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Identification of Chaetognaths as Protostomes Is Supported by the Analysis of Their Mitochondrial Genome
     * Centre d'Océanologie de Marseille, Marseille, France; Laboratoire de Biologie Animale (Plancton), Université Aix-Marseille I, Marseille, France; and Institut de Biologie du Développement de Marseille, Laboratoire de Génétique et Physiologie du Développement, Campus de Luminy, Marseille, France

    E-mail: papillon@com.univ-mrs.fr.

    Abstract

    Determining the phylogenetic position of enigmatic phyla such as Chaetognatha is a longstanding challenge for biologists. Chaetognaths (or arrow worms) are small, bilaterally symmetrical metazoans. In the past decades, their relationships within the metazoans have been strongly debated because of embryological and morphological features shared with the two main branches of Bilateria: the deuterostomes and protostomes. Despite recent attempts based on molecular data, the Chaetognatha affinities have not yet been convincingly defined. To answer this fundamental question, we determined the complete mitochondrial DNA genome of Spadella cephaloptera. We report three unique features: it is the smallest metazoan mitochondrial genome known and lacks both atp8 and atp6 and all tRNA genes. Furthermore phylogenetic reconstructions show that Chaetognatha belongs to protostomes. This implies that some embryological characters observed in chaetognaths, such as a gut with a mouth not arising from blastopore (deuterostomy) and a mesoderm derived from archenteron (enterocoely), could be ancestral features (plesiomorphies) of bilaterians.

    Key Words: Chaetognatha ? mitochondria ? gene loss ? evolution ? phylogeny ? Spadella cephaloptera

    Introduction

    Chaetognaths (commonly named arrow worms) are transparent marine metazoans, ranging in size from 2 to 120 mm. They are mostly planktonic and display a very simple body plan, divided into three regions separated by two transverse septa: the head, the trunk, and the tail. In addition to the nervous system and muscles, the main internal organs observed are the gut and ovaries localized in the trunk and the testes in the tail. In the past century, Chaetognatha relationships within the metazoans have not been determined on the basis of their embryological and morphological features (Hyman 1959; Beklemishev 1969; Ghirardelli 1994). Indeed, chaetognaths display some embryological characters considered typical of deuterostomes (a gut with a mouth not arising from blastopore and a mesoderm derived from the archenteron by enterocoely), whereas their morphology recalls the organization of the protostomes (ventral nerve cord and chitinous structures) (Nielsen 2001). Such a division of characters strongly isolates the chaetognaths from the majority of bilaterians (Hyman 1959) and allowed Beklemishev (1969) to put the chaetognaths and the brachiopods in a separate group, as he thought that both of them belonged neither to protostomes nor to deuterostomes. Despite several proposed relatives (Ghirardelli 1968), it has been traditionally espoused that chaetognaths are allied to deuterostomes on the basis of their embryological features (for review see Willmer [1990]). Comparison of their photoreceptive cells with those of the chordates corroborated such a phylogenetic position (Eakin and Westfall 1964).

    However, the affinities with the deuterostomes have not been supported by most recent authors. Indeed, in the following years, Casanova (1986) has revived the idea of a relationship with mollusks, after the discovery of a hermaphrodite gland in deep benthoplanktonic species belonging to the Archeterokrohnia genus. Finally, Nielsen (2001) has regarded the phylum as a sister group of the gnathostomulids and the rotifers on the basis of the chitinous spines surrounding the mouth (hooks) and the innervations of the muscles from a vestibular ganglion that resemble the ganglion in rotifers and gnathostomulids.

    More recently, the metazoan phylogeny surveys using molecular data, sometimes combined with morphological characters, suggested various affinities, mostly among protostomes: within lophotrochozoans with intermediate filament sequences (Erber et al. 1998) or with ribosomal RNA 18S sequences (18S) within ecdysozoans (Halanych 1996; Littlewood et al. 1998; Zrzàvy et al. 1998), basal ecdysozoans (Peterson and Eernisse 2001), between deuterostomes and protostomes (Giribet et al. 2000), or as an early offshoot of the bilaterian lineage (Telford and Holland 1993; Wadah and Satoh 1994). However, the sequences used in these works are extremely divergent, and most of these authors admitted that the phylogenetic position of Chaetognatha remains dubious because of long-branch attraction artifacts. Landmark studies of 18S in nematodes (Aguinaldo et al. 1997) and acoels (Ruiz-Trillo et al. 1999) clearly excluded Chaetognatha sequences from the analyses, and this long-branch attraction problem is well documented and reviewed in the study by Mallatt and Winchell (2002). Finally, the Hox gene survey we performed emphasized a basal position among Bilateria (Papillon et al. 2003), whereas other nuclear markers we isolated (myosin, elongation factor, and slow-evolving 18S from several genera [unpublished data]) did not allow us to propose an unambiguous position.

