Fadoua Bouchikhi

Emilie Rossignol

Martine Sancelme

Bettina Aboab

Fabrice Anizon

Doriano Fabbro

Michelle Prudhomme

Pascale Moreau

Synthesis and biological evaluation of diversely substituted indolin-2-ones

Keywords: Indolin-2-ones, Kinases' inhibitors, Antimicrobial agents

Introduction

Increased emergence of bacterial resistance to antibiotic therapy has created an urgent need for the development of new antibacterial agents. As part of our ongoing studies concerning the preparation of potential biologically active compounds, we were interested in the synthesis of indolin-2-one derivatives [1] and [2]. Since various kinases are involved in the growth of microorganisms and because many oxindoles are kinases' inhibitors [3], [4], [5], [6], [7] and [8], these compounds could exert antibacterial activities as it was shown previously for several oxindole derivatives (Fig. 1) [9], [10] and [11].

Fig. 1 We have reported, in a previous paper, the synthesis and antiproliferative activities, toward various human tumor cell lines, of indolin-2-one derivatives substituted in the 3-position by an aminomethylene group bearing different amino acid moieties (Scheme 1) [2]. The amino acid side chains were either flexible (compound A) or more rigid and differently oriented (compounds B and C). The results obtained in these previous structure-activity relationship studies have shown that only compound A, bearing a flexible side chain which allows the folding of this chain, has exhibited antiproliferative activities toward a panel of various human solid tumor cell lines [2]. Therefore, in order to evaluate the influence of the chain length on the biological activities, indolin-2-one derivatives substituted in the 3-position by an aminomethylene group bearing either an ornithine or a lysine residue were synthesized (Fig. 2). Moreover, as already shown in the literature, the substitution of the aromatic moiety by a halogen atom could affect the pharmacological profile in this series [8] and [9]. Therefore, to get an insight on the effect of bromination of the aromatic moiety, on the biological activities, compounds 6 and 8, brominated in the 5-position were also prepared. The inhibitory potencies of the prepared compounds toward a panel of eight kinases were examined. The kinases tested were either receptor tyrosine kinase (RTKs), non-receptor tyrosine kinases (CTKs) or serine/threonine kinases (STKs). The RTKs tested were KDR, IGF-1R, c-Met, and Ret. The CTKs tested were Src and Abl. The STK tested were PKA and CDK2. Finally, the antibacterial activities of the various compounds against two Gram-positive bacteria Bacillus cereus and Streptomyces chartreusis, a Gram-negative bacterium Escherichia coli and a yeast Candida albicans are reported. Fig. 2.

Chemistry

As described in our previous paper [2], the first indolin-2-one derivatives substituted in the 3-position by an aminomethylene group bearing different amino acid moieties were synthesized, in moderate yields, in three steps via the chloromethylene analogue D. Compound D was prepared in 51% yield by a Wittig reaction performed with chloromethyl triphenylphosphonium iodide in the presence of isatine E (Scheme 1). In order to improve the synthesis of these compounds, different approaches requiring either the corresponding 3-(N,N-dimethylaminomethylene)indolin-2-ones or 3-ethoxymethyleneindolin-2-ones were investigated. Compound A which was prepared in 21% overall yield via the chloromethylene derivative [2] was obtained in 70% overall yield from 3-(N,N-dimethylaminomethylene)indolin-2-one 2 (Scheme 2). Compound 2 was prepared as a mixture of Z/E isomers in 89% yield by treatment of indolin-2-one (1) with dimethylformamide-N,N-dimethylacetal (DMF-DMA) in toluene as described in the literature for 4hydroxyquinolinone derivatives [12]. Compound 2 Z/E isomeric ratio (52:48) was determined from the 1 H NMR spectrum, in DMSO-d 6 , on the H 6 signal (E isomer) and the H 4 signal (Z isomer). The Z/E assignment was made according to the NMR data described for compound 2 by Joseph-Nathan and coworkers [13]. The coupling reaction with N-α-Boc-l-lysine was performed in the presence of HCl in ethanol. After deprotection of the α-amino group in acidic medium, compound A was obtained in 78% yield as a Z/E isomeric mixture with the same Z/E isomer ratio (70:30) as the one obtained in the previously described synthetic pathway [2]. Scheme 2.

