Juba Ghouilem

Dr Morgane De Robichon

Dr Franck Le Bideau

Angélique Ferry
email: frangelique.ferry@cyu.fr

Samir Messaoudi
email: samir.messaoudi@universite-paris-saclay.fr

Emerging Organometallic Methods for the Synthesis of C-Branched (Hetero)aryl, Alkenyl, and Alkyl Glycosides: C-H Functionalization and Dual Photoredox Approaches

Keywords: C-glycosides, C-H functionalization, Photoredox dual catalysis, glycals, glycosylation

HAL is

Introduction

Vitexin was the first known C-glycoside extracted at the turn of the 20 th century by Perkin (Figure 1), 1 from a New Zealand tree, Vitex littoralis. This discovery followed a request of an Auckland resident for an expertise on the commercial value of a yellow substance as a dye contained in this wood. Nevertheless, the full determination of its structure was established only in 1964 by Horowitz and Gentili, 2 although Perkin had already guessed 3 that this stable compound consisted of a sugar residue linked in an abnormal way to apigenin. This stability results from the presence of an unnatural C-C bond between the carbohydrate and its aglycone, the very definition of a C-branched glycoside, making them less sensitive to acidic or enzymatic hydrolysis than the usual C-O and C-N glycosidic bonds. Since the discovery of vitexin, other Cglycosides have been extracted from the living world, as illustrated by the few examples reported in Figure 1. Isovitexin, an isomer of vitexin, differs from the vitexin by the position where the sugar is linked to the apigenin aglycone. Both were found to possess various biological effects including anti-cancer, anti-oxidant, anti-inflammatory, anti-nociceptive, anti-AD (AD, Alzheimer's disease), anti-hypertensive, anti-spasmodic, antihypoxia/ischemia injury, anti-depressant-like actions and antiviral activities. 4 In the same vein, orientin and isoorientin, which incorporate an extra hydroxyl group, were also the subject of diverse pharmacological studies. 5 Apart from these flavonoids, C-glycosides are also found in the angucycline family, antibiotics of microbial origin, whose aquayamycin is one of the members. 6 The latter was found to inhibit tyrosine hydroxylase and is active against Gram-positive bacteria, Ehrlich ascites carcinoma, and Yoshida rat sarcoma cells but showed no particular effect on a wide range of human tumor cells. 7 Pyrazomycin, also called pyrazofurin, found in Streptomyces candidus, is a pyrazolelinked C-furanoside,which is a potent inhibitor of orotidine 5'monophosphate decarboxylase and which possesses broadspectrum antiviral and antitumor activities. 8 Showdomycin, whose gene cluster for its biosynthesis was recently discovered, 9 is an other example of a natural C-furanoside bearing a maleimide moiety. Gilvocarcin V, the main product of Streptomyces griseoflavus Gö 3592, showed strong anti-tumor activities in comparison with its analogs gilvocarcins E and M lacking the aglycone vinyl group. 10 The antibiotic granaticin, first isolated in 1957 11 from the culture of Streptomyces olivaceus, is a rare example of fused C-aryl glycoside incorporating two C-C bonds between the sugar and its aglycon. Surprisingly, despite its originality, it has only been the subject of one racemic and one enantioselective syntheses to date. 12 Papulacandin D is the simplest member of a series of naturally occurring anti-fungal agents, displaying a benzannulated spiroketal unit. 13  -C-Mannosyltryptophan, based on an indole motif, observed in Ribonuclease 2 from human urine, is the first example of a C-glycosyl amino acid found in proteins. 14 Eventually, cassialoin, like showdomycin, is a rare example of a C-glycoside linked to a Csp 3 carbon of the aglycone. 15 It was isolated from the heartwood of Cassia garrettiana Craib and shows anti-cancer effects. Anthracycline-type C-glycosides 16 such as marmycin, kidamycin and saptomycin exhibit very promising antiproliferative properties and have been proposed to operate via similar mechanism than doxorobucin as DNA intercalants. In addition, it was reported recently that these Cglycosides may act through other original mechanisms including direct genome targeting. 16a The existence of many natural C-glycosides possessing various biological properties has stimulated the interest of medicinal chemists, which led to the development of several commercial synthetic C-glycoside drugs (Figure 2) bearing mainly anti-diabetic properties. Among them, empagliflozin, dapagliflozin, canagliflozin, ipragliflozin, ertugliflozin and tofogliflozin have been approved for the treatment of type 2 diabetes mellitus recently. 17 Empagliflozin, dapagliflozin and canagliflozin were also found to possess cardio-renal benefits in adults with type 2 diabetes mellitus who either have established cardiovascular disease or multiple cardiovascular risk factors. 18 Tofogliflozin displays a benzannulated spiroketal unit found in natural papulacandin D. Pro-xylane TM is a rare example of C-alkylglycoside, synthesized from renewable feedstock, which is used in skin anti-aging applications in cosmetic. 19 Synthetic galactosylceramide (-GalCer also named KRN7000) has shown interesting anti-cancer activities among others. 20 The corresponding C-glycoside (-C-GalCer) was shown to remain active in a mouse malaria model for a longer period of time than its O-glycosyl counterpart, one reasonable explanation being the highest stability of the former toward degradation by endogenous -galactosidases. -C-GalCer was also found to be 100 times more effective than -GalCer against B16 melanoma cells in C57BL/6 mice. However, none of these compounds were developed as drugs up to date. Owing to the interesting pharmacological properties of numerous C-glycosides, their synthesis 21,22 and the synthesis of their analogues are a field of intense studies. Besides organic synthetic methods, the development of organometallic catalyzed processes under sub-stoichiometric conditions has emerged in the eighties. The Tsuji-Trost approach, firstly reported in 1985, 23 allowed a stereospecific access to Csp 3or Csp 2 -glycosides (Scheme 1) where the outcome of the reaction is controlled by the nature of the nucleophile: stabilized nucleophiles favor the formation of products with retention of configuration whereas non-stabilized nucleophiles led to an inversion of configuration. Nevertheless, loss of stereoselectivity may be observed via an inversion of configuration of the -allyl intermediate. Due to the electron-rich vinyl ether motif, the process is totally regioselective, the attack of the nucleophile proceeds at the C1 position of the -allyl glucal and needs the presence of a good leaving group. 24 , 25 Heck 26 or Heck type 27 reactions were also applied to C-aryl-glycosides syntheses. 28 The reaction is fully regioselective at the C1 position and stereospecific, an antichirality transfer from the initial stereocenter C3 of the reactant governing the anomeric configuration in the products. 29 Recently, 2,3-pseudoglycal building blocks were also used for the synthesis of C2-aryl glycosides through transfer chirality approach. 30 Stille, Suzuki-Miyaura, Hiyama, Denmark and Negishi cross-couplings were useful tools to deliver vinyl-or aryl-Csp 2 -glycosides since the seminal work of Friesen and Sturino. 31 These methods were both successfully applied to glycal-or glycoside-type substrates 32,33,34,35,36,37,38 (Scheme 1) in mild conditions. In the former case, a major disadvantage lies in the preparation of C1 functionalized glycals which generally require the use of a strong base excess, not compatible with a wide range of functional groups. In an inverted strategy, 2-iodoglycals were widely used as electrophilic partners for the synthesis of 2-C-branched glycosides (Scheme 1). Various Pdcatalyzed reactions such as Suzuki-Miyaura, 39 Heck, 40 Sonogashira, 41 and carbonylative cross couplings 42 have been applied successfully in this context.

These last few years, organometallic catalysis evolved toward C-H functionalization 43 , 44 , 45 and dual photoredox 46 , 47 processes with the concern of limiting as much as possible number of steps in a synthetic route. In view of these different advantages, this review will deal with the synthetic successes of these two approaches.

Juba

. Directed C-H functionalization of aromatic and aliphatic substrates

One of the established strategies for the synthesis of Cbranched glycosides through C-H functionalization is based on the use of aryl or alkyl substrates bearing a designed directing group (DG). In this case, the link between the sugar and the aglycone was performed via a C-H functionalization using sugar halides as electrophilic partners.

Csp 2 -H functionalization

In 2016, Ye et al. published 48 the synthesis of aryl-C- 1,2 glycosides from 1-iodoglycals 2 based on a palladium-catalyzed C-H activation of benzamides 1 (Scheme 2) directed by an ortho N-quinolyl (N-Q) motif, leading to mono-and di-arylated products 3 and 4. A thorough screening of the reaction conditions allowed a reversal in the proportion of the two possible products in favor of the monoglycosylated adduct 3. Thus, the use of NaHCO3 as an inorganic base, t-amylOH/H2O (10:1) as a solvent mixture and a L-proline derivative as an external ligand led preferentially to the formation of a wide array of monoglycosylated products 3, bearing electron-donor or electron-withdrawing groups in different positions of the aryl moieties in yields ranging from 60 to 91%. However, when the second ortho position is not substituted, up to 16% of di-arylated compounds 4 were observed despite the use of two equivalents of expensive glycals 2 for one equivalent of the aryls 1. Removal of the DG was accomplished using two equivalents of the expensive Schwartz's reagent (bis(cyclopentadienyl)-zirconium(IV) chloride hydride) followed by transformation in situ of the unstable resulting aldehydes into alkenes or alcohols in good yields (68-70%).

Recently, Ackermann et al. reported 49 on the same type of transformation involving a N-triazolyldimethylmethyl (N-TAM) DG (5, Scheme 3). Milder reaction conditions (80°C) were used to afford, without the need of an additional ligand, monoglycosylated derivatives 6 in excellent yields (78-95%), except in the case of the furan adduct (54%). A reasonable catalytic cycle (Scheme 4) starts with the complexation and the C-H palladation of the benzamide, delivering the Pd(II) palladacycle I. Oxidative addition of the latter in the C-I bond of the glucal then leads to Pd(IV) intermediate II, which undergoes a reductive elimination to release the aryl-C- 1,2 glycoside moiety in the Pd(II) complex III. The strong electron donor effect of the N,N-bidentate N-Q ligand in I facilitates the oxidative addition step. Eventually, decomplexation in the presence of AcOH and a base regenerates Pd(OAc)2 concomitantly with the formation of the product of the reaction. Scheme 4. A possible mechanism for the Pd(II) catalyzed synthesis of aryl-C- 1,2 glycosides (X = no atom) according to Ye and Ackermann reports. Except for the oxidative addition step, this mechanism is also reasonable for the synthesis of C-aryl pyranosides (X = no atom, CH2 or O) relatively to Chen's work (vide infra).

In 2019, Chen et al. disclosed 50 the synthesis of C-aryl glycosides (Scheme 5) in reaction conditions close to Ye's work, using KOAc as base and toluene as the solvent. Despite excellent results (90-95% yield, except in one case (R = OAc, 40%)) obtained with the inexpensive co-ligand , the more expensive N-acetyl protected isoleucine (Ac-Ile-OH) was used in most cases. The exact role of this co-ligand is not clear, but appears to be essential to obtain excellent yields (90%) at low temperatures (up to 45°C). A wide range of products was synthesized in good to excellent yields (62-95%) varying the nature of the benzamide 1 (Scheme 5 and 6) or the nature of the pyranose/furanose (Scheme 6 and 7). The presence of a strong electron withdrawing group (CF3) on the aryl ring of 1 led to a low 27% yield of the desired product. A moderate yield (53%) was also obtained in the case of L-xylose with no particular explanation. Heteroarenes such as furans, pyrroles, thiophenes and indoles were used successfully in this reaction. Formation of di-arylated substrates was prevented by steric hindrance found in 1,3substituted benzamides. Functional groups such as tertbutyldiphenylsilyl (TBDPS), methoxymethyl (MOM), acetate, 4,6benzylidene, isopropylidene and allyl were tolerated. This reaction was also applied to more complex substrates such as tri-and tetra-saccharides and to the mannosylation of repaglinide, an antidiabetic drug. Removal of the NHQ DG was studied on a few examples, using Schwartz's reagent or saponification conditions.

A few other DG were tested in the C-H activating process (Scheme 8) showing that the presence of a bidentate moiety as well as a non-substituted N-amide atom were essential, leading to the coupled products in moderate to excellent yields (57-93%). The nature of the different amidoquinolyl-type DG were also studied as shown in Figure 3.In the case of aryl acetamide (X = CH2 in scheme 4) and phenol derivatives (X = O in scheme 4). The corresponding aryl and heteroaryl substrates were obtained in moderate to excellent yields (42-95%). Removal of the DG and recovery of the phenolic group was quantitatively realized on one derivative in the presence of ethylamine in methanol at room temperature, opening the route to further modification of such compounds.