    The mitochondrial DNA (mtDNA) genome is now widely used to infer deep phylogenetic relationships (Boore 1999). It is, then, of particular interest in deciphering the Chaetognatha phylogenetic position, and it gives us the opportunity to add phylogenetically useful genes to a growing metazoan data base. Metazoan mtDNA genome is usually considered a small extrachromosomal circular molecule ranging in size from about 13 to 18 kb and almost always containing the same 37 genes: two genes coding for rRNAs, 22 coding for tRNAs, and 13 coding for proteins. This gene content has been shown to vary in several metazoans (Boore 1999), such as nematodes (lack of atp8), mollusks (lack of atp8 and presence of one extra tRNA), or cnidarians (lost of nearly all tRNA genes and gain of genes not found in other mtDNAs). Phylogenetic analyses of the primary sequences of mitochondrial genes (Boore and Staton 2002; Stechmann and Schlegel 1999) support the new animal phylogeny based on 18S (Aguinaldo et al. 1997) and Hox genes (de Rosa et al. 1999). Moreover, mitochondrial gene arrangement is a powerful phylogenetic tool for several reasons. First, the gene content is almost invariant in the metazoans and, thus, provides a unique and universal pool of information. Second, stable structural gene rearrangements are rare events because functional genomes must be maintained. Third, the great number of potential gene rearrangements makes it very unlikely that different lineages would independently adopt the same gene order or that any gene would move back to a previous location (Boore and Brown 1998).

    Here we report the complete mitochondrial genome sequence of Spadella cephaloptera in which we identified three major striking and unique features: (1) the lack of both atp8 and atp6, (2) the absence of all tRNA genes, and (3) the smallest size among metazoan mtDNA genomes.

    Materials and Methods

    Specimens and DNA Extraction

    Specimens of a benthic chaetognath species, Spadella cephaloptera Busch 1851, were collected in the bay of La Ciotat near Marseilles (South of France) on the Posidonia oceanica bed, at a depth of 3 to 5 m. In the laboratory, samples were kept in large aquaria containing natural seawater and placed in a constant temperature at 21°C ± 1°C. DNA was prepared from adult specimens starved for 2 days in filtered seawater to prevent contamination by ingested preys, mainly represented by crustaceans and even small chaetognaths. Then, the selected specimens were dried on pieces of filter paper; homogenized in lysis buffer containing 100 mM Tris-HCl, 100 mM EDTA, and 0.1% SDS; and incubated with proteinase K at 50 mg/ml for 4 h at 50°C. After successive extractions with phenol, phenol/chloroform, and chloroform, DNA was ethanol-precipitated in presence of 0.3 M NaAc.

    Isolation of the Mitochondrial Genome

    The ribosomal RNA 16S (rrnL) gene was the first mitochondrial sequence isolated in S. cephaloptera, using the following primers: 16SF (5'-CCTGTTTANCAAAAACAT-3') and 16SR (5'-GGTCCAACCAAAGATAGA-3'). We isolated the cytochrome b gene (cob) while randomly sequencing a cDNA library of S. cephaloptera (see Papillon et al. [2003]). Using rrnL and cob as anchor genes, we tried to amplify regions of mitochondrial genomes lying between these two genes with the Advantage 2 PCR Kit (CLONTECH) dedicated to long-range PCR. All combinations of genome organization were tested, and we were able to amplify a 2-kb region containing the NADH dehydrogenase subunit 6 (nad6) gene and a 10kb region, possibly spanning the almost complete mitochondrial genome. This last fragment was used as a template for the following PCR experiments. We amplified the cytochrome oxydase 1 (cox1) gene with universal degenerate primers COI5A2 (5'-TAATWGGTGGNTTYGGNA-3') and COI3A (5'-TCAGGRTGNCCRAARAAYCA-3'). This sequence and the rrnL-nad6-cob fragment were again used as anchor genes to isolate missing regions. Two 5-kb bands were obtained. Both fragments were cut with EcoR1 and resulting subfragments were cloned into pBC SK+ (Promega) and sequenced by "primer-walk."

