J Perthuison

P Schaeffer

P Debels

P Galant

P Adam

BETULIN-RELATED ESTERS FROM BIRCH BARK TAR: IDENTIFICATION, ORIGIN AND ARCHAEOLOGICAL SIGNIFICANCE

Keywords: Archaeology, Birch bark tar, Lupane-related triterpenoids, Triterpenoid esters, Neolithic

Birch bark tar, an organic material frequently encountered during archaeological excavations, has been identified from its lipid composition on the cracks of a ceramic dated to the late Neolithic. Lipids of this black substance were dominated by a characteristic triterpenoid assemblage of lupane-related triterpenoids from birch bark together with their thermal degradation products formed during preparation of the tar. Among the latter, four main series of unusual triterpenoid esters have been detected and were postulated to correspond to esters of D 2 -betulin and D 2dihydrobetulin based on their mass spectra and hydrolysis experiments.

Their conclusive identification has been achieved by synthesis of reference compounds. These compounds most likely originate from the esterification between triterpenoid alcohols related to betulin and fatty acids from suberin formed upon heating of birch bark tar. They could be considered as markers of intense heating during birch bark tar preparation using the "single pot" procedure.

Introduction

Natural plant resins and resinous materials have played an important role in the daily life since ancient times, explaining their frequent occurrence at Version avant révision archaeological sites. Among them, birch bark tar is a vegetal substance produced by dry distillation of birch bark [START_REF] Aveling | Identification of birch bark tar at the Mesolithic site of Star Carr[END_REF][START_REF] Urem-Kotsou | Birch-bark tar at Neolithic Makriyalos, Greece[END_REF][START_REF] Rageot | Exploitation of beehive products, plant exudates and tars in Corsica during the early Iron Age[END_REF][START_REF] Courel | Birch bark tar and jewellery: The case study of a necklace from the Iron Age (Eckwersheim, NE France)[END_REF]. This sticky and hydrophobic material has been used since the Paleolithic [START_REF] Koller | High-tech in the Middle Palaeolithic: Neandertal-manufactured pitch identified[END_REF][START_REF] Grünberg | Middle Palaeolithic birch-bark pitch[END_REF] for various purposes, like the hafting of lithic tools or jewellery (Sauter et al., 2000;[START_REF] Bosquet | Identification de brai de bouleau sur quatre vases du site rubané de Fexhe-le-Haut-Clocher "Potri l'Cortri[END_REF][START_REF] Koller | High-tech in the Middle Palaeolithic: Neandertal-manufactured pitch identified[END_REF][START_REF] Courel | Birch bark tar and jewellery: The case study of a necklace from the Iron Age (Eckwersheim, NE France)[END_REF], the repairing or caulking of ceramics [START_REF] Charters | Identification of an adhesive used to repair a Roman jar[END_REF][START_REF] Connan | Apport de la chimie moléculaire à la connaissance des enduits utilisés pour le traitement des bois et le calfatage des bateaux romains de la Saône[END_REF][START_REF] Bosquet | Identification de brai de bouleau sur quatre vases du site rubané de Fexhe-le-Haut-Clocher "Potri l'Cortri[END_REF][START_REF] Rageot | Exploitation of beehive products, plant exudates and tars in Corsica during the early Iron Age[END_REF][START_REF] Reunanen | Analysis of archaeological birch bark pitches[END_REF][START_REF] Urem-Kotsou | Birch-bark tar at Neolithic Makriyalos, Greece[END_REF], the decorating of ceramic vessels [START_REF] Vogt | The Birch as a source of raw material during the Stone Age[END_REF] and has even been used as chewing-gum [START_REF] Aveling | Chewing tar in the early Holocene: an archaeological and ethnographic evaluation[END_REF][START_REF] Karg | Vegetal grave goods in a female burial on Bornholm (Denmark) from the Late Roman Iron Age period interpreted in a comparative European perspective[END_REF]. Birch bark tar has a typical molecular composition, making it easy to identify by detection of predominant lupane-related triterpenoids using gas chromatography coupled to mass spectrometry -GC-MS - [START_REF] Hayek | A bicentennial of betulin[END_REF][START_REF] Hayek | Identification of archaeological and recent wood tar pitches using gas chromatography/mass spectrometry and pattern recognition[END_REF][START_REF] Reunanen | Composition of tars from softwoods and birch[END_REF][START_REF] Aveling | Identification of birch bark tar at the Mesolithic site of Star Carr[END_REF]Schnell et al., 2014).

In the frame of a study dedicated to the use of pottery during the Neolithic, we have investigated an organic black substance having most likely served to repair different ceramics (Fig. 1) found at the site of La Rouvière (Rogues, Gard, Occitanie, Southern France) and dated to the late Neolithic (ca. 3000 BCE). GC-MS analysis of this substance led to its identification as birch bark tar. Along with typical triterpenoid markers from the lupane series, four series of uncommon compounds eluted late on the gas chromatogram were detected. Based on MS interpretations, they were postulated to

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correspond to triterpenes ester-linked to monocarboxylic or dicarboxylic acids. One of this compound series was previously reported to occur in birch bark tar samples from Roman archaeological sites in Great Britain, but the identification, based on MS interpretations and hydrolysis experiments, remained tentative [START_REF] Dudd | Unusual triterpenoid fatty acyl ester components of archaeological birch bark tars[END_REF]. We report here the conclusive identification of the four series of compounds by synthesis of one homologue of each series. Their origin as well as their significance with regard to the mode of preparation of the organic material are discussed.

The archaeological site of La Rouvière

The site of La Rouvière corresponds to a settlement unearthed in the city of Rogues (Southern France). It has been discovered in 1989 by J. Halgand and colleagues (members of the « Groupe de Recherches et d'Explorations Souterraines du Vigan ») and excavated by P. Galant from 1989[START_REF] Galant | Rogues -Aven de la Rouvière[END_REF][START_REF] Galant | L'aven de la Rouvière: de la découverte spéléologique à l'étude archéologique[END_REF]. The archaeological site is composed of a sinkhole (discovered in 1989) and an outdoor establishment (discovered in 1999) linked by a narrow corridor (Fig. 1a). The site was used to collect and store water. Two main phases of occupation both dated to the late Neolithic (ca. 3000 BCE) were clearly distinguished, with the latest ending with a fire event. More than 40 jars related to the last occupation stage were discovered inside the establishment. One of them, named I15, was found in the corridor at the entrance of the sinkhole and was probably used for water storage [START_REF] Galant | Rogues -Aven de la Rouvière[END_REF]Fig. 1a). In this jar, black organic residues were

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discovered along the edges of ancient cracks on the inner surface, suggesting that this material was used to repair/waterproof the ceramic. The archaeological sample investigated corresponds to a black organic residue collected on the jar I15 (Fig. 1b).

Experimental

Extraction

The archaeological sample was extracted by sonication (20 min) using a mixture of CH2Cl2/MeOH (1:1 v/v) followed by filtration of the supernatant through celite and removal of the solvent under reduced pressure. 25.0 mg of organic extract were obtained from 204.7 mg of starting material.

Fractionation of the solvent extract

Protocol A

An aliquot of the extract was acetylated (Ac2O, Pyridine, 2 h, 60 °C) and, after removal of the solvents and excess reagents under a stream of Ar, esterified with a solution of diazomethane in diethyl ether. The derivatized extract was fractionated by liquid chromatography (LC) on silica gel into an apolar fraction (FA.1) eluted with CH2Cl2/EtOAc (8:2, v/v; 3 dead volumes -Dvol -) and a more polar fraction (FA.2) eluted with CH2Cl2/MeOH (1:1, v/v; 2 Dvol). FA.1 was analyzed by GC-MS.