Using methyl iodide (MeI) as electrophilic partner in the presence of 1 (Scheme 5, R = 3-methyl) under the standard reaction conditions, resulted in an absence of reactivity, suggesting that a more reactive species than starting glycosides 7 could be involved in the process. Moreover, the stereoretention in favor of the  configuration observed in most cases, as well as the exclusive formation of this  product starting from a 2:1 / mixture of glycosyl chloride led the authors to propose the existence of an oxocarbenium ion IV (Scheme 9) prior to oxidative addition. Such an intermediate, which could be formed from 7 in the presence of Pd(OAc)2, which acts as a Lewis acid, would be preferentially attacked through its less hindered side, explaining the large observed  selectivity. This suggestion was also supported by the example involving 2-phenylethanol and mannose (Scheme 9).

Complex I includes a 5,5-or 5,6-bicyclic structure (in green in scheme 4 and 9; 5,5: X = no atom, 5,6: X = CH2 or O) incorporating the metal, which could explain a C-H activation in favor of the ortho position of the aryl ring. To demonstrate its role in the catalytic cycle, Palladacycle I (Scheme 9) was prepared by reacting Pd(OAc)2 with 2methylbenzamide in acetonitrile and submitted to standard conditions in the presence of 7 to furnish the C-aryl glycoside 8 in excellent yield (91%) Kinetic isotope effect experiments showed that the formation of I is not the rate-limiting step. Iridium complexes were also used in hydroarylation of glycals via a C-H functionalization process as recently reported by Nishimura et al. (Scheme 10). 51 Thus, compounds 13 and 14 were reacted in the presence of [IrCl(cod)]2 (5 mol% Ir), NaBAr F 4 (10 mol%) and (S)-binap (6 mol%) to afford mostly, after 24 hours in toluene, the  adducts 15. By the same way, (R)-binap led to the  adducts 16. The high -selectivity was observed at higher temperature (80°C) than the -selectivity, and several Caryl glycosides 15 and 16 were obtained in good to excellent yields (61-92%) and excellent diastereoselectivities (up to 98% de) except in four cases (28%-70% de) from diversely protected or non-protected glycals 14. This method shortens the access to the desired products, when compared to Pd(II) catalyzed coupling (vide supra), since glucals 14 are precursors of the corresponding halogenated compounds 2 (Scheme 2). Moreover, it allows an access to both  and  configurations starting from the same precursor, just by changing the configuration of the additional catalytic ligand ((S)-or (R)-binap) and no di-arylated products were observed. Alkene insertion of glycal 14 into Ir-H bond affords preferentially VII or VIII according to the absolute configuration of the chosen (S)-or (R)-binap. Intermediate VII eventually delivers the -aryl glycoside 15 (VIII led respectively to -aryl glycoside 16) through an irreversible reductive elimination, whose rate might determine the observed diastereoselectivity. An alternative pathway was proposed in which the alkene insertion occurs through the irreversible carbometalation of the aryl(hydrido)iridium complex to glycal to form an alkyliridium intermediate.

Csp 3 -H functionalization

Csp 3 -H functionalization is far less common in the synthesis of C-glycosides. Ackermann et al. recently reported 49 the pallado-catalyzed C-H activation of -phthalimide substituted amides 21 and subsequent coupling with iodoglycals 2, which delivered products 22 in moderate to excellent yields and good diastereoselectivities (Scheme 14). TAM R and Q assisting groups gave similar results and rhamnose derived glycals were found effective in the process. Based on a DFT study, a mechanistic pathway was proposed but no explanation was furnished about 1-adamantane carboxyl acid additive (AdCO2H), which is used in substoichiometric amount (30 mol%) in the reaction. As previously mentioned (Scheme 4), the catalytic cycle should imply a favored 5,5-bicyclic structure around the metal, and a Pd(II)-Pd(IV)-Pd(II) pathway. This methodology was applied with success to the synthesis of small peptides as illustrated in figure 4, replacing Pd(OAc)2 by Pd(TFA)2 on a longer reaction time (16 h). Only traces of unprotected peptide Phth-Ala-Tzl-Tyr-OMe were obtained while the reaction allowed the access to other congested glycopeptides.

Other biomolecules and BODIPY labeled glycoamides were also synthesized following this methodology in moderate to excellent yields (Figure 5). ) and moderate to excellent diastereoselectivities (3:1 to >20:1 (dr)). Both teams showed through deuterium kinetic studies that the Csp 3 -H bond cleavage was not the rate-limiting step of the catalytic cycle and the mechanism depicted in scheme 4 seems to be reasonable in that case too.

Non-directed C-H functionalization of (hetero)aromatics substrates

In 2017, Ye et al. described 54 an access to five-membered nitrogen-containing heteroaryl C-glycosides based on a Pd(II)/Cu(I) co-catalyzed methodology (Scheme 16). Heteroaryl-C-Δ 1,2 -glycosides, incorporating indole, thiazole, benzothiazole, imidazole, benzimidazole, and benzoxazole moieties as aglycones were obtained in 43-99% yields starting from 1iodoglycals in the presence of t-BuOLi (4 equiv.) and 1,10phenanthroline (0.2 equiv.) in 24 hours at 100°C in 1,4-dioxane.

Scheme 16. Synthesis of N-heteroaryl C-glycosides following Ye's methodology.

The tosyl-protecting group on the indole ring was chosen to decrease the nucleophilicity of its C3 position owing to its electron withdrawing ability. The reaction is effective with electron-withdrawing and donating groups except in the case of an indole ring bearing a nitro substituent. 1-iodoglycals derived from D-galactose and L-rhamnose were successfully coupled in these conditions. The main limitation of the method is the use of 6 equivalents of t-BuOLi, which is not compatible with base sensitive compounds. Scheme 17. A reasonable mechanism for the synthesis of N-heteroaryl Cglycosides following Ye's methodology.

The most reasonable catalytic cycle for this transformation depicted in scheme 17 begins with the complexation of the 1,10-phenanthroline to Pd(OAc)2 follows by oxidative addition of the latter in the C-I bond of the 1-iodoglucal 2 delivering the Pd(IV) intermediate IX. Copper assisted deprotonation of the Nheteroaryl reactant by t-BuOLi and subsequent lithium-copper exchange then furnishes the organo-copper species X whose transmetalation with IX provides the Pd(IV) species XI. Desired products 26 are finally obtained by reductive elimination of XI with concomitant regeneration of Pd(II). 1,10-Phenanthroline could favor the oxidative step through Pd(II) electronic enrichment.

A second example of a C-glycoside synthesis via a nondirected C-H activation was reported in 2017 by Rasool and Mukherjee (Scheme 18). 55 Scheme 18. Synthesis of uracil C-glycosides following Mukherjee's methodology.

Several uracil-substituted pseudo glycals were synthesized reacting compounds 27 and pyranoid glycals 28 catalyzed by Pd(OAc)2 in the presence of Ag2O and PivOH (2 equiv.) in moderate to good yields. Unprotected uracil failed to react due to poor solubility and uracil bearing longer protective alkyl chains gave better yields. Contrary to C3--substituted glycals, which delivered the corresponding C1--products, glycal 30 derived from rhamnose was converted into the C1--pseudo glycal 31, thus showing that the stereoselectivity of the process is governed by the orientation of the C3 substituent in the starting material, possibly explained by the mechanism depicted in scheme 19 (vide infra). A screening of the reaction conditions allowed the exclusive formation of products 32 in moderate yields (Scheme 20) using AgOAc in polar DMF solvent.

Where the sugar is the C-H activated partner

Directed C-H functionalization of sugar substrates

Csp 3 -H functionalization

One another approach for the synthesis of C-glycosides is based on directed C-H functionalization of a C-H sugar bond where the DG was placed on the sugar core. The C-H functionalization process in this case may occur in a convergent way with aryl halides. The main advantage of this approach compared to the previous one is the commercial availability of one of the reaction partners (aryl halides)..

In this context, Messaoudi and co-workers 56 reported for the first time the use of this approach to the diastereoselective Csp 3 -H arylation of glycosides. As a proof of concept, the authors designed at first the β-amidosugar substrate 33 bearing a 8-aminoquinoline DG and showed that the use of Pd(OAc)2 (20 mol%) and Ag2CO3 (2 equiv.) in dioxane at 120 °C led to the C3-aryl-glycosides 34 in good to excellent yields (Scheme 21). Interestingly, the authors reported that the Csp 3 -H arylation occurs in an exclusive 2,3-trans diastereoselectivity, which contrasts with the reported cis one in the case of unsubstituted cyclic substrates. 57 Different functional groups on the (hetero)aromatic ring are tolerated such as methoxy, nitro, methyl ester, ketone, cyano, fluoride and chloride. Although the acetyl and benzyl protecting groups were used successfully in this study, silyl groups led to low yield (~ 10%). The authors examined the influence of the anomeric configuration in this C-H activation process, and showed that amidoglycosides 35 (Scheme 22) necessitates higher reaction temperature (up to 160°C) to observe the arylation process. In this case, C-aryl pseudoglucals 36 arising from Csp 3 -H arylation followed by AcOH elimination were obtained in moderate yields ranging from 23% to 41%. Corresponding benzylated glycoside failed to yield an arylated product. In the aim to get further insights into this exclusive equatorial Csp 3 -H activation, the authors performed DFT calculations. They thus showed that the lowest transition state found corresponds to a trans-CMD with a +13.1 kcal.mol -1 activation barrier (Figure 6), where the C4 acetate is equatorial, yielding the trans palladacycle min-tri-a with a release of 6.9 kcal/mol of free energy. To achieve the cis palladacycle, the conformational shape of a pyranose switches from the 4 C1 conformation to 1 C4 one in which the C4 acetate substituent occupies an axial position by demanding higher value of the free energy of activation of +15.7 kcal.mol -1 . Based on these calculations, the authors proposed the mechanism depicted in scheme 23. Initial deprotonation of the quinolyl amide leads to the formation of complex XV, which undergoes C-H metalation through a concerted metalation deprotonation (CMD) mechanism, to give -organopalladium complex XVI. Oxidative addition of the aryl iodide generates palladium(IV) complex XVII followed by C-C bond formation by reductive elimination to XVIII.

Scheme 23. Proposed Mechanism for trans-arylation reaction reported by Messaoudi et al.

After ligand exchange, the arylated product 34 is formed and the active palladium species XV is regenerated.

One important point to solve in this approach is the ability to remove the DG at the end of the process. The authors demonstrated in their study that the aminoquinoline DG could be removed from the final C3-aryl-glycoside under Maulide's conditions (oxidative cleavage with O3 followed by ammoniac aminolysis) to deliver the free amide C3-aryl-glycoside. 58

Csp 2 -H functionalization

By using the same approach, Ferry and co-workers 59 reported recently a novel route to C-aryl/alkenyl-glycosides 38 through a directed palladium-catalyzed Csp 2 -H functionalisation of C2-amido glycals 37 onto the anomeric position (Scheme 24). The 8-aminoquinoline DG was used to successfully introduce diverse (hetero)aryl and alkenyl groups at the C1 position of various protected glycals. The reaction conditions involve the use of Pd(cod)Cl2 (20 mol%) as the catalyst, 6 equivalents of the aryl-iodide in the presence of a mixture of K2CO3 (3.6 equiv.), AgOAc (1.5 equiv.) and citric acid (0.85 equiv.) in toluene at 130 °C. The exact role of citric acid is not clear but it probably intervenes in the CMD step of the catalytic cycle.

Different functional groups on the aromatic ring are tolerated such as methoxy, trifluoromethyl, nitro, methyl ester, cyano and chloride with moderate to good yields (38 to 78%). Moreover, this methodology tolerates N-Boc-indole and protected iodo phenylalanine. Interestingly, alkenylation also occurred under these conditions when activated alkene iodide was used. In addition, the synthesis of a disaccharide amino acid could be performed between the D-lactal and the phenylalanine iodide (41%). This method was applied successfully to the synthesis of an analogue of the dapagliflozin drug (FORXIGAs), commercialized to treat type 2 diabetes. In addition to benzyl-protecting groups, the sensitive isopropylidene-protecting group is tolerated in this process, however, ester-type protecting groups on glycal such as -OAc inhibit completely the reactivity due a possible "disarming-like effect". Scheme 24. Pd(II)-catalyzed arylation of glycal 37 reported by Ferry et al.

