboratory analysis; M. Bréret: macroalgae taxonomy, laboratory analysis; A. Viricel: analysis of molecular data, drafting and editing manuscript; E. Pante: original concept, molecular analysis, analysis of molecular data, drafting and editing manuscript.


Jean-Philippe Karr

)]. Further, they pave the way towards precision calculations of the g-factor through calculation of higher-order QED corrections and hence to an alternative precision route to obtaining the proton-electron mass ratio.

I. INTRODUCTION

Spectroscopic measurements of the antihydrogen molecular ion H-2 , compared with its normal matter counterpart, have been recently proposed as a new avenue towards improved tests of the CP T symmetry [START_REF] Myers | CP T tests with the antihydrogen molecular ion[END_REF]. This perspective relies on the possibility to store a single H-2 or H + 2 ion in a Penning trap and identify its ro-vibrational state in a nondestructive way. The envisaged experiments use similar methods to those developed for high-precision measurements of bound-electron g-factors (see e.g. [START_REF] Sturm | High-precision measurement of the atomic mass of the electron[END_REF]). They would be performed in a double Penning trap consisting of a "precision trap" with a highly uniform magnetic field, where spectroscopic measurements are carried out, and an "analysis trap" with an inhomogeneous magnetic field allowing the positron or electron spin state to be determined via the so-called continuous Stern-Gerlach technique. The ion's internal state can then be determined using the fact that the spinflip frequencies depend in a resolvable and calculable way on the ro-vibrational and hyperfine state. This detection technique requires knowledge of a large number of Zeeman transition frequencies in a ∼ 5 Tesla magnetic field, at a precision level of ∼ 10 -6 -10 -7 . Since the dominant contribution to the Zeeman shift stems from the interaction of the magnetic field with the positron/electron spin (through the term g s•B), this implies that the bound positron/electron g factors should be determined theoretically with similar precision for an extensive range of rovibrational levels.

Beyond its importance for nondestructive internal state detection, the g-factor itself could be measured with high precision from the ratio of the cyclotron and spin-flip frequencies [START_REF] Sturm | High-precision measurement of the atomic mass of the electron[END_REF]. As discussed in [START_REF] Myers | CP T tests with the antihydrogen molecular ion[END_REF], measuring this frequency ratio in H-2 and H + 2 provides a way to compare m(e -)/m(p) with m(e + )/m(p) at a competitive precision level, under the assumptions that charges and g-factors have opposite signs in matter and antimatter. Further, if the theoretical g-factor of H + 2 is calculated with sufficiently high accuracy, the comparison between theory and experiment would lead to a stringent test of bound-state QED or to an independent determination of m(e -)/m(p).

The theoretical g-factors of hydrogen molecular ions have been calculated by Hegstrom [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF]. In that work, the author derived an effective Hamiltonian describing leading-order relativistic (α 2 ) and radiative (α 3 ) corrections in the nonrecoil limit, and performed numerical calculations of the α 2 -order correction in the adiabatic approximation, for 43 rovibrational levels of H + 2 . The theoretical uncertainty of the g-factor was estimated to about 10 -7 due to uncalculated α 3 -order corrections. It is worth noting that the g-factor of H + 2 has so far been measured in only one experiment [START_REF] Loch | Measurement of the electronic g factor of H + 2[END_REF] for a mixture of three (unresolved) vibrational levels, with a relative uncertainty of 0.9 ppm. The experimental result was found to be in good agreement with the theoretical predictions of [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF].

