Cécile Gateau

Guilherme Dias De Melo

Philippe Uriac

Olivier Tasseau

Jacques Renault
email: jacques.renault@univ-rennes1.fr

Arnaud Blondel

Nicolas Gouault
email: nicolas.gouault@univ-rennes1.fr.

Frédéric Barbut

Paola Minoprio

Guilherme D Melo

Irreversible inhibitors of the proline racemase (PRAC) unveil innovative mechanism of action as antibacterial against Clostridioides difficile

Keywords: Structure-activity relationship, Proline racemase, Enzyme inhibitors, Clostridioides difficile, Antibacterial

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Racemases, including proline racemases (PRAC), enzymes which catalyze the interconversion of L-proline and D-proline are frequently found as essential factors in pathogenes. [START_REF] Conti | Drug Discovery Targeting Amino Acid Racemases[END_REF] The first eukaryotic proline racemase was discovered and isolated by Minoprio's team from the human parasitic pathogen Trypanosoma cruzi. [START_REF] Conti | Drug Discovery Targeting Amino Acid Racemases[END_REF][START_REF] Reina-San-Martín | A B-cell mitogen from a pathogenic trypanosome is a eukaryotic proline racemase[END_REF] The group used the Trypanosoma cruzi proline racemase (TcPRAC) to design PRAC signatures and search for homologs in other microorganisms. They discovered novels PRACs from pathogenic microorganisms and characterized them; one had already been isolated in 1957 from Clostridioides stricklandii (CsPRAC), and more interestingly, two new functional PRACs, were respectively identified and isolated from Trypanosoma vivax (TvPRAC) and Clostridioides difficile (CdPRAC). [START_REF] Buschiazzo | Crystal structure, catalytic mechanism, and mitogenic properties of Trypanosoma cruzi proline racemase[END_REF][START_REF] Chamond | Biochemical characterization of proline racemases from the human protozoan parasite Trypanosoma cruzi and definition of putative protein signatures[END_REF] CdPRAC closely resembled to TcPRAC and is also a strong B-lymphocyte mitogen. TvPRAC is an PLP-independent enzyme which kinetic properties were fully characterized. [START_REF] Chamond | Biochemical characterization of proline racemases from the human protozoan parasite Trypanosoma cruzi and definition of putative protein signatures[END_REF][START_REF] Chamond | Proline racemases are conserved mitogens: characterization of a Trypanosoma vivax proline racemase[END_REF] Importantly, genes

Accepted Article

presenting simularities in human and other bacterial genomes lack the essential catalytic residues or the whole domain. [START_REF] Chamond | Proline racemases are conserved mitogens: characterization of a Trypanosoma vivax proline racemase[END_REF] They are not functional or encode other enzymes (e.g. hydroxyproline isomerase), which conforts CdPRAC as an interesting and specific target for drug development against Clostridioides difficile. [START_REF] Goytia | Molecular and structural discrimination of proline racemase and hydroxyproline-2epimerase from nosocomial and bacterial pathogens[END_REF] Clostridioides difficile is considered as the major cause of healthcare-associated diarrhea in hospitalized adults. This anaerobic spore-forming bacterium is responsible for 15-25% of antibiotic-associated diarrhea and for virtually all cases of pseudomembranous colitis in humans. [START_REF] Guh | Clostridioides difficile Infection[END_REF] It has also recently emerged as an important enteropathogen both in the community and in animals. The epidemiology of Clostridioides difficile infections (CDI) has been characterized by an increase of its incidence over the last 15 years and the emergence of hypervirulent clones (e.g. 027) is responsible for large outbreaks of severe cases worldwide. The medical treatment of CDI currently relies on three oral drugs: metronidazole, vancomycin and fidaxomicin. [START_REF] Guery | Clostridioides difficile: diagnosis and treatments[END_REF] However, recent guidelines from the Infectious Diseases Society of America recommend excluding metronidazole as the first line therapy of mild CDI due to the high rate of clinical failures and recurrences following this treatment. [START_REF] Mcdonald | Clinical Practice Guidelines for Clostridioides difficile Infection in Adults and Children: 2017 Update by the Infectious Diseases Society of America (IDSA) and Society for Healthcare Epidemiology of America (SHEA)[END_REF] We previously reported the discovery of irreversible inhibitors of the Trypanosoma cruzi PRAC (TcPRAC) using a combination of computational and a classical drug design approaches. This led, in particular to compound NG-P27, which could be co-crystallized with the TcPRAC (Fig 1). [START_REF] Berneman | Combined Approaches for Drug Design Points the Way to Novel Proline Racemase Inhibitor Candidates to Fight Chagas[END_REF][START_REF] Amaral | Designed mono-and di-covalent inhibitors trap modeled functional motions for Trypanosoma cruzi proline racemase in crystallography[END_REF] Using the structural data, more efficient inhibitors, bearing an aryl substituent in C-4 (Fig1), were subsequently designed. The activity of the inhibitors on both promastigote and epimastigote forms of the parasite was established in vitro. Recently, the efficacy of the association of one of these aromatic inhibitors in prodrug form with benznidazole was described in mice. [START_REF] De Melo | Prodrugs as new therapy against Chagas disease: in vivo synergy between Trypanosoma cruzi proline racemase inhibitors and benznidazole[END_REF]

Accepted Article

General method B1 (Compounds 14, 15, 17)

To a suspension of glyoxylic acid monohydrate (1.2 g, 13 mmol) in a mixture of acetic acid (15 mL) and concentrated aqueous HCl solution (2.5 mL) was added slowly an arylmethylketone (13 mmol). The reactional mixture was then heated to reflux for 18 hrs. Water was added and the mixture was extracted with diethyl ether (3 x 30 mL). After drying over MgSO 4 and evaporation of diethyl ether, the resulting oil slowly crystallized. It was filtered and crystallized from ethyl acetate to yield the pure compound (25% to 27%).

Accepted Article

General method B2 (Compounds 5, 7, 12, 18)

A mixture of an arylmethylketone (15 mmol) and glyoxylic acid monohydrate (0.92 g, 10 mmol) in 1,4-dioxane (15 mL) an concentrated H 2 SO 4 (2 mL) was refluxed for 1.5 hrs. Water (50 mL) was added and the mixture was extracted with 3 x 30 mL of ethyl acetate. The solvent was dried over Na 2 SO 4 and evaporated under reduced pressure to yield a solid that was chromatographed over silica gel (CH 2 Cl 2 / Et 2 O/ 1% AcOH) to yield the desired compound (47% to 63%).

(E)-4-oxo-4-phenylbut-2-enoic acid 1

(E)-4-(4-(2-benzamidoethyl)phenyl)-4-oxobut-2-enoic acid 9

Compound 9 was obtained from N-phenethylbenzamide 14 using general method A as a white solid (1.00 g, 70%).

(E)-4-(4-(2-(4-methoxybenzamido)ethyl)phenyl)-4-oxobut-2-enoic acid 11

A solution of 4-methoxybenzoic acid (3.0 g, 19.7 mmol) was solved and refluxed in thionyl chloride (15 mL) for 1.5 hrs. Excessive thionyl chloride was evaporated under reduced pressure then phenethylamine (1.55 g, 12.8 mmol) was slowly added at 0° C under stirring. An aqueous saturated solution of NaHCO 3 (20 mL) was then added dropwise. The solid residue was collected by filtration, washed with water and crystallized from cyclohexane/ethyl acetate (80/20) to yield 4methoxy-N-phenethylbenzamide [START_REF] Yamada | Developpement of Thiazinones-Based Condensing Reagent for Amide Formation[END_REF] as a pale solid (0.82 g, 25 %).

