Alaa Sweidan

Marylene Chollet-Krugler

Pierre Van De Weghe

Ali Chokr

Sophie Tomasi

Martine Bonnaure-Mallet

Latifa Bousarghin

R= Ldh

C9H19, C13H27, C15H31, C16H33 Y= H, CH3 X=COOH Y=CH3 C7h15

Design, Synthesis and Biological Evaluation of Potential Antibacterial Butyrolactones

Keywords:

Novel butyrolactone analogues were designed and synthesized based on the known lichen antibacterial compounds, lichesterinic acids (B-10 and B-11), by substituting different functional groups on the butyrolactone ring trying to enhance its activity. All synthesized butyrolactone analogues were evaluated for their in vitro antibacterial activity against Streptococcus gordonii. Among the derivatives, B-12 and B-13 had the lowest MIC of 9.38 µg/mL where they have shown to be stronger bactericidals, by 2-3 times, than the reference antibiotic, doxycycline. These two compounds were then checked for their cytotoxicity against human gingival epithelial cell lines, Ca9-22, and macrophages, THP-1, by MTT and LDH assays which confirmed their safety against the tested cell lines. A preliminary study of the structure-activity relationships unveiled that the functional groups at the C4 position had an important influence on the antibacterial activity. An optimum length of the alkyl chain at the C5 position registered the best antibacterial inhibitory activity however as its length increased the bactericidal effect increased as well. This efficiency was attained by a carboxyl group substitution at the C4 position indicating the important dual role contributed by these two substituents which might be involved in their mechanism of action.

Introduction

The usages of antibiotics on a large scale alongside their misapplication have lead to the emergence of resistant pathogenic bacteria. 1 Both, the infection of these re-emergent strains which has increased the global mortality rate to be a growing concern and the global reduction in antibiotics production open a new era where other potent candidates should be found to fight against bacteria. 2 Indeed, an infinite number of plant species have been tested against a huge number of bacterial strains in vitro. In addition, many phytochemicals found effective against a broad spectrum of microorganisms comprising fungi, yeast and bacteria were uncovered. 3 Throughout the last 2 decades, plants are becoming a famous rich source of antimicrobial substances. 4 Furthermore, many other promising drug sources still need to be explored. 5 Lichens which are symbiotic organisms comprising a fungus and a photosynthetic alga and/or cyanobacterium constitutes a potential source of over 1000 distinct secondary metabolites. 6 They comprise antitumor, antiviral and antimicrobial activities. [6][7][8][9] Sensitive as well as several multi-drug resistant bacterial strains were shown to be susceptible to these lichen compounds. 6 Streptococcus gordonii (S. gordonii) is an eminent member of the viridans streptococci large category. Not only was this bacteria described as an agent of septic arthritis but also it can colonise damaged heart valves and represents the primary etiological agent of subacute bacterial endocarditis. 10 In the oral cavity, S. gordonii adhere to the salivary pellicle which coats the teeth, proliferate and excrete an extracellular polysaccharide matrix protecting their developing microcolony on which secondary colonizers will adhere. 11 The late colonizing strains such as Porphyromonas gingivalis bind the sites provided by S. gordonii and form a highly pathogenic complex microbial community. 12,13 S. gordonii as a pioneer initial colonizer initiates the formation of dental plaques contributing in turn to the onset of dental caries and periodontal diseases as well as their progression. 14,15 Inhibiting S. gordonii might block the successive steps leading to acute oral diseases and this may constitute prevention rather than a risky cure after biofilm formation. To address this oral issue, we synthesized a natural butyrolactone, L-lichesterinic acid. Cavalito et al have extracted it from the lichen, Cetraria islandica, and shown to have an activity against Streptococcus hemolyticus and Staphylococcus aureus. 16 Our goal in this study is to evaluate its antibacterial activity against S. gordonii in solid and liquid media under anaerobic conditions. Trying to enhance its activity, some derivatives were synthesized and tested (Fig. 1). Finally, the cytotoxic effect of the most active compounds was

A RT I C L E I N F O

A BS T RA C T evaluated on two human cell lines, gingival epithelial cells, Ca9-22, and macrophages, THP-1. To the best of our knowledge, it is the first study to describe some of these synthetic derivatives, their antibacterial activity against S. gordonii and their cytotoxic effects.

Results and Discussion

Chemistry

Scheme 1 illustrates the synthesis of enantiopure (-)-lichesterinic acid B-10 and its derivatives. This straightforward asymmetric synthesis has already been described by Braukmüller and Brückner in 2006 for the preparation of paraconic acids. 17 To the best of our knowledge, this strategy had been used only for the synthesis of naturally aliphatic -methylene butyrolactone (+)-methylenolactocin (R = C 5 H 11 ) and (+)-protolichesterinic acid (R = C 13 H 27 ) and their (-) enantiomers (Fig. 2). Based on a six steps method, one additional step is required to obtain a series of lichesterinic acid derivatives by isomerization of the double bond. Moreover, this lactone strategy has been extended to include different alkyl chain lengths R (C 7 H 15 , C 9 H 19 , C 15 H 31 , and C 16 H 33 ). Briefly, it began with the preparation of hydroxyl lactones 1a-e where the enantiocontrol was imposed by the asymmetric dihydroxylation of trans-configured -unsaturated carboxylic ester with AD mix-® or AD mix-®. The resulting lactones were deshydrated giving butenolides 2a-e. For the two next steps we modified the approach according to Perepogu et al. 18 A Gilman addition of a vinyl group was added trans-selectively to the C=C bond giving vinyl lactones 3a-e, followed by an oxidation of the double bond allowing access to HO 2 C-substituted lactones 4a-e. Activation by Stiles' reagent, followed by amino-methylation in situ fragmentation provided the -methylene butyrolactones 5a-e. Then, the target enantiopure lichesterinic acid derivatives 6a-e were obtained by isomerization of the double bond using NEt 3 in DMF. This synthesis is achieved in seven steps and around 10% overall yield with good enantioselective excess determined by chiral HPLC.

Agar dilution

Seven out of the thirteen butyrolactones screened (Fig. 1) showed an activity with the concentrations tested against S. gordonii under anaerobic conditions (Table 1). Compounds B-2, B-4, B-5, B-6, B-7, and B-13 didn't exhibit any activity. The least active compounds were B-1 and B-3 showing the highest Minimal Inhibitory Concentration (MIC) of 300 µg/mL. Then, the MIC decreased to be 200 µg/mL for B-12 and continued decreasing to pass by 150 µg/mL for B-8 and B-9 and reaches the lowest value with B-10 and B-11 registering 90 µg/mL (Fig. 3A, Table 1). Alongside, doxycycline displayed an MIC of 0.41 µg/mL which was fixed and used always as a positive control (Fig. 3B, Table 1). In addition, the mixture of the solvents (DMSO + methanol) used to dissolute our compounds was found inactive at the highest concentration tested. These results were taken into the liquid medium to confirm and compare. The highest inhibitory activity was for B-10 and B-11 which registered the same results with MIC = 4.69 µg/mL and Minimal Bactericidal Concentration (MBC) = 18.75 µg/mL. While MIC increased to be 9.38 µg/mL for B-12, its MBC remained at the same value. B-13 showed the same MIC as B-12 and it was also its MBC exhibiting the strongest killing effect. Doxycycline activity decreased here to have MIC = 0.51 µg/mL and MBC = 32.8 µg/mL which were fixed and used always as positive control. If we compare butyrolactones to doxycycline antibiotic, we can notice that the latter's MBC was higher than that of B-12 and B-13 by 2 or 3 times, respectively (Table 2). For the next experiments we have selected B-12 and B-13 because B-10 and B-11 are already known natural compounds.

Table 1. Minimal Inhibitory Concentration of butyrolactones against S. gordonii by agar dilution

Cytotoxicity

B-10 and B-11 were already described so we have chosen B-12 and B-13 to check their cytotoxicity. The viability of gingival epithelial cells, Ca9-22, and macrophage-like cells, THP-1, was evaluated by LDH and MTT assays (Fig. 4A,4B).

LDH assay

The positive control, Triton 1%, was considered as the maximum with 100 percent cytotoxicity (Fig. 4A). Cells alone displayed 9 and 18% of LDH release from Ca9-22 and THP-1, respectively. No significant difference was displayed between the cells treated with the compounds and the cells alone showing around the same percentages of cell death. Regarding Ca9-22, 9.6 and 12% were found for B-12, B-13, respectively. With the same order, 20 and 10% of cell death were registered for THP-1 (Fig. 4A).

MTT assay

Cells alone were considered the maximum with 100% cell viability (Fig. 4B). Triton 1% was the positive control which showed a significant different result decreasing the cell population into around 3% for both types of cells. The compounds were fluctuating in a very close range around 100% and their activities were not significantly different from those against the cells alone. With respect to Ca9-22, 100 and 96% were displayed by B-12, B-13, respectively. Following the same pattern, 107 and 106% were found for THP-1 (Fig. 4B).

Structure-activity relationships

To analyze structure-activity relationships, four structural components were considered: the saturation of the C 3 -C 4 bond, the nature of the substituent (X) and (Y) at the C 3 and C 4 position respectively, and the length of the alkyl chain (R) (Fig. 5). Starting with the Agar dilution assay, the MIC was controlled by the length of the R alkyl chain. As the length of R increased, the activity increased. This was observed between C 9 H 19 chain compounds B-8 and B-9, which had MIC = 150 µg/mL and C 13 H 27 chain counterparts or lichesterinic acids (B-10 and B-11), which had MIC = 90 µg/mL (Table 1). Hence, we decided to test shorter and longer chains to confirm our hypothesis. The C 7 H 15 chain compound B-7, as well as B-13, with the longest chain C 16 H 33 , didn't show any activity at the highest concentration tested. But, B-12 which was shorter than B-13 by only 1 carbon atom, showed an MIC = 200 µg/mL. This was interpreted as that the length of the chain plays an important role in its activity where there is an optimum length of 13 carbon atoms which has the highest potential. As the chain length increases or decreases, the activity decreases as well (Table 1).