The present work pursues a double aim. Firstly, in order to enable state identification of (anti-)hydrogen molecular ions in Penning trap experiments, it is important to extend g-factor calculations to a wider range of rovibrational states, covering all possible states the ions may be found in. H + 2 ions are conveniently produced by electron-impact ionization of H 2 , which creates ions predominantly in v = 0 -12, L = 0 -4 [START_REF] Busch | Photodissociation of H + 2 and D + 2 : Experiment[END_REF] or may be produced in a selected rovibrational state using resonance-enhanced multi-photon ionization (REMPI) [START_REF] O'halloran | Photoionization dynamics of H2C 1 Πu: Vibrational and rotational branching ratios[END_REF][START_REF] Schmidt | Trapping, Cooling, and Photodissociation Analysis of State-Selected H + 2 Ions Produced by (3 + 1) Multiphoton Ionization[END_REF]. Formation of H-2 ions through the reaction H+ + p → H-2 + e + , leading to production in v = 0 -8, L = 0 -27, has been proposed in [START_REF] Myers | CP T tests with the antihydrogen molecular ion[END_REF]. Assuming these production schemes, and taking into account the possible use of the Stark quenching induced by the ion's motion in the trap's magnetic field to accelerate vibrational decay [START_REF] Karr | Stark quenching of rovibrational states of H + 2 due to motion in a magnetic field[END_REF], 201 rovibrational levels (out of 481 bound levels in total [START_REF] Korobov | Ro-vibrational states of H + 2 . Variational calculations[END_REF]) have been identified as the most experimentally relevant. Other mechanisms to produce H-2 , using collisions between laser-excited H atoms, have been explored in [START_REF] Zammit | Laser-driven production of the antihydrogen molecular ion[END_REF], but resulting ro-vibrational distributions were not discussed in that work.

The second aim is to provide complete and accurate calculations of the α 2 -order relativistic correction, that may serve as a reliable basis for future high-precision calculations of the g-factor through inclusion of higher-order corrections. To this end, we improve the calculations of Ref. [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF] by performing extensive nonadiabatic (three-body) calculations, and by including recoil corrections, which had been neglected in [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF]. This allows us to compute the α 2 correction with an (absolute) numerical uncertainty of about 10 -13 .

II. THEORY

In this section, we write the theoretical expressions of corrections to the g-factor in the general case of a one-electron diatomic molecule. We thus consider a three-body system made of two nuclei, with masses m 1 , m 2 and charges Z 1 e, Z 2 e, and one electron (mass m 3 ≡ m e , charge -e). Particle positions are denoted by R 1 , R 2 , R 3 ≡ R e , and we use the internal coordinates r 1 = R e -R 1 , r 2 = R e -R 2 , and r 12 = R 1 -R 2 = r 2 -r 1 . The nuclear and electronic momenta are denoted by P 1 , P 2 and p e , respectively.

The α 2 -order relativistic correction to the g-factor including recoil terms can be described in an effective Hamiltonian approach [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF][START_REF] Grotch | Hydrogenic Atoms in a Magnetic Field[END_REF][START_REF] Hegstrom | Nuclear-Mass and Anomalous-Moment Corrections to the Hamiltonian for an Atom in a Constant External Magnetic Field[END_REF][START_REF] Eides | Gyromagnetic Ratios of Bound Particles[END_REF][START_REF] Pachucki | Long-wavelength quantum electrodynamics[END_REF][START_REF] Pachucki | Nuclear mass correction to the magnetic interaction of atomic systems[END_REF][START_REF] Eides | Universal Binding and Recoil Corrections to Bound State g Factors in Hydrogenlike Ions[END_REF]. The first contribution at this order to the interaction of the electron spin s e with an external magnetic field B is

H 1 = -g e e 2m e (s e •B) p 2 e 2m 2 e , (1)
where g e is the free electron's g-factor. A second term comes from the electronic spin-orbit Hamiltonian H so in the external field, which can be written in the center-of-mass frame as

H so = g e -1 2m 2 e Z 1 r 3 1 (r 1 ×π e ) + Z 2 r 3 2 (r 2 ×π e ) •s e - g e 2m e Z 1 m 1 r 3 1 (r 1 ×Π 1 ) + Z 2 m 2 r 3 2 (r 2 ×Π 2 ) •s e , (2)
where [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF]

π e = p e + eA(r eC ) -e m e M (Z 1 A(r 1C ) + Z 2 A(r 2C ) -A(r eC )) , Π 1 = P 1 -Z 1 eA(r 1C ) -e m 1 M (Z 1 A(r 1C ) + Z 2 A(r 2C ) -A(r eC )) , (3)
Π 2 = P 2 -Z 2 eA(r 2C ) -e m 2 M (Z 1 A(r 1C ) + Z 2 A(r 2C ) -A(r eC )) , A(r) = 1 2
B×r.