To get an insight into the influence of the chain length on the biological activity, an ornithyl analogue 3 was prepared using a similar procedure as the one described above for compound A (Scheme 2). The required N-α-Boc-l-ornithine was prepared by hydrogenolysis of the commercially available N-α-Boc-Nδ-benzyloxycarbonyl-l-ornithine [14]. In this case, after the coupling reaction, the final deprotection step of the α-amino function was carried out in the presence of HCl gas generated from concentrated sulphuric acid and sodium chloride. Compound 3 was obtained as a mixture of Z/E isomer in a 65:35 ratio. This ratio was determined from the 1 H NMR spectrum on the signal of the vinylic protons, respectively, at 7.93 and 7.37 ppm. Unfortunately, due to purification problems the coupling/deprotection sequence was performed in only 29% yield of pure isolated 3. Accordingly, we decided to attempt another approach for the preparation of the bromo-analogue 6, involving the 5-bromo-3-ethoxymethyleneindolin-2-one (5) instead of the dimethylaminomethylene intermediate [7]. 5-Bromoindolin-2-one (4) was obtained by bromination of the corresponding indolin-2-one as previously described [15]. The required 5-bromo-3ethoxymethyleneindolin-2-one (5) was obtained as a single isomer in 75% yield by heating the 5bromoindolin-2-one (4) in the presence of triethylorthoformate in acetic acid. The E configuration of the exocyclic double bond was determined from the value of the long range 1 H- 13 C coupling constant between the vinylic proton and the 13 C nucleus of the carbonyl function. Indeed, the configuration of the double bond of α,β-unsaturated carbonyl compounds can be determined on the basis of this long range 3 J ( 1 H, 13 C O) heteronuclear coupling constant. The 3 J(H,C) of 1 H and 13 C O nuclei with a cis orientation with respect to the double bond are usually smaller (2-6 Hz) than those with a trans orientation (8-12 Hz) [16]. In our case, the 3 J(H,C) value of 4 Hz indicated a cis orientation of 1 H and 13 C nuclei corresponding to the E configuration of the exocyclic double bond. Compound 6 was then obtained in 68% yield by coupling 5 with N-α-Boc-L-ornithine in CH 2 Cl 2 followed by deprotection of the α-amino function with HCl gas as described for compound 3. Since the signals of the vinylic protons were superimposed on other signals, compound 6 Z/E isomeric ratio (66:34) was determined from the 1 H NMR spectrum on the signal of H 6 protons, respectively, at 7.00 and 7.07 ppm. Then we planned to use this method for optimizing the preparation of compound 3. Unfortunately, when indolin-2-one (1) was treated with triethylorthoformate in acetic acid, 3-ethoxymethyleneindolin-2-one was obtained in a poor yield excluding further use of this synthetic intermediate. For the preparation of compound 8, the brominated analogue of A, both approaches were carried out. Unfortunately, compound 8 could not be isolated when the coupling/deprotection sequence was performed with the ethoxymethylene intermediate 5 and N-α-Boc-l-lysine. Therefore, compound 8 had to be prepared via the corresponding N,N-dimethylaminomethylene intermediate. So, compound 4, by reaction with DMF-DMA in toluene gave the dimethylaminomethylene derivative 7 in 72% yield as a mixture of Z/E isomers in a 73:27 ratio. The Z/E isomeric ratio was determined from the 1 H NMR spectrum on the signal of the H 6 protons, respectively, at 6.95 and 7.05 ppm. The exocyclic double bond configuration for the two isomers of compound 7 was determined on the basis of the 1 H- 13 C coupling constant between the vinylic proton and the 13 C nucleus of the carbonyl group as described for compound 5. For compound 7 isomeric mixture, a coupling constant value of 9 Hz was determined for the major isomer and of 5 Hz for the minor isomer. Therefore, the major isomer had a Z configuration, as previously described for its analogue 2. The coupling step between compound 7 and N-α-Boc-L-lysine was carried out with HCl in ethanol to lead, after removal of the tert-butoxycarbonyl protective group, to compound 8 in 38% yield. Compound 8 was obtained as a mixture of Z/E isomer a 69/31 ratio. This ratio was determined from the 1 H NMR spectrum on the signal of the H 6 protons, respectively, at 6.99 and 7.06 ppm. To shorten the side chain length, we also tried the reaction between N-α-Boc-2,4-diaminobutyric acid and 3-(N,N-dimethylaminomethylene)indolin-2-one (2). Nevertheless, despite several trials, we never obtained the attempted product. The Z/E isomeric ratios for compounds 3, 6 and 8 were also evaluated by SAM1 [17] quantum semiempirical calculations, using Ampac8.0 program [18], on both isomers' global minima geometries which were found previously by conformational analysis using Monte-Carlo Multiple Method [19] with MM3 force field [20] of Macromodel 7.0 program [START_REF]Shroedinger Inc, 1500 SW First Ave. Suite 1180[END_REF]. The comparison of the E a activation energy (energy difference between the reactants and the transition state) of Z and E isomers for compounds 3, 6 and 8 showed that the Z configuration should be the major one (Table 1). As it was previously described for compound A, the results showed that, for these three compounds, the Z isomer was stabilized by an intramolecular hydrogen bond between the NH hydrogen and the oxygen of the oxindole carbonyl group and that the folding of the flexible side chain was stabilized by an additional hydrogen bond between the α-amino function and the oxygen atom of the oxindole moiety (Table 1).

Table 1.

Computational results of Z/E isomers for compounds 3, 6 and 8

Z 3E

Results and discussion

Kinase inhibition

The in vitro inhibitory potencies of compounds A, 3, 6 and 8 were tested toward a panel of eight kinases. KDR, IGF-1R, c-Met, Ret, Src, Abl, and PKA were tested using the same method as the one described by Traxler et al. with AEE788 as control [START_REF] Traxler | [END_REF] whereas CDK2/Cyclin A assay was performed as previously described [23].

Compounds A, 3 and 8 did not inhibit the kinases tested (IC 50 > 10 µM) whereas compound 6 was active toward CDK2 with an IC 50 value of 6.6 µM. Compound 6 was inactive toward the other kinases tested (IC 50 > 10 µM).

