Loïs Monnier
email: loismonnier@yahoo.fr

Stefano Salvi
email: stefano.salvi@get.omp.eu

Anthony Pochon
email: pochon.anthony@gmail.com.

Jérémie Melleton
email: j.melleton@brgm.fr

Didier Béziat
email: didier.beziat@get.omp.eu.

Philippe Lach
email: p.lach@brgm.fr

Laurent Bailly
email: l.bailly@brgm.fr.

Antimony in quartz as a vector to mineralization: A statistical approach from five Variscan Sb occurrences (France)

Keywords: LA-ICP-MS, Trace-elements signature, Quartz, Antimony, Mineralization, Footprint

Using in-situ LA-ICP-MS, the trace-elements compositions of hydrothermal quartz from five Sb mineralized Variscan districts, from different parts of France, were compared. Each of these districts is characterized by the presence of stibnite, with quartz being an ubiquitous gangue mineral. The resulting dataset was processed using a set of multivariate statistical analyses (i.e. principal component analysis, hierarchical cluster analysis, and permutational multivariate analysis of variance). This approach highlights differences in ore-forming processes (e.g., fluid chemistry), recorded in the traceelements pattern of the gangue minerals. Using this methodology, we could distinguish different mineralizing events with intricate paragenesis, such as those in the Sioule nappe, which appear to have recorded two different events of Sb precipitation. Results indicate that Sb incorporation in quartz is weakly correlated to other trace elements, but that its contents increase sharply in quartz from stibnitebearing veins. In this case, Sb shows values >10 ppm and up to 200 ppm, considerably higher than the J o u r n a l P r e -p r o o f Journal Pre-proof values measured in quartz unrelated to Sb mineralization (mainly ≤ 1 ppm). Intermediate Sb contents in quartz would point to the potential for a fluid to precipitate Sb ore. It is concluded that incorporation of Sb in quartz can be confidentially correlated to its content in the mineralizing fluid. Based on our data from five Sb districts of the French Variscan belt, supported by published Sb trace-elements data in quartz from a number of other Sb deposits worldwide, this study highlights the strong potential of the Sb content in quartz as a proxy for tracing Sb mineralization.

Introduction

During the last couple of decades, many studies have undertaken to determine the concentration of trace elements in quartz in order to better constrain magmatic and/or hydrothermal processes [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF]references therein, Müller et al., 2018;Wertich et al., 2018;Breiter et al., 2019;Hong (Anderson, 2001) was only employed for the first time in geochemistry in the recent study of [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF]. The final purpose of this exercise is to understand whether or not the chemical signature of different quartz generations is an efficient means for distinguishing the five districts considered, and whether it may represent an efficient marker to fingerprint Sb mineralizations.

Geological Settings

Bournac district

Bournac is an epithermal deposit located in the North-East of the Montagne Noire, a metamorphic core complex which exposed gneissic and migmatitic Variscan basement during orogenic collapse at ca. 290 Ma (Rey et al. 2011;Roger et al. 2015;Whitney et al. 2013;Trap et al. 2017). Mineralization occurs as large veins (80 cm thick) oriented N 20° E, hosted within the Brusque nappe at the level of an anticline composed of Lower Cambrian rock (various metasediments and rhyolite-trachyte). These formations are intersected by the Faoulat granite, which form a network of interconnected dykes to the south of the mineralized zone, emplaced about 290 Ma (dating by the Rb-Sr method on total rock; [START_REF] Hamet | french. Étude systématique par la méthode 87Rb-87Sr des processus orogéniques: exemple de la Montagne Noire[END_REF]. Mineral assemblages related to contact metamorphism of the granite indicate a setting pressure of less than 1 kbar, consistent with a pressure of about 0.5 kb estimated from fluid inclusions [START_REF] Munoz | Fluid inclusion study of the bournac polymetallic (Sb-As-Pb-Zn-Fe-Cu...) vein deposit (montagne noire, France)[END_REF]. The ore paragenesis consists of a succession of crystallization stages, accompanied by quartz and/or ankerite (for more details, see [START_REF] Munoz | In french, abstract in english. Etude paragénétique de la minéralisation sulfurée complexe (Sb-Pb-Zn...) de Bournac (Hérault, France)[END_REF][START_REF] Munoz | Fluid inclusion study of the bournac polymetallic (Sb-As-Pb-Zn-Fe-Cu...) vein deposit (montagne noire, France)[END_REF]. The first and second stages comprise As-Fe and Zn-Cu massive suphides (e.g., arsenopyrite, pyrite, sphalerite, chalcopyrite) formed above 300 °C (homogenization temperature of fluid inclusions from 300 to 350 °C). The third stage consists of Pb-Sb ore (galena and antimony sulfosalts) and occurred at approximately 250°C, while Sb ore (stibnite plus quartz) makes up the fourth, and last, stage and was recorded by fluid inclusions displaying homogenization temperatures around 200°C.

La Lucette district

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Journal Pre-proof At La Lucette, Sb mineralization is hosted in the Laval basin, located along the North Armorican shear zone, a structure that delimits the Northern from the Central Armorican Domain. During the Variscan orogeny, this area underwent low-grade regional metamorphism and moderate deformation marked by north-verging open folds [START_REF] Brun | Cadomian tectonics in northern Brittany: a contribution of 3-D crustal-scale modelling[END_REF][START_REF] Gumiaux | Tectonic history of the Hercynian Armorican shear belt (Brittany, France)[END_REF]. Mineralized quartz and carbonate veins of significant width (up to several meters) are hosted by sandstone and pelites, within an anticline oriented N 120° E. The metal stock is estimated at 42 kt antimony and 8.4 t gold [START_REF] Chauris | Metallogeny of the Armorican massif[END_REF]. The vein mineralogy was studied by [START_REF] Pochon | In french. Magmatisme mafique et minéralisations Sb-Au dans le domaine Centre Armoricain: contrôles spatio-temporels et implications metallogéniques[END_REF], who describes a first barren episode, consisting mostly of massive quartz, followed by an As-Zn-Au stage accompanied by minor Sb-sulphosalts, and a last Fe-Pb-Cu-Au stage. Important precipitation of stibnite took place during this last stage. Each of these four episodes was accompanied by quartz precipitation, trapping fluid inclusions which record temperature decrease of the mineralizing fluid would have recorded widerange temperature of 300 ± 80 °C (microthermometric data on fluid inclusions, [START_REF] Pochon | In french. Magmatisme mafique et minéralisations Sb-Au dans le domaine Centre Armoricain: contrôles spatio-temporels et implications metallogéniques[END_REF]. In the Armorican Massif, an important Sb-Au mineralization event seems to be spatially correlated with mafic rocks around 360 Ma (Pochon et al., 2016a(Pochon et al., , 2016b(Pochon et al., , 2019)), nevertheless, the emplacement age of the La Lucette deposit remains unclear and could be late Variscan [START_REF] Chauris | Metallogeny of the Armorican massif[END_REF].

Le Semnon district

The Le Semnon deposit is located in the central part of Variscan Armorican massif. Here, the central armorican domain consists of Late Neoproterozoic to Late Paleozoic sedimentary rocks metamorphosed to low grade facies moderately deformed. Main structures consist of upright folds with sub-vertical axial planes and E-W sub-horizontal axes, produced by a N120-125-striking dextral wrenching of the CAD [START_REF] Gapais | Is the Hercynian belt of Brittany a major shear zone?[END_REF][START_REF] Gumiaux | Tectonic history of the Hercynian Armorican shear belt (Brittany, France)[END_REF]. [START_REF] Gloaguen | In french. Revalorisation du potentiel minier français: le district antimonifère du Semnon (Ille-et-Vilaine, France)[END_REF] estimated the stock at ca. 45 kt of antimony and 3.3 t of gold. The Sb-Au mineralization consists of a network of irregular quartz-carbonate veins, up to 80 cm thick, crosscutting the contact between the dolerite dyke and Lower Ordovician slates. The mineralization would be genetically related to the emplacement of these mafic dykes, which are dated at ca. 360 Ma (U-Pb on apatite; [START_REF] Pochon | In french. Magmatisme mafique et minéralisations Sb-Au dans le domaine Centre Armoricain: contrôles spatio-temporels et implications metallogéniques[END_REF]al., 2016b, 2018). A first pervasive alteration of the wall rock, induced by orthomagmatic fluids from the dolerites, led to the crystallization of early minerals enriched in Sb such as rutile (ca. 100 ppm). A

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Journal Pre-proof second As-Fe stage (with minor Au-Sb) occurred during the initiation of fracturing, happening at approximately 360 °C (As-geothermometer in arsenopyrite). The last Sb-Au episode occurred during a crack-sealing process with a fluid temperature around 250 to 300 °C (microthermometric data on fluid inclusions, [START_REF] Pochon | Variscan Sb-Au mineralization in Central Brittany (France): A new metallogenic model derived from[END_REF]. All of these episodes took place at shallow depth, probably less than 2 km. For more details about petrographic and genetic considerations about the Le Semnon deposit, see [START_REF] Pochon | Variscan Sb-Au mineralization in Central Brittany (France): A new metallogenic model derived from[END_REF].