Protocol B

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An aliquot of the extract was fractionated by liquid chromatography (LC) on silica gel without derivatization. A first fraction (FB.1) containing the fatty acyl esters from series E1 and E2 was recovered by elution with CH2Cl2 (1.4 Dvol). A second fraction (FB.2), eluted with a mixture of CH2Cl2/MeOH (1:1 v/v; 2 Dvol), was treated with a solution of diazomethane in diethyl ether. The resulting methylated fraction FB.2 was re-fractionated into three fractions.

The first fraction (FB.2.1 eluted with CH2Cl2 (1.25 Dvol) was shown to contain mainly methylated fatty acids, and the second fraction (FB.2.2), also eluted with CH2Cl2 (2 Dvol), comprised dicarboxylic esters from series E3 and E4. The last eluted one (FB.2.3), recovered using a mixture of CH2Cl2/MeOH (1:1 v/v; 2 Dvol), corresponded to more polar material not further investigated.

GC-MS

GC-MS analyses were carried out using a Thermo Trace gas chromatograph (Thermo Scientific) coupled to a Thermo Scientific TSQ Quantum mass spectrometer equipped with an autosampler Tri Plus and a programmed temperature vaporizing (PTV) injector. The temperature of the source was set at 220 °C. The mass spectrometer was operating in the electron impact (EI) mode at 70 eV and scanning m/z 50 to 700 or 50 to 900. Gas chromatographic separations were performed on a HP5-MS column (30 m x 0.25 mm ; 0.25 µm film thickness) using He as carrier gas.

Two oven temperature programs were used:

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Program 2 : 70 °C (1 min), 70 °C-320 °C (10 °C/min), isothermal at 320 °C for 80 min.

NMR

The NMR spectra were recorded on a Bruker Avance I 500 MHz spectrometer.

The chemical shifts are reported in ppm relative to tetramethylsilane with the solvent used as internal standard (CDCl3 : δ 1 H 7.26 ppm ; δ 13 C 77.16 ppm).

LiAlH4 hydrogenolysis of ester bonds

Fraction FB.1 (cf. § 3.2.2.) was submitted to hydrogenolysis using LiAlH4. A small amount of LiAlH4 (in powder) prewashed with distilled cyclohexane was put under stirring in a vial containing the fraction FB.1 dissolved in THF (1 h, room temperature). The mixture was transferred dropwise into a roundbottom flask containing MeOH in order to remove the excess of reagent. After removal of the solvents under reduced pressure, the crude mixture was transferred into a separatory funnel and extracted using EtOAc after addition of distilled water. The solvent extract was filtered on a small silica gel column and acetylated (Ac2O/Pyr, 1:1 v/v, 2 h, 60 °C) before analysis using GC-MS.

Synthesis of reference compounds

Synthesis of the reference compounds was based on the articles from Sun et al. (1998aSun et al. ( , 1998b)).

3.6.1. Dehydration of betulin derivatives (Fig. 2a) Diethylazodicarboxylate (0.15 mL, 4 eq) was added dropwise to a solution containing 100 mg of betulin T1 (respectively 100 mg of dihydrobetulin T9)

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in THF (3-4 mL), triphenylphosphine (237 mg, 4 eq) and 3,3dimethylglutarimide (129 mg, 4 eq) at 0 °C under Ar atmosphere. The mixture was left for 3 h at room temperature. After removal of the solvent under reduced pressure, the crude residue was fractionated by silica gel chromatography with a mixture of EtOAc/CH2Cl2 (5:95 v/v) to obtain 68 mg of pure Δ 2 -betulin T5 (respectively 46 mg of Δ 2 -dihydrobetulin T7) with a yield of 71% (respectively 48%).

Δ 2 -betulin T5 (Fig. 3a): GC-MS (acetate derivative of T5) (EI, 70 eV) m/z (rel. intensity) 466 (M +. , 10%), 406 (8), 393 (20), 229 (12), 216 (21), 203 (40), 189 (100), 187 (55), 173 (27), 159 (32), 147 (35), 133 (42), 119 (56), 107 (48), 93 (40).

1 H NMR (500 MHz; CDCl3): 0.86 (3H, s), 0.87 (3H, s), 0.94 (3H, s), 0.99 (3H, s), 1.05 (3H, s), 1.69 (3H, s, H-30), 2.40 (1H, td, J = 5.5; 11.0 Hz, H-19), 3.34 (1H, d, J = 10.5 Hz, H-28), 3.82 (1H, d, J = 10.5 Hz, H-28), 4.59 (1H, s, H-29), 4.69 (1H, s, H-29), 5.35 (1H, dd, J = 2.0; 10.0 Hz, H-3), 5.40 (1H, ddd, J = 1.0; 5.5; 10.0 Hz, H-2). 13 C NMR (500 MHz; CHCl3): 14.9, 15.8, 16.5, 19.2, 19.6, 21.4, 22.7, 25.5, 27.2, 29.3, 29.9, 31.9, 33.5, 34.1, 34.8, 36.5, 37.6, 41.1, 41.4, 42.9, 47.9, 48.0, 48.9, 49.2, 52.2, 60.7, 109.8, 121.7, 138.1, 150.7.

Δ 2 -dihydrobetulin T7 (Fig. 3b): GC-MS (acetate derivative of T7) (EI, 70 eV) m/z (rel. intensity) 468 (M +. , 9%), 408 (3), 395 (54), 229 (10), 217 (13), 204 (26), 191(41), 189 (100), 177 (36), 161 (22), 147 (27), 135 (47), 122 (62), 119 (47), 107 (43), 95 (35). 1 H NMR (500 MHz; CDCl3): 0.77 (3H, d, J = 7.0 Hz, H-29 or H-30), 0.84 (3H, d, J = 7.0 Hz, H-29 or H-30), 0.87 (3H, s), 0.88 (3H, s), 0.95 (3H, s), 0.97 (3H, s), 1.06 (3H, s), 3.31 (1H, d, J = 11.0 Hz, H-28), 3.79 (1H, d,

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J= 11.0 Hz, H-28), 5.36 (1H, dd, J = 2.5; 10.0 Hz, H-3), 5.40 (1H, ddd, J = 1.5; 5.5; 10.0 Hz, H-2). 13 C NMR (500 MHz; CHCl3): 14.8, 15.1, 15.8, 16.5, 19.6, 21.4, 21.9, 22.8, 23.1, 27.1, 27.1, 29.4, 29.6, 31.9, 33.6, 34.2, 34.8, 36.5, 37.1, 41.2, 41.4, 43.0, 44.7, 48.1, 48.2, 48.9, 52.2, 60.8, 121.7, 138.1.

Esterification of betulin derivatives

Synthesis of E1a and E2a (Fig. 2b) A solution containing Δ 2 -betulin T5 (14.7 mg) (respectively Δ 2dihydrobetulin T7, 17.9 mg), nonanoyl chloride (50 µL, ~7 eq), N-methyl imidazole (N-Me Im, one drop) and pyridine (0.6 mL) was placed in a vial at 90 °C for 4 h. After transfer of the mixture into a separatory funnel and addition of an aqueous solution of CuSO4 (3 mL, 9/1 wt:wt), the organic layer was recovered using CH2Cl2. The organic phase was washed with distilled water, and the solvent removed under reduced pressure. The crude residue was fractionated by silica gel chromatography with toluene to obtain 4.3 mg of pure E1a with a yield of 22% (resp. 4.4 mg of E2a; 19%, yield).