    Genome Assembling

    After sequencing, mtDNA genome was assembled using MacVector version 7.1 program (Oxford Molecular Group). Protein-encoding genes and start codons were identified by Blast matching to other animal mtDNAs. Transfer RNA searching was done by hand and using the tRNA scan-SE Search Server (www.genetics.wustl.edu/eddy/tRNAscan-SE). Sequence of the mtDNA genome of S. cephaloptera has been submitted to GenBank under the accession number AY545549.

    Alignments

    We chose a broad representation of taxa from available complete mtDNA sequences for this study (table S1 in Supplementary Material online). Inferred amino acid sequences of 11 mitochondrial protein genes (cox1 to cox3, cob, nad1 to nad6, and nad4L) from 28 bilaterian ingroup and two outgroup taxa were concatenated and aligned using ClustalW in MacVector 7.1, and alignments were refined by eye. Phyla with long branches were constantly excluded from the analyses (notably the nematodes and the platyhelminths). Among ecdysozoans, except arthropods, only nematode mtDNA genomes are fully sequenced, but these sequences are problematic for phylogenetic reconstructions because of dramatically accelerated substitution rates (data not shown), even with the mtDNA genome of Trichinella, considered as the less derived of the nematodes phylum (Lavrov and Brown 2001). Thus, arthropods are the only ecdysozoan representatives. To avoid subjectivity in excluding unreliably aligned positions from phylogenetic analysis, the program Gblocks version 0.91b (Castresana 2000) was used with the least-stringent settings.

    Phylogenetic Reconstruction

    Sequence alignments were analyzed by Neighbor-Joining (NJ) (Gamma model of distances and sites pairwise deletion) and maximum-parsimony (MP) with MEGA version 2.1 (Kumar, Tamura, and Nei 1994). Confidence estimates included bootstrap analysis with 1,000 replicates. The full alignment comprised 2,511 amino acid positions, of which 1,671 were parsimony-informative. Maximum-likelihood (ML) analysis employed 10,000 quartet-puzzling steps, an mtREV24 model of substitution, and eight Gamma rate categories in Tree-Puzzle version 5.0 (Schmidt et al. 2002).

    In another analysis, two set of trees were compared so that support for specific phylogenetic hypotheses could be assessed. We produced an exhaustive search of ML trees in ProtML (Molphy 2.3b3 [Jun Adachi and Masami Hasegawa 1992–1996]) using the mtREV24 model with various constraints. First, we tested hypotheses of Chaetognatha being a deuterostome, a protostome, a basal bilaterian, or between protostomes and deuterostomes. Then, four positions within protostomes were also assessed: chaetognaths included in the lophotrochozoans, in the ecdysozoans, as a third protostomian branch, or as a basal protostome. Two different approaches were applied in tree selection to obtain two sets of trees. First, we built consensus trees (PHYLIP version 3.5c [Felsenstein 1993]) for each phylogenetic hypothesis from the entire set of generated trees or from the first 1,000 top-ranking trees when more trees were generated. Second, we retained the best trees for each constraint using the Approximately Unbiased (AU), Kishino-Hasegawa (KH), and Shimodaira-Hasegawa (SH) tests (Shimodaira 2002, and references therein), as implemented in the CONSEL program (Shimodaira and Hasegawa 2001). Then, to chose among these phylogenetic hypotheses, the selected trees of each set ("consensus" or "best tree strategies") were compared using CONSEL, as described above to test whether the difference between the log-likelihood scores (LnL) of the ML trees obtained was statistically significant. The "best-tree" strategy was also conducted with a second substitution model, mtREV24-F, to test its influence on the results.

    For phylogenetic analysis of gene arrangements, two different methods were used to construct pairwise distance matrices. The first is based on minimum-breakpoint analysis (Blanchette, Kunisawa, and Sankoff 1999). The program is given a set of gene orders, and it finds the tree and the ancestral gene orders that relate them. The optimization criterion used is the minimal number of breakpoints. The second method is used in the GRIMM program (Tesler 2002), which computes the minimum possible number of rearrangement steps and determines a possible scenario taking this number of steps. This method takes only into account gene inversions. Pairwise distance matrices obtained with these two methods (figure S2 in Supplementary Material online) were used to construct phylogenetic trees with the NJ algorithm in MEGA version 2.1.

