Yoan Brissonnet

Romulo Araoz

Rui Sousa

Lucie Percevault

Sami Brument

David Deniaud

Denis Servent

Jean-Yves Le Questel

Jacques Lebreton

Sébastien G Gouin

Di-and heptavalent nicotinic analogues to interfere with α7 nicotinic acetylcholine receptors

Keywords:

Introduction

Synthetic multivalent systems, bearing several copies of a given ligand, have been widely developed to interfere with multimeric proteins. Affinity enhancements of several orders of magnitudes can be reached when the multivalent ligands span across the different recognition domains of the receptor. These chelate interactions are distinct from positive cooperativity or allosteric controls. They are driven by high local concentrations of the cognate ligands when the first is bound and by the lower entropic costs paid due to ligand preorganization at the protein surface. Most of the successful examples are so far reported with synthetic glycoclusters targeting carbohydrate binding proteins (lectins), 1,2 which are particularly prone to multivalent interactions due to their generally multimeric nature and shallow binding domains. Recently, the concept of multivalency has been extended to carbohydrate processing enzymes such as glycosidases and glycosyltransferases. [3][4][5][6][7] The multivalent effects were studied and ascribed to different binding modes such as interactions in secondary subsites, 8,9 in additional carbohydrate binding modules, 10 and the formation of stabilized cross-linked aggregates. 11,12 Although mostly exemplified in the field of glycosciences, 13,14 multivalent ligands were also reported for proteins and enzymes involved in cellular signal transduction pathways. [15][16][17] Strong increased affinity were reported with homo and heterobivalent ligands of members of the G-protein coupled receptors (GPCRs) including the oxytocin receptor, 18 muscarinic acetylcholine receptor, 19 and -aminobutyric acid type A receptors. 20 Kramer and Karpen have designed dimeric agonists of cyclic-nucleotide-gated channels (CNG) allowing ions to flow across cell membranes for signal transduction in retinal photoreceptors and olfactory neurons. 21 Cyclic guanosyl monophosphate (cGMP) was linked through polyethyleneglycols spacers (PEGs) of different lengths. Divalent cGMP with optimal PEG lengths to match the distance between the CNG binding sites were shown to be up to a thousand times more potent activators than monovalent cGMP. More recently, Capelli and coworkers designed dendrimeric tetravalent ligands with picomolar affinity for the serotonin-gated ion channel 5-HT3. 22 Altogether these results highlight the benefits of targeting ligand-gated ion channels (LGIC) in their orthosteric and/or allosteric binding sites with multivalent ligands. Nicotinic acetylcholine receptors (nAChRs) are important members of the LGIC superfamily, involved in cellular signalling of the peripheral and central nervous system. Agonists or antagonists of neuronal nAChRs have been extensively developed as analgesic targets or as potential drugs to treat neurological disorders such as schizophrenia, depression and Alzheimer's disease. [23][24][25] The main difficulty in this approach is to develop selective drugs to target the pharmacologically distinct homo-and heteropentamer nAChR subtypes resulting from the assembly of diverse subunits (α1-α10, β1-β4, γ, δ, ε). The heterogeneity of nAChR subtypes and subunits greatly complicates the design of potent monovalent agonists and antagonists with selective profiles. Acetylcholine binding proteins (AChBP)

A R T I C L E I N F O A B S T R A CT

In the field of nicotinic acetylcholine receptors (nAChRs), recognized as important therapeutic targets, much effort has been dedicated to the development of nicotinic analogues to agonize or antagonize distinct homo-and heteropentamers nAChR subtypes, selectively. In this work we developed di-and heptavalent nicotinic derivatives based on ethylene glycol (EG) and cyclodextrin cores, respectively. The compounds showed a concentration dependent inhibition of acetylcholine-induced currents on α7 nAChR expressed by Xenopus oocytes. Interesting features were observed with the divalent nicotinic derivatives, acting as antagonists with varied inhibitory concentrations (IC50) in function of the spacer arm length. The best divalent compounds showed a 16-fold lowered IC50 compared to the monovalent reference (12 vs 195 µM). Docking investigations provide guidelines to rationalize these experimental findings.

are valuable structural surrogates of the extracellular domain of α7 nAChRs albeit their low sequence identity to nAChRs. 26 A systematic modification of the sequence of AChBP toward a closer sequence identity with the homomeric human α7 nAChR was conducted by Taylor and Nemecz and the corresponding structured chimeric proteins have been shown to resemble the binding characteristics of the native receptor. 27 From a structural point of view, the crystallographic structure of the complex between methyllycaconitine and one α7/Ac-AChBP (3SIO pdb entry) may therefore be used as a relevant model to investigate the interactions of antagonists to α7 nAChR. In the same vein, an α7-AChBP chimera was designed by Lin Chen and coworkers and X-ray crystal structures of the resulting pentamer and its complex with epibatidine (PDB code 3SQ6), an efficient agonist, were determined. 28 These crystallographic structures can therefore be used as useful templates in the design of α7 nAChRs selective drugs and the study of the behavior of ligands of special interest.

The aim of this study is to assess if multivalent nicotinic derivatives could function as modulators of the nAChRs.

Although the concept has previously been evoked, 29 there has been, to our knowledge, no report of such multivalent ligands in the literature. The binding properties of monoand multivalent nicotinic derivatives 1-8 were investigated on α7 nAChRs. The α7 nAChR subtype is a major neuronal receptor, abundantly present in the hippocampus and prefrontal cortex. It forms homopentameric structures displaying five potential nicotinic binding sites at the α subunits interfaces. 24,30 In the present work, the α7/Ac-AChBP x-ray structure (3SIO pdb entry) has been used for a preliminary molecular modeling investigation of the binding characteristics of some nicotinic derivatives (1, 2 and 3) in a first attempt to rationalize some of the experimental observations.

Results and Discussion

To simplify the chemical synthesis, the multivalent nicotinic analogues were designed racemic, from the monovalent nicotinic synthon altinicline (SIB-1508Y) that reached phase II for the treatment of Parkinson's disease. Its racemic synthesis is relatively straightforward, and the R,S enantiomers were shown to agonize multimeric neuronal nAChR bearing a β-subunit. 31 Preliminary results showed the SIB-1508Y behaved as an antagonist of α7 nAChR. This observation, together with the fact that α7 nAChRs are homopentameric, possessing five ACh-binding sites, allowed us to use this receptor as a model for our study of mutivalent altinicline compounds. The presence of a triple bond also offered the opportunity of a direct grafting to azide-armed scaffolds by copper-catalyzed azide-alkyne cycloaddition (CuAAC). A monovalent (1) and five divalent (2-6) compounds were designed from ethylene glycol (EG) cores of different lengths. EGs were selected for their high flexibility, hydrophilicity and because they do not interact non-specifically with proteins. 32 Indeed, EG conserve a conformational mobility even when the attached ligand is bound to the protein, 33 allowing the second ligand to reach a secondary binding sites. An heptavalent nicotinic 8 and its monovalent analogue 7 were also obtained from mono-and heptazido-β-cyclodextrins, a scaffold that we previously reported to design multivalent E.coli antiadhesives that prove effective in vitro and in vivo. We suggest that a chelate interaction between nAChRs and the multivalent nicotinic design would not be favoured (Figure 2A). The various structural data available at the atomic resolution from molluscean AChBP for example, Lymnaea stagnalis, 26 have shown that the nicotinic binding sites are buried from the solvent at the protomers interface. 35 As a consequence, a chelate interaction with a divalent nicotinic analog would require linkers embracing a high distance, of around 80Å. We roughly estimated that the required EG units to span this distance would be of 18 when considering the extended EG conformation. Compounds 5 and 6 designed in the present study would fit with this requirement. However, it was previously shown that the average PEG length in solution, which is proportional to the root mean square of the PEG molecular weight, is much shorter than the extended conformation and more representative of the effective PEG length. 36 For example, Kramer and Karpen found that a 2000 PEG (44 EG unit) was optimum for a divalent cGMP to activate binding sites of CNG channels separated by 39Å. 21 Instead of potentially targeting the receptor in a chelate fashion, we foresee that compounds 2-6 and 8 may interact with α7 nAChRs by other binding mechanisms previously observed with different classes of proteins. This includes the rebinding processes where the tethered ligands can "bind and jump" in a single site due to their high effective concentration (Figure 2C), 37 the interaction in both orthosteric and lower affinity peripheral sites simultaneously by a heterochelate-type interaction (Figure 2B), 16 and the formation of stabilized cross-linked receptors at the membrane surface (Figure 2D). 38 The alcohol group of commercially available EGs were first activated in O-mesylate group with methanesulfonyl chloride and triethylamine. 39 After substitution using sodium azide in DMF, the corresponding functionalized EGs 9-14 were obtained with good yields. 40 Racemic altinicline was obtained in nine steps from bromonicotinic acid, by a procedure previously described by Cosford and co-workers. 31 This protocol was reproducible, allowing the obtention of gram quantities of altinicline. The triple bond of the ligand was used as an anchor for its grafting to the azido-functionalized linkers 9-14 and to the β-cyclodextrin scaffolds 15 41 and 16 42 by a CuAAC protocol (Scheme 1). The compounds were dissolved in dioxane and water with sodium ascorbate and copper sulfate as catalysts. After a night at 60°C, the mono-(1 and 17), di-(2-6), and heptavalent (18) compounds were obtained with high yields. As generally observed with CuAAC, 1,2,3-triazol 1,4-regioisomers were formed in all cases as evidenced by the large Δ(δC-4-δC-5) values (>20 ppm) observed by 13 C NMR. 43 Prior to chromatography purification, the crude mixtures were washed with an EDTA solution in order to avoid copper contamination. The acetate groups of compounds 17 and 18 were deprotected on a basic resin (IRN78) to give 7 an 8. Finally, the compounds were purified by size-exclusion chromatography.