In broth microdilution assay, all of the compounds were found active except B-7 and exhibited better activity than in solid medium assay (Table 2). This discrepancy between the two media was also shown by Guzman et al. They screened natural products from Columbian plants against Mycobacterium tuberculosis and they obtained different activity between the two media. 19 Therefore, we can propose that butyrolactones can move more freely due to their lipophilicity and inhibit more efficiently in liquid medium compared to the other different physiological solid state, where the bacteria are confined to the surface. The liquid results confirm the hypothesis of being C 13 H 27 chain is the optimum length. B-7 with a C 7 H 15 alkyl group did not show any activity and the effect decreased when the chain length increased. Comparing to B-10, MIC increased to be 9.38 µg/mL for B-12 but the MBC remains the same. Then, when the chain length increases more to be 16 carbon atoms, B-13 showed the same MIC as B-12 but its MBC was the strongest to exhibit a surprising effect that the liquid medium owns. The latter elects an optimum chain length of 13 carbon atoms for the best inhibition but the killing effect increases as the R chain increases in length since their lipophilicity increases as well. This chain length contribution was discussed by Yang et al where they tested the derivatives of 8-alkylberberine against gram-positive and gram-negative strains to find an optimum length of 8 carbon atoms. Shorter or longer chains showed lowering in the antibacterial activity. They also mentioned that gram-positive strains were more susceptible to these derivatives. 20 It can be proposed that the saturated chain, R, may be involved in the butyrolactones mechanism of action. The optimum length hypothesis was also supported by comparing other compounds which can be grouped into 3 couples having the same main structure differing only in the length of the R chain, B-1/B-2, B-3/B-4 and B-5/B-6, where when the length increased to 13 carbon atoms the activity increased. MIC and MBC decreased by half comparing B-1 to B-2, from 150 and 300 µg/mL for B-1 into 75 and 150 µg/mL for B-2, respectively. Also, the other couples were displaying the same effect. We can see clearly that MIC and MBC decreased from 150 and 300 µg/mL for B-3 and B-5 into 75 µg/mL for B-4 and B-6, respectively. So, the activity is better regarding the couples, B-3/B-4 and B-5/B-6, than the first couple, B-1/B-2, highlighting the drawback of the introduction of a hydrophilic group for the antibacterial activity of these butyrolactones. We can also notice that the enantiomers showed the same antibacterial effect comparing the value between B-8 and B-9 and B-10 and B-11.

In addition to the R chain, substitution of different functional groups at C 4 position constitutes a second factor affecting the antibacterial activity. These groups divided our compounds into four classes. The first class, B-1 and B-2, comprised a hydroxyl group, OH, the second class, B-3 and B-4, comprised a double bond in the ring with no substitutions, the third class, B-5 and B-6, possessed a vinyl group with a saturated ring and finally, the fourth class, B-8, B-9, B-10, B-11, B-12, and B-13, contained an unsaturated ring with two substituents, carboxyl and methyl groups. Since the highest activity was demonstrated for the last class, this suggests that the carboxyl group may stand behind this potency. The importance of this functional group was mentioned by Sebastianes et al who tested the antibacterial activity of a fungal compound, 3-hydroxy propionic acid, 3-HPA, against Staphylococcus aureus and Salmonella typhi. Indeed, 3-HPA showed relevant antibacterial activity against the tested strains. When it was esterified to produce 3-hydroxypropanoic ethyl ester, no antimicrobial activity was registered. 21 This gives a complementary idea for the probable underpinning mechanism of action in which the carboxylic group and the R chain could be implicated (Fig. 5). Moreover, these butyrolactones have a similar structure to the -butyrolactone autoregulators described formerly. The latter are produced by the gram-positive Streptomyces genus and they regulate the DNA binding activity of cognate receptor proteins triggering antibiotic production as mentioned by Kitani et al. [START_REF] Kitani | Proc. Natl. Acad. Sci[END_REF] Hence, lichesterinic acid and its analogues may modulate the DNA binding activity of some proteins.

Conclusions

To conclude, all butyrolactone derivatives were synthesized in good yield with an efficient enantioselective strategy. All compounds were then screened for their antibacterial activity against S. gordonii in solid and liquid media using agar dilution and broth microdilution methods, respectively. The compounds have shown a stronger activity in the liquid medium than in the solid one where only B-7 was found not active. The alkyl chain of 13 carbons showed the best inhibitory activity with an MIC of 4.69 µg/mL. Among the derivatives, B-12 and B-13 were the best promising compounds registering a better bactericidal activity than the reference antibiotic used, doxycycline, by 2 or 3 times, respectively. This chain alongside the carboxyl functional group may be involved in their mechanism of action. Finally, B-12 and B-13 were evaluated for their cytotoxicity against human gingival epithelial cells, Ca9-22, and macrophages, THP-1, and found not toxic. This gives a bright hope to continue with these two butyrolactones into their antibiofilm activity for their graduation as new oral antibiotic agents. These new compounds are capable to inhibit S. gordonii which may block the successive steps leading to oral complications, thus, a safe prevention rather than a risky late treatment after biofilm formation.

Experimental

Chemistry

All reagents of high quality were purchased from commercial suppliers and used without further purification. Melting points were recorded on a Kofler Leica VMHB melting point apparatus and are uncorrected. IR spectra were obtained with PerkinElmer UATR Two infrared spectrophotometer. 1 H (300 MHz) and 13 C (75 MHz) NMR spectra were performed on a Bruker DMX 300 spectrometer.

Chemical shifts were referenced to the residual solvent signal (CDCl 3 : H = 7.26,  C = 77.0). The  values are given in parts per million (ppm), and the coupling constants (J values) are given in Hertz (Hz). The multiplicity of the signals is reported as s (singlet), d (doublet), t (triplet), q (quadruplet), m (multiplet). ESI-HRMS were carried out on a MICROMASS ZabspecTOF spectrometer for electrospray ionization at the CRMPO (Centre Régional de Mesures Physiques de l'Ouest), University of Rennes 1. Elemental analyses were performed on a microanalysor Flash EA1112 CHNS/O Thermo Electron at the CRMPO. Optical rotations were measured on a Perkin Elmer Model 341 polarimeter at 20 °C using thermostable optical glass cell (1 dm path length and c in g/100 mL). The ee values were determined by chiral LC with a TSP Spectra System UV2000 and P1000 XR apparatus with a CHIRALPACK® IC or IA column. Reactions were monitored by TLC on Merk 60 F254 (0.25 mm) plates which were visualized by UV detection or sprayed with vanilline or KMNO 4 solutions, then heated.

General procedures for compounds 1a-e

A mixture of the appropriate aldehyde (28.1 mmol), monoethyl malonate (3.32 mL, 3.72 g, 28.1 mmol, 1.0 eq.) and NEt 3 (3.92 mL, 2.85 g, 28.1 mmol, 1.0 eq.) was heated at 90-95 °C under argon atmosphere. After stirring overnight, the reaction mixture was cooled at room temperature and poured at 0 °C into an aq. H 2 SO 4 solution (20%, 100 mL). The organic phase was separated and the aqueous phase was extracted with diethyl ether (3 x 100 mL). The combined organic phases were dried with MgSO 4 . After filtration, the solution was concentrated under vacuum to give quantitatively the -unsaturated carboxylic ester. Then a mixture of -unsaturated carboxylic ester (28.1 mmol), AD mix-® (39.34 g), methanesulfonamide (2.67 g, 28.1 mmol, 1.0 eq.) was added to a 1:1 mixture of tBuOH and H 2 O (180 mL) at 0 °C. After 40 h the reaction was quenched by adding a satd. aqueous solution of Na 2 SO 3 (100 mL), this solution was stirred for 1 h before extraction with diethyl ether (3 x 50 mL). The combined organic layers were dried with MgSO 4 and concentrated under vacuum. The residue was purified by chromatography on silicagel using ethyl acetate/petroleum ether 2:8 and then 4:6 as eluents.

(4S,5S)-5-heptyl-4-hydroxy-dihydrofuran-2(3H)-one (4S,5S-1a).

Yield: 64%; white solid; m.p. 75 °C. R f = 0.26 (petroleum ether/ethyl acetate 6:4). [α] D = -47.6 (c 1.07, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.89 (t, J = 6.2 Hz, 3H), 1.28-1.93 (m, 12H), 2.55 (d, J = 17.7 Hz, 1H), 2.80 (dd, J = 5.4 Hz and J = 17.7 Hz, 1H), 4.34-4.40 (m, 1H), 4.46-4.48 (m, 1H) ppm. 13

(4S,5S)-4-hydroxy-5-nonyl-dihydrofuran-2(3H)-one (4S,5S-1b).

This compound was prepared as published [START_REF] Berkenbusch | [END_REF] . Yield = 63%. [α] D = -38.9 (c 1.15, CHCl 3 ). HRMS (ESI, m/z): Calcd. for C 13 H 24 O 3 Na: 251.16231, found [M+Na] + : 251.1618.

(4R,5R)-4-hydroxy-5-nonyl-dihydrofuran-2(3H)-one (4R,5R-1b) (B-1).

Yield = 63%; white solid; m.p. 69 °C. R f (petroleum ether/ethyl acetate 8:2) = 0.10. [α] D = + 44.2 (c 1.07, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.27-1.95 (m, 16H), 2.55 (dd, J = 0.9 Hz and J = 17.7 Hz, 1H), 2.80 (dd, J = 5.4 Hz and J = 17.7 Hz, 1H), 4.33-4.39 (m, 1H), 4.46-4.51 (m, 1H) ppm. 13

(4S,5S)-4-hydroxy-5-tridecyl-dihydrofuran-2(3H)-one (4S,5S-1c).

This compound was prepared as published. 17

(4R,5R)-4-hydroxy-5-tridecyl-dihydrofuran-2(3H)-one (4R,5R-1c) (B-2).