    Results and Discussion

    The mtDNA genome of S. cephaloptera (GenBank accession number AY545549) is very unusual both in size and in composition (fig. 1). With 11,905 bp, it is the smallest known metazoan mtDNA genome and contains only 13 of the 37 genes usually found. Genome composition, usage codon and initiation codons (9 ATG and 2 ATA) are described in the table S3 of Supplementary Material online.

    FIG. 1.— The complete mtDNA gene map of the chaetognath Spadella cephaloptera. Genesare abbreviated as in the text. Genes coding on different strands are indicated by arrow direction. Numbers represent, respectively, in base pairs (bp), the gene lengths and the size of the noncoding regions separating each gene. Asterisks (*) indicate that length of the ribosomal RNA genes rrnL and rrnS could not be precisely determined.

    The most striking feature is the failure in detecting any region homologous to one of the 22 bilaterian mitochondrial tRNAs. Among the usual 13 protein-encoding genes, ATP synthetase subunit 6 (atp6) and atp8 are lacking. The remaining 11 protein-encoding genes (cox1 to cox3, cob, nad1 to nad6, and nad4L) use the classical invertebrate mitochondrial genetic code. Six genes are transcribed from one strand (nad4, 5, 6, cob, rrnL, and cox1), and seven are transcribed from the other strand (nad1, 2, and 3; 4L; cox2 and 3, and rrnS). No intron is found, as expected, given that in metazoans, introns have only been characterized in cnidarians (Boore 1999). Intergenic spaces are not longer than 100 bp and have a total length of only 265 bp. In these noncoding regions, the signaling elements found in some animals (Boore 1999) cannot be identified. Although the genome is very compact, a single overlap (2 nucleotides long) between the neighboring genes nad4L and cox1 is observed. It has been proposed that differences in size of mtDNA are mostly caused by marked variations in the length and organization of intergenic regions (Burger, Gray, and Lang 2003). This is corroborated by the comparison of the size of Spadella cephaloptera mtDNA genes with those of other bilaterians (table S2 in Supplementary Material online). The length of the genes in Spadella cephaloptera are sometimes a bit smaller than those of mouse or fly but can be longer than those of the smallest metazoan mtDNA genome known until now (Taenia crassiceps, 13,503 bp), all of these organisms having a longer mtDNA genome. This shows that the small length of the Chaetognatha mtDNA genome is, then, mostly caused by the lack of genes (tRNAs, atp6, and atp8) and compact noncoding regions.

    The loss of atp8 can be observed not only in S. cephaloptera but also in a few other species of mollusks, nematodes, and platyhelminths. In these last organisms, it is not clear whether atp8 (1) moved to the nucleus, (2) became dispensable entirely, or (3) had its function coopted by one of the other ATPase subunits (Boore 1999). Unlike the atp8 loss, the absence of atp6 in S. cephaloptera is unique among metazoan mtDNAs.

    Chaetognaths are the first metazoans that could possess an mtDNA genome with a complete absence of tRNA genes. Total lack of tRNA genes has previously been reported only for some protozoans (Plasmodium), and some tRNA genes are lacking in green alga (Pedinomonas) and angiosperm plants (Arabidopsis) (Burger, Gray, and Lang 2003). The metazoans closest to this situation are the cnidarians, for instance, Metridium senile (Beagley, Okimoto, and Wolstenholme 1998) and Acropora tenuis (Van Oppen et al. 2002), where only Tryptophan and Methionine tRNAs are observed and the other necessary tRNAs are imported nuclear products (Boore 1999). As in cnidarians, paucity of mtRNA genes in S. cephaloptera is probably a derived condition rather than a conserved primitive state for multicellular animals.

    Finally, given the very peculiar features of the S. cephaloptera mtDNA genome, the existence of other mtDNA molecules containing the lacking genes cannot be excluded (linear concatemers or supplementary circular mtDNA), as it has been reported in few unrelated organisms (for review see Burger, Gray, and Lang [2003]).