Biological assay

Nicotinic agonists and antagonists can compete for binding the orthosteric site located at the extracellular domain of the nAChR. It should be noted that subtle structural modifications of a given ligand can dramatically impact its selectivity profile for nAChR subtypes and the nature of the current response. In contrast to nicotine (Figure 1, X=H) which is the natural agonist of human α7 receptor, altinicline (X= -C≡CH) did not elicit inwards currents in Xenopus oocytes expressing α7. 31 Thus, the oocyte was exposed to ACh at its EC50 concentration (100 µM) for three times spaced by 4 min washing with buffer, afterwards, the oocytes were incubated for 1 min to a given concentration of a given multivalent compound and immediately shifted to a well containing a mixture of 100 µM ACh and the multivalent compound at the same concentration. The exposure of oocytes expressing α7 nAChRs did not elicite any changes at the oocyte membrane potential, however, when exposed to the mixture ACh-multivalent compounds, the evoked AChinward current was reduced when compared to the AChcontrol currents obtained at the beginning of each experiment. The multivalent nicotinic ligands showed a concentration dependent inhibition of ACh-induced currents on α7 nAChRs expressed by Xenopus oocytes. Here, the introduction of a triazol moiety formed during CuAAc leads to compounds with an electrophysiological response that may significantly differ between nicotine and SIB-15087. Thus, to quantify a multivalent effect, it is more relevant to compare multivalent 2-6 and 8 with their monovalent analogues 1 and 7, respectively, than with nicotine or SIB-15087 that do not carry the triazol ring. Compounds 1-8 were all shown to exert antagonist effects on human α7 nAChRs with inhibition constants ranging from 12 to 611 µM (Table 1). The multivalent effects were quantified by dividing the IC50 value of the compound by the IC50 of the corresponding monovalent reference (i.e.; 1 for 2-6 and 7 for 8). This relative potency number (Rp) was further corrected (divided) by the valency of the compound to lead to the relative inhibitory potency (RIP) factor representing the enhancement per nicotinic ligand (Table 1). Most of the compounds showed negative multivalent effects with RIP values < 1. However, we were pleased to see that positive multivalent effects were observed for divalent compounds 2 and 6, which are respectively 16 (RIP = 8) and 9 (RIP = 4.6) fold more potent than reference 1. These two compounds likely interact with their target by different binding modes as their spacer arm length differs significantly in size and because compounds 3-5 of intermediate EG numbers showed β values<1. Rebinding mechanisms (Figure 1A) or additional interactions in close binding site proximity (Figure 1B) may prevail for compounds 2 bearing short spacers. Interestingly, a gradual extension of the EG numbers (n = 7→11→19→27) leads to a gradual decrease of the IC50 values (611→300→160→21 µM). Similar evolutions of the affinity or inhibitory constants in function of EG length were previously reported with multivalent ligands bearing shorter EG spacers than required to bridge protein binding domains. 21,36 This suggests that bivalent compounds with higher numbers of EG units may perform better in terms of antagonists.

Molecular Modeling

Table 2 reports the energetic parameters (Glide energies and docking Scores) computed with Glide for altinicline derivatives 1-3. We have limited the theoretical study to these first three compounds because our aim was mainly to provide a possible interpretation to the highest potency of compound 2 in terms of IC50. For this investigation, we have therefore used 1 as the reference ligand and 3 as the next compound in the series.

Table 2. Relative energetic parameters (kJ/mole, using 1 as the reference compound) for the docking of altinicline derivatives 1-3 in the binding site of α7/Ac-AChBP.

The values have been calculated considering 1 as the reference compound; in other words, they correspond to relative energies (difference between energy of nicotinic derivative 2 or 3 and energy of derivative 1). Negative values are therefore indicative of an energy stabilization in comparison with 1 whereas positive values mean destabilization of the corresponding complex. It is worth noting that 16, 8 and 4 docking poses have respectively been obtained for 1, 2 and 3. For the calculation of the relative energies of Table 2, we have selected the pose obtained for 1, the closest to the experimental structure, that is to say having the closest orientation of the nicotine fragment with respect to the crystallographic structure of the nicotine-AChBP complex (1UW6). This pose corresponds to the one ranked in ninth position (from about 16 and 2 kJ/mol in energy and score, respectively). We were pleasantly surprised by the agreement between the docking trends and the experimental data. Thus, compound 2 appears stabilized in the binding site with respect to 1 whereas, as observed experimentally, the 3 derivative seems to have fewer interactions in the binding site. . Of course, these results have to be considered cautiously since they are limited to docking investigations but they nevertheless provide guidelines to rationalize the experimental trends. To complete this analysis, we then considered some geometric parameters of the interactions between the three nicotinic derivatives and their surroundings in the α7/Ac-AChBP 3D model designed. The corresponding values are reported in Supporting Information (Table S1). Figure 2 shows a global view allowing to catch the position of the binding site in α7/Ac-AChBP and the orientation of the three ligands. The orientation of the pyrrolidine ring of the nicotinic fragment of the three compounds is similar and close to the relevant nicotine-Ac-AChBP complex experimental structure (Figure 3). The difference of positioning becomes apparent for the 3 derivative, starting from the pyridine ring. It is clear from Figure 2 that the surroundings of the nicotinic fragments are very different in derivatives 2, 3 owing to the lengthening of the linker. It is however worth noting that the conformations adopted by the two derivatives 2 and 3 in the α7/Ac-AChBP environment are not extended. Figure 4 and5 show the surroundings of the two fragments of the three derivatives, their interactions within the α7-AChBP environment being reported in Table S1 of the Supporting Information. Cation π interactions of the pyrrolidine ammonium moiety of nicotine with the indole ring of Trp 143, a hydrogen bond between the N11H + and the main chain carbonyl of Trp 143 and an apolar contact between one carbon of the pyrrolidine ring (C9) and a carbon of the indole ring of Trp53 are key interactions of the complex between nicotine with Ls-AChBP in the 1UW6 crystallographic structure. 44 Table 1 and Figure 4a-c show that these contacts are conserved in the models predicted by our docking simulations for fragment 1 of the three derivatives, allowing to check the consistency of these theoretical data. Thus the π interactions between the electron deficient methylene groups of the pyrrolidine ring (C9 and C10) and the six membered ring of Trp147 are systematically obtained, as well as the key hydrogen bond interaction between the N11H + ammonium group and the main chain carbonyl of Trp147. The binding pocket of fragment 1 of the three derivatives shows a significant aromatic character, several aromatic groups of lateral chains of amino acid residues being involved in the contact with the ligands. These trends are in agreement with the known behavior of such an heteroaromatic fragment in the binding site of nAChRs. Table S1 shows that these variations of contacts for the three derivatives do not rationalize the docking energy parameters even if slight modifications are obtained.

Examination of the results predicted by the docking (Figure 5 and Table S1 in the Supporting Information) for the second fragment of the three derivatives provide guidelines to rationalize the relative stabilization of the three ligands in the α7-AChBP surroundings. Thus, it is worth noting that the ending OH group of 1 is anchored through a hydrogen bond interaction with the main chain carbonyl of Ser94 (d(O)H…O) of 1.9 Å). The greater stabilization of 2 compared to 3 can be explained by the number of contacts, which is significantly more important (4 compared to 2) and by their features. Thus, for the complex with 2, a hydrogen bond interaction between the electron deficient C12' methyl group of the pyrrolidine ring and the main chain carbonyl of Met126 is predicted (Figure 5a) together with a cation π interaction between the ammonium group of Lys143 and the triazole ring of fragment 2. Furthermore, two additional contacts involving the pyrrolidine moiety of fragment 2 (C8' and C12') are predicted by the docking. It is interesting to note that despite the difference in length of the linker between derivatives 2 and 3, the only amino acid residue involved in the binding of the second fragment of 3 is also involved in non-covalent interactions in 2. As already mentioned above, this feature highlights the fact that 3 adopts a rather compact conformation in the α7-AChBP environment instead of an extended one. Figure 5, which shows the binding of the two relevant fragments of the 2 and 3 derivatives allows a better comprehension of this behavior. In fact, in 3, the OH group of Thr139 is engaged in a hydrogen bond interaction with a nitrogen atom of the triazole ring (N17') whereas its main chain carbonyl accepts a hydrogen bond from the positively polarized C12' methyl group. On the whole, the results of the docking simulations are in line with the experimental trends and reinforce the hypothesis according to which the variation affinity observed can be attributed to a pseudochelate effect, the second fragment of the various derivatives interacting with amino acid residues from peripheral sites.

Conclusion

In summary, multivalent (di-and heptavalent) nicotinic derivatives were synthesized for the first time and interesting profiles on inwards currents with α7 nAChRs were observed. The spacer arm length of divalent 2-8 was shown to significantly impact on the extent of signal inhibition. Due to the large distance between the nicotinic binding sites of α7 (around 80Å), a chelate binding mode is discarded to explain the improved inhibitory activities that should be ascribed to subsite interactions as suggested from the docking investigations. The enhanced inhibitory effect observed for 2 (n=3) remains modest compared to the improved level of inhibition observed with other channel receptors. Nevertheless, these first hints should foster nAChR subtypes evaluations of a second generation of multivalent nicotinic receptors with enhanced valency (polymers) and spacer arm length.