Here, M = m 1 + m 2 + m e , and r eC , r 1C , r 2C are the positions of the electron and nuclei with respect to the center of mass, which are given by

r 1C = -(m 2 + m e )r 1 + m 2 r 2 M , r 2C = m 1 r 1 -(m 1 + m e )r 2 M , (4)
r eC = m 1 r 1 + m 2 r 2 M .
As shown in [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF], the expressions of the momenta in Eq. ( 3) result from careful separation of the center-of-mass motion, following a procedure that was first proposed in [START_REF] Grotch | Hydrogenic Atoms in a Magnetic Field[END_REF][START_REF] Hegstrom | Nuclear-Mass and Anomalous-Moment Corrections to the Hamiltonian for an Atom in a Constant External Magnetic Field[END_REF] in atomic systems. The latter results have been confirmed using the NRQED approach [START_REF] Pachucki | Long-wavelength quantum electrodynamics[END_REF][START_REF] Pachucki | Nuclear mass correction to the magnetic interaction of atomic systems[END_REF][START_REF] Eides | Universal Binding and Recoil Corrections to Bound State g Factors in Hydrogenlike Ions[END_REF]. In the present work, we improve the treatment of Ref. [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF] by keeping all recoil terms in Eqs. [START_REF] Sturm | High-precision measurement of the atomic mass of the electron[END_REF][START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF][START_REF] Loch | Measurement of the electronic g factor of H + 2[END_REF]. The full contribution from H so to the g-factor is then given by the following effective Hamiltonian (taking g e = 2 in order to retain only the α 2 -order contribution):

H 2 = e 2m e σ ij s i e B j , (5)
σ ij = 1 2m e c 1 r 2 1 δ ij -r i 1 r j 1 r 3 1 + c 2 r 2 2 δ ij -r i 2 r j 2 r 3 2 + c (1) 12 r 1 •r 2 δ ij -r i 1 r j 2 r 3 1 + c (2) 12 r 1 •r 2 δ ij -r i 1 r j 2 r 3 2 , c 1 = 1 M 2 (M -m e )m 1 Z 1 + m 1 m e Z 1 Z 2 - (2M + m 1 )(m 2 + m e )m e Z 2 1 m 1 , c 2 = 1 M 2 (M -m e )m 2 Z 2 + m 2 m e Z 1 Z 2 - (2M + m 2 )(m 1 + m e )m e Z 2 2 m 2 , c (1) 12
= 1 M 2 (M -m e )m 2 Z 1 -(m 1 + m e )m e Z 1 Z 2 + (2M + m 1 )m 2 m e Z 2 1 m 1 , c (2)
12 = 1 M 2 (M -m e )m 1 Z 2 -(m 2 + m e )m e Z 1 Z 2 + (2M + m 2 )m 1 m e Z 2 2 m 2 .
The last contribution comes from the second-order energy shift induced by the orbital Zeeman term H Z and the spin-orbit coupling term:

∆E so-Z = 2 ψ 0 |H so Q(E 0 -H 0 ) -1 QH Z |ψ 0 , (6)
H Z = e 2m e L eC - Z 1 e 2m 1 L 1C - Z 2 e 2m 2 L 2C •B
Here, H 0 is the nonrelativistic (Schrödinger) Hamiltonian of the three-body system, ψ 0 the wave function for the rovibrational state under consideration, E 0 the corresponding nonrelativistic energy level, and Q is a projection operator on a subspace orthogonal to ψ 0 . L eC , L 1C , L 2C are the angular momenta of the electron and nuclei about the center of mass. Again, in our calculations we take into account all the recoil terms in H so and H Z .

Corrections to the electronic g-factor can be deduced from Eqs. ( 1), ( 5) and [START_REF] O'halloran | Photoionization dynamics of H2C 1 Πu: Vibrational and rotational branching ratios[END_REF]. The H 1 term [Eq. ( 1)] induces a correction for a rovibrational state (v, L),

∆g 1 (v, L) g e = - v, L|p 2 e |v, L 2m e . (7)
The other terms, H 2 [Eq. [START_REF] Busch | Photodissociation of H + 2 and D + 2 : Experiment[END_REF] and ∆E so-Z [Eq. ( 6)] are anisotropic. This is linked to the departure from spherical symmetry in a molecule, which led Hegstrom to introduce a g tensor in Ref. [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF]. Alternatively, the results can be expressed in terms of a g-factor similarly to the atomic case, the difference being that in a molecule, the g-factor acquires a dependence on the magnetic quantum number M .