This compound was prepared as published. 17 13

(4S,5S)-5-hexadecyl-4-hydroxy-dihydrofuran-2(3H)-one (4S,5S-1e).

Yield = 52%; white solid; m.p. 99 °C. R f (petroleum ether/ethyl acetate 8:2) = 0.10. [α] D = -34.6 (c 1.04, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.6 Hz, 3H), 1.25-1.93 (m, 30H), 2.56 (d, J = 17.7 Hz, 1H), 2.80 (dd, J = 5.4 Hz and J = 17.7 Hz, 1H), 4.34-4.39 (m, 1H), 4.46-4.49 (m, 1H) ppm. 13

General procedure for compounds 2a-e

At 0 °C under argon atmosphere NEt 3 (5.33 mL, 3.87 g, 38.2 mmol, 2.2 eq.) and methanesulfonyl chloride (1.48 mL, 2.19 g, 19.1 mmol, 1.1 eq.) were added dropwise to a solution of hydroxylactone 1 (17.4 mmol) in anhydrous CH 2 Cl 2 (85 mL). After sttirring for 1 h the reaction was quenched by adding a satd. aqueous solution of NH 4 Cl (150 mL). The organic phase was separated and the aqueous phase was extracted with diethyl ether (3 x 50 mL). The combined organic layers were dried with MgSO 4 and concentrated under vacuum. The residue was purified by chromatography on silicagel using diethyl ether/petroleum ether 1:4 as eluents.

(S)-5-heptylfuran-2(5H)-one (5S-2a).

Yield=94%; colorless oil. R f (petroleum ether/diethyl ether 8:2) = 0.17.

[α] D = + 77.8 (c 1.21, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.28-1.83 (m, 12H), 5.02-5.07 (m, 1H), 6.11 (dd, J = 2.0 Hz and J = 5.7 Hz, 1H), 7.46 (dd, J = 1.45 Hz and J = 5.7 Hz, 1H) ppm. 13

(S)-5-nonylfuran-2(5H)-one (5S-2b).

This compound was prepared as published. [START_REF] Berkenbusch | [END_REF]

(R)-5-nonylfuran-2(5H)-one (5R-2b) (B-3).

Yield=91%; colorless oil. R f (petroleum ether/diethyl ether 8:2) = 0.16. [α] D = -62.7 (c 1.17, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.82 (m, 16H), 5.01-5.07 (m, 1H), 6.11 (dd, J = 2.0 Hz and J = 5.7 Hz, 1H), 7.46 (dd, J = 1.5 Hz and J = 5.7 Hz, 1H) ppm. 13

(S)-5-tridecylfuran-2(5H)-one (5S-2c).

This compound was prepared as published. 17

(R)-5-tridecylfuran-2(5H)-one (5R-2c) (B-4

). This compound was prepared as published. 17

(S)-5-pentadecylfuran-2(5H)-one (5S-2d

). Yield = 75%; white solid; m.p. 67 °C. R f (petroleum ether/ethyl acetate 7:3) = 0.78. [α] D = + 48.9 (c 1.17, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.83 (m, 28H), 5.01-5.07 (m, 1H), 6.10 (dd, J = 1.9 Hz and J = 5.7 Hz, 1H), 7.46 (dd, J = 1.4 Hz and J = 5.7 Hz, 1H) ppm. 13

(S)-5-hexadecylfuran-2(5H)-one (5S-2e).

Yield = 91%; white solid; m.p. 74 °C. R f (petroleum ether/diethyl ether 8:2) = 0.26. [α] D = + 50.7 (c 1.04, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.5 Hz, 3H), 1.25-1.83 (m, 30H), 5.01-5.06 (m, 1H), 6.11 (dd, J = 2.0 Hz and J = 5.7 Hz, 1H), 7.45 (dd, J = 1.4 Hz and J = 5.7 Hz, 1H)ppm. 13

General procedure for compounds 3a-e

To a -78 °C solution of CuI (15.41 g, 80.9 mmol, 5.0 eq.) in anhydrous tetrahydrofuran (THF, 120 mL) was slowly added a solution of methyl lithium (50.6 mL, 80.9 mmol of 1.6 M, 5.0 eq.). The suspension was allowed to reach rt during 15 min and then cooled to -78 °C and vinyl magnesium bromide (80.9 mL, 80.9 mmol of 1 M, 5.0 eq.) was added. The mixture was stirred for 15 min at -78 °C and then butenolide 2 (16.2 mmol) in THF (20 mL) was added dropwise. The resulting black solution was allowed to warm at rt and was stirred for 2 h. Then the mixture was poured into 500 mL of vigorously stirred satd. aqueous NH 4 Cl solution. The pH of solution was adjusted to 8-10 by addition of conc NH 4 OH. The mixture was stirred at rt for 1.5 h until all the copper salts had dissolved. The blue solution was extracted with diethyl ether, the organic phase was filtered through celite and then washed with a solution of ethylenediamine (50 mL with 450 mL H 2 O). The organic layer was dried with MgSO 4 and concentrated under vacuum. The residue was purified by chromatography on silicagel using petroleum ether/ethyl acetate 9.5:0.5 as eluents.

(4S,5S)-5-heptyl-4-vinyl-dihydrofuran-2(3H)-one (4S,5S-3a

). Yield = 81%; colorless oil. R f (petroleum ether/ethyl acetate 9.5:0.5) = 0.27. [α] D = -63.1 (c 1.13, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.6 Hz, 3H), 1.28-1.76 (m, 12H), 2.45 (dd, J = 10.5 Hz and J = 17.6 Hz, 1H), 2.68 (dd, J = 8.2 Hz and J = 17.0 Hz, 1H), 2.73-2.85 (m, 1H), 4.11-4.18 (m, 1H), 5.14-5.21 (m, 2H), 5.67-5.79 (m, 1H) ppm. 13

(4S,5S)-5-nonyl-4-vinyl-dihydrofuran-2(3H)-one (4S,5S-3b

). Yield = 62%; colorless oil. R f (petroleum ether/ethyl acetate 9.5:0.5) = 0.14. [α] D = -54.5 (c 1.19, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.78 (m, 16H), 2.45 (dd, J = 10.3 Hz and J = 16.9 Hz, 1H), 2.68 (dd, J = 8.2 Hz and J = 16.9 Hz, 1H), 2.73-2.84 (m, 1H), 4.11-4.17 (m, 1H), 5.15-5.21 (m, 2H), 5.67-5.78 (m, 1H) ppm. 13

(4R,5R)-5-nonyl-4-vinyl-dihydrofuran-2(3H)-one (4R,5R-3b) (B-5).

Yield = 70%; colorless oil. R f (petroleum ether/ethyl acetate 9.5:0.5) = 0.14. [α] D = + 55.6 (c 1.27, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.78 (m, 16H), 2.45 (dd, J = 10.3 Hz and J = 16.9 Hz, 1H), 2.68 (dd, J = 8.2 Hz and J = 16.9 Hz, 1H), 2.73-2.84 (m, 1H), 4.11-4.17 (m, 1H), 5.15-5.21 (m, 2H), 5.67-5.78 (m, 1H) ppm. 13

(4S,5S

)-5-tridecyl-4-vinyl-dihydrofuran-2(3H)-one (4S,5S-3c). Yield = 77%; white solid; m.p. 48 °C. R f (petroleum ether/ethyl acetate 9:1) = 0.51. [α] D = -42.9 (c 1.20, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.5 Hz, 3H), 1.26-1.80 (m, 24H), 2.44 (dd, J = 10.3 Hz and J = 16.9 Hz, 1H), 2.68 (dd, J = 8.2 Hz and J = 16.9 Hz, 1H), 2.72-2.81 (m, 1H), 4.11-4.17 (m, 1H), 5.15-5.21 (m, 2H), 5.66-5.78 (m, 1H) ppm. 13

(4R,5R)-5-tridecyl-4-vinyl-dihydrofuran-2(3H)-one (4R,5R-3c) (B-6

). This compound was prepared as published. 24

(4S,5S)-5-pentadecyl-4-vinyl-dihydrofuran-2(3H)-one (4S,5S-3d).

Yield = 80%; white solid; m.p. 61 °C. R f (petroleum ether/ethyl acetate 9:1) = 0.43. [α] D = -34.6 (c 1.33, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.71 (m, 28H), 2.45 (dd, J = 10.4 Hz and J = 17.0 Hz, 1H), 2.69 (dd, J = 8.2 Hz and J = 16.9 Hz, 1H), 2.75-2.81 (m, 1H), 4.11-4.17 (m, 1H), 5.15-5.21 (m, 2H), 5.66-5.78 (m, 1H) ppm. 13

(4S,5S)-5-hexadecyl-4-vinyl-dihydrofuran-2(3H)-one (4S,5S-3e).

Yield = 65%; white solid; m.p. 65 °C. R f (petroleum ether/ethyl acetate 9:1) = 0.38. [α] D = -41.3 (c 1.04, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.76 (m, 30H), 2.45 (dd, J = 10.3 Hz and J = 16.9 Hz, 1H), 2.68 (dd, J = 8.2 Hz and J = 17.0 Hz, 1H), 2.75-2.84 (m, 1H), 4.11-4.17 (m, 1H), 5.15-5.21 (m, 2H), 5.67-5.78 (m, 1H) ppm. 13

General procedure for compounds 4a-e

To a stirred solution at rt of vinyl lactone 3 (13.1 mmol) in a solvent mixture of CH 3 CN/CCl 4 /H 2 O (14:14:21 mL) were added NaIO 4 (11.19 g, 52.3 mmol, 4.0 eq;) and RuCl 3 (0.27 g, 1.31 mmol, 0.1 eq.). After 3 h at rt, CH 2 Cl 2 was added and the aq. phase was separated and extracted with CH 2 Cl 2 . The combined organic layers were filtered once through celite and then through silicagel+celite. The filtrate was concentrated under vacuum and the residue was diluted with diethyl ether (200 mL) and satd. NaHCO 3 solution (200 mL) was added. After separation of the two phases, the aq. phase was acidified with HCl 1M until pH = 2. The product was extracted with CH 2 Cl 2 , dried with MgSO 4 and concentrated under vacuum to yield the desired compounds in pure form.