Compound E1a (Fig. 3c): GC-MS (EI, 70 eV) m/z (rel. intensity) 564 (M +. , 6%), 406 (18), 393 (19), 229 (14), 216 (22), 203 (46), 189 (100), 187 (64), 175 (25), 173 (26), 159 (34), 147 (34) 135 (34), 133 (40), 121 (44), 119 (48), 107 (45), 95 (37), 81 (27). 1 H NMR (500 MHz; CDCl3): 0.94 (3H, s), 0.99 (3H, s), 1.06 (3H, s), 1.69 (3H, s, H-30), 2.32 (2H, t, J = 7.5 Hz, H-2'), 2.46 (1H,td,J = 6.0;11.5 Hz,3.86 (1H,d,J = 11.0 Hz,4.26 (1H,d,J = 11.0 Hz, H-28), 4.59 (1H, s, H-29), 4.69 (1H, s, H-29), 5.35 (1H, dd, J = 2.0; 10.0 Hz, H-3), 5.40 (1H, ddd, J = 1.0; 5.5; 10.0 Hz, H-2). 13 C NMR (500 MHz;

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CHCl3): 14.2, 14.9, 15.9, 16.5, 19.3, 19.6, 21.3, 22.8, 22.8, 25.3, 25.5, 27.2, 29.3, 29.4, 29.4, 29.8, 29.9, 31.9, 32.0, 33.5, 34.7, 34.8, 34.8, 36.5, 37.8, 41.1, 41.4, 42.9, 46.6, 47.9, 48.9, 49.2, 52.3, 62.7, 109.9, 121.7, 138.1, 150.4, 174.5.

Compound E2a (Fig. 3d): GC-MS (EI, 70 eV) m/z (rel. intensity) 566 (M +. , 5%), 408 (8),395 (47),229 (16),217 (14),204 (39),191 (43),189 (100),177 (37), 161 (23), 159 (24), 149 (28), 147 (30), 135 (50), 122 (51), 107 (42), 95 (42), 81 (30). 1 H NMR (500 MHz; CDCl3): 0.77 (3H, d, J = 6.5 Hz, H-29 or H-30), 0.84 (3H, d, J = 7.0 Hz, H-29 or H-30), 0.87 (3H, s), 0.88 (3H, s), 1.07 (3H,s), 2.31 (2H, t, J = 8.0 Hz, H-2'), 3.83 (1H, d, J = 11.0 Hz, H-28), 4.26 (1H, d, J = 11.0 Hz, H-28), 5.36 (1H, dd, J = 2.0; 10.0 Hz, H-3), 5.41 (1H, ddd, J= 1.0; 6.0; 10.0 Hz, H-2). 13 C NMR (500 MHz; CHCl3): 14.2, 14.8, 15.1, 15.9, 16.5, 19.6, 21.3, 21.8, 22.8, 22.8, 23.1, 25.3, 27.1, 27.1, 29.3, 29.4, 29.4, 29.6, 30.1, 31.9, 32.0, 33.6, 34.7, 34.8, 34.9, 36.5, 37.4, 41.1, 41.4, 43.0, 44.7, 46.7, 48.3, 48.9, 52.2, 62.7, 121.7, 138.1, 174.5.

Synthesis of E3a and E4a (Fig. 2b) A solution containing 12.9 mg of Δ 2 -betulin T5 (resp. 13 mg of Δ 2dihydrobetulin T7), methyl 8-chloro-8-oxooctanoate (50 µL, 11 eq), N-methyl imidazole (N-Me Im) (one drop) and pyridine (0.6 mL, was placed in a vial at 90 °C for 4 h. To remove pyridine and N-Me Im, the mixture was transferred into a separatory funnel, shaken with 3 mL of an aqueous solution of CuSO4 (9/1 wt:wt) and extracted 2x with CH2Cl2 and 2x with EtOAc. The combined organic extracts were evaporated under reduced pressure, dissolved in EtOAc and washed with distilled water. The crude residue obtained after

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removal of the solvent under reduced pressure was fractionated by silica gel chromatography with a mixture of EtOAc/CH2Cl2 (5:95 v/v) to obtain 8.9 mg (49 % yield) of pure E3a (respectively 10.1 mg of E4a, 55 % yield).

Compound E3a (Fig. 3e): GC-MS (EI, 70 eV) m/z (rel. intensity) 594 (M +. , 2%), 406 (31),391 (13),363 (7),229 (15),215 (23),203 (41),202 (41),189 (100), 187 (62), 173 (28), 159 (32), 147 (36), 133 (41), 119 (51), 107 (47), 95 (39), 81 (26). 1 H NMR (500 MHz; CDCl3): 0.86 (3H, s), 0.87 (3H, s), 0.94 (3H, s), 0.98 (3H, s), 1.06 (3H, s), 1.68 (3H, s, H-30), 2.30 (2H, t, J = 7.5 Hz, H-2' or H-7'), 2.32 (2H, t, J = 7.5 Hz, H-2' or H-7'), 2.45 (1H, td, J = 5.5; 11.5 Hz, H-19), 3.66 (3H, s, OCH3), 3.86 (1H, d, J = 11.5 Hz, H-28), 4.27 (1H, d, J = 11.0 Hz, H-28), 4.59 (1H, s, H-29), 4.69 (1H, s, H-29), 5.35 (1H, dd, J = 2.0; 10.0 Hz, H-3), 5.39 (1H, ddd, J = 1.0; 5.5; 10.0 Hz, H-2). 13 C NMR (500 MHz; CHCl3): 14.9, 15.9, 16.5, 19.3, 19.6, 21.3, 22.8, 24.9, 25.0, 25.5, 27.2, 28.9, 28.9, 29.8, 29.9, 31.9, 33.5, 34.1, 34.5, 34.7, 34.8, 36.5, 37.8, 41.1, 41.4, 42.9, 46.6, 47.9, 48.9, 49.2, 51.6, 52.3, 62.7, 110.0, 121.7, 138.1, 150.4, 174.3, 174. 0.84 (3H, d, J = 7.0 Hz, H-29 or H-30), 0.87 (3H, s), 0.88 (3H, s), 0.94 (3H, s), 0.96 (3H, s), 1.07 (3H, s), 2.30 (2H, t, J = 7.5 Hz, H-2' or H-7'), 2.31 (2H, t, J = 7.5 Hz, H-2' or H-7'), 3.66 (3H, s, OCH3), 3.83 (1H, d, J = 11.0 Hz, H-28),

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4.26 (1H, d, J = 11.0 Hz, H-28), 5.36 (1H, dd, J = 2.0; 10.0 Hz, H-3), 5.40 (1H, ddd, J = 1.0; 5.5; 10.0 Hz, H-2). 13 C NMR (500 MHz; CHCl3): 14.8, 15.1, 15.9, 16.5, 19.6, 21.3, 21.8, 22.8, 23.1, 24.9, 25.0, 27.1, 27.1, 28.9, 28.9, 29.6, 30.0, 31.9, 33.5, 34.1, 34.5, 34.8, 34.9, 36.5, 37.4, 41.1, 41.4, 43.0, 44.7, 46.7, 48.3, 48.9, 51.6, 52.2, 62.8, 121.7, 138.1, 174.3, 174.3.