The term H 2 may be decomposed into irreducible tensor components as follows:

H 2 = σ (0) (s e •B) + σ (2) • (s e ⊗ B) (2) , (8)
σ (0) = 1 2m e 2 3 c 1 r 1 + c 2 r 2 + c (1) 12 r 1 •r 2 r 3 1 + c (2) 12 r 1 •r 2 r 3 2 , (9)
σ (2) = 1 2m e 1 3 c 1 Q (2) 11 r 3 1 + c 2 Q (2) 22 r 2 + c (1) 12 Q
(2) 12

r 3 1 + c (2) 12 Q
(2) 12

r 3 2 , ( 10
)
where

Q (2)
ab (a, b = 1, 2) is the tensor having the Cartesian components

Q (2)ij ab = r a •r b δ ij -3r i a r j b . (11)
The second-order term ∆E so-Z can also be decomposed into irreducible tensor components following the Appendix B of [START_REF] Korobov | Hyperfine structure in the H + 2 and HD + molecular ions at order mα 6[END_REF]. One obtains

∆E so-Z = vL T (0) vL √ 2L + 1 s e •B + vL T (2) vL L (L ⊗ L) (2) L (L ⊗ L) (2) • (s e ⊗ B) (2) , (12)
where the orbital reduced matrix elements are given by [START_REF] Korobov | Hyperfine structure in the H + 2 and HD + molecular ions at order mα 6[END_REF] T

s = vL T (0) vL √ 2L + 1 = 1 3 (a -+ a 0 + a + ) , (13)
T t = vL T (2) vL √ 2L + 1 = L(L + 1)(2L -1)(2L + 3) 3 - a - L(2L -1) + a 0 L(L + 1) - a + (L + 1)(2L + 3) . (14)
Here, a -, a 0 , and a + are the contributions to the second-order perturbation term from intermediate states of angular momentum L -1, L, and L + 1, respectively:

a -= - 1 2L + 1 n =0 vL O (1) Z v n L -1 v n L -1 O (1) so vL E 0 -E n , (15)
a 0 = 1 2L + 1 n =0 vL O (1) Z v n L v n L O (1) so vL E 0 -E n , (16)
a + = - 1 2L + 1 n =0 vL O (1) Z v n L + 1 v n L + 1 O (1) so vL E 0 -E n , (17)
with

O (1) Z = e m e L eC - Z 1 e m 1 L 1C - Z 2 e m 2 L 2C , (18)
O (1) so = 1 2m 2 e Z 1 r 3 1 (r 1 ×p e ) + Z 2 r 3 2 (r 2 ×p e ) - 1 m e Z 1 m 1 r 3 1 (r 1 ×P 1 ) + Z 2 m 2 r 3 2 (r 2 ×P 2 ) . (19)
Finally, the g-factor including the complete α 2 -order relativistic correction is given by

g(v, L, M ) = g s (v, L) + 3M 2 -L(L + 1) L(L + 1)(2L -1)(2L + 3) g t (v, L), (20)
where the scalar part of the g-factor is

g s (v, L) g e = 1 - vL|p 2 e |vL 2m 2 e + σ s + T s , (21)
and the tensor part is

g t (v, L) g e = σ t + T t , (22)
with the definitions:

σ s = vL σ (0) vL √ 2L + 1 , σ t = vL σ (2) vL √ 2L + 1 . (23)
The expressions [START_REF] Korobov | Hyperfine structure in the hydrogen molecular ion[END_REF][START_REF] Karr | Higher-order corrections to spin-spin scalar interactions in HD + and H + 2[END_REF] are correct to order α 2 . The approximation g e 2 has been used in the last two terms of Eq. ( 21) (which comes to neglecting terms of order (α/π)(σ s + T s )), and in Eq. ( 22). Now, we can relate the above expressions to those given in Ref. [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF]. In that work, the g tensor is defined by writing the interaction of the electron spin with the magnetic field in the form