Compound 11 was obtained from 4-methoxy-N-phenethylbenzamide using general method A as a white solid (0.35 g, 32%).

Accepted Article

(m, 3H), 7.49 ( Compound 12 was obtained from general method B2 as a pale solid (0.65 g, 47%).

(E)-4-(2,5-difluorophenyl)-4-oxobut-2-enoic acid 14

Compound 14 was obtained from general method B1 as a white solid (0.28 g, 25%).

(E)-4-oxo-4-(3-(trifluoromethyl)phenyl)but-2-enoic acid 15

Compound 15 was obtained from general method B1 as a white solid (0.35 g, 27%). (q, J = 3.5 Hz), 129.7, 125.6 (q, J = 3.7 Hz), 123.6 (q, J = 272.

(E)-4-(3-chlorophenyl)-4-oxobut-2-enoic acid 17

Compound 17 was obtained from general method B1 as a white solid (0.75 g, 27%).

(E)-4-(3-chlorothiophen-2-yl)-4-oxobut-2-enoic acid 19

Compound 19 was obtained from general method A as a white solid (1.10 g, 61%).

(E)-4-(3-bromothiophen-2-yl)-4-oxobut-2-enoic acid 20

Compound 20 was obtained from general method A as a white solid (1.30 g, 81%).

Bacteriology

Strains Eleven clinical strains (10 toxigenic, 1 non toxigenic) of C. difficile were selected to perform antimicrobial screening assays of the 21 selected TcPRAC inhibitors.

Accepted Article

Villebon-sur-Yvette, France) was used to analyze the banding patterns. PCR ribotypes (RTs)

were identified using the webribo software (https://webribo.ages.at/).

Screening of compound activity by the disk diffusion technique on agar

Antimicrobial susceptibility to the 21 selected TcPRAC inhibitors was determined by the disk diffusion method as described elsewhere [F.

Minimum inhibitory concentration (MIC) determination

MICs were determined by the agar dilution method described by the National Committee for

Biology

Preparation of recombinant CdPRAC: Recombinant Clostridioides difficile proline racemase (EC 5.1.1.4) was produced in E. coli BL21 (DE3) (Invitrogen) and purified by immobilized metal affinity chromatography on nickel columns, as previously described for Trypanosoma cruzi proline racemase (1).

Racemization of L-Proline and inhibition assays: Proline racemization conditions for CdPRAC

were determined as previously described for TcPRAC (3). Briefly, L-to D-proline conversion by

CdPRAC and TcPRAC took place in 1.0 mL reaction containing 40 mM of L-Proline in 0.2 M sodium acetate pH 6.0 and 5 µg/mL of CdPRAC. The concentrations of D-proline were determined by optical rotation of the solution at 365 nm with a 10 cm optical path cell, at 37°C

during one hour using a polarimeter (Jasco, P-2000, Bouguenais, France). Inhibition assays were performed in the same conditions as above with 5 µM of potential CdPRAC/TcPRAC inhibitors during 500 seconds (exponential phase of L-proline racemization).

In vivo toxicity of CdPRAC inhibitor prodrugs in Zebrafish embryos: AB wild-type zebrafish (Danio rerio) embryos were raised at 28°C in embryo water composed of mineral water supplemented with 280 µg/mL methylene blue (M-4159, Sigma-Aldrich) and 30 µg/mL 1-phenyl-2-thiourea (P-7629, Sigma-Aldrich), and handled according to the institutional approved guidelines. 72 hours post-fertilization (hpf) embryos were distributed in 24-well plates (10 embryos/well) in 500 µL of embryo water containing freshly-diluted drugs and incubated at 28°C during 72 hours in the dark.

The quantification of live embryos and possible malformations was achieved using a stereomicroscope.

In vivo activity of the CdPRAC inhibitor NG-P68 in the golden Syrian hamster model: Male golden

Syrian hamsters (Mesocricetus auratus) of 6 weeks of age (average weight 60-80 grams) were purchased from Janvier Laboratories (RjHan:AURA) and handled under specific pathogen-free conditions. The animals were housed with ad libitum access to water and food. Five days before infection, the animals received a subcutaneous injection of clindamycin (30 mg/kg) (5). The infection was performed by oral gavage with approximatively 1x10

RESULTS AND DISCUSSION

The structural similarity between TcPRAC and CdPRAC enzymes led us to evaluate the activity of selected aromatic TcPRAC irreversible inhibitors against Clostridioides difficile.

We report in this work the evaluation of their biochemical and biological effects as antibacterial agent, both in vitro and in vivo.

Synthesis of selected TcPRAC irreversible inhibitors (Scheme 1)

Following the hypothesis that TcPRAC inhibitors could have an antibacterial activity against Clostridioides difficile, we selected twenty 4-aryl substituted derivatives of the (E)

4-oxopent-2-enoic acid (1-20) efficiently inhibiting TcPRAC, as well as NG-P27 (21) for testing. These compounds were prepared according to two strategies reported in the literature [START_REF] Bianchi | Gastric anti-secretory, anti-ulcer and cytoprotective properties of substituted (E)-4-phenyland heteroaryl-4-oxo-2butenoic acids[END_REF] and following the fundamental rules of drug design: [START_REF] Wermuth | The Practice of Medicinal Chemistry[END_REF] -A Friedel and Crafts reaction (A) involving an aromatic or heteroaromatic compound and the maleic anhydride, -A crotonic condensation (B) between an arylmethylketone and the glyoxylic acid monohydrate. The (E) configuration of the double bond was established by 1 H NMR from the values of the 3 J coupling constants. Eight compounds have been previously reported. [START_REF] Bianchi | Gastric anti-secretory, anti-ulcer and cytoprotective properties of substituted (E)-4-phenyland heteroaryl-4-oxo-2butenoic acids[END_REF] Relevant data (chemical formula, molecular weight, LogD at pH 5, 7 and 9, number  of mol in g) are reported in the experimental section (Table 1).

In vitro Antibacterial Activity of 21 TcPRAC irreversible inhibitors against

Clostridioides difficile species

The compound inhibitory potential against Clostridioides difficile was tested in vitro against eleven strains of different PCR-ribotypes. Susceptibility of these strains to different antimicrobial drugs is reported in Table 2. Depending on the nature of the substituent in 4-position the 21 compounds were divided in three families. The data corresponding to the inhibition diameter values () induced by

Strain

Accepted Article

g of each compound are reported in the Tables 345. In order to compare and select inhibitors, the total of diameter values, , and the diameter cumulative increment, , were also calculated. Absence of activity of all compounds against E. coli and E. faecalis, lacking a PRAC enzyme, was also confirmed with  values = 12mm (contact).

Clostridioides difficile

Compounds As reported in Table 3, non-aromatic compound NG-P27 (21) was completely inactive but introduction of an aromatic substituent in position 4 induced an antibacterial activity (compounds 1-3). In particular, phenyl and thiophenyl groups were quite equivalent with  = 88-95, whereas the presence of a more hydrophilic furanyl substituent decreased the activity ( = 41). No activity was observed against strains with 126 and 78 PCRribotypes.

Accepted Article

Further inverstigating the positive impact of the aromatic ring in 4-position, eight substituted derivatives were prepared (Table 4). Polar substituents such as NO 2 , CN, and OH in para position (compounds 4-6) gave overall a low antibacterial activity as compared to compound 1 ( = 40, 3, 14, v.s. 95, respectively). Interestingly, methylation of the OH (compound 7) induced a strong activity increasing  = 112 versus 14 Addition of a long lipophilic chain in para position -compounds 9-11-possibly able to bind in the canal entrance was poorly efficient ( = 35-37). As observed with compounds 1-3, these eight derivatives were also inactive against strains with PCRribotypes 126 and 78. Noticeably, compound 8 possessing an acetamido group in para position presented an interesting profile with ( = 100 with detectable activity against the latter strains.