Results and discussion

Triterpenoids as markers of birch bark tar

The gas chromatogram of fraction FA.1 (cf. § 3.2.1.) isolated from the archaeological sample is shown in Fig. 4. Its lipid distribution was dominated by triterpenoids from the lupane series, indicating a major contribution from a vegetal source derived from angiosperms (Fig. 4). The triterpenoids notably comprised betulin T1, lupeol T2 and betulone T3 which are native triterpenoids occurring in birch bark [START_REF] Hayek | A bicentennial of betulin[END_REF][START_REF] Hayek | Identification of archaeological and recent wood tar pitches using gas chromatography/mass spectrometry and pattern recognition[END_REF]Schnell et al., 2014). However, these compounds represent generally by far the predominant triterpenoids in the case of fresh (or even weathered) birch bark [START_REF] Hayek | Identification of archaeological and recent wood tar pitches using gas chromatography/mass spectrometry and pattern recognition[END_REF][START_REF] Aveling | Identification of birch bark tar at the Mesolithic site of Star Carr[END_REF]Schnell et al., 2014;[START_REF] Courel | Birch bark tar and jewellery: The case study of a necklace from the Iron Age (Eckwersheim, NE France)[END_REF], whereas they appear only as minor constituents in the archaeological sample. In the latter case, the distribution is dominated by compounds T4, T5, T10, T7 and T11, which all belong to the series of D 2 -betulin-related triterpenoids. These compounds, together with the allobetulane derivatives T10-T13, are known from the literature to be

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biomarkers of birch bark tar [START_REF] Bosquet | Identification de brai de bouleau sur quatre vases du site rubané de Fexhe-le-Haut-Clocher "Potri l'Cortri[END_REF][START_REF] Modugno | Chemical study of triterpenoid resinous materials in archaeological findings by means of direct exposure electron ionisation mass spectrometry and gas chromatography/mass spectrometry[END_REF][START_REF] Regert | Molecular characterisation of birch bark tar by headspace solid-phase microextraction gas chromatography-mass spectrometry: A new way for identifying archaeological glues[END_REF][START_REF] Courel | Birch bark tar and jewellery: The case study of a necklace from the Iron Age (Eckwersheim, NE France)[END_REF][START_REF] Révision Rageot | Birch bark tar production: Experimental and biomolecular approaches to the study of a common and widely used prehistoric adhesive[END_REF].

Δ 2 -triterpenoids are formed by elimination of the oxygenated function at C-3 of lupeol T2 and betulin T1, this reaction being generally induced by a thermal treatment as is the case for the preparation of birch bark tar [START_REF] Courel | Birch bark tar and jewellery: The case study of a necklace from the Iron Age (Eckwersheim, NE France)[END_REF][START_REF] Révision Rageot | Birch bark tar production: Experimental and biomolecular approaches to the study of a common and widely used prehistoric adhesive[END_REF]. Similarly, it was shown that the same thermal treatment leads to the formation of allobetulane derivatives (T10-T13) which results from an acid-catalyzed intramolecular rearrangement of the ring E of betulin and by-products (Davy et al., 1951a(Davy et al., , 1951b;;[START_REF] Green | Isolation of betulin and rearrangement to allobetulin. A biomimetic natural product synthesis[END_REF]Salvador et al., 2009). In some cases, both reactions may co-occur, resulting in the formation of D 2 -allobetulin derivatives such as T10 and T11.

Similar observations regarding birch bark tar composition, in which alteration products dominate over genuine ones, are reported in the literature in the case of archaeological samples (e.g., [START_REF] Urem-Kotsou | Birch-bark tar at Neolithic Makriyalos, Greece[END_REF][START_REF] Regert | Adhesive production and pottery function during the Iron Age at the site of Grand Aunay (Sarthe, France)[END_REF], and were interpreted as being the result of an intense heating of the material upon tar preparation. Such seems to be the case with our sample as well which was likely submitted to an abnormally high thermal stress, resulting in the almost complete transformation of the genuine triterpenoids (Fig. 4). In addition to these triterpenoids closely related to birch bark tar, four unusual late-eluted compound series were detected (named "series E1-E4"; Fig. 4), which present fragmentation patterns in MS (Fig. 3c-3f) very similar to those of Δ 2 -betulin T5 (Fig. 3a)

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and Δ 2 -dihydrobetulin T7 (Fig. 3b) but with quite higher molecular masses in the range 494-748 Da. In order to have a more detailed look to these different compounds, a fractionation procedure of the lipid extract was developed and led to obtaining chromatographic fractions considerably enriched in triterpenoids from series E1-E4 (cf. § 3.2.2).

Late-eluted triterpenoid esters from series E1-E4

Identification of triterpenoid esters from series E1-E2

Detailed investigation of fraction FB.1 (cf. § 3.2.2.) revealed the occurrence of compounds belonging to the series E1 and E2 (Fig. 5). Homologues from series E1 have a mass fragmentation pattern very similar to that of Δ 2 betulin T5 (Fig. 3a and3c) and a molecular weight of 494 + 𝑛 × 14 (n = 0-18), while those from series E2 have mass spectra close to that of Δ 2 dihydrobetulin T7 (Fig. 3b and3d) and with a molecular weight shifted upwards by 2 mass units compared to series E1 (i.e., M + of 496 + 𝑛 × 14, n = 0-18). According to the literature, the generic structure of compounds from series E1 has been previously proposed to correspond to Δ 2 betulin T5 esterified at C-28 with saturated monocarboxylic acids of different chain lengths [START_REF] Dudd | Unusual triterpenoid fatty acyl ester components of archaeological birch bark tars[END_REF]. By analogy, it can be proposed that compounds from the series E2 have the same generic structure, but without the D 20(29) unsaturation. For both series, the fatty acyl moiety is ranging from C4 to at least C22 (Fig. 5). However, since the original structural identification of compounds from series E1 was based on MS interpretations and hydrolysis experiments [START_REF] Dudd | Unusual triterpenoid fatty acyl ester components of archaeological birch bark tars[END_REF], thus remaining

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tentative, we have carried out the synthesis of one reference compound from each series E1 and E2 (Fig. 2 and § 3.6.) for firm identification. Briefly, the hydroxy group at C-3 from betulin T1 (resp. dihydrobetulin T9) was selectively dehydrated following the method reported by Sun et al. (1998b), leading to the formation of D 2 -betulin T5 (resp. D 2 -dihydrobetulin T7). The remaining hydroxy group at C-28 from T5 (resp. T7) was then esterified with nonanoyl chloride to yield the C9 ester of D 2 -betulin E1a (resp. E2a).

Since the mass spectra and retention times in GC of the synthetic and naturally-occurring compound E1a (resp. E2a) were identical, it can be considered that both compound series have been successfully identified.

Identification of the triterpenoid esters from series E3-E4

The fraction FB.2.2 recovered after purification of the solvent extract (cf. § 3.2.2.) revealed the occurrence of late eluting compounds from series E3 and E4 (Fig. 6). Like compounds from series E1, the mass spectra of the homologues from series E3 (Fig. 3e) showed close similarities with those of Δ 2 -betulin T5 (Fig. 3a) but with molecular ions at 538 + 𝑛 × 14 (n = 0-6).

Similarly, the mass spectra of compounds from series E4 (Fig. 3f) closely resemble that of Δ 2 -dihydrobetulin T7 (Fig. 3b), with a molecular weight of 540 + 𝑛 × 14 (n = 0-6). Based on these data, it was proposed that compounds from series E3 (resp. E4) could correspond to dicarboxylic acids esterified at the C-28 hydroxy group of Δ 2 -betulin T5 (resp. Δ 2 -dihydrobetulin T7). For both series, the dicarboxylic acid moieties comprised C4-C10 homologues with a predominance of the C8 and C9 homologues. In order to confirm these

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hypotheses, the synthesis of one homologue of each series was performed following the same synthetic scheme as for compounds E1a and E2a, except for the acylating agent which was methyl-8-chloro-8-oxooctanoate (Fig. 2b).

Mass spectra and retention times in GC of the synthetic references and of the archaeological sample were in good agreement, confirming our structural hypotheses.