(2S,3R)-2-heptyl-5-oxo-tetrahydrofuran-3-carboxylic acid (4R,5S-4a

). Yield = 80%; white solid; m.p. 110 °C. R f (CH 2 Cl 2 /MeOH 9:1) = 0.48. [α] D = -29.2 (c 1.13; CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.4 Hz, 3H), 1.28-1.83 (m, 12H), 2.78-3.00 (m, 2H), 3.07-3.15 (m, 1H), 4.60-4.69 (m, 1H) ppm. 13

(2S,3R)-5-oxo-2-tridecyl-tetrahydrofuran-3-carboxylic acid (4R,5S-4c

). Yield = 71%; white solid; m.p. 114 °C; ref. 17

(2R,3S)-5-oxo-2-tridecyl-tetrahydrofuran-3-carboxylic acid (4S,5R-4c

). Yield = 66%; white solid; m.p. 114 °C; ref. 17 Yield = 92%; white solid; m.p. 108 °C. R f (CH 2 Cl 2 /MeOH 9:1) = 0.33. [α] D = -29.9 (c 1.12, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.82 (m, 26H), 1.72-1.82 (m, 2H), 2.77-2.99 (m, 2H), 3.06-3.14 (m,1H), 4.59-4.65 (m, 1H) ppm. 13

(2S,3R)-5-oxo-2-hexadecyl-tetrahydrofuran-3-carboxylic acid (4S,5R-4e).

Yield = 95%; brownish solid; m.p. 114 °C. R f (CH 2 Cl 2 /MeOH 9:1) = 0.35. [α] D = -29.0 (c 0.98, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.6 Hz, 3H), 1.27-1.56 (m, 28H), 1.72-1.81 (m, 2H), 2.79 (dd, J = 9.8 Hz and J = 17.8 Hz, 1H), 2.90 (dd, J = 8.3 Hz and J = 17.8 Hz, 1H), 3.04-3.12 (m, 1H), 4.58-4.64 (m, 1H) ppm. 13

General procedure for compounds 5a-e

A mixture of carboxy lactone 4 (7.88 mmol) in a solution of methoxymagnesium monomethylcarbonate MMC (150 mL, 299.6 mmol of 2M in DMF, 38 eq.) was heated at 135-140 °C for 70 h under argon atmosphere. After the system had cooled to rt, the reaction was quenched by adding a solution of HCl 10% (150 mL), this solution was then extracted with CH 2 Cl 2 (3 x 100 mL). The combined organic layers were dried with MgSO 4 and concentrated under vacuum. The resulting brown oil was dissolved in a solvent mixture of acetic acid (38 mL), formaldehyde (28 mL), N-methylaniline (9.8 mL) and NaOAc (1,13 g). After stirring for 2 h at rt, the mixture was poured into a solution of HCl 10%. The solution was extracted with tert-butyl methyl ether (3x100 mL), the combined organic layers were washed once with brine and then three times with H 2 O. The organic layer was dried with MgSO 4 and concentrated under vacuum. The residue was purified over silicagel Geduran®Si 60 (diethyl ether/petroleum ether/acetic acid 3:7:0.2) to yield compounds 5.

(2S,3R)-2-heptyl-4-methylene-5-oxo-tetrahydrofuran-3-carboxylic acid (4R,5S-5a).

Yield = 30%; white solid; m.p. 71 °C. R f (petroleum ether/diethyl ether/acetic acid 8:2:0.2) = 0.13. [α] D = -11.4 (c 1.10, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.4 Hz, 3H), 1.28-1.79 (m, 12H), 3.62-3.66 (m, 1H), 4.82 (d, J = 5.3 Hz, 1H), 6.03 (d, J = 2.4 Hz, 1H), 6.46 (d, J = 2.7 Hz, 1H) ppm. 13

(2R,3S)-4-methylene-2-nonyl-5-oxo-tetrahydrofuran-3-carboxylic acid (4S,5R-5b).

Yield = 41%; white solid; m.p. 93 °C. 25 R f (petroleum ether/diethyl ether/acetic acid 8:2:0.2) = 0.20. [α] D = + 12.82 (c 0.975, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.87 (t, J = 6.6 Hz, 3H), 1.26-1.50 (m,14H), 1.71-1.76 (m, 2H), 3.61-3.63 (m, 1H), 4.81 (dt, J = 5.8 Hz and J = 7 Hz, 1H), 6.02 (d, J = 2.6 Hz, 1H), 6.46 (d, J = 3.0 Hz, 1H), 10.58 (s, 1H) ppm. 13

(2S,3R)-4-methylene-2-nonyl-5-oxo-tetrahydrofuran-3-carboxylic acid (4R,5S-5b).

Yield = 54%; white solid; m.p. 93 °C. R f (petroleum ether/diethyl ether/acetic acid 8:2:0.2) = 0.20. [α] D = -6.3 (c 0.98, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.87 (t, J = 6.6 Hz, 3H), 1.26-1.50 (m, 14H), 1.71-1.76 (m, 2H), 3.61-3.63 (m, 1H), 4.81 (dt, J = 5.8 Hz and J = 7 Hz, 1H), 6.02 (d, J = 2.6 Hz, 1H), 6.46 (d, J = 3.0 Hz, 1H), 10.58 (s, 1H) ppm. 13 This compound was prepared as published. 17 Yield = 36%. m.p. 108 °C, ref. 17 This compound was prepared as published. 17 Yield = 28%. m.p. 108 °C, ref. 17 : 104-105 °C. [α] D = -7.9 (c 0.99, CHCl 3 ), ref. 17

General procedure for compounds 6a-e

To a solution of compound 5 (1.99 mmol) in anhydrous DMF (17 mL) under argon atmosphere, was added NEt 3 (279 µL, 1.99 mmol, 1 eq.). After stirring overnight at rt, the reaction was quenched by adding a solution of HCl 1M, this solution was then extracted with diethyl ether (3 x 50 mL). The combined organic layers were washed with H 2 O, dried with MgSO 4 and concentrated under vacuum. The residue was purified over silicagel Geduran®Si 60 (diethyl ether/petroleum ether/acetic acid 3:7:0.2) to yield the desired compound 6.

(S)-2-heptyl-4-methyl-5-oxo-2,5-dihydrofuran-3-carboxylic acid (5S-6a) (B-7

). Yield = 71%; white solid; m.p. 120 °C. R f (petroleum ether/diethyl ether/acetic acid 8:2:0.2) = 0.21. [α] D = -11.4 (c 1.10, CHCl 3 ), 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.27-1.67 (m, 12H), 2.08-2.19 (m, 1H), 2.25 (d, J = 2.1 Hz, 3H), 5.11-5.15 (m, 1H) ppm. 13 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.67 (m, 16H), 2.09-2.16 (m, 1H), 2.24 (d, J = 2.1 Hz, 3H), 5.12-5.13 (m, 1H) ppm. 13 C NMR (CDCl 3 , 75 MHz) δ 11.0, 14.1, 22.6, 24.7, 29.1, 29.2, 29.3, 29.4, 31.8, 32.7, 81.4, 139.7, 146.7, 165.5, 172.6 26 Yield = 67%; white solid; m.p. 117 °C. R f (petroleum ether/diethyl ether/acetic acid 8:2:0.2) = 0.29. [α] D = + 37.3 (c 1.04, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.67 (m, 16H), 2.09-2.16 (m, 1H), 2.24 (d, J = 2.1 Hz, 3H), 5.12-5.13 (m, 1H) ppm. 13 + 0.1% HCOOH, 210 nm, 1 mL/min, 0.5 mg/mL. T R = 6.91 min, ee = 91%.

(S)-4-methyl-5-oxo-2-pentadecyl-2,5-dihydrofuran-3-carboxylic acid (5S-6d) (B-12).

Yield = 89%; white solid; m.p. 120 °C. R f (petroleum ether/diethyl ether/acetic acid 6:3.8:0.2) = 0.38. [α] D = -21.8 (c 0.98, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.5 Hz, 3H), 1.25-1.64 (m, 28H), 2.09-2.20 (m, 1H), 2.25 (d, J = 1.8 Hz, 3H), 5.12-5.14 (m, 1H) ppm. 13 Chiralpak IA, 100% ACN + 0.1% HCOOH, 210 nm, 1 mL/min, 0.5 mg/mL. T R = 7.58 min, ee = 86%.

(S)-2-hexadecyl-4-methyl-5-oxo-2,5-dihydrofuran-3-carboxylic acid (5S-6e) (B-13).

Yield = 62%; white solid; m.p. 125 °C. R f (petroleum ether/diethyl ether/acetic acid 6:3.8:0.2) = 0.38. [α] D = -25.9 (c 1.015, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.67 (m, 30H), 2.07-2.18 (m, 1H), 2.24 (d, J = 2.1 Hz, 3H), 5.09-5.13 (m, 1H) ppm. 13 Chiralpak IA, 100% ACN + 0.1% HCOOH, 210 nm, 1 mL/min, 0.5 mg/mL. T R = 8.09 min, ee > 99%.

Chemical compounds

All butyrolactones were dissolved in pure DMSO and then diluted with pure methanol to get 3 µg/mL and to reach a final concentration of DMSO less than 25%. After that, they were filter sterilized through a 0.22-μm-pore-size filter.