R

N H

Accepted Article

Finally, introduction of a halogen substituent (F, Cl and Br) on the aromatic ring was successful (Table 5). This pharmacomodulation increased the overall lipophilicity in a limited volume. Thus, mono-or di-fluorination (compounds 12-14) slightly improved the activity ( = 240-250) except for strains with ribotypes 126 and 78 as already observed.

However, introduction of CF 3 in meta position (compound 15) induced one of the best antibacterial effect with  = 272 but also an interesting activity against strains with ribosomes 126 and 78. Except on these strains, the nature (F, Cl, Br) and the position (o, m, p) of the halogen atom did not induce important differences between compounds 12-14 and 16-18 ( = 240-250). In this case, a chlorine or a bromine atom is needed (compounds 16-18). Finally, the best profiles were obtained with the chloroand -bromothiophen compounds 19 and 20. In conclusion, this in vitro study highlighted six compounds, namely compounds 15-20, as promising candidates against Clostridioides difficile species. These derivatives possess a (E) 4-oxobut-2-enoïc acid core substituted in position 4 by a halogeno phenyl or thiophenyl ring. In addition, the para-acetamido derivative 8 represented also an interesting candidate. Indeed, they presented the best inhibition diameters and were active on the PCR-ribotypes 078 and 126. It should be noticed that these compounds presented a Hammett  and Hantzch  positive constant [START_REF] Ertl | Craig plot 2.0: an interactive navigation in the substituent bioisosteric space[END_REF] Finally, it appeared that lipophilic inhibitors were more efficient to inhibit CdPRAC. This data could be synergistic with a better penetration of the compounds into the bacteria. Therefore, compound 20 was chosen for complementary studies and in vivo test.

In vivo study of compound 20

Before the in vivo evaluation of compound 20 on infected hamsters, minimum inhibitory concentrations (MICs) were determined and the data are reported on Figure 3. MICs Accepted Article ranged from 16 mg/l to 64 mg/l (Figure 3). Consistently with previous results, strains belonging to PCR ribotype 126 or 78, except for CD17-019 to a lower extent, displayed the highest MIC among all strains. Furthermore, the in vivo toxicity test using zebrafish embryos [START_REF] Zon | In vivo drug discovery in the zebrafish[END_REF] was performed. No mortality was observed for 20 up to 100 µM (Figure 4). Interestingly compound RU 38086, analog of 20 was tested for its anti-secretory, antiulcer and cytoprotective properties [START_REF] Bianchi | Gastric anti-secretory, anti-ulcer and cytoprotective properties of substituted (E)-4-phenyland heteroaryl-4-oxo-2butenoic acids[END_REF] and clinically evaluated [START_REF] Christidis | Treatment of gastric and gastro-duodenal disorders with derivatives of phenyl aliphatic carboxylic acids[END_REF] indicating the good tolerance of this family.

Finally, as an in vivo proof of concept, 1x10 4 C. difficile spores were orally inoculated to clindamycin-pretreated golden hamsters. [START_REF] Chang | Clindamycin-induced enterocolitis in hamsters as a model of pseudomembranous colitis in patients[END_REF] After 24 hours, the infected animals received the CdPRAC inhibitor 20 (also called NG-P68) orally 100 mg/kg, once a day during four

Accepted Article

Conclusion

In this study we demonstrated that PRAC enzyme constituted an interesting and specific target for the development of new antibacterial agents against Clostridioides difficile infections. The use of local or gastro-resistant galenic forms as well as the administration frequency are certainly to explore. Finally, prevention of Clostridioides difficile infections by use of this new specific class of antibacterial agents could represent a promising indication that should be explored. Therefore, its co-administration in association with typical antibiotics used for primary infection could inhibit the development of Clostridioides difficile.

FIGURE 1 :

 1 FIGURE 1: Co-crystallization of 21 (NG-P27) with TcPRAC and general structure of inhibitors 1-20

1 H

 1 NMR (300MHz, Acetone-D6) δ (ppm): 7.96 (m, 1H), 7.94 (d, J = 15.5 Hz, 1H), 7.81 (m, 1H), 7.68 (m, 1H), 7.51 (m, 1H), 6.82 (d, J = 15.5 Hz, 1H). 13 C NMR (75MHz, Acetone-D6) δ (ppm): 188.2, 165.5, 162.9 (J C-F = 246.2 Hz), 138.9 (J C-F = 6.4 Hz), 136.1, 132.8, 131.1 (J C-F = 7.8 Hz), 124.9 (J C-F = 2.9 Hz), 120.6 (J C-F = 21.5 Hz), 115.0 (J C-F = 22.6 Hz). HRMS (ESI) calcd. for C 10 H 6 O 3 F [M-H] -: 193.03065; found: 193.0307. (E)-4-(4-fluorophenyl)-4-oxobut-2-enoic acid 13 1 H NMR (300MHz, CDCl 3 ) δ (ppm): 8.06 (m, 2H), 7.98 (d, J = 15.5 Hz, 1H), 7.22 (m, 2H), 6.90 (d, J = 15.5 Hz, 1H). 13 C NMR (75MHz, CDCl 3 ) δ (ppm): 187.5, 170.5, 166.3 (d, J = 257.0 Hz), 138.1, 132.8 (d, J = 2.9 Hz), 131.8, 131.6 (d, J = 5.0 Hz), 116.3 (d, J = 22.1 Hz). HRMS (ESI) calcd. for C 10 H 6 O 3 F [M-H] -: 193.03065; found: 193.0307.

1 H

 1 NMR (300MHz, CDCl 3 ) δ (ppm): 7.84 (dd, J = 15.5 Hz, J = 3.5 Hz, 1H), 7.55 (m, 1H), 7.29 (m, 1H), 7.18 (m, 1H), 6.87 (dd, J = 15.5 Hz, J = 1.4 Hz, 1H). 13 C NMR (75MHz, CDCl 3 ) δ (ppm): 186.3 (d, J = 4.5 Hz), 170.3, 158.8 (d, J = 245.6 Hz), 157.9 (d, J = 249.1 Hz), 140.4 (d, J = 8.0 Hz), 131.6, 126.3 (dd, J = 14.9 Hz, J = 6.7 Hz) 122.3 (dd, J = 24.5 Hz, J = 9.3 Hz), 118.4 (dd, J = 26.3 Hz, J = 8.0 Hz) 117.0 (dd, J = 25.1 Hz, J = 2.7 Hz). HRMS (ESI) calcd. for C 10 H 5 O 3 F 2 [M-H] -: 211.02123; found: 211.0211.