H eff = e 2m e i,j g ij s i e B j , (24)
and the components g ⊥ = g xx = g yy and g = g zz are calculated with the z axis taken to be along the internuclear axis (let us recall that all calculations were done in the Born-Oppenheimer approximation). The g-factor of a (v, L, M ) state can be obtained from these quantities through the relationship [START_REF] Ramsey | Dependence of Magnetic Shielding of Nuclei upon Molecular Orientation[END_REF] g

(v, L, M ) = 2 3 g ⊥ + 1 3 g + 2 3 3M 2 -L(L + 1) (2L -1)(2L + 3) g ⊥ -g . (25)
Comparing Eqs. ( 25) and ( 20) one gets

g s = 2 3 g ⊥ + 1 3 g , ( 26
)
g t = 2 3 L(L + 1) (2L -1)(2L + 3) g ⊥ -g . ( 27
)

III. NUMERICAL RESULTS

In order to calculate the scalar [Eq. ( 21)] and tensor [Eq. ( 22)] corrections to the g-factor, the three-body Schrödinger equation is solved using a variational expansion of the wavefunction involving exponentials of interparticle distances [START_REF] Karr | Stark quenching of rovibrational states of H + 2 due to motion in a magnetic field[END_REF][START_REF] Korobov | Ro-vibrational states of H + 2 . Variational calculations[END_REF][START_REF] Korobov | Coulomb three-body bound-state problem: Variational calculations of nonrelativistic energies[END_REF]:

Ψ (vL) 0 (R, r 1 ) = l1+l2=L Y l1l2 LM ( R, r1 )G l1l2 (R, r 1 , r 2 ), Y l1l2 LM ( R, r1 ) = R l1 r l2 1 Y l1 ( R) ⊗ Y l2 (r 1 ) LM , G l1l2 (R, r 1 , r 2 ) = N/2 n=1 C n Re e -αnR-βnr1-γnr2 + D n Im e -αnR-βnr1-γnr2 . ( 28
)
where R is the internuclear vector, and r 1 , r 2 the electron's position with respect to both nuclei. The complex exponents α n , β n and γ n are generated pseudorandomly in several intervals. The interval bounds as well as the number of basis functions N i,l1 in each interval i and angular momentum subset {l 1 , l 2 } (keeping the total basis length N constant), have been optimized for a few tens of rovibrational states. This was sufficient to have good convergence for all the states considered in this work, because the wave functions (and therefore the optimal values of the parameters) evolve only slowly with the rotational quantum number. The expectation values of p 2 e , σ (0) , and σ (2) [Eqs. [START_REF] Korobov | Ro-vibrational states of H + 2 . Variational calculations[END_REF][START_REF] Zammit | Laser-driven production of the antihydrogen molecular ion[END_REF]] are obtained with 9-10 digits of accuracy, using basis lengths N between 2000 and 5600, depending on the operator and on the rovibrational state. The second-order terms T s and T t [Eqs. [START_REF] Eides | Gyromagnetic Ratios of Bound Particles[END_REF][START_REF] Pachucki | Long-wavelength quantum electrodynamics[END_REF]] are more challenging to calculate with high accuracy. However, they are still simpler than the singular second-order terms discussed in [START_REF] Korobov | Hyperfine structure in the H + 2 and HD + molecular ions at order mα 6[END_REF]. The basis set used for intermediate states includes "regular" subsets where the interval bounds for the exponents α n , β n , γ n are the same as those used for to obtain the zero-order wavefunction Ψ (vL) 0

. In contradistinction with the singular terms evaluated in [START_REF] Korobov | Hyperfine structure in the H + 2 and HD + molecular ions at order mα 6[END_REF], it is not strictly necessary to add "singular" subsets containing higher exponents, but we found that the inclusion of two additional subsets with exponents β n , γ n up to 10 improves the convergence. Overall, a 9-digit accuracy is achieved for all rovibrational states using intermediate basis sets of length N ∼ 4000 -12000.

Detailed numerical results for g s and g t are shown in Table I and compared with those of [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF] for 38 rovibrational states. Differences with respect to Hegstrom's values amount to a few 10 -8 , or a few 10 -3 in relative value, which is consistent with the order of magnitude of nonadiabatic and recoil corrections. Complete results for the 201 states identified as the most experimentally relevant are given in Tables II andIII. All digits are converged, so that the uncertainty of the α 2 -order relativistic correction to the g-factor is smaller than 10 -13 .