Mode of formation of the triterpenoid esters E1-E4

As proposed by [START_REF] Dudd | Unusual triterpenoid fatty acyl ester components of archaeological birch bark tars[END_REF], the esters from series E1 and E2 may originate from esterification reactions involving compounds T5 and T7 and monocarboxylic acids during the heating of birch bark or birch bark tar with fat in a process aimed at producing birch bark tar with modified properties. However, at least in our case, we propose that the monocarboxylic acids may rather originate from the thermal degradation of suberin, a biopolymer of birch bark (Fig. 7), and not from the input of fat.

Our hypothesis is notably based on the presence in the lipid extract of free saturated fatty acids dominated by the C16, C18 and C22 homologues, together with that of C20-C22 a,w-hydroxyacids and C16-C22 a,w-diacids (Fig. 8a). Such a distribution closely resembles that of bound fatty acids released from birch bark suberin [START_REF] Holloway | The composition of suberin from the corks of Quercus suber L. and Betula pendula roth[END_REF][START_REF] Ekman | The suberin monomers and triterpenoids from the outer bark of Betula verrucosa Ehrh[END_REF][START_REF] Ferreira | Isolation of suberin from birch outer bark and cork using ionic liquids: A new source of macromonomers[END_REF] and from birch bark tar [START_REF] Charters | Identification of an adhesive used to repair a Roman jar[END_REF][START_REF] Reunanen | Analysis of archaeological birch bark pitches[END_REF][START_REF] Reunanen | Composition of tars from softwoods and birch[END_REF][START_REF] Courel | Caractérisation de substances naturelles en contexte archéologique: apport des études moléculaires, isotopiques et de la datation au 14[END_REF][START_REF] Révision Rageot | Birch bark tar production: Experimental and biomolecular approaches to the study of a common and widely used prehistoric adhesive[END_REF]. Furthermore, fraction FB.1 containing the triterpenoid esters from series E1 and E2 was treated with LiAlH4, and the distribution of the resulting alcohols released by

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hydrogenolysis of the acyl chains (Fig. 8c) was compared to that of the free fatty acids from the same sample (Fig. 8b). It appeared from this experiment that the free fatty acids and the alcohols shared the same type of predominance, with C16 > C18 > C22 > C20 = C21, confirming an origin from fatty acids from suberin for the esterifying moieties.

The esters E3 and E4 might correspond to the oxidation products of triterpenoids T5 and T7 originally esterified with mid-chain unsaturated fatty acid such as those occurring in birch bark suberin [START_REF] Ekman | The suberin monomers and triterpenoids from the outer bark of Betula verrucosa Ehrh[END_REF][START_REF] Ferreira | Isolation of suberin from birch outer bark and cork using ionic liquids: A new source of macromonomers[END_REF] and which have been oxidized during ageing of the material or during tar preparation (Fig. 7). In this respect, it is worth noting that oleic and linoleic acids (both bearing a D 9 double bond) are predominant among the unsaturated acids present in suberin. Interestingly, it has been shown that oleic acid can form C2-C12 dicarboxylic acids during photochemical degradation [START_REF] Rontani | Photodegradation of unsaturated fatty acids in senescent cells of phytoplankton: photoproduct structural identification and mechanistic aspects[END_REF][START_REF] Révision Tedetti | Hydroxyl radical-induced photochemical formation of dicarboxylic acids from unsaturated fatty acid (oleic acid) in aqueous solution[END_REF], with azelaic acid (C9) predominating [START_REF] Passi | Saturated dicarboxylic acids as products of unsaturated fatty acid oxidation[END_REF][START_REF] Révision Tedetti | Hydroxyl radical-induced photochemical formation of dicarboxylic acids from unsaturated fatty acid (oleic acid) in aqueous solution[END_REF], as is the case with the distribution of compounds from series E3 and E4 dominated by homologues esterified with a C9 diacid moiety.

Mode of preparation of the archaeological birch bark tar

The presence of lupane-related biomarkers and of their thermal degradation products, like Δ 2 -betulin derivatives, in the archaeological sample collected from the ceramic I15, clearly indicates that the organic residue corresponds to birch bark tar. As it was found on ancient cracks, it is very likely that it has been used to repair the ceramic [START_REF] Charters | Identification of an adhesive used to repair a Roman jar[END_REF]Connan et al.,

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2000; [START_REF] Bosquet | Identification de brai de bouleau sur quatre vases du site rubané de Fexhe-le-Haut-Clocher "Potri l'Cortri[END_REF][START_REF] Rageot | Exploitation of beehive products, plant exudates and tars in Corsica during the early Iron Age[END_REF][START_REF] Urem-Kotsou | Birch-bark tar at Neolithic Makriyalos, Greece[END_REF].

However, compared to the distributions generally reported in the literature (Aveling andHeron, 1998, 1999;[START_REF] Courel | Birch bark tar and jewellery: The case study of a necklace from the Iron Age (Eckwersheim, NE France)[END_REF][START_REF] Koller | High-tech in the Middle Palaeolithic: Neandertal-manufactured pitch identified[END_REF], the triterpenoid distribution was unusually dominated by thermal degradation products, whereas genuine compounds from bark, like betulin T1, were present in very low abundance. Such a situation has been seldom reported [START_REF] Urem-Kotsou | Birch-bark tar at Neolithic Makriyalos, Greece[END_REF][START_REF] Regert | Adhesive production and pottery function during the Iron Age at the site of Grand Aunay (Sarthe, France)[END_REF], and was interpreted as being the result of a drastic heating during tar preparation. In this respect, birch bark tar making procedures used in the past are still little documented, but thanks to archaeological findings and experimental archaeological research on tar-making, two main procedures known as autothermic and allothermic procedures [START_REF] Révision Rageot | Birch bark tar production: Experimental and biomolecular approaches to the study of a common and widely used prehistoric adhesive[END_REF] have been identified. For the former, the raw material is exposed directly to the heat source [START_REF] Révision Kurt | Traditional tar production from Cedrus libani A. Rich on the Taurus mountains in Southern Turkey[END_REF], whereas an indirect heat transfer by a conductor is required for the latter [START_REF] Révision Rageot | Birch bark tar production: Experimental and biomolecular approaches to the study of a common and widely used prehistoric adhesive[END_REF]. However, according to [START_REF] Révision Rageot | Birch bark tar production: Experimental and biomolecular approaches to the study of a common and widely used prehistoric adhesive[END_REF], it seems that the autothermic process has been used mainly for preparing conifer tar and not birch bark tar since birch bark is easily flammable. Within the allothermic systems, two main ways of tar production are described: one without separation, and the other with separation, this technique being named "per descensum" [START_REF] Révision Rageot | Birch bark tar production: Experimental and biomolecular approaches to the study of a common and widely used prehistoric adhesive[END_REF]. In the first case, the tar remains in the reaction chamber with the bark until the end of the process.