Districts from the Sioule nappe: Pouzol-Servant and Nades

To date, there are few data about Sb mineralizations from Pouzol-Servant and Nades. [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF] describe the petrography of the Sb-bearing veins at Nades and provides trace-elements chemistry of quartz from one sample. The most detailed study about the Nades and the Pouzol-Servant deposits focusses on the structural features [START_REF] Aubert | In french. Concession d'antimoine de Nades (Allier)[END_REF]. The district of Nades contains several veins, (up to meter in size), made up mostly of quartz, stibnite, and lesser arsenopyrite and pyrite. This district is located a few kilometers east of the Sillon Houiller, in the southern zone of an antiform centered on the granitic complex of Colettes/Beauvoir (Fig. 2). The formation of proximal veins and of the Nades veins in the mica schists led to intense alteration of the vein selvages, which consists mostly of muscovitisation (greisen facies). Peak of metamorphism of the host unit has been dated at approximately 360 Ma [START_REF] Couto | Monazite U-Th-Pb EPMA and zircon U-Pb SIMS chronological constraints on the tectonic, metamorphic, and thermal events in the inner part of the Variscan orogen, example from the J o u r n a l P r e -p r o o f Journal Pre-proof Sioule series, French Massif Central[END_REF]. Located about 500 m south of Nades, the granite of Pouzol-Servant intruded the host metasediments at 330 ± 9 Ma (Rb-Sr method on whole rock; Pin 1991). On the western edge, its contact with the schist is underlined by a fault, while to the North, laccolitic contacts with the schist can be observed [START_REF] Aubert | In french. Concession d'antimoine de Nades (Allier)[END_REF]. These authors describe several Sb occurrences in the Pouzol-Servant granite, consisting of quartz and stibnite veins with no other metals. The two main veins are those of Cros and Capitraux; the former, referred to as the Fontabourgnon vein in the study by [START_REF] Aubert | In french. Concession d'antimoine de Nades (Allier)[END_REF], was prospected at the beginning of the twentieth century and has a maximum thickness of 0.7 m with stibnite modal composition of the order of 10 to 20 %. No information is available on the Capitraux vein. [START_REF] Aubert | In french. Concession d'antimoine de Nades (Allier)[END_REF]

Petrography

The following chapters contain brief description of the main Sb mineralizing episodes for the five districts. These generally occur during the last hydrothermal stage and consist mostly of quartz and stibnite deposition. For more comprehensive information of each deposit, the reader is invited to consult the studies cited in the geological setting section above.

Bournac district

Although mineralized quartz veins in the Bournac district contain a wide variety of sulfide and sulfosalt, stibnite occurs in substantial larger proportion. This sulfide precipitated as (i) massive clusters of up to several mm in thickness around quartz crystals (Fig. 3-A), (ii) within fractures in quartz, in sealed fracture plains in the continuity of secondary fluid inclusion trails (Fig. 3-B), (iii) acicular grains within quartz crystals, orthogonal to the quartz facets, underlining their simultaneous growths (Fig. 3-C). Quartz is mostly euhedral, crystals measuring up to 1 mm in width. It commonly contains a large number of inclusions consisting of a majority of secondary fluid inclusions and stibnite acicular crystals (occurrence iii, above). Only quartz rimmed by substantial thickness of massive stibnite (Fig. 3-A) display large patches without inclusions, probably due to the protective role of the stibnite envelop during late fluid circulation.

La Lucette district

At La Lucette, quartz in veins forms euhedral crystals of millimeter size. Stibnite forms massive interconnected grains around the quartz. An important number of secondary inclusions (fluids and acicular stibnite) can be found in quartz crystals although they are preferentially located in the rims.

Primary fluid inclusions (organized hexagonally and parallel to the quartz facets, indicating trapping J o u r n a l P r e -p r o o f Journal Pre-proof during crystal growth) were also observed, and these are not accompanied by acicular stibnite (Fig. E,F).

Le Semnon district

In this district, Sb-bearing extensional veins formed after an episode of pervasive alteration at selvages of dolerite dikes [START_REF] Pochon | Variscan Sb-Au mineralization in Central Brittany (France): A new metallogenic model derived from[END_REF]. The extensional veins occur as lens shapes of mainly a few cm in size, which opened and filled during cyclic fracturing of the host rock. Quartz seals most parts of the veins, though locally calcite and ankerite also occur. The last mineralizing episode is marked by a crack-and-seal texture of quartz, growing perpendicular to the elongation of the lenses (Fig. H,I).

Locally, this quartz displays rather unique elongation along the c axis, with a very high length to width ratio (more than 10:1). Stibnite occurs as in primary inclusion in quartz, or as acicular grain at the edges of elongated quartz crystals. It can also be found in the metasomatized host rock, albeit in smaller amounts (Fig. H,I).

Pouzol-Servant district

The veins at Pouzol-Servant, Cros and Capitraux, are similar in texture. They are composed of "star" quartz aggregates of ca. one millimeter in diameter which seem to have grown radially (Fig. 3-J,K) around a iron oxide nucleus, of few microns in size. In the periphery of these star quartz crystals, rare acicular stibnite crystals may grow included in the quartz (Fig. 3 L). However, most of the mineralization occurs outside the star quartz (Fig. 3-J,K). Beside quartz and stibnite crystals, porosity can be significant or even predominant in the total volume of the vein in the case of Capitraux veins, in the form of large vacuoles around the star quartz network. Late clay commonly fills these vacuoles.

Nades district

Veins in the Sioule metasediments of the Nades district can contain stibnite with or without quartz.

Tiny stibnite-only veinlets intersect the shale foliation (Fig. 3-M), as do mm-to cm-sized quartzstibnite veinlets (Fig. 3-N). Stibnite can also occur within flakes of newly formed muscovite, next to these veinlet (Fig. 3-N). Both veinlet types form a dense interconnected network that intersects the

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Journal Pre-proof foliation in the mica schist. In the quartz-stibnite veinlets, stibnite mostly envelops euhedral to subhedral quartz grains, or fills fractures in quartz (Fig. 3-N), much as the stibnite found in Bournac quartz. Locally, small stibnite crystals are found included in the border of quartz crystals, forming a cloudy area in the crystal (Fig. 3-O), indicating synchronous growth of these two minerals. Few euhedral pyrite crystals also occur in the veinlets and in the schist, texturally predating quartz and stibnite crystallization.

Based on the petrographic evidences detailed above, relationships between stibnite and quartz common to the five districts are as follows: stibnite precipitated towards the end of quartz crystallization and mostly after it at Nades, Pouzol-Servant, and in the La Lucette district. With regards to the Bournac district, stibnite crystallizes mainly contemporaneously to and, to a lesser extent, after quartz crystallization. Lastly, stibnite precipitation in the Le Semnon district mainly took place before and during quartz crystallization.

Materials and methods

Sampling strategy and preparation

Samples from the Le Semnon were collected in the mine's adit (-17 m level) and correspond to samples that were studied by [START_REF] Pochon | Variscan Sb-Au mineralization in Central Brittany (France): A new metallogenic model derived from[END_REF]. Collection samples from Bournac are part of a batch collected for the studies of [START_REF] Munoz | In french, abstract in english. Etude paragénétique de la minéralisation sulfurée complexe (Sb-Pb-Zn...) de Bournac (Hérault, France)[END_REF] and [START_REF] Munoz | Fluid inclusion study of the bournac polymetallic (Sb-As-Pb-Zn-Fe-Cu...) vein deposit (montagne noire, France)[END_REF]. Finally, quartz veins from Nades and Pouzol-Servant districts were sampled on outcrop during the autumn 2016 and 2017, respectively [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF][START_REF] Monnier | Contrasting fluid behavior during two styles of greisen alteration leading to distinct wolframite mineralizations: the Echassières district (Massif Central, France)[END_REF]. Samples from La Lucette, which were taken from different parts on the main mineralized vein, come from the rock collection of the French Geological Survey (BRGM). For each district, several samples were collected from different veins or veinlets, excepted for La Lucette where only the main vein was sampled. All samples were sliced and selected areas prepared into polished thick sections (approximately 200 µm thick). In addition to the above localities, we also used trace-elements data from quartz collected from the Sb-poor Suchot, Mazet and Colettes kaolin quarry vein sets, in order to provide a comparison with

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Journal Pre-proof similar quartz veins that do not contain Sb mineralization. These veins have been characterized in detail in previous studies (cf. [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF].

LA-ICP-MS analyses protocol

LA-ICP-MS analyses were performed using a UV laser-ablation microsampler CETAC® Excite (193 nm) coupled to a ThermoScientific® X serie II quadrupole plasma mass spectrometer, at the BRGM facility in Orleans (France). The laser is equipped with a HelEx® 2 volume ablation chamber. It was used with frequency of 8 Hz, fluency of 7 J.cm -2 , and an ablation spot diameter of 85 μm. Acquisitions run 40 s (320 pulse), separated, before and after, by 20 s of background recording. The Glitter® software was used to quantify absolute values, using external standard NIST SRM 612 (concentration from [START_REF] Pearce | A compilation of new and published major and trace element data for NIST SRM 610 and NIST SRM 612 glass reference materials[END_REF] for calibration. Si was used as internal standard with a value fixed at 99 % of SiO 2 in quartz for all analyses. More details can be found in [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF], as the analytical protocol was exactly the same. A total of 8 elements (Li, B, Na, Al, Ti, Ge, As, Sb) were measured in quartz from the districts of Bournac, Nades, the La Lucette and the Le Semnon. To these elements, in order to compare our data with other quartz veins of the Sioule nappe studied by [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF], we added Be, Rb and Sr for the district of Pouzol-Servant, as these elements were also measured in that study. Detection limits (d.l.) are given in Table 1 for each element. Acquisition error are inferior to 10 %, for concentrations at least 1 order of magnitude superior to d.l. Sulfur was monitored (but not calculated) to verify for the presence of micro inclusions of stibnite that may not have been detected during microscopic observation. Although quartz from the Nades district had already been analyzed by [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF], we perform new analyses from different veins in order to obtain As concentrations, and to obviate to the limited number of analyses (i.e., 7) obtained in the older work.

This also allowed comparing the data obtained during the two analytical sessions (cf., Supplementary Materials 1). The two datasets show very good reproducibility; however, we observed a moderate tough significant drop in Sb and Na values in the new analyses. Such slight differences may be attributed to limited chemical variation between different veins, which is commonly observed in multiple expressions of a single hydrothermal episode, as pointed out in [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF].

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To investigate the trace-elements signature of quartz from each district we used a set of multivariate statistical analyses, namely, principal component analyses (PCA), hierarchical cluster Analyses (HCA), and permutational multivariate analysis of variance (PERMANOVA). Distance measures for HCA and PERMANOVA were calculated using the Bray Curtis similarity index [START_REF] Bray | An ordination of the upland forest communities of southern Wisconsin[END_REF], which well describes the statistical characteristics (e.g., presence/absence, broad variations) of trace-elements variables [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF]. All multivariate analyses were performed with the software Past 3.17 [START_REF] Hammer | PAST: Paleontological Statistics software package for education and data analysis[END_REF].