Antimicrobial properties

The antimicrobial activities of compounds A, 3, 6 and 8 were tested against two Gram-positive bacteria (B. cereus and S. chartreusis), a Gram-negative bacterium (E. coli) and a yeast (C. albicans) (Table 2). Ampicillin and Amphotericin B were used as a positive control for the bacterial strains and the yeast, respectively. The most efficient was compound 3 bearing an ornithyl side chain. This compound inhibited clearly the growth of the Gram-positive bacteria tested, particularly that of S. chartreusis. Moreover compound 3 inhibited weakly the growth of E. coli and C. albicans. Compound 6, the brominated analogue of 3, was only active on the Gram-positive bacteria tested. Compounds A and 8, substituted in the 3-position by a lysine residue, had similar inhibition profiles: they both inhibited the growth of all the microorganisms tested except E. coli. Therefore, these preliminary results indicate that the presence of a side chain containing three methylene groups seems to be favourable to the antimicrobial activities of these compounds. In conclusion, indolin-2-one derivatives substituted in the 3-position by an aminomethylene group bearing either an ornithine or a lysine residue were synthesized using different synthetic approaches requiring either the corresponding 3-(N,N-dimethylaminomethylene)indolin-2-one or 3ethoxymethyleneindolin-2-one intermediates. The methods described here are compatible with the presence of a bromine atom in the 5-position. Furthermore, these synthetic approaches are more efficient than the one previously described via the chloromethylene intermediate.

The biological results have shown that compounds 3 and 6 were the most promising, indicating that the presence of an ornithine residue is favourable for the biological activities in this series. The preparation of various analogues of compounds 3 and 6, diversely substituted on the aromatic ring, is currently under investigation, especially to improve their mild inhibitory potencies, compared to that of known CDK2 inhibitors (e.g.: IC 50 roscovitine CDK2/cyclinA = 0.7 µM).

Experimental 4.1. Chemistry

IR spectra were recorded on a Perkin-Elmer Paragon 500 spectrometer (ν in cm -1 ). NMR spectra were performed on a Bruker AVANCE 400 ( 1 H: 400 MHz, 13 C: 100 MHz) or AVANCE 500 ( 1 H: 500 MHz, 13 C: 125 MHz), chemical shifts δ in ppm, the following abbreviations are used: singlet (s), doublet (d), triplet (t), quadruplet (q), doubled doublet (dd), sextuplet (sext), multiplet (m), broad signal (br s). When necessary to identify all carbon atoms, complementary NMR experiments (HSQC, HMBC) were performed on a Bruker AVANCE 500. Mass spectra (ES) were determined on a high resolution Waters Micro Q-toff apparatus. Chromatographic purifications were performed by flash silica gel Geduran SI 60 (Merck) 0.040-0.063 mm column chromatography. For purity tests, TLC were performed on fluorescent silica gel plates (60 F254 from Merck).

(Z/E)-(2S)-2-amino-6-([(2-oxo-indolin-3-ylidene)methyl]amino)hexanoic acid, hydrochloride (A)

To a suspension of (Z/E)-3-(N,N-dimethylaminomethylene)indolin-2-one (2) (100 mg, 0.531 mmol) in ethanol (1.8 mL) were added 37% hydrochloric acid (44 µL, 0.531 mmo1) and N-(α)-Boc-l-lysine (138 mg, 0.560 mmol). The mixture was refluxed for 36 h. The solvent was evaporated and the residue was purified by flash chromatography (ethyl acetate then ethyl acetate/methanol 98:2 to 7:3) to give the coupling product (167 mg) as a grey solid. Ethyl acetate (2 mL) was added to the residue (31.6 mg) and 3 M aqueous hydrochloric acid (1.26 mL) was added. The mixture was stirred at 50 °C for 4 h and then, after evaporation, the residue was precipitated in a minimum of diethylether. Elimination of the liquid phase led to compound A (25.6 mg, 0.0785 mmol, 78% yield for the two steps) as a grey solid.

(Z/E)-(2S)-2-amino-5-([(2-oxo-indolin-3-ylidene)methyl]amino)pentanoic acid, hydrochloride (3)

To a suspension of (Z/E)-3-(N,N-dimethylaminomethylene)indolin-2-one (2) (156 mg, 0.83 mmol) in ethanol (3 mL) were added 37% hydrochloric acid (69 µL, 0.83 mmol) and N-(α)-Boc-l-ornithine (231 mg, 0.995 mmol). The mixture was stirred at 50 °C for 48 h. The solvent was evaporated and the residue was purified by flash chromatography (ethyl acetate then ethyl acetate/methanol 98:2 to 8:2). Dichloromethane (3 mL) was added to the residue and the mixture was stirred at room temperature for 1.5 h while bubbling HCl through the mixture. Solvent was evaporated in an argon flux and the residue was precipitated in a minimum of diethylether. Elimination of the liquid phase led to compound 3 (74.