1 H

 1 NMR (300MHz, CDCl 3 ) δ (ppm): 8.26 (s, 1H), 8.20 (d, J = 7.9 Hz, 1H), 8.00 (d, J = 15.5 Hz, 1H), 7.91 (d, J = 7.8 Hz, 1H), 7.70 (t, J = 7.8 Hz, 1H), 6.95 (d, J = 15.5 Hz, 1H). 13 C NMR (75MHz, CDCl 3 ) δ (ppm): 188.0, 170.4, 137.4, 136.9, 132.5, 132.0, 131.4 (q, J = 33.2 Hz), 130.4

SCHEME 1 :

 1 SCHEME 1: Synthesis of the PRAC inhibitors

4 : 7 :

 47 X= p-NO 2 , 5: X= p-CN, 6: X= p-OH, X= m-OCH 3 , 8: X= p-NHCOCH 3

12 :

 12 X= m-F, 13: X= p-F, 14: X= o,p-F 2 , 15: X= m-CF 3 , 16: X= p-Cl, 17: X= m-Cl, 18: X= m-Br

FIGURE 2

 2 FIGURE 2 Inhibition of proline racemases by specific inhibitors. (A) Kinetic curves of 40 mM L-proline racemization catalyzed by 15 µg of CdPRAC. (B) Kinetic curves of 40 mM L-proline racemization catalyzed by 15 µg of CdPRAC in the presence of 5 µM of potential CdPRAC inhibitors. (C) Kinetic curves of 40 mM L-proline racemization catalyzed by 15 µg of TcPRAC. (D) Kinetic curves of 40 mM L-proline racemization catalyzed by 15 µg of TcPRAC in the presence of 5 µM of potential TcPRAC inhibitors. Curves of 15 and 18 idem as 16. The red crosshatched areas indicate the activity of the enzyme alone.

FIGURE 3 :

 3 FIGURE 3: Minimum inhibitory concentration (MIC) values of 20

FIGURE 4 : 20 .

 420 FIGURE 4: CdPRAC inhibitor 20 does not induce mortality in zebrafish embryos. Dose-response curves for zebrafish embryos mortality after 24-, 48-and 72-hours exposition to 20. The graphs show data obtained from three independent experiments.

FIGURE 5 :

 5 FIGURE 5: Treatment with 20 in Clostridioides difficile infected hamsters. Cumulative Kaplan-Meier survival curves of hamsters treated with the compound 20 (NG-P68) or not. Treatment in non-infected control hamsters (left panel) presented no effect. Treatment in infected hamsters (right panel) produced an increase in the survival rate. Log-rank (Mantel-Cox) test to compare the treated group with the infected, non-treated group. The graphs show data obtained from two independent experiments.

(E)-4-oxo-4-(thiophen-2-yl)but-2-enoic acid 3

	1 H NMR (300MHz, CDCl 3 ) δ (ppm): 8.01 (m, 3H), 7.65 (m, 1H), 7.53 (m, 2H), 6.90 (d, J = 15.5
	Hz, 1H). 13 C NMR (75MHz, CDCl 3 ) δ (ppm): 189.2, 170.0, 138.4, 136.4, 134.1, 131.3, 129.0,
	128.9. HRMS (ESI) calcd. for C 10 H 7 O 3 [M-H] -: 175.04007; found: 175.0399
	(E)-4-(furan-2-yl)-4-oxobut-2-enoic acid 2
	1 H NMR (300MHz, CDCl 3 ) δ (ppm): 7.84 (d, J = 15.6 Hz, 1H), 7.71 (m, 1H), 7.40 (m, 1H), 6.99 (d,
	1 H NMR (300MHz, Acetone-D6) δ (ppm): 8.17 (dd, J = 3.8 Hz, J = 1.1 Hz, 1H), 8.05 (dd, J = 4.9
	Hz, J = 1.1 Hz, 1H), 7.86 (d, J = 15.5 Hz, 1H), 7.32 (dd, J = 4.9 Hz, J = 3.8 Hz, 1H), 6.83 (m, 1H).
	13 C NMR (75MHz, Acetone-D6) δ (ppm): 181.9, 166.4, 145.3, 137.1, 137.0, 135.2, 132.6, 129.8.
	HRMS (ESI) calcd. for C 8 H 5 O 3 S [M-H] -: 180.9965; found: 180.9963.
	(E)

J = 15.6 Hz, 1H), 6.64 (m, 1H).

[START_REF] Bianchi | Gastric anti-secretory, anti-ulcer and cytoprotective properties of substituted (E)-4-phenyland heteroaryl-4-oxo-2butenoic acids[END_REF]

C NMR (75MHz, CDCl 3 ) δ (ppm): 176.2, 169.3, 152.8, 148.0, 137.4, 130.9, 119.8, 113.1. HRMS (ESI) calcd. for C 8 H 5 O 4 [M-H] -: 165.01933; found: 165.0191.

-4-(4-nitrophenyl)-4-oxobut-2-enoic acid 4

	1 H NMR (300MHz, Acetone-D6) δ (ppm): 8.42 (m, 2H), 8.33 (m, 2H), 7.95 (d, J = 15.6 Hz, 1H),
	6.83 (d, J = 15.6 Hz, 1H). 13 C NMR (75MHz, Acetone-D6) δ (ppm): 189.2, 165.9, 151.3, 141.8,
	136.7, 133.9, 130.6, 124.5. HRMS (ESI) calcd. for C 10 H 6 NO 5 [M-H] -: 220.02515; found: 220.0250

(E)-4-(4-cyanophenyl)-4-oxobut-2-enoic acid 5 Compound 5 was obtained from general method B2 as

	a pale solid (0.84 g, 42%).
	1 H NMR (300MHz, Acetone-D6) δ (ppm): 8.25 (d, J = 8.7 Hz, 2H), 8.02 (d, J = 8.7 Hz, 2H), 7.93
	(d, J = 15.6 Hz, 1H), 6.83 (d, J = 15.6 Hz, 1H). 13 C NMR (75MHz, Acetone-D6) δ (ppm): 188.7,
	165.4, 139.7, 136.0, 132.8, 129.3, 117.7, 116.5. HRMS (ESI) calcd. for C 11 H 6 NO 3 [M-H] -:
	200.03532; found: 200.0352.
	(E)

-4-(4-hydroxyphenyl)-4-oxobut-2-enoic acid 6

	1 H NMR (300MHz, Acetone-D6) δ (ppm): 8.02 (m, 2H), 7.95 (d, J = 15.5 Hz, 1H), 7.00 (m, 2H),
	6.74 (d, J = 15.5 Hz, 1H). 13 C NMR (75MHz, Acetone-D6) δ (ppm): 187.8, 166.7, 163.6, 137.7,
	132.4, 132.0, 129.8, 116.5. HRMS (ESI) calcd. for C 10 H 7 O 4 [M-H] -: 191.03498; found: 191.0348.
	(E)

-4-(3-methoxyphenyl)-4-oxobut-2-enoic acid 7 Compound 7 was

  obtained from general method B2 as a pale solid (1.27 g, 62%).

	1 H NMR (300MHz, CDCl 3 ) δ (ppm): 7.99 (d, J = 15.8 Hz, 1H), 7.60 (m, 2H), 7.54 (m, 1H), 7.46 (t,
	J = 7.9 Hz, 1H), 7.21 (m, 1H), 6.92 (d, J = 15.8 Hz, 1H), 3.91 (s, 3H). 13 C NMR (75MHz, CDCl 3 ) δ (ppm): 189.0, 169.6, 160.1, 138.3, 137.7, 131.3, 129.9, 121.7, 120.9, 112.7, 55.6. HRMS (ESI) calcd. for C 11 H 9 O 4 [M-H] -: 205.05063; found: 205.0505. Accepted Article (E)

-4-(4-acetamidophenyl)-4-oxobut-2-enoic acid 8

	1 H NMR (300MHz, DMSO-D6) δ (ppm): 10.37 (s, 1H), 8.02 (d, J = 8.8 Hz, 2H), 7.88 (d, J = 15.5
	Hz, 1H), 7.76 (d, J = 8.8 Hz, 2H), 6.66 (d, J = 15.5 Hz, 1H), 2.10 (s, 3H). 13 C NMR (75MHz,
	DMSO-D6) δ (ppm): 187.6, 169.1, 166.4, 144.5, 136.2, 132.4, 130.7, 130.3, 118.4, 24.2. HRMS
	(ESI) calcd. for C 12 H 10 NO 4 [M-H] -: 232.0615; found: 232.0615.

