Frédéric Justaud

Hippolyte Paysant

Louis Bastien Weiswald

Abdelghani Jebahi

Marie Jouanne

Nicolas Elie

Anne Sophie Voisin-Chiret

Thierry Roisnel

Clément Orionne

Nicolas Levoin

Laurent Poulain

René Grée

Synthesis and biological evaluation of FJ-809, a compound originally described as MIM1 and inhibitor of the anti-apoptotic protein Mcl-1

Keywords: cancer, anti-apoptotic proteins, Mcl-1, MIM1, thiazoles, bioassays, molecular modelling

The development of inhibitors of anti-apoptotic proteins, such as Mcl-1, is currently a very active area in the field of cancer research. One of the very first reported inhibitor of Mcl-1 was the molecule MIM1, but we have demonstrated recently that the structure of this compound had to be revised from 2 to the derivative 1 (FJ-809). In this paper we first develop a strategy to prepare unambiguously molecules such as 1 with a thiazol-3(2H)-yl)imino core, instead of the [2(3H)thiazolylidene]hydrazine previously found in MIM1 (2). Next a series of biological studies have been performed on 1, using IGROV1-R10 ovarian cancer cells as models, and they have been complemented by Fluorescence Polarisation Assays. These studies demonstrated that the new compound FJ-809 (1) was devoid of any significant activity on Mcl-1, contrary to 2. Then molecular modelling and molecular dynamic studies have been performed in order to elucidate the differences between FJ-809 and MIM1 in their interaction with the Mcl-1 protein.

Introduction

Cancers are responsible for nearly 10 million deaths in 2020, 1 and overcoming resistance of cancer cells to conventional treatment is a major challenge in oncology. As one of the hallmarks of cancer, 2 deregulation of apoptosis is frequent, both in carcinogenesis and chemoresistance, thus the development of innovative pro-apoptotic strategies appears as a key issue in cancer research. Bcl-2 family proteins have critical roles in the control of apoptosis and its anti-apoptotic relatives i.e. Bcl-xl, Mcl-1, Bcl-w and Bfl1/A1, are overexpressed in various hematopoietic and solid tumors. 3 They are known to contribute to cancer development and therapy resistance and development of strategies aiming at inhibiting these antiapoptotic proteins is therefore an intensive area of research in medicinal chemistry. This led to the design of small molecules able to target these proteins and the development of Venetoclax (ABT-199), the first-in-class targeted medicine designed to selectively inhibit Bcl-2 and approved by the Food and Drug Administration (FDA). 4 Considerable efforts are currently being made to develop inhibitors of Mcl-1 since its overexpression is often detected in many tumor types and is associated with tumorigenesis, poor prognosis and drug resistance, 5a including resistance to Bcl-2/Bcl-xL inhibitors. Several compounds [S64315 (Servier), AMG176 (Amgen) and AZD5991 (AstraZeneca)] have entered phase 1 clinical trials involving patients with hematological malignancies. 6 Among Mcl-1 inhibitors already described, 7 MIM1 (2, Figure 1) is a small molecule first identified by virtual screening of a 71296 members chemical library and then by a stapled peptide-based Fig. 1 Structures of FJ-809 [START_REF] Ferlay | Global Cancer Observatory: Cancer Today[END_REF] and MIM1 [START_REF] Hanahan | [END_REF] competitive screen. Further, this molecule was proved to selectively bind and inhibit the BH3-binding groove of Mcl-1 8 and it has often been used as a reference in biological studies. We have demonstrated recently that the structure 1, initially proposed for MIM1, was not exact and we have revised it to compound 2 (Fig. 1). Further, we have designed and synthetized a focused library of analogues of 2. Then in vitro and in cellulo biological experiments, combined with extensive molecular modelling studies, allowed us to propose preliminary Structure-Activity Relationships for these series of molecules. 9 To complete previous studies we decided to prepare, by an independent route, the compound with the structure attributed to 1 and study its chemical and biological properties, in comparison with MIM1 [START_REF] Hanahan | [END_REF]. Thus, the goal of this paper is to report an unambiguous synthesis of compound 1, that we will name FJ-809 to avoid any confusion with corrected MIM1 [START_REF] Hanahan | [END_REF]. Besides showing analytical and physical properties clearly different from 2, this molecule FJ-809 exhibit also different biological properties as described in the second part of this paper. A rationale for such differences will be proposed based on molecular modelling and molecular dynamic studies.

Results and discussion

-

Chemical Synthesis

The reaction of thiosemicarbazides with -halogeno ketones is the classical method employed for the preparation of this type of thiazole-derived heterocycles. As indicated in Scheme 1, it could lead to two types of structures, either a thiazol-3(2H)yl)imino core 6 if the nitrogen of the hydrazone group is concerned by the cyclization (pathway 1) or a [2(3H)thiazolylidene]hydrazone 7 if the terminal nitrogen is involved (pathway 2). We have shown that, in the case of MIM1 synthesis where R 2 = aryl groups, the second pathway is involved affording the type-7 structures. This was the background for the correction of MIM1 structure. On the other hand, it has been described in the literature that when R 2 is H or an alkyl group the pathway 1 is occurring, 10 and similar result was obtained if R 2 is a COPh. 11 Based on this information, for the synthesis of compound FJ-809 with the originally assigned structure 1, we designed a strategy using a Boc group in R 2 position and this was performed according to Scheme 2:

Scheme 2 Synthesis of the target molecule FJ-809.

The known thiourea 9, obtained by condensation of cyclohexylisothiocyanate 8 with aqueous hydrazine, was protected as the N-Boc derivative 10 in a moderate yield. The heterocyclization was performed by reaction of 10 with chloroacetone in an ethanol-chloroform mixture at 65°C, affording in 88% yield the derivative 11, as an hydrochloride salt. Deprotection by zinc bromide in dichloromethane at room temperature, followed by an alcaline wash gave the neutral intermediate 12 in 95% yield. Acidic conditions have to be avoided since they led to degradation. The direct condensation of 12 with 2,3,4trihydroxybenzaldehyde proved to be unsuccessfull and in most cases, no reaction took place. Different Lewis acids, molecular sieves, drying agents and/or different solvents were tried without success and increase of the temperature led to degradation. Therefore, we replaced the triphenol intermediate by the known, 12 trisOTMBS protected derivative 13. Then the condensation went on smoothly, affording 14 in 98% yield. The final deprotection step was not straightforward but after trying various conditions, it was found that using an excess of KHF2.H2O in methanol at room temperature, the target molecule 1 (FJ-809) could be isolated in 92% yield. Thus, FJ-809 was obtained under mild conditions in 6 steps and 19% overall yield from 8.

All these molecules have spectral and analytical data (see experimental section and ESI) consistent with their structure. Most informative are the comparison of the NMR data of 1 (FJ-809) with those (already reported) 9 of the MIM1 molecule 2.

The 1 H NMR spectrum of a 1:1 mixture of both compounds (Fig. 2) clearly illustrates these differences. In particular, the signals of the protons of the CH=N-groups are very different: they appear at 10.2 ppm for FJ-809 and at 8.2 ppm in the case of 2.

Further, contrary to the results obtained in the case of 2, no NOESY correlation was observed between the methyl group on the heterocycle and the CH and the CH2 protons of the cyclohexyl substituent of FJ-809 (see ESI). This was anticipated, taking into account the long distance between these groups in such a thiazol-3(2H)-yl)imino core.

Fig. 2 1 H NMR spectrum of a 1:1 mixture of 1 (FJ-809) and 2 (MIM1).

In spite of many experiments, it was not possible to grew crystals of FJ-809 suitable for X-Ray analysis. However, starting from the key intermediate 12, we could prepare two analogues as indicated in Scheme 3.

Scheme 3 Synthesis of the analogues 15 and 16.

The reaction with 2,3,4-trimethoxy-benzaldehyde and with the 4-nitrobenzaldehyde gave the molecules 15 and 16 in 76 and 71% yields, respectively. The physical and analytical data of these derivatives are similar to those of FJ-809 and consistent with the indicated structures. Further, in the case of 15, they are clearly different from those of 17 (see for instance the 1 H NMR spectrum of the 1:1 mixture of 15 and 17, Fig. 1 in ESI), the trimethoxy analogue of 2 that we have described recently. 9 Finally, in the case of 16, the structure was unambiguously confirmed by X-Ray diffraction analysis as indicated in Figure 3. 13 Therefore, we have designed a new strategy, based on the use of a N-Boc-protected thiourea, which should allow the preparation of many heterocyclic molecules bearing a thiazol-3(2H)-yl)imino core.

-Biological studies

We compared the activity of the two molecules 1 and 2 on IGROV1-R10 ovarian cancer cells, whose survival is dependent on both Bcl-xL and Mcl-1, as described previously. 9 In these cells, the concomitant inhibition of these two anti-apoptotic proteins leads to massive cell apoptotic death. In contrast, none of these events were observed after exposure to FJ-809 (1) (Fig. 4), even at a high concentration (25µM, Fig. S3 in ESI), when combined to ABT-737, suggesting that this molecule is inactive.

Comparison of FJ-809 and MIM1 by molecular simulations

As we already described, 9 the binding mode of FJ-809 and MIM1 are very similar (Fig. 6), with conserved interaction features (van der Waals contacts for the core, hydrogen bonds for the triphenol). Even if the ligand chemical structure was erroneous, the binding mode suggested by Cohen et al, 8a based on docking and NMR chemical shifts is still valid for both inhibitors. In particular, amino acids located at the C-t end of helix α3 (K234, D236), the N-t part of helix α4 (D241), in the crevice between the two helices (M231, L235, F270), and D256, are either in direct interaction or very close to the ligands. The main difference is that the methyl of FJ-809 is partially solvent exposed, while the methyl of MIM1 is more favourably buried in the hydrophobic cavity. In agreement with this observation, interaction energy of the docking pose calculated by a MM-GBSA method showed that MIM1 has clearly a higher affinity than FJ-809:

-56.6 vs. -34 kcal/mol. By running molecular dynamics experiments, we noticed that the ligands were movable within the binding pocket, particularly the triphenol group which appeared to be able to form different hydrogen bonds with several amino acids. To reach a more quantitative understanding of the inhibitors interaction with Mcl-1, we performed extensive molecular simulations, by means of five independent µs molecular dynamics of each complex (see SI Fig. S22). Contrary to our intuition, MIM1 appeared less stationary than FJ-809, or rather to fluctuate between a few alternate binding modes (Fig. 7). In Fig. 7A for example, root-mean-square deviation (RMSD) of the ligand showed recurrent fast shifts of the binding mode.

During another experiment (Fig. 7B), the initial (t < 100 ns) and late sates (t = 800-900 ns) seemed similar, while the ligand was stable in another binding mode for most the simulation (t = 150-650 ns). On the contrary, FJ-809 appeared static (Fig. 7C). A detailed analysis of the trajectories suggested that this is probably due to the solvent exposure of the FJ-809 methyl group, which creates an indirect additional attachment point in the pocket. This is the consequence of its first hydration shell which disrupts the hydrogen bond network of water molecules above the binding site and freeze them as a clathrate, as illustrated by the radial distribution function (RDF) between the methyl and water molecules. This behavior is not possible for MIM1's methyl group since it is buried in the binding pocket (Fig. 8).

Hence the flexibility of MIM1, for which the entropic cost of binding is reduced, as well as the perturbation of the solvent near FJ-809 explain the higher affinity of MIM1.

Experimental

Chemistry

General informations

NMR spectra were performed on a Bruker AVANCE 300, AVANCE 400 ( 1 H at 300 or 400 MHz; 13 C at 75 or 100 MHz). Solvent peaks were used as reference values with CDCl3 at 7.26 ppm for 1 H NMR and 77.16 ppm for 13 C NMR, with MeOD at 4.78 and 3.31 ppm for 1 H NMR and 49.00 ppm for 13 C NMR, Chemical shifts δ are given in ppm, and the following abbreviations are used: singlet (s), broad singlet (bs), doublet (d), doublet of doublet (dd), doublet of doublet of doublet (ddd), triplet (t), triplet of doublet (td), quadruplet (q), doublet of quadruplet (dq), and multiplet (m).