PCA allows reduction of the dimensionality of the variable space by reporting it with a series of new orthogonal (uncorrelated) variables, called principal components, which capture most of the total variability. Results correspond to a number of components recording the total variation of the initial data. The principal components represent a large proportion of the total information and, consequently, the last components have a negligible significance. In other terms, PCA reduces the dimensionality, i.e. the number of variables, of a multivariate data to few principal components; that is, newlycalculated variables that can be visualized graphically, with maximal preserved information. In this study, PCA highlights whether the incorporation of one element in quartz is constrained by the presence of any other given element; it also characterizes the main chemical correlations that typify each district. In this way, in order not to give too much weight to the correlations between elements marked in the most represented groups (districts with the highest population analyses), we operate simple random sampling without replacement for 17 analyses in each group (17 corresponds to the number of analyses in the smallest district, i.e., Nades) using the Mersenne Twister algorithm [START_REF] Matsumoto | Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator[END_REF]. To ensure the representativeness of the data with "only" 17 analyses per group, we repeated the protocol three times, to provide comparable results. This ensued in three PCAs, constructed with 17 (random sample analyses) x 5 (districts) total measures. These three PCAs are strongly similar; consequently, we decide to only depict the first multivariate analyses in the main body of the manuscript while PCA # 2 and PCA # 3 can be found in Supplementary Materials 2.

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Journal Pre-proof HCA builds the hierarchy from each individual chemical analysis by merging clusters as a function of distance measures among data. Any given district where quartz was sampled will not constrain the cluster building (all analyses were equally compared). Common HCA representation is done using dendrograms. Linkage criteria are defined by the unweighted pair group average (UPGMA) algorithm.

Clusters are joined based on the average distance between all members in the two groups (Sokal and Michener 1958). Stated differently, the more a cluster emerges at low dissimilarity, the more chemical analyses grouped in this cluster are similar. We used HCA to determine whether the several districts display homogeneous and specific signatures, and to test similarities between genetically unconstrained sample groups.

PERMANOVA quantifies the dissimilarity between any two objects (district, vein, etc.) included in the test, considering all trace-elements analyses of the two groups, using an F-ratio constructed by an inter-point geometric approach (scalar correlations based on the distance measure); the lower the Fvalue, the more the hypothesis of no statistical difference between groups is true [START_REF] Anderson | A new method for non-parametric multivariate analysis of variance[END_REF].

Significance of the variables was tested by 9999 permutation. In other words, PERMANOVA permits to underline the more similar groups among all. In our case, PERMANOVA was performed to test the resemblances between the districts and veins in the Sioule nappe in order to highlight potential genetic links.

Results

The eight chemical elements (Li, B, Na, Al, Ti, Ge, As, Sb) measured in all samples were clearly detected (i.e., well above the detection limit, d.l.) for the majority of the districts studied. The additional elements (Be, Rb and Sr) quantified in samples from Pouzol-Servant also resulted largely above d.l. (Table 1). For each district, a specific signature could be delineated, based on correlations among element pairs (Fig. 4). In the following section, we describe the characteristics of each district as revealed by common descriptive statistics and multivariate statistical analyses. Quartz traceelements data are available in Supplementary Materials 3. 1). Germanium is relatively low whereas Sb is present in higher concentrations than in other districts (2.6 ppm and 86.5 ppm, respectively). Quartz from Bournac has the lowest Ge:Ti ratio of all districts (0.18 on average) (Fig. 4). Quartz from Nades displays intermediate contents for all elements (with median values of 206, 18.1, 65, 2342, 6.6 and 49.4 ppm, respectively), except for Ge which is relatively high (median value of 65 ppm). As it can be visualized in Figure 4, the La Lucette quartz displays low concentration in Na (with a maximum value of 95 ppm) and in B (maximum value of 5.5 ppm). The contents of Li, Al and Ti are intermediate, (median of 116 ppm, 2598 ppm and 6 ppm respectively) with high variability compared to other districts. In the Li vs Al plot (Fig. 4), the La Lucette data stands out from the general correlation displayed by the other districts, with a higher concentration of Al. Concentrations of As and Ge are relatively high (7.7 and 6.1 ppm in median concentrations) and Sb contents are the highest of all districts (median value of 97 ppm). Quartz from the Pouzol-Servant district exhibits high concentrations of trace-elements, with the highest values being for Li, B, Na, Al and Ge (respective median value of 370, 51, 205, 4453, 8.7 ppm), low and unvarying Sb contents (median value of 14 ppm) and intermediate Ti contents (median equals to 4.8 ppm). The concentrations of Be, Rb and Sr are very high (median value of 1.9, 3.3 and 11.3 ppm, respectively) and correspond roughly to the highest concentrations measured for all types of quartz monitored in the Sioule nappe [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF]. Quartz originating from the Le Semnon deposit contains the lowest amounts of trace-elements (Li, B, Na, Al, Ti, Ge), with the exception of As and Sb, which occur at intermediate concentrations (respective median value of 0.7 and 29 ppm).

Although Sb may vary among districts by about an order of magnitude (e.g., median values of the La Lucette and Pouzol-Servant districts), its concentration remains high for all districts (Fig. 5). For the Le Semnon and Pouzol-Servant district, more than 75% of the analyses show Sb levels above 10 ppm.

For Nades, Sb concentrations are always above 10 ppm, and 8 out of 17 analyses are greater than 50

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Journal Pre-proof ppm. globally, the highest Sb concentration is recorded in quartz from the Bournac (23 of the 24 analyses are superior to 50 ppm) and the La Lucette districts (75% of the analyses are superior to 75 ppm). Such a high Sb content in quartz has only been rarely reported in the literature, and when so, it is related to Sb mineralization (Rusk et al. 2011;[START_REF] Pacák | Trace-element chemistry of barren and ore-bearing quartz of selected Au, Au-Ag and Sb-Au deposits from the Bohemian Massif[END_REF][START_REF] Feng | Ore-forming mechanism and physicochemical evolution of Gutaishan Au deposit, South China: Perspective from quartz geochemistry and fluid inclusions[END_REF][START_REF] Fu | Trace element chemistry of hydrothermal quartz and its genetic significance: A case study from the Xikuangshan and Woxi giant Sb deposits in southern China[END_REF][START_REF] Li | Genesis of gold and antimony deposits in the Youjiang metallogenic province, SW China: Evidence from in situ oxygen isotopic and trace element compositions of quartz[END_REF].

Multivariate analyses

Hierarchical cluster analysis allowed confirmation of homogeneous and specific quartz trace-elements signature for each district (Fig. 6). Overall HCA results for the entire dataset are remarkable, in that they allow a very clear distinction of each district from the others, with the exception of the Nades and La Lucette, which are only poorly distinguished from one another. The first group emerging entirely from the rest of the analyses is the Le Semnon district, probably because of its very low contents of Li, B, Na, Al and Ge. However, a small number of analyses (which represent extreme values in relation to the distribution of elements in the group) are scattered throughout the rest of the HCA. The second cluster which emerges to the rest of the data consists of quartz from Pouzol-Servant district, probably because of its enrichment in most trace elements. A third cluster emerges clearly and corresponds to the Bournac district which is characterized by high concentration of As, Ti and Sb. The rest of the HCA is less straightforward, including the few artefacts of the Le Semnon, and the districts of the La Lucette and Nades. The latter two groups share many similar elemental concentrations, including Na, Al, Ge and Ti, which can explain their ambiguous and similar position in HCA. Nevertheless, the amounts of B and As easily distinguish these two district (Fig. 4).

Principal component analyses were performed to highlight the principal correlations between the different elements and, at a second time, characterize the role of each district on these correlations (Fig. 7). Component 1 of PCA, corresponding to the horizontal axis, represents 45 % of the entire data and reveals that Al, Li, B, Na and Ge show good correlation while As, Sb and Ti have independent behavior. Using component 2, which represents, in addition to component 1, 75 % of the total analyses, we can distinguish two groups of elements already correlated by the first component: Al and

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Journal Pre-proof Li on one hand and Ge, Na and B on the other. Consequently, these two groups display strong correlations for three-quarters of the analyses. As, Sb and Ti are almost exclusively projected onto component-2's axis, meaning that their correlation is effective for ca. 30 % of the data and displays an independent behavior in regard to elements projected mostly in component 1 (i.e. Al,Li,Ge,B and Na). If we use the third component, less representative but not negligible (14 % of the total data), the couple As-Ti shows a negative correlation with Sb, becoming negatively correlated. To resume, Sb variation is independent of most of the elements for 45 % of the data, positively correlated to As and Ti for 30 % of them, while it is negatively correlated to the same couple of elements (As and Ti) for 14 % of the analyses. Consequently, we can conclude that the behavior of Sb in quartz is mainly independent of the presence of other elements. To resume, for most analyses, some trace elements show strong positive correlations, such as Al with Li, Na with B and Ge, and, to a lesser extent, As with Ti. Others vary mostly independently, as Sb, while no element shows a clearly negative correlation to another. This last result can be explained by the fact that trace-element incorporation in quartz is a global process, i.e., the more a quartz crystal incorporates a given element, the higher its capacity to incorporate others elements [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF]. Once this parameter is met, the nature of the fluid chemistry also comes into play, particularly in controlling the relative proportions of the different elements incorporated.