Experimental details

All reagents were purchased from Acros Organics, Alfa Aesar, or Aldrich and were used without further purification. General method A (CuAAC).The azide derivative and the alkyne derivative (1.1 eq./azide function) were dissolved in dioxane (2mL/mmol). A solution of copper sulfate (0.2 eq./azide function) and sodium ascorbate (0.4 eq./azide function) in water (0.5mL/mmol) was added and the mixture was heated to 60°C until completion. The mixture was dissolved in DCM (50mL/mmol), washed with a solution of EDTA (50 mL/mmol) and the aqueous layer was extracted twice with DCM (2 x 50mL/mmol). The organic layer was dried over MgSO4, filtered, concentrated under reduced pressure, and purified.

General method B (deprotection).The protected compound (1eq.) was dissolved in MeOH/H2O (1:1, 1 mL/mmol). Amberlite resin IRN 78 OH-1.25 meq/mL (150 mg/mmol) was added, and the mixture was stirred overnight at rt. The resin was filtered off and washed with methanol and water. The solvent was evaporated under reduced pressure.

Compound 1. Obtained following the CuAAC method A. The crude product was chromatographed on a silica gel column with 9/1 (DCM/MeOH) as eluent to afford 1 (84% yield) as a white solid.

Molecular Modeling

In the present work, the crystallographic structure of the complex between methyllycaconitine and one α7/Ac-AChBP (3SIO pdb entry) was used for a preliminary molecular modeling investigation of the binding characteristics of altinicline derivatives with human α7 nAChR. The crystal structure of the α7/Ac-AChBP complex with the methyllycaconitine antagonist (3SIO pdb entry) were downloaded from the Protein Data Bank (www.pdb.org). 45 Addition of H atoms, adjustment of ionization states of ionizable residues were carried out using the Schrödinger's protein preparation wizard. All the crystallized water molecules were kept. The 3D ligand molecules, protonated using the LigPrep v3. The docking was performed using the Glide v6.3 program of the Schrodinger suite 2017-1. 46 The residues around 6˚A of the ligand were defined as the active site and were selected for the receptor grid generation. The standard-precision (SP) mode of the docking algorithm was employed to dock the altinicline derivatives. It is worth remembering that nAChRs are organized as pentamers; that is, there are five identical ligand binding sites and the ligand binds between the cleft of the two subunits. We docked the ligands in one interface (AB) of the pentameric structure, the five interfaces having close structural features.

Two-electrode voltage clamp electrophysiology on Xenopus laevis oocytes expressing human α7 nAChR

We purchased adult X. laevis females from the Biological Resource Center (University of Rennes, France). Animal care and experimental procedures were performed according to the Ethical Committee for Animal Experimentation C2EA-59 of Paris Center-South. Detailed experimental procedures for oocytes harvesting, heterologous expression of human α7 mRNA and manual two-microelectrode voltage clamp recordings (TEVC) were previously described. 47, 48 For automated TEVC we used a HiClamp system (MCS GmbH., Reutlingen, Germany): single oocytes are withdrawn from the 96-wells oocyte-microplate and put it in a silver wire basket that acts also as reference bath electrode. The oocytes were automatically clamped at a holding potential of -60 mV using glass microelectrodes (50-60 MΩ resistance) filled with a mix of 1 M KCl and 1 M ammonium acetate. The clamped oocyte was transferred into a well of the sample-microplate filled with 200 μL OR2 (88 mM Na Cl, 1 mM MgCl2, 5 mM Hepes; pH 7.6) containing 100 µM acetylcholine (Ach) or the tested molecules. The amplifier allows continuous recording even during the transfer of the oocyte from the washing-station to the test-well back and forth. The micro-stirrers disposed in each test-well ensured a rapid solution exchange. Doseresponse curves for agonist activation were analyzed using the equation: nH , where I is the measured agonistevoked current, [L] is the agonist concentration, EC50 is the agonist concentration that evoked half the maximal current (Imax), and nH is the Hill coefficient. For antagonist inhibition, current (I) values were normalized to the Imax value recorded from the same oocyte to yield fractional (%) response data. IC50 values were determined from dose-response curves by fitting to the equation: F = 1/[1 + ([X]/IC50) nH ], where F is the fractional response obtained in the presence of the inhibitor at concentration [X] and IC50 is the inhibitor concentration that reduced the ACh-evoked amplitude by half.

I = Imax[L] nH /(EC50 + [L])

34

 34

Figure 1 .

 1 Figure 1. Structure of (S)-nicotine, altinicline and the racemic mono-(1, 7), di-(2-6) and heptavalent (8) nicotinic analogues.

Figure 2 .Scheme 1 .

 21 Figure 2. Potential binding modes between di or tetravalent ligands and pentameric nAChRs. A) Chelate binding mode at two protomers interfaces; B) Binding of the cognate nicotinic ligand in a subsite located in proximity of the primary binding domain; C) effects of the high ligands concentration favouring a rebinding process; D) Formation of cross-linked aggregates at the membrane surface.

Figure 2 .

 2 Figure 2. Global view of the orientation and position of the three altinicline derivatives (1 (grey), 2 (green), 3 (pink)) in the α7/Ac-AChBP binding site.

Figure 3 .

 3 Figure 3. Detailed view of the first fragment of the three nicotinic derivatives 1 (grey), 2 (green), 3 (pink)) in the α7/Ac-AChBP binding site.

Figure 4 .

 4 Figure 4. Environment and interactions of fragment 1 of the three altinicline derivatives: a) 1 (grey); b) 2 (green); c) 3 (pink) in the α7/Ac-AChBP binding site. The numbering of the key interacting amino acid residues is indicated in a) using their one letter code for better clarity (W=Trp).

Figure 5 .

 5 Figure 5. Environment and interactions of the fragment 2 of: a) 2 (green); b) 3 (pink)) of the nicotinic derivatives in the α7/Ac-AChBP binding site. The numbering of the key interacting amino acid residues is specified using their one letter code for better clarity (K=Lys; M=Met; T=Thr).

  Dichloromethane, ethyl acetate and petroleum ether were distilled on a Buchi rotavapor R-220-SE. Reactions requiring anhydrous conditions were performed under argon. Column chromatography was conducted on silica gel Kieselgel SI60 (40-63 μm) from Merck, or on Silica cartridge from Interchim and eluted via a Puriflash 430 with an UV and ELSD detection. Thin layer chromatography (TLC): Merck Silica gel 60 F254 analytical plates, detection either with UV (254 nm) or dipped in a solution of potassium permanganate, ninhydrin and subsequently heated.1 H and13 C NMR spectra were recorded on Bruker Avance 300 spectrometer, on a Bruker Avance 400 spectrometer. The spectra are referenced to the solvent in which they were run. Chemical shifts (δ) are given in parts per million (ppm) and coupling constants (J) are given in Hz. Low resolution mass spectrometry (MS) was recorded on a ThermoFinnigan DSQII quadripolar spectrometer (coupled with a TracUltra GC apparatus) for Chemical Ionization (CI), on a ThermoFinnigan LCQ Advantage spectrometer for ElectroSpray Ionization (ESI). High resolution mass spectrometry (HRMS) was recorded on a ThermoFisher Scientific LTQ-Orbitrap spectrometer (for ESI+).

Table 1 .

 1 Inhibition Constants for ligands 1-8 on ACh-Evoked Nicotinic Currents in Xenopus Oocytes

a 95% confidence intervals are shown in brackets. Results from triplicates.

  0 module of the Schrodinger suite 2017-1 [Schrödinger Release 2017-1, Schrödinger, LLC, New York, NY, 2014] were then subjected to the ConfGen [Schrödinger Release 2017-1: ConfGen, version 2.7, Schrödinger, LLC, New York,NY, 2017] program to retrieve the lowest energy conformer for docking. Only the "eutomer" (S) configuration of the nicotine-like multivalent ligands were docked, based on the configuration of nicotine in the 1UW6 crystal structure.

H NMR (500 MHz, DMSO-d6): δ(ppm): 8.85 (s, 1H, H-8), 8.74 (s, 1H, H-9), 8.46 (s, 1H, H-12), 8.23 (s, 1H, H-6), 5.94-

Acknowledgments

This work was carried out with financial support from the Conseil Régional des Pays de La Loire (MAGGIC project), the Centre National de la Recherche Scientifique (CNRS) and the Ministère de l'Enseignement Supérieur et de la Recherche in France.

Supplementary Material

Supplementary data associated with this article can be found, in the online version.


domain: sdv chim

text:
Sahla Belhadj
email: belhadjsahla@hotmail.com

Henda Keskes

Cécile Apel

Fanny Roussi

Marc Litaudon

Olfa Hentati

Noureddine Allouche

Anti-diabetic and lipid-lowering effects of drimane sesquiterpenoids isolated from Zygogynum pancheri

Keywords: T2D, type 2 diabetes, STZ, streptozotocine, DMSO, dimethyl sulfoxide, FBG, Fasting Blood Glucose, Winteraceae, Zygogynum pancheri subsp. arrhantum, streptozotocin, hypoglycemia, hypolipidemia

Background: Recently, it has been shown that drimane-type sesquiterpenoids isolated from Zygogynum pancheri, a species native to New Caledonia, possessed significant α-amylase inhibitory activities.