IR spectra were recorded on a Perkin Elmer apparatus using a Universal ATR Sampling Accessory and spectra were analysed with spectragryph 1,2,4 version. High resolution mass spectra were recorded in the Centre Régional de Mesures Physiques de l'Ouest, Rennes (CRMPO), on a Maxis 4G. Reaction courses and product mixtures were routinely monitored by TLC on silica gel (precoated F254 Merck plates), and compounds were visualised under a UVP Mineralight UVGL-58 lamp (254 nm) and with p-anisaldehyde or KMnO4/. Column chromatography was performed using silica gel 60 (40-63 mm, 230-400 mesh). Solvents were used as received from commercial sources. Reagents were purchased from Sigma-Aldrich, Alfa Aesar and Acros. All corresponding products showed 1 H, and 13 C NMR spectra in agreement with the assigned structures.

Compound 8 was purchased from Alfa Aesar supplier and used as received.

N-cyclohexylhydrazinecarbothioamide (9)

This compound 9 has been prepared according to reported procedures and its spectral data are in agreement with literature. 14

tert-butyl 2-(cyclohexylcarbamothioyl)hydrazinecarboxylate (10)

A solution of Boc2O (358 mg, 1.64 ml, 1.0 equiv.) in THF (8.0 mL) was added to a solution of thiosemicarbazide 9 (346 mg, 1.64 ml, 1.0 equiv.) in THF (8.0 mL) cooled at 0 °C. After complete dissolution within 2 min, the solution was stirred for 12 h. Then, saturated aqueous NaHCO3 solution (15 mL) was added and the mixture extracted with DCM (3 x 50 mL). The combined organic extracts were washed with 0.1 N aqueous HCl (3 x 100 mL), dried over MgSO4 and the volatiles removed by rotary evaporator. The crude mixture was purified by column chromatography using Silica 60-M with DCM/MeOH: 98/2 as eluant to afford 9 (178 mg, 39% yield) of colourless liquescent solid. FTIR (thin film) νmax/cm -1 3333, 3295, 3139, 2926, 2852, 1624, 1522, 1499, 1232, 931. 1 H NMR (300 MHz, Chloroform-d) δ 7.88 (s, 1H), 6.97 (s, 1H), 6.67 (d, d, J = 8.3 Hz, 1H), 4.34 -4.13 (m, 1H), 2.12 -1.99 (m, 2H), 1.81 -1.59 (m, 4H), 1.50 (s, 9H), 1.47 -1.34 (m, 2H), 1.29 -1.15 (m, 2H). 13 C NMR (75 MHz, CDCl3) δ 181. 63, 155.44, 82.74, 53.35, 32.51, 28.11, 25.40, 24.78. HRMS (ESI): calcd for C12H23N3O2SNa: [M + Na] + : m/z 296.14032. Found 296.1402 (0 ppm).

(Z)-tert-butyl (2-(cyclohexylimino)-4-methylthiazol-3(2H)-yl) carbamate hydrochloride (11)

A mixture of the thiosemicarbazide 10 (124 mg, 0.454 mmol, 1.0 equiv.) and chloroacetone (72 µL, 0.910 mmol, 2.0 equiv.) in 7:3 ethanol/chloroform mixture (2.0 ml) was refluxed for 6 h. Then, the mixture was concentrated under vacuum. The foam obtained was dissolved in diethyl ether before evaporation to remove excess of chloroacetone and the process repeated 3 times. The brown foam obtained was used without further purification (140 mg, 88%). FTIR (thin film) νmax/cm -1 3298, 3134, 2928, 2853, 1713, 1532, 1448, 1366, 1220, 1182. 1 H NMR (300 MHz, Chloroform-d) δ 11.07 (s, 1H), 10.52 (d, J = 7.6 Hz, 1H), 6.33 (s, 1H), 3.27 -3.02 (m, 1H), 2.13 (s, 3H), 2.09 -2.01 (m, 1H), 1.96 -1.50 (m, 6H), 1.46 (s, 9H), 1.34 -1.13 (m, 4H). 13 C NMR (75 MHz, CDCl3) δ 167. 67, 153.75, 139.96, 97.52, 83.77, 58.84, 31.61, 30.73, 24.67, 24.40, 13.07. HRMS (ESI): calcd for C15H26N3O2S: [M + H] + : m/z 312.17402. Found 312.1740 (0 ppm).

Synthesis of 2-cyclohexylimino-4-methyl-2,3-dihydro-1,3thiazol-3-amine (12)

8.2 H 2 O at 4 Å 0.12 H 2 O at 4 Å To a stirred solution of 11 (765 mg, 2.17 mmol, 1.0 equiv.) in DCM (25 mL) was added by portion zinc bromide (2.44 g, 10.85 mmol, 5.0 equiv.) with rapid solubilisation and decoloration of the solution, further stirred for 3 h at RT. Then, water (20 mL) was added followed by NaOH solution, (2.0 N, 25 mL). The organic layer was collected, dried over MgSO4 and concentrated under vacuo to yield 12 (434 mg, 95% yield) of a pale yellow oil. FTIR (thin film) νmax/cm -1 3244, 3096, 3094, 2922, 2849, 1639, 1610, 1586, 1397, 967. 1 H NMR (300 MHz, Chloroform-d) δ 6.33 -5.87 (m, 3H), 3.23 -3.08 (m, 1H), 2.34 (d, J = 1.3 Hz, 3H), 2.09 (d, J = 13.2 Hz, 2H), 1.95 -1.57 (m, 4H), 1.37 -1.21 (m, 4H). 13 C NMR (75 MHz, Chloroform-d) δ 157. 97, 136.91, 87.68, 63.41, 33.69, 25.84, 25.29, 14.23. HRMS (ESI): calcd for C10H17N3SNa: [M + Na] + : m/z 234.10409. Found 234.1037 (1 ppm).

Synthesis of 2,3,4-tris((tert-butyldimethylsilyl)oxy)benzalde hyde (13)

t-Butyldimethylsilylchloride (1.64 g, 10.9 mmol, 3.0 equiv.) was added dropwise over 5 min to a solution of 2,3,4trihydroxybenzaldehyde (0.5 g, 3.25 mmol, 1.0 equiv.), and diisopropylethylamine (2.5 ml, 14.6 mmol, 4.5 equiv.) in DMF (20 mL) and stirred for an additional 16 h at RT. A solution of NaHCO3 (100 mL) was added and the reaction mixture was extracted with Diethyl Ether (3 x 100 ml). The combined organic layers were dried on MgSO4 and concentrated under vacuo. The solid obtained was washed with cold methanol (2 x 5 mL) and cold diethyl ether (10 ml) prior to be dried under vacuo to give 1.81 g of product. Recristallisation from MeOH afforded 13 (1.37 g, 85 % yield) as a colorless cristalline powder. 1 H NMR (300 MHz, Chloroform-d) δ 10.21 (d, J = 0.8 Hz, 1H), 7.38 (d, J = 8.7 Hz, 1H), 6.62 (dd, J = 8.6, 0.8 Hz, 1H), 1.05 (s, 9H), 1.00 (s, 9H), 0.94 (s, 9H), 0.27 (s, 6H), 0.14 (s, 6H), 0.12 (s, 6H). HRMS (ESI): calcd for C25 H48 O4 Na Si3: [M + Na] + : m/z 519.27527. Found 519.2759 (1 ppm).

Synthesis of (2Z,NE)-2-(cyclohexylimino)-4-methyl-N-(2,3,4tris((tert-butyldimethylsilyl)oxy) benzylidene)thiazol-3(2H)-amine (14)

In a small Schlenk tube equiped with a magnet stirrer, 12 (247 mg, 1.0 mmol, 1.0 equiv.) and 2,3,4-tri-(tbutyldimethylsilyloxy)benzaldehyd (13) (502 mg, 1.01 mmol, 1.0 equiv.) were dissolved in 10 mL of a 1:1 mixture of MeOH/CHCl3. The orange mixture was heated at 60 °C for 2 h and the volatiles were removed under vacuo. The crude mixture was purified by column chromatography on neutral alumina with cyclohexane/diethyl ether 80:20 as eluent to give of product 14 (677 mg, 98% yield) as a brownorange solid. FTIR (thin film) νmax/cm -1 2929, 2856, 1630, 1591 ; 1302, 1253, 1066, 826, 801, 757 . 1 H NMR (300 MHz, Chloroform-d) δ 10.25 (s, 1H), 7.40 (d, J = 8.7 Hz, 1H), 6.57 (d, J = 8.7 Hz, 1H), 5.57 (s, 1H), 2.82 (s, 1H), 2.20 (d, J = 1.3 Hz, 3H), 1.90 -1.48 -1.14 (m, 10H), 1.06 -0.99 (m, 17H), 0.97 (s, 9H), 0.25 (s, 6H), 0.15 (s, 6H), 0.13 (s, 6H). 13 C NMR (75 MHz,CDCl3) δ 152.68,150.20,148.37,147.46,138.81,137.61,123.10,118.40,115.00,90.38,65.62,33.18,26.65,26.4,26.15,26.00,25.01,18.83,18.55,18.52,16.07,. HRMS (ESI): calcd for C35H64N3O3Si3SNa : [M + Na] + : m/z 690.39708. Found 690.3971 (0 ppm).

Synthesis of 4-((E)-(((Z)-2-(cyclohexylimino)-4-methylthiazol-3(2H)-yl)imino)methyl)benzene-1,2,3-triol 1 (FJ-809)

To a stirred solution of 14 (463 mg, 0.637 mmol, 1.0 equiv.) in MeOH (6.0 mL) was added at once KHF2.2H2O (248 mg, 3.18 mmol, 5.0 equiv.) and left at RT for 5 h. The crude mixture was filtrated over a coton plug to discard the precipitate and rinsed with cold MeOH. Then, volatiles were removed under vacuo at r. t. and the crude mixture was purified by a short column chromatography on silica gel eluting with Chloroform/MeOH 99:1 to give product 1 (FJ-809) (203 mg, 92% yield) as a brown orange solid. FTIR (thin film) νmax/cm -1 3124, 3005, 2931, 2856, 1580, 1209, 1038, 1005, 726. 1 H NMR (300 MHz, Methanol-d4) δ 9.80 (s, 1H), 6.85 (d, J = 8.5 Hz, 1H), 6.46 (d, J = 8.5 Hz, 1H), 6.16 (d, J = 1.4 Hz, 1H), 3.03 (m, 1H), 2.18 (d, J = 1.4 Hz, 3H), 2H),2H),1H),J = 19.2,9.7 Hz,5H). 13 C NMR (75 MHz, Methanol-d4) δ 165. 65, 158.17, 151.17, 148.70, 136.43, 132.86, 122.61, 110.54, 108.26, 95.78, 61.42, 32.10, 25.14, 24.51, 13.31. HRMS (ESI): calcd for C17H22N3O3SNa: [M + Na] + : m/z 348.13764. Found 348.1377 (0 ppm).