-4-oxo-4-(4-(2-(phenylsulfonamido)ethyl)phenyl)but-2-enoic acid 10

	1 H NMR (300MHz, Acetone-D6) δ (ppm): 8.02 (m, 2H), 7.94 (d, J = 15.5 Hz, 1H), 7.85 (m, 2H),
	7.47 (m, 5H), 6.77 (d, J = 15.5 Hz, 1H), 3.70 (m, 2H), 3.05 (t, J = 7.2 Hz, 2H). 13 C NMR (75MHz,
	Acetone-D6) δ (ppm): 189.6, 167.5, 166.6, 147.5, 137.6, 136.0, 135.9, 132.9, 132.0, 130.4,
	129.9, 129.2, 128.0, 41.6, 36.5. HRMS (ESI) calcd. for C 19 H 16 NO 4 [M-H] -: 322.10848; found:
	322.1083.
	(E)Compound 10 was obtained from N-phenethylbenzenesulfonamide 15 using general method A as
	a pale solid (0.85 g, 53%).
	1 H NMR (300MHz, CDCl 3 ) δ (ppm): 9.59 (s, 1H), 7.86 (m, 5H), 7.49 (m, 4H), 7.25 (m, 1H), 6.83

(d, J = 15.5 Hz, 1H), 5.32 (m, 1H), 3.25 (m, 2H), 2.86 (t, J = 6.9 Hz, 2H).

[START_REF] Bianchi | Gastric anti-secretory, anti-ulcer and cytoprotective properties of substituted (E)-4-phenyland heteroaryl-4-oxo-2butenoic acids[END_REF]

C NMR (75MHz, CDCl 3 ) δ (ppm): 188.7, 169.9, 144.8, 139.7, 138.1, 134.9, 132.8, 131.4, 129.4, 129.3, 129.2, 129.1, 126.9, 126.3, 43.7, 36.0. HRMS (ESI) calcd. for C 18 H 16 NO 5 S [M-H] -: 358.07547; found: 358.0756.

  d, J = 8.3 Hz, 2H), 6.96 (m, 2H), 6.77 (d, J = 15.5 Hz, 1H), 3.84 (s, 3H), 3.68 (m, 2H), 3.04 (t, J = 7.2 Hz, 2H). 13 C NMR (75MHz, Acetone-D6) δ (ppm): 189.6, 166.6, 162.9, 147.6, 137.5, 135.8, 132.9, 130.4, 129.9, 129.7, 128.1, 114.3, 55.7, 41.5, 41.4, 36.6. HRMS (ESI) calcd. for C 20 H 18 NO 5 [M-H]

-: 352.11905; found: 352.1191.

  Hz, 1H).[START_REF] Bianchi | Gastric anti-secretory, anti-ulcer and cytoprotective properties of substituted (E)-4-phenyland heteroaryl-4-oxo-2butenoic acids[END_REF] C NMR (75MHz, CDCl 3 ) δ (ppm): 187.9, 169.8, 140.8, 137.8, 134.7, 131.7, 130.3, 129.4. HRMS (ESI) calcd. for C 10 H 6 O 3 Cl [M-H] -: 209.0011; found: 209.0009.

	1 H NMR (300MHz, CDCl 3 ) δ (ppm): 7.98 (m, 3H), 7.54 (m, 1H), 7.52 (m, 1H), 6.93 (d, J = 15.5
	C 11 H 6 O 3 F 3 [M-H] -: 243.02745; found: 243.0273. (E)-4-(4-chlorophenyl)-4-oxobut-2-enoic acid 16 Accepted Article	6 Hz) HRMS (ESI) calcd. for

TABLE 1 :

 1 Calculated

LogD of compounds 1-21.

  PCR ribotyping Strains were characterized by capillary gel electrophoresis PCR ribotyping as previously described by Bidet et al. [P. Bidet, F. Barbut, V. Lalande, B. Burghoffer, J.C. Petit.

	These strains
	belonged to PCR ribotype 002, 10 (non toxigenic), 014/020/077, 015, 018, 027, 056, 078, 106,
	126, and FR056. For MIC determination, 14 strains of C. difficile (from PCR ribotypes 002 (n=1),
	10 (n=1), 014/20/77 (n=1), 015 (n=1), 027 (n=1), 056 (n=1), 078 (n=3) 106 (n=1), 126 (n=3) and
	FR 056 (n=1). were used.
	Development of a new PCR-ribotyping method for Clostridioides difficile based on ribosomal RNA
	gene sequencing. FEMS Microbiol. Lett.,1999;175:261-266]. After DNA amplification, 1 μL of a
	1/200 dilution of each PCR product was mixed with 10.5 μL formamide and 0.5 μL GeneScan
	LIZ600 (Applied Biosystems®, Foster City, USA) as an internal marker. After 30 s of denaturation
	at 90 °C, capillary electrophoresis was performed on an 8-capillary 3500 Genetic Analyzer
	(Applied Biosystems®, Foster City, USA). The GeneMapper software (Thermo Fisher Scientific,

  Barbut, B. Gariazzo, L. Bonne, V. Lalande, B. Briefly, colonies of C. difficile were suspended in sterile saline buffer (no. 1 McFarland standard) and were swabbed on prereduced Brucella agar. A 12 mm-diameter disk was placed on the agar and 50µl of each TcPRAC inhibitors (1 mg/ml) were deposited on the disk. The diameter of inhibition was read after 48h-incubation in anaerobic jars.

	Burghoffer, R. Luiuz, et al. Clinical features of Clostridioides difficile-associated infections and
	molecular characterization of strains: results of a retrospective study, 2000-2004. Infect. Control
	Hosp. Epidemiol. 2007;28:131-139].

Other antimicrobials including vancomycin (30 μ g), metronidazole (4 μ g [Sanofi Diagnostics Pasteur, Marnes la Coquette, France] and 16 μg [Rosco, Taastrup, Denmark])), erythromycin (15 IU), clindamycin (2 IU), moxifloxaxin (5 μ g), chloramphenicol (30 μ g), imipenem (10 μ g) and tetracycline (30 IU) were also tested. The results were interpreted according to the French Society of Microbiology's " Comit é de l ' Antibiogramme " (2013, available at: https://resapath.anses.fr/resapath_uploadfiles/files/Documents/2013_CASFM.pdf). Strains were considered susceptible if the inhibition diameters for vancomycin (VA), metronidazole (MZ), erythromycin (ERY), clindamycin (CM), moxifloxacin (MXF), chloramphenicol (C), imipenem (IMI) and tetracycline (TE) were greater than or equal to 17 mm, 21 mm, 22 mm, 15 mm, 23 mm, 23 mm, 24 mm and 23 mm respectively.

  -infection, the animals were treated by oral gavage with 100 mg/kg of the compound 20 diluted in PBS + 6.7% DMSO in a final volume of 300 µL/animal. Treatment was administered once a day, during four days. Control animals received the PBS+DMSO only. All hamsters were followed-up twice a day.