Purpose:

To further explore their antidiabetic potential, we investigated the effect of 1β-O-(Ecinnamoyl)-6α-hydroxy-9epi-polygodial (D) and 1β-E-O-p-methoxycinnamoylbemadienolide (L), two of the most active compounds of the series, on diabetic model rats. Methods: Compounds D and L (2 mg kg/day) were daily and orally administrated for 30 days to streptozotocin (STZ) (150 mg/kg) induced male diabetic Wistar rats. Animals were allocated into five groups of six rats. Results: Comparatively to diabetic rats, treatments with D and L compounds were able to significantly (P < 0.05) decrease Fasting Blood Glucose (FBG) (70.15%, 71.02%), serum total cholesterol (46.27% and 39.38%), triglycerides (56.60% and 58.15%), creatinine (37.31% and 36.49%) and uric acid levels (67.76% and 69.68%), respectively. Compounds D and L also restored the altered plasma enzyme (aspartate aminotransferase, AST (47.83% and 43.20%), alanine aminotransferase, ALT (49.76% and 48.35%, alkaline phosphatase, ALP (72.78% and 73.21%)) and lactate dehydrogenase, LDH (47.95% and 53.93%) levels to near normal, respectively. Administration of Glymepiride, significantly (p<0.05) reduced FBG (73.94%) in STZ induced diabetic rats. Additionally, the compounds D and L exhibited inhibitory effects in vivo on lipase activity of diabetic rats (54.83% and 52.25%), respectively.

Conclusion:

The outcomes of this study suggested that these two drimanes could be considered as efficient hypoglycemic, hypolipidemic and antiobesity agents for diabetes management and its complications.

Introduction

Diabetes mellitus is a major and growing public health problem throughout the world, with an estimated worldwide prevalence in 2030 to 578 million and to 700 million in 2045 [START_REF] Saeedi | Global and regional diabetes prevalence estimates for 2019 and projections for 2030 and 2045: Results from the International Diabetes Federation Diabetes Atlas, 9th edition[END_REF]. It is a heterogeneous disorder with varying prevalence among different ethnic groups and is reported to constitute the 16th leading cause of global mortality [START_REF] Danaei | National, regional, and global trends in fasting plasma glucose and diabetes prevalence since 1980: systematic analysis of health examination surveys and epidemiological studies with 370 country-years and 2•7 million participants[END_REF], [START_REF] Sundaram | Ameliorating effect of chromium administration on hepatic glucose metabolism in streptozotocin-induced experimental diabetes[END_REF]. It is characterized by abnormalities in carbohydrate, lipid and lipoprotein metabolisms, which not only lead to hyperglycemia but also cause many complications, such hyperinsulinemia, hyperlipidemia, atherosclerosis and hypertension [START_REF] Ozkan | Potential Use of Turkish Medicinal Plants in the Treatment of Various Diseases[END_REF].

There are two main types of diabetes characterized by insulin defects: type 1 diabetes (T1D) defined by insulin deficiency and type 2 diabetes (T2D) characterized by insulin resistance [START_REF] Verma | Obesity and diabetes: An update[END_REF]. The conventional pharmacological treatments for diabetes have a number of limitations, such as adverse effects and high rates of secondary failure. However, medicinal herbs are expected to have a similar degree of efficacy without the troublesome side effects associated with conventional synthetic drug treatments. This explains the growing interest in herbal remedies for diabetes treatment [START_REF] Kim | Anti-diabetic effect of cinnamon extract on blood glucose in db/db mice[END_REF].

Previously, it has been reported that drimane sesquiterpenoids were the main constituents present in the bark and the leaves of various New Caledonian species of the genus Zygogynum, one of the two genera of the family Winteraceae occurring on mainland Australia and offshore islands [START_REF] Allouche | Cytotoxic sesquiterpenoids from Winteraceae of Caledonian rainforest[END_REF][START_REF] Keskes | Antioxidant and α-amylase inhibitory activities of extract and isolates from Zygogynum pancheri subsp. arrhantum[END_REF][START_REF] Fomekong Fotsop | Isolation and characterization of two new drimanes from Zygogynum baillonii and synthesis of analogues[END_REF]. Many of them, which have the particularity to bear a cinnamoyl ester moiety in position 1, have been isolated in large quantities and as such are considered to be good chemotaxonomic markers of this genus [START_REF] Allouche | Cytotoxic sesquiterpenoids from Winteraceae of Caledonian rainforest[END_REF]. The Zygogynum genus consists of about 41 species, of which 18 species are endemic to New Caledonia [START_REF] White | Winteracées[END_REF].

These drimane sesquiterpenoids are known to possess antifungal, antibacterial, molluscicidal, antifeedant, piscicidal, phytotoxic, cytotoxic [START_REF] Jansen | Occurrence, biological activity and synthesis of drimane sesquiterpenoids[END_REF]Groot, 2004, Allouche et al., 2009) and glucosidase inhibiting properties [START_REF] Harinantenaina | Cinnamacrins A-C, cinnafragrin D, and cytostatic metabolites with alpha-glucosidase inhibitory activity from Cinnamosma macrocarpa[END_REF].

Recently, [START_REF] Keskes | Antioxidant and α-amylase inhibitory activities of extract and isolates from Zygogynum pancheri subsp. arrhantum[END_REF] showed that drimane-type sesquiterpenoids bearing a cinnamoyl ester group isolated from a Zygogynum pancheri subsp. arrhantum EtOAc bark extract possessed significant α-amylase inhibitory activities ( [START_REF] Keskes | Antioxidant and α-amylase inhibitory activities of extract and isolates from Zygogynum pancheri subsp. arrhantum[END_REF]. Among the compounds isolated, the interesting to note that drimane D can spontaneously be rearranged under acidic conditions to give the lactone L [START_REF] Fomekong Fotsop | Isolation and characterization of two new drimanes from Zygogynum baillonii and synthesis of analogues[END_REF].

Therefore, in order to compare the effect of the two drimanes on diabetes. Their antidiabetic effects, were evaluated and compared in streptozotocin induced diabetic rats.

Materials and methods

Reagents

1β-O-(E-cinnamoyl)-6α-hydroxy-9epi-polygodial (D) and 1β-E-O-p-methoxycinnamoylbemadienolide (L) (Figure 1) have been isolated from the bark t of Zygogynum pancheri subsp.

arrhantum as previously described [START_REF] Allouche | Cytotoxic sesquiterpenoids from Winteraceae of Caledonian rainforest[END_REF].

Determination of lipase activity in vitro

The method was modified from the assay reported by [START_REF] Nakai | Inhibitory effects of oolong tea polyphenols on pancreatic lipase in vitro[END_REF] who used 4ethylumbelliferyl oleate (4-MU oleate) (Biolabo, France) as a substrate to measure the pancreatic lipase inhibitory activity of plant extracts. The assay consisted to mix the pancreatic lipase (Biolabo, France) solution with 4-MU oleate solution. The plate was immediately placed in the 37 • C preheating FL × 800 micro plate fluorescence reader to measure the amount of 4-methylumbelliferone released by lipase every minute for 30min at an excitation wavelength of 360 nm and an emission wavelength of 455 nm. The lipase inhibitory activity was determined by measuring the effect on the enzyme reaction rate after adding extracts, compared with the positive control: Glymepiride (Amarel 3 mg, Pharmacy of the medical herbalist Mohamed Dammak, Sfax-Tunisia) [START_REF] Belhadj | Metabolic impairments and tissue disorders in alloxan-induced diabetic rats are alleviated by Salvia officinalis L. essential oil[END_REF]. The inhibition percentage (IP) is expressed according the equation 1.

  control test

Absorbance -Absorbance IP = *100 1 Absorbance control Routine breeding of animals fed with a pellet diet (SICO, Sfax, Tunisia) and water was allowed ad libitum under strict hygienic conditions. Before initiation of experiment, the rats were maintained in the quarantine for a period of 7 days under standard abiotic defined by a temperature (22 + 2 • C), a relative humidity (45-55%) and a 12/12 h dark/light cycle. The Tunisian Ethical Committee for the care and use of laboratory animals approved the handling of the animals.

Induction of diabetes in experimental animals

Rats were injected intraperitoneally with a freshly prepared solution of STZ at a dose of 40 mg kg bw for 15 days [START_REF] Arokiyaraj | Antihyperglycemic effect of Hypericum perforatum ethyl acetate extract on streptozotocin-induced diabetic rats[END_REF].

At the end of the induction period, rats displaying glycosuria and hyperglycemia (> 2 g/l) were chosen for the experiments.

Experimental procedure

Five groups of rats, six in each received the following treatment schedule.

Group I: Normal rats (Ctr); Group II: STZ-induced-diabetic rats (Diab.); Group III: STZ-induced-diabetic rats treated with compound D which was administrated at 2 mg/kg bw (Diab+D); Group IV: STZ-induced-diabetic rats treated with compound L at 2 mg/kg bw (Diab+L); Group V: STZ-induced-diabetic rats treated with Glimepiride 3 mg/kg bw (Diab+Gly).

The two compounds were dissolved in drinking water and dimethyl sulfoxide (DMSO) at a concentration of 3%. After 1 month of treatment, the animals were sacrificed, and the blood was collected. Liver, kidney and pancreas were rapidly removed and cleaned of fat. All these samples were stored at -80 • C until used.