Non-directed C-H functionalization of sugar substrates

Csp 2 -H functionalization

Metal-catalyzed C-H functionalization processes are wellknown methods for the construction of C-C bonds on glycal type substrates. Besides, without any DG, the addition of electrophilic partners occurs exclusively at the nucleophilic position 2 of the glycal. The first example of this type of reactivity was reported by Liu et al. in 2011 and deals with the coupling of glycals with substituted alkenes. 60 The best conditions require the use of catalytic quantities of Pd(OAc)2, the presence of Cu(OTf)2, in a mixture of solvents (DMA/AcOH = 1:1) under an atmosphere of O2 (Scheme 25, A). They thus described 18 examples in moderate to good yields (43-87%). Furthermore, they investigated the use of a disubstituted alkene (ethyl methacrylate), which gave a mixture of compounds 41b and 41c with a ratio of 1:2.5 in 74% total yield (Scheme 26). A plausible mechanism was proposed for this reaction (Scheme 27): the Pd complex XXI is formed after steps 1 and 2 through complexation and subsequent electrophilic substitution on the glycal at the C2 position.

Scheme 26. Liu et al. C-H functionalization with a disubstituted alkene.

A carbopalladation of the acrylate occurs delivering the more stable conformation XXII with the two bulky groups situated in anti (Scheme 27, step 3). The formation of the antiproduct justifies the observation of the E configuration for all the synthesized compounds. Then the final compound 3 is formed via a β-hydride elimination during the step 4. Finally, the Pd(0) complex obtained through reductive elimination (step 5) is oxidized by Cu(II) to regenerate the active Pd(II) catalyst species (Scheme 27, step 6). Concerning products 41b and 41c, the conformation is explained regarding the β-hydride elimination step, which could occur on the two possible carbons in β of the Pd. Inspired by this work, Yu et al. used these C-H functionalization conditions in 2015 in a total synthesis of a monosaccharide, the bradyrhizose. 61 This process was achieved from the tri-O-acetyl-D-glucal in 26 steps with a global yield of 9%. The pallado-catalyzed Csp 2 -H activation was applied at the first step of this synthesis, which permits to combine glucal with a methyl acrylate moiety (Scheme 25, A). They optimized the previous conditions to increase the yield to 76%. Another example of alkenylation on glycals was reported in 2013 by Bäckvall et al.. 62 Their conditions were close to those described by Liu et al. but they aimed to develop a biomimetic aerobic oxidative coupling between two Cvinyl-H bonds for the synthesis of conjugated dienes. Indeed, they engaged smaller quantities of Pd(OAc)2 in combination with iron phthalocyanine and pbenzoquinone in catalytic amounts as electron-transfer mediators, and O2 as an oxidant to regenerate the catalyst in its active form (Scheme 25, B). In this case, they established that the key step is the reoxidation of Pd(0) to Pd(II) by this biomimetic approach. Thereby, five C2-substituted glycals with yield ranging from 43% to 81% were synthetized. Additionally, a scale up was done with the methyl acrylate as the electrophilic partner on the tri-O-acetyl-D-glucal and 2.1g of the corresponding product was obtained in a good 81% yield. Afterward, Mukherjee's group developed in 2016 a C-H allylation reaction between glycals 39 and inactivated cycloalkenes 42 (Scheme 28). 63 Pd(OAc)2 was used as a catalyst and AgOAc as an oxidant in the process. They thus obtained 22 compounds 43 in yields ranging between 65 and 85%. The same group implemented two years after a coupling between glycals and alkenes using the same previous conditions with some modifications (Scheme 25, C and D). 64 Scheme 28. Mukherjee's group C-H functionalization on glycals with cycloalkenes substrates. Scheme 29. Mukherjee's group proposed mechanism for C-H functionalization with cycloalkenes on glycal type substrates by two pathways.

They also developed other conditions for the same reaction, giving similar results. In view of the equal efficiency for both conditions, they used principally conditions B for economic reasons, and they could obtain 22 in 59 to 91% yield. On top of that, a mechanism was proposed for this coupling implying two pathways (Scheme 29).

The first one begins with an electrophilic palladation at the C2 position of the glucal to give a vinyl Pd(II) complex XIII, followed by a cycloalkene insertion leading to the species XXIV. Additionally, a Wacker type nucleopalladation reaction could take place, where the cycloalkene-Pd(II)π-complex XXV formed previously, is attacked by the sugar enol ether which acts as a nucleophile, forming the intermediate XXIV. The latter then undergoes a β-H elimination delivering the desired product 5 and a Pd(0) complex. The second pathway involves the formation of a π-allyl-Pd(II) complex XXVI in the first step, reacting Pd(II) with cycloalkane via an electrophilic allylic C-H bond cleavage. Then, XXVI is attacked by the glycal via its nucleophilic C2 position leading to the formation of species XXVII after reductive elimination which undergoes dissociation of Pd(0) to generates the desired product 43. Eventually, silver acetate is supposed to help the oxidative addition of Pd(0) to Pd(II) in both pathways.

Finally, Pannecoucke et al. developed a copper-catalyzed C-H process to introduce fluorinated ester moieties on the C2 position of glycal-type substrates (Scheme 30). 65 First of all, the optimal conditions were developed on simple 1,2-dihydropyrans and were then extended to glycals. Thanks to the addition of ethyl bromo(difluoro)acetate as electrophilic partner, 13 examples were reported in modest to good yields (39 and 67%). Moreover, a mechanism was proposed based on experimental studies (Scheme 31). Firstly, a radical pathway was excluded thanks to the addition of radical inhibitors and radical scavengers on the reaction mixture, which led to the same yields. Secondly, the standard conditions were applied on enol ether, which allowed to show an isomerization of the double bond and to indicate the presence of an oxonium intermediate. Thereby, this transformation takes place exclusively on the less hindered position, between the two sites of deprotonation available, leading to the formation of the trisubstituted enol ether as the only compound.

According to these results, the following mechanism was proposed. Firstly, an oxidative addition of the Cu(I) into the carbon-halogen bond of 44 occurs resulting in the formation of a highly electrophilic Cu(III) species XXVIII (Scheme 31, step 1). The latter reacts (step 2) with glycal 39 to form an oxonium intermediate XXIX, which is transformed into the enol ether XXX by the base K2CO3 (Scheme 31, step 3). Finally, reductive elimination (step 4) regenerates the copper catalyst delivering the desired product 45.

Csp 3 -H functionalization

The current literature dealing with non-directed C-H functionalization of a glycosidic Csp 3 -H bond into a C-C bond proceeds with no exception through intramolecular pathways. Most of the examples use a rhodium-catalyzed carbene insertion process. The pioneers in this domain were Kim et al., in 2000, who used this reactivity for the synthesis of C2-branched nucleosides. They developed a diastereoselective and regioselective [1,5]-C-H insertion of 2-deoxy-nucleosides bearing a diazoacetate moiety on the hydroxyl in position 3 (compounds 46, Eq1, Scheme 32). 66 In the presence of dirhodium tetraacetate, 46 was transformed into fused butyrolactones 47 in moderate to good yields (58% to 80%) and diastereoselectivities (90 to 96%) in favor of the exo isomer. The preference for this exo isomer is explained by the authors by a potential steric hindrance between proton in position 1 and R 1 during the process. These conditions were successfully applied to a constraint nucleoside presenting a tertiary Csp 3 -H bond in position 2. In 2002, Norris et al. extended this reactivity to a glycofuranose compound 48 presenting a similar diazo moiety in position 3 (Eq2, Scheme 32). 67 The tertiary C2-H bond was functionalized to form bicycle 49 in a modest 47% yield. The possible isomer where a six-membered ring would be formed by functionalization of the activated C1-H, for example, was not observed.

In the 2010s, Lecourt et al. took advantage of this rhodiumcatalyzed carbene insertion reactivity to propose a stereospecific route to both α-and β-ketopyranosides. 68 Starting from an O2diazoacetate α-mannopyranose 50, they regioselectively quaternized the anomeric C-H bond via an intramolecular 1,5-C-H insertion process leading to bicyclic compound 51 in modest to excellent yields (Eq1, Scheme 33) depending on the nature of the protecting groups. By the same way, they proposed an access to β-ketopyranosides 53 by the intramolecular cyclization of an O2-diazoacetate β-gluco-or β-galacto-pyranose 52 (Eq 1, Scheme 33). They showed that a low catalyst loading (0.5 to 2 mol%), a highly diluted suspension of the catalyst, a precise amount of freshly activated molecular sieves as well as vigorous stirring and a slow addition of the substrate are essential to the results without undesired dimerization byproducts. Moreover, the influence of the protecting group at position 3, close to the reactive site, was clearly demonstrated since a sterically-demanded silylated or ester group led in good to excellent yields, whereas products bearing a benzyl group were generated in modest yields due to side reactivities (1,7-C-H insertion into the benzylic position or competitive dimerization). Five disaccharide-type α-and β-ketopyranosides were successfully obtained via this method. The authors could also adapt this protocol on a gram-scale level. Mechanistically, the process is expected to run via the decomposition of the diazo functional group mediated by the rhodium(II) salts, which then delivers a metal intermediate. Further ring-opening functionalization under Lewis-acid conditions of compounds 51 and 53 were finally presented. Very recently, Lecourt et al. applied this methodology to access C-branched glucopyranoses, quaternized in position 5 (Eq 2, Scheme 33). 69 By the same strategy, they placed the diazoacetate functionality on the position 6. They revealed that the choice of the protecting groups of the position 2, 3 and 4 was crucial both to avoid side reactions and to deactivate the non-desired adjacent C-H bonds. Nevertheless, they observed that the success of this reactivity was dependent on the anomeric configuration. Indeed, whereas a β configuration led to 55 in 70% yield, the α compound led only to mixtures where low amount of 55 was accompanied to dimers and 1,4-C-H insertion product. They explain this trend by the possibility that anomeric effect in the α compound could deactivate the C5-H bond. In 2017, Messaoudi et al. investigated the intramolecular arylation of 2-bromophenyl glycosylcarboxamide substrates (Scheme 35) in the presence of palladium catalysts. 70 In the case of a peracetylated 2-deoxy glucoanalogue 56, they described an intramolecular Pd-catalyzed selective anomeric C-H activation leading to glucosylspirooxindole 57 in very good yield as a mixture of two diastereoisomers α and β (d.r. = 3:2). Acetyl protecting groups appear to be crucial in the success of the process since a perbenzylated-or a protecting group-free analogue failed to furnish the corresponding spirocompound in the same conditions. Supported by DFT calculations, the authors proposed that the process starts by an oxidative addition of Pd(0) into the C-Br bond of the carboxamide moiety, followed by ligand exchange leading to intermediate XXXI (Scheme 35) where the palladium atom is spatially close to the anomeric C-H bond. Via a concerted metalation deprotonation (CMD) mechanism, a palladium C-enolate is then formed (equilibrium between the two diastereoisomers XXXII and XIII). From these species, a reductive elimination process furnishes 57 in the case of a 2-deoxy starting substrate (R = H). Indeed, when R = OAc, XXXII evolved to glucal 58 through a three-step process involving a 1,2 Pd-migration, followed by a reductive elimination, and finally an anti-HOAc elimination (Scheme 35).