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Deciphering polyphase Sb events, the case of the Sioule Nappe

We have seen above how the five districts studied display different quartz trace-element signatures, including the two neighbour districts in the Sioule Nappe, Pouzol-Servant and Nades. [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF][START_REF] Monnier | Multiple Generations of Wolframite Mineralization in the Echassieres District (Massif Central, France)[END_REF][START_REF] Monnier | Contrasting fluid behavior during two styles of greisen alteration leading to distinct wolframite mineralizations: the Echassières district (Massif Central, France)[END_REF] have shown that greisen alteration of the Beauvoir granite developed a regionalscale fluid circulation that gave rise to distal mineralizations, among which is the Nades district. A particular aspect of the latter is that it was the only site showing Sb mineralization, although quartz from all manifestation of the greisen event(s) at Beauvoir showed Sb enrichment in their traceelements signatures. In the vicinity of the Nades district, occur the Sb-rich veins of Pouzol-Servant, emplaced within the homonymous granite. A question that needs answering at this stage, is whether the latter system is related, or not, to the Beauvoir greisen episode responsible for mineralization at Nades. The statistical data presented above indicate that each of the Sb mineralized sites in the five districts has a specific signature, suggesting that Pouzol-Servant and Nades, and by extrapolation the greisen system, also have distinct origins. This is consistent with the petrographic observations and fluid inclusion data presented in [START_REF] Monnier | Contrasting fluid behavior during two styles of greisen alteration leading to distinct wolframite mineralizations: the Echassières district (Massif Central, France)[END_REF] showing clear differences in textures (euhedral to subhedral quartz for the Nades veins vs an uncommonly abundant star quartz filling at Pouzol-Servant) and fluid inclusion homogenization temperatures between the two sites (ca. 170 °C and 210 °C for the median Th of the quartz of the Pouzol-Servant and Nades veins, respectively). To verify this hypothesis, we performed new multivariate analyses (HCA and PERMANOVA) using an enlarged signature for quartz from Pouzol-Servant, by adding analyses of Be, Rb and Sr. This allows a direct comparison with the quartz chemistry obtained from the proximal vein system at Beauvoir, and from the distal Nades veins (data from Nades, including Be, Rb and Sr, are from [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF]. The results of HCA analysis show only a weak similarity between the Pouzol-Servant group and the veins from the proximal systems and Nades (Fig. 8). The PERMANOVA analysis also points to the same conclusion (Fig. 9), with the quartz from Pouzol-Servant district showing significant higher F-values when compared to those of Nades and proximal veins. On the contrary, the quartz of the Nades veins and the group of the three proximal veins (Mazet, Colettes kaolin quarry and Suchot veins) show

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Journal Pre-proof moderate F-values between each of these groups, suggesting a common origin. In summary, it appears clear that two distinct hydrothermal events occurred in the Sioule Nappe, both at the origin of a distinct stibnite mineralization episode. This observation suggests a local fertility for occurrences of Sb mineralization.

Sb in quartz as vector to mineralization

As already revealed by chemical mapping at the crystal scale (Rusk et al., 2011), PCA shows that there are almost no correlations between Sb and other elements (Fig. 6). This means that Sb incorporation in quartz is mainly controlled by the content of Sb in the mineralizing fluid, and not by the possible presence of other elements, which is crucial for its use as a potential proxy for Sb deposits. Moreover, the different timing of quartz crystallization in the five Sb districts (mainly pre-, syn-or post-dating stibnite crystallization) did not constrain the incorporation of Sb in quartz in trace amounts, suggesting that the competition with stibnite does not represent an inhibiting factor.

Considering the high content of Sb (>10 ppm) in quartz from our five Sb districts and the low Sb concentration (not detected or ≤ 1ppm) reported in the literature for hydrothermal or magmatic quartz unrelated to Sb mineralization [START_REF] Götze | Trace element incorporation into quartz: a combined study by ICP-MS, electron spin resonance, cathodoluminescence, capillary ion analysis, and gas chromatography 1[END_REF]Rusk et al., 2011;[START_REF] Audétat | Origin of Ti-rich rims in quartz phenocrysts from the Upper Bandelier Tuff and the Tunnel Spring Tuff, southwestern USA[END_REF]Muller et al., 2018;[START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF][START_REF] Peterková | agmatic-hydrothermal transition of Mo-W-mineralized granitepegmatite-greisen system recorded by trace elements in quartz: rupka district, Eastern rušné hory/Erzgebirge[END_REF][START_REF] Hong | Cathodoluminescence features, trace elements, and oxygen isotopes of quartz in unidirectional solidification textures from the Snmineralized Heemskirk Granite, western Tasmania[END_REF][START_REF] Pacák | Trace-element chemistry of barren and ore-bearing quartz of selected Au, Au-Ag and Sb-Au deposits from the Bohemian Massif[END_REF][START_REF] Li | Genesis of gold and antimony deposits in the Youjiang metallogenic province, SW China: Evidence from in situ oxygen isotopic and trace element compositions of quartz[END_REF], it can be inferred that the presence of Sb in hydrothermal quartz is an excellent tracer of Sb mineralization. Moreover, our results agree with preliminary data from Rusk et al. ( 2011) which show high Sb contents in quartz from epithermal deposits in Comstock (Nevada) and Creede (Colorado), both mineralized in Sb (miargyrite and tetrahedrite for Comstock; [START_REF] Hudson | Epithermal alteration and mineralization in the Comstock district, Nevada[END_REF]; and tetrahedrite for Creede; Plumlee and Whitehouse-Veaux, 1994). In addition, Rusk et al. (2011) show that quartz from hydrothermal deposits of other metals (i.e., Carlin, MVT and porphyry deposits, all of which do not show Sb mineralization) contain very little Sb. A recent study of several Au (-Sb) mineralizations in the Bohemian massif came to the same conclusion, i.e., quartz with high trace contents of Sb was only found in Sb-mineralized veins [START_REF] Pacák | Trace-element chemistry of barren and ore-bearing quartz of selected Au, Au-Ag and Sb-Au deposits from the Bohemian Massif[END_REF]. In China, several recent studies have analyzed trace-element in quartz in a number of Sb districts. Quartz from the giant Woxi and Xikuangshan Sb deposits exhibits extraordinary Sb contents, always superior to 10 ppm and commonly higher than J o u r n a l P r e -p r o o f have median Sb concentrations ranging from ca. 10 to 70 ppm [START_REF] Feng | Ore-forming mechanism and physicochemical evolution of Gutaishan Au deposit, South China: Perspective from quartz geochemistry and fluid inclusions[END_REF]. Finally, in the Youjiang metallogenic province in China, which hosts both Sb and Au deposits, Sb was only detected in quartz from the Sb deposits [START_REF] Li | Genesis of gold and antimony deposits in the Youjiang metallogenic province, SW China: Evidence from in situ oxygen isotopic and trace element compositions of quartz[END_REF].

If we use the results from our study, we can define a quartz chemistry that could be considered a potential vector to Sb mineralization. Different generations of quartz from the fluid that led to stibnite mineralization in the Nades district (greisen quartz, proximal veins) have a similar signature marked by unusually high Sb contents (mainly > 1 ppm and < to 10 ppm), even if these quartz are not directly associated to Sb mineralization. In comparison, quartz completely disconnected from Sb mineralizing fluids mostly exhibits less than 1 ppm Sb. Conversely, quartz directly associated to Sb mineralization has Sb content commonly greater than 10 ppm. Therefore, we can conclude that quartz crystallizing from a Sb-enriched fluid can record this pre-enrichment, even if no Sb-bearing minerals occurs in the immediate proximity of this quartz (Fig. 10). This observation can be critical to target potential mineralization along the circulation pathway of a cooling Sb-bearing fluid.

To get an idea of the effectiveness of the protocol for Sb prospecting, at the rate of ca. five LA-ICP-MS analyses per vein tested, the Sb mineralization potential can be apprehended by monitoring 10 to 50 quartz veins per day, and much more in case of automatized LA-ICP-MS. This protocol can be used in strategical zone, such as potential epithermal or orogenic quartz veins, with bad outcrop conditions or poorly constrained samples from drill core. Even if preliminary observations may not be conclusive as to the presence of Sb mineralization, reproducible concentrations of more than 10 ppm Sb in quartz are likely to signify that a vein network has very elevated potential for Sb mineralization in the immediate proximity. Much in the same way as LA-ICP-MS analyses of pyrite and arsenopyrite can help targeting potential Au deposit (e.g., [START_REF] Morey | Bimodal distribution of gold in pyrite and arsenopyrite: examples from the Archean Boorara and Bardoc shear systems, Yilgarn Craton, Western Australia[END_REF][START_REF] Cook | Arsenopyrite-pyrite association in an orogenic gold ore: Tracing mineralization history from textures and trace elements[END_REF]Velasquez et al., 2014;Traore et al., 2016;[START_REF] Gourcerol | Results of LA-ICP-MS sulfide mapping from Algoma-type BIF gold systems with implications for the nature of mineralizing fluids, J o u r n a l P r e -p r o o f Journal Pre-proof metal sources, and deposit models[END_REF][START_REF] Gourcerol | Application of LA ICP-MS analysis of arsenopyrite to gold metallogeny of the Meguma Terrane, Nova Scotia, Canada[END_REF][START_REF] Feng | Ore-forming mechanism and physicochemical evolution of Gutaishan Au deposit, South China: Perspective from quartz geochemistry and fluid inclusions[END_REF], our approach may help tracking Sb mineralization and improve prospecting during mineral exploration.

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Conclusion

The results of a study on five Sb mineralized occurrences from five districts in the Variscan belt in Concentrations mainly inferior to ca. 1 ppm indicate that quartz is not associated with Sb mineralized systems. It is also possible to differentiate quartz from different mineralization event, and therefore coming from different fluids, using their chemicals signature. The latter may turn useful to distinguish different mineralizing episodes in a same locality. Thus, we have seen that the veins of Pouzol-Servant and Nades, which are located at proximity in the same tectonic nappe, originated from two distinct mineralizing fluids. With regards to the fluid source, new perspectives appear concerning the Sb metallogenic paradigm: in the case of the Echassières district, we highlight the genetical link between greisen alteration, formation of peripheral veins rich in W, Sn, Nb-Ta, and, further away, Sb occurrences. This trace-element signature, specific to each hydrothermal event, can also be used in the identification of the origin of mineralization. Finally, the trace chemistry of quartz could denote the specific signature of many deposits, thereby allowing tracing the original source of an ore sample of controversial origin, provided it contained a few small quartz crystals.

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Journal Pre-proof Zhang, Y., Chu, F., Li, Z., Dong, Y., Wang, H., Li, X., & Long, J. (2020). Gold enrichment in hydrothermal sulfides from the Okinawa Trough: An in situ LA-ICP-MS study. Ore Geology Reviews, 116, 103255. https://doi.org/10.1016/j.oregeorev.2019.103255 hydrothermal quartz in stibnite-free veins from the Sioule nappe (in orange; data from [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF]. The latter are divided into two groups: to the left, hydrothermal quartz completely unrelated to Sb mineralization, originating from orthomagmatic fluids (i.e., the La Bosse stockwork), alkali alteration (granite apex veins), and supergene alteration (late quartz), while, to the right, quartz from the Beauvoir greisen, i.e., a fluid that transported Sb but did not produce Sb mineralization in the proximity of the Beauvoir granite (pervasive and greisen vein quartz in the granite, and quartz from the proximal veins). More distally, the greisen fluid led to the formation of the Nades veins. (Names in brackets refer to the terminology in [START_REF] Monnier | Quartz trace-element composition by LA-ICP-MS as proxy for granite differentiation, hydrothermal episodes, and related mineralization: The Beauvoir Granite (Echassières district), France[END_REF].