(E)-5-bromo-3-ethoxymethyleneindolin-2-one (5)

A suspension of 5-bromooxindole (4) (1.05 g, 4.95 mmol) in acetic acid (20 mL) was warmed until complete dissolution and triethylorthoformate (8.5 mL, 51.1 mmol) was added at this temperature. The reaction mixture was stirred at reflux for 5 h. The solvent was removed under reduced pressure and the residue was co-evaporated twice with toluene. The residue was dissolved in warm toluene (20 mL), cyclohexane was added (400 mL) and the solution was left to precipitate for 12 h. The precipitate was collected by filtration to give 5 (997 mg, 3.72 mmol, 75% yield) as a yellow powder. Mp 180-182 °C. IR (KBr): 3440, 1700, 1680, 1650, 1625. HRMS (ES+) calcd for C 11 H 11 79 BrNO 2 (M + H) + 267.9973, found 267.9967. 1 H NMR (400 MHz, DMSO-d 6 ): 1.37 (3H, t, J = 7.0 Hz), 4.40 (2H, q, J = 7.0 Hz), 6.77 (1H, d, J = 8.0 Hz), 7.27 (1H, dd, J 1 = 8.0 Hz, J 2 = 2.0 Hz), 7.52 (1H, d, J = 2.0 Hz), 7.74 (1H, s), 10.36 (1H, s, NH). 13

(Z/E)-(2S)-2-amino-5-([(5-bromo-2-oxo-indolin-3-ylidene)methyl]amino)pentanoic acid, hydrochloride (6)

To a suspension of (E)-5-bromo-3-ethoxymethyleneindolin-2-one (5) (500 mg, 1.86 mmol) in dichloromethane (10 mL) was added N-(α)-Boc-l-ornithine (269 mg, 1.16 mmol). The mixture was stirred at reflux for one night. The solvent was evaporated and a mixture CH 2 Cl 2 /cyclohexane 1:1 was added to the residue before precipitation with a few drops of diethylether. The precipitate was filtered off to give the coupling product (415 mg) as a yellow-orange solid. Dichloromethane (8 mL) was added to the residue (227 mg) and the mixture was stirred at room temperature for 1.5 h while bubbling HCl through the mixture. Solvent was evaporated in an argon flux and the residue was precipitated in a minimum of diethylether. Elimination of the liquid phase led to compound 6 (168 mg, 0.43 mmol, 68% yield for the two steps) as a yellow-orange solid. HRMS (ES-) calcd for C 14 H 15 79 BrN 3 O 3 (M -H) -352.0297, found 352.0307. IR (KBr): 3000-3600, 1737, 1654. NMR of the major Z isomer: 1

(Z/E)-(2S)-2-amino-6-([(5-bromo-2-oxo-indolin-3-ylidene)methyl]amino)hexanoic acid, hydrochloride (8)

To a mixture of (Z/E)-5-bromo-3-(N,N-dimethylaminomethylene)indolin-2-one (7) (57 mg, 0.21 mmol) in ethanol (2 mL) were added 37% hydrochloric acid (18 µL, 0.21 mmol) and N-(α)-Boc-l-lysine (64 mg, 0.26 mmol). The mixture was refluxed for 48 h. The solvent was evaporated and the residue was purified by flash chromatography (ethyl acetate/methanol 9:1). The residue was dissolved in ethyl acetate, precipitated with cyclohexane, filtered and washed with dichloromethane. Diethylether (3 mL) was added to the residue and the mixture was stirred at room temperature for 1.5 h while bubbling HCl through the mixture. The precipitate was collected by filtration under argon, washed with dichloromethane to give

Kinases' inhibition

The in vitro kinase assays were performed in 96-well plates (30 µl) at ambient temperature for 15-45 min using the recombinant glutathione S-transferase-fused kinase domains (4-100 ng, depending on specific activity) prepared previously [24] and [25]. [γ 33 P]ATP was used as phosphate donor and polyGluTyr-(4:1) peptide as acceptor. With the exception of protein kinase A for which the heptapeptide Leu-Arg-Arg-Ala-Ser-Leu-Gly (known as Kemptide Bachem; Bubendorf, Switzerland) was used as peptide substrate. Assays were optimized for each kinase using the following ATP concentrations:

EE

  a activation energy (kcal mol -1 ) 26.0 34.0 E a (E) -E a (Z) (kcal mol -1 ) 8.0 Theoretical ratio Z/E 99.9 0.1 Experimental ratio Z/E 65 35 6 Z 6E E a activation energy (kcal mol -1 ) 30.0 38.0 E a (E) -E a (Z) (kcal mol -1 ) 8.0 Theoretical ratio Z/E 99.9 0.1 Experimental ratio Z/E 66 34 a activation energy (kcal mol -1 ) 23.0 26.0 E a (E) -E a (Z) (kcal mol -1 ) are represented by grey lines.

  C NMR (100 MHz, DMSO-d 6 ): 15.4 (CH 3 ), 72.1 (CH 2 ), 110.8, 124.2, 129.1, 158.3 (CH), 106.2, 112.5, 138.4 (C), 168.9 (C O).