Dual photoredox

Csp 2 -H functionalization

Photoredox catalysis became very attractive these last decades and is described as a mild, efficient and practical transformation. In 2015, a process used by Ye et al. for the direct C-H trifluoromethylation of glycals (Scheme 36). 71 This photocatalytic reaction needs the presence of fac-Ir 3+ (ppy)3 as photocatalyst, Umemoto's reagent (CF3 source) and a visible light source. This trifluoromethylation was applied on a wide range of glycals, with both electron-withdrawing anddonating protecting groups, and allows the formation of 17 products in moderate to good yields (49-78%). On top of that, two glycal derived drugs, namely Neu2en derivatives, were synthesized in 43 and 45% yields. Furthermore, they investigated the reaction mechanism with a series of experiments. In this respect, a possible mechanism illustrated in Scheme 37 was proposed. Firstly, the emission of a photon from the light source, accepted by fac-Ir 3+ (ppy)3, induces the excited state [fac-Ir 3+ (ppy)3]*. Then a single electron from latter is transferred to the CF3 reagent and generates the electron deficient intermediate fac-Ir 4+ (ppy)3, CF3 radical and 61. A small amount of trifluoromethylated dibenzothiophene 62 was observed, resulting from the freeradical aromatic substitution reaction of dibenzothiophene with the CF3 radical. The addition of CF3 radical to the double bond of the glycal leads to the glycal XXXIV, which undergoes an oxidation by fac-Ir 4+ (ppy)3, regenerating fac-Ir 3+ (ppy)3 in the catalytic cycle, and forming the glycosyl oxocarbenium ion XXXV. Finally, the intermediate XXXV or XXXVI furnishes the desired compound 60 by deprotonation.

Csp 3 -H functionalization

In the 2010s, Gagné et al. used the well-known capacity to form anomeric radical species from glycosyl bromides 63 to perform intermolecular additions with electron-deficient alkenes mediated by visible light. 72 Indeed, [Ru(bpy)3](BF4) as photocatalyst in combination with visible light led to the formation of C1 sugar radicals which can react with alkenes 64 and furnish exclusively α-C-glycosides 66 in modest to excellent yields (Scheme 38). A key parameter to suppress side reactivities (oligomerization, hydrolysis) was to use Hantzsch ester 65 as a proton donor in highly concentrated dichloromethane solution. N,N-diisopropylethylamine was chosen as the best reductant to effectively run the photocatalyst cycle. Diverse acetate or benzoate protected glucosyl-, mannosyl-or galactosyl-type bromides were successfully used in this process. In terms of alkene partners, several mono-or 1,1-disubstituted compounds were involved but over-conjugate addition was more competitive in the case of highly electron deficient alkenes. Moreover, this method was scaled up to 1.2 mmol. Several control experiments were performed to understand mechanistic aspects of the reaction. Based on control experiments and EPR measurements, the authors thus proposed the following mechanism: the reaction starts by the irradiation of the photocatalyst which enters in an excited state (*Ru(bpy)3). This species is then reduced by the amine to generate the reactive strongly reductant ruthenium complex (Ru(bpy)3 + ) capable to form the C1 radical in the presence of 63. Addition of this radical XXXVII to the alkene and reduction by Hantzsch ester provide C-glycosides 66 (Scheme 38). Via several experiments, the authors investigated the key parameters of the reaction. 73 Indeed, the conditions described in Scheme 38 need long reaction time, witnessing a possible inefficient photoredox cycle. They demonstrated that, at low concentrations, the reductant (amine) and the catalyst concentration influence the reaction rate, whereas a saturation effect is observed at high concentrations. Moreover, they observed that the presence of water and the use of a hydrophobic catalyst (Ru(dmb)3) are beneficial to obtain an efficient conversion.

In 2012, Gagné et al. developed a continuous flow photoredox equipment which led to an improvement of the TOF value of the process. 74 They observed that a thin diameter of the reaction vessel increases the flux of photons, inducing an increase of the concentration of active catalyst. Their optimized reactor using a 1.6 mm inner diameter tubing and a concentration of 1 mM. By this way, they synthesized two Cglycosides on almost 5 g scale using acrolein as alkene partner in 85% yield in the presence of only 1 mol% of ruthenium photocatalyst (Scheme 38). These gram-scale aldehydes were proved to be versatile building blocks for the synthesis of a series of C-linked glycoconujugates such as C-glycopeptides and C-glycolipids.

MacMillan and coworkers reported in 2015 a photoredox αalkylation/lactonization of alcohols with methyl acrylate via a hydrogen atom transfer mechanism. 75 The study was not focused in the reactivity of glycosides but the authors reported in their scope one example with protected sugar 67. Under the optimized reaction conditions, the lactone alkyl C-glycoside 68 was isolated in 85% yield and 1:1 diastereoselectivity (Scheme 39). More recently, Dumoulin et al. developed a new route for the construction of C-aryl/(hetero)arylglycosides via a nickelcatalyzed photoredox cross-coupling using aryl/heteroaryl bromides and an organic photocatalyst (Scheme 40). 76 They synthesized 30 products 70 in mild conditions in yields ranging between 24 and 94% and diastereoisomeric ratios between 1.5:1 and >20:1. They used dihydropyridyl saccharides as radical precursors initiated by the organic photocatalyst in its excited form. Besides, a plausible mechanism was proposed where the photoexcitation of the photocatalyst by blue LEDS initiates the reaction (Scheme 41). Then the resulting radical cation XXXVIII generates by fragmentation an aromatized pyridine derivative and the corresponding radical XXXIX. The active Ni(0) catalyst is complexed to this last species forming a Ni(I)-saccharyl complex XXXX and the aryl bromide is added by oxidative addition yielding the Ni(III) complex XXXXI. Finally, reductive elimination gives both the desired compound 70 but also generates a Ni(I) species which is then reduced to Ni(0) thanks to the reduced form of the photocatalyst allowing in this case generation of both active catalysts. Very recently, Wang and Zhang proposed a combination of photoredox and nickel catalysis capable to promote arylation of the anomeric position of ribosyl/deoxyribosyl acids via a decarboxylative cross-coupling reaction with aryl/heteroaryl bromides. 77 They developed mild reaction conditions using visible-light irradiation in the presence of an organic photocatalyst (4CzIPN), which led to the synthesis of diverse Caryl/heteroaryl-nucleosides in a highly stereoselective manner in moderate to high yields (Scheme 42).

Their conditions were compatible with both electron-rich and electron-poor aryl-heteroaryl bromide partners and benzyl protected starting substrates, but did not tolerate benzoyl protected ribosides. A para-strongly electron-donating group on the bromide decreases the reactivity, as well as a substituent in ortho-position. In term of stereoselectivity, both α and β starting materials led to almost exclusively the β products (>99:1) in high yields, except for β-deoxyribosyl acid which led to low β diastereoselectivity and 2-bromopyridine partners which furnished the completely opposite diastereoselectivity α (75, Scheme 41). Moreover, a non-anomeric furanosyl acid was also successfully converted to 74 in the same conditions (Scheme 41). Finally, the authors applied this reactivity to the synthesis of various C-vinyl-nucleosides 76. They could observe the formation of the trans-product exclusively and independently of the conformation of the starting bromide. From a mechanistic point of view, based on control experiments, the authors suggested a photocatalytic cycle starting from the generation of an excited state of the photocatalyst (PS*), which promotes a photo-oxidative decarboxylation reaction of 69, leading to the formation of a key anomeric radical XXXXII. Concomitantly, nickel (0) species, generated by a photoredox process, undergoes an oxidative addition into the bromide partner to form XXXXIII, which reacts with radical XXXXII to furnish intermediate XXXXIV. Finally, reductive elimination process produces product 71 and regenerates the active Ni(0) species and the ground state of the photocatalyst after photoredox reduction (Scheme 43).

Conclusion

This review has highlighted the recent progresses achieved on complex sugar substrates using C-H functionalisation and photoredox approaches as two useful methods for the synthesis of C-branched glycosides. These new and sustainable processes have emerged as powerful methodologies, in comparison with former organometallic catalytic routes, offering the advantage of reducing the generation of wastes from the time-consuming preparation of often unstable prefunctionalized substrates. In the span of approximately two years, these processes have become a workhorse for the construction of an array of C-C bonds on the sugar nucleus.

The illustrated methodologies showed the considerable interest that these two emerging tools took in the past years and proved their success for the construction of complex C-branched glycosides. The protocols presented in this review show high efficiency as well as stereo-and regioselectivity, good functional group tolerance and good scalabilities. Future developpement in this field include the development of additional photoredox reactivity involving, for example, more visible-light catalyzed process, electrochemical methodologies, general protocols for C-H activation on tertiary centers, and alkyl-alkyl couplings. The continuation of efforts in these areas will serve to increase further the impact of this rapidly growing field of research and to reach complex glycosides of biological interest.

Lit of abbreviation

AD

Figure 1 .

 1 Figure 1. Selected examples of natural products including a C-glycoside unit.

Figure 2 .

 2 Figure 2. Selected examples of synthetic C-glycosides with noticeable biological activities.

Scheme 1 .

 1 Scheme 1. Previous organometallic catalyzed methods for the synthesis of Cglycosides.

  Ghouilem received his M.Sc. in BioOrganic Chemistry at Lille University in 2018, France. The same year, he joined the CosMIT team at Paris Saclay University as a Ph.D student under the supervision of Dr. Samir Messaoudi. His ANR research project "SelFSuCHi" is focused on the synthesis of CMorgane de Robichon was born in Maisons-Laffitte (78 -France) in 1995. She obtained a Master degree in Molecular and Macromolecular Chemistry at CY Cergy-Paris University. She is currently a second year PhD student at the School of CY Cergy-Paris University under the supervision of Dr. Jacques Uziel and Dr. Angélique Ferry. She studied carbonylative metallo-catalyzed cross-coupling reactions on iodoglycals and is currently working on metallo-catalyzed directed C-H functionalisation of glycoside substrates. Franck Le Bideau graduated with a Ph.D in 1995 at the Université Pierre et Marie Curie (Paris, France), under the supervision of Professor Max Malacria, working on palladium-catalyzed rearrangement of silicon substituted vinyloxiranes. In 1996, he joined the group of Professor Léon Ghosez in Belgium at Université Catholique de Louvain (Louvain-la-Neuve) as a postdoctoral researcher and worked on [2+2] cycloaddition reactions. In 1997, he joined the CNRS as a researcher at the Ecole Nationale Supérieure de Chimie de Paris (France) where his research mainly focused on the synthesis of organometallic species of potential biological interest. In 2011, he moved to the Université Paris Sud (Châtenay-Malabry, France) where he is currently working on organocatalysis and subsequent total synthesis of natural products. Angélique Ferry was born in Mulhouse (France) in 1987. She obtained her Master degree in 2010. She then started her PhD on the synthesis of phostone and hydroxylamine-type glycomimetics (Prof. Crich, Dr. Guinchard, ICSN) and received, in 2014, the Dina Surdin PhD Award of the French Chemical Society. Then, she joined Prof. Glorius team (Germany) to work on heterogeneous C-H functionalization and on N-heterocyclic carbenes-stabilized metallic nanoparticles. In 2015, she became associate professor at the CY Cergy-Paris University, where she develops metal-catalyzed functionalization on glycosides. She received, in 2018, the "women and science" award of the CY Cergy-Paris University and a "Young Researchers" ANR funding as project coordinator. In 2020, she obtained the Marc Julia Award of the French Chemical Society and the GFG Award of the French Group of Glycosciences. Samir Messaoudi completed his PhD in 2004 at Blaise Pascal University, Clermont Ferrand, France. After two years as postdoctoral fellow, he joined the group of "Therapeutic Chemistry" in 2007 as a junior CNRS-researcher at BioCIS UMR 8076, university of Paris Saclay. In 2017, he was promoted as Director of Research DR-CNRS. His main research concern deals with glycochemistry (functionalization of carbohydrates under transition-metal catalysis), as well as with the preparation of molecules and molecular scaffolds of biological and therapeutic interest.

Scheme 2 .

 2 Scheme 2. Pd(II) catalyzed synthesis of aryl-C- 1,2 glycosides according to Ye et al..

Scheme 3 .

 3 Scheme 3. Pd(II) catalyzed synthesis of aryl-C- 1,2 glycosides according to Ackermann et al..

Scheme 5 .Scheme 6 .

 56 Scheme 5. Pd(II) catalyzed synthesis of C-aryl pyranosides varying the aryl moiety. All compounds have an  configuration and adopt a 1 C4 conformation. Di-arylation products are observed only when mentioned.

Scheme 7 .

 7 Scheme 7. Pd(II) catalyzed synthesis of C-aryl furanosides varying the sugar and the aryl moieties.

Scheme 8 .

 8 Scheme 8. Other DG tested in the Pd(II) catalyzed synthesis of C-aryl pyranosides.

Figure 3 .

 3 Figure 3. Pd(II) catalyzed synthesis of C-aryl pyranosides and furanosides bearing an amidoquinolyl-DG. Reaction conditions are the same as reported in Scheme 5 using Ac-Ile-OH (30%) as the additional ligand.