In the second case, the fresh tar formed gives drops that fall into a second receptacle isolated from the fire. The containers used for tar making could

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have been ceramics leading to the so called "single-pot" and "double-pot" systems or, for aceramic societies, could be made of clay, sand, ash, turf [START_REF] Kozowyk | Experimental methods for the Palaeolithic dry distillation of birch bark: implications for the origin and development of Neandertal adhesive technology[END_REF]Schenck and Groom, 2018;[START_REF] Révision Rageot | Birch bark tar production: Experimental and biomolecular approaches to the study of a common and widely used prehistoric adhesive[END_REF]. To our knowledge, the oldest allothermic ceramic system known for tar making is dated to the final Bronze Age and was a double-pot system [START_REF] Dal Ri | Una distilleria per la pece della fine dell'età del Bronzo dal lago di Ledro nel Trentino[END_REF]. Ceramic systems and especially the "double-pot" are more frequently encountered in Roman times and in the Middle Age in Europe [START_REF] Balsan | L'industrie de la résine dans les Causses et son extension dans l'Empire romain[END_REF][START_REF] Connan | Identification de poix et de résine de conifère dans des échantillons archéologiques du lac de Sanguinet : exportation de poix en Atlantique à l'époque gallo-romaine[END_REF][START_REF] Regert | Adhesive production and pottery function during the Iron Age at the site of Grand Aunay (Sarthe, France)[END_REF][START_REF] Trintignac | La production de poix dans la cité des Gabales (Lozère) à l'époque gallo-romaine[END_REF][START_REF] Burri | La fouille programmée d'un site de distillation du bois à la Roche Redonne (Le Castellet, Var)[END_REF][START_REF] Burri | Production et commerce de la poix et de l'huile de cade en basse Provence au Moyen Âge[END_REF][START_REF] Burri | Des ressources naturelles à la santé: Approche interdisciplinaire de la production des goudrons de conifères et de leur usage médicinal en Méditerranée sur la longue durée[END_REF]. According to the distribution of the triterpenoids, and particularly given the almost absence of genuine triterpenoids from birch bark, it is likely that the archaeological birch bark tar investigated in the present study was prepared using a system without separation. With this method, the produced tar is staying with the bark in the reaction chamber until the end of the heating phase, thus explaining the presence of thermal degradation products in high proportions and the low amounts of betulin T1 in the sample. This hypothesis is in agreement with the findings of [START_REF] Révision Rageot | Birch bark tar production: Experimental and biomolecular approaches to the study of a common and widely used prehistoric adhesive[END_REF] who provided new molecular data on tar production by single or double-pot processes and showed that the birch bark tars with low amount of betulin were exclusively produced using the "singlepot" procedure. However, caution is needed since the use of ceramics for tar production is unclear during the Neolithic time [START_REF] Pietrzak | Wood tar in the Dnieper and Elbe communities: VI -II millenium BC[END_REF].

Triterpenoid esters as new biomarkers of birch bark tar

Version avant révision [START_REF] Dudd | Unusual triterpenoid fatty acyl ester components of archaeological birch bark tars[END_REF] originally reported on the presence of fatty acyl esters of betulin-related triterpenoids from archaeological samples. These samples, found in Roman archaeological sites in Great Britain, were interpreted as being the result of the condensation between birch bark tar triterpenoids and fatty acids from animal fat upon strong heating. Since such compounds were never previously reported in birch bark tar, they were considered by the authors as being molecular indicators of a composite material made of birch bark tar and fat. Based on the present work, it seems that esters of triterpenoids can (also) be formed during the preparation of birch bark tar by esterification reactions between birch triterpenoids and thermal degradation products of suberin (i.e., fatty acids, a,w-dicarboxylic acids) during heating, without the presence of an additional ingredient (i.e., fat; Fig. 7). In this respect, it is interesting to note that recently, [START_REF] Urem-Kotsou | Following their tears: Production and use of plant exudates in the Neolithic of North Aegean and the Balkans[END_REF] reported the occurrence of the same fatty acyl triterpenoids than those from the study of [START_REF] Dudd | Unusual triterpenoid fatty acyl ester components of archaeological birch bark tars[END_REF], which were interpreted as the result of a mixture between birch bark tar and fat.

However, these authors also reported, in the same samples, on the presence of free fatty acids, as well as long and short chain dicarboxylic acids with distributions typical of thermal degradation products of suberin, which could indicate that triterpenoid esters occurring in these samples may also have been directly formed from birch bark during tar preparation. In this case, triterpenoid esters E1-E4 should be considered as molecular indicators of strong thermal processes during birch bark tar preparation, such as those

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expected to occur in the "single-pot" procedure. The condensation/esterification reactions between the triterpenoid alcohols and the compounds bearing a carboxylic acid functionality may have been favored by an acid catalysis, possibly provided by phenols formed during the pyrolytic degradation of lignin [START_REF] Faix | Thermal degradation products of wood[END_REF][START_REF] Reunanen | Composition of tars from softwoods and birch[END_REF][START_REF] Dudd | Unusual triterpenoid fatty acyl ester components of archaeological birch bark tars[END_REF][START_REF] Regert | Molecular characterisation of birch bark tar by headspace solid-phase microextraction gas chromatography-mass spectrometry: A new way for identifying archaeological glues[END_REF][START_REF] Colombini | A multi-analytical study of degradation of lignin in archaeological waterlogged wood[END_REF][START_REF] Orsini | Micromorphological and chemical elucidation of the degradation mechanisms of birch bark archaeological artefacts[END_REF]. Further chemical investigations of archaeological samples and birch bark tar production experiments should be undertaken to further clarify under which conditions esters of triterpenoids may be formed, the latters representing potentially new molecular tools allowing different modes of preparation of birch bark tar to be distinguished.

Conclusion

The set of lipids identified in an organic material found on an ancient crack on a jar from the late Neolithic led to its identification as birch bark tar that has been used to repair the ceramic. Further chemical investigation of the sample allowed identification of four unusual series of triterpenoid esters by synthesis of reference compounds, three of them being reported here for the first time. The investigation of these compounds allowed a new interpretation of their mode of formation to be proposed. We suggest that these markers represent the esterification products between triterpenoid alcohols related to betulin and carboxylic acids from suberin formed upon heating of birch bark tar. They could be indicators of a high level of heating

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during birch bark tar preparation using the "single pot" procedure rather than biomarkers resulting from the mixture between birch bark tar and animal fat, as envisaged previously (cf. [START_REF] Dudd | Unusual triterpenoid fatty acyl ester components of archaeological birch bark tars[END_REF][START_REF] Urem-Kotsou | Following their tears: Production and use of plant exudates in the Neolithic of North Aegean and the Balkans[END_REF].

Figure Captions

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Program 1 :

 1 70 °C (1 min), 70 °C-200 °C (10 °C/min), 200 °C-320 °C (4 °C/min), isothermal at 320 °C for 40 min.

  3.Compound E4a (Fig.3f): GC-MS (EI, 70 eV) m/z (rel. intensity) 596 (M +. , 2%), 408 (15), 395 (16), 365 (8), 326 (7), 229 (13), 217 (11), 204 (39), 191 (38), 189 (100), 177 (27), 159 (22), 149 (24), 147 (25), 135 (43), 119 (40), 107 (37), 95 (38), 1 H NMR (500 MHz; CDCl3): 0.77 (3H, d, J = 7.0 Hz, H-29 or H-30),

Fig. 1 .

 1 Fig. 1. Map of the site of La Rouvière (a), Jar I15 discovered at La Rouvière

Fig. 2 .Fig. 3 .

 23 Fig. 2. Synthesis of (a) D 2 -belulin T5 and D 2 -dihydrobetulin T7; (b)

Fig. 4 .

 4 Fig. 4. Gas chromatogram of fraction FA.1. Bold numbers refer to structures

Fig. 5 .

 5 Fig. 5. Partial gas chromatogram (RIC) showing the distribution of the

Fig. 6 .

 6 Fig. 6. Partial gas chromatogram (RIC) showing the distribution of the

Fig. 7 .

 7 Fig. 7. Hypothetical pathway leading to the formation of the triterpenoid

Fig. 8 .

 8 Fig. 8. (a) Partial gas chromatogram (RIC) showing the aliphatic

Acknowledgements

Mrs E. Motsch (University of Strasbourg) is thanked for the GC-MS analyses and Dr L. Allouche (University of Strasbourg) for the measurement of the NMR spectra. J.P. has benefited from a doctoral fellowship from the "French Ministère de l'Enseignement Supérieur et de la Recherche".