Captions

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Journal Pre-proof

  propose a common hydrothermal event explaining both Sb mineralizations in Pouzol-Servant and Nades districts. Recently, Monnier et al. (2020) described the main petrographic J o u r n a l P r e -p r o o f Journal Pre-proof features and provided fluid inclusion data for these two districts. Minimal temperatures for Sb mineralization were estimated at ca. 175 °C and 215 °C for Pouzol-Servant and Nades district, respectively.

  Quartz from Bournac is the most enriched in Ti and As (median values are 14.1 ppm and 19.3 ppm, respectively) and, compared to other districts, has intermediateLi, B, Na, Al values (respective median values: 295, 14.1, 31 and 3497 ppm; cf. Table

  Regarding the position of the five Sb districts in the PCA diagrams, we can observe that the Pouzol-Servant district is characterized mostly by B, Na, Ge, Al and Li. The Bournac district is constrained by the behaviors of Sb, As and Ti. The intermediate content of most of the elements in the Nades quartz puts this group close to the center of the diagram. Finally, the projection of the La Lucette and Le Semnon districts on opposite sides of Al, Li, B and Na likely results from their low content of these elements and their differences in Sb contents, leading to different vertical positions of these two groups in the diagram.

France

  , indicate that the trace-elements composition of vein quartz has an excellent potential as a proxy for Sb mineralization. Concentrations of Sb in quartz predominantly superior to 10 ppm indicate occurrence of Sb mineralization within the considered vein or in close proximity. Intermediate concentrations (between ca. 1 and 10 ppm) would indicate that the magmatic or hydrothermal system is pre-enriched in Sb and could lead to the formation of possible associated mineralization.

Figure 1 :

 1 Figure 1: A sketch map of France locating the Sb districts in Variscan terranes (in gray) that were the

Figure 2 :

 2 Figure 2: Geological map of the central part of the Sioule nappe, locating the mineralized occurrences

Figure 3 :

 3 Figure 3: Photographs and Photomicrographs illustrating textural relationships between stibnite and

Figure 4 :

 4 Figure 4: Binary diagrams depicting quartz trace-elements data for each of the five districts.

Figure 5 :

 5 Figure 5: Box plot diagrams illustrating variations in Sb content for quartz from the five districts.

Figure 6 :

 6 Figure 6: Hierarchical cluster analysis (HCA) dendrogram illustrating the similarities between each

Figure 7 :

 7 Figure 7: A diagram depicting the results of principal component analysis (PCA) of trace-elements

Figure 8 :

 8 Figure 8: HCA dendrogram comparing the trace-elements data for quartz from the proximal veins and

Figure 9 :

 9 Figure 9: F-value statistics of a permutational analysis of variance (PERMANOVA) concerning the

Figure 10 :

 10 Figure 10: Boxplot diagram illustrating the concentration of Sb in quartz as a function of their genetic

Table 1 :

 1 Descriptive statistics of quartz trace-elements concentrations from the different Sb districts (n.a.: not analyzed; d.l.: detection limit).

			The Lucette	The Semnon		Bournac			Nades		Pouzol-Servant
				n = 93			n = 46			n = 24			n = 17			n = 51
	pp	d.l	Mi	Me	Ma	Mi	Me	Ma	Mi	Me	Ma	Mi	Me	Ma	Mi	Me	Ma
	m	.	n.	d.	x.	n.	d.	x.	n.	d.	x.	n.	d.	x.	n.	d.	x.
	Li 0.5 31 116 244	3	8	192 16	295 368 87 206 325 24	370 453
									6						8
	Be 0.5 n.a. n.a. n.a. n.a	n.a. n.a. n.a. n.a. n.a. n.a	n.a. n.a. 0.3	1.9 2.9
						.						.			7		3
	B 0.5 1.1 3.6 5.5 0.7 1.2	4	9.6 14.	22.	10.	18.	25.	31.	51.	61
										1	1	6	1	2	9	0
	Na 5	5	13	95	5	5	45	6	31 215 5	65 163 19 205 518
	Al 10 10	259	442	63 192 579	21	349	416	95	234	382	32	445	509
			29	8	7			7	02	7	6	6	2	9	51	3	6
	Ti	1 2.3 6.0 9.6 1.4 3.3 10.	7.7 14.	24.	4.5 6.6 13.	3.5 4.8 6.7
								9		1	5			2
	Ge 0.5 4.3 7.7 15.	1.5 2.2 11.	1.4 2.6 8.6 4.9	8	16.	5.3 8.7 11.
					1			8						4

Acknowledgements

We would like to acknowledge the BRGM (French Geological Survey), the CNRS (French National Center for Scientifc Research) as well as the University of Toulouse for providing financial support to this study. we also wish to thank M. Munoz for providing us with some sample and for her valuable pearls of wisdom on Sb mineralizations in the study area. The associate editor and reviewers and are warmly thanked for helping in obtaining a better manuscript.

Declaration of interests

☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

J o u r n a l P r e -p r o o f


Dr Sophie Vichier-Guerre

Dr Therese C Ku

Dr Sylvie Pochet
email: sylvie.pochet@pasteur.fr

Katherine L Seley-Radtke

Prof K Seley-Radtke

An expedient synthesis of flexible nucleosides through enzymatic glycosylation of proximal and distal fleximer bases

Keywords: biocatalysis, fleximer, nucleoside, nucleoside 2'deoxyribosyltransferase, purine nucleoside phosphorylase

The structurally unique "fleximer" nucleosides were originally designed to investigate how flexibility in the nucleobase could potentially affect receptor-ligand recognition and function.

Recently they have shown low to sub-micromolar levels of activity against a number of viruses including coronaviruses, filoviruses and flaviviruses. The synthesis of distal fleximers in particular, has thus far been quite tedious and low yielding. As a potential solution to this issue, a series of proximal fleximer bases (flex-bases) were successfully coupled to both ribose and 2'-deoxyribose sugars using the N-deoxyribosyltransferase II of Lactobacillus leichmannii (LlNDT) and E. coli purine nucleoside phosphorylase (PNP). To explore the range of this facile approach, transglycosylation experiments using a thieno-expanded tricyclic heterocyclic base, as well as several distal and proximal flex-bases were performed to determine whether the corresponding fleximer nucleosides could be obtained in this fashion, thus potentially significantly shortening the route to these biologically significant compounds. The results of those studies are reported herein.

The synthesis of nucleoside analogues has been classically achieved through various chemical methodologies and wellknown coupling reactions. [START_REF] Townsend | Nucleic acid chemistry: improved and new synthetic procedures, methods, and techniques[END_REF][START_REF] Merino | Chemical synthesis of nucleoside analogues[END_REF][START_REF] Vorbrüggen | Handbook of Nucleoside Synthesis[END_REF] However, chemical synthesis typically involves difficult and time-consuming multistep processes. Moreover, suitable protection of various functional groups is typically required on the nucleos(t)ide sugars and/or on the heterocyclic bases, and subsequent deprotection steps often result in low overall yields. Another significant problem often encountered is the stereospecific control of configuration at the anomeric center (α versus β). [START_REF] Merino | Chemical synthesis of nucleoside analogues[END_REF][START_REF] Vorbrüggen | Handbook of Nucleoside Synthesis[END_REF][START_REF] Downey | [END_REF] In contrast, enzymatic syntheses of nucleoside analogues do not typically require protecting groups and are highly stereospecific. [START_REF] Fernández-Lucas | Enzymatic and Chemical Synthesis of Nucleic Acid Derivatives[END_REF] Nucleoside phosphorylases (NPs) and N-deoxyribosyltransferases (NDTs) are the predominant classes of enzymes used in the synthesis of nucleosides by mediating the transglycosylation reaction of a nucleoside sugar to a free heterocyclic base. [START_REF] Mikhailopulo | [END_REF][7][START_REF] Condezo | Biocatalysis in the Pharmaceutical and Biotechnology Industries[END_REF][START_REF] Pérez | [END_REF][START_REF] Fresco-Taboada | [END_REF] NPs (EC 2.4.2.1) catalyse the reversible phosphorolysis of riboor 2′-deoxyribonucleosides to generate a free nucleobase and ribose-or 2-deoxyribose-1-phosphate in the presence of inorganic ortho-phosphate. Addition of a second nucleobase to the reaction mixture can promote the formation of a new nucleoside with the equilibrium in favor of nucleoside formation. [11- 12] NPs have been reported to accept modified bases as well as unnatural glycosyl donors. [START_REF] Mikhailopulo | [END_REF][7][START_REF] Condezo | Biocatalysis in the Pharmaceutical and Biotechnology Industries[END_REF][START_REF] Pérez | [END_REF][13][14][15] NDTs (EC 2.4.2.6) catalyse the transfer of 2-deoxyribose between purine and pyrimidine bases. [16] Two NDT-types have been defined based on their substrate specificity: NDT-I (also named PTD) specific for purines [17] and NDT-II that accepts either purine or pyrimidine, with a strong preference for 2′-deoxyribosylpyrimidine as the donor substrate. [18] Despite its relatively low acceptance for modified sugar moieties, NDT-II has served to synthesize some nucleoside analogues of biological interest. [19,20] Interestingly, NDTs tolerate a wide range of modified nucleobases from azole derivatives [21,22] to expanded-size purines [23][24][25] with an increased regioselectivity as compared to PNP (i.e. N9 versus N7 for purines). [START_REF] Mikhailopulo | [END_REF] The fleximers were originally designed to investigate how flexibility in the nucleobase could affect receptor-ligand recognition and function. [26][27][28][29][30][31] These interesting molecules have demonstrated several advantages over their analogous natural rigid purine nucleosides. For instance, the distal guanosine fleximer (Figure 1) was found to be an inhibitor of S-adenosyl-Lhomocysteine hydrolase (SAHase), an adenosine-metabolizing enzyme. [26] By rotation of the hemiaminal bond to place the heterocyclic moieties into a syn-like conformation rather than the thermodynamically favored anti-conformation, the flexguanosine's base was able to reposition the amino group to mimic the adenosine nucleobase. [26,32] Furthermore, the flex-guanosine triphosphate (Flex-GTP) was shown to be a superior substrate of human GDP-L-fucose pyrophosphorylase compared to the natural substrate guanosine triphosphate (GTP), [33] likely due to the fleximer's ability to interact with amino acids in the active site not accessible by the natural substrate. [34] This also allowed Flex-GTP to retain all activity when essential catalytic residues needed for GTP binding were mutated. [33,34] While the biological results for these compounds have been ground-breaking in some cases, their syntheses are nontrivial. To