  H NMR (400 MHz, DMSOd 6 ): 1.59-1.91 (m, 4H, CH 2 ), 3.33-3.44 (m, 2H, CH 2 ), 3.89-3.98 (m, 1H, C α H), 6.73 (d, 1H, J = 8.0 Hz), 7.00 (dd, 1H, J 1 = 8.0 Hz, J 2 = 2.0 Hz), 7.48 (d, 1H, J = 1.5 Hz), 8.06 (d, 1H, J = 13.5 Hz, CH vinylic ), 8.38-8.53 (br s, 3H, NH 3 Cl), 8.82-8.92 (m, 1H, NH), 10.32 (s, 1H, NH indole ). The signal of the carboxylic acid proton is not visible on the spectrum. 13 C NMR (100 MHz, DMSO-d 6 ): 26.2, 26.9, 47.7 (CH 2 ), 51.5 (CH), 110.3, 117.5, 124.0, 149.1 (CH), 93.7, 112.0, 127.9, 134.5 (C), 169.2, 170.9 (C O).4.1.5. (Z/E)-5-bromo-3-(N,N-dimethylaminomethylene)indolin-2-one (7)DMF-DMA (2.66 mL, 20.0 mmol) was added to a solution of 5-bromooxindole (4) (2.12 g, 10.0 mmol) in toluene (40 mL). The mixture was stirred for 48 h, the precipitate was collected by filtration and washed with dichloromethane to give 7(1.91 g, 7.2 mmol, 72% yield) as a beige solid. HRMS (ES+) calcd for C 11 H 12 79 BrN 2 O (M + H) + 267.0133, found 267.0132. IR (KBr): 3445, 1675, 1615, 1590. NMR of the major Z isomer: 1 H NMR (400 MHz, DMSO-d 6 ): 3.23-3.31 (br s, 3H), 3.61-3.69 (br s, 3H), 6.62 (d, 1H, J = 8.0 Hz), 6.95 (dd, 1H, J 1 = 8.0 Hz, J 2 = 2.0 Hz), 7.48 (d, 1H, J = 1.5 Hz), 7.67 (s, 1H), 10.07 (s, 1H). 13 C NMR (100 MHz, DMSO-d 6 ): 42.3, 46.5 (CH 3 ), 109.3, 116.8, 123.5, 149.0 (CH), 92.1, 111.7, 131.4, 134.5 (C), 165.6 (C O).

4 . 3 .

 43 1.0 µM (c-Met and RET), 5.0 µM (c-Abl), 8.0 µM (KDR, IGF-1R) and 20.0 mM (c-Src and PKA). The reaction was terminated by the addition of 20 µl 125 mM EDTA. Thirty µl (c-Abl, c-Src, IGF-1R, RET) or 40 µl (all other kinases) of the reaction mixture was transferred onto Immobilon-polyvinylidene difluoride membrane (Millipore, Bedford, MA), pre-soaked with 0.5% H 3 PO 4 and mounted on a vacuum manifold. Vacuum was then applied and each well rinsed with 200 µl 0.5% H 3 PO 4 . Membranes were removed and washed four times with 1.0% H 3 PO 4 and once with ethanol. Dried membranes were counted after mounting in a Packard TopCount 96-well frame and with the addition of 10 µl/well of Microscint. For RET kinase assay, either glutathione S-transferase-wild-type RET (15 ng) or glutathione Stransferase-RET-Men2B protein (15 ng) were used. The CDK2/Cyclin A assay was performed as previously described[23]. Antibiogram tests Four strains were tested, two Gram-positive bacteria (B. cereus ATCC 14579, S. chartreusis NRRL 11407), a Gram-negative bacterium (E. coli ATCC 11303) and a yeast (C. albicans 444 from the Pasteur Institute, Paris). The antimicrobial activity was determined by the conventional paper disk (Durieux No. 268; 6 mm in diameter) diffusion method using the following nutrient media: Mueller-Hinton (Difco) for B. cereus and E. coli, Sabouraud agar (Difco) for C. albicans and Emerson agar (0.4% beef extract, 0.4% peptone, 1% dextrose, 0.25% NaCl, 2% agar, pH 7.0) for the Streptomyces strains. Paper disks impregnated with solutions in DMSO (300 µg of drug per disk) were placed on Petri dishes. Growth inhibition was examined after 24 h incubation at 27 °C.

Table 2 .

 2

	Cpds	B. cereus	S. chartreusis	E. coli	C. albicans
		ATCC 14579 NRRL 11407	ATCC 11303	IP 444
	Controls Amp. ++++	Amp. >++++ (25) Amp. >++++ (31) Amph. B. ++++
	A	±	±	-	±
	3	+	+++	±	±
	6	+	++	-	-
	8	±	+	-	±

Antimicrobial activities of compounds A, 3, 6, 8 (300 µg per disk) against two Gram-positive bacteria (B. cereus and S. chartreusis), a Gram-negative bacterium (E. coli) and a yeast (C. albicans)

  The signal of the carboxylic acid proton is not visible on the spectrum.13 C NMR (100 MHz, DMSO-d 6 ): 26.4, 27.0, 47.5 (CH 2 ), 51.6 (C α H), 108.6, 115.0, 119.7, 122.0, 147.8 (CH), 94.5, 125.3, 135.6 (C), 169.5, 170.8 (C O).

1 mg, 0.238 mmol, 29% yield for the two steps) as a grey solid. HRMS (ES-) calcd for C 14 H 16 N 3 O 3 (M -H) -274.1192, found 274.1192. IR (KBr): 3420, 1740, 1665, 1620. NMR of the major Z isomer: 1 H NMR (400 MHz, DMSO-d 6 ): 1.59-1.92 (m, 4H, CH 2 ), 3.31-3.42 (m, 2H, CH 2 ), 3.83-3.91 (m, 1H, C α H), 6.73-6.89 (m, 3H), 7.27 (d, 1H, J = 7.0 Hz), 7.93 (d, 1H, J = 13.0 Hz, CH vinylic ), 8.35-8.69 (br s, 3H, NH 3 Cl), 8.70-8.79 (m, 1H, NH), 10.20 (s, 1H, NH indole ).