(2Z,NE)-2-(cyclohexylimino)-4-methyl-N-(2,3,4-trimethoxy benzylidene)thiazol-3(2H)-amine (15)

In a Schlenk tube with a magnetic stirrer, 12 (32 mg, 0.12 mmol, 1.0 equiv.) and 2,3,4-trimethoxy benzaldehyde (20 mg, 0.095 mmol, 0.7 equiv.) were dissolved in 1.0 mL of MeOH. The mixture was heated at 75 °C for 12 h, filtrated through a coton plug at r.t. and volatiles evacuated under vacuo. The crude mixture was purified by column chromatography eluting with Cyclohexane/Diethyl Ether 1:1 to afford of 15 (30 mg, 92% yield) as an orange oil. FTIR (thin film) νmax/cm -1 2925, 2951, 1593, 1284, 1222, 1093, 804, 776 1 H NMR (400 MHz, Chloroform-d) δ 10.81 (s, 1H), 7.70 (d, J = 8.8 Hz, 1H), 6.75 (d, J = 8.8 Hz, 1H), 5.60 (s, 1H), 3.94 (s, 3H), 3.91 (s, 6H), 2.94 (t, J = 8.1 Hz, 1H), 2.22 (s, 3H), 1.90 -1.24 (m, 10H). 13 C NMR (101 MHz, CDCl3) δ 154. 85, 153.43, 152.33, 144.64, 142.18, 137.33, 124.24, 123.05, 120.16, 108.01, 90.83, 64.26, 61.79, 610.95, 56.18, 33.29, 225.80, 24.26, 15.85. HRMS (ESI): calcd for C20H27N3O3SNa: [M + Na] + : m/z 412.16653. Found 412.1670 (1 ppm).

(2Z,NE)-2-(cyclohexylimino)-4-methyl-N-(4-nitrobenzylidene) thiazol-3(2H)-amine (16)

In a Schlenk tube with a magnet stirrer, 12 (44 mg, 0.17 mmol, 1.0 equiv.) and 4-nitrobenzaldehyde (20 mg, 0.135 mmol, 0.7 equiv.) were dissolved in 1.0 mL of MeOH. The mixture was heated at 75 °C for 2 h, filtrated through a coton plug at r.t. and the volatiles were evacuated under vacuo. Product 16 (33 mg, 71% yield) was obtained and subjected to 1 H NMR analysis which gave satisfactory purity. Crystals suitable for X-ray analysis were grown by layer/layer diffusion of an ethyl acetate solution of product 16 into chloroform one. FTIR (thin film) νmax/cm -1 2923, 2920, 1604, 1565, 1510, 1340, 772, 684. 1 H NMR (300 MHz, DMSO-d6) δ 10.65 (s, 1H), 8.28 (d,J = 8.8 Hz,2H),2H),6.18 (d,J = 1.4 Hz,1H),2.89 (s,1H),2.20 (d,J = 1.3 Hz,3H),10H). 13 C NMR (75 MHz, DMSO) δ 152. 84, 148.17, 145.21, 142.41, 136.63, 128.10, 124.56, 93.98, 64.54, 33.19, 25.92, 24.36, 15.58

Biology Material &Method

FPA assay

Carboxy-fluorescein labelled peptide (5-FAM-EDIIRNIARHLAQVGDSMDR-NH2) and Bim-WT peptide (H-Ahx-DMRPEIWIAQELRRIGDEFNAYYAR-OH) were purchased from GENEPEP and used without further purification. was produced in collaboration with Biomolecules laboratory of Pierre and Marie Curie University. 15 FPA measurements were carried out in 96-well, black, flat-bottom plates (Greiner Bio-One) using the Bioteck microplate reader Synergy 2. All assays were conducted in assay buffer containing 20mM Na2HPO4 (pH 7.4), 50mM NaCl, 2μM EDTA, 0.05% Pluronic F-68. 16 For IC50 determination, compounds were diluted in DMSO in an 8-point, serial dilution scheme, added to assay plates and were incubated with 100nM Mcl-1 for two hours. Then, to measure inhibition of Mcl-1/FAM-Bid interaction, 15nM 5-FAM-Bid peptide was added and plates were incubated for two hours at room temperature. For each experiment, a negative control containing Mcl-1 and 5-FAM-Bid peptide (equivalent to 0% inhibition), and a positive control containing Mcl-1, 5-FAM-Bid peptide, and 10μM Bim-WT peptide (equivalent to 100% inhibition), were included on each assay plate. Each point was duplicated and each experiment was performed in two biological replicates. The change in polarization was measured and used to calculate an IC50 (inhibitor concentration at which 50% of bound peptide is displaced), by fitting the inhibition data using GraphPad Prism 6 software to a sigmoïdal, 4PL, X is log(concentration). This was converted into a binding dissociation constant (Ki) according to the formula described by Nikolovska-Coleska et al. 17 Cell culture and treatment IGROV1-R10 cell line was established as described previously 18 from the IGROV1 cell line, itself kindly provided by Dr. Jean Bénard (Institut Gustave Roussy, Paris, France). It was grown in RPMI1640 (Gibco) medium supplemented with 2 mM Glutamax™, 25mM HEPES (4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid), 10% decomplemented FBS (Fetal Bovine Serum) (Gibco) and 33mM sodium bicarbonate (Gibco) and maintained in a 5% CO2 humidified atmosphere at 37 °C. Ovarian cancer cell line was certified mycoplasma-free thanks to a MycoAlert test. Cells were seeded in 25 cm2 flasks 24 hours before their treatment and exposed to MIM1 or FJ-809 as single agents or in combination with ABT-737 or S63845 for 24 hours at the indicated concentrations.

Nuclei staining by DAPI

After treatment, both detached and adherent cells were pooled after trypsinization, applied to a polylysine-coated glass slide by cytocentrifugation and fixed with a solution of ethanol/chloroform/acetic acid (6:3:1). The preparations were then incubated for 15min at room temperature with 1μg/ml DAPI solution (Boehringer Mannheim-Roche, Mannheim, Germany), washed in distilled water, mounted under a coverslip in Mowiol (Calbiochem) and analysed under a fluorescence microscope (BX51, Olympus, Rungis, France). Then, a closing morphological operator was applied to regroup the clusters of small fragments. Each cluster was counted as one fragmented nucleus.

DNA content analysis by flow cytometry

After treatment, both detached and adherent cells were pooled after trypsinization, washed with 1X PBS and fixed with ethanol 70°. Cells were centrifuged at 2000 rpm for 5 min and incubated for 30 min at 37°C in PBS to allow the release of lowmolecular weight DNA (characteristic of apoptotic cells). Cell pellets were stained with propidium iodide (PI) using the DNA Prep Coulter Reagent Kit (Beckman-Coulter, Villepinte France). Samples were thereafter analysed using a Gallios flow cytometer (Beckman-Coulter) and cell cycle distribution was determined using Kaluza acquisition software (Beckman-Coulter).

Western blot analyses

After treatment, both detached and adherent cells were pooled after trypsinization. Cells were rinsed with ice-cold PBS, suspended in a lysis buffer [RIPA : NaCl 150mM, Tris (pH 8) 50mM, Triton X100 1%, PMSF 4mM, EDTA 5mM, NaF 10mM, NaPPi 10mM, Na3OV4 1mM, aprotinin 0.5µL/mL and 4.6 mL ultra-pure water] and incubated on ice for 30min. Lysates were collected after centrifugation (13200g, 10 min, 4°C) and protein concentrations were determined using the Bradford assay (Bio-Rad, Hercules, USA). 25µg of protein were separated by SDS-PAGE on a 4-15% gradient polyacrylamide gel (Invitrogen, Cergy-Pontoise, France) and transferred to Hybond-PVDF membranes (Amersham, Orsay, France). After blocking nonspecific binding sites for 1 hour at RT by 5% (w/v) non-fat dry milk in TBS with 0.1% (v/v) Tween20 (T-TBS), the membranes were incubated overnight at 4°C with the following rabbit monoclonal antibody: PARP, caspase-3 (Cell Signaling Technology, Ozyme, Saint-Quentin-en-Yvelines, France) or mouse monoclonal antibody: actin (EMD, Millipore, France). Membranes were then washed with T-TBS and incubated for 1 hour with the appropriate horseradish peroxidase-conjugated anti-rabbit (Cell Signaling Technology, France) or anti-mouse (Amersham, Orsay, France) secondary antibodies. Revelation was performed exposing membranes to Clarity Max Western ECL Substrate (BioRad #1705062) using a luminescent Image Analyzer (GE Healthcare, Orsay, France).

Molecular modelling

Material &Method

We used a two-step protocol for molecular docking. First, a raw complex was obtained with Glide, using the standard precision mode (enhanced sampling for ligand conformers, enhanced planarity for conjugated π-group, post-docking minimization and strain correction term for final scoring). Then an induced-fit docking was performed with the best Glide pose, refining amino acids located within 7 Å of the ligand. The OPLS3e force field was used, with redocking in the standard precision mode. Finally, MM-GBSA was calculated on the final pose (VSGB solvation model and OPLS3e force field). Molecular dynamics experiments were a three-step process: after insertion of the Mcl-1/MIM1 or Mcl-1/FJ-809 complex in a solvent box (2 Cl -and nearly 8600 water molecules), (i) energy minimization for 0.3 ns with restraints for protein and ligand (only ions and water were allowed to move (ii) 0.1 ns equilibration in NPT conditions (300 K, 1 bar), using the RESPA integrator (2 fs / 6 fs time steps for near / far atoms), and a cutoff radius of 9 Å for Coulombic interactions, keeping only protein backbone restraints (iii) production for 1 µs in the same conditions, without any restraint. Molecular dynamics and RDF calculation were performed with Desmond, as implemented in Maestro v. 12.4 (Schrödinger LLC, San Diego, CA). The Mcl-1 structure PDB:3KJ0 was used for all simulations. 19

Conclusions

In summary we have designed a strategy to prepare unambiguously molecules with a thiazol-3(2H)-yl)imino core, exemplified here by the synthesis of the FJ-809 compound (1) which has the structure originally ascribed to MIM1 [START_REF] Hanahan | [END_REF]. Biological studies demonstrated that this molecule FJ-809 is not an inhibitor of the antiapoptotic protein Mcl-1, contrary to 2, and these results have been confirmed by FPA assays. Extensive molecular modelling and molecular dynamic studies have been performed to understand the higher affinity of MIM1 [START_REF] Hanahan | [END_REF] for Mcl-1.

Fig. 3

 3 Fig. 3 Structure of compound 16 by X-Ray analysis.

  It thus constitutes a pertinent model in the context of Bcl-xL/Mcl-1 inhibitors screening studies. The cells are exposed simultaneously to the candidate molecules and to a known Bcl-xL inhibitor (ABT-737). When apoptosis is observed in presence of ABT-737, the molecule is identified as a possible Mcl-1 inhibitor (to be confirmed by other approaches).Here we showed (Fig.4) in this model that MIM1 (2) exerts a modest activity on IGROV1-R10 cells at 10µM in combination with ABT-737, as previously described,9 suggesting its Mcl-1 inhibitor properties, as expected. Cell detachment and nuclear condensations or fragmentations are observed after a 24h exposure to MIM1 (2) (13% at 10µM, Fig.4, and 18% at 25µM, see Fig.S1in ESI), as well as a slight caspase 3 cleavage and a strong PARP cleavage in western blot experiments, observed at 10µM.

Fig. 4

 4 Fig.4Evaluation of the biological effect of MIM1 (2) and FJ-809 (1) on IGROV1-R10 cell line IGROV1-R10 chemoresistant ovarian cancer cell line. This cell line was exposed to either MIM1 or FJ-809 (10µM) combined, or not, with ABT-737 (5µM). Effect on cellular and nuclear morphology, as well as cell cycle repartition, is represented (A) and nuclei fragmentation is quantified as described in the experimental section (B). PARP and Caspase 3 cleavage is evaluated by western blot (C).

Fig. 6

 6 Fig. 6 MIM1 (2, left) and FJ-809 (1, right) bound to Mcl-1, as suggested by molecular docking.

Fig. 7

 7 Fig. 7 Fluctuation of MIM1 and FJ-809 bound to Mcl-1 (RMSD in Å vs. time in ns). For MIM1, the occurrence of different binding mode is highlighted by the RMSD calculated during two independent molecular dynamics experiments (A, B). By contrast, FJ-809 appears nearly motionless in the binding site (C).