Distal fleximer

Proximal fleximer date, the only approach to realize the adenosine, guanosine and inosine distal fleximer nucleosides requires the construction of a tricyclic expanded purine nucleoside, which involves more than 7 steps to realize the base itself, with an additional 5-7 steps after the coupling reaction to convert the tricyclic nucleoside to the fleximer. [28][29][START_REF] Ku | Enzymatic and Chemical Synthesis of Nucleic Acid Derivatives[END_REF][START_REF] Chen | [END_REF][37][38][39][40][41] Attempts to synthesize the tricyclic bases followed by coupling of the base to a ribose resulted only in the undesired N7 isomer, likely due to the sterics caused by the arched shape. [29,41] Previously, a series of imidazole nucleosides were successfully synthesized via transglycosylation methodologies. [42,43] Of those, four products resembled the Seley-Radtke proximal fleximers (I-IV, Scheme). The flex-bases were synthesized using microwaveassisted Suzuki-Miyaura cross-coupling of 4( 5)-iodoimidazole and the boronic acid pyrimidine partners. [44] Subsequent transglycosylation using E. coli PNP for ribonucleosides and LlNDT for 2'-deoxyribonucleosides produced the corresponding proximal fleximers as major products.

Scheme 1. Enzymatic transglycosylation of proximal fleximers. [42,43] As a few proximal fleximer nucleosides have been realized via enzymatic transglycosylation, the expectation was that the same methodology could produce the desired distal and tricyclic nucleosides from their respective bases. Thus, a series of tricyclic [41] and flex-bases [45] (shown in Figure 2) were synthesized as previously reported and their transglycosylation products were studied.

Enzymatic glycosylation mediated by LlNDT

First, we tested the capability of NDT from LINDT to use these five unnatural nucleobases as substrate. Typically, the flex-base (1 µmol) and thymidine used as the 2'-deoxyribose donor (4 µmol) were incubated at 37°C in the presence of LlNDT (different amounts of enzyme) in a 10 mM citrate buffer (0.1 mL) at the optimum pH of 6.5. Due to the low water solubility of flex-bases, the transferase reactions were carried out in the presence of 5% v/v DMSO in the media. [42] The rate of glycosylation (% conversion) was monitored by analytical reverse phase HPLC as a function of time and enzyme concentration (Table 1). The incubation temperature (37°C in place of 50°C) was selected in an attempt to optimize the enzyme activity and further isolate all the possible glycosylated products. Indeed, we previously showed that it is achievable to manage the regioselectivity of the transferase reaction (N1 versus N3) of some 4-substituted imidazole derivatives. [43] All the glycosylation products were purified by reverse phase HPLC and their chemical structures, particularly the glycosylation site, were confirmed by NMR analysis.

As reported previously, [42] the NDT-catalyzed transfer reaction between 1 and thymidine led to the simultaneous formation of N1 and N3 glycosylated products (1a and 1b, Scheme 2) in a 1:1 ratio after 3 h at 37°C using NDT at 1.25 µL/µmol acceptor (Table 1, entry 1). Nucleoside 1b (i.e. C5 arylimidazole derivative) was progressively converted into the thermodynamically more stable N1 isomer 1a (i.e. C4 aryl-imidazole nucleoside) (see Supporting Information, Figure S1A). By increasing enzyme concentration (from 1.25 to 10.0 µL/µmol acceptor), the conversion was nearly complete, reaching a plateau at 90%, and nucleoside 1a was formed as the sole product (Table 1, entries 7 & 8). A catalytic mechanism for LlNDT-catalyzed glycosylation involving the formation of an oxocarbonium intermediate has recently been proposed. [46] Scheme 2. NDT-catalyzed transglycosylation of flex-bases 1-4.

A similar trend was observed in the case of flex-base 2 with the formation of both N1 and N3 glycosylated products (2a and 2b, respectively) in a ratio depending on incubation time or enzyme concentration (Figure S1B). Compared to flex-base 1, flex-base 2 appeared to be a better substrate since lower enzyme concentration was needed to achieve complete conversion (Table 1, entries 11-14). These results confirmed the easy accessibility of proximal 2'-deoxy-fleximers using trans-glycosylation reactions.

In the case of the flex-bases 3 and 4, HPLC monitoring showed that the glycosylation reactions were more effective than in the case of 1 (Figure S2 and Table 1, entries 15-21). Only one isomer, the N-1 glycosylated product 3a or 4a was formed, even if the enzyme concentration was reduced 10-fold (entries 19-21). However, the target distal 2'-deoxynucleosides (glycosylation at the N3 position of the imidazole ring) were not detected under the conditions tested. Finally, we tested the tricyclic nucleobase 5 (Scheme 3) which contains, in addition to the imidazole ring N1 and N3, a nitrogen on the pyrimidine ring susceptible to be glycosylated, as previously observed in the transglycosylation of 5-(imidazol-4yl)pyrimidine-2,4-(1H,3H)-dione. [42] Heterocycle 5 was rapidly converted into glycosylated products under the conditions used (Table 1, entries 22-25), confirming that hindered nucleobases can serve as acceptors for NDT. HPLC monitoring of the reaction showed the formation of two main products 5a and 5b, and a third minor one 5c (Figure 3). Structures of 5a (glycosylation at imidazole N1, noted N1'-Im in Scheme 3), 5b (glycosylation at both imidazole N1 and pyrimidine N1, noted N3' Im and N1'), and 5c (glycosylation at imidazole N3, noted N3' Im) were assigned based on NMR spectra (Figure 4). By reducing incubation time and enzyme concentration, it was possible to control further conversion of 5a into 5b, and thus isolate 5a as the main product (Table 1, entry 24). Interestingly, this enzymatic procedure is an attractive route to 5a over the previously published ones, [29,37] based on the coupling of 4,5diiodoimidazole to protected 2-deoxysugar chloride, followed by the formation of the tricyclic inosine derivative and sugar deprotection. [a] Reaction conditions: 1 μmol acceptor in 5% v/v DMSO, 4 μmol thymidine in 10 mM citrate buffer pH 6.5 (0.1 mL) in the presence of LlNDT (μL as indicated) at 37 °C. [b] Percent Conversion was determined by reverse-phase HPLC analysis of an aliquot of the incubation mixture monitored at 254 nm. [c] Product from glycosylation at both imidazole and pyrimidine nitrogens of 5.

Enzymatic glycosylation mediated by PNP.

We also tested the ability of purine nucleoside phosphorylase (PNP) to use flex-bases 1, 3 and the thieno-expanded tricyclic base 5 for ribonucleoside synthesis (Scheme 4). Reaction mixtures (0.1 mL) contained the flex-base (10 mM), adenosine (20 mM) as a donor and bacterial PNP (0.2 U/μmol) in 10 mM phosphate buffer (pH 7.4). Incubation was initially performed at 50°C. Conversion of 3 into the corresponding ribonucleoside 3c is slow and reached a plateau at 75% after 16 h at 50°C, as previously observed for PNP-catalyzed ribosylation of 1. [42] In the case of flex-base 5, conversion is much more effective (80% after 2 h at 50°C) and two mono-glycosylated products (5d and 5e) were formed, the target compound 5d being the major one (Scheme 4). When incubation was performed at 37°C, the formation of 5e increased as compared to 50°C, allowing further characterization (Figure S3). The glycosylation sites of 5d and 5e at the N1' or N3' position of imidazole ring, respectively, were assigned based on NMR analysis (see Supporting Information). It should be noted that chemical glycosylation of 5 using classical procedures yielded nucleoside 5e (glycosylation at N3' position of imidazole ring) as the sole product. [29] Scheme 4. PNP mediated transglycosylation of flex bases 1, 3 and 5

In conclusion, the four flex-bases 1-4 and the tricyclic nucleobase 5 were found to be substrates for the nucleoside 2'deoxyribosyltransferase from L. leichmannii (LlNDT) and bacterial purine nucleoside phosphorylase (PNP). Experimental conditions were designed to manage the regioselectivity in the NDTcatalyzed transfer reaction (i.e. at N1 versus N3 position of the imidazole ring) and also the bis-glycosylation side reaction (at the pyrimidine nitrogen). Thus, proximal fleximers in both deoxy and ribose series are easily obtained from flex-bases 1 and 2 by transglycosylation reaction catalyzed by NDT and PNPase, respectively. On the other hand, distal fleximers are not accessible from the corresponding flex-bases 3 and 4, since glycosylation catalyzed by LlNDT or PNP occurs at the N1 position of the imidazole ring. Importantly, enzymatic glycosylation of the tricyclic base 5 into the deoxy-and ribonucleoside 5a and 5d should provide a simple and efficient chemoenzymatic access to the target distal deoxy-and riboinosine fleximers, thereby overcoming the tedious and lowyielding approach that has been used to date. The chemoenzymatic approach compares very well with the classical chemical routes to distal and proximal fleximer nucleosides, as only one anomer and one regioisomer were obtained. This strategy will now be useful for the preparation of other distal fleximers, i.e. A and G analogues.

Entry for the Table of Contents

Thieno-expanded tricyclic base and several distal and proximal flex-bases were converted into their corresponding fleximer nucleosides by enzymatic transglycosylation reactions catalyzed by N-deoxyribosyltransferase from L. leichmannii (LlNDT) or E. coli purine nucleoside phosphorylase (PNP).