IV. CONCLUSION

The complete relativistic corrections of order α 2 to the g-factor have been calculated with high accuracy for a wide range of rovibrational states. For the time being, the accuracy gain is not relevant for experiments since the theoretical uncertainty due to uncalculated α 3 -order radiative corrections is about 0.1 ppm [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF]. However, these results are a first step towards high-precision calculation of the g-factor; in this perspective, it was important to show that the numerically challenging second-order contribution induced by the Zeeman and spin-orbit Hamiltonians can be evaluated with high precision, so that they do not represent a serious limitation regarding the achievable accuracy level.

These results may now be readily used to calculate spin-flip transition frequencies in the magnetic field of a Penning trap. Precise knowledge of these frequencies is required for nondestructive identification of the molecule's internal state in future experiments with H + 2 and H-2 [START_REF] Myers | CP T tests with the antihydrogen molecular ion[END_REF]. To achieve this, one should diagonalize the Hamiltonian H Z + H hfs , where H Z and H hfs are respectively the Zeeman and hyperfine structure Hamiltonians. The Zeeman effect has been studied in [START_REF] Karr | Vibrational spectroscopy of H + 2 : Precise evaluation of the Zeeman effect[END_REF], and the hyperfine structure has been investigated in detail in [START_REF] Korobov | Hyperfine structure in the H + 2 and HD + molecular ions at order mα 6[END_REF][START_REF] Korobov | Hyperfine structure in the hydrogen molecular ion[END_REF][START_REF] Karr | Higher-order corrections to spin-spin scalar interactions in HD + and H + 2[END_REF]. Using the results of those [START_REF] Ramsey | Dependence of Magnetic Shielding of Nuclei upon Molecular Orientation[END_REF].323 -0.040576987 0.007699943 0.3102274 0.309 6 0 1.045243047 0.174081855 0.015007517 17.7610388 17.767 0.000000000 0.000000000 0.0000000 0.000 8 0 1.013989580 0.168879593 0.014587468 17.2282898 17.236 0.000000000 0.000000000 0.0000000 0.000 10 0 0.990865938 0.165032806 0.013643118 16.8677417 16.878 0.000000000 0.000000000 0.0000000 0.000 12 0 0.976285366 0.162610698 0.012022406 16.6948098 16.705 0.000000000 0.000000000 0.0000000 0.000 TABLE I: Relativistic corrections to the g-factor for rovibrational states (v, L) of H + 2 , and comparison with previous calculations. Column 3-5 (resp. 8-9) are the contributions to the scalar (tensor) part of the g-factor (see Eqs. ( 21) and [START_REF] Karr | Higher-order corrections to spin-spin scalar interactions in HD + and H + 2[END_REF], respectively), in atomic units. The values of 1 -gs/ge (resp. -gt/ge) obtained in this work and in Ref. [START_REF] Hegstrom | g factors and related magnetic properties of molecules. Formulation of theory and calculations for H + 2 , HD + , and D + 2[END_REF] are given in columns 6-7 (resp. [START_REF] Zammit | Laser-driven production of the antihydrogen molecular ion[END_REF][START_REF] Grotch | Hydrogenic Atoms in a Magnetic Field[END_REF]; they should be multiplied by 10 -6 .

works, the spin-flip transition frequencies can be obtained with a relative uncertainty of 0.1 ppm, limited by the uncertainty of the g-factor, which is expected to be sufficient for unambiguous identification of the internal state.

  .102238625 0.183572071 0.014934166 18.7771239 18.782 -0.037534640 0.007730789 0.3437892 0.342 3 4 1.098932954 0.183021406 0.014963189 18.7168863 18.722 -0.041918645 0.007644688 0.3369849 0.335 4 0 1.084559830 0.180628298 0.015014469 18.4588976 18.465 0.000000000 0.000000000 0.0000000 0.000 4 1 1.083769176 0.180496594 0.015020357 18.4445458 18.450 -0.029223896 0.009522682 0.3913838 0.389 4 2 1.082198272 0.180234919 0.015031993 18.4160343 18.421 -0.031849229 0.008054448 0.3295705 0.328 4 3 1.079867562 0.179846681 0.015049100 18.3737409 18.379 -0.036343726 0.007790306 0.3166502 0.315 4 4 1.076806815 0.179336839 0.015071268 18.3182157