Scheme 9 .

 9 Scheme 9. Mechanism of the Pd(II) catalyzed synthesis of C-aryl pyranosides and furanosides: modification of the general catalytic cycle proposed in scheme 7.

Scheme 10 .

 10 Scheme 10. Ir(I) catalyzed hydroarylation of glycals via C-H functionnalisation: influence of the absolute configuration of the additional ligand binap on the diastereoselectivity of the reaction.

Scheme 11 .

 11 Scheme 11. Ir(I) catalyzed hydroarylation of glycal 17 in the presence of (S)binap, varying the nature of the aryl: access to compounds 18, predominantly under their  configuration.

Scheme 14 .

 14 Scheme 14. Pd(II) catalyzed Csp 3 -H activation process according to Ackermann et al.

Figure 4 .

 4 Figure 4. Synthesis of small peptides following Ackermann's methodology.

Figure 5 .

 5 Figure 5. Synthesis of BODIPY labeled glycoamides and other bio-ligands following Ackermann's methodology.

Scheme 15 .

 15 Scheme 15. Pd(II) catalyzed Csp 3 -H activation process according to Liu et al.. Concomitantly to Ackermann's report, Liu et al. published 53 a similar protocol (Scheme 15) to synthesize close compounds 24, replacing AdCO2H by trifluoroacetic acid (TFA) in good yields (63-83%) and moderate to excellent diastereoselectivities (3:1 to >20:1 (dr)). Both teams showed through deuterium kinetic studies that the Csp 3 -H bond cleavage was not the rate-limiting step of the catalytic cycle and the mechanism depicted in scheme 4 seems to be reasonable in that case too.

Scheme 19 .Scheme 20 .

 1920 Scheme 19. A reasonable mechanism for Mukherjee's methodology

Scheme 21 .

 21 Scheme 21. Pd(II)-catalyzed diastereoselective 2,3-trans Csp 3 -H arylation of β-amidoglycosides 33 according to Messaoudi et al.

Scheme 22 .

 22 Scheme 22. Pd(II)-catalyzed arylation of α-amidoglycosides 35 according to Messaoudi's procedure.

Figure 6 .

 6 Figure 6. Free energy profiles of the cis and trans CMDs.

Scheme 25 .

 25 Scheme 25. Pd(II) catalyzed C2-vinyl glycosides according to Liu et al..

Scheme 27 .

 27 Scheme 27. Liu et al. plausible mechanism for C-H functionalisation with alkenes on glycal type substrates.

Scheme 30 .Scheme 31 .

 3031 Scheme 30. Pannecoucke et al. C-H functionalization on glycals with fluorinated compounds.

Scheme 32 . 3 -

 323 Scheme 32. Non-directed Csp 3 -H functionalization of furanose-type glycosides.

Scheme 33 .

 33 Scheme 33. Lecourt et al. non-directed Csp 3 -H quaternization of glycopyranosides.

Scheme 34 .

 34 Scheme 34. Lecourt et al. non-directed Csp 3 -H quaternization of the position 5 on glycopyranosides.

  Scheme35.Messaoudi's intramolecular arylation of 2-deoxyglucosylcarboxamide 56.

Scheme 36 .Scheme 37 .

 3637 Scheme 36. Ye et al. C-H functionalization on glycals via Photoredox trifluoromethylation.

Scheme 38 .

 38 Scheme 38. Gagné's photoredox synthesis of C-glycosides.

Scheme 39 .

 39 Scheme 39. MacMillan et al. synthesis of lactone C-glycosides viaIr/Photoredox Dual Catalysis.

Scheme 40 .

 40 Scheme 40. Dumoulin et al. synthesis of C-aryl/(hetero)arylglycosides via Nickel/Photoredox Dual Catalysis.

Scheme 43 .

 43 Scheme 43. Proposed mechanism for the formation of C-aryl-nucleosides 18.

Scheme 41 .

 41 Dumoulin al. proposed mechanism for synthesis of Caryl/(hetero)arylglycosides via Nickel/Photoredox Dual Catalysis. Wang and Zhang photoredox/nickel dual catalyzed synthesis of C-aryl/vinyl-nucleosides.

	Scheme 42.

Acknowledgements ((optional))

(SelFSuCHi ANR-18-CE07-0012) and (SuCH_Fun, JCJC ANR-18-CE07-0030-01), la Ligue Contre le Cancer through an Equipe Labellisée 2014 grant. The ANR is gratefully acknowledged for a doctoral fellowship to MDR and J G.

Entry for the Table of Contents

This review highlight the major progresses on C-H functionalization and dual photoredox catalysis as two powerful approaches for the synthesis of C-branched glycosides.

Institute and/or researcher Twitter: @samuccen


Mohammed Hanaki

Jérôme Harmand

Zoubida Mghazli

Alain Rapaport

Tewfik Sari

Pablo Ugalde

Mathematical study of a two-stage anaerobic model when the hydrolysis is the limiting step

Keywords: Commensal system, Anaerobic digestion, Steady state, Mortality, Stability, Operating diagrams

A two-step model of the anaerobic digestion process is mathematically and numerically studied. The focus of the paper is put on the hydrolysis and methanogenesis phases when applied to the digestion of waste with a high content in solid matter: existence and stability properties of the equilibrium points are investigated. The hydrolysis step is considered as a limiting step in this process using the Contois growth function for the bacteria responsible for the first degradation step. The methanogenesis step being inhibited by the product of the first reaction (which is also the substrate for the second one), the Haldane growth rate is used for the second reaction step. The operating diagrams with respect to the dilution rate and the input substrate concentrations are established and discussed.

Introduction

Two-step models are very common in the environmental engineering literature to describe engineered biological systems. They are of particular interest to design feedback control laws since they are usually enough complex to capture the most important process dynamics while mathematically tractable.

The most common two-step model is used to describe the so called 'commensal ecological relationship': it takes the form of a cascade of two biological reactions where one limiting substrate S 1 is consumed by one microorganism/ecosystem X 1 to produce S 2 which serves as the main limiting substrate for a second microorganism/ecosystem X 2 as schematically represented by the following reaction :

S 1 µ1(.) -→ X 1 + S 2 , S 2 µ2(.)
-→ X 2 + CO 2 + CH 4 CONTACT Jérôme Harmand. Email: jerome.harmand@inrae.fr This model is given by the following general dynamical system:

                 Ṡ1 = D(S in 1 -S 1 ) -µ 1 (.) X1 Y1 , Ẋ1 = [µ 1 (.) -αD -k 1 ]X 1 , Ṡ2 = D(S in 2 -S 2 ) + µ 1 (.) X1 Y3 -µ 2 (.) X2 Y2 , Ẋ2 = [µ 2 (.) -αD -k 2 ]X 2 (1)
where D is the dilution rate, while S in 1 and S in 2 are the input substrate concentrations, respectively. Parameters Y i are yield coefficients associated to the bio-reactions, k i are the mortality terms while α ∈ [0, 1] is a term allowing to decouple the retention time applied to substrates (supposed to be soluble) and biomass (supposed to be particulate). The kinetics µ 1 and µ 2 are the growth rate functions associated to X 1 and X 2 , respectively.

The different analyses of the class of models (1) available in the literature essentially differ on the growth rate functions used and whether a specific input for S 2 is considered or not (i.e. if there is a source term S in 2 in the dynamic equation of S 2 or not). They differ also on the values i) of the coefficient α (allowing to decouple the solid and liquid retention times) and ii) of the mortality terms k i . For details on the various models of this type considered in the literature the reader can refer to Tables 2 and3 of the review paper [START_REF] Wade | Perspectives in mathematical modelling for microbial ecology[END_REF].

Among the numerous biological systems used in environmental engineering, the anaerobic systems are among those which are the most studied. Anaerobic digestion (AD) is the biological degradation of organic matter in the absence of oxygen. The final product is methane, a renewable energy source. This is why this process is more and more used for the treatment of liquid and solid wastes. Because of its relative instability due to the possible accumulation of intermediate products however, notably the volatile fatty acids (VFA), the modeling of this process has been extensively studied over these last years. Their complexity highly depends on the objectives pursued by the modeler. On the one hand, when the objective is to develop models for integrating and formalize the available knowledge -typically to better understand bioprocessesmodels are generally of high order and not tractable from a mathematical viewpoint, cf. for instance the ADM1 developed by the International Water Association or IWA [START_REF] Batstone | The iwa anaerobic digestion model no 1 (adm1)[END_REF]. On the other hand, as already mentioned hereabove, when the aim of the modeling is to develop decision tools or control systems, low order models are better suited, as for instance the well known 'AM2', cf. [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]. Considering a system where the second step was limiting and subject to inhibition, the authors considered a Monod function for µ 1 and a Haldane function for µ 2 . In [START_REF] Sbarciog | Determination of appropriate operating strategies for anaerobic digestion systems[END_REF], this model was studied for α = 1 while the most interesting case where 0 < α < 1 and where growth functions were characterized by qualitative properties was studied in Benyahia, Sari, Cherki, and Harmand (2012).

On the one hand, it is particularly important to underline here that such a model has been validated many times on experimental data and that it has been used in a large number of studies related to the AD of wastes notably as a basis for control design, cf. [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF], [START_REF] Lopez | Modelling a full scale uasb reactor using a cod global balance approach and state observers[END_REF] or still [START_REF] Arzate | Anaerobic digestion model (am2) for the description of biogas processes at dynamic feedstock loading rates[END_REF] and related references. On the other hand, using real data, several authors have shown that such low-order models are appropriate to describe the AD processes (cf. [START_REF] Garcia-Dieguez | Reducing the anaerobic digestion modelno. 1 for its application to an industrial wastewater treatment plant treatingwinery effluent wastewater[END_REF] or [START_REF] Weedermann | Mathematical model of anaerobic digestion in a chemostat: Effects of syntrophy and inhibition[END_REF]). Following this idea, a method has even been proposed to easily, directly and systematically calibrate AM2 parameters from real data or data obtained by simulating a complex model like the ADM1, cf. [START_REF] Hassam | A generic and systematic procedure to derive a simplified model from the anaerobic digestion model no. 1 (adm1)[END_REF].

Most of the wastes treated in the anaerobic systems studied in the above-mentioned studies are liquid wastes. Depending on the nature of the wastes, and in particular whether they are liquid or solid, the limiting step of the AD is not the same. If the treated waste is liquid, the main limiting step is usually considered to be the methanogenesis: in such a case, simple models including only acidogenesis and methanogenesis can be used as in the AM2. If the waste contains a high proportion of solid matter however, it is the rule rather than the exception to consider that the hydrolysis is the main limiting step of the AD. In such a case, a model including only hydrolysis and methanogenesis can be used. As proposed in [START_REF] Vavilin | Hydrolysis kinetics in anaerobic degradation of particulate organic material: An overview[END_REF], when hydrolysis is the limiting step, rates depending only on substrate concentrations, such as the Monod function, are not the most appropriate. It is better to describe such complex phenomena by density-dependent models, such as density dependent kinetics, a family in which Contois models falls, cf. [START_REF] Ramirez | Modified adm1 disintegration/hydrolysis structures for modeling batch thermophilic anaerobic digestion of thermally pretreated waste activated sludge[END_REF]. Using this model, the rate of the hydrolysis step is modeled as

µ 1 (S 1 , X 1 ) = m 1 S 1 K 1 X 1 + S 1 = m 1 S1 X1 K 1 + S1 X1
which exhibits the following properties specific to hydrolysis, cf. [START_REF] Ramirez | Modified adm1 disintegration/hydrolysis structures for modeling batch thermophilic anaerobic digestion of thermally pretreated waste activated sludge[END_REF]:

   S1 X1 K 1 ⇒ µ 1 (S 1 , X 1 )X 1 ≈ m 1 X 1 ∝ X 1 , S1 X1 K 1 ⇒ µ 1 (S 1 , X 1 )X 1 ≈ m1 K1 S 1 ∝ S 1 (2)
While the analysis of the general model of AD initially purposed in [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF] (representing acidogenesis and methanogenesis steps) has been realized in [START_REF] Benyahia | Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion prosesses[END_REF], from the best of authors knowledge, a two-step model where the kinetic of the first step is modeled by a generic density-dependent kinetics while the second step exhibits a Haldane-type function has never been studied in the literature. It is the aim of the actual paper to study such a generic model. This analysis in realized in taking advantage of the fact that the system has a cascade structure: known results are then applied to study the whole 4th order system as the coupling of two 2nd order chemostat models. The main contribution of the paper is the set of operating diagrams of the 4th order system that are provided in section 4.