    Despite this very unusual organization, mitochondrial data analyses based on deduced proteins sequences allowed us to investigate the phylogenetic position of chaetognaths. The search strategies for inferring phylogenetic trees included Neighbor-Joining (NJ), maximum parsimony (MP), and maximum likelihood (ML). In each analysis, deuterostome and protostome monophylies are highly supported (fig. 2). Discrepancies among these analyses are in the deuterostome relationships and the monophyly of lophotrochozoans. First, B. lanceolatum, P. marinus, and E. burgeri change phylogenetic positions, depending on the methods used. Second, lophotrochozoans monophyly is supported in the NJ and ML trees (support values 99% and 88%, respectively) but not in MP analyses (39%). Nevertheless, the three methods yielded unambiguously inclusion of Chaetognatha within the protostomes (100% branch-support values in each tree) either among (in ML tree [fig. 2] or outside the lophotrochozoans (in MP and NJ tree [figure S1a and b in Supplementary Material online]). Tree topology presented in figure 2 also indicates that chaetognaths are distant from arthropods. This hypothesis is supported by a recent study in which a specific tissue marker showed that chaetognaths are excluded from ecdysozoans (Haase et al. 2001).

    FIG. 2.— Position of Spadella cephaloptera within the bilaterians as suggested by maximum-likelihood (ML) phylogenetic analyses based on comparisons of primary mitochondrial sequences. The tree was rooted with the cnidarian sequences. Numbers at various nodes indicate, respectively, from left to right, percentage of quartet-puzzling replicates for ML and bootstrap support values for Neighbor-Joining (NJ) and maximum parsimony (MP). Values are indicated only when percentage of quartet-puzzling replicates are in excess of 75% in the ML tree. The phylogenetic relationships for which the methods give differing results are discussed in the text. Minus signs (–) indicate that a node is absent in the corresponding method; asterisks (*) indicate that in the NJ and MP trees, S. cephaloptera is within protostomes but excluded from lophotrochozoans and ecdysozoans (figure S1a and b in Supplementary Material online).

    To test this positioning, we compared four alternative topologies using statistical tests: Chaetognatha within protostomes, deuterostomes, as a third bilaterian branch, or basal bilaterian. In both strategies used ("best tree" and "consensus"), the three latter phylogenetic hypotheses were strongly rejected in favor of the proposed belonging of Chaetognatha to protostomes (table 1). Another set of tests was performed to precisely determine the positioning of chaetognaths within protostomes. They showed that relationships of Chaetognatha in the protostomian group cannot be accurately determined. Indeed, although the hypothesis of the Chaetognatha affinities with lophotrochozoans gives the best tree in both strategies, the positions as basal protostome and within ecdysozoans cannot be rejected in the "best-tree strategy" (only the hypothesis as a basal protostome cannot be excluded in the "consensus-tree strategy") (table 1). Similar results have been obtained in the analysis based on the other substitution model (data not shown).

    Table 1 Tests of Significance for Several Competing Phylogenetic Hypotheses Within Metazoans and Within Protostomes

    In addition, although the Chaetognatha branch is longer than the average bilaterian branch length, long-branch attraction artifacts can be rejected for two reasons: (1) the cnidarian long branch does not attract the Chaetognatha branch outside the bilaterians, and (2) bootstrap supports for the belonging of Chaetognatha to the protostomes are very strong in all phylogenetic analyses. Finally, it is noteworthy that alignments of S. cephaloptera mitochondrial proteins with those of various metazoans reveal the presence of residues characteristic to protostomes in the Chaetognatha nad5 sequence (fig. 3).

    FIG. 3.— Alignment of previously identified nad5 proteins of various metazoans with Spadella cephaloptera nad5 sequence. S. cephaloptera displays several protostome signature residues.

    As an independent source of phylogenetic information, S. cephaloptera mitochondrial gene arrangement was compared with those of various representative metazoans. We built pairwise distance matrices based on two methods of genome comparison, GRIMM and minimum breakpoint, and constructed phylograms from these matrices. In both analyses, the three main branches of bilaterians are recovered (fig. 4a and b), and S. cephaloptera clusters within the lophotrochozoan protostomes, supporting the results obtained from primary sequences.