  8 (33 mg, 0.082 mmol, 38% yield) as a black solid. HRMS (ES+) calcd for C 15 H 19 79 BrN 3 O 3 (M + H) + 368.0610, found 368.0607. IR (KBr): 3410, 1735, 1665, 1615. NMR spectra of the major Z isomer: 1 H NMR (400 MHz, DMSO-d 6 ): 1.30-1.52 (m, 2H, CH 2 ), 1.53-1.67 (m, 2H, CH 2 ), 1.77-1.89 (m, 2H, CH 2 ), 3.30-3.50 (m, 2H, CH 2 ), 3.84-3.93 (m, 1H, C α H), 6.72 (d, 1H, J = 8.0 Hz), 6.99 (d, 1H, J = 7.5 Hz), 7.48 (s, 1H), 8.04 (d, 1H, J = 13.5 Hz), 8.19-8.50 (br s, 3H, NH 3 Cl), 8.82-8.93 (m, 1H), 10.24-10.34 (br s, 1H, NH). The signal of the carboxylic acid proton is not visible on the spectrum. 13 C NMR (125 MHz, DMSO-d 6 ): 21.2, 29.4, 29.9, 47.9 (CH 2 ), 51.7 (C α H), 110.1, 117.3, 123.7, 149.0 (CH), 93.4, 111.8, 127.8, 134.3 (C), 169.1, 170.8 (C O).

Acknowledgements

This work was partly supported by a postdoctoral fellowship (Emilie Rossignol) from the 6th European Programme Protein Kinase Research.


domain: math info

text:
Selcuk Muhammed
email: muhammed-selcuk.kok@inria.frinria

Kok

Is Okun's law still valid in France?

This paper analyzes the Okun's law in France, over the 1982-2021 period. We find that the variation in Gross Domestic Product (GDP) implies a variation in unemployment with a coefficient of 1 : 3. We furthermore analyze the impact of a GDP shock on unemployment with a VAR model.

Introduction

This paper is interested in analyzing the relationship between unemployment and GDP, according to the notorious Okun's law. The so called Okun's law was named after Arthur Okun in 1962, and highlights a relationship of 1 : 3; this means that when the GDP increases by 3%, unemployment drops by 1%.

Contributions We analyze the 1982-2021 period. The first objective is to determine the optimal model for the fit. The second is to analyze Okun's coefficient over the period from 1992 to 2021, in order to observe the trend modification of this coefficient and during crises.

This paper is organized as follows: Section 2 will review all the literature on Okun's law necessary to develop this paper. Section 3 will present the data and their sources. Section 4 will present the methods used in the development of the model. Section 5 will present the results. Finally, Section 6 concludes the paper.

Related work: Okun's law

The seminal paper is entitled "Potential GNP: its measurement and significance" and is by Arthur Okun in 1962; he develops a correlation model between GDP and unemployment in the United States. He finds a coefficient of 1:3, a 3% increase in GDP corresponds to a 1% decrease in unemployment [START_REF] Arthur M Okun | Potential GNP: its measurement and significance[END_REF].

In 1993 Prachowny analyzed the theoretical foundations of Okun's law and the validity of Okun's coefficient, he concluded that the ratio 1:3 is actually 1:1.5. "Changes in weekly hours and capacity utilization have independent effects on the output gap" [START_REF] Martin | Okun's law: theoretical foundations and revised estimates[END_REF].

In 2000, Lee measures the robustness of Okun's law after the war in 16 OECD countries. Okun's law is statistically valid for most countries. Quantitative estimates -as opposed to qualitative estimates-are far from uniform [START_REF] Lee | The robustness of okun's law: Evidence from oecd countries[END_REF].

In 2007, Knotek performed a dynamic analysis of Okun's law and showed that Okun's coefficient was not stable over time [START_REF] Edward | How useful is okun's law?[END_REF].

In 2013, Binet analyzes Okun's law in the regions of France, he concludes that Okun's law is valid for the 14 regions analyzed [START_REF] Binet | Okun's law in the french regions: a cross-regional comparison[END_REF].

In 2014, Karfakis verifies Okun's law for Greece and analyzes with a VAR model the response of unemployment to a shock on GDP [START_REF] Karfakis | Does output predict unemployment? a look at okun's law in greece[END_REF].

In 2017, Ball analyzes Okun's law in 20 countries on the adaptability of short-term unemployment. He finds that Okun's law exhibits a strong relationship in most countries and quite stable over time. Okun's coefficient varies from one country to another, depending on the country's economic situation [START_REF] Ball | Okun's law: Fit at 50?[END_REF].

Data

The data we leveraged in this study comes from INSEE 1 Let us briefly analyze the evolution of GDP and unemployment. GDP is a series with a positive trend, we can easily imagine a straight line representing the trend component of GDP and see the differences. Between 1989 and 1993, we observe that the GDP shifts positively from its trend, it is exactly over this same period that we observe the drop in unemployment. Thereafter, in 1993 to 1995 the GDP resumes its level of equilibrium, there is a return to the trend. Yet, unemployment reacts to this variation with an increase. The same thing can be observed during the periods of financial crisis 2009 and covid crisis in 2020.