The paper is organized as follows. In section 2 the two-step model with two input substrate concentrations is presented and the general hypotheses on the growth functions are given. In section 3 the expressions of the steady states are given and their stability properties are established. In section 4, the effect of the second input substrate concentration on the steady states is illustrated in designing the operating diagrams, firstly, with respect to the first input substrate concentration and the dilution rate and secondly, with respect to the second input substrate concentration and the dilution rate.

Mathematical model

The two-step model reads:

                 Ṡ1 = D(S in 1 -S 1 ) -µ 1 (S 1 , X 1 ) X1 Y1 , Ẋ1 = [µ 1 (S 1 , X 1 ) -D 1 ]X 1 , Ṡ2 = D(S in 2 -S 2 ) + µ 1 (S 1 , X 1 ) X1 Y3 -µ 2 (S 2 ) X2 Y2 , Ẋ2 = [µ 2 (S 2 ) -D 2 ]X 2 (3)
where S 1 and S 2 are the substrate concentrations introduced in the chemostat with input concentrations S in 1 and S in 2 . D 1 = αD + k 1 and D 2 = αD + k 2 are the sink terms of biomass dynamics where D is the dilution rate, k 1 and k 2 represent maintenance terms and parameter α ∈ [0, 1] represents the fraction of the biomass affected by the dilution rate while Y i are the yield coefficients. X 1 and X 2 are the hydrolytic bacteria and methanogenic bacteria concentrations, respectively. The functions µ 1 : (S 1 , X 1 ) → µ 1 (S 1 , X 1 ) and µ 2 : (S 2 ) → µ 2 (S 2 ) are the specific growth rates of the bacteria.

To ease the mathematical analysis of the system, it is rescaled. Notice that it is simply equivalent to changing units of variables:

s 1 = S 1 , x 1 = 1 Y 1 X 1 , s 2 = Y 3 Y 1 S 2 , x 2 = Y 3 Y 1 Y 2 X 2
The following system is obtained:

                 ṡ1 = D(s in 1 -s 1 ) -f 1 (s 1 , x 1 )x 1 , ẋ1 = [f 1 (s 1 , x 1 ) -D 1 ]x 1 , ṡ2 = D(s in 2 -s 2 ) + f 1 (s 1 , x 1 )x 1 -f 2 (s 2 )x 2 , ẋ2 = [f 2 (s 2 ) -D 2 ]x 2 (4)
where s in 2 = Y3 Y1 S in 2 , f 1 and f 2 are defined by

f 1 (s 1 , x 1 ) = µ 1 (s 1 , Y 1 x 1 ) and f 2 (s 2 ) = µ 2 Y 1 Y 3 s 2
It is assumed that the functions µ 1 (., .) and µ 2 (.) satisfy the following hypotheses.

H1. µ 1 (s 1 , x 1 ) is positive for s 1 > 0, x 1 > 0, and satisfies µ 1 (0, x 1 ) = 0 and µ 1 (+∞, x 1 ) = m 1 (x 1 ). Moreover µ 1 (s 1 , x 1 ) is strictly increasing in s 1 , and decreasing in x 1 that is to say ∂µ1 ∂s1 > 0 and ∂µ1 ∂x1 ≤ 0 for s 1 > 0, x 1 > 0. H2. µ 2 (s 2 ) is positive for s 2 > 0, and satisfies µ 2 (0) = 0 and µ 2 (+∞) = 0. Moreover µ 2 (s 2 ) increases until a concentration s M 2 and then decreases; thus µ 2 (s 2 ) > 0 for 0 ≤ s 2 < s M 2 , and µ 2 (s 2 ) < 0 for s 2 > s M 2 . As underlined in the introduction, particular kinetics models as Contois function verify H1 while the Haldane function verifies H2. Since the functions µ 1 and µ 2 satisfy the hypotheses H1 and H2, it follows from the above that functions f 1 and f 2 satisfy the following assumptions.

A1. f 1 (s 1 , x 1 ) is positive for s 1 > 0, x 1 > 0, and satisfies f 1 (0, x 1 ) = 0 and f 1 (+∞, x 1 ) = m 1 (x 1 ). Moreover ∂f1 ∂s1 > 0 and ∂f1 ∂x1 ≤ 0 for andsatisfies f 2 (0) = 0 andf 2

s 1 > 0, x 1 > 0. A2. f 2 (s 2 ) is positive for s 2 > 0,
(+∞) = 0. Moreover f 2 (s 2 ) increases until a concentration s M
2 and then decreases, with f 2 (s 2 ) > 0 for 0 < s 2 < s M 2 , and f 2 (s 2 ) < 0 for s 2 > s M 2 .

Analysis of the model

3.1. The dynamics of s 1 and x 1 3.1.1. Study of the steady states of system (5)

Model (4) has a cascade structure which renders its analysis easier. In other terms s 1 and x 1 are not influenced by variables s 2 and x 2 and their dynamics are given by:

   ṡ1 = D(s in 1 -s 1 ) -f 1 (s 1 , x 1 )x 1 , ẋ1 = [f 1 (s 1 , x 1 ) -D 1 ]x 1 .
(5)

The behaviour of this system is well-known, cf. Harmand, J. and C. Lobry and A. Rapaport and T. Sari (2016). A steady state (s * 1 , x * 1 ) must be solution of the system

   0 = D(s in 1 -s 1 ) -f 1 (s 1 , x 1 )x 1 , 0 = [f 1 (s 1 , x 1 ) -D 1 ]x 1 (6)
From the second equation it is deduced that x * 1 = 0, which corresponds to the washout E 0 = (s in 1 , 0), or s * 1 and x * 1 must satisfy both equations

f 1 (s * 1 , x * 1 ) = D 1 and x * 1 = D D 1 (s in 1 -s * 1 ). (7)
Let γ a function defined by :

γ(s 1 ) = f 1 s 1 , D D 1 (s in 1 -s 1 ) , so s * 1 is a solution of γ(s 1 ) = D 1 , and it is noticed that γ (s 1 ) = ∂f 1 ∂s 1 - D D 1 ∂f 1 ∂x 1 .
According to the hypothesis A1, γ(s 1 ) is strictly increasing over the interval ]0, s in 1 [, with γ(0) = 0 and γ(s in 1 ) = f 1 (s in 1 , 0). According to the theorem of intermediate values, the equation γ(s 1 ) = D 1 has a solution between 0 and s in 1 if and only if

D 1 < γ(s in 1 ), that is to say if D 1 < f 1 (s in 1 , 0), cf. Figure 1. Hence, for x * 1 = 0, the equilibrium E 1 (s * 1 , x * 1 ) exists if and only if D 1 < f 1 (s in 1 , 0)
. The local stability of the steady state is given by the sign of the real part of eigenvalues of the Jacobian matrix evaluated at this steady state. In the following, the abbreviations LES for locally exponentially stable is used.

Proposition 3.1. Assume that Assumptions A1 and A2 hold. Then, the local stability of steady states of (5) is given by :

(1)

E 0 = (s in 1 , 0) is LES if and only if f 1 (s in 1 , 0) < D 1 ( i.e. s in 1 < s * 1 ). (2) E 1 = (s * 1 , x * 1 ) is LES if and only if f 1 (s in 1 , 0) > D 1 ( i.e. s in 1 > s * 1 ), (E 1 is stable if it exists)
The reader may refer to Harmand, J. and C. Lobry and A. Rapaport and T. Sari (2016) for the proof of this proposition. In the same book, notice that global stability results for system (5) are also provided. When E 0 and E 1 coincide, the equilibrium is attractive (the eigenvalues are equal to zero). The results of Proposition 3.1 are summarized in the following table. Always exists

f 1 (s in 1 , 0) < D 1 E 1 f 1 (s in 1 , 0) > D 1
Stable when it exists 3.1.2. Operating diagram of the system (5)

Apart from the two operating (or control) parameters, which are the input substrate concentration s in 1 and the dilution rate D -that can vary -all others parameters (α, k 1 and the parameters of the growth function f 1 (s 1 , x 1 )) have biological meaning and are fixed depending on the organisms and substrate considered. The operating diagram shows how the steady states of the system behave when the two control parameters s in 1 and D are varied. The operating diagram for the system (5) is shown in Figure 2. The condition

f 1 (s in 1 , 0) > D 1 of existence of E 1 is equivalent to D < 1 α [f 1 (s in 1 , 0) -k 1 ]. Therefore, the curve Γ : (s in 1 , D) : D = 1 α [f 1 (s in 1 , 0) -k 1 ]
separates the operating plan in two regions as defined in Figure 2. The curve Γ is the border which makes E 0 unstable and at the same time E 1 exists. Table 2 indicates the stability properties of steady states of the system (5) in each region where S and U read for LES and unstable respectively and no letter means that the steady state does not exist.

Region

Equ.

E 0 Equ. E 1 (s in 1 , D) ∈ R 0 S (s in 1 , D) ∈ R 1 U S
Except for small values of D and s in 1 , notice that the operating diagram of this first part of the twop-step system under study is qualitatively similar to that one of the first part of the AM2 model, that is when a Monod-like growth function is considered, cf. [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF].

3.2. The dynamics of s 2 and x 2 3.2.1. Study of the steady states of system (8) Due to the cascade structure of (4), the dynamics of the state variables s 2 and x 2 are given by

   ṡ2 = D(F (t) -s 2 ) -f 2 (s 2 )x 2 , ẋ2 = [f 2 (s 2 ) -D 2 ]x 2 , (8) with F (t) = s in 2 + 1 D f 1 (s 1 (t), x 1 (t))x 1 (t)
where s 1 (t), x 1 (t) are a solution of (5). A steady state (s * 1 , x * 1 , s * 2 , x * 2 ) of (4) corresponds to a steady state (s * 2 , x * 2 ) of ( 8) where either (s

1 (t), x 1 (t)) = E 0 or (s 1 (t), x 1 (t)) = E 1 .
Therefore (s * 2 , x * 2 ) must be a steady state of the system

   ṡ2 = D(s in * 2 -s 2 ) -f 2 (s 2 )x 2 , ẋ2 = [f 2 (s 2 ) -D 2 ]x 2 (9)
where

s in * 2 = s in 2 or s in * 2 = s in 2 + D 1 D x * 1 . (10)
The first case corresponds to (s * 1 , x * 1 ) = E 0 and the second to (s * 1 , x * 1 ) = E 1 . The system (9) corresponds to a classical chemostat model with Haldane-type kinetics, including a mortality term for x 2 and an input substrate concentration. Notice that s in * 2 , given by ( 10), depends explicitly on the input flow rate. For a given D, the long term behaviour of such a system is well-know, cf. Harmand, J. and C. Lobry and A. [START_REF] Harmand | Chemostat: Theory mathematics of continuous culture of microorganisms[END_REF].

A steady state (s * 2 , x * 2 ) must be a solution of the system

   0 = D(s in * 2 -s 2 ) -f 2 (s 2 )x 2 , 0 = [f 2 (s 2 ) -D 2 ]x 2 . ( 11
)
From the second equation it is deduced that x * 2 = 0, which correspond to the washout

F 0 = (s in * 2 , 0) or s * 2 must satisfy the equation f 2 (s 2 ) = D 2 . ( 12
)
Under hypothesis A2, and if

D 2 < f 2 (s M 2 ) ( 13
)
this equation has two solutions which satisfy s 1 2 < s 2 2 . Therefore the system has two positive steady states F 1 = (s 1 2 , x 1 * 2 ) and F 2 = (s 2 2 , x 2 * 2 ), where

x i * 2 = D D 2 (s in * 2 -s i 2 ), i = 1, 2. ( 14
)
For i = 1, 2, the steady states F i exist if and only if s in * 2 > s i 2 .

Proposition 3.2. Assume that Assumptions A1, A2 and condition (13) hold. Then, the local stability of steady states of (9) is given by :

(1)

F 0 is LES if and only if s in * 2 < s 1 2 or s in * 2 > s 2 2 . (2) F 1 is LES if and only if s in * 2 > s 1 2 (stable if it exists ). (3) F 2 is unstable if it exists (unstable if s in * 2 > s 2 2 ).
The reader may refer to Harmand, J. and C. Lobry and A. Rapaport and T. Sari (2016) for the proof of this proposition.