Figure 1 .

 1 Figure 1. Seley-Radtke's guanosine fleximers.

Figure 2 .

 2 Figure 2. Flex-and tricyclic bases used for transglycosylation.

Scheme 3 .

 3 Scheme 3. NDT-catalyzed transglycosylation of flex-base 5.

Figure 3 .

 3 Figure 3. HPLC analysis of the transglycosylation reaction of 5 in the presence of thymidine (4 equiv) and NDT (0.63 µL/µmol acceptor) at t = 0, 3 h and 12 h at 37°C. HPLC conditions: 0-20% linear gradient of acetonitrile in 10 mM TEAA buffer pH 6.0 over 20 min at a flow rate of 1mL min -1 . Detection at 254 nm.

Figure 4 .

 4 Figure 4. Expansion of the 1 H-[ 13 C]HMBC NMR spectra of compounds 5a (panel A) and 5c (panel B).

  protected by copyright. All rights reserved.

  protected by copyright. All rights reserved.

  protected by copyright. All rights reserved.

Table 1 .

 1 Transglycosylation reactions of flex-bases (1-5) catalyzed by LlNDT[a] .

	Product (%) b

Acknowledgements

The authors from IP would like to thank Frédéric Bonhomme (UMR3523) for assisting with HRMS analysis, Dr Bruno Vitorge and Dr Inaki Guijarro (BioNMR plateform) for providing access to Avance 600 and Avance Neo 800 spectrometers. This work was funded by the Institut Pasteur and the Centre National de la Recherche Scientifique (CNRS) (S.V.-G. and S.P.) and the National Institutes of Health/NIAID R21 AI118470-01 (KSR) and T32 GM066706 (KSR and TK).


Didier Clamond
email: didier.clamond@univ-cotedazur.fr.

REMARKS ON DISPERSION-IMPROVED SHALLOW WATER EQUATIONS WITH UNEVEN BOTTOM

Keywords: AMS Classification: 35Q35, 35Q86, 35S99, 76B03, 76B07, 76B15 Shallow water, varying bottom, improved dispersion, well-posedness

It is shown that asymptotically consistent modifications of (Boussinesq-like) shallow water approximations, in order to improve their dispersive properties, can fail for uneven bottoms (i.e., the dispersion is actually not improved). It is also shown that these modifications can lead to ill-posed equations when the water depth is not constant. These drawbacks are illustrated with the (fully nonlinear, weakly dispersive) Serre equations. We also derive asymptotically consistent, well-posed, modified Serre equations with improved dispersive properties for constant slopes of the bottom.

Introduction

Shallow water models are weakly dispersive approximations of the Euler equations with a free surface. For many applications, it is necessary to improve their dispersive and shoaling properties, for instance modifying the equations in an asymptotically consistent way. This possibility is investigated here for the Serre [START_REF] Serre | Contribution to the study of permanent and non-permanent flows in channels[END_REF] equations with uneven bottom, this special case being of general interest and it is sufficient to illustrate the problem for any other shallow water approximation (e.g., Boussinesq-like equations [START_REF] Kirby | Boussinesq models and their application to coastal processes across a wide range of scales[END_REF][START_REF] Madsen | A review of Boussinesq-type equations for surface gravity waves[END_REF]).

We then consider two-dimensional surface gravity waves in irrotational motion propagating at the surface of perfect incompressible fluid. The free surface at y = η(x, t) and the bottom at y = -d(x, t) are both impermeable, with x the horizontal variable, y the vertical upper one and t the time; the still water level is at y = 0. Assuming long waves in shallow water (i.e., ∂ x and ∂ t are "small" operators) without restriction on their amplitudes (i.e., fully nonlinear), Serre [START_REF] Serre | Contribution to the study of permanent and non-permanent flows in channels[END_REF] derived a set of approximate equations for constant depth. In presence of a varying bottom, these equations can be written [START_REF] Seabra-Santos | Contribution à l'étude des ondes de gravité bidimensionnelles en eau peu profonde[END_REF]

h t + [ h ū ] x = 0, (1)
ūt + ū ūx + g η x = 1 2 (γ + γ)

d x -1 3 h -1 h 2 γ + 1 2 h 2 γ x , (2)
where h = η + d is the total water depth, ū is the depth-averaged horizontal velocity, g > 0 is the acceleration due to gravity, γ and γ being the (approximate) vertical accelerations at, respectively, the bottom and the free surface, i.e.,

γ = -d tt -2 ū d xt -ū2 d xx -( ūt + ū ūx ) d x , γ = γ + h ū 2 x -ūxt -ū ūxx . (3)
The equation [START_REF] Antunes | Boussinesq and Serre type models with improved linear dispersion characteristics: Applications[END_REF] for the mass conservation is exact, while the momentum equation ( 2) is an approximation: its left-hand side involves first-order terms (in shallowness parameter, i.e., in derivatives) and its right-hand side involves third-order terms. This approximation yields a fully nonlinear but only weakly dispersive model of water waves [START_REF] Wu | A unified theory for modeling water waves[END_REF]. It is then desirable to improve these models for a better description of dispersive effects without increasing the mathematical complexity, that is without introducing higher-order derivatives because they are computationally demanding and prone to trigger numerical instabilities. Thus, in order to improve the dispersive properties of shallow water models -i.e., to extend their validity to deeper water -some asymptotically consistent modifications of the momentum equation have been proposed. These modifications involve free parameters that can be chosen to tune the (weakly dispersive) linear dispersion relation such that it better matches the (fully dispersive) exact relation [START_REF] Antunes | Boussinesq and Serre type models with improved linear dispersion characteristics: Applications[END_REF][START_REF] Clamond | Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics[END_REF][START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics[END_REF][START_REF] Nwogu | Alternative form of Boussinesq equations for nearshore wave propagation[END_REF][START_REF] Witting | A unified model for the evolution of nonlinear water waves[END_REF]. For the Serre equations, such modified equations can be obtained replacing the momentum equation ( 2) by [START_REF] Cienfuegos | A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-like equations. Part I: Model development and analysis[END_REF][START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics[END_REF] 1

-αd 2 ∂ 2 x (ū t + ūū x + gη x ) = 1 2 (γ + γ) d x -1 3 h -1 ∂ x h 2 γ + 1 2 h 2 γ , ( 4
)
where α is a free parameter at our disposal. One can easily check that α = 0 yields [START_REF] Cienfuegos | A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-like equations. Part I: Model development and analysis[END_REF] and that equation ( 4) with α = 0 is asymptotically consistent with (2). The parameter α is generally chosen considering a travelling wave of permanent form in constant depth. In such a case, the linear dispersion relation (relating the angular frequency ω and the wavenumber k) of the linearised Serre equations is a (2, 2)-Padé approximation (in the wavenumber) of the exact relation, while when α = 0 one gets the (4, 2)-Padé approximation

d g -1 ω 2 = (kd) 2 + α (kd) 4 1 + ( 1 3 + α) (kd) 2 -1 . (5)
For all α, the linear dispersion relation ( 5) matches the exact one ω 2 = gk tanh(kd) at least up to the fourth-order in its Maclaurin expansion in terms of the wavenumber, but for α = 1/15 the matching is up to the sixth-order. Thus, α = 1/15 is the best choice to improve the dispersive properties (according to the criterion considered here). However, this improvement occurs only for horizontal bottoms. Indeed, in presence of a bottom slope (d x = 0), the dispersive properties of the modified equations are of fourth-order only for all α. This means that the modification (4) cannot improve the model in presence of a varying seabed, as one can check (see below). (In practice, the dispersive properties are nevertheless somewhat improved for very mild slopes (|d x | ≪ 1), but it is of practical interest to remove this restriction.) Worse, we show below that (4) generally yields ill-posed models for varying bottoms.

In this note, we propose an alternative modification of the Serre (and others) momentum equation such that the dispersive effect are improved for finite constant slopes. To this aim, we first reduce, in section 2, the exact (i.e., fully dispersive) linear equations to a single pseudodifferential equation for the free surface only. In section 3, we consistently modify the Serre equations with an unknown operator that is determined by identification with the shallow water approximation of the exact linear equation. These modified Serre equations for varying depth thus provide an improvement, at least for mild curvature of the seabed (i.e., when the gradient of the bottom slope is very small). In section 4, we show that these modified Serre equations are linearly well-posed for constant slopes, while the modified equations using (4) with α = 1/15 are ill-posed for varying bottoms.

Linear waves on sloping beach

The (fully dispersive) linearised equations for an irrotational motion [START_REF] Stoker | Water Waves[END_REF] are

φ xx + φ yy = 0 for -d y 0, (6)
φ y + d t + d x φ x = 0 at y = -d, (7)
φ t + g η = 0 at y = 0, (8)
φ y -η t = 0 at y = 0, ( 9
)
where φ is a velocity potential. Here, we consider constant slopes, i.e., d(x, t) = d 0 + sx where d 0 0 is the depth at x = 0 and s def = tan(θ 0 ) 0 is the constant slope of the seabed (-θ 0 the seabed angle of inclination). Using the conformal mapping z = x + iy → Z = X + iY where

Z def = d 0 θ 0 log z d 0 + 1 s , (10)
the wedge domain {sx

-d 0 ; -d 0 -sx y 0} is mapped onto the strip -d 0 Y 0 with x = -d 0 /s → X = -∞ and x = +∞ → X = +∞.
In the mapped variables, after elimination of η(x, t) = -g -1 φ t (x, 0, t), the equations ( 6)-( 9) become

Φ XX + Φ Y Y = 0 for -d 0 Y 0, (11)
Φ Y = 0 at Y = -d 0 , (12)
Φ Y + g -1 θ 0 exp(θ 0 X/d 0 ) Φ tt = 0 at Y = 0, (13)
where Φ(X, Y, t)

def = φ(x(X, Y ), y(X, Y ), t)
is the velocity potential in the mapped plane. The general solution of the Laplace equation ( 11) satisfying the lower boundary condition ( 12) is (see [START_REF] Clamond | Steady finite-amplitude waves on a horizontal seabed of arbitrary depth[END_REF][START_REF] Clamond | Cnoidal-type surface waves in deep water[END_REF] for details) Φ(X, Y, t) = Re Φ(Z + id 0 , t) = 1 2 Φ(Z + id 0 , t)