Biochemical variables and blood glucose determination

Fasting Blood Glucose (FBG) was assayed by glucose oxidase, using a commercial kit (Biomaghreb, Tunisia).

The analyses of plasma lipase and plasma triglycerides (TGs), total cholesterol (T-Ch), high-density lipoprotein cholesterol (HDL-c), low-density lipoprotein cholesterol (LDL-c) levels, were measured using the corresponding commercial kits (Biolabo, France).

Plasma levels of creatinine, aspartate aminotransferase (AST), alanine aminotransferase (ALT), alkaline phosphatase (ALP), lactate dehydrogenase (LDH), creatinine and uric acid were measured in frozen aliquots of plasma by standardized enzymatic procedures using commercial kits (Biolabo, France).

Statistical Analysis

Statistical analysis was performed using the statistical Package for Statistica. Data are presented as means ± Standard Error of the Mean (SEM). Statistical significance was assessed using one-way analysis of variance (ANOVA) followed by the Fisher test (at P < 0.05 the differences were considered statistically significant).

Results

Alpha-amylase activity in plasma and FBG

The results revealed that diabetes induced a significant (P < 0.001) increase in the α-amylase activity in the plasma by 42.19% which leads to an increase rate by 73.54 % of the FBG. In drimanes treated diabetic rats, the activity of this enzyme underwent a considerable improvement. Indeed, as compared to Diab group, the compounds D, L and glymepiride succeeded to reduce by 44.57%, 44.93% and 42.20% this enzyme activity, respectively. Hence, this potential inhibitory effect of the compounds D, L and Glimepiride was confirmed by a strong significant (P < 0.001) reduction of FBG by 70.15%, 71.02% and 73.94%, respectively (Fig. 2B).

The histology of the pancreas in diabetic rats (Fig. 3B) showed an atrophication of Langerhans islets and a reduction in the number of cells by comparison with Langerhans islets in control rats (Fig. 3A). In contrast, treatment of diabetic rats with drimanes and glymepiride (Fig. 3C, Fig. 3D andFig. 3E) has shown potential protection of pancreatic tissues with regeneration of Langerhans islet cells.

Effect of compounds D and L on pancreatic lipase levels

Figure 4 evidenced that compared to the control rats, there was a significant (P < 0.05) rise in the activity of pancreatic lipase in plasma of untreated diabetic rats by 52.68 %. However, the administration of compound D and compound L induced a considerable reduction in plasma lipase activity by 54.83% and 52.25%, respectively. As expected, a significant (P < 0.001) improvement (55.70%) in the drug treated group (Diab+Gly) was registered compared to untreated one (Diab).

Effect of compounds D and L on lipid profiles

In diabetic rats, significant (P < 0. However, the administration of compound D (Diab+D) to diabetic rats significantly (P < 0.001) reduced T-Ch (Fig. 5A) and TGs (Fig. 5A) levels in plasma (37.63% and 50%, respectively). This reduction was even better after the administration of the compound L (43.31% Fig. 5A and 53.79% Fig. 5B, respectively) as compared to Diab groups.

Even though the compound D (Diab+D) succeeded to significantly (P < 0.05) increase plasmatic HDL-c, the glymepiride and compound L did not exhibit significant differences with diabetic rats.

Effect of compounds D and L on liver parameters

Data in Table 1 showed an increase of the liver digestive parameters in the Diab group.

Comparatively to Ctr group, AST, ALT and ALP activities increased by 42.41%, 80.32% and 75.80% in the Diab group, respectively. The appearance of fat cells (Fig. 6B) in liver tissue confirmed the installation of the metabolic disorder. However, treated rats with compound D (Diab+D) and L (Diab+L), a notable protective effect was observed in hepatic function and tissue integrity. In fact, their administration to diabetic rats significantly decreased AST and ALT and ALP rates (57.27%, 51.42%, 65.85% respectively with the compound D and 47.66%, 45.08%, 53.78% respectively with the compound L).

Histological findings (Fig. 6C and6D) confirmed the positive effects of these molecules. The cinnamates administration to diabetic rats reduced the appearance of fat cells in the liver (Fig. 6B) and was as efficient as glymepiride (Fig. 6E).

Effect of compounds D and L on atherogenic index (LDH)

This current study indicated a significant (P < 0.001) elevated levels of LDH activities (48.14%), in plasma of Diab group as compared to Ctr group. This higher level of atherogenic index observed in Diab group was significantly reverted towards normal levels after treatment with naturel resources (Diab+D: 885.21UI/l ± 6.71 and Diab+L: 786.24UI/l ± 63.22) and Glimepiride (Diab+Gly: 1013.67UI/l ± 96.55) (Table 1).

Effect of compounds D and L on kidney

Results in table 1 showed that hyperglycemia (Diab group) induced kidney toxicity as indicated by a significant increase of creatinine and uric acid rates in plasma by 38.44% and 69.03%, respectively, when compared to Ctr group. In addition, kidney sections in Diab group showed a glomerular condensation (Fig. 6B). Interestingly, the administration of Compound D (Diab+D), L (Diab+L) and Glimepiride to diabetic rats reverted towards normal levels the above biochemical indices, improved Bowman's capsule thickness and cellular contiguity in glomerulus (Fig. 7).

Discussion

The purpose of this study was to evaluate for the first time the antihyperglycemic, hypolipidemic and anti-obesity effects of Sesquiterpene dialdehyde cinnamates in vitro and in vivo and to compare these effects between two particular compounds 1β-O-(E-cinnamoyl)-6αhydroxy-9epi-polygodial (compound D) and 1β-E-O-p-methoxycinnamoyl-bemadienolide (compound L).

Our results showed a protective effect of these natural substances against streptozotocin-induced diabetes in Wister rats through the modulation of α-amylase, lipase, glycemia, lipid profile, liver parameters and atherogenic index. In fact, the effect of the Sesquiterpene dialdehyde (D) cinnamates was comparable to that of lactone (L).

In our experiments, the STZ-induced diabetic model have been used for many reasons, including pancreatic swelling, causing degeneration in Langerhans islet beta cells and resulting in hyperglycemia and hyperlipidemia [START_REF] Akbarzadeh | Induction of diabetes by Streptozotocin in rats[END_REF]. The diabetic state caused by STZ is due to its generation of nitric oxide and free radicals that disrupt the mitochondrial function of beta cells [START_REF] Rakieten | Studies on the diabetogenic action of streptozotocin (NSC-37917)[END_REF].

Alpha-amylase is a key enzyme in the treatment of diabetes. It breaks down the polysaccharides' complex and consequently it limits the glucose uptake by inhibiting the intestinal enzyme, αglucosidase [START_REF] Saito | Effect of an alpha-glucosidase inhibitor (voglibose), in combination with sulphonylureas, on glycaemic control in type 2 diabetes patients[END_REF].

Both drimanes similarly inhibited α-amylase resulting in decreased FBG in STZ-induced diabetic rats.

Our findings confirm those of [START_REF] Keskes | Antioxidant and α-amylase inhibitory activities of extract and isolates from Zygogynum pancheri subsp. arrhantum[END_REF] who reported the inhibitory effect on α-amylase activity in vitro of these two compounds. Indeed, several researchers [START_REF] Qin | Cinnamon extract (traditional herb) potentiates in vivo insulin-regulated glucose utilization via enhancing insulin signaling in rats[END_REF][START_REF] Khan | Cinnamon improves glucose and lipids of people with type 2 diabetes[END_REF] have proved that the presence of cinnamoyl acid function in compound D and L was responsible for the efficient reduction of glycemia.

The cinnamoyl acid function might also inhibit hepatic glucogenesis by inducing glucose uptake or stimulating insulin secretion by the remaining pancreatic beta cells and generation of other cells as confirmed by the pancreas sections.

More, the anti-hyperglycemic cinnamaldehyde effect was significantly comparable to glymepiride.

To stimulate insulin secretion by pancreatic β-cells, this standard hypoglycemic drug is a conventional medicine to treat diabetes since many years. The similarity between blood glucose levels obtained after treatments by the drug and the two compounds, strongly suggest that the tested drimanes (D and L) in this study would act through the same mechanism as glimepiride, which is potentiation of insulin secretion and action by the remaining β cells in the Langerhans islets.

Pancreatic lipase is a key enzyme in the treatment of obesity, considered among the most important targets for the development of anti-obesity drugs. As a primordial lipolytic enzyme, pancreatic lipase is responsible for the hydrolysis of 50-70% of the total dietary fat in the intestinal tract [START_REF] Ha | Potential pancreatic lipase inhibitory activity of phenolic constituents from the root bark of Morus alba L[END_REF].

Our results showed a good lipase activity by both drimanes extracted from Zygogynum. pancheri in vivo. There are no research works that explain the mechanism of inhibition of these drimanes on the pancreatic lipase activity.

An important outcome of our study is that the compound D and L boosted pancreatic lipase to break down dietary fat so that it is well absorbed in the intestine.

Dyslipidemia is one of the secondary complications of diabetes that accompanies long-term diabetes [START_REF] Mooradian | Dyslipidemia in type 2 diabetes mellitus[END_REF]. This imbalance is characterized by an increase of T-Ch, TGs and LDL-c levels and a decrease of the good cholesterol (HDL-c) level [START_REF] Arvind | Diabetes & coronary artery disease[END_REF]. In diabetes, these abnormal blood lipid levels have been reported to be the intimate consequence of the excessive action of the hormone-sensitive enzyme, the pancreatic lipase [START_REF] Adedeji | Role of lipase in glucose and lipid metabolisms[END_REF].