Biological activity

Bacterial culture

Streptococccus gordonii DL1 was used in this study 29 . Brain-heart infusion broth (BHI) (DIFCO, France) and/or blood Columbia agar plates (AES Chemunex, France) supplemented with hemin (5 µg/mL) and menadione (1 µg/mL) (Sigma Aldrich, France) were used for its growth. S. gordonii was grown under anaerobic conditions (N 2 -H 2 -CO 2 [80:10:10]) at 37 °C to mimic the conditions created by the microorganisms colonizing the tooth surface rendering it rapidly anaerobic. 30

Cell lines

Two different human cell lines were chosen: a gingival epithelial carcinoma cell line, Ca9-22 (Health Science Research Resources Bank, Osaka, Japan) and a macrophage-like monocytic leukemia cell line, THP-1. Ca9-22 cells were grown in Dulbecco's Modified Eagle Medium (DMEM) (Lonza, France) whereas RPMI 1640 medium with sodium pyruvate (1 mM) and Hepes buffer (1M) (Sigma Aldrich) was used for THP-1 growth. Both lines were grown in a 5% CO 2 atmosphere at 37 °C and their media were supplemented with L-Glutamine (2 mM), 10% heat-inactivated fetal bovine serum (FBS, Lonza, France) and antibiotics (penicillin 100 mg/mL and streptomycine 50 mg/mL) (Sigma Aldrich). For THP-1 differentiation into macrophages, Phorbol 12-myristate 13-acetate (PMA) (Sigma Aldrich) was used at 10 ng/mL for 72 hours.

Antibacterial Assay

Agar dilution

Agar dilution assay was chosen to test the antibacterial activity against S. gordonii strain under anaerobic conditions as recommended by the Clinical and Laboratory Standards Institute (CLSI). [START_REF] Hecht | National Committee for Clinical Laboratory Standards, Methods for antimicrobial susceptibility testing of anaerobic bacteria: approved standard[END_REF] Briefly, Columbia agar is mixed with hoarse blood and different concentrations of butyrolactones (300, 250, 200, 150, 100, 90, 80 and 70 µg/mL), or doxycycline (1:2 serial dilutions from 1.31 to 2x10 -5 µg/mL) as a positive control or the mixture of solvents used to dissolute the compounds (DMSO + methanol) or distilled water as negative controls and left to solidify. The agar mixture is then inoculated with 2 µl spot containing 10 5 CFU/mL before its incubation for 24 hours under anaerobic conditions. The lowest concentration of the mixture that prevented the growth of the bacteria was then determined and the corresponding concentration was defined as the Minimal Inhibitory Concentration (MIC). This was repeated three times.

Broth microdilution

Broth microdilution test was done to confirm the results in broth as described by CLSI. [START_REF] Hecht | National Committee for Clinical Laboratory Standards, Methods for antimicrobial susceptibility testing of anaerobic bacteria: approved standard[END_REF] In brief, starting with 300 or 32.8 µg/mL as an initial concentration for the compounds to be tested or the positive control, doxycycline, respectively, 1:2 serial dilutions were made in BHI in a 96-well microtiter plate (Sterile, Flat bottom, with lid, Greiner Bio-one, Germany). Each well was then inoculated by 3x10 7 CFU/mL of S. gordonii. In addition, the mixture of the solvents (DMSO + methanol) used to dissolute the compounds was 1:2 serially diluted to check their activity. Then, the plate is incubated for 24 hours under anaerobic conditions after which the clear wells will be spreaded on Columbia Petri plates to be incubated for another 24 hours. The clear well with the lowest concentration represents the MIC which has inhibited the visible bacterial growth and the Petri plate showing no colonial growth will be the MBC defined as the lowest concentration that killed ≥99% of the initial inoculum.

Cytotoxicity

Each well of a 96-well plate (Sterile, Flat bottom, with lid, Greiner Bio-one, Germany) was seeded with 70 000 cells after their trypsination and counting in case of Ca9-22 cells or only counting for THP-1. Ca9-22 cells were incubated for 24 hours whereas THP-1 cells were incubated with PMA for 72 hours. After that and for the two cell lines, the contents of the wells were removed and the compounds or only media as negative controls were added to be incubated for 24 hours. The compounds best inhibitory concentrations, MICs, were chosen to test whether they have a cytotoxic effect or not and Triton 1% was used as positive control. Finally, LDH and MTT assays were done to investigate the cytotoxicity. The experiments were done three times in triplicate.

LDH

According to Promega protocol, 50 µL of the supernatant from each well was transferred into a new 96-well plate. Then, 50 µL of the CytoTox Reagent was added to each well and the plate was incubated for 30 minutes in the dark at room temperature. Finally, 50 µL of the stop solution was added and the O.D was then read at 490 nm.

MTT

Ten µL of 5 mg/mL MTT, 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyl tetrazolium bromide (Sigma Aldrich, France), prepared in PBS and filter sterilized through a 0.22 µm filter, was added to the wells containing 100 µL of medium. The 96-well plate was then incubated for 4 hours at 37 °C under 5% CO 2 . After that, 100 µL of acid-isopropanol, 0.04 N HCL in isopropanol, was added to the wells and mixed very well to dissolve the formazan crystals. Finally, the O.D was read after a few minutes at 595 nm and at 655 nm (measurement and reference, respectively). [START_REF] Mosmann | [END_REF] The results were presented as percent MTT activity where the readings for the untreated control cells were considered as 100%.

Figure 1 .

 1 Figure 1. Chemical structures of butyrolactones

Figure 2 . 1 .

 21 Figure 2. Chemical structures of two aliphatic -methylene butyrolactones

Figure 3 . 9 .

 39 Figure 3. Effect of butyrolactones on S. gordonii in agar dilution

Figure 4 .

 4 Figure 4. Evaluation of butyrolactones cytotoxicity on Ca9-22 and THP-1 cells by LDH (A) and MTT (B) assays. *, P<0.05; ***, P<0.001

Figure 5 .

 5 Figure 5. Pharmacomodulation

  Yield = 61%. [α] D = -37.1 (c 1.07, CHCl 3 ). HRMS (ESI, m/z): Calcd. for C 17 H 32 O 3 Na: 307.2249, found [M+Na] + : 307.2244.

  Yield = 61%. [α] D = + 18.5 (c 1.04, CHCl 3 ). HRMS (ESI, m/z): Calcd. for C 17 H 32 O 3 Na: 307.2249, found [M+Na] + : 307.2249. 5.1.1.6. (4S,5S)-4-hydroxy-5-pentadecyl-dihydrofuran-2(3H)-one (4S,5S-1d). Yield = 53%; white solid; m.p. 96 °C. R f (petroleum ether/ethyl acetate 8:2) = 0.11. [α] D = -32.9 (c 0.93, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.93 (m, 29H), 2.56 (dd, J = 0.8 Hz and J = 17.7 Hz, 1H), 2.80 (dd, J = 5.4 Hz and J = 17.5 Hz, 1H), 4.34-4.40 (m, 1H), 4.46-4.49 (m, 1H) ppm.

  Yield = 94%. [α] D = + 63.5 (c 0.94, CHCl 3 ). HRMS (ESI, m/z): Calcd. for C 13 H 22 O 2 Na: 233.1518, found [M+Na] + : 233.1517.

  Yield = 77%. [α] D = + 54.2 (c 1.01, CHCl 3 ). HRMS (ESI, m/z): Calcd. for C 17 H 30 O 2 Na : 289.2144, found [M+Na] + : 289.2138.

  Yield = 81%. [α] D = -57.2 (c 1.09, CHCl 3 ), HRMS (ESI, m/z): Calcd. for C 17 H 30 O 2 Na : 289.2144, found [M+Na] + : 289.2144.

  Yield = 71%. [α] D = + 45.0 (c 1.11, CHCl 3 ). HRMS (ESI, m/z) : Calcd. for C 19 H 34 O 2 Na : 317.2457, found [M+Na] + : 317.2456.

  C NMR (CDCl 3 , 75 MHz) δ 14.2, 22.7, 25.3, 29.2, 29.3, 31.8, 32.0, 35.5, 45.5, 82.0, 174.5, 176.4 ppm. HRMS (ESI, m/z) : Calcd. for C 12 H 19 O 4 : 227.1289, found [M-H] -: 227.1290. 5.1.4.2. (2S,3R)-2-nonyl-5-oxo-tetrahydrofuran-3-carboxylic acid (4R,5S-4b). Yield = 81%; white solid; m.p. 115 °C. R f (CH 2 Cl 2 /MeOH 9:1) = 0.45. [α] D = -44.7 (c 1.05, CHCl 3 ). 1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.27-1.60 (m, 14H), 1.72-1.82 (m,2H), 2.82 (dd, J = 9.7 Hz and J = 17.8 Hz, 1H), 2.95 (dd, J = 8.3 Hz and J = 17.8 Hz, 1H), 3.06-3.15 (m, 1H), 4.59-4.65 (m, 1H) ppm. 13 C NMR (CDCl 3 , 75 MHz) δ 14.1, 22.6, 25.1, 29.1, 29.2, 29.3, 29.4, 31.8, 31.9, 35.3, 45.3, 81.8, 174.4, 175.5 ppm. HRMS (ESI, m/z) : Calcd. for C 14 H 23 O 4 : 255.1596, found [M-H] -: 255.1602.5.1.4.3. (2R,3S)-2-nonyl-5-oxo-tetrahydrofuran-3-carboxylic acid (4S,5R-4b). Yield = 78%; white solid; m.p. 115 °C. R f (CH 2 Cl 2 /MeOH 9:1) = 0.45. [α] D = + 44.7 (c 1.05, CHCl 3 ).1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.27-1.60 (m, 14H), 1.72-1.82 (m,2H), 2.82 (dd, J = 9.7 Hz and J = 17.8 Hz, 1H), 2.95 (dd, J = 8.3 Hz and J = 17.8 Hz, 1H), 3.06-3.15 (m, 1H), 4.59-4.65 (m, 1H) ppm.13 C NMR (CDCl 3 , 75 MHz) δ 14.1, 22.6, 25.1, 29.1, 29.2, 29.3, 29.4, 31.8, 31.9, 35.3, 45.3, 81.8, 174.4, 175.5 ppm. HRMS (ESI, m/z) : Calcd. for C 14 H 23 O 4 : 255.1596, found [M-H] -: 255.1604.