+ 1 2 Φ(Z * -id 0 , t) = cos((Y + d 0 ) ∂ X ) Φ(X, t) = ∞ n=0 (-1) n (2n)! (Y + d 0 ) 2n ∂ 2n Φ(X, t) ∂X 2n , (14)
where a star denotes the complex conjugate and Φ(X, t) def = Φ(X, -d 0 , t) is the velocity potential at the bottom. Substituting [START_REF] Roseau | Short waves parallel to the shore over a sloping beach[END_REF] into the upper boundary condition [START_REF] Porter | An extended linear shallow-water equation[END_REF], one gets the complex difference-differential equation

ΦX (X +id 0 , t) -ΦX (X -id 0 , t) + θ 0 ig exp θ 0 X d 0 Φtt (X +id 0 , t) + Φtt (X -id 0 , t) = 0. ( 15
)
Exploiting the relation Φ(X±id 0 , t) = exp(±i d 0 ∂ X ) Φ(X, t) (Taylor expansion around d 0 = 0) and using the velocity potential at the free surface Φ(X, t)

def = Φ(X, 0, t) = cos(d 0 ∂ X ) Φ(X, t), the difference-differential (15) equation is rewritten as the real pseudo-differential equation tan(d 0 ∂ X ) ΦX (X, t) -g -1 θ 0 exp(θ 0 X/d 0 ) Φtt (X, t) = 0. ( 16
)
Note that H(X, t)

def = η(x(X, 0), t) = -g -1
Φt (X, t) also satisfies the equation ( 16) because the latter is linear and its coefficients are independent of t.

With the change of independent variable ξ def = d 0 exp(θ 0 X/d 0 ) = x + d 0 /s (hence dξ/dX = θ 0 ξ/d 0 ) and since Φ(X, t) = φ(ξ -d 0 /s, 0, t) is the velocity potential at the free surface, the equation ( 16) becomes

∂ ∂ξ tan θ 0 ξ ∂ ∂ξ - 1 g ∂ 2 ∂t 2 φ(ξ -d 0 /s, 0, t) = 0, (17)
where

tan θ 0 ξ ∂ ∂ξ def = θ 0 ξ ∂ dξ + θ 3 0 3 ξ ∂ ∂ξ ξ ∂ ∂ξ ξ ∂ ∂ξ + 2 θ 5 0 15 ξ ∂ ∂ξ ξ ∂ ∂ξ ξ ∂ ∂ξ ξ ∂ ∂ξ ξ ∂ ∂ξ + • • • . (18)
Finally, the equation ( 17) differentiated with respect of t is rewritten -since

∂ ξ = ∂ x , d = sξ = d 0 + sx, s = tan(θ 0 ) = d x and φ t (x, 0, t) = -gη(x, t) -in terms of the original variables ∂ 2 ∂t 2 -g ∂ ∂x tan arctan(d x ) d x d ∂ ∂x η(x, t) = 0. ( 19
)
It should be emphasised that the equation ( 19) is exact for (fully dispersive) linear waves provided that d xx = d t = 0, i.e., for flat static bottoms of arbitrary inclination, in particular for a constant depth. For more general seabeds, ( 19) provides a reasonable (linear) approximation if the bottom varies very slowly in time and if its curvature is small, such assumptions being made to derive shallow water approximations. The relation [START_REF] Wu | A unified theory for modeling water waves[END_REF] suggests to introduce an "apparent" or "effective" water depth D as

D def = d -1 x arctan(d x ) d d. (20)
This shows that a bottom slope creates a slowdown of the wave compared to a horizontal bottom of the same depth, an effect conjectured in [START_REF] Dutykh | Shallow water equations for large bathymetry variations[END_REF] from a non-dispersive shallow water model. In shallow water ∂ x is "small" and, since d xx = 0, the operator tan(D ∂ x ) can be expanded up to the fifth-order as

tan(D ∂ x ) def = D ∂ x + 1 3 D ∂ x D ∂ x D ∂ x + 2 15 D ∂ x D ∂ x D ∂ x D ∂ x D ∂ x + • • • ≈ d ∂ x + 1 3 ∂ x d 3 ∂ 2 x + 2 15 d 5 ∂ 5 x + 4 3 d x d 4 ∂ 4 x + 3 d 2 x d 3 ∂ 3 x + d 3 x d 2 ∂ 2 x .
(21) This expansion should be compared with the (approximate) shallow water equations in order to derive suitable improvements. We note in passing that some approximate and empirical relations used in engineering for water waves propagating over mild slopes may be somewhat improved replacing d by D. Similar modifications have already been proposed [START_REF] Ehrenmark | An alternative dispersion equation for water waves over an inclined bed[END_REF][START_REF] Porter | The mild-slope equations: a unified theory[END_REF][START_REF] Porter | An extended linear shallow-water equation[END_REF] and computations therein suggest that the effective depth can improve significantly the performance of a model.

It should be noted that the equation ( 19) can be solved analytically for various wave fields over constant slope [START_REF] Ehrenmark | An alternative dispersion equation for water waves over an inclined bed[END_REF][START_REF] Roseau | Short waves parallel to the shore over a sloping beach[END_REF]. Then, in order to improve a shallow water model, similar solutions must be derived for this approximation. Such special analytic solutions, often complicated or even impossible to write in closed-form, are actually not needed for improving the linear dispersion. Indeed, it is sufficient, simpler and more general to compare the equations directly via their shallow water expansions, i.e., expansions such as (21) obtained assuming that ∂ x and ∂ t are "small" operators, as illustrated below.

Modified Serre's equations

In order to address the drawback of (4), a better modification is sought replacing -α d 2 ∂ 2

x by another second-order differential operator D to be defined such that the linearised equations match the (fully dispersive) exact ones for constant slopes up to the highest possible order in the shallowness expansion. Of course, this alternative modification of the momentum equation is also asymptotically consistent with the original equation, as one can easily check.

We thus consider infinitesimal waves with a fluid motion close to rest -i.e., η and ū are small -with d = d(x) and d xx = 0. The linearised modified Serre-like equations are then

η t + [ d ū ] x = 0, (22)
(1 + D) ( ūt + g η x ) -1 3 d 2 ūxxt -d x d ūxt = 0, (23)
and eliminating ū between these two relations, one gets after some algebra

g -1 ∂ 2 t -∂ x d d + d D -1 3 ∂ x d 3 ∂ x -1 ( d + d D ) ∂ x η = 0. ( 24
)
The sixth-order shallow water expansion (i.e., assuming that ∂ x is "small") of this linear equation

g -1 ∂ 2 t -∂ x d∂ x -1 3 ∂ 2 x d 3 ∂ 2 x + 1 3 ∂ x dDd -1 ∂ x d 3 ∂ 2 x -1 9 ∂ 2 x d 3 ∂ x d -1 ∂ x d 3 ∂ 2 x η ≈ 0, ( 25
)
is to be compared with the exact (i.e., fully dispersive) linear relations ( 19)-( 21) for constant slopes. Thus, the expansion (25) matches the exact one up to the sixth-order only if

D = 1 5 d 2 x -1 15 d -1 ∂ x d 3 ∂ x . (26)
The classical improvement is recovered on constant depth, as it should be. This choice is an improvement for constant slopes but, in practice, it should also improve the model when the bottom curvature is small (i.e., if |dd xx | ≪ 1) and, at least, when the bottom varies very slowly in time.

The operator D derived here in the expression (26) is, obviously, also suitable for any shallow water approximations having the same linear approximation as the Serre equations.

Linear well-posedness

With (26), the differential operator d + dD is a self-adjoint and positive-definite, and the linear equation (24) yields

g -1 ∂ 2 t -∂ x d d + 1 5 d d 2 x -2 5 ∂ x d 3 ∂ x -1 ( d + d D ) ∂ x η = 0. ( 27
)
The differential operator inside the square bracket of ( 27) is also self-adjoint and positivedefinite (since d > 0), so it is invertible and the equation ( 27) is therefore well defined. Conversely, with the operator D 0

def = -1 15 d 2 ∂ 2
x commonly used to improve the dispersion, d + dD 0 is not self-adjoint and not positive-definite if d x = 0, and the equation (24) becomes

g -1 ∂ 2 t -∂ x d d + 1 5 d 2 d x ∂ x -2 5 ∂ x d 3 ∂ x -1 ( d + d D 0 ) ∂ x η = 0. ( 28
)
The differential operator inside the square bracket of (28) is not positive-definite (nor negativedefinite) if d x = 0, so it is not invertible and the equation ( 28) is ill-posed, in general. Thus, depending to the problem at hand, unphysical phenomena and numerical instabilities may appear as a consequence of "improving" the shallow water equations using D 0 . This simple analysis shows that the modification (4) is generally leading to ill-posed models for uneven bottoms, that is a worse drawback than not being a dispersion improvement.

Discussion

We have shown that attempts to improve shallow water models, by tweaking the equations in an asymptotically consistent way, does not always lead to sound improved models. Worse, one can get ill-posed equations. These points were illustrated with the Serre equations where their classical modification (4) improves the dispersion and it is well-posed only in contant depth; for varying bottoms, (4) does not improve the dispersion and it is generally linearly ill-posed.

Considering fully-dispersive linear waves on constant slopes, we proposed another modification of the Serre equations such that their dispersive properties are improved for constant slopes. We also showed that this modification is linearly well-posed.

Here, we focused on Serre's equations but similar modifications hold for any variant of shallow water (Boussinesq-like) approximations. The way the equations are modified to consistently improve the dispersion is of course not unique. Whatever approach is considered, suitable modifications can be derived along the lines illustrated here for the Serre equations.

The effectiveness, for practical applications, of this approach could be investigated via numerical simulations. These are not the purpose of the present short note where we focussed on demonstrating theoretically the drawbacks of (4) and how they can be addressed. Further theoretical investigations could be performed. For instance, one can look for: (i) dispersion improvements for arbitrary (i.e., not only constant) slopes; (ii) a three-dimensional extension (i.e., two horizontal spacial dimensions); (iii) variational derivations of modified equations. These will be the purpose of future investigations.