	v L	p 2 e	σs	Ts	1-gs/ge 1-gs/ge (this work) [3]	σt	Tt	-gt/ge (this work)	-gt/ge [3]
	0 0 1.188584982 0.197953422 0.014090083 20.3552762 20.359 0.000000000 0.000000000 0.0000000 0.000
	0 1 1.187531896 0.197777979 0.014101909 20.3359500 20.340 -0.032682214 0.008941067 0.5286804 0.526
	0 2 1.185438336 0.197429194 0.014125424 20.2975286 20.302 -0.035634946 0.007569228 0.4455650 0.444
	0 3 1.182329094 0.196911198 0.014160360 20.2404665 20.245 -0.040692364 0.007330723 0.4286492 0.427
	0 4 1.178240299 0.196230013 0.014206324 20.1654259 20.169 -0.045474983 0.007258466 0.4206783 0.419
	0 6 1.167317949 0.194410381 0.014329191 19.9649660 19.969 -0.053897721 0.007247903 0.4100693 0.408
	0 8 1.153136628 0.192047838 0.014488785 19.7046887 19.709 -0.061075213 0.007301354 0.4000008 0.398
	0 10 1.136248917 0.189234465 0.014678648 19.3947475 19.399 -0.067289815 0.007384038 0.3887244 0.387
	0 12 1.117244006 0.186068437 0.014891527 19.0459880 19.051 -0.072714188 0.007484065 0.3758892 0.374
	0 14 1.096705506 0.182647005 0.015119723 18.6691807 18.674 -0.077452803 0.007594643 0.3614692 0.360
	0 16 1.075181550 0.179061504 0.015355330 18.2744772 18.280 -0.081569610 0.007710556 0.3455422 0.344
	0 18 1.053167409 0.175394464 0.015590341 17.8710960 17.877 -0.085102995 0.007827188 0.3282231 0.327
	0 20 1.031098644 0.171718462 0.015816634 17.4672018 17.473 -0.088073738 0.007940156 0.3096381 0.308
	0 26 0.968081716 0.161222808 0.016356395 16.3194982 16.326 -0.093621519 0.008212290 0.2474918 0.247
	1 0 1.159234438 0.193064762 0.014421001 19.8165041 19.821 0.000000000 0.000000000 0.0000000 0.000
	1 1 1.158250614 0.192900862 0.014431537 19.7984759 19.803 -0.031894491 0.009149724 0.4933508 0.491
	1 2 1.156294965 0.192575061 0.014452475 19.7626398 19.767 -0.034772716 0.007744201 0.4157136 0.414
	1 3 1.153391076 0.192091290 0.014483546 19.7094287 19.714 -0.039702043 0.007497771 0.3998213 0.398
	1 4 1.149573307 0.191455274 0.014524357 19.6394735 19.644 -0.044359752 0.007420710 0.3922428 0.390
	2 0 1.132170502 0.188557196 0.014689251 19.3216577 19.327 0.000000000 0.000000000 0.0000000 0.000
	2 1 1.131253178 0.188404379 0.014698386 19.3048846 19.310 -0.031058959 0.009318676 0.4586657 0.457
	2 2 1.129429961 0.188100651 0.014716519 19.2715486 19.276 -0.033858126 0.007885496 0.3864088 0.385
	2 3 1.126723309 0.187649753 0.014743382 19.2220626 19.227 -0.038651541 0.007632107 0.3715242 0.370
	2 4 1.123165936 0.187057137 0.014778579 19.1570284 19.162 -0.043176666 0.007550435 0.3643344 0.363
	3 0 1.107303081 0.184415723 0.014889282 18.8694330 18.875 0.000000000 0.000000000 0.0000000 0.000
	3 1 1.106449987 0.184273612 0.014896877 18.8538819 18.859 -0.030170737 0.009444398 0.4246628 0.422
	3 2 1.104754698 0.183991205 0.014911929 18.8229807 18.828 -0.032885813 0.007990107 0.3576827 0.356
	3 3 1

Acknowledgments

I am grateful to E. G. Myers and L. Hilico for useful discussions and remarks on the manuscript.