  : 112 °C. [α] D = -25.3 (c 1.01, CHCl 3 ); ref. 17 : [α] D = -42.8 (c 1.76, CHCl 3 ). HRMS (ESI, m/z) : Calcd. for C 18 H 31 O 4 : 311.2222, found [M-H] -: 311.2225.

  : 110 °C. [α] D = + 34.4 (c 0.81, CHCl 3 ). HRMS (ESI, m/z) : Calcd. for C 18 H 31 O 4 : 311.2222, found [M-H] -: 311.2228. 5.1.4.6. (2S,3R)-5-oxo-2-pentadecyl-tetrahydrofuran-3-carboxylic acid (4R,5S-4d).

  C NMR (CDCl 3 , 75 MHz) δ14.0, 22.6, 24.7, 29.1, 29.2, 29.4, 31.8, 35.7, 49.5, 79.0, 126.2, 132.5, 168.4, 174.9 ppm. HRMS (ESI, m/z) : Calcd. for C 15 H 23 O 4 : 267.1596, found [M-H] -: 267.1603.

  C NMR (CDCl 3 , 75 MHz) δ 14.0, 22.6, 24.7, 29.1, 29.2, 29.4, 31.8, 35.7, 49.5, 79.0, 126.2, 132.5, 168.4, 174.9 ppm. HRMS (ESI, m/z) : Calcd. for C 15 H 23 O 4 : 267.1596, found [M-H] -: 267.1603. 5.1.5.4. (2R,3S)-4-methylene-5-oxo-2-tridecyl-tetrahydrofuran-3-carboxylic acid (4S,5R-5c).

  : 104-105 °C. [α] D = + 18.3 (c 0.24, CH 2 Cl 2 ), ref. 17 : [α] D = + 13.6 (c 1.72, CHCl 3 ). HRMS (ESI, m/z) : Calcd. for C 19 H 31 O 4 : 323.22223, found [M-H] -: 324.2228. 5.1.5.5. (2S,3R)-4-methylene-5-oxo-2-tridecyl-tetrahydrofuran-3-carboxylic acid (4R,5S-5c).

  : [α] D = -13.2 (c 1.52, CHCl 3 ). ). HRMS (ESI, m/z) : Calcd. for C 19 H 32 O 4 : 324.2301, found [M] + : 324.2291.5.1.5.6. (2S,3R)-4-methylene-5-oxo-2-pentadecyl-tetrahydrofuran-3-carboxylic acid (4R,5S-5d). Yield = 56%; white solid; m.p. 108 °C. R f (petroleum ether/diethyl ether/acetic acid 6:3.8:0.2) = 0.33. [α] D = -9.4 (c 1.13, CHCl 3 ).1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.5 Hz, 3H), 1.26-1.75 (m, 28H), 3.62-3.64 (m, 1H), 4.81 (dt, J = 6.0 Hz and J = 12.3 Hz, 1H), 6.02 (d, J = 2.4 Hz, 1H), 6.47 (d, J = 2.8 Hz, 1H) ppm.13 C NMR (CDCl 3 , 75 MHz) δ 14.3, 22.8, 24.9, 29.3, 29.5, 29.6, 29.8, 32.1, 35.9, 49.6, 79.0, 126.2, 132.5, 168.4, 174.8 ppm. HRMS (ESI, m/z) : Calcd. for C 21 H 35 O 4 Na: 375.2111, found [M+Na] + : 375.2518.5.1.5.7. (2S,3R)-2-hexadecyl-4-methylene-5-oxo-tetrahydrofuran-3-carboxylic acid (4R,5S-5e). Yield = 45%; white solid; m.p. 107 °C. R f (petroleum ether/diethyl ether/acetic acid 6:3.8:0.2) = 0.33. [α] D = -2.3 (c 1.01, CHCl 3 ).1 H NMR (CDCl 3 , 300 MHz) δ 0.88 (t, J = 6.7 Hz, 3H), 1.26-1.55 (m, 28H), 1.68-1.81 (m, 2H), 3.63 (ddd, J = 5.6 Hz, J = 2.5 Hz and J = 2.8 Hz, 1H), 4.81 (dt, J = 5.8 Hz and J = 7 Hz, 1H), 6.03 (d, J = 2.7 Hz, 1H), 6.47 (d, J = 3.0 Hz, 1H) ppm.

  ; H, 8.25. HPLC : Chiralpak IC, nheptane/MtBE/TFA 80:20:0.1, 250 nm, 1 mL/min, 0.5 mg/mL. T R = 15.09 min, ee = 91%. 5.1.6.2. (S)-4-methyl-2-nonyl-5-oxo-2,5-dihydrofuran-3-carboxylic acid (5S-6b) (B-8). Yield = 65%; white solid; m.p. 120 °C. R f (petroleum ether/diethyl ether/acetic acid 8:2:0.2) = 0.29. [α] D = -36.3 (c 1.30, CHCl 3

5. 1 . 6 . 3 .

 163 (R)-4-methyl-2-nonyl-5-oxo-2,5-dihydrofuran-3-carboxylic acid (5R-6b) (B-9).

Table 2 .

 2 Minimal inhibitory and bactericidal concentrations of butyrolactones against S. gordonii by broth microdilution

	Compound	MIC (µg/mL)	MBC (µg/mL)
	B-1	150	300
	B-2	75	150
	B-3	150	300
	B-4	75	75
	B-5	75	150
	B-6	75	75
	B-7	>300	>300
	B-8	150	>300
	B-9	150	>300
	B-10	4.69	18.75
	B-11	4.69	18.75
	B-12	9.38	18.75
	B-13	9.38	9.38
	Doxycycline	0.51	32.80

  C NMR (CDCl 3 , 75 MHz) δ 14.2, 22.7, 25.7, 28.4, 29.2, 29.5, 31.9, 39.6, 69.1, 85.3, 176.4 ppm. HRMS (ESI, m/z) : Calcd. for C 11 H 20 O 3 Na : 223.13101, found [M+Na] + : 223.1309.

  C NMR (CDCl 3 , 75 MHz) δ 13.4, 22.0, 24.9, 27.6, 28.6, 28.7, 28.8, 31.2, 38.8, 68.3, 84.4, 175.4 ppm. IR (ATR) 3466, 2952, 2922, 2850, 1740. HRMS (ESI, m/z): Calcd. for C 13 H 24 O 3 Na: 251.1623, found [M+Na] + : 251.1626.

  C NMR (CDCl 3 , 75 MHz) δ 14.3, 22.8, 25.7, 28.4, 29.5, 29.6, 29.7, 29.8, 29.9 32.1, 39.6, 69.2, 85.1, 176.05 ppm. HRMS (ESI, m/z) : Calcd. for C 19 H 36 O 3 Na : 335.25622, found [M+Na] + : 335.2559.

  C NMR (CDCl 3 , 75 MHz) δ 14.3, 22.8, 25.7, 28.4, 29.5, 29.6, 29.7, 29.8, 32.1, 39.6, 43.6, 69.2, 85.0, 175.8 ; HRMS (ESI, m/z): Calcd. for C 20 H 38 O 3 Na : 349.2719, found [M+Na] + : 349.2719.

  C NMR (CDCl 3 , 75 MHz) δ 14.2, 22.7, 25.1, 29.1, 29.3, 31.8, 33.3, 83.6, 121.6, 156.5, 173.3 ppm. HRMS (ESI, m/z): Calcd. for C 11 H 18 O 2 Na : 205.1205, found [M+Na] + : 205.1203.

  C NMR (CDCl 3 , 75 MHz) δ 14.4, 23.0, 25.3, 29.6, 29.62, 29.7, 29.8, 32.2, 33.5, 83.8, 121.8, 156.7, 173.5 ppm. IR (ATR) 2923, 2853, 1744. HRMS (ESI, m/z): Calcd. for C 13 H 22 O 2 Na: 233.1518, found [M+Na] + : 233.1519.

  C NMR (CDCl 3 , 75 MHz) δ 14.2, 22.8, 25.1, 29.4, 29.5, 29.6, 29.7, 29.8, 32.0, 33.3, 83.6, 121.6, 156.5, 173.3 ppm. HRMS (ESI, m/z) : Calcd. for C 19 H 34 O 2 Na : 317.2457, found [M+Na] + : 317.2456.

  C NMR (CDCl 3 , 75 MHz) δ 14.1, 22.6, 25.8, 29.1, 29.3, 31.7, 33.6, 35.5, 46.4, 84.8, 118.0, 135.8, 175.8 ppm. HRMS (ESI, m/z) : Calcd. for C 13 H 22 O 2 Na: 233.1518, found [M+Na]

+ : 233.1517.

  C NMR (CDCl 3 , 75 MHz) δ 14.1, 22.6, 25.7, 29.2, 29.3, 29.4, 29.45, 31.8, 33.6, 35.4, 46.3, 84.8, 117.9, 135.8, 175.8 ppm. HRMS (ESI, m/z) : Calcd. for C 15 H 26 O 2 Na : 261.1831, found [M+Na] + : 261.1832.

  C NMR (CDCl 3 , 75 MHz) δ 14.1, 22.6, 25.7, 29.2, 29.3, 29.4, 29.5, 31.8, 33.6, 35.4, 46.3, 84.8, 117.9, 135.8, 175.8 ppm. HRMS (ESI, m/z) : Calcd. for C 15 H 26 O 2 Na : 261.1831, found [M+Na] + : 261.1832.

  C NMR (CDCl 3 , 75 MHz) δ 14.2, 22.8, 25.8, 29.5, 29.6, 29.7, 29.8, 32.0, 33.8, 35.6, 46.5, 85.0, 118.1, 135.95, 175.9 ppm. HRMS (ESI, m/z) : Calcd. for C 19 H 34 O 2 Na : 317.2457, found [M+Na] + : 317.2456.

  C NMR (CDCl 3 , 75 MHz) δ 14.2, 22.8, 25.8, 29.5, 29.6, 29.7, 29.8, 32.0, 33.7, 35.6, 46.5, 84.9, 118.1, 135.9, 175.9. HRMS (ESI, m/z) : Calcd. for C 21 H 38 O 2 Na : 345.2770, found [M+Na] + : 345.2767.