  Fig. 8 Example of the RDF of water molecules around the methyl group of FJ-809 (top) and MIM1 (bottom). For other molecular dynamics examples, see ESI, Fig. S5 and S7. Integration of the RDF give an approximation of the average coordination number, here 8.8 H2O for FJ-809 and 0.12 H2O for MIM1.

  . HRMS (ESI): calcd for C17H20N4O2S: [M + H] + : m/z 345.13797. Found 345.1383 (1 ppm).

  On each image a deep-learning-based method of 2D nucleus detection was applied [Uwe Schmidt, Martin Weigert, Coleman Broaddus, and Gene Myers. Cell Detection with Star-convex Polygons. International Conference on Medical Image Computing and Computer-Assisted Intervention [MICCAI, Granada, Spain, September 2018] with a python program. A size filter was used to class intact nuclei and fragmented nuclei.

Acknowledgments

We thank CNRS and University of Rennes 1 for their support. Financial support by the "Ligue contre le Cancer, Conseil Interrégional Grand Ouest" and "Ligue Contre le Cancer, Conseil Interrégional de Normandie et Comité du Calvados" are gratefully acknowledged. This work was also supported by the "Conseil Régional de Normandie" and European Union (The ONCOTHERA European project "Normandy Network for innovative therapeutics in oncology" which is co-funded by the Normandy County Council, the European Union within the framework of the Operational Programme ERDF/ESF 2014-2020), by the French State (CPER Innovons 2), the University of Caen Normandie and Inserm. We are very grateful to the Centre Régional de Mesures Physiques de l'Ouest, Rennes (CRMPO) team for the HRMS analyses plus various NMR studies and for very fruitful discussions. We thank European FEDER funds for acquisition of D8Venture X-ray diffractometer used for crystal structure determination. HP is recipient of a doctoral fellowship from the "Ministère de l'Enseignement Supérieur, de la Recherche et de l'Innovation" granted by Normandy Doctoral School 497 "EDN Bise". We acknowledge Maryline Guillamin (Flow cytometry accommodation, SF 4206 ICORE, University of Caen Normandie), for her helpful technical support and Pr Marc-André Mahé, head of the Comprehensive Cancer Center François Baclesse (Caen, France), for his constant support.

Notes and References.

Conflicts of interest

There are no conflict of interest to declare.


Dr Michele Tomasini

Lucia Caporaso

Dr Jonathan Trouvé

Jordi Poater Teixidor

Rafael Gramage-Doria
email: rafael.gramage-doria@univ-rennes1.fr

Albert Poater
email: albert.poater@udg.edu

Dr Jordi Poater
email: jordi.poater@ub.edu

Dr ] M Tomasini

Unravelling Enzymatic Features in a Supramolecular Iridium Catalyst by Computational Calculations

Non-biological catalysts following the governing principles of enzymes are attractive systems to disclose unprecedented reactivities. Most of those existing catalysts feature an adaptable molecular recognition site for substrate binding that are prone to undergo conformational selection pathways. Herein, we present a non-biological catalyst able to bind substrates via the induced fit model according to in-depth computational calculations. The system, which is constituted by an inflexible substrate-recognition site derived from a zinc-porphyrin in the second coordination sphere, features destabilization of ground states as well as stabilization of transition states for the relevant iridium-catalyzed C-H bond borylation of pyridine. In addition, this catalyst appears to be most suited to tightly bind the transition state rather than the substrate. Besides these features, which are reminiscent from the action mode of enzymes, new elementary catalytic steps (i.e. C-B bond formation and catalyst regeneration) have been disclosed owing to the unique distortions encountered in the different intermediates and transition states.

Introduction

Chemical transformations catalyzed by enzymes, Nature's catalysts, exhibit highly active levels of activity and excellent levels of chemo-, regio-and stereo-selectivity. [1] Although not fully understood, the main reasons for these superb performances appear to be the (1) physicochemical protection of the active site in a confined space within the enzyme, [2] (2) the pre-organization of the substrate around the active site by means of weak interactions occurring in the second coordination sphere, [3] (3) the exploitation of cooperative reaction mechanisms (redox active, proton shuttles, etc.) including allosteric behavior by means of cofactors, [4] and (4) the subtle motions and distortions taking place near the active site to destabilize ground states and stabilize key transition states. [5] Consequently, many of these strategies have been successfully implemented in abiological catalysts to improve state-of-the-art catalysts. [6] These so-called supramolecular catalysts aim at mimicking such enzymatic properties in view to control the reaction outcome. [START_REF]Supramolecular catalysis[END_REF] For instance, catalysts confined in discrete (supra)molecular cages [8] as well as those featuring substrate pre-organization [9] have been elegantly demonstrated powerful to achieve unique reactivities since the seminal studies by Breslow and Crabtree, independently. [10] On the other hand, from the many supramolecular catalysts known to date, none of the studies involving them managed to identify the relevance of distortions occurring between the substrate and the catalyst. As currently known for biological enzymes, such distortions are known to play a central role in the activities and selectivities observed. [5] The absence of these studies is likely based on the fact that most supramolecular catalysts are equipped with an hydrogen-bonding or ion-pairing substrate-recognition site that are still flexible enough to accommodate substrates via conformational selection pathways. [11] Although this translates into catalytic systems able to reach an atom-precise reactivity in a predictive fashion, [9] they fail to provide a model to identify and understand any distortion taking place between the catalyst and the substrate. [START_REF] Vladimirova | For examples of distortions affecting the reactivity within enzymes[END_REF] Herein, we provide an in-depth mechanistic study by purely computational means that highlights the importance of such distortions in a nonbiological catalyst appended with a rigid and unbendable substrate-recognition site, aimed at destabilizing ground states and stabilizing transition states. [START_REF] Phillips | For examples of ground-state destabilization in enzymes[END_REF] It has recently been shown by some of us that a supramolecular iridium catalyst equipped with a zinc-porphyrin as a substrate-recognition site led to meta-selective C-H bond borylation of pyridines following the enzymatic Michaelis-Menten rate equation law as enzymes do. [14] The remote, kinetically labile Zn•••N interaction enables the pyridine substrate to be located at a given distance from the catalytically active iridium site to reach the observed regio-selectivity (Figure 1). This supramolecular catalyst features a zinc-porphyrin moiety as a conformationally restricted, substrate-recognition site since the zinc center can exclusively bind pyridines through the axial vacant site remaining almost unchanged before and after binding to pyridine substrates as shown by Sanders in organocatalysis. [15] In addition, the catalytically active iridium intermediates should tightly activate the C-H bond from the pyridine substrate, thus imposing also a restricted conformation on the pyridine substrate while binding to the zinc center. As such, any changes deviating from the ideal linearity between the pyridine and the zinc binding can be followed as distortion effects between the substrate and the overall catalyst structure (Figure 1). We reasoned that this catalytic system could be an appropriate model to study catalyst-substrate distortions at the molecular level by density functional theory (DFT) calculations. Moreover, owing to the rigidity of the porphyrin backbone, the distances between the zinc center in the substrate-recognition site and the nitrogen atom from the substrate can directly be correlated to the strength of such non-covalent interaction (Figure 1).

Results and Discussion

We initially focused on the formation of the active iridium(III) species and its compatibility with the binding of a pyridine substrate to the remote zinc-porphyrin site (Scheme 1). For all the computed iridium complexes a singlet ground state has been considered throughout the reaction pathway, since test calculations on the triplet state lie at least 20 kcal/mol higher in energy (see Figure S1 in the Supporting Information). [16] The Gibbs energy results were obtained at the B3LYP-D3/Def2TZVP~sdd(smd-p-xylene)//BP86-D3/Def2SVP~sdd level of theory.

The starting point of this study is the Ir-1 complex that experimentally forms upon reaction of the supramolecular ligand L with the [Ir(COD)Cl]2 (COD = 1,5-cyclooctadiene) precursor under borylating reaction conditions. [14] Ir-1 species contains an octahedral iridium(III) atom coordinated to the peripheral N,Nchelating moiety, three (pinacolato)boryl ligands and a COD ligand (Figure 1). From the many possibilities, the faccoordination of the boryl ligands to the iridium atom is the more plausible as discussed by Hartwig. [17] At this stage, the binding of pyridine to zinc below the porphyrin plane, that is the opposite site from the catalytically active iridium site, appears energetically more feasible (ΔG = -3.2 kcal/mol, Figure 1) than the binding above the plane (ΔG = +5.1 kcal/mol, Figure 1). This is hardly surprising since the binding of pyridine in the same face that the active iridium species will result in a highly sterically demanding system (Figure 1). Interestingly, such large difference in energy regarding the binding of pyridine above and below the porphyrin plane of ΔΔG = 8.3 kcal/mol for Ir-1 is reduced to only ΔΔG = 0.4 kcal/mol (A vs A', Figure 1), when the binding of pyridine to zinc occurs after the release of the labile COD ligand previously coordinated to iridium, that is from species A0 (Figure 1). As such, once the COD ligand is released from the iridium center (A0), the binding of pyridine from the same face (A) or the opposite one (A') is energetically similar. For comparison purposes, we calculated the energy associated to the binding of pyridine to zinc when the system lacks the iridium species (see Figure S2 in the Supporting Information), which is known to occur in solution and in the solid state as shown previously. [14] In this case, a ΔG = -2.3 kcal/mol is obtained that is significantly higher in energy than that observed with the binding of pyridine to the iridium-coordinated species (A0 → A) with a ΔG = -5.1 kcal/mol (Figure 1). [16] Figure 1. Probing pyridine substrate binding to the molecular recognition site of the supramolecular catalyst starting from the catalytically productive Ir-1 species. B = (pinacolato)boron, pyr = pyridine. Relative Gibbs free energies in kcal/mol. 10.1002/chem.202201970

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This article is protected by copyright. All rights reserved. Table 1. Summary of the key parameters relevant for substrate-binding strength and distortion effects between the catalyst and the ligand encountered for intermediates and transition states involved in the catalytic cycle from A→E.

Intermediates or transition states

Zn•••N distance (Å)
MBO [a] Angle Zn-N-Cpara (°) Deviation angle (°) [b] A [a] Mayer Bond Order calculated for the Zn•••N interaction. [b] Defined as the difference between 180° (ideal Zn-N-Cpara) and the observed Zn-N-Cpara angle for a given species.

A careful study of the molecular structure of the catalytically productive intermediate A (Figure 2) revealed a Zn•••N distance of 2.191 Å with a Mayer Bond Order (MBO) of 0.122 (Table 1). The Zn•••N distance in A is larger than that found experimentally in the X-ray structure of the binding of pyridine to the supramolecular catalyst lacking the iridium site (dZn•••N = 2.135 Å) which also displays an almost ideal binding of the pyridine substrate to the zinc center with an angle Zn-N-Cpara = 179°, [14] as it could be expected. [18] Notably, the weakening of the Zn•••N binding in A is additionally followed by a strong deviation from the linearity in the binding of pyridine to the zinc center with an angle Zn-N-Cpara = 151° (Table 1). This highly distorted binding, that deviates almost 30° from the ideal case (see deviation angle definition in Table 1), is compensated by π•••π interactions [19] between the pyridine substrate and the iridium-coordinated pyridine moiety from the supramolecular catalyst in the periphery (Figure 2 the %VBur descriptor developed by Cavallo (76.7% for meta vs 94.6% for ortho, see Figure S8 in the Supporting Information). [20] In other words, the ortho-C-H bonds from pyridine are inaccessible for iridium due to the important steric shields provided by the porphyrin backbone. Overall, intermediate A appears as an example of a ground state highly destabilized by the synergy of the substrate-recognition site and the steric effects encountered at the active site. In fact, these findings contrast with the common scenario in which a substrate S and a catalyst C lead to a more stable system S+C. [START_REF]Enzyme Catalysis in Organic Synthesis: A Comprehensive Handbook[END_REF] Figure 3. Probing the iridium-mediated meta-C-H bond activation step in a pyridine substrate. B = (pinacolato)boron. Relative Gibbs free energies in kcal/mol, selected distances in Å.