The insulin deficiency and consequently the mobilization of free fatty acids highly contribute to increase the lipid levels in diabetes [START_REF] Schofield | Diabetes Dyslipidemia[END_REF]. Generally, in mammals, the pancreas, liver, kidney and adipose tissue are the key organs simultaneously concerned by the metabolic syndrome of diabetes.

The liver plays a very important role in the maintenance of a balanced glycemia through the regulation between glyneocogenesis (decreasing glycemia) and the glycogenolysis gluconeogenesis increasing glycemia), and in the post-absorption state [START_REF] Sherwin | Role of Liver in Glucose Homeostasis[END_REF].

In the current work, the carbohydrate metabolism disorder observed in STZ-induced diabetic rodents, lead to the increase of ALT, AST and LDH blood levels, creatinine and uric acid.

LDH, ALT and AST are enzymes mainly found in the cytosol and the cell membrane of liver cells. These are the markers often used to evaluate liver function and more specifically hepatotoxicity. Their high records reflected the liver dysfunction in Diab group. Similarly, [START_REF] Larcan | Light and electron microscopic study of hepatic lesions in the course of hyperlactatemia in diabetic patients (author's transl)[END_REF] reported that liver tissue was necrotic in diabetic patients. The disruption of the hepatocyte membranes and the leakage of LDH and transaminases out of the cells to the blood circulation are the main reasons for their elevated plasmatic concentrations [START_REF] Bouché | The cellular fate of glucose and its relevance in type 2 diabetes[END_REF]. This clearly shows the hepatotoxicity as an evidenced action of hyperglycemia.

Oral administration of the natural compounds D and L and glymepiride to the Diab group, has shown an improvement of the LDH and transaminases to normal levels. These latter allowed a clear disappearance of fat lipid droplets in the liver parenchyma with well-sharpened borders between contiguous hepatocytes.

The excessive excretion of urine called polyuria, the increase of creatinine and uric acid levels compared to healthy being observed during the assays, were among the expected complication symptoms of the stably installed diabetes.

As major excretory organs in diabetes, the kidneys were profoundly affected by hyperglycemia forcing them to an intensive filtration from the micro vessels, and a failure of the nephric tubules to reabsorb the glucose again.

The excess of glucose passes in the urine (glycosuria) thus proving an osmotic diuresis and consequently the polyuria [START_REF] Andrianesis | The renal effects of SGLT2 inhibitors and a minireview of the literature[END_REF].

The creatinine content in the internal medium reflects a balance between the production of waste and elimination by the glomeruli filtration [START_REF] Andrianesis | The renal effects of SGLT2 inhibitors and a minireview of the literature[END_REF].

Treatment with compound D and L succeeded to reestablish normal rates of creatinine and uric acid translating the positive effect of these two molecules on the reuptake of glucose molecules in the proximal glomeruli tubules. This significant decrease was equal to the one noticed in Ctr and Diab+Gly groups.

The profits on excretion function brought by the tested drimanes were also confirmed by renal sections where they proved to be as efficient as glymepiride and similar to control group.

Conclusion

The novelty of this study consisted in comparing the effects of the drimanes D and L isolated from Zygogynum pancheri: O-(E-cinnamoyl)-6α-hydroxy-9epi-polygodial (D), a compound that is quantitatively transformed into 1β-E-O-p-methoxycinnamoylbemadienolide (L) in acidic medium.

Gathered the results of this work demonstrated that the compounds D and L revealed a protective effect against STZ-induced diabetes in Wistar rats.

Administered at 2 mg kg -1 bw, these two molecules (dialdehyde D and lactone L) showed comparable effects and acted as antidiabetic and anti-obesity agents by catching the digestion of carbohydrates and lipids through the pancreatic α-amylase and pancreatic lipase inhibition, respectively, thus decreasing glycemia.

The tested molecules could modulate the liver parameters, renal biomarker concentrations and atherogenic index.

Both molecules were not toxic in diabetic rats. Indeed, the histological sections of pancreas, liver and kidney showed a normal-like protection of the tissue targeted organs thus reducing or avoiding As it stands, this study is conclusive but the mechanism of actions of these two molecules is still unknown.

  drimanes 1β-E-O-p-methoxycinnamoyl-bemadienolide, 1β-O-(E-cinnamoyl)-6α-hydroxy-9-epi-polygodial, 1β-O-(E-cinnamoyl)-6α-hydroxypolygodial, and 1β-O-E-cinnamoylpolygodial exhibited highest α-amylase inhibitory activities, while cinnamolide, a structurally close analogue of 1β-E-O-pmethoxycinnamoyl-bemadienolide but without the ester moiety, was found inactive. For this reason and given that several of the above-mentioned compounds have been isolated in large quantities we undertook an in-depth study of the antidiabetic properties of 1β-O-(E-cinnamoyl)-6α-hydroxy-9-epipolygodial (denoted D) and E-O-p-methoxycinnamoyl-bemadienolide (denoted L). Moreover, it is

  001) increases of plasmatic T-Ch (Fig 5A) and TGs (Fig 5B) levels (34.40% and 60.19%, respectively) as well as a significant decrease of plasmatic HDL-c level (20.22 %, Fig 5C) were observed as compared to Ctr group.

  their toxicities. Corroborating results handled by this research, it was possible to stipulate that 1β-O-(Ecinnamoyl)-6α-hydroxy-9epi-polygodial (D) and 1β-E-O-p-methoxycinnamoylbemadienolide (L) represented bioactive supplements for oral use to treat diabetes and metabolic complications including obesity.


Kaouther Moussa
email: kaouther.moussa@gipsa-lab.fr

Mirko Fiacchini
email: mirko.fiacchini@gipsa-lab.fr

Mazen Alamir
email: mazen.alamir@gipsa-lab.fr

Robust Optimal Control-based Design of Combined Chemo-and Immunotherapy Delivery Profiles

Keywords: Biomedical control, robust optimal control, uncertain dynamic systems, generalized moment problem, cancer treatment scheduling, chemotherapy, immunotherapy

Robust optimal control-based design of a combined chemo and immunotherapy delivery profiles

Kaouther Moussa, Mirko Fiacchini, Mazen Alamir

INTRODUCTION

Control design for biological systems is a very promising topic in the sense that robust control theory can help to handle the uncertainties which are due to the system dynamics complexity, the different neglected phenomena and the lack of knowledge on model parameters.

It is well known that biological phenomena, in particular tumors, are highly uncertain systems. Both evolution and treatment effects are patient-dependent. In the context of cancer treatment, control theory can help to design new drug delivery schedules and to combine effectively many therapies, in the presence of parametric uncertainties, in order to achieve better treatment performances.

The advances in genetics that have taken place recently led to considerable progress in experimental and clinical immunology [START_REF] Eftimie | Mathematical Models for Immunology: Current State of the Art and Future Research Directions[END_REF] and many researches on modeling the immune system dynamics had been carried out. Since the mathematical modeling of the entire immune system can be a very complex task, researchers focus on the elements of the immune system that are known to be significant in controlling the tumor growth [START_REF] De Pillis | Mixed Immunotherapy and Chemotherapy of Tumors : Modeling , Applications and Biological Interpretations[END_REF]. The readers interested in tumor-immune interactions modeling can refer to De [START_REF] De Pillis | Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls[END_REF],De [START_REF] De Pillis | Mathematical Model Creation for Cancer Chemo-Immunotherapy[END_REF]), d'Onofrio et al. (2012), [START_REF] Eftimie | Mathematical Models for Immunology: Current State of the Art and Future Research Directions[END_REF] and references therein. The model considered in this paper describes the interactions between two populations, tumor cells and immune effector cells. It also includes explicitly two therapies delivery, cytotoxic chemotherapy and immunostimulation. This model takes into account the chemotherapy-induced loss of tumor cells and incorporates the benefical effects of the immune system on controlling the tumor growth.

Cancer treatment usually implies handling state and input (drugs amounts) constraints, some nonlinearities, many uncertainties and optimality issues. It is definitely the collection of all complexity ingredients in terms of control design. We can find in the literature many works regarding the application of optimal control methods on cancer treatment problems. For instance, [START_REF] Ledzewicz | Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis[END_REF], [START_REF] Schättler | Robustness of Optimal Controls For a Class of Mathematical Models for Tumor Anti-Angiogenesis[END_REF]d'Onofrio and[START_REF] Onofrio | A Bi-parametric Model for The Tumour Angiogenesis and Antiangiogenesis Therapy[END_REF], where optimal protocols for anti-angiogenic treatment were investigated, or De [START_REF] De Pillis | Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls[END_REF] who designed quadratic and linear controls for a tumorimmune interactions model with chemotherapy delivery. However, robust control is not yet extensively studied for cancer treatment. We can cite for example, [START_REF] Alamir | Optimal Control Applications And Methods Robust Feedback Design For Combined Therapy of Cancer[END_REF], where a robust feedback scheme is proposed to schedule antiangiogenic treatment combined with chemotherapy. [START_REF] Kovacs | Model-based angiogenic inhibition of tumor growth using modern robust control method[END_REF] applied an H ∞ based robust control to the same model.

In this paper, we propose to formulate robust optimal control problems in the moment optimization framework by lifting a polynomial nonlinear dynamical system to the infinite-dimensional space of measures. Since the problem is reformulated in the space of probability measures, it is straightforward to define states and parameters as random variables characterized by their probability distributions.