The results of Proposition 3.2 are summarized in the following table.

= s 1 2 or s in * 2 = s 2 2 are equivalent to f 2 (s in * 2 ) = D 2 , that is to say D = 1 α (f 2 (s in * 2 ) -k 2 )
. Therefore, the horizontal line

Γ 1 : (s in * 2 , D) : D = 1 α (f 2 (s M 2 ) -k 2 ), s in * 2 > s M 2
together with the curve

Γ 2 : (s in * 2 , D) : D = 1 α (f 2 (s in * 2 ) -k 2 )
separate the operating diagram plane in three regions as defined in Figure 3. The following table indicates the stability properties of steady states of system (8).

F 0 Equ. F 1 Equ. F 2 (s in * 2 , D) ∈ R 2 S (s in * 2 , D) ∈ R 3 U S (s in * 2 , D) ∈ R 4 S S U
3.3. Analysis of the whole system (4)

Steady states

The aim of this section is to study the dependence of the steady state of (4) with respect to the operating parameters D, s in 1 and s in 2 . Let (s * 1 , x * 1 , s * 2 , x * 2 ) be a steady state of (4), then (s * 1 , x * 1 ) is a steady state of (5) and (s * 2 , x * 2 ) is a steady state of (9) where s in * 2 is given by (10).

If (s *

1 , x * 1 ) = E 0 = (s in 1 , 0) then s in * 2 = s in 2 and three possibilities can occur (1) (s * 2 , x * 2 ) = (s in 2 , 0), and

E 0 1 := (s in 1 , 0, s in 2 , 0). (2) (s * 2 , x * 2 ) = (s 1 2 , x 1
2 ), and

E 1 1 := (s in 1 , 0, s 1 2 , x 1 2 ) (3) (s * 2 , x * 2 ) = (s 2 2 , x 2
2 ), and

E 2 1 := (s in 1 , 0, s 2 2 , x 2 2 ). If (s * 1 , x * 1 ) = E 1 = (s * 1 , x * 1
) then three others possibilities can occur (1) (s * 2 , x * 2 ) = (s in * 2 , 0), and

E 0 2 := (s * 1 , x * 1 , s in * 2 , 0). (2) (s * 2 , x * 2 ) = (s 1 2 , x 1 * 2 )
, and

E 1 2 := (s * 1 , x * 1 , s 1 2 , x 1 * 2 ). (3) (s * 2 , x * 2 ) = (s 2 2 , x 2 * 2 )
, and

E 2 2 := (s * 1 , x * 1 , s 2 2 , x 2 * 2
). These results are summarized in the following proposition.

Proposition 3.3. The system (4) has at most six steady states :

• E 0 1 = (s in 1 , 0, s in 2 , 0), always exists. • E 1 1 = (s in 1 , 0, s 1 2 , x 1 2 ), exists if and only if s in 2 > s 1 2 . • E 2 1 = (s in 1 , 0, s 2 2 , x 2 2 ), exists if and only if s in 2 > s 2 2 . • E 0 2 = (s * 1 , x * 1 , s in * 2 , 0), exists if and only if f 1 (s in 1 , 0) > D 1 . • E 1 2 = (s * 1 , x * 1 , s 1 2 , x 1 * 2 ), exists if and only if f 1 (s in 1 , 0) > D 1 and s in * 2 > s 1 2 . • E 2 2 = (s * 1 , x * 1 , s 2 2 , x 2 * 2 ), exists if and only if f 1 (s in 1 , 0) > D 1 and s in * 2 > s 2 2 .
3.3.2. Steady states stability of the system (4)

In this section, the stability of the steady states given in the Proposition 3.3 is studied.

For this the following Jacobian matrix is considered:

J = J 11 J 12 0 J 22 ,
where J 11 and J 22 are defined by ( 15) and ( 16), respectively given by :

J =   -D -M x 1 -N x 1 -f 1 (s 1 , x 1 ) M x 1 N x 1 + [f 1 (s 1 , x 1 ) -D 1 ]   , (15)
and

J =   -D -f 2 (s 2 )x 2 -f 2 (s 2 ) f 2 (s 2 )x 2 f 2 (s 2 ) -D 2   . (16)
This matrix has a block-triangular structure. Hence, the eigenvalues of J are the eigenvalues of J 11 and the eigenvalues of J 22 . The existence of steady states depends only on the relative positions of the two numbers s in 1 and s * 1 defined by ( 7) with respect to the four numbers s 1 2 and s 2 2 , defined by ( 12) on the one hand, and s in 2 and s in * 2 , solutions of (10) on the other hand. Equilibrium stability is summarized in Table 5 while the different regions of the operating diagram are synthesized in Tables 6 and7.

Table 5. The stability conditions for the steady states of the system (4)

Equ. Matrices J 11 and J 22 Conditions of stability

E 0 1 T r(J 11 ) < 0 if f 1 (s in 1 , 0) < D 1 , J 11 = -D -f 1 (s in 1 , 0) 0 f 1 (s in 1 , 0) -D 1 T r(J 22 ) < 0 if s in 2 < s 1 2 or s in 2 > s 2 2 , det(J 11 ) > 0 and det(J 22 ) > 0 J 22 = -D -f 2 (s in 2 ) 0 f 2 (s in 2 ) -D 2 ⇒        E 0 1 is stable if f 1 (s in 1 , 0) ≤ D 1 and (s in 2 < s 1 2 or s in 2 > s 2 2 ), E 0 1 is unstable if f 1 (s in 1 ) > D 1 or s 1 2 < s in 2 < s 2 2 ). E i 1 (i = 1, 2) T r(J 11 ) < 0 and det(J 11 ) > 0 if f 1 (s in 1 , 0) < D 1 J 11 = -D -f 1 (s in 1 , 0) 0 f 1 (s in 1 , 0) -D 1 T r(J 22 ) < 0 and det(J 22 ) > 0 at E 1 1 , det(J 22 ) < 0 at E 2 1 J 22 = -[D + f 2 (s i 2 )x i 2 ] -D 2 f 2 (s i 2 )x i 2 0 ⇒    E 1 1 is stable, E 2 1 is unstable E i 1 are both unstable if s in 1 > s * 1 . E 0 2
T r(J 11 ) < 0 and det(J 11 ) > 0 by A1

J 11 = -[D + ( ∂f 1 ∂s 1 )x * 1 ] -[D 1 + ( ∂f 1 ∂x 1 )x * 1 ] ( ∂f 1 ∂s 1 )x * 1 ( ∂f 1 ∂x 1 )x * 1 T r(J 22 ) < 0 and det(J 22 ) > 0 if s in * 2 < s 1 2 or s in * 2 > s 2 2 J 22 = -D -f 2 (s in * 2 ) 0 [f 2 (s in * 2 ) -D 2 ] ⇒        E 0 2 is stable if s in 1 ≥ s * 1 and (s in * 2 < s 1 2 or s in * 2 > s 2 2 ), E 0 2 is unstable if s in 1 > s * 1 and s 1 2 < s in * 2 < s 2 2 . E i 2 (i = 1, 2)
T r(J 11 ) < 0 and det(J 11 ) > 0

J 11 = -[D + ( ∂f 1 ∂s 1 )x * 1 ] -[D 1 + ( ∂f 1 ∂x 1 )x * 1 ] ( ∂f 1 ∂s 1 )x * 1 ( ∂f 1 ∂x 1 )x * 1 T r(J 22 ) < 0 and det(J 22 ) > 0 at E 1 2 , det(J 22 ) < 0 at E 2 2 J 22 =    -[D + f 2 (s i 2 )x i * 2 ] -D 2 f 2 (s i 2 )x i * 2 0    ⇒ E 1 2 is stable, E 2 2 is unstable Table 6. The three cases when s in 1 < s * 1 Case Area Condition E 0 1 E 1 1 E 2 1 E 0 2 E 1 2 E 2 2 1.1 A 1 s in 2 < s 1 2 < s 2 2 S 1.2 A 2 s 1 2 < s in 2 < s 2 2 U S 1.3 A 3 s 1 2 < s 2 2 < s in 2 S S U Table 7. The six cases when s in 1 > s * 1 Case Area Condition E 0 1 E 1 1 E 2 1 E 0 2 E 1 2 E 2 2 2.1 A 4 s in 2 < s in * 2 < s 1 2 < s 2 2 U S 2.2 A 5 s in 2 < s 1 2 < s in * 2 < s 2 2 U U S 2.3 A 6 s in 2 < s 1 2 < s 2 2 < s in * 2 U S S U 2.4 A 7 s 1 2 < s in 2 < s in * 2 < s 2 2 U U U S 2.5 A 8 s 1 2 < s in 2 < s 2 2 < s in * 2 U U S S U 2.6 A 9 s 1 2 < s 2 2 < s in 2 < s in * 2 U U U S S U
Remark 1. Here the limit values in the stability condition (Ex:

s in 2 = s 1 2 or s in 2 = s 2 2 )
are excluded. These limit cases are related to the eigenvalues of the Jacobian matrix with a real part equal to 0 in which case the corresponding states are named nonhyperbolic stationary states. Otherwise, they are named hyperbolic steady states.

The different possible cases of non-hyperbolic (NH) equilibrium are summarized in the theorem 3.4.

Theorem 3.4. If s in 1 < s * 1 , then there are three sub-cases:

Case Condition NH S U 1.4 s in 2 = s 1 2 < s 2 2 E 0 1 = E 1 1 1.5 s 1 2 < s in 2 = s 2 2 E 0 1 = E 2 1 E 1 1 1.6 s 1 2 = s 2 2 < s in 2 E 1 1 = E 2 1 E 0 1 If s in 1 > s * 1 , then there are nine sub-cases Case Condition NH S U 2.7 s in 2 < s in * 2 = s 1 2 < s 2 2 E 0 2 = E 1 2 E 0 1 2.8 s in 2 < s 1 2 < s 2 2 = s in * 2 E 0 2 = E 2 2 E 1 2 E 0 1 2.9 s in 2 = s 1 2 < s in * 2 < s 2 2 E 0 1 = E 1 1 E 1 2 E 0 2 2.10 s in 2 = s 1 2 < s 2 2 < s in * 2 E 0 1 = E 1 1 E 1 2 ,E 0 2 E 2 2 2.11 s in 2 = s 1 2 < s 2 2 = s in * 2 E 0 1 = E 1 1 ,E 0 2 = E 1 2 E 1 2 2.12 s 1 2 < s in 2 = s 2 2 < s in * 2 E 1 2 = E 2 2 E 0 2 E 0 1 2.13 s 1 2 < s in 2 < s 2 2 = s in * 2 E 0 2 = E 2 2 E 1 2 E 0 1 ,E 1 1 2.14 s 1 2 < s in 2 = s 2 2 < s in * 2 E 0 1 = E 2 1 E 0 2 ,E 1 2 E 0 1 ,E 2 2 2.15 s 1 2 = s 2 2 < s in 2 < s in * 2 E 1 1 = E 2 1 ,E 1 2 = E 2 2 E 0 2 E 0 1
Proof. Let us give the details of the proof in the case 2.9. The other cases can be studied similarly. Assume that s in 2 = s 1 2 < s in * 2 < s 2 2 , then x 1 2 = 0, x * 1 > 0 and x 1 * 2 > 0 by (14). Therefore (cf. Proposition 3.3), the system has three equilibria

E 0 1 = E 1 1 , E 0 2
and E 1 2 . Using the linearization, it is established that E 0 2 and E 1 2 are hyperbolic. Remark 2. In each case among those cited in the Tables 6, 7 and in the previous Theorem 3.4, a corresponding figure which represents the relative position of s 1 2 , s 2 2 , s in 2 and s in * 2 Benyahia et al. ( 2012) can be associated.

Simulations

To illustrate these results, the operating diagrams of the system (3) under hypothesis H1 and H2 in a number of situations are plotted. Recall that operating diagram summarizes the existence and the nature of the the steady states of a dynamical system as a function of its input variables. Here, the control inputs are D, S in 1 and S in 2 . More particularly, either the operating diagrams in the plan {S in 1 , D} for a fixed value of S in 2 or in the plan {S in 2 , D} for a fixed value of S in 1 are plotted. All simulations are performed with the following growth functions:

µ 1 (S 1 , X 1 ) = m 1 S 1 K 1 X 1 + S 1 , µ 2 (S 2 ) = m 2 S 2 S 2 2 I + S 2 + K 2 .
Of course, these operating diagrams depend on the model parameters. The choice of their values is a difficult task. Indeed, our objective here is not to match a set of data but rather to highlight interesting qualitative properties of the model under interest.