Next, we evaluated the C-H bond activation step (Figure 3). In this case, the tight Zn•••N interaction anchors the pyridine to the substrate-recognition site and so, unlike in the mechanism reported by Hartwig, [17g] the pyridine cannot change its orientation but only slightly rotate around the axes along the Zn•••N interaction. Therefore, when the pyridine rotates around the Zn•••N axes, the C-H bond at the meta position is activated and cleaved (C-H = 1.010 Å in A vs 1.722 Å in TSAB) and an Ir-H bond is formed (Ir-H = 1.765 Å in B) through the transition state TSAB while the Zn•••N interaction strengthens by 0.092 Å from A to B (Zn•••N distance in B = 2.099 Å, Figure 3 and Table 1). The overall process requires to overcome an energy barrier of 32.3 kcal/mol (Figure 3), becoming the rate determining step (rds) of the whole reaction pathway (vide infra). The validity of the computational method in the rds was checked with M06 and M06-D3 instead of B3LYP-D3, with inappreciable differences of only 0.5 and 1.0 kcal/mol, respectively. [22] In fact, from the pre-catalyst Ir-1, the TSAB increases by 1.4 kcal/mol up to 33.7 kcal/mol. Thermodynamically, the latter transition state TSAB leads to a relatively unstable hepta-coordinated species B (ΔG = 8.4 kcal/mol), in which the Zn•••N distance remains constant (2.099 Å) with respect to that found in TSAB with a MBO of 0.253 (Table 1). The angles Zn-N-Cpara were found to be 176° for TSAB and 171° for B indicating that these systems imply significantly less distortion systems compared to A (Table 1). Importantly, the fact 10.1002/chem.202201970

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that the distance is shorter in the TSAB and B rather than in A clearly indicates that the pyridine prefers to bind to the molecularrecognition site during the rds rather than as a free substrate. Consequently, this observation strongly suggests that the supramolecular catalyst follows an induced fit mechanism with differential binding rather than a conformational selection one. [23] For comparison purposes, the alternative reaction pathway in para was also calculated. Even though TSAB para requires to pay only an energetic cost by 20.6 kcal/mol (see Figure S3 in the Supporting Information), with additional weakening of the Zn•••N interaction to 2.125 Å compared to the meta intermediate, [16] at 80 °C the intermediate B para (formed after activation of the Cpara-H bond in pyridine) is 6.9 kcal/mol less stable than the intermediate B. Next, the transition state TSBC para leading to the iridiummediated activation of the C-H bond in para position of the pyridine (see Figure S4 in the Supporting Information) becomes kinetically disfavored by 22.3 kcal/mol with respect to TSBC meta , which blocks the attack in para because of not only the larger sterical hindrance, but also the constrained angles. In addition, B para can overcome a very low kinetic barrier of only 5.3 kcal/mol to come back to A, [START_REF]and break the π•••π interaction with the porphyrin-linked triazolopyridine moiety to be further borylated in para position[END_REF] thus we can assume a reversible process, while the energy barrier for the C-B bond formation in para is highly kinetically demanding, 34.1 kcal/mol. To further explain why the para C-H activation results less stable than the preferred meta one, structurally Ir-C para -N angle is 160.8° and the planarity of the ring is partially lost as in an iridium catalysis not assisted by zinc. The same distortion is present in the TSAB para . As additional computations, we calculated the acidity of each pyridine proton upon binding to zinc for a model system lacking the iridium site (see Figure S5 in the Supporting Information). Using this simplification, the meta proton of the pyridine bonded to zinc was found to be the most reactive since its removal is favored by 1.0 and 2.7 kcal/mol with respect to the para and ortho ones, respectively. [16] In contrast, in the absence of Zn•••N binding, the removal of the para proton of the pyridine is favored by 1.7 and 7.8 kcal/mol with respect to the meta and ortho ones, respectively. [16] These findings indicate that there is minimal (almost negligible) modification of the acidity of the hydrogen atoms from pyridine upon binding to the zinc-porphyrin. [START_REF] Barckholtz | Bond dissociation energies for unfunctionalized pyridine have been similarly computed[END_REF] In addition, we also performed the calculation for the corresponding ortho C-H activation and the energy barrier went up by 16.1 kcal/mol. However, it is known from experimental results that no C-H borylation takes place if the iridium site is not covalently linked to the substrate-recognition site. [14] This was further supported by additional computational calculations on the energy costs associated to overcome the rate-determining step, i.e. TSAB, using zinc-porphyrinoids lacking the covalently attached iridium site, namely zinc-tetraphenylporphyrin and zinc-salphen (see Figure S6 in the Supporting Information). The calculations show larger activation energy barriers by 9.2 and 11.9 kcal/mol higher in energy, respectively. [16] In addition, we also computed the TSAB with a system lacking the zinc-porphyrin substrate-recognition site, and the energy barrier increased by 3.1 kcal/mol, which is a significant value considering that the homogeneous catalysis takes place at 80 °C. [16] Unexpectedly, the formation of the hydride species in intermediate B results in an elongation of one of the Ir-B bonds from 2.048 Å to 2.179 Å with a formal B-H bonding interaction with the hydride ligand (Figure 4). The hydride interacts with the empty p orbital of the boron moiety, and it makes less reactive this boryl group compared to the other two ones. The next step of the reaction mechanism occurs upon the rotation of the other boryl group located trans to the one involved in the B-H interaction leading to the formation of a C-B bond in TSBC, overcoming an energy barrier of 18.7 kcal/mol, with concomitant weakening (almost cleavage) of the H-B interaction from 1.383 Å to 2.070 Å. In TSBC, the B-Cmeta distance shortens to 1.927 Å while the Ir-B bond results only quite activated (2.145 Å in TSBC vs 2.105 Å in B, Figure 4). As a result, the pyridine substrate is borylated at the meta position and intermediate C is formed, even though the interaction with zinc becomes slightly weaker again (Zn•••N distance = 2.141 Å) but still significant, with an associated MBO of 0.264. The intermediate C presents an octahedral geometry at the iridium center with a  2 -borylated pyridine as ligand and a weakening of the Zn•••N distance to 2.193 Å comparable to that found in intermediate A (Figure 4). Similarly, a remarkable bending of 16 degrees in the deviation angle was found for the binding of the meta-borylated pyridine to zinc (Table 1), which is also stabilized by π•••π interactions between the borylated pyridine substrate and the iridium-coordinated pyridine moiety from the supramolecular catalyst in the periphery. Analogously to TSBC, no H•••B interaction is present and the intermediate C results 3.7 kcal/mol less stable than B. We then focused on the catalyst regeneration. This latter consists mainly of two different elementary steps: (1) the oxidative addition of a molecule of B2pin2 and (2) the reductive elimination of HBpin at the iridium site (Figure 5). Initially, we considered these two steps to occur without the borylated pyridine product binding to the molecular-recognition site, namely with an iridium center that has a vacant coordination site since the prior intermediate C displays  2 -coordination to iridium. [16] The corresponding intermediate formed after product decoordination releases only 2.8 kcal/mol, and the resulting oxidative process is highly kinetically demanding, with an energy barrier of 22.0 kcal/mol, as it could be expected. In fact, the resulting intermediate is rather unstable since this step is thermodynamically endergonic by 20.0 kcal/mol. [16] On the other hand, in the case that the borylated pyridine product is still binding to zinc as well as to iridium, steps (1) and ( 2) are energetically 10.1002/chem.202201970 accessible (Figure 5). Indeed, when B2pin2 reacts with the iridium center the process yields a highly stable iridium(V)tetrakis(boryl)hydride complex (intermediate D, ΔG = -0.5 kcal/mol) overcoming an energy barrier of only 5.4 kcal/mol. Comparing this system with the one analyzed by Sakaki, [26] it is noted that the two systems are comparable in energy (5.4 vs 8.0 kcal/mol). However, while the diboron molecule is added at the vacancy left by the borylated benzene in Sakaki's system, [26] in the present supramolecular system, this process takes place on the opposite side. In fact, the change on this crucial elementary step occurs because the borylated pyridine product remains interacting with the zinc center from the molecular recognition site; and the reactive system cannot rotate along the bond that connects the porphyrin backbone to the iridium site. Therefore, the B2pin2 is added on the side that results to be less hindered. At the approach of the diboron towards the iridium complex, this last one isomerizes into an octahedral hydride complex with a  2 -B2pin2 ligand as shown in the transition state TSCD (Figure 5 and Table S1 in the Supporting Information) in order to minimize the steric hindrance of the boryl group. Next, a molecule of HBpin is released into solution leading to the formation of intermediate E (G = 1.1 kcal/mol) with the process facilitated by the fact that one of the Ir-B bond results already activated (2.185 Å in D vs 2.048 Å in A) in intermediate D and by interaction of the borylated pyridine product to the iridium center. The intermediates and transition states from C→E, reveal weaker Zn•••N interactions compared to the one observed in the key transition state TSAB (vide supra). In other words, this supramolecular catalyst appears to be designed to better fit the key rate-determining step rather than the other intermediates or transition states of the catalytic cycle, a feature reminiscent from enzymes, but observed herein for a transition metal catalyst. The catalytic cycle is closed with formation of intermediate A after product release and substrate binding to the molecular recognition site of the supramolecular catalyst (Figure 5). As it could be expected from all the abovestated discussion, we found a linear correlation between the two key descriptors of this study: the Zn•••N distance and the deviation angle (see Figure S7 in the Supporting Information). For completing this study, and with the aim to better understand the role of pyridine, the aromaticity of the pyridine ring was analyzed throughout the full catalytic cycle. [16,27] Importantly, its binding to zinc first, and later functionalization with iridium, makes the pyridine ring to become less aromatic. Next, consequently, we wondered whether the inclusion of either an electron-donor or electron-acceptor group in the pyridine substrate could affect the reaction barrier of the iridium-catalyzed meta-C-H bond borylation. For such, we embarked in the analysis of the rate-determining step, namely the energy associated to TSAB, for a series of 3-functionalized pyridines as the substrates for the supramolecular iridium-catalyzed C-H bond borylation reaction (Table 2). Overall, the inclusion of either electron-donor or electron-acceptor substituents (Me, OMe, CF3, t Bu) in the initial substrate led to no major changes associated to the energy of TSAB (Table 2). However, a trend was roughly established by comparing the Grds and the MBOs (a higher Grds corresponds to a higher MBO), thereby confirming the key role played by the zinc center in its interaction with the pyridine substrates. And such minor role of the inclusion of a substituent is also supported by the analysis of the aromaticity of a series of both substituted intermediate A and TSAB systems (see Table S1 in the Supporting Information). [16] The electron-withdrawing trifluoromethyl group caused the largest decrease of aromaticity in A (MCICF3 = 0.027), whereas the other systems presented a more constant aromatic pyridine ring regardless of the substituent. Interestingly, the reactivity of the different substituents on 3functionalized pyridine substrates, defined by the bottleneck of the energy barrier for the rate-determining step, can be directly related to the energy of the LUMO of the rds. Actually, there is a linear correlation among both variables (R 2 = 0.909). [16] Even though the electronics seem fundamental, they are equally important as the sterics, because the calculation on the metal center of the %VBur, developed by Cavallo [20] , led to an identical good agreement (R 2 = 0.901). [16] The additional effect of both contributions led us to apply a multilinear regression to found the equation: Grds = -0.122 * %VBur -314.259 * ELUMO + 11.992, [16] even though the short set of data does not allow a clear confirmation of this good correlation (R 2 = 0.940). In summary, there is a clear trend in which the larger the substituent is, the lower is the kinetic cost, whereas from an electronic point of view, the electron-withdrawing groups enlarge the energy barrier of the rate-determining step. For instance, the pyridine substrate with a tert-butyl substituent is the most sterically hindered, and despite not being the most electron-donating group, it is the one that requires a lower reaction barrier. Thus, it can be concluded that steric effects are more important than the electronic ones, bearing in mind the methoxy group is the most electron-donating one. However, the electronics also drive the kinetics. To better exemplify this case, the LUMO orbitals for intermediate A with pyridine and 3-trifluoromethyl-pyridine substrates, respectively, were calculated (Table 2) and displayed in Figure 6 showing significant differences (for the other substituents, see the Supporting Information). [16] In the case of 3-trifluoromethylpyridine substrate the LUMO is slightly delocalized in the triazolopyridine moiety from the supramolecular catalyst, while in the case of unfunctionalized pyridine, no delocalization beyond the porphyrin backbone is observed. For the sake of comparison, we placed the substituents in relative meta-position because (1) otherwise experimentally a second borylation in the second meta-position of pyridine could occur, and (2) ortho-and para-functionalized pyridine substrates are unreactive with this catalytic system. [14] Nevertheless, we calculated the energy of the transition state of the C-H activation step when a boryl group is present in meta position and it is only 22.4 kcal/mol higher than the isolated reactants since the sterical hindrance plays a key role to support the best substitution.