We can say just by looking at these two figures, Figure 1 and Figure 2 that when GDP increases unemployment decreases. But there is an important point if we look at stable periods, the GDP increases due to its positive trend, but the unemployment does not vary or varies little. It is for this reason that we will do stationary transformations the series thereafter to have only the effect of the variation and not the effect of the trend. To remove the effect of the trend, the GDP is transformed into a growth rate, and unemployment is transformed into a change in percentage points. We can see these transformed series in Figure 3 and Figure 4. Figure 4 is not too readable because of the two crises that have taken place which lead to extreme variations. But in Figure 3 we can very well observe the interaction between the two series when the growth rate of GDP is high (i.e. GDP increases more than its trend) the change in unemployment is negative (i.e. unemployment decreases) and when the growth rate is low (i.e. GDP increases less than the trend or falls) the change in unemployment is positive (unemployment is increasing).

Methodology

After the transformations carried out on the GDP and unemployment series, detailed in the previous paragraph. We observe in Figure 3 and Figure 4 that the series is stationary, that is to say that they have no trend. We check the presence of a unit root in the series, which amounts to checking if the series have a trend.

We test the unit root of the series, that is to say if the series is stationary and have no trend. In Table 1, we use the augmented Dickey Fuller test and the Kwiatkowski-Phillips-Schmidt-Shin test. These tests confirm the stationarity.

Details of the tests :

• ADF : The null hypothesis for the ADF test is that there is a unit root, i.e. non-stationary series. The P -value is less than 5% for ∆GDP and ∆unemployment, we reject the H0 hypothesis so the series has no unit root and they are stationary, with no trend.

• KPSS : The null hypothesis for the KPSS test is that there is no unit root, i.e. a stationary series(the reverse of the ADF test). P-value is greater than 5% for ∆GDP and ∆unemployment, we accept the H0 hypothesis so the series has no unit root and they are stationary, with no trend (same result with two different tests).

Correlated variables means they are dependent, here we calculate the correlation coefficient i.e. the degree of dependence. A correlation coefficient is between -1 and 1. A correlation coefficient equal to -1 means a perfect negative dependence, 1 means perfect dependence, and 0 means no dependence. So the greater the absolute value of the correlation coefficient, the greater the dependence between the variables concerned. We choose the lag for the GDP, in the Table 2 we have the correlation values between ∆u and ∆GDP t+i i ranging from -4 to 4. We select lag q = 2 for ∆GDP because it is the highest correlation value. The correlation value between GDP and unemployment and negative is in line with Okun's theory Autocorrelation measures the correlation of a variable with itself, when the observations are considered with a time lag. Autocorrelation makes it possible to detect regularities, repeated patterns in a signal such as a periodic signal disturbed by a lot of noise, or a fundamental frequency of a signal which does not actually contain this fundamental. We calculate the auto-correlation of unemployment, the highest correlation value is ∆u t-2 so we choose a lag p = 2. Unemployment has an autocorrelation coefficient equal to 0.29 for a lag = 2. This means that unemployment is self-fulling, the rise in unemployment in the past will induces the rise in unemployment in the present and in the future. The highest correlation is to lag = 2, our data being quartered, this means that the effect of the variation in unemployment impacts unemployment over the next 6 months.

We define a model with q lag for GDP and p lag for unemployment and the dummies for the crash : Lag q for GDP is equally at 2 and lag p for unemployment is also equal at 2. We implement two dummy variable, DFC for dummy financial crisis and DCC for the dummy covid crisis.The regression equation is : In the Table 5 we have some statistic and test on residuals :

∆u t = α 0 + β 1 ∆u t-1 + β 2 ∆u t-2 + γ 1 ∆GDP t-1 + γ 2 ∆GDP t-
• Omnibus p-value < 0.05 means the data does not follow a normal law. A linear regression approach would probably be better than random guessing, but likely not as good as a nonlinear approach.

• Durbin Watson is a test for homoscedasticity, statistic close to 2 means there is no linear autocorrelation on residuals.

• Jarque-Bera p-value < 0.05 means the data does not follow a normal law, it tests more precisely if the Skewness and the Kurtosis are respectively different from 0 and 3.

• Condition number value is larger than 30, the multicollinearity is regarded as strong.

In the Figure 8 we can see a distribution plot of residuals, residues appear to be white noise.The tests in Table 5 we can draw this conclusion, part of the variance is unexplained, by using other exogenous variables we could have a more powerful predictive capacity, but in the context of the GDP and unemployment relationship we have a robust model. We did Granger causality test between unemployment and GDP with lag equal to 2. Since the p-value is less than 0.05, we can reject the null hypothesis of the test and conclude that knowing the value of GDP is useful for predicting the future value of unemployment.

Results

The Okun's ratio for the period 1982 to 2021 is equal to -0.3035. Which means that when the GDP increases by 3%, unemployment drops by 1%, inversely.This ratio is composed of a short term effect equal to the sum of the GDP coefficients, -0.1328.And a long-term effect which is the sum of the unemployment coefficients, 0.5622.The short-term effect is the direct effect of GDP on unemployment, it is negative, it confirms Okun's law which highlights the negative relationship between GDP and unemployment. The long-term effect is the self-fulfilling effect of unemployment. It is positive, which means that the effect of the increase in unemployment in Q-2 and Q-1 will have a significant impact on unemployment in Q.