  C NMR (CDCl 3 , 75 MHz) δ 14.2, 22.8, 25.8, 29.5, 29.6, 29.7, 29.8, 32.0, 33.8, 35.6, 46.5, 84.9, 118.1, 135.9, 175.9 ppm. HRMS (ESI, m/z) : Calcd. for C 22 H 40 O 2 Na : 359.2926, found [M+Na] + : 359.2925.

  C NMR (CDCl 3 , 75 MHz) δ 14.3, 22.8, 25.3, 29.3, 29.5, 29.6, 29.8, 32.0, 32.1, 35.5, 45.4, 81.9, 174.4, 176.0 ppm. HRMS (ESI, m/z) : Calcd. for C 20 H 35 O 4 : 339.2541, found [M-H] -: 339.2543.

  C NMR (CDCl 3 , 75 MHz) δ 14.1, 22.8, 25.3, 29.4, 29.5, 29.6, 29.7, 29.8, 32.1, 35.6, 35.7, 45.5, 52.7, 82.0, 174.3, 175.4 ; HRMS (ESI, m/z) : Calcd. for C 21 H 37 O 4 : 353.2697, found [M-H] -: 353.2701.

  C NMR (CDCl 3 , 75 MHz) δ 14.2, 22.7, 24.8, 29.1, 29.2, 31.8, 35.8, 49.6, 79.1, 126.2, 132.5, 168.6, 174.7 ppm. HRMS (ESI, m/z) : Calcd. for C 13 H 19 O 4 : 239.12888, found [M-H] -: 239.1292.

  C NMR (CDCl 3 , 75 MHz) δ 11.2, 14.2, 22.7, 24.9, 29.2, 29.3, 31.8, 32.9, 81.5, 140.3, 146.7, 166.7, 172.7 ppm. HRMS (ESI, m/z) : Calcd. for C 13 H 19 O 4 : 239.1289, found [M-H] -: 239.1291. Anal. Calcd. For C 13 H 20 O 4 : C, 64.98 ; H, 8.39. Found : C, 64.17

  ppm. HRMS (ESI, m/z) : Calcd. for C 15 H 24 O 4 Na : 291.15723, found [M+Na] + : 291.1572. Anal. Calcd. For C 15 H 24 O 4 : C, 67.14 ; H, 9.01. Found : C, 67.97 ; H, 9.07. HPLC : Chiralpak IC, n-heptane/MtBE/TFA 80:20:0.1, 250 nm, 1 mL/min, 0.5 mg/mL. T R = 16.92 min, ee = 97%.

.6.4. (-)-Lichesterinic acid (5S-6c) (B-10).

  C NMR (CDCl 3 , 75 MHz) δ 11.0, 14.1, 22.6, 24.7, 29.1, 29.2, 29.3, 29.4, 31.8, 32.7, 81.4, 139.7, 146.7, 165.5, 172.6 ppm. HRMS (ESI, m/z) : Calcd. for C 15 H 23 O 4 : 267.15963, found [M-H] -: 267.1602. Anal. Calcd. For C 15 H 24 O 4 : C, 67.14 ; H, 9.01. Found : C, 66.90 ; H, 8.98. HPLC : Chiralpak IC, n-heptane/MtBE/TFA 80:20:0.1, 250 nm, 1 mL/min, 0.5 mg/mL. T R = 18.92 min, ee = 99%. Yield = 69%; white solid; m.p. 122 °C; ref. 27 : 120-121 °C. [α] D = -23.5 (c 1.055, CHCl 3 ); ref. 27 [α] D = -35 (c 0.6, CHCl 3 ). HRMS (ESI, m/z) : Calcd. for C 19 H 32 O 4 : 324.2301, found [M] + : 324.2288. Anal. Calcd. For C 19 H 32 O 4 : C, 70.33 ; H, 9.94. Found : C, 71.48 ; H, 10.03. HPLC : Chiralpak IA, 100% ACN + 0.1% HCOOH, 210 nm, 1 mL/min, 0.5 mg/mL. T R = 6.16 min, ee = 98%.

	5.15.1

.6.5. (+)-Lichesterinic acid (5R-6c) (B-11).

  Yield = 71%; white solid; m.p. 122 °C; ref.28 : 120-122 °C. [α] D + 24.9 (c 1.03, CHCl 3 ); ref.28 : [α] D = + 31.9. HRMS (ESI, m/z) : Calcd. for C 19 H 32 O 4 : 324.2301, found [M] + : 324.2288. Anal. Calcd. For C 19 H 32 O 4 : C, 70.33 ; H, 9.94. Found : C, 71.20 ; H, 9.95. HPLC : Chiralpak IA, 100% ACN

  C NMR (CDCl 3 , 75 MHz) δ 11.2, 14.3, 22.8, 24.9, 29.4, 29.5, 29.7, 29.8, 32.1, 32.9, 81.5, 140.3, 146.8, 167.0, 172.8 ppm. HRMS (ESI, m/z) : Calcd. for C 21 H 35 O 4 : 351.25408, found [M-H] -: 351.2545. Anal. Calcd. For C 21 H 36 O 4 : C, 71.55 ; H, 10.29. Found : C, 71.80 ; H, 10.37. HPLC :

  C NMR (CDCl 3 , 75 MHz) δ 11.2, 14.3, 22.8, 24.9, 29.4, 29.5, 29.7, 29.8, 32.1, 32.9, 81.5, 140.3, 146.8, 166.7, 172.8 ppm. HRMS (ESI, m/z) : Calcd. for C 22 H 37 O 4 : 365.26974, found [M-H] -: 365.2699. Anal. Calcd. For C 22 H 38 O 4 : C, 72.09 ; H, 10.45. Found : C, 72.24 ; H, 10.39. HPLC :

Acknowledgments

We acknowledge Pr J.Boustie (Head of PNSCM team) for his helpful discussion. We would also like to thank C. Le Lann and N. Oliviero (EA 1254 -Rennes I university), and Nathalie Legrave (UMR CNRS 6226 -Rennes I University) for their technical assistance.

Funding

The research was supported by Rennes I University, CNRS, Association of Specialization and Scientific Orientation and Melanolichen Grant.


Christophe Giraud-Audine

Michel Amberg

Frédéric Giraud

Charles Mangeot

Betty Lemaire-Semail

Dynamic model of a PAD actuator : dynamic operations and pull-off at high speed

Keywords: PAD motor, pull-off, dynamic model

The PAD motor has remarkable abilities for ultra-precise positioning at very low speed. Recent works have shown the possibility to extend its speed range, but some challengess remain such as pull-off. This paper proposes a model that highlights the parameters of the voltage supply that act on the torque and the contact, and can be used for control. Experimental results demonstrate the role of higher vibration modes, and show that typical signature of the imminence of pull-off can be detected thanks to some of the harmonic of the displacement of the actuators, giving some indications on means to address this problem.

1-Introduction

PAD motors are low speed, precision motors manufactured by NOLIAC. Initially, the operations being slow, this type of motor can be considered as quasi-static motors. Recent studies have shown that the PAD could be driven at higher speed up to and beyond resonance. However, in [START_REF] Mangeot | Operation of a quasi-static piezomotor in transitory frequency range up to resonance[END_REF] some difficulties were observed in the vicinity of resonance due the relative phase of the vibration resulting from the sudden phase rotation near the resonant frequencies of the actuators. This paper proposes a simplified model of the PAD dynamics including the contact to address this issue. It provides some insight in the phenomena affecting the contact forces and the torque, and can describe the dynamic at higher frequencies. Furthermore, an experimental study shows that the motor can operate at frequencies higher than the free resonant frequencies of the actuators. Besides, the results indicate that the pull off resulting in the stalling of the motor is not only due the voltages applied but also to vibratory behaviour at the contact. In the first part, the dynamic model is explained. The pull-out and the torque generation mechanism is explained in quasi static operation, and the dynamic equations are modified to yield a simplified model. In the second part, some experiments are presented. Using a frequency analysis, some insight of the vibratory behaviour of the actuators is gained, and a typical signature of the imminence of stalling is identified.

2-Theoretical study Equilibrium of the crowned wheel

The PAD motor features two sets of actuators acting along two perpendicular directions, so as to generate a circular translation applied to a crown wheel (Fig. 1, top). When brought into contact with a geared shaft, this wobbling movement is converted into a rotation of the shaft. The micro gear adds a high gear ratio thus allowing ultra low speed operation and exceptional positioning capabilities. To obtain the wobbling, the actuators are driven using two sinusoidal voltages shifted by 90°.

On the same figure, the motor is depicted during operation. Voltages are applied to the actuators, inducing their deformations. The forces applied to the crown wheel are given by :

F a =2 ( N V x -K s u x N V y -K s u y ) (1)
where N and Ks are the piezoelectric actuator force factor and rigidity respectively. ux, uy, Vx and Vy are the displacement of the piezoelectric actuator tips and the voltages applied for the x and y direction respectively.

The normal force at the contact point C is modelled as an elastic contact (rigidity k) such that :

F N =-k ( 1-R d ) ( u x u y ) (2)
where d= √ u x 2 + u y 2 and R is the radius of the shaft.

Finally, the tangential force at the contact point can be related to the torque T applied by the crowned wheel on the shaft by:

F T = T R ( -sin θ cos θ ) (3)
where θ is the direction of the displacement vector.

It should be noted that the simplified contact model supposes that d>R that is, strictly speaking, that the crowned wheel intersect the shaft. However, k being large the error can be considered to be small, and moreover, R represent the addendum circle of the shaft.

To obtain the dynamic model, in contact condition, one writes Newton's first law applied to the crowned wheel yielding:

-(2 K s +k )u x +k R cos θ- T R sin θ+2 N V x =M üx -(2 K s +k )u y + k R sin θ- T R cos θ+2 N V y =M ü y (4)
The equation above can be extended to the non contact operations by letting k=0 and cancelling T.