    FIG. 4.— Unrooted phylogram of Neighbor-Joining analysis obtained from metazoan mtDNA genome arrangements. Nine taxa were chosen among the three main branches of bilaterians. The pairwise distance matrices were constructed using the GRIMM algorithm (a) and the minimum-breakpoint method (b). Both of them were implemented in MEGA to reconstruct a phylogenetic tree. The two encoding protein genes, atp6 and atp8, and all tRNAs were not considered in the GRIMM analysis, whereas only the tRNAs were excluded in the minimum-breakpoint method.

    In addition, we could identify in the S. cephaloptera mtDNA genome several gene boundaries that are likely to be ancestral bilaterian features, because they can be observed both in triploblasts and diploblasts. The nad1-nad3-nad2 block present in chaetognaths is shared by the annelid/pogonophore and cnidarian clades. In addition, the nad1-nad3 association is also present in another lophotrochozoan group, the platyhelminths. Another proposed plesiomorphy, the nad3-nad2 block, is conserved in several taxa, including annelids/pogonophores, mollusks, brachiopods, and cnidarians. Neither of these blocks has been reported in ecdysozoans or deuterostomes. The nad6-cob couple can be found in representative taxa of lophotrochozoans, ecdysozoans, cnidarians, but not in deuterostomes. These analyses have several implications with respect to the Chaetognatha phylogenetic position. Indeed, there is no gene boundary in common between S. cephaloptera and deuterostomes. Thus, in terms of gene pair arrangements, there is no derived and specific feature (synapomorphy) shared between Chaetognatha and deuterostomes. However, given that all common gene boundaries observed in Chaetognatha, protostomes and cnidarians are probable plesiomorphies, the affinities of arrow worms within protostomes are still unclear on the basis of mtDNA gene organization. Therefore, all mitochondrial data (primary sequences, gene order, and boundaries) support the identification of chaetognaths as protostomes. However, none of these data allows a more precise positioning within the protostomian group.

    It should be noticed that the inclusion of Chaetognatha within protostomes, based on mitochondrial data, is conflicting with previous conclusions, based on Hox genes (Papillon et al. 2003). This could imply that the proposed basal bilaterian feature conserved (a mosaic Hox gene that has retained characteristics of both central and posterior classes of Hox genes) could rather be a derived feature of the phylum.

    The way to divide the bilaterians is traditionally based on two characters: the origin of the coelom and the fate of the blastopore. In the new phylogeny (Adoutte et al. 2000), several phyla initially described as deuterostomes have been placed into protostomes. This is, for instance, the case for brachiopods (de Rosa 2001; Cohen 1998; Stechmann and Schlegel 1999) and phoronids (Helfenbein and Boore 2004), even though in these species, the coelomic cavities form by enterocoely and the mouth does not arise from the blastopore, like in deuterostomes. Valentine (1997), supported by Peterson and Eernisse (2001), previously advocated that traditional characters placing the lophophorates into the deuterostomes are plesiomorphies of bilaterians. Chaetognaths exhibit such plesiomorphies: a complete gut with a mouth not arising from the blastopore and coelomic cavities forming by enterocoely. Moreover, chaetognaths do not display any of the typical ecdysozoan and lophotrochozoan synapomorphies (possession of a molting habit and presence of lophophores or trochophore larvae, respectively). As Chaetognatha clearly exhibit plesiomorphies and lack synapomorphies, they cannot be accurately placed among the current protostomes, but this could bring them closer to the common ancestor, excluded from lophotrochozoan and ecdysozoan groups. Another embryological character traditionally used to link Chaetognatha with deuterostomes is the radial cleavage of the egg. However, recent cell fate analysis indicates that the four-cell embryo displays similarities with classic spiralians (Shimotori and Goto 2001), supporting the conclusion of the work presented here. Nevertheless, more precise phylogenetic investigations on arrow worms will need further sequencing efforts, for example, EST programs, to isolate new nuclear molecular markers.

    The acceptance that Chaetognatha are genealogically allied to undoubted protostomes strengthens the phylogenetic validity of some morphological characters such as the presence of a ventral nerve cord and chitinous structures but stresses that ontological criteria (i.e., embryological location of mouth and anus and mode of coelom formation) that traditionally define the deuterostome and protostome lineages can be misleading and raises once more the question of their relevance to define a bilaterian phylogeny.

    Acknowledgements

    We wish to thank Céline Brochier and André Gilles for phylogenetic advices. Also, we are grateful to Patrick Lemaire, Stephen Kerridge, and Laurent Fasano for helpful discussions and corrections.

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