	Ethical statement: Animals were housed in the Institut Pasteur animal facilities accredited by the
	French Ministry of Agriculture for performing experiments on live rodents. Work on animals was
	performed in compliance with French and European regulations on care and protection of
	laboratory animals (EC Directive 2010/63, French Law 2013-118, February 6th, 2013). All
	experiments were approved by the Ethics Committee #89 and registered under the reference
	#2013-0047.
	4 spores of C. difficile (strain
	K027) in 300 µL of PBS. Mock-infected animals received the PBS only. Infected and mock-

infected animals were housed in separated cages in a BSL-3 environment. Starting on day 1

Accepted Article

post

  Strains were considered susceptible if the inhibition diameters for vancomycin, metronidazole, erythromycin, clindamycin, moxifloxacin, chloramphenicol, imipenem and tetracycline were greater than or equal to 17 mm, 21 mm, 22 mm, 15 mm, 23 mm, 23 mm, 24 mm and 23 mm respectively.

		PCR-				MO	MTZ	MTZ
			ERY CIP CM			VA TE IMP C
	Number	Ribotype				XI	5µg	16µg
	CD17-027	002 25	6	8 25	36	60 30 50	28 30
	CD17-029	106 30	6	8 24	34	50 35 50	30 35
	CD17-159	027	6	6	6	6	22	43 36 50	22 36
	CD17-033	015 25	6	13 21	33	46 30 42	24 29
	CD17-118	010	6	8	6 20	6	28 30 50	24 27
	CD17-035	FR056 30	6	6 24	34	51 34 46	24 32
	CD17-037	014/020/07	23	6	6	6	29	44 30 46	30 25
	CD17-038	018	6	6	6	6	35	50 33 46	14 28
	CD17-049	056 27	9	13 24	36	52 30 54	26 34
	CD17-142	126	6	6	6 21	38	55 32 21	25 30
	CD16-220	078 31	6	13 28	34	66 32 20	26 32

ERY: Erythromycin, CIP: Ciprofloxacin, CM: Clindamycin, MOXI: Moxifloxacin, MTZ: metronidazole, VA: Vancomycin, TE: Tetracyclin, IMP: imipenem, C: chloramphenicol.

Table 2 :

 2 Antimicrobial susceptibility of the studied strains

TABLE 3 :

 3 Activities of compounds 1

	(inhibition diameter at 50L)

-3, 21

9 :

 9 R= C 6 H 5 CO-, 10: R= C 6 H 5 SO 2 -, 11: R= p-MeO-C 6 H 4 CO-

		4-8					9-11
	Clostridioides difficile	Compounds (inhibition diameter at 50L)
	strain	PCR-ribo	4	5	6	7	8	9	10 11
	CD17-027	2	17	12	13	23 22 16	16 16
	CD17-029	106	12	12	12	25 22 17	12 17
	CD17-159	27	19	12	16	26 23 18	18 15
	CD17-033	15	17	12	12	24 20 16	16 17
	CD17-118	10	17	12	14	24 22 16	15 17
	CD17-035	FR056	16	12	12	24 22 15	16 16
	CD17-037	014/020/077 18	12	14	25 24 16	19 15
	CD17-038	18	18	12	12	24 20 12	14 15
	CD17-049	56	14	15	17	25 23 17	17 17
	CD17-142	126	12	12	12	12 17 12	12 12
	CD16-220	78	12	12	12	12 17 12	12 12
			172	135	146	247	238	167	167	169
			40	3	14	112	100	35	35	37

TABLE 4 :

 4 Activities of compounds 4-11

TABLE 5 :

 5 Activities of compounds 12-20

	Clostridioides difficile	Compounds (inhibition diameter at 50L)
	strain	PCR-ribo	12	13	14	15	16	17	18	19	20
	CD17-027	2	23	24	24	27	24	24	23	24	26
	CD17-029	106	24	25	25	13	25	25	24	25	27
	CD17-159	27	25	25	24	26	25	25	25	/	26
	CD17-033	15	23	26	23	26	24	24	23	25	25
	CD17-118	10	22	24	23	26	23	24	22	/	25
	CD17-035	FR056	23	25	23	25	23	24	23	24	25
	CD17-037	014/020/077	23	26	24	27	25	25	24	27	26
	CD17-038	18	22	24	23	24	22	23	22	23	24
	CD17-049	56	23	26	25	27	25	24	24	29	25
	CD17-142	126	14	14	15	21	19	17	19	22	19
	CD16-220	78	17	16	15	21	20	20	19	22	20
			239	255
			107

  giving similar Log D values as illustrated in Table6.

	Compound	Log D (pH 5)	Log D (pH 7)	Log D (pH 9)
	1	1,71	-1.68	-1.81
	8	0.95	-2.49	-2.57
	15	2.59	-1.04	-1.05
	16	2.32	-1.16	-1.21
	17	2.32	-1.16	-1.21
	18	2.48	-1.04	-1.05
	19	2.23	-1.29	-1.30
	20	2;39	-1.14	-1.13

TABLE 6 :

 6 Log D at pH 5, 7 and 9

3.3 Inhibiting Activity of compounds 8, 15-20 on CdPRAC In

  Of note, the level of TcPRAC inhibition was higher than for CdPRAC (for example 20 inhibitted 76% of TcPRAC activity versus 38% for CdPRAC) and that the activity sequence differed (Figure2). Derivative 8, the most hydrophilic compound, was the best inhibitor for TcPRAC but the worse for CdPRAC. If the catalytic site sequence in both

	enzymes is conserved, this result indicated that variations of other amino-acids may
	Accepted Article induce recognition modifications.

order to identify a hit for further development, evaluation of the cdPRAC inhibition in connection with the antibacterial activity was assessed on the seven selected compounds (8, 15-20) (Figures

2 A, B

). The highest inhibition activity (38%) was observed in the presence of compound 20. For comparison inhibition obtained with the Trypanosoma cruzi proline racemase

[6]

with derivatives 8, 16, 17, 19 and 20 are reported on Figures 2 C, D.

Data availability statement

Data available on request from the authors: the data that support the findings of this study are available from the corresponding author upon reasonable request.


Ilhem Slama
email: ilhem.slama@imt-atlantique.fr

Oussama Ben-Ammar
email: oussama.ben-ammar@emse.fr

Alexandre Dolgui
email: alexandre.dolgui@imt-atlantique.fr

A model for a multi-level disassembly system under random disassembly lead times

Keywords: Reverse supply chain, Disassemble-to-order, Stochastic lead times

This paper deals with the problem of planned disassembly lead time calculation in a Reverse Material Requirement Planning (RMRP) environment under uncertainty. A multi-level disassembly system with one type of end-of-life product and several types of components at each level is considered for the first time in the disassembly planning problem under uncertainty. The paper presents a mathematical model with corresponding proofs. The objective of the proposed model is to minimise the total expected cost which is composed of holding and backlogging costs. Some advantages of the proposed model and perspectives of this research are discussed.

Introduction and related publications

With accumulated environmental pressure on industrial activities, the reverse flow has become indispensable. The remanufacturing process is a crucial step in the reverse logistics network, and it concerns all tasks associated with the collection, disassembly, refurbishing, repair, recycling, disposal, etc. of end-oflife (EoL) products [START_REF] Kim | Multi-period disassembly levelling and lot-sizing for multiple product types with parts commonality[END_REF]. In the scientific literature, the disassembly process has recently received a lot of attention due to its importance in the recovery of products. It allows the selective separation of parts in order to recover materials, isolate hazardous substances and separate reusable items [START_REF] Slama | New mixed integer approach to solve a multi-level capacitated disassembly lot-sizing problem with defective items and backlogging[END_REF].