Conclusion

In conclusion, we have thoroughly studied the reaction mechanism for the meta-selective C-H bond borylation of pyridines using a supramolecular iridium catalyst by means of DFT calculations underlying the relevance of distortion effects and substrate binding strength as key descriptors. The reaction mechanism for this supramolecular system displayed several features reminiscent from enzymatic behaviors such as the uncommon ground-state destabilization upon binding the substrate to the catalyst (uphill formation of intermediate A) as well as the tight binding of the substrate in the rate-determining step over the other elementary steps, thereby indicating that the overall structure of the supramolecular catalyst fits better the transition state rather than the substrate and/or the product which is in line with an induced fit mechanism rather than a conformational selection one. [START_REF] Vladimirova | For examples of distortions affecting the reactivity within enzymes[END_REF][START_REF] Phillips | For examples of ground-state destabilization in enzymes[END_REF]22] In addition, the exquisite levels for meta-selectivity were rationalized by the precise atomic distance between the active site and the substrate binding site that leaves a single C-H bond at close proximity enough to react with the catalytically productive iridium site. Furthermore, we managed to identify an unprecedented pathway for the regeneration of the iridium catalyst thanks to the additional stabilization ensured by the presence of a  2 -coordinated product to iridium as well as a new type of C-B bond-forming elementary step. Additional studies indicate the subtleties associated for correlating the Grds with the different substitution patterns of different pyridine substrates. Overall, this work shed lights on the unexpected reaction mechanisms that might be encountered in supramolecular catalysts aiming at mimicking enzymes, thereby upgrading the rational for the further development of more powerful catalysts tunable at the second coordination sphere.

Experimental Section

XYZ coordinates and energies of all computed species discussed in this contribution are provided in the Supporting Information.

Figure 1 .

 1 Figure 1. Supramolecular catalysis considered in this work in order to study how distortions effects are at play during the catalytic cycle in a similar manner as enzymes do for biological catalysis. B = (pinacolato)boron.

Figure 2 .

 2 Figure 2. Molecular structure of the computed intermediate A (left, and with the fragments involved in π-π stacking in spacefill representation, right). For the sake of clarity, all hydrogen atoms have been omitted except those belonging to the pyridine substrate, and the zinc and iridium atoms are shown in ball & stick representation.

  , right). Intermediate A also indicated that the closest pyridinic C-H bond to iridium was the one in meta position with a distance Ir•••Cmeta of 2.620 Å compared to those in ortho (dIr•••Cortho = 2.696 Å) and para (dIr•••Cpara = 3.337 Å) positions. Although, at first glance both meta and ortho C-H bonds seem accessible regarding the Ir•••C-H distances, the C-H bond in meta results less sterically hindered than the C-H bond in ortho as shown by

Figure 4 .

 4 Figure 4. Probing the iridium-mediated meta-carbon-borylation step in a pyridine substrate. B = (pinacolato)boron. Relative Gibbs free energies in kcal/mol, selected distances in Å.

Figure 5 .

 5 Figure 5. Catalyst regeneration from a η 2 -coordinated borylated pyridine to iridium and closing of the catalytic cycle. B = (pinacolato)boron, pyr = pyridine, prod = 3-borylated pyridine product. Relative Gibbs free energies in kcal/mol.

  This article is protected by copyright. All rights reserved.

Figure 6 .

 6 Figure 6. LUMO orbitals for intermediate A with pyridine (left) and 3-(trifluoromethyl)pyridine (right) as the substrates.

  This article is protected by copyright. All rights reserved.

Table 2 .

 2 Selected parameters found for the transition state TSAB considering different 3-functionalized-pyridine substrates. Mayer Bond Order calculated for the Zn•••N interaction.

		Grds (kcal/mol)	MBO [a]	%Vbur	HOMO (a.u.)	LUMO (a.u.)
	R = H	33.7	0.122	58.8	-0.186	-0.092
	R = Me	33.2	0.139	63.2	-0.184	-0.090
	R = OMe	31.2	0.097	72.6	-0.182	-0.088
	R= CF3	32.0	0.112	72.9	-0.181	-0.088
	R = t Bu	30.9	0.115	82.6	-0.180	-0.087
	[a]

Acknowledgements A.P. is a Serra Húnter Professor and thanks ICREA Academia 2019 and the Spanish MINECO for project ref. PGC2018-097722-B-I00. J.T. and R.G.-D. acknowledges financial support from the CNRS, Université de Rennes 1 and the Agence Nationale de la Recherche (ANR-19-CE07-0039). J.P. also acknowledges the MINECO (PID2019-106830GB-I00, PID2021-127423NB-I00 and MDM-2017-0767). Keywords: C-H functionalization • DFT calculations • enzyme mimics • iridium • supramolecular catalysis

Entry for the Table of Contents

Computational calculations aiming at elucidating the reaction mechanism of a supramolecular iridium-catalyzed C-H bond borylation enabled the identification of unique intermediates and transition similar to those found in enzyme catalysis. The unexpected energies associated to the elementary steps are reached thanks to the relevant distortion effects at play, which are upgraded by the molecular-recognition pocket from the catalyst.

Institute and/or researcher Twitter usernames: @Rafa_gramage @chimie_ISCR @INC_CNRS @UnivRennes1 @jordipoater @APoater @DIMOCAT_iqcc @IQCCUdG ((max. width: 11.5 cm; max. height: 2.5 cm))


Dinil Sebastien Soudan

Mon Divakaran

Eitan Altman

Pascale Vicat-Blanc

Primet

Equilibrium in size-based scheduling systems

Keywords: Size-based scheduling, equilibrium Equilibre, ordonnancement basé sur la taille

Size-based scheduling is advocated to improve response times of small flows. While researchers continue to explore different ways of giving preferential treatment to small flows without causing starvation to other flows, little focus has been paid to the study of stability of systems that deploy size-based scheduling mechanisms. The question on stability arises from the fact that, users of such a system can exploit the scheduling mechanism to their advantage and split large flows into multiple small flows. Consequently, a large flow in the disguise of small flows, may get the advantage aimed for small flows. As the number of misbehaving users can grow to a large number, an operator would like to learn about the system stability before deploying size-based scheduling mechanism, to ensure that it won't lead to an unstable system. In this paper, we analyse the criteria for the existence of equilibria and reveal the constraints that must be satisfied for the stability of equilibrium points. Our study exposes that, in a two-player game, where the operator strives for a stable system, and users of large flows behave to improve delay, size-based scheduling doesn't achieve the goal of improving response time of small flows.

Introduction

Scheduling based on flow size (or flow age) has been gaining importance in the recent times. Researchers have proposed different ways of scheduling based on size, ranging from SRPT (Shortest Remaining Processing Time) to LAS (Least Attained Service) to MLPS (Multi-level Processor Sharing) scheduling mechanisms [START_REF] Kleinrock | Queueing Systems[END_REF][START_REF] Avrachenkov | Differentiation Between Short and Long TCP Flows: Predictability of the Response Time[END_REF][START_REF] Sun | DRR-SFF: A Practical Scheduling Algorithm to Improve the Performance of Short Flows[END_REF]. These scheduling strategies differ from the general model for flow scheduling in the Internet. The queues in the Internet nodes, though are served in an FCFS order at packet level, can be modeled using an M/G/1-PS (processor sharing) queue at flow level. The motivation to deviate from this norm, and schedule flows based on size, is to give better completion time to small flows. Strictly speaking, the aim has been to improve the conditional mean response time of small flows, at negligible cost to large flows. LAS, for example, always gives highest priority to the flow that has attained the least service. More details on size-based scheduling policies and the advantages they bring, can be found in [START_REF] Nuyens | The foreground-background queue: A survey[END_REF] and [START_REF] Avrachenkov | Differentiation Between Short and Long TCP Flows: Predictability of the Response Time[END_REF]. Note that, researchers use age-based scheduling to refer to the scheduling schemes that are blind, in the sense that, they do not have information about the size of the flow when it arrives, and hence uses its age (the number of bytes/packets already scheduled) to make scheduling decision. Whereas, in this paper, we use the broader phrase size-based scheduling to include all the policies that use age or size to make scheduling decisions.

A user (an end-user or an application) sends a file as a single flow across the Internet. We take this as a normal behaviour. If size-based scheduling is deployed by an operator, there is a clear motivation for one or more users to deviate from the normal behaviour. Indeed, there is an incentive in splitting a flow (possibly large, but more precisely, one that is not small) into multiple small flows to exploit the advantage (say, priority in scheduling) given to small flows to improve the response time. If a considerable number of users deviate from the normal behaviour, then the operator's aim of giving shorter response time to small flows might well be deceived. More importantly, an operator would like to know if such user manipulations would lead to an unstable system behaviour. This poses an important problem in the context of size-based scheduling systems which, to the best of our knowledge, has not been addressed yet. This is the problem we address in this work. In the scenario where users do not misbehave, the stability issue (for network of queues) has been addressed in [START_REF] Brown | Stability of networks with age-based scheduling[END_REF] recently.

The focus of this work is to study the equilibria in size-based scheduling system where users misbehave. We believe this would lead to better understanding of the implication of deploying a sizebased scheduling mechanism. More description of the problem is given in Section 2. The model is elaborated in Section 3. The existence of equilibria are studied for two kinds of system behaviours: one in which the service rates are fixed, is studied in Section 4; and the other in which the service rates are varying, is studied in Sections 5 and 6. We summarize our analysis as a game between the operator and users, in Section 7.

Problem statement and assumptions

We study the problem that arises when an operator deploys a size-based scheduling mechanism. Though there are different ways of scheduling based on size, our focus is on size-based scheduling using two queues. Here, flows are classified based on their sizes. Small flows are sent to one queue, and large flows to another 1 . Each queue is assigned a specific service rate, such that the total service rate equals the line capacity. The aim of operator in setting such a mechanism is to give to reduce the average response times of small flows.

To formulate the objective of the operator, we assume Poisson flow arrivals. Arrivals and service rates are in units of small flow. λ x and λ y are the arrival rates for small and large flows respectively. Each large is F times a small flow. The service rates at small and large queues are φ x and φ y respectively, such that if C denotes the line capacity, φ x + φ y = C. Each queue is served using the PS discipline; hence it is an M/G/1 -P S queue.

We study the existence of equilibria under the scenario where users cheat by splitting a large flow into multiple small flows to improve their delay. This is explored in two cases: (i) where the service rates assigned are static, (ii) where the operators exhibits control by dynamically changing the service rates. In the latter case, we explore the existence of interesting equilibria, and state the conditions required for stability, under the assumption that the incentive for players to migrate is to minimize the delay the flow will incur. Note that, by 'players', we consider only the users who migrate. The fluid model used in this work is inspired by the one used in [START_REF] Altman | Bandwidth allocation for guaranteed versus best effort service categories[END_REF], where the authors analyse dynamic bandwidth resource allocation and migration between guaranteed performance and best effort traffic classes.