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Figure 8

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Gabriel Morgado

Bogdan Nowakowski

Annie Lemarchand

DSMC Simulations Of Turing Patterns In Concentrated Growing Systems

The control of self-organized structures such as Turing patterns is a challenge for chemical engineers. We propose a reaction-diffusion model that includes the reactive role of the solvent and admits Turing patterns of adaptable wavelength. Changing the concentration of the solute is sufficient to tune the wavelength. Particles dynamics simulations are successfully performed, illustrating wavelength control down to the submicrometric scale.

INTRODUCTION

The formation of a periodic spatial structure along an axis is a typical example of self-organisation encountered in biology in the plant kingdom as in the animal kingdom. Palm leaves, the segments of many insects or crustaceans, and the spine of vertebrates give some examples of structures resulting from axial segmentation. It is tempting to use models from biology in the context of chemical engineering, with the goal of designing periodic spatial structures by biomimetism. In particular, the model of morphogenesis introduced by Turing has the advantage of being compatible with a microscopic approach [1]. A model with a small number of elementary processes of reaction and diffusion type can be used to simulate submicrometric Turing patterns [2]. However, the wavelength of the structure is imposed by the dynamical parameters whereas an application to chemical engineering would require a flexible pattern with a tunable wavelength. We propose to control the value of the wavelength, simply by playing on the dilution of the system. Indeed, sufficiently far from the high dilution limit, the solvent can be explicitly involved in the reaction scheme, in such a way that varying the total concentration could be sufficient to tune the wavelength of Turing pattern [3]. Our goal is to show the possibility of controlling the wavelength of submicrometric Turing patterns by adapting the Direct Simulation Monte Carlo (DSMC) method to the simulation of concentrated solutions. DSMC has been introduced as a direct simulation of the Boltzmann equation in order to efficiently simulate the dynamics of rarefied gases with large Knudsen numbers for which the continuous description is not valid. The principles of DSMC extend over the entire range of validity of the Boltzmann equation. We apply the Bird simulation method to a reactive system said to be concentrated because of the relatively high proportion of the solute with respect to the solvent. Hence, the departure from dilute conditions is characterized by the ratio of the solute density to the total density. However, the total density is supposed to remain sufficiently low to ensure that dynamics is correctly described in the framework of the Boltzmann equation. When binary reactive collisions are involved, the kinetic Monte Carlo simulation method is straightforwardly extended to reactive systems, the transition rates being deduced from the reactive collision integrals of the corresponding Boltzmann equation. We use a trick to simulate ternary reactions that already proved to be satisfactory [4,5]: The trimolecular interaction is decomposed into a standard binary collision and reaction is accepted depending on the presence of the third reactive species in the neighborhood of the colliding pair. We refer to reference [6] for a detailed presentation of the adapted DSMC algorithm. As a result, the diffusion coefficients of the components and the rate constant of the ternary reaction depend on the total density and consequently the departure from dilute conditions, implying a scaling of the simulation results before the comparison with those of the macroscopic description.

In addition to simulating the Boltzmann equation, DSMC includes the description of the internal fluctuations. Using DSMC to simulate Turing patterns allows us to check the robustness of the spatial structure to fluctuations in small systems down to the nanoscale.

The paper is organized as follows. Section 2 is devoted to the presentation of the reaction-diffusion model including an explicit role for the solvent. Section 3 gives the specificities of DSMC algorithm used to reproduce a Turing pattern in a concentrated system. We give the results in section 4. Section 5 contains conclusion.

A MICROSCOPIC MODEL OF TURING PATTERNS WITH POSSIBLE SOLVENT ROLE

We consider the following reaction scheme involving an activator A with a diffusion coefficient D A , an inhibitor B with a larger diffusion coefficient D B > D A , and interaction with two reservoirs of chemical species or chemostats R 1 and R 2 which maintain the system far from equilibrium [7,8,9]:

A k 1 -→ R 1 (1) 2A + B k 2 -→ 3A (2)
B

k 3 k -3 R 2 (3)
This system is known to be associated with reaction-diffusion equations admitting spatial structures of Turing type.

As explained by Gierer and Meinhardt [10], stabilization of a Turing pattern requires local activation through the autocatalytic formation of the activator A and long-range inhibition through the fast-diffusive inhibitor B. We already proved at the macroscopic scale that considering the solvent S as an actual reactant in a concentrated system according to

A + S -→ R 1 (4) 2A + B -→ 3A (5) B + S R 2 (6)
leads to a Turing pattern whose wavelength can be tuned simply by varying the total concentration C = A + B + S , e.g. the quantity of solvent S at constant A + B value [3]. In view of DSMC simulations, it is tempting to assign the role of reservoir particles to solvent particles. In particular, the first step given in Eq. ( 4) is replaced by A+S -→ 2S, which interestingly maintains the total number of particles constant. However, the third step of the scheme introduced in the framework of a macroscopic description cannot be straightforwardly used in particle dynamics simulations in which the solvent would play the role of the reservoir R 2 . Indeed, considering the step B + S 2S would lead to the correct term proportional to -BS but to an erroneous term proportional to S 2 instead of a constant in the rate equation of species B. Eventually, the chemical scheme is modified as follows [6] A

+ S k S 1 -→ 2S (7) 2A + B k 2 -→ 3A (8) 070008-2 B + S k S 3 -→ 2S (9)              R k -3 -→ B S -→ R (10)
where the two last steps mean that B concentration is increased at the constant rate k -3 = k -3 R while S concentration is decreased by the same amount to keep the total concentration C constant. It is worth noting that the total concentration C = A + B + S is conserved. The concentration A and B obey the partial differential equations:

∂A ∂t = -k 1 A 1 -A+B C + k 2 A 2 B + D A ∂ 2 A ∂x 2 (11)
∂B ∂t = k -3 -k 3 B 1 -A+B C -k 2 A 2 B + D B ∂ 2 B ∂x 2 (12)
where the relation of conservation has been used to eliminate S and the relations k 1 = k S 1 C and k 3 = k S 3 C between the rate constants introduced in Eqs. (1-3) and Eqs. (7-10) have been introduced. We straightforwardly verify that Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] converge towards the equations associated with the system not involving the solvent given in Eqs. (1-3) in the high dilution limit A+B C -→ 0. In a range of rate constants, the system given in Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] has three stationary states, two of which denoted by (A T , B T ) and (A 0 = 0, B 0 ) are stable in the homogeneous case. When the ratio of the diffusion coefficients, d = D B /D A , is sufficiently large and for appropriate rate constant values, the steady state (A T , B T ) is destabilized by inhomogeneous perturbations and replaced by a periodic spatial structure of Turing type. The steady state (A 0 = 0, B 0 ) remains stable and can be used to define a criterion evaluating the deviation from the dilution limit:

δ = A 0 + B 0 C (13)
which remains unchanged for given parameter values, whereas A + B may vary in space and time.

ADAPTATION OF DSMC ALGORITHM

The standard Direct Simulation Monte Carlo (DSMC) method [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] has to be adapted to the simulation of a concentrated solution including two species A and B with sufficiently different diffusion coefficients, D A and D B , and a solvent S. The three kinds of particles P = A, B, S are hard spheres and we use their radius r P as discriminating parameter. Collisions between three species are necessary to obtain different values of D A and D B . Even in the most concentrated case considered, S is in excess with respect to A and B and the collisions between A and B can be neglected with respect to the collisions between A and S and the collisions between B and S as regards diffusion.

Qualitatively, we impose r B r S r A and S A + B to obtain D B > D A , as required to stabilize the Turing structure. The precise conditions are given in references [2,6]. The diffusion coefficient D P of species P = A, B obeys:

D P = 3 8(P + S )(r P + r S ) 2 k B T πm ( 14
)
where k B is Boltzmann constant, T is temperature, and m is the mass of the hard spheres. It is worth noting that D P depends on S , i.e. on the dilution of the system. Roughly speaking, one has D P ∝ δ, where δ is the deviation from the dilution limit defined in Eq. ( 13).