The resulting infinite-dimensional problem is solved using truncations known as finite-dimensional semidefinite (SD) hierarchies [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF], providing a converging sequence of lower bounds on the optimal solution.

In Section 2, we present the dynamical model used for numerical simulations. Section 3 recalls the main key points of the generalized moment problem for optimal control. The optimal control problems to be solved and the numerical results are presented in Section 4. Finally, in Section 5, we discuss advantages and limitations of this approach and we set some perspectives for future works.

DYNAMICAL MODEL

In this paper, we will consider a modified version of the two dimensional model presented in d [START_REF] Onofrio | On the Dynamics of Tumor-Immune System Interactions and Combined Chemo-and Immunotherapy[END_REF], which describes the interaction dynamics between the tumor and the immune system. This model had been intensively used in the literature in order to investigate its equilibria and propose some optimal control strategies. For instance, [START_REF] Sharifi | Multiple model predictive control for optimal drug administration of mixed immunotherapy and chemotherapy of tumours[END_REF] proposed a multiple model predictive control scheme to design chemo-and immunotherapy injection schedules.

We propose to replace the Gompertzian tumor growth model in d [START_REF] Onofrio | On the Dynamics of Tumor-Immune System Interactions and Combined Chemo-and Immunotherapy[END_REF] with a logistic one

µ C x 1 1 - x 1
x ∞ , which leads to the following polynomial dynamics :

ẋ1 =µ C x 1 - µ C x ∞ x 1 2 -γx 1 x 2 -κ X x 1 u 1 , ẋ2 =µ I x 1 -βx 1 2 x 2 -δx 2 + κ Y x 2 u 2 + α, (1)
where x 1 and x 2 denote, respectively, the number of tumor cells and the density of effector immune cells (ECs), u 1 and u 2 are respectively, the delivery profiles of a cytotoxic agent and an immunostimulator. Model (1) has two locally asymptotically stable equilibria.

The macroscopic malignant equilibrium is (x m , y m ) (735.9, 0.032) and the benign one is (x b , y b ) (34.98, 0.53).

It is important to notice that the treatment performance depends highly on the initial conditions, since there is a coexistence of macro-and microscopic equilibria.

OVERVIEW ON GENERALIZED MOMENT PROBLEM FOR OPTIMAL CONTROL

In this section we will provide an overview on the basic concepts and computational tools for solving optimal control problems by resorting to the generalized moment problem approach. This method, developed by Lasserre [START_REF] Lasserre | Optimisation globale et théorie des moments[END_REF], is based on the fact that polynomial optimization problems (thus, a class of nonconvex finite dimensional problems), are equivalent, under mild assumptions, to linear problems in the space of measure, that are infinite dimensional. Although the infinite dimensional nature makes the latter problems hardly manageable, relaxations can be obtained, providing a converging sequence of lower bounds on the global optima, under some compactness assumptions. Therefore, generating and solving such sequence of relaxations, referred to as Lasserre hierarchy [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF], allows to approach the exact solution of the original polynomial optimization problem with arbitrary precision.

This approach has been recently extended to optimal control problems with a polynomial structure and bounded constraints [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF], for which sequences of convex problems can be determined to obtain suboptimal solutions converging to the exact optimal control. Moreover, as the linear infinite dimensional problems are defined over the space of measures, which can be naturally employed to model probability distributions, this approach allows to address and solve optimal control problems involving probability distributions as states and parameters.

Consider the polynomial optimal control problem inf

u(•) T 0 L(t, x(t), u(t)) dt + Φ (x(T )) s.t. ẋ(t) = f (t, x(t), u(t)), x(t) ∈ X, u(t) ∈ U, t ∈ [0, T ], x(0) ∈ X 0 , x(T ) ∈ X T , (2)
where

x ∈ R n is the state, u ∈ R m is the input, functions f : R × R n × R m → R, L : R × R n × R m → R and Φ :
R n → R are polynomials and the constraints sets X 0 , X, X T and U are compact basic semi-algebraic sets, i.e. defined as the union of finitely many intersections of closed polynomial superlevel sets. Under some additional assumption on the convexity of the set f (t, x, U ) for all t and x, the optimum of ( 2) is the same as the optimum of a linear infinite dimensional optimization problem defined over the space of probability measures, given below.

For this, we briefly introduce the concept of measure and moments, the reader interested in a more detailed formalization is referred to the related literature, e.g. Lasserre (2010) and [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF]. Given a compact set X ∈ R n and the Borel σ-algebra B(X), that is a particular set of subsets of X containing all the open subsets of X, a Borel measure µ on X is a function that associates a real value to every element of B(X). A measure is nonnegative if it takes only nonnegative values, is a probability measure if it is nonnegative and µ(X) = 1. An example of a positive measure is the Lebesgue measure, that associates the surface to sets in R 2 , the volume to sets in R 3 etc., while the Dirac measure δ z (x) with z ∈ X is a probability measure that assigns the value 1 to every subset of X containing z and 0 otherwise. The spaces of measures and positive measures are denoted M(X) and M + (X), respectively. Measures can also be defined as the space of linear functionals acting on the space of functions that are continuous on X, i.e. by the action they have over the elements of the dual space C(X) through integration

v, µ = X v(x)µ(dx),
for all v ∈ C(X). The following linear problem in the space of measures inf

µ0,µ,µ T L, µ + Φ, µ T s.t. [0,T ]×X×U ∂v(t, x) ∂t + ∇ x (v(t, x)) f (t, x, u) dµ = v, µ T -v, µ 0 , ∀v ∈ C 1 ([0, T ] × X) µ 0 ∈ M + ({0} × X 0 ), µ T ∈ M + ({T } × X T ) µ ∈ M + ([0, T ] × X × U ), 1, µ 0 = 1,
(3) is infinite dimensional, has uncountable many constraints and has the same optimum value as the original optimal control problem (2), under mild assumptions, see [START_REF] Lasserre | Nonlinear optimal control via occupation measures and LMI relaxations[END_REF]. Such a problem remains highly complex and unmanageably large, however, Lasserre hierarchy [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF] of relaxed LMI problems can be determined to obtain suboptimal solutions that converge, under certain compactness and convexity assumptions, to the optimal solution of the original optimal control problem. To obtain the relaxations, one has first to consider the relation between the measure µ 0 , µ and µ T and their moments. Given x ∈ R n and σ ∈ N n , the moment of order σ ∈ N n of µ ∈ M(X) is defined as :

y σ = X x σ µ(dx) = x σ , µ , (4)
where x σ = n k=1 x σ k k . LMI conditions can be given in terms of the moments of µ that are equivalent to the constraint µ ∈ M + (X), conditions that still involve infinite dimensional matrices which are functions of the infinitely many variables y σ for all σ ∈ N n . The relaxations consist, in practice, in considering the matrix structures obtained by appropriately truncating the vector of moments to a finite maximal degree (r) and imposing in (3) constraints over polynomials of a finite maximal degree in spite of all v ∈ C 1 ([0, T ] × X). This leads to a hierarchy of finitedimensional semidefinite programming problems whose solutions converge to the solution of the optimal control problem as the relaxation degree grows.

The particularly interesting feature of this approach is the fact that, even in case of deterministic dynamical systems, the initial state as well as the final one and the state along trajectories, are dealt with by defining measures on the state space, see (3). The same holds for the input. For instance, if x 0 = x(0) ∈ X 0 is a singleton, then the initial measure µ 0 in (3) should be imposed by fixing, for all σ ∈ N n , its moments given as :

t ι x σ , µ 0 = x σ 0 if ι = 0 0 if ι ∈ N + \{0}
Therefore, the polynomial optimization method based on measures is particularly suitable for dealing with states and inputs that are characterized by probability distributions, by simply imposing the moments of the related probability density functions.

In the particular case under study, we aim at designing a robust optimal control for a dynamical model describing the tumor growth, the parameters of which are supposed to be not perfectly known. This lack of knowledge can be modeled through uncertain parameters characterized by probability distributions, with compact support. Then, in practice, it is sufficient to define an extended state containing both tumor and immune cell populations and the uncertain parameters, i.e. µ C and α, see (1), and to impose their time invariant characteristic through the dynamics μC (t) = 0 and α(t) = 0. Thus, supposing that η(µ c ) and ν(α) denote the probability distributions of parameters µ C and α, the optimal control problem (3) to be solved should have as initial condition

µ 0 = δ 0 (t) × δ x1(0) (x 1 ) × δ x2(0) (x 2 ) × η(µ c ) × ν(α)
, imposed through moments of the initial measure.