The two-queues model is depicted in Fig. 1. The queue for small flows is called small queue and is referred to as Q x . The other queue is called the large queue which is denoted by Q y . The number of flows at Q x is represented by x. At the large queue, this number (in number of small flows) is denoted by y. We assume infinite queues. The service rates, φ x and φ y , are also in number of small flows. They are both assumed to take non-zero values.

The system parameters φ x and φ y are set by the operator. System state is modeled using averaged queue sizes: x and y. Depending on the measured delay values, a user might decide to split a large flow into multiple small flows. Therefore, a fraction of the flows arriving at the large queue might be migrated to the small queue. This migration function, which is a result of aggregate user behaviour, is represented as m(x, y). It is linear in λ y F as a result of the integration of individual user that send dλ y each: mdλ y = λ y m. We take m to be a non-negative and continuous function of x and y. m represents the fraction of λ y which goes to

Q x . 0 ≤ m(x, y) ≤ 1 (1)
For every large flow that migrates, it adds an overhead of η (e.g. connection establishment cost, slow-start cost). The rate equations can now be written as:

dx dt = λ x -φ x + λ y F m(x, y)(1 + η), x > 0 (2)
dy dt = λ y F -φ y -λ y F m(x, y), y > 0 (3)
The rate equations are different at the borders. For x = 0,

dx dt x=0 = [λ x -φ x + λ y F m(x, y)(1 + η)] + (4)
and for y = 0,

dy dt y=0 = [λ y F -φ y -λ y F m(x, y)] + . ( 5
)
4 System analysis for static service rates

This section details the analysis of a system where the service rates at both the queues are fixed.

Proposition 4.1. An interior point (x, y) is an equilibrium iff φ x -λ x = λ y F -φ y and m is such that m(x, y) = φx-λx λyF and 0 ≤ m(x, y) ≤ 1. Proof of Prop. 4.1. Let (x, y) be an interior point. It is an equilibrium if and only if :    dx dt = 0 dy dt = 0 0 ≤ m(x, y) ≤ 1 ⇐⇒      m(x, y) = φx-λx λyF (1+η) m(x, y) = λyF -φy λyF 0 ≤ m(x, y) ≤ 1 Remark 4.2.
Existence of interior equilibrium does not only depend on m function but also on the arrival rates and service rates. Meaning that they can only exist in very specific cases.

Proposition 4.3. (0, 0) is an equilibrium point if and only if: 4) and ( 5), we obtain that (0, 0) is an equilibrium point if and only if:

     m(0, 0) ≤ φx-λx λyF (1+η) λyF -φy λyF ≤ m(0, 0) 0 ≤ m(0, 0) ≤ 1 Proof of Prop. 4.3. Using equations (
     dx dt x=0 = 0 dy dt y=0 = 0 0 ≤ m(0, 0) ≤ 1 ⇐⇒    λx-φx 1+η + λ y F m(0, 0) ≤ 0 λ y F -φ y -λ y F m(0, 0) ≤ 0 0 ≤ m(0, 0) ≤ 1 ⇐⇒      m(0, 0) ≤ φx-λx λyF (1+η) λyF -φy λyF ≤ m(0, 0) 0 ≤ m(0, 0) ≤ 1
     m(0, y) ≤ φx-λx λyF (1+η) m(0, y) = λyF -φy λyF 0 ≤ m(0, y) ≤ 1
Proof of Prop. 4.4. Using equations ( 4) and ( 3), we obtain that (0, y) is an equilibrium point if and only if:

   dx dt x=0 = 0 dy dt = 0 0 ≤ m(0, y) ≤ 1 ⇐⇒    λx-φx 1+η + λ y F m(0, y) ≤ 0 λ y F -φ y -λ y F m(0, y) = 0 0 ≤ m(0, y) ≤ 1 ⇐⇒      m(0, y) ≤ φx-λx λyF (1+η) m(0, y) = λyF -φy λyF 0 ≤ m(0, y) ≤ 1
Proposition 4.5. (x, 0) with x > 0 is an equilibrium point if and only if:

     m(x, 0) = φx-λx λyF (1+η) λyF -φy λyF ≤ m(x, 0) 0 ≤ m(x, 0) ≤ 1
Proof of Prop. 4.5. Using equations ( 2) and ( 5), we obtain that (x, 0) is an equilibrium point if and only if:

     dx dt = 0 dy dt y=0 = 0 0 ≤ m(0, y) ≤ 1 ⇐⇒    λx-φx 1+η + λ y F m(x, 0) = 0 λ y F -φ y -λ y F m(x, 0) ≤ 0 0 ≤ m(x, 0) ≤ 1 ⇐⇒      m(x, 0) = φx-λx λyF (1+η) λyF -φy λyF ≤ m(x, 0) 0 ≤ m(x, 0) ≤ 1

Discussion

The aim of a network operator in deploying such a scheduling mechanism is to give shorter delays to small flows, at negligible cost to large flows. With this in mind, we can now evaluate which among the equilibrium points are interesting and useful (from the perspective of a network operator).

To start with, let us consider the equilibrium point (0, 0). The inequalities of Prop. 4.3 give the shaded region of Fig. 2, where one m can exist to make (0, 0) an equilibrium. This region is dominated by the line λ x + λ y F = C, which defines the region where a single queue system would have empty queue equilibrium. Thus, this equilibrium (in the two queue system) is not of great interest for the network operator. The lines (x, 0) and (0, y) constitute the remaining border point equilibria. (x, 0) is the set of those points where there is queueing in the small queue, but not at the large queue. For this reason, these are not desirable equilibria from operator's point of view. Similarly existence of (0, y) means, there is nothing queueing at Q x . So, there is incentive for users to migrate to Q x . Hence (0, y) will not be stable.

seen in previous section, interior point equilibrium are only possible in limiting cases where the surplus rate at the large queue is exactly equal to the surplus of service of x, with the additional constraint that m transfers exactly this. This situation is too constrained to happen in a real scenario. To introduce more flexibility, the operator can control the service rate. But this requires the use of some observable parameters of the system. In this system, the only observable parameters are x and y as arrival rates λ x and λ y are not separable at the queues.

Control on φ x using parameter x

In this section we study the system when operator controls the service rates using a single parameter. Let f be the control function, and x be the control parameter. In the remaining of this section we use the following definition for φ x (x) and φ y (x). Definition 5.1. φ x (x) and φ y (x)

φ x (x) = f (x) φ y (x) = C -f (x)
C being the maximum link capacity (or service rate), let:

0 < f (x) < C (6)
so that the service rate at any queue doesn't vanish.

Delay condition

We introduce the delay condition which is satisfied at equilibrium, as the users have no incentive to migrate once the delays at both queues are equal. Let us look the delay a large flow will incur Q x , if it is split into F small flows. For a service rate of φ x at x , each small flow gets φx x+F of service. Hence the time to transfer a large flow through Q x is T x = x+F F φx (1 + η). On the other hand, if the arriving large flow decides to queue at Q y , the delay experienced will be T y = y+1 φy . At equilibrium, T x = T y ; thus,

(x + F )(1 + η) F φ x = y + 1 φ y (7)

Analysis of equilibrium

For equilibrium to exist, the equations ( 2) and ( 3) should be equated to zero.

Proposition 5.2. If η is zero, no equilibrium will exist unless C = λ x + λ y F .

Proof of Prop. 5.2. From the combination (2) + (3) at equilibrium, when η is 0, we get C = λ x + λ y F .

In the remaining, η is taken to be strictly positive.

Using equations ( 2) and (3) at equilibrium, gives the constraints (8) on f for the existence of such an equilibrium point.

f (x e ) = (1 + η)(C -λ y F ) -λ x η (8)
There can be multiple such points x e or no depending on f . Proposition 5.3. For a given set of parameters (λ x , λ y , C, η, f, m) with η > 0, the system has inner equilibrium points (x e , y e ) where:

x e ∈ f -1 ( (1 + η)(C -λ y F ) -λ x η ) (9)
and:

y e = C -f (x e ) F f (x e ) (x e + F )(1 + η) -1 iff:          λ x + λ y F ≤ C λ x + λ y F (1 + η) > C f -1 ( (1+η)(C-λyF )-λx η ) = ∅ m(x e , y e ) = C-(λx+λyF ) ηλyF (10)
Proof of Prop. 5.3. From the combination, (2) + (1 + η)(3), at equilibrium, we obtain Eq. (8). The system has equilibriums iff there is point x e satisfying this equation, meaning f -1 (

(1+η)(C-λyF )-λx η
) is not empty (η = 0). From Eq. ( 7), we get corresponding y e . Then from Eq. (2) at equilibrium, we have m as defined in Eq. (10).

Due to constraint (1) on m, and constraint (6) on f , we have the existence of this equilibrium iff:

λ x + λ y F ≤ C C < λ x + λ y F (1 + η) (11)
Second inequality is strict because of Eq. ( 6).

INRIA Fig. 3 shows the region of arrival rates where equilibrium can exist, dashed-line is excluded from this.

Corollary 5.4. If f is strictly monotonic. For every 2-tuple of (λ x , λ y ) satisfying the line equation:

(1 + η)(C -λ y F ) -λ x = k (for a constant k)
, there is maximum one equilibrium point.

Proof. Corollary 5.4 If f is strictly monotonic, there is utmost one pre-image by f -1 . As potential equilibria are determined by Eq. (9) (and y e which only depends on x e ), all points of the line of arrival rates:

(1 + η)(Cλ y F )λ x = k have the same potential equilibrium. Since m(x e , y e ) has to satisfy Eq. ( 10), which gives a different line in λ x and λ y , there is at most one equilibrium point (the intersection). From the above, it can be observed that, for a monotonic f , there exists utmost one equilibrium point for the whole line of arrival rates. This gives only a few equilibrium points for a wide range of arrival rates. A non-monotonic f will give more equilibrium points. But still, it is not feasible to obtain equilibrium points for all values of (λ x , λ y ) satisfying the line of arrival rates, as it would require an infinite queue or an infinite variability of f . Hence, we conclude that control using a function of x alone, is not of any use to the operator.

λ x λ y F C C C (1+η)
6 Control on φ x using parameters x and y

As seen in previous section, using only one parameter is not enough to stabilize the system as the control space is too small. We thus use a control function with two parameters: x and y. Definition 6.1. φ x (x, y) and φ y (x, y) φ x (x, y) = g(x, y) φ y (x, y) = Cg(x, y)

Similar to what have been done with f , if C is the maximum link capacity (or service rate), let:

0 < g(x, y) < C (12)
Note that, definition of delay equation at equilibrium as given in [START_REF] Sastry | Nonlinear Systems. Analysis, Stability and Control[END_REF] remains the same and so we directly proceed to the analysis of potential equilibria.

Analysis of equilibrium

For equilibrium to exist, the equations ( 2) and ( 3) should be equated to zero. Prop. 5.2 still holds in this case as φ x and φ y also sum to C; therefore from equations ( 2) and ( 3) we can prove the same. Hence η is also taken strictly positive here.

Similarly to what have been done for Prop.5.3, at equilibrium, using Eq. ( 2) and ( 3), we obtain the following constraint on g:

g(x e , y e ) = (1 + η)(C -λ y F ) -λ x η (13)
Proposition 6.2. For a given set of parameters (λ x , λ y , C, η, g, m) with η > 0, the system has inner equilibrium points (x e , y e ) iff:

               g(x e , y e ) = (1+η)(C-λyF )-λx η m(x e , y e ) = C-(λx+λyF ) ηλyF (x+F )(1+η) F φx = y+1 φy λ x + λ y F ≤ C λ x + λ y F (1 + η) > C (14)
Proof of Prop. 6.2. Same as Prop. 5.3 except that (8) has been replaced by (13).