Static characteristics

Cancelling the accelerations in the previous equations and replacing u x =d cosθ and u y =d sin θ , one can get the following static equations thanks to trigonometrical identities :

-(2 K s +k ) d +k R+2 N ( V u cos θ+V y sin θ ) =0 T R =2 N (V u sin θ-V y cos θ) (5)
Since the voltages applied are two phase in quadrature, we write V x =V cosϕ and V y =V sin ϕ and finally the torque can be rewritten as:

T =2 NRV sin (θ-ϕ) (6)
and the displacement of the crowned gear is:

d = kR+ 2 NV cos (θ-ϕ) 2 K s + k (7)
These equations show that the static performances are driven by : 1. the amplitude of the voltage 2. the "phase shift" between the actual position of the crowned gear and the voltages phase It is possible to derive some conditions. First the contact must exist as a premise to the operations, that is R=d defines a limit. Thus, replacing this condition in (7), and expressing the sine of the phase shift as a function of the applied torque and voltage amplitude, the limit equation is:

V = 1 2 N √ (2 K s R) 2 + ( T R ) 2 (8)
together with the limit phase shift:

θ-ϕ=arctan T 2 K s R 2 (9)
Actually, (8) and (9) represent limit cases, and thus the actual feasible regions are depicted on Figure 2 and3 using normalized values for the sake of generality. The voltage amplitude should be in the area enclosed at the intersection of the voltage greater than the limit given by (8) and below the maximum voltage of the piezoelectric actuators. The corners of this area are the maximum realizable torque (note however that on the figure the indicated values are only indicative and not related to the actual actuator).

Figure 2 : Theoretical operating area of the model of the PAD motor

Figure 3 shows that the phase shift should be greater than the limit given by (9). On the graph the limit due to voltage limitation is also represented and, again, is only indicative. An interesting point on this figure is that it becomes clear that the phase shift will control motor or braking operations.

Dynamic model

To describe the dynamic of the shaft, the following equation has to be added to equations (4):

J ψ=T m -T l (10)
where ψ is the shaft angle, J is the shaft inertia, Tm and Tl are the motor and load torque respectively. Due to the gearing, the crowned wheel angle and the shaft angle are linked by the following relationship if the contact is ensured: ψ=γ θ (11) where γ is the reduction ratio of the gearing. Therefore , the following relation between the torques holds:

T =γ T m . u=û e iω t . In this case, the inertial force will be

-M ü= M ω 2 u
where ω is the angular frequency of the voltages and, since synchronism is imposed by the gearing, the rotor speed i.e. θ=ω . From this diagram, it can be deduced that as the speed increases, the forces applied by the actuators Fax and Fay should be increased to maintain the normal and tangential forces FN and FT respectively. In order to do so, the voltages amplitude should be increased. In other words, as the speed increases, assuming constant voltages, and provided that to maintain the contact a limit normal force is required, the motor will be more prone to "pull off" that is the transmission between the gear will be lost. In order to estimate this, a more elaborated model of the contact should be developed. In the following, an experimental study is presented to estimate the occurrence of pull off in the case of the motor under investigation.

Experimental set up

The experiment was designed to investigate the dynamical behaviour of the motor, with a special emphasis on the pull off. It was therefore necessary to have a measurement of the actual displacement of the actuator. In the model at hand, the actuators are piezoelectric multilayer benders. In these actuators the outermost layer is usually not electroded to provide some isolation. The PAD motor actuators were modified by creating an additional electrode (Fig. 5). As deformation occurs, due to the direct piezoelectric effect, a voltage will be induced which is proportional to the difference of slopes at the end of the electrode [START_REF] Bogolioubov | Méthodes asymptotiques en théorie des oscillations non linéaires[END_REF] . Therefore, for a given mode, the measured voltage is proportional to the amplitude of the vibration. On Fig. 6, the electrical circuit of the set up is depicted. The capacitances Ca represent the capacitance of a section of an actuator. The bender must be polarized to 100 V, and the motion is obtained by imposing the voltage Va. Due to the internal structure of the actuators, the last layer used for the sensor (represented by the capacitance Cs) is connected to Va. A floating measurement using an isolation amplifier is thus necessary. The voltages delivered by the sensing piezoelectric layer can vary up to 60 V, hence, it was preferred to measure the current through a shunt resistor on the high voltage side, that is the motional current rather than the voltage Vs was measured. On Fig . 7, the measurement of the motional current vs frequency is presented for the motor without the shaft. The resonances of the first mode are correctly detected, and the cross talk between the two axes is small. It can be noted that the resonant frequencies are 20 Hz apart as mentioned in [START_REF] Mangeot | Operation of a quasi-static piezomotor in transitory frequency range up to resonance[END_REF], which may cause the problem described in the introduction.

Figure 7 : Measurement vs frequency of the motional current sensors

Experimental results

As represented on Fig. 6, the sensors signals were processed thanks to a DSP. A DFT algorithm synchronised on the supply voltages was then implemented to analyse the dynamical behaviour. The tests consist in varying the frequency at the end of each period thus approaching a sweep signal, so the speed is varying linearly.

Figure 8: Frequency response of the displacement with mounted shaft for small amplitude radial vibrations

The first experiment consists in reiterating the frequency response once the shaft is mounted back. The actuators were polarized to ensure contact at θ=0°, and to this DC voltage a sinusoidal voltage was superimposed at 10% of the rated voltage. Hence no rotation occurs, but a small amplitude vibration is generated. The frequency response is modified (Fig. 8) and is very different for each axis. Examining the 3 rd and 5 th harmonics for axis X, it can be observed that their amplitude tend to increase as the fundamentals amplitude of axis X and Y vary in the vicinity of 380 Hz, 620 Hz, 710 Hz and most dramatically near 1000 Hz. This indicates that the vibration regime changes and possibly shocks occur. It should be noted though that the amplitude varies by a small amount so it can hardly be considered as resonances. A similar test was conducted at θ=135°, the same kind of vibration were applied, radially as before but also tangentially. Indeed the radial cross section of the crowned wheel is not constant as suggested by Fig. 1, and it can be expected that the small signal behaviour might be dependent on the combined rigidities of the shaft and the crowned wheel.

Figure 9: frequency responses of the two axis for small amplitude vibration along the radial and tangential directions

The position of the contact does not have real effect (Fig. 9), but the direction of the vibration does modify the response in the case of the X axis. As a matter of fact, an amplitude jump is clearly visible which can be attributed to intermittent contact of the teeth. Indeed such jumps are typical of oscillators with change of elasticity dependent on the amplitude [START_REF] Bogolioubov | Méthodes asymptotiques en théorie des oscillations non linéaires[END_REF]. To conclude with these low amplitude tests, the motor was supplied by two low amplitude voltages in quadrature, thus inducing the circular translation of the crowned wheel. The frequency response (Fig. 10) exhibits several resonances. Since several mode shapes are excited and the sensitivity of the sensors are dependent on the slope of the shapes at the electrodes, the amplitudes should be considered carefully. Indeed, the sensor electrodes are not identical on each actuator. Hence, although the signals may appear to be larger in some frequency bandwidth, this could also be due to discrepancies of the sensors sensitivity to the different modes excited. The third harmonic is also excited near 1 kHz, and might cause the contact conditions to be degraded. In order to assess the possibility to achieve high speed positioning, the motor was supplied in working conditions at various amplitudes. Pull-off was observed at various frequencies, near 400 Hz and 1000 Hz depending on the amplitude of the voltage applied.

Figure 10: wide band frequency response of displacements for small vibration in circular translation motion

Figure 11: test at 60% of the maximum rated voltage Some of the results are presented on fig. 11 and 12 for voltage at 60% and 85% of the maximum rated voltages. On Fig. 11 the pull-off can clearly be seen at the sudden change in amplitude of the fundamental. Interestingly enough, it can be observed that the motor stops before 360 Hz then starts again at 420 Hz. Finally at 970 Hz, control is completely lost. The most striking feature of this test is the progressive growth of the 2 nd and 4 th harmonics, before they suddenly collapse when the pull-off occurs. For larger voltages, the pull-off only occurs at 980 Hz which is about 30% higher than the resonance of the motor without shaft. The same trend can be observed on the 4 th harmonic.

Figure 12: test at 85% of the maximum rated voltage

Discussion and conclusion

This paper has proposed a dynamic model to describe the operation of the PAD actuator and points out the role of the voltage and the phase shift between voltage and contact position. An experimental study was presented to understand the issues that can trigger pull-off. Thanks to the insight offered by a DFT analysis, it has been demonstrated that higher modes, excited by harmonics induced by the contact, could cause the pull off. Moreover, the pull-off is preceded by a variation of the 2 nd and 4 th harmonics which provides a way to anticipate it. Therefore, future work will extend the model for control purposes. A special attention will be paid to the damping of higher modes. This could be realized by techniques such as the one proposed in [START_REF] Fairbairn | Q Control of an Atomic Force Microscope Microcantilever: A Sensorless Approach[END_REF].

Figure 1 :

 1 Figure 1 : schematic of the structure of the Pad motor (up) and during operation (bottom)

Figure 3 :Figure 4

 34 Figure 3 : Theoretical phase shift vs normalized torque and corresponding operation area Figure 4 depict the diagram of the dynamic equilibrium of the crowned wheel in sinusoidal steady state i.e.

Figure 4 :

 4 Figure 4 : vector diagram of the dynamic equilibrium of the crowned wheel.

Figure 5 :

 5 Figure 5 : Picture of the PAD motor and detail of one of the implemented sensor electrodes (upper left side)

ACTUATOR 2016, MESSEBREMEN 2/5

ACTUATOR 2016, MESSEBREMEN 3/5

ACTUATOR 2016, MESSEBREMEN 4/5

ACTUATOR 2016, MESSE BREMEN 5/5

Acknowledgements

This work has been carried out within the framework of the project StimTac of IRCICA (institut de recherche sur les composants logiciels et matériel pour la communication avancée), and the Project Mint of Inria.