NUMERICAL RESULTS

Nominal optimal control problem

Similarly to d [START_REF] Onofrio | On the Dynamics of Tumor-Immune System Interactions and Combined Chemo-and Immunotherapy[END_REF] and [START_REF] Sharifi | Multiple model predictive control for optimal drug administration of mixed immunotherapy and chemotherapy of tumours[END_REF], we assume that the initial state of the system dynamics ( 1) is (x 10 , x 20 ) = (600, 0.1), we also consider that the maximum drug dose is 1 for both chemotherapy and immunotherapy. Furthermore, we add constraints on the immune cells density and the number of tumor cells in order to ensure the compactness of the state set X. Another constraint on the final tumor size is imposed in order to drive the tumor to the benign region. The nominal (i.e. considering nominal values of model parameters) optimal control problem that we propose to solve for t ∈ [0, 60] is the following: min

u1(•),u2(•) J(x 1 , x 2 , u 1 , u 2 ) s.t. ẋ1 = µ C x 1 1 - x 1 x ∞ -γx 1 x 2 -κ X x 1 u 1 , ẋ2 = µ I x 1 -βx 1 2 x 2 -δx 2 + α + κ Y x 2 u 2 , x 1 (0) = 600, x 2 (0) = 0.1, x 1 (60) 100, 0 u 1 1 , 0 u 2 1, 0 x 1 780 , 0 x 2 5, t ∈ [0, 60]. ( 5
)
The cost J is chosen according to the objectives that one seeks to achieve. It can contain many terms such as final states, integrals of state trajectories and control inputs, with different penalties in order to achieve a trade-off between the different control objectives. Problem (5) can be reformulated in the framework of moment optimization via GloptiPoly 3 [START_REF] Henrion | Gloptipoly 3: Moments, optimization and semidefinite programming[END_REF], as explained in Section 3, and can be solved using YALMIP and the semidefinite programming solver MOSEK. The control inputs are approximated, based on the knowledge of their moments, using Christoffel-Darboux kernel, for more details, see [START_REF] Marx | Tractable semi-algebraic approximation using Christoffel-Darboux kernel[END_REF]. For practical reasons, time and states trajectories are scaled to [0, 1], therefore, the control inputs presented in this paper are computed for scaled dynamics.

Let's consider for instance the minimization of the cost J 1 = x 1 (60). Figure 1 shows the approximations of control inputs that we obtained after solving the reformulated problem corresponding to (5) with J = J 1 . The evolution of state trajectories with those controls is presented in Figure 2, we can see that the tumor burden decreases slowly to reach the final value that lies in the benign region. Assuming that we want the tumor size to decrease faster, we can minimize J 2 = x 1 (t)dt. The approximated control inputs are presented in Figure 3, we can notice that the chemotherapy profile is aggressive and persistent, this is due to the choice of the cost which considers only the minimization of the integral of x(t). Such controls are not allowed practically because of the high toxicity of the cytotoxic agent. Figure 4 shows that the state corresponding to the tumor cells, x 1 , goes to 0 faster than in Figure 2. Furthermore, we can see that the immune cells density goes up rapidly to reach very high values (thus, violating the imposed constraint 0 x 2 5).

Since chemotherapy has damaging side effects on the human body, it is common to frame an optimal control problem so that the total amount of drugs is minimized [START_REF] De Pillis | Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls[END_REF]. It is also important to take a look at the evolution of the immune system, because the immune-weakening has damaging effects on the human body. Thereby, one can easily notice that the choice of the cost J, to be minimized, is very important in order to meet the control objectives. Now, we propose to minimize J 3 = x 1 (60)+0.4 x 2 (t)dt+ 0.01 u 1 (t)dt + 0.01 u 2 (t)dt. As we can see in Figure 5, penalizing the control inputs integrals allows to reduce considerably the injected drugs amounts. In Figure 5, we show the graphs in the time interval [0,5] to emphasis the differences between the two profiles, since for t ∈ [5, 60], u 1 (t) = 0 and u 2 (t) = 0. Figure 6 shows that the states converge to the benign equilibrium at around 30 days. Therefore, the control profiles approximated by minimizing J 3 allow to satisfy perfectly the standard control objectives, since they drive the states to the benign equilibrium (x b , y b ). Furthermore, we can notice that those drug injection profiles minimize rapidly the tumor while maintaining a relatively strong immune system. Even though controls in Figure 5 satisfy standard objectives in the context of nominal optimal control, we will show that when the dynamics are subject to parameters uncertainties, those controls will not meet the goals set in the optimal control problem.

Let's assume that µ C ∼ N (0.5599, 0.1) truncated in [0, 1.1198] and α ∼ N (0.1181, 0.05) truncated in [0, 0.2362].

Remark 1: The considered distributions are not based on practical knowledge of the system parameters, they are chosen only to illustrate the problem of handling parametric uncertainties. The robust schedules will be designed considering truncated distributions in order to satisfy compactness conditions.

Figure 7 presents 100 Monte-Carlo simulations with random values of µ C and α (the random selection is carried out according to their corresponding probability distributions). It shows that there is a probability of 19% for the states to converge to the malignant equilibrium (x m , y m ) (i.e. leading to patients death). Therefore, it is crucial to consider the potential uncertainties on model parameters.

Robust optimal control problem

Let's extend system (1) to the following dynamics:

ẋ1 = µ C x 1 - µ C x ∞ x 1 2 -γx 1 x 2 -κ X x 1 u 1 , ẋ2 = µ I x 1 -βx 1 2 x 2 -δx 2 + κ Y x 2 u 2 + α, μC = 0, α = 0. (6)
The state augmentation in (6) allows to characterize µ C and α by their probability distributions, as explained in Section 3. Similarly to problem (5), one can reformulate the robust optimal control problem with dynamics (6) by including the moments of the parameters distributions.

Figure 8 presents a comparison between nominal and robust injection schedules, approximated after minimizing the cost J 3 in the nominal case and E (J 3 ) in the robust case, since we have a flow of trajectories, generated by the parameters distributions . We can notice that similarly to the nominal profiles, the robust ones are also single doses injected at the begining of the treatment. However, we can see that the robust profiles use more amounts of drugs which highlights the importance of taking into account parametric uncertainties description. Figure 9 presents 100 Monte-Carlo simulations using the approximated robust injection profiles. We can notice that the probability of convergence to the malignant equilibrium has been reduced from 19%, using the nominal profiles, to 8%, in the case of robust schedules.

Cost-based performance comparison

Problem (5) can be written in a compact form as follows : In order to effectively compare the performance of nominal and robust schedules, we write the equivalent problem of (7) as : min

min u1(•),u2(•) J(x 1 , x 2 , u 1 , u 2 ) s.t. g(x 1 , x 2 , u 1 , u 2 ) ≤ 0. ( 7
u1(•),u2(•) J(x 1 , x 2 , u 1 , u 2 ) + ρ max(g(x 1 , x 2 , u 1 , u 2 ), 0), (8)
where ρ ∈ R is arbitrary big.

Using the asymptotic equivalence between ( 7) and ( 8), we computed the costs corresponding to nominal and robust profiles, based on Monte-Carlo simulations that we carried out for both schedules. Table 2 shows that the mean and variance of the costs corresponding to robust schedules, are considerably less than those of the nominal costs. This is mainly due to the excessive number of constraints violations that occur when applying nominal controls. Table 3 presents a comparison between the computational times of the nominal optimal control problem (OCP) and the robust one. We notice a considerable difference in the computational cost, it is mainly due to the increase of the problem dimension, after performing dynamics extension to solve the robust OCP. Increasing the relaxation order r allows to have better approximations on the moments, however, it increases the problem dimension and therefore, the computational time.

CONCLUSION AND DISCUSSION

We presented in this paper preliminary results on the application of moment optimization theory to schedule cancer treatment. We highlighted the importance of taking into account parametric uncertainties in the optimal control problem. Furthermore, we designed robust and optimal combined chemo-and immunotherapy injection profiles that allow to meet specific objectives.

The moment optimization approach can be very promising for many applications, since it allows to reconstruct optimal injection schedules for a class of nonlinear systems with parametric uncertainties considerations. However, it may have some limitations, mainly the restriction on polynomial dynamics and the limited dimension (state and control variables) that can be handled. Although the required computational time is high in the case of solving a robust OCP, in some applications, it remains crucial to guarantee robust performances.

Finally, furture work will be focused on exploring the uncertainties effects of other model parameters on cancer schedules treatment design, including uncertainties on initial states and investigating the consequences of adding a minimal immune cells density constraint in the OCP.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Control inputs chemotherapy (u 1 ) and immunotherapy (u 2 ), for J 1 .

Fig. 3 .Fig. 4 .

 34 Fig.3. Control inputs chemotherapy (u 1 ) and immunotherapy (u 2 ), for J 2 .

Fig. 5 .

 5 Fig. 5. Control inputs chemotherapy (u 1 ) and immunotherapy (u 2 ), for J 3 .

Fig. 7 .

 7 Fig. 6. States trajectories for J 3 .

  Fig. 8. Chemo-and immunotherapy schedules (robust and nominal), for J 3 .

Fig. 9 .

 9 Fig. 9. Monte-Carlo tests on the robust schedules

Table 1

 1

			contains the
	definitions of the model parameters and their numerical
	values.
	Table 1. Numerical values and definitions of
	the parameters used in model (1) (d'Onofrio
		et al., 2012)
	Parameter	Definition	Numerical value
	µ C	tumor growth rate	0.5599 •10 7 cells/day
	µ I	tumor stimulated	0.00484 day -1
		proliferation rate
	α	rate of immune	0.1181 day -1
		cells influx
	β	inverse threshold	0.00264
	γ	interaction rate	1 •10 7 cells/day
	δ	death rate	0.37451 day -1
	κ X	chemotherapeutic	1 •10 7 cells/day
		killing parameter
	κ Y	immunotherapy	1 •10 7 cells/day
		injection parameter
	x∞	fixed carrying capacity	780 •10 6 cells

Table 2 .

 2 Statistics of the normalized costs (nominal and robust)

		Mean Variance
	Nominal cost	0.20	0.14
	Robust cost	0.12	0.07

Table 3 .

 3 Computation times on hp EliteBook 2.60GHz Intel Core i7

		Relaxation order r	Computation time (mn)
	Nominal OCP	8	1
	Robust OCP	8	62