Note that the region of arrival rates where equilibrium points can exist is the same. We define an equivalent load Γ:

Definition 6.3. Γ(λ x , λ y ) = λx 1+η + λ y F . Definition 6.4. D(Γ)
is the set of (x, y) satisfying:

y = a(Γ)x + b(Γ) ( 15
)
where

a(Γ) = (1 + η)Γ -C F (C -Γ) and b(Γ) = (2 + η)Γ -2C C -Γ Proposition 6.5.
For a given setting of arrival rates (λ x , λ y ) satisfying

Γ(λ x , λ y ) = k (16)
and the two inequalities of Prop. 6.2, equilibria (x e , y e ) under this load are on D(k). Besides, for all the equilibrium points in D(k), g satisfies (13) and is constant:

g(x e , y e ) = 1 + η η (C -Γ) (17)
Proof of Prop. 6.5. Let (λ x , λ y ) be a setting of arrival rates satisfying Eq. ( 16) and the two inequalities of Prop. 6.2. We first show that D(k) contains all the potential equilibrium points. By replacing g using Eq. ( 13) in the delay equation [START_REF] Sastry | Nonlinear Systems. Analysis, Stability and Control[END_REF], we obtain Eq. ( 15).

All equilibrium points of arrival settings satisfying ( 16) have the same value of g as Eq. ( 13) holds and gives Eq. ( 17) which is constant in Γ(λ x , λ y ).

Fig. 4 shows Γ-lines in the λ x λ y -plane and their corresponding D(Γ)-lines in the xy-plane. On each such line in the xy-plane, g is constant and thus gradient is orthogonal. From Eq. ( 17), we also know ∂g ∂Γ is negative which justifies the orientation of gradient on the figure. Proposition 6.6. For any (λ x , λ y ) verifying the two inequalities of Prop. 6.2, D(Γ(λ x , λ y )) for does not intersect in first quadrant.

Proof of Prop. 6.6. For Γ(λ x , λ y ) = Γ satisfying the two inequalities of Prop. 6.2, satisfy:

(1+η)Γ > C and C ≥ λ x + λ y F > Γ.
Under this, da dΓ and db dΓ are strictly positive and a is strictly positive. Thus, D(Γ(λ x , λ y )) do not intersect in the first quadrant.

This basically means g is 'feasible'. As a corollary of Prop. 6.6, we give: Corollary 6.7. g can exist in the sense that there are no incompatible constraints resulting from Prop. 6.2.

Proof of Corollary 6.7. Prop. 6.5 gives the value g must have on D(Γ) in order to have equilibria on it and according to Prop. 6.6 these lines do not intersect in the first quadrant where g can be defined. It proves there is no incompatibility in the definition of g. Proposition 6.8. lim

Γ→C - a(Γ) = +∞ ; lim Γ→ C 1+η + a(Γ) = 0 lim Γ→C - b(Γ) = +∞ ; lim Γ→ C 1+η + b(Γ) = -1
Proof of Prop. 6.8. Trivial.

In particular, this last proposition implies that g can be defined in the whole first quadrant using Eq. ( 13) and lines D(Γ).

As of now, we demonstrated that it is feasible to define g so that point of D(Γ) can be equilibria for (λ x , λ y ) on Γ line. Next, we study the stability of the potential equilibria in order to define the additional constraints on m. The only constraint on m coming from existence of equilibrium (Prop. 6.2) is that m(x e , y e ) =

C-(λx+λyF ) ηλyF

. The point where this will hold is not specified and depends on m. Defining m will thus define a mapping of arrival rates (λ x , λ y ) to the actual equilibrium point.

Stability of the equilibria and definition of m

As demonstrated in the previous section, g that doesn't prevent existence of equilibrium is feasible.

We now like to have the constraints that m has to satisfy. We already know from the previous section the range m must cover, but we don't know where they have to be located in xy-plane. In order to get more constraints on m, we study the conditions for stable equilibriums. To do so we rely on Hartman Grobman theorem and the study of the stability of the linearized system. Proposition 6.9. For an equilibrium point (x e , y e ) as defined by Prop. Inequalities of the proposition are obtained using combination of equations ( 21) and ( 22). Proposition 6.2 and 6.9 give sufficient conditions on m to define stable equilibria. Next, we prove that there exists m which stabilizes the system for any arrival setting. Proposition 6.10. There exists an m satisfying the constraints of propositions 6.2 and 6.9 which stabilizes the system for any arrival rates in the shaded region of the Fig. 4. Proof of Prop. 6.10. We prove this by exhibiting one such m. Let m be such that

m(x, y) = e -xy
The m satisfies the constraints of Eq. ( 18) for any λ y in (0, C) as ∂m ∂x = -ye -xy and ∂m ∂y = -xe -xy , are both strictly negative on the interior. Besides, as m ranges from 1 to 0, from the borders (y = 0 and x = 0) to infinity, thus by continuity, there exists an equilibrium point (x e , y e ) where m(x e , y e ) = C-(λx+λyF ) ηλyF for any arrival rates as all D(Γ)-lines enter the first quadrant by one its borders.

Note that if m is strictly monotonic, there is only one equilibrium point for any arrival rate setting in the equilibrium existence region (refer Fig. 3) located at the intersection of the level set of g and m. In addition, it is not possible to apply this for all setting of arrival rates in order to get equilibria for all of them, unless queue are infinite. Proposition 6.11. If queues are finite, some setting of arrival rates can't have equilibrium.

Proof of Prop. 6.11. As a(Γ) tends to 0 when Γ tends to C/1 + η + , and b(Γ) tends to -1, intersection of y = 0 and D(Γ) tends to infinity. Hence, for any x max , it is possible to find Γ close enough to C/1 + η so that equilibrium which have to be on D(Γ) (due to Prop. 6.5) would have to be after x max .

Using limits of a(Γ) and b(Γ) when Γ tends to C, it is possible to pursue the same reasoning and prove that for some settings of arrival rates, there can't be equilibrium under finite queue for large flows.

Thus we see that, the system can attain stability depending on the decision of users, and the control function used by operator.

Game

We summarize our results in the form of a game with two players: operator and user (with a large flow to send). Here, we make the fair assumption that T x < T y . The operator can take one of the two actions:

• AFP: Assume fair play, and not use a g.

• AUP: Assume unfair play, and use a g.

From the users, we consider a collective behaviour.

• UC: Users cheat,

• UR: Users rightful Under AUP, T x = T y . We use preferential ordering of payoffs for both players. That is (a o , a u ) ≺ p (a ′ o , a ′ u ), if player p prefers second strategy over the first. The letter o is used to refer to operator, and u to refer to users.

• (AFP, UR) ≺ u (AFP, UC): Users prefer to cheat when the operator does nothing to stop them from cheating, as this would give them shorter response time in the small queue (when T x < T y ).

• (AUP, UR) ≺ u (AUP, UC): Users also prefer to cheat when the operators are aware and are setting service rates dynamically to achieve stability, as this would ensure a finite queue; hence a finite delay. Observe that, if the don't cheat (and stay in Q y ), there is no equilibrium (from Prop. 6.2); hence the queue will build up without bound.

Therefore, it can be drawn that UC strictly dominates UR under any action of the operator (AFP or AUP). Hence, the action UR can be eliminated [START_REF] Osborne | A Course in Game Theory[END_REF]. So, what lefts to be analysed is the preference of operator under this user action (UR). We see, (AFP, UC) ≺ o (AUP, UC), as there is no equilibrium for general arrival rates (from Prop. 6.2, and if T x remains less than T y , migration will create additional load due to η) leading to overflow.

From the above analysis, (AUP, UC) is a Nash equilibrium in the two-players game. That is, assuming operators and users are rational, users will tend to cheat, and operators will look to stabilize the system to maintain finite queues (when the system is operating near to saturation, depending on η).

Note that if the operator's setting of service rates is such that T x > T y , then migrating to small queue is no more an incentive for large flows. This doesn't preclude operator from favoring small flows as x φx < y φy can still hold. In such a scenario, it can be seen that (AFP, UR) will be a Nash equilibrium. This situation can happen if η is large enough and φy φx can be maintained such that: Second constraint will not be satisfied if operator want to favor small flows too much, say, as in the priority based scheduling proposed in [START_REF] Avrachenkov | Differentiation Between Short and Long TCP Flows: Predictability of the Response Time[END_REF]; meaning that it will be of interest for users to cheat.

Conclusions

Starting from the setting of static service rates, and moving to dynamic service rate settings, we analysed the existence of equilibria. For the existence of equilibria that is of interest to the operator, it is necessary to have control over the service rate as a function of the queue lengths. Even then, not all the stable equilibrium points are of interest to the operator, as they give the same delay to small and large flows. Therefore, if a large number of users cheat, the operator has no visible incentive in deploying a size-based scheduling system.

INRIA

The focus of our study revolved around saturation (of the line capacity) as we assumed that there is some cost η incurred due to migration. In the future, we plan to analyse the system in overload. Similarly, it would be interesting to understand what happens if the operator deploys a mechanism to detect and shift some of the disguised large flows from the small queue to the large queue.

Figure 1 :

 1 Figure 1: Two-queues model.

Proposition 4 . 4 .

 44 (0, y) with y > 0 is an equilibrium point if and only if:

Figure 2 :

 2 Figure 2: Existence region of equilibrium (0, 0) under static service rate.

Figure 3 :

 3 Figure 3: Interior equilibrium existence region under φ x (x) = f (x).

Figure 4 :

 4 Figure 4: Interior equilibrium existence region and mapping of Γ(λ x , λ y ) lines to D(Γ), level sets and gradient field of g(x, y).

  To analyse of the equilibrium point (x e , y e ), we take the Jacobian J of the rate equations (2) and (3) at this point. The partial derivatives ∂g ∂x and ∂g ∂y at (x e , y e ) are obtained from the delay equation, Eq. (7).The equilibrium point (x e , y e ) is asymptotically stable if the eigenvalues of the J at (x e , y e ) have strictly negative real parts[START_REF] Sastry | Nonlinear Systems. Analysis, Stability and Control[END_REF] Ch. 2 & 5]. Characteristic polynomial of J is: From this and equations (19) and (20), real parts of the roots are strictly negative iff:

	(1 + η)	∂m ∂x	-	∂m ∂y		<	(1 + η)C(1 + x e + y e + F ) λ y (F (2 + y e + η) + x e (1 + η)) 2	(21)
	and				∂m ∂x	+	y e + 1 x e + F	∂m ∂y	< 0	(22)
								6.2 if the following equations
	hold:
	    	∂m ∂x	+	y e + 1 x e + F	∂m ∂x ∂y ∂m	(1+ye)C λy(F (2+η+ye)+xe(1+η)) 2 < 0 <	(18)
	then (x e , y e ) is asymptotically stable.
	Proof of Prop. 6.9. ∂g ∂x	(x e , y e ) =	(1 + η)(y e + 1)F C (F y e + x e (1 + η) + F (2 + η)) 2	(19)
		∂g ∂y	(x e , y e ) = -	(1 + η)(x e + F )F C (F y e + x e (1 + η) + F (2 + η)) 2	(20)
								λ 2 +
		(λ y F ( ∂m ∂y -(1 + η) ∂m ∂x ) + ∂g ∂x -∂g ∂y ) λ + η(λ y F ( ∂m ∂x ∂g ∂y -∂m ∂y ∂g ∂x ))

A flow is called small if its size is less than a threshold, θ. In practice, θ bytes of every large flow also go to the small queue. But, we ignore this to keep the model simple. Besides, this affects neither the analysis nor the results given here.RR n°6888

RR n°6888

Acknowledgement

This work was done in the framework of the INRIA and Alcatel-Lucent Bell Labs Joint Research Lab on Self Organized Networks.

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