The current paper proposes to model the disassembly planning problem, which is one of the main problems associated with product recovery. The motivations for initiating tactical disassembly planning are very varied and multiple. On a large scale, optimal planning of disassembly operations is necessary to efficiently process a large volume of products to be upgraded. It also aims to organise a succession of operations over time in order to meet the demands for components on predefined delivery dates.

Making decisions in an uncertain environment is a difficult task for many industrial sectors. In fact, some parameters are often considered deterministic, when in reality they are inherently uncertain. There are many sources of uncertainty related to demand and the quantity and quality of parts that can disrupt the disassembly process. For example, a machine failure can interrupt the disassembly process and subsequently, component disassembly times become longer than expected. Consequently, these uncertainties disrupt the level of stock, and thus, create disruption and unnecessary storage. Several methods have been developed in the literature to take into account the effect of uncertainty in the disassembly planning problem. Without trying to do an exhaustive review of the literature, we discuss the previous works on disassembly planning problem under uncertainty. For a more exhaustive literature review, readers can refer to [START_REF] Slama | Disassembly scheduling problem: literature review and future research directions[END_REF][START_REF] Kim | Disassembly scheduling: literature review and future research directions[END_REF].

The models proposed in the literature can be classified as a single or multiperiod planning problem, two or multi echelon bill of materials (BOM) (see Table 1). Also, they can be divided into four different categories depending on the type of uncertainty: (1) demand (see [START_REF] Kim | Capacitated disassembly scheduling with random demand[END_REF][START_REF] Fang | A stochastic production planning problem in hybrid manufacturing and remanufacturing systems with resource capacity planning[END_REF]), (2) disassembly yield (see for example the works of [START_REF] Inderfurth | How yield process misspecification affects the solution of disassemble-to-order problems[END_REF][START_REF] Inderfurth | Heuristics for solving disassemble-to-order problems with stochastic yields[END_REF][START_REF] Kongar | Disassembly to order system under uncertainty[END_REF]), (3) yield and demand [START_REF] Liu | Capacitated disassembly scheduling under stochastic yield and demand[END_REF] and (4) disassembly lead time (DLT). The latter is defined as the time difference between placing a disassembly order and receiving the disassembled item at each level of the BOM. For different reasons (machine breakdowns, absenteeism, quality problems, etc.), the disassembly lead times are often stochastic. To minimise the effects of these random factors, companies can implement safety lead times (or safety stock ), but theses stocks are very expensive. On the other hand, if there are not enough stocks, we can observe stockout and the corresponding backlog cost. So, the main goal is to minimise the total cost which is composed of holding and backlog costs.

As far as can be determined from the literature, for the uncertainty of timing, the number of publications that study disassembly systems is very modest compared to those that deal with assembly systems [START_REF] Ben-Ammar | A hybrid genetic algorithm for a multilevel assembly replenishment planning problem with stochastic lead times[END_REF][START_REF] Slama | Genetic algorithm and monte carlo simulation for a stochastic capacitated disassembly lot-sizing problem under random lead times[END_REF]. In the paper of [START_REF] Slama | Scenario-based stochastic linear programming model for multi-period disassembly lot-sizing problems under random lead time[END_REF], only the uncertainty of DLT of the EoL product is studied. The case of multiperiod, single product type and two-level disassembly system is investigated. The disassemble capacity is supposed infinite. The Scenario-based stochastic Linear Programming model (S-LP) is developed. The proposed approach is used to determine the optimal EoL products to be disassembled in order to minimise the Average Total Cost (ATC) over the planning horizon. In the work of [START_REF] Guiras | Optimization of two-level disassembly/remanufacturing/assembly system with an integrated maintenance strategy[END_REF], the authors developed a mathematical model to found the optimal plan of disassembly/assembly system under breakdowns machine and stochastic lead times. A heuristic is developed to determine the optimal ordered date of the EoL product as well as the optimum release dates of new external components. Later, [START_REF] Slama | A newsboy formulae to optimize planned lead times for two-level disassembly systems[END_REF] proposed a generalisation of the discrete Newsboy formulae to find the optimal release date when the time of disassembling the EoL product is random variable. The case of one-period, single product type and two-level disassembly system is treated. In the same year, [START_REF] Slama | Newsboy problem with twolevel disassembly system and stochastic lead time[END_REF] extend the work of [START_REF] Slama | A newsboy formulae to optimize planned lead times for two-level disassembly systems[END_REF] by considering the uncertainty of disassembly lead times of each component. In the same context, [START_REF] Slama | A stochastic model for a two-level disassembly lot-sizing problem under random lead time[END_REF] extend the work proposed by [START_REF] Slama | Scenario-based stochastic linear programming model for multi-period disassembly lot-sizing problems under random lead time[END_REF] by considering the time limit of disassembly capacity in each period of the planning horizon. The problem is formulated as a two-stage stochastic Mixed Integer Linear Programming (2S-MILP) model. The Sample Average Approximation (SAA) approach is developed in order to minimise the ATC. Recently, [START_REF] Slama | Approches d'optimisation pour un problème de planification de désassemblage sous incertitude des délais de désassemblage[END_REF] addressed a multi-period disassembly lot-sizing problem. The case of single product type and two-level disassembly system is studied. The problem is formulated as 2S-MILP model through all possible scenarios. To solve large scale problems, the authors proposed a basic genetic algorithm (GA).

As presented above, all the previous works under uncertainty of DLT are confined to the two-level disassembly system. Solving the studied problem with a multi-level BOM is more complex because of dependencies between the different sub-assemblies and components at each level. For this reason, this paper extends the work of [START_REF] Slama | A newsboy formulae to optimize planned lead times for two-level disassembly systems[END_REF] and [START_REF] Slama | Newsboy problem with twolevel disassembly system and stochastic lead time[END_REF] by considering a multi-level disassembly system to study a one-period planning for the disassemble-to-order (DTO) problem. A mathematical model is developed to calculate the total expected costs composed of the component backlog and holding costs.

The rest of this paper is organised as follows. Section 2 describes the problem. Section 3 presents the stochastic model of the multi-level disassembly to-order problem. Section 4 presents the conclusions with avenues for future studies.

Problem description

We consider the problem with the multi-level disassembly system (with several types of components at each level) as shown in Figure 1. The non-leaf item i (can be root or sub-assembly (i ∈ A 1 )) represents the item to disassemble. It has more than one child and a child element denotes non-root item (can be part or sub-assembly) that has only one parent. Customers have a fixed demand for all disassembled leaf item (i ∈ A 0 ). Theses demands must be delivered with certain delivery day. Once the disassembly process is started, the items recovery pro- Fig. 1: A multi-level disassembly system. cess provides no information about the state of components until the end of the disassembly process. Then, each item can be received after a stochastic disassembly lead time where the items undergo several renovation processes (such as repair and cleaning). The DLT of each item L i is a random discrete variable with a known probability distribution and bounded over known intervals

(L i ∈ [L - i , L + i ]
). Here, we supposed that the probability is not identically distributed, that's to say, the DLT for each item don't following the same probability distribution.

The assumptions of the studied problem can be summarised as follows:

1. The one-period planning for the DTO environment and a single demand for each type of component is considered; 2. At each level, all parent items must proceed to disassembly once they are ready; 3. If the date of obtaining an item from its parent is less than its scheduled disassembly start date, this item will be kept until the scheduled release time, which incurs a holding cost; 4. If, at a certain level in the BOM, a component is not received as expected, a backlog cost is incurred.

Problem formulation

The aim of this research is to develop a mathematical model for multi-level disassembly systems under a fixed part's demand and uncertainty of parts disassembly lead times.