To do so, most parameters are taken from [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF], while others are changed significantly as the inhibition coefficient I of the Haldane function. Indeed, as underlined in [START_REF] Khedim | Effect of control parameters on biogas production during the anaerobic digestion of protein-rich substrates[END_REF], inhibition of the second reaction is not visible if original parameters proposed in [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF] are used considering reasonable ranges of variations for S 1 and S 2 . With respect to this later, the Haldane parameter I was thus significantly decreased to willingly increase the inhibition effect of S 2 on the growth of X 2 . Finally, parameter values used are summarized in the following table To compute the different regions

1 d -1 0.5 K 1 g/L 2.1 m 2 d -1 1 I mmol/L 60 K 2 mmol/L 24 k 1 d -1 0.1 k 2 d -1 0.06 α in [0, 1) 0.5 Y 1 g/g 1/25 Y 2 g/mmol 1/250 Y 3 g/mmol 1/268
of the operating diagrams, the numerical method reported in [START_REF] Khedim | Effect of control parameters on biogas production during the anaerobic digestion of protein-rich substrates[END_REF] is used. The algorithm is recalled hereafter.

Algorithm for the determination of the operating diagrams

The algorithm is as follows: for each value of input variables chosen on a grid, the equilibria are computed. The eigenvalues of the Jacobian matrix are then calculated for each equilibrium. Finally, according to the conditions of existence and the sign of the real parts of the eigenvalues, a 'flag' is assigned to each of the 6 equilibria: 'S' for stable, 'U' for unstable or nothing if the steady state does not exist. This procedure stops when all the values of the grid {S in , D} have been scanned. As a result, a number of 'signatures' composed of sequences of 'S', 'U' or 'nothing' are obtained. These cases code for the existence and stability of the equilibria that are grouped into regions as summarized in the tables at the end of sections 3.1 and 3.2, respectively. This algorithm may be formalized as follows: let N 1 , N 2 be two integers in N * and h 1 = D N1 and h 2 = Sin N2 the two iteration steps:

Operating diagrams

In this part the results are illustrated by plotting the operating diagrams and are discussed.

Figure 4 represents the operating diagram of model (1) in the plan {S in 1 , D} for S in 2 = 1.5 mmol/L. The regions are defined as follows. A 1 (in green) is the stability region of the washout E 0 1 , A 5 (in blue) is the stability region of steady-state E 1 2 , A 6 (in red) is the bi-stability region of the steady-states E 0 2 and E 1 2 , and A 7 (in dark blue) is the stability region of steady-state E 1 2 , the difference between the regions A 5 and A 7 being that the equilibrium E 1 1 does not exist in the region A 5 but exists and is unstable in A 7 , cf. Tables 6 and7. is numerically linked to the value of S in 1 . Indeed, when reducing the value of S in 1 , the region A 4 appears while the size of the region A 8 is getting smaller and smaller (compare Figures 5 and6). In other words, in decreasing S in 1 , the attraction basin of the positive stable equilibrium of A 7 (E 1 2 ) increases. It is equivalent to say that, given two values of S in 1 , say S in 1 1 and S in 2 1 where S in 2 1 > S in 1 1 , a greater dilution rate is needed to destabilize the process if S in 1 = S in 2 1 than if S in 1 = S in 1 1 , thus the shard decrease of A 8 observed in Figure 6.

Practical interpretations of the operating diagrams

Here below, it is explained how the operating diagrams may be used in practice. First, notice that the operator must avoid the process to operate in regions where either E 0 1 and/or E 0 2 would be the only stable equilibrium points. Indeed, in such regions, x * 2 = 0 and the process does not produce any methane: then A 1 and A 4 must be avoided. In addition, a particular attention must be paid to operating conditions in which there is bi-stability with one of the stable equilibrium point is E 0 1 or E 0 2 , that are A 3 , A 6 , A 8 and A 9 . In the following, what may happen within regions A 6 and A 7 that are methane-producing areas is analyzed. To illustrate their practical interest, let us consider the operating diagram pictured in Figure 4 (that is for S in 2 = 1.5 g/L) and let us browse it for increasing values of D at a fixed value of S in 1 . Example 1: Let S in 1 = 18 g/L. In such a situation, the following regions are browsed This last diagram allows us to see the appearance/disappearance of steady states as a function of the input variable D (recall that S in 1 and S in 2 are fixed). As long as D is small enough (i.e. such that αD + k 2 < d 1 ), the quantity of substrate entering the second step of the reaction is very important: the system is in the region A 7 where the positive equilibrium is the only stable equilibrium. As D increases the size of the attraction basin of this equilibrium decreases until D reaches a critical value (corresponding to the point P 1 in Figure 7, which is situated on the frontier between regions A 7 and A 6 ). This critical value corresponds to that one for which the term S in * 2 becomes exactly the largest solution of the equation µ 2 (S 2 ) = D 2 (equivalent to equation (12) for the system (4)): the system enters then in the region A 6 . From a biological point of view, the interpretation is as follows: as D increases, X * 1 decreases and thus S in * 2 decreases as can be seen from equation (10). When D 2 = d 2 , the quantity of available resource necessary to X 2 to grow may become limiting for some initial conditions, leading the system to enter a bi-stability zone. With the values of the parameters chosen, further increasing D leads definitely X 2 to the washout: the system enters into A 5 in crossing the point P 2 of Figure 7. Finally, if D is such that D 1 = d 2 (the critical value corresponding to the maximum growth rate of X 1 ) X 1 goes also to extinction and the system enters into A 1 .

Example 2: Let S in 1 = 14 g/L. This case is even more interesting since, when D Figure 9. Biological interpretation for S in 1 = 14 g/L increases, the system goes back to A 5 once before leaving it definitely in browsing the following regions: A 7 A 6 A 7 A 5 A 1 . While D is small enough (i.e. such that αD+k 2 < d 3 , cf. Figure 9), the reasoning remains the same than before. The only difference is that the value of D leading the system to enter into A 6 through P 3 is a little bit higher than in the previous case (d 3 > d 1 ). It is due to the fact that the second step of the reaction receives less input from the first step when compared to the case where S in 1 = 18g/L, thus enlarging the attraction basin of the stable positive equilibrium. Then, when D is further increased, it may happen an interesting phenomenon: the system enters back into A 7 through point P 4 instead of entering A 5 as it was the case before. In fact, this strongly depends on model parameters and in particular on the relative rate at which S in * 2 and the largest solution of the equation (10) vary as functions of D, cf. Figure 10. In other words, it depends on how the input concentration of the second step S in * other hand, if D is such that D 2 crosses µ 2 in a sharper zone of the Haldane function (typically around the inflexion point, still considering S 2 evolves at concentrations such as the system is on the right of the maximum of µ 2 ), a small change on D (and thus on D 2 ) will affect much more the solution of the equation (10) than before. In any of these situations, the relative positions of the largest solution of the equation ( 10) and of S in * 2 determine whether the system will evolve in the region A 7 or A 6 and, as D increases to a value such αD + k 2 = d 4 a return of the system from A 7 into A 6 can be observed. It goes through P 3 and then goes back into A 7 through P 4 . It is well illustrated in the bifurcation diagram plotted in Figure 10.

Conclusions

In this paper, a model of the AD has been studied. A two stages mass-balance model, corresponding to hydrolysis and methanogenesis phases, is considered. A non usual growth function for the hydrolysis step that is a generic density-dependent growth rate has been used while a Haldane-type function is considered for the methanogenesis step. From the best of authors knowledge, it is the first time such a model including the association of Contois-Haldane-type growth functions in a two-step model of the AD is considered. In this analysis it has been shown that this model has six steady states (E 0 1 , E 1 1 , E 2 1 , E 0 2 , E 1 2 , E 2 2 ). Conditions under which they exist and are stable or unstable have been highlighted. The regions of stability of these equilibria have been established with the help of the operating diagrams and their practical use in a number of situations has been discussed. Finally, it should be underlined that the proposed change in the growth rate function of the first reaction step when considering the AD of solid wastes -using a density function instead of a substrate-dependent functionchanges the qualitative properties of the system when compared to systems treating liquid wastes.

Figure 1 .

 1 Figure 1. The existence of the solution of γ(s 1 ) = D 1 .

Figure 2 .

 2 Figure 2. Operating diagram of (5)

Figure 3 .

 3 Figure 3. Operating diagram of (9)

Figure 5 .

 5 Figure 5. The operating diagram for S in 1 = 0.8 g/L.

Figure 6 .

 6 Figure 6. The operating diagram for S in 1 = 0.03 g/L.

Figure 7 .

 7 Figure 7. Biological interpretation for S in 1 = 18 g/Lin considering successive equilibrium when increasing D:A 7 A 6 A 5 A 1 , cf.Figure 7. To better interpret whose 'steady states the system passes through', the bifurcation diagram is plotted in Figure 8.

Figure 7 .Figure 8 .

 78 Figure 8. The bifurcation diagram for the input control D for S in 1 = 18g/L.

Figure 10 .

 10 Figure 10. The bifurcation diagram of the operating diagram shown in Figure 4 for S in 1 = 14g/L.

Table 1 .

 1 Summary of the results of Proposition 3.1

	Steady state Existence condition Stability condition
	E 0

Table 2 .

 2 Stability properties of steady states of system (5) in each region

Table 3 .

 3 Summary of the results of Proposition 3.2

		Steady-state Existence condition Stability condition
		F 0 F 1 F 2	Always exists s in * 2 > s 1 2 s in * 2 > s 2 2	s in * 2 Stable if it exists < s 1 2 or s in * 2 Unstable if it exists > s 2 2
	3.2.2. Operating diagram of the system (8)
	Now, the operating diagram shows how the system behaves when the two control
	parameters s in * 2 conditions s in * 2	and D are varied. The operating diagram is shown in Figure 3. The

Table 4 .

 4 Stability properties of the steady states of system (8)

	Region	Equ.

Table 8 .

 8 Parameters values for the simulations

	Parameter	Unit	Nominal value
	m

-which includes the part of S 1 transformed into S 2 during the first reaction -is affected by D. On the one hand, if the system is in a 'flat' zone of the growth rate µ 2 assuming concentrations at the right of the maximum of µ 2 are considered, a small variation of D (and thus of D 2 ) will change very much the largest solution of the equation (10) while S in * 2 may almost remain constant. On the

Acknowledgements

The authors thank the projects PHC TOUBKAL TBK 17/47-36773WE and the TREASURE Euro-Mediterranean research network (www.inra.fr/treasure) for their financial supports.

Algorithm 1 Operating diagram for i varying from 1 to N 1 do;

for j varying from 1 to N 2 ; determine 6 equilibria of the model E 1 ...E 6 for k varying from 1 to 6 do calculate the Jacobian matrix at E k (J Ek ) calculate the eigenvalues of (J Ek ) if all the conditions of existence of E k are fulfilled and all real parts of the eigenvalues of (J Ek ) are non-positive then E k is stable else if all conditions of existence of E k are fulfilled and at least one real part of eigenvalue of (J Ek ) is positive then E k is unstable else E k does not exist end if end for (k) end for (j) end for (i) and the steady-state E 1 1 , A 8 and A 9 (in pink) and (in dark pink), respectively, are the bi-stability regions of the steady-states E 0 2 and E 1 2 , the difference between A 8 and A 9 being that the equilibrium E 2 1 does not exist in the region A 8 but exists and is unstable in A 9 , cf. Tables 6 and7.

Figure 6 represents the operating diagram of model (1) in the plan {S in 2 , D} for S in 1 = 0.03 g/L., that is a smaller value of S in 1 than before. The differences with the previous case are i) the appearance of a little region A 4 (in dark orange) which is the stability region of steady-state E 0 2 and ii) a sharp decrease of the size of region A 8 which almost disappears (it is reduced to a very narrow surface along the frontier with the region A 9 as can seen in Figure 6).

The region A 8 becomes very small and narrow compared to Figure 5 because it