Another specificity of the system is the ternary reactive step given in Eq. ( 8) and associated with the autocatalytic formation of the activator A. We already address this issue in DSMC simulations of Schlögl model [5] and solved the problem as follows. A binary collision between A and B is performed with the usual condition on the relative velocity of the colliding pair leading to the formation of the short-lived complex AB. Then, the reaction between the complex AB and the nearest molecule A is accepted with a probability A/C, leading to the release of three A as expected. Hence, the rate constant k 2 obeys:

k 2 = 4(r A + r B ) 2 C πk B T m (15)
The specific expression of k 2 induces a dependence on C, i.e. on the deviation δ from the dilution limit. As for the diffusion coefficient D P with P = A, B, we have k 2 ∝ δ.

The ternary step leads to a cubic term in the reaction-diffusion equations given in Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] which is necessary for the system to have three steady states. This last condition is required to observe trigger waves propagating between two homogeneous stable steady states, one of which may be destabilized by inhomogeneous perturbations leading to a Turing pattern. We exploit this property in the choice of original boundary conditions inspired by the formation of the spine in vertebrate embryos and introduced here by biomimicry [9,13,14,3]. The great advantage of considering a propagating front in a growing system is to avoid fixed boundary conditions, known to perturb the wavelength of the Turing pattern. Indeed, using DSMC for zero-flux boundary conditions in a small system with a length slightly smaller than two wavelengths, transitions between Turing patterns developing over one and a half wavelength or two wavelengths can be observed [2]. This phenomenon could affect the estimation of the wavelength of the structure, whereas considering a Turing pattern developing between zero-flux boundary conditions at one end and a propagating wave front ensures the reproducibility of wavelength determination.

TURING PATTERNS WITH TUNABLE WAVELENGTH

We choose a step function as initial condition with the first 50 cells on the left prepared in the steady state (A T , B T ) and the remaining cells prepared in the steady state (A 0 , B 0 ). Typical profiles for A and B species deduced from the numerical integration of Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] and DSMC simulations are given in Fig. 1. A Turing structure with a well- ), for a given deviation δ = 0.05 from the dilution limit, at scaled time k 2 t = 6986. Snapshot of the profiles for species A (a) and B (b). The parameters take the following values:

k 1 k 2 = 2.9 × 10 4 , k 3 k 2 = 2.2 × 10 4 , k -3 k 2 = 8.8 × 10 6 , d = 10.
defined wavelength λ develops behind a wave front propagating to the right for both approaches. The main question is: Does λ vary as the deviation δ from the dilution limit or equivalently the total concentration C varies ? Before answering the question, we indirectly check the ability of DSMC simulations to properly reproduce the rate constant values for different deviations from the dilution limit. Figure 2 gives the variation of the steady state (A T , B T ) versus the deviation δ from the dilution limit. The prediction of the macroscopic approach based on Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] compares well with the spatially-averaged numbers N A T and N B T of particles A and B deduced from DSMC. The average is performed over four wavelengths. The simulation results are not displayed for δ = 0.01 due to the computation cost: In this case, the total number of particles per cell would typically reach 40400 for the chosen parameter values. We compare the variations of the scaled wavelength λ k 2 D A of Turing pattern versus δ deduced from Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] and DSMC results in Fig. 3. Indeed, the scaled wavelength depends on ratios of rate constants and diffusion coefficients. These ratios are set in all the simulations to the same values given in the caption of Fig. 1 and do not vary with the deviation δ from the dilution limit. Knowing that D A and k 2 are constant in the macroscopic approach, we introduce the scaled wavelength to isolate the intrinsic dependence on δ introduced by the role of the solvent in the chemical mechanism, regardless of the specific expressions of the parameters D A (Eq. ( 14)) and k 2 (Eq. ( 15)) in the microscopic approach. As shown in Fig. 3, the agreement between the microscopic and the macroscopic descriptions is very good, which validates that DSMC correctly reproduce the ratio of diffusion coefficients. Simulation results become more noisy as δ increases, in agreement with the loss of stability of Turing structure predicted by Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] [6]. DSMC results are obtained for δ = 0 by considering the chemical mechanism without explicit role of the solvent given in Eqs. (1)(2)(3). We have checked that, for δ = 0, the same result was obtained for all the considered values of S . Figure 4 gives the answer to the main question raised in this work. The wavelength of a Turing pattern may be tuned in particle dynamics simulations by varying the total concentration of solvent, i.e. the deviation δ from the dilution limit. Typically, for the chosen parameter values, the wavelength goes from 26 to 17 when δ goes from 0.02 to 0.04. Hence, increasing the deviation from the dilution limit by a factor 2 leads to a decrease of the wavelength by a factor greater than 1.5. The simulation results are given in arbitrary units. The conversion of the wavelength in nanometers is easy. For example, for δ = 0.05 and small molecules A and B in water S, we find a wavelength of the order of 200 nm 070008-5 [6]. Hence, DSMC reveals the possibility to tune the wavelength of Turing pattern on a submicrometric scale, simply by playing on the dilution of the system. The application to chemical engineering nevertheless requires the control of a chemical system whose kinetics explicitly involves the solvent, for example as made precise in the mechanism given in Eqs. (4)(5)(6).

070008-4

CONCLUSION

We have reported on a reaction-diffusion model involving the reactive role of the solvent and proved that dilution could be used to control the wavelength of a Turing pattern. The direct simulation Monte Carlo (DSMC) method has been successfully extended to concentrated solutions. The simulation results show that it is possible to monitor Turing patterns at a submicrometric scale, simply by changing the total concentration: It is sufficient to increase concentration by a factor 2 to obtain a wavelength reduction by the same factor. The results give some indications to design a chemical scheme allowing experimental validation in the framework of chemical engineering.

FIGURE 1 .

 1 FIGURE 1. Concentration profiles deduced from the numerical solution of Eqs. (11,12) (dashed line) and number of particles per cell in simulations (solid line) versus scaled spatial coordinate x k 2 D A for Eqs. (7-10), for a given deviation δ = 0.05 from the dilution limit, at scaled time k 2 t = 6986. Snapshot of the profiles for species A (a) and B (b). The parameters take the following values: k 1 k 2 = 2.9 × 10 4 , k 3 k 2 = 2.2 × 10 4 , k -3 k 2 = 8.8 × 10 6 , d = 10.

FIGURE 2 .

 2 FIGURE 2. Macroscopic prediction of the steady state concentrations A T (solid line) and B T (dotted line) versus the deviation δ from the dilution limit. Circles (diamonds, resp.) give the mean number of A (B, resp.) particles per cell in the region of the simulated Turing pattern. For δ 0, the mechanism given in Eqs. (7-10) is used whereas the mechanism given in Eqs. (1-3) is used for δ = 0. The value of the other parameters is given in caption of Fig.1.

FIGURE 3 .

 3 FIGURE 3. Macroscopic prediction of the scaled wavelength of the Turing pattern (solid line) and scaled wavelength deduced from the simulations (circles) versus deviation δ from the dilution limit. For δ 0, the mechanism given in Eqs.(7)(8)(9)(10) is used whereas the mechanism given in Eqs. (1-3) is used for δ = 0. The value of the other parameters is given in caption of Fig.1.

FIGURE 4 .

 4 FIGURE 4. Same caption as in Fig.3without scaling of the wavelength.

ACKNOWLEDGMENTS

This publication is part of a project that has received funding from the European Union's Horizon 2020 (H2020-EU.1.3.4.) research and innovation program under the Marie Skłodowska-Curie Actions (MSCA-COFUND ID 711859) and from the Polish Ministry of Science and Higher Education for the implementation of an international cofinanced project.