Let i = 1 be the index of the EoL product, n the index of the last disassembled item and the following three sets: (i) A the set of all elements i = 1, . . . , n, (ii) A 0 the set of all obtained items that we can not disassemble and to be delivered to clients and (iii) A 1 the set of all obtained items that we can disassemble. Therefore A = {1} ∪ A 0 ∪ A 1 Here, we have only demands for items i from the last level. i.e. i ∈ A 0 . The notation used in the paper are presented in Table 2.

Table 2: Model notation and definition

Parameters

i Index of items, i = 1, ..., n p(i) Index of the parent of item i, ∀i ∈ A 0 ∪ A 1 T i Delivery date for item i, ∀i ∈ A 0 h i Unit inventory holding cost for item i, ∀i ∈ A 0 ∪ A 1 b i Unit backlogging cost for item i, ∀i ∈ A 0 ∪ A 1 L i Actual disassembly lead time of item i, ∀i ∈ A 0 ∪ A 1 Variables X i Decision variable: Planned date of availability of item i, ∀i ∈ A 1 Functions E[[.]] Excepted value F i [[.]] Distribution function of the random variable L i , ∀i ∈ A 0 ∪ A 1 T + i max(M i ; T i ), ∀i ∈ A 0 T - i min(M i ; T i ), ∀i ∈ A 0
The total cost is equal to the sum of backlogging and inventory holding costs for all components. Proposition 1. For the system described in the previous section, the total cost, noted by T C(X, L), is as follows:

T C(X, L) = i∈A 0 (b i + h i ) × max(T i ; M i ) + i∈A 1 (b i + h i ) × max(X i ; M i ) - i∈A 0 b i T i - i∈A 1 b i X i - i∈A 0 ∪A 1 h i M i (1)
Proof. In this prof, several costs will be detailed. Let M i be the actual availability date of item i; M i = X 1 + L i ∀i ∈ A 1 and p(i) = 1. We first calculate C h1 (X, L) the inventory holding cost of components disassembled from EoL product (∀i

∈ A 1 , p(i) = 1): C h1 (X, L) = i∈A 1 p(i)=1 h i (X i -min(X i ; M i )) (2)
Let M i be the actual availability date of item i; M i = max(X p(i) ; M p(i) ) + L i ∀i ∈ A 1 and p(i) > 1. Then, we can calculate C h (X, L) the inventory holding cost of components disassembled from items:

C h (X, L) = i∈A 1 p(i)>1 h i (X i -min(X i ; M i )) (3)
Let M i be the actual availability date of item i; M i = max(T i ; M p(i) ) + L i ∀i ∈ A 0 . We calculate C h0 (X, L) the inventory holding cost of items that we have to deliver to clients:

C h0 (X, L) = i∈A 0 h i (T i -min(T i ; M i )) (4)
From Expressions ( 2)-( 4), we can deduce the total holding cost:

C H (X, L) = i∈A 0 h i (T i -min(T i ; M i )) + i∈A 1 h i (X i -min(X i ; M i )) (5)
By the same way, we can easily deduce the total backlogging cost:

C B (X, L) = i∈A 0 b i (max(T i ; M i ) -T i ) + i∈A 1 b i (max(X i ; M i ) -X i ) (6)
Finally, the total cost can be formulated from Expressions ( 5)-( 6).

Proposition 2. The mathematical expectation of the total cost is given by the following expression:

E[[C(X, L)]] = i∈A 0 (b i + h i ) × s≥Ti (1 -F i (s)) + i∈A 1 (b i + h i ) × s≥Xi (1 -F i (s)) + i∈A 0 h i T i - i∈A \1 h i (E[[L i ]] + s≥X p(i) (1 -F p(i) (s))) + i∈A 1 h i X i - i∈A \1 h i X p(i)
where:

F i (s) =          1 if i = 1 F i (-X p(i) + s) if p(i) = 1 F i (-X p(i) + s) vi+wi=s vi+wi∈N P[[L i = v i ]] × F p(i) (w i ) otherwise Proof. First, we formulate E[[M i ]], E[[max(X i ; M i )]] and E[[max(T i ; M i )]]. For all i ∈ A and p(i) = 1, M i is the actual availability date of item i; M i = X 1 + L i . So E[[M i ]] = E[[L i ]] + X 1 .
For all i ∈ A 1 and p(p(i)) = 1, M i is the actual availability date of item i. So M i is equal to max(X p(i) ; M p(i) ) + L i and:

E[[M i ]] = E[[L i ]] + s≥0 (1 -P[[X p(i) ≤ s]]P[[M p(i) ≤ s)]] Moreover, knowing that P r[[X p(i) ≤ s]] = 0 ∀s ∈ [0, X p(i) [ and P r[[X p(i) ≤ s]] = 1 ∀s ≥ X p(i) , then: E[[M i ]] = E[[L i ]] + X p(i) + s≥X p(i) (1 -P[[M p(i) ≤ s)]] Knowing that M p(i) = X 1 + L p(i) , so: E[[M i ]] = E[[L i ]] + X p(i) + s≥X p(i) (1 -F p(i) (-X 1 + s)) (7)
For all i ∈ A and p(p(p(i)))) = 1, M i is the actual availability date of item i. So M i is equal to max(X p(i) ; M p(i) ) + L i and:

E[[M i ]] = E[[L i ]] + X p(i) + s≥X p(i) (1 -F p(i) (-X p(p(i)) + s) o 1 +o 2 =s o1+o2∈N (P[[L p(i) = o 1 ]]P[[M p(p(i)) ≤ o 2 )]])
By using the recursive function, we can easily formulate E

[[M i ]], E[[max(X i ; M i )]] and E[[max(T i ; M i )]] as follows: E[[M i ]] = E[[L i ]] + X p(i) + s≥X p(i) (1 -F p(i) (s)) (8) E[[max(X i ; M i )]] = X i + s≥Xi (1 -F i (s)) (9)
E[[max(T i ;

M i )]] = T i + s≥Ti (1 -F i (s)) (10)
Then, the expression of the expected total cost can be deduced.

Conclusion and perspectives

In this preliminary work, we model a multi-level disassembly problem in a DTO environment. We consider a one type of EoL product and several types of components at each level. The disassembly lead times at each level are discrete independent random variables, and the items demand at the finished level is fixed. The disassembly of the components is carried out as soon as their parents are available. The demand of the leaf items should be satisfied on predefined delivery dates. Otherwise a backlogging cost is incurred. If a given component is available before the delivery date, we stored it until this period. The developed mathematical model calculates the expected value of the sum of backlogging and inventory costs.

The expected total cost is not linear. In the future, we will develop an optimisation approach to optimise the release dates for the non-leaf items. Research into solving this problem is in progress. The developed approach is based on genetic algorithm developed for assembly systems under uncertainty of lead times [START_REF] Ben-Ammar | Optimal order release dates for two-level assembly systems with stochastic lead times at each level[END_REF].

Table 1 :

 1 Summary of relevant literature under uncertainly

	Authors	Resolution	Uncertainty
	[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] Current Paper Analytical model MILP, Lagrangian heuristics MILP, Lagrangian heuristics Lagrangian heuristics Heuristics Fuzzy goal programming MILP, Outer-approximation Analytical model, Heuristic S-LP Analytical model, Newsboy formulae Analytical model, Newsboy formulae 2S-MILP, SAA 2S-MILP, GA	Yield √ √ √ √ √ Demand √ √	DLT √ √ √ √ √ √ √	#Periods m 2 #Levels m 2 1 2 1 2 1 2 m 2 1 2 m 2 1 2 1 2 m 2 m 2 1 m
	m: multi-