We calculated the Okun's ratio over 117 periods using a sliding window of 10 years (40 quarters).

The formula of extended version (Figure 9):

Regression model :

∆u t = α 0 + β 1 ∆u t-1 + β 2 ∆u t-2 + γ 1 ∆GDP t-1 + γ 2 ∆GDP t-2 + t Short term effect = γ 1 + γ 2 Long term effect β 1 + β 2 Okun's ratio = γ1+γ2 1-(β1+β2)
The formula of restricted version (Okun's correlation model):

Regression model : ∆u t = α 0 + γ∆GDP t + t

Okun's ratio = γ

We can indeed see that the Okun ratio is not stable over time, Figure 8 illustrates the evolution of this ratio since 1992. However, we can see -0.3 as an equilibrium level. The Okun's ratio is very strongly impacted by the crisis (2009 financial crisis and 2020 covid crisis). The impacts, according to the two crises are different. During the financial crisis of 2009 this ratio reached more than -0.6. A 1% increase in GDP reduced unemployment by 1.66%. During the covid crisis the ratio reached -0.1 which means that the 1% increase in GDP generated a 10% drop in unemployment, which is 6 times more.

We can draw this conclusion, the drop in GDP during the covid crisis had a more negative impact on unemployment than the financial crisis. But the post-crisis recovery was much more beneficial for unemployment during the covid crisis than the financial crisis. In Figure 9 we can see the components of Okun's ratio with the short term and long term component. The variation of its components is more spectacular than the variation of the Okun's coefficient, which varies little over time. So the effect of the change in GDP on the change in unemployment varies little or not at all over time. What really varies is the response time of unemployment to GDP. Which seems logical if we think in the context of a company. Depending on the nature of the event (crisis or post-crisis) the company will take more or less time to lay off or recruit, depending on its future needs.

Figure 9: The Okun ratio and its components

In the Figure 10 we can see how unemployment responds to a shock to GDP. We can see that a positive shock to GDP causes unemployment to fall over the next 10 quarters. The major effect is between the first and the second quarter to come. And we do not see any inverse effect, which means that a positive shock to GDP has a constant positive effect of unemployment. In the Figure 11 we can see the cumulative effect. We can interpret these two curves as follows: GDP is a variable with a positive trend, i.e. GDP is constantly increasing. This trend is called potential GDP. In this context, if the real GDP goes beyond the potential GDP, then unemployment will go below its current equilibrium level.

Conclusion

The two major crises, the financial crisis and the covid crisis, have indicated us that the Okun coefficient in France is very sensitive to these. The effects are different depending on the type of economic crises: we saw Okun's coefficient being multiplied by two during the financial crisis, and divided by three during the covid crisis. Okun's law is certainly valid even during crises, but we can no longer afford to set a rate as before, as for example 1:3, and make our predictions and forecast with such a fixed ratio. It is also important to note the lasting effect on unemployment induced by a shock to GDP. For the financial crisis, unemployment in France was 7% with the crisis, unemployment reached to 9.25% after the crisis, and it never reached its pre-crisis level again (same for the covid crisis).

In conclusion, Okun's law is still valid in France, and a coefficient of 1:3 can be used in normal times to easily predict the evolution of unemployment. Yet in times of crisis, a statistical model will have to be used to calculate the effect of that crisis for deducing from the Okun's coefficient, and make a more precise prediction of unemployment.

  and FRED DATA 2 . GDP and unemployment in France are in quarterly series. The first two figures illustrate the series between 1982 to 2000 and 2000 to 2021. The two following figures are the quarterly evolution of GDP and unemployment between 1982 to 2000 and 2000 to 2021.

Figure 1 :Figure 2 :

 12 Figure 1: Real GDP and unemployment rate in France (1982:2000)

Figure 3 :Figure 4 :

 34 Figure 3: Changes in France's Real GDP and unemployment rate (1982:2000)

Figure 5 :

 5 Figure 5: Actual and predicted unemployment change (in sample)

Figure 6 :

 6 Figure 6: Distribution of residuals of the regression model

Figure 7 :

 7 Figure 7: Actual and predicted unemployment change (out sample : Covid crisis period)

Figure 8 :

 8 Figure 8: Okun's ratio for France between 1992 and 2021 (Extended version)

Figure 10 :

 10 Figure 10: Response of ∆unemployment to a shock on ∆GDP

Figure 11 :

 11 Figure 11: Cumulative response of ∆unemployment to a shock on ∆GDP

Table 1 :

 1 Unit root tests

	Variable	ADF	KPSS
	∆GDP	-12.313703	0.202515
		p-value: 0.0000001	p-value: 0.1
		Critical Values:	Critical Values:
		1% : -3.473	1% : 0.739
		5% : -2.880	5% : 0.463
		10%: -2.577	10%: 0.347
	∆unemployment	-4.955604	0.135499
		p-value: 0.000017	p-value: 0.1
		Critical Values:	Critical Values:
		1% : -3.474	1% : 0.739
		5% : -2.881	5% : 0.463
		10%: -2.577	10%: 0.347

Table 2 :

 2 Cross correlations of unemployment with GDP

	∆GDPt-4	∆GDPt-3	∆GDPt-2	∆GDPt-1	∆GDPt
	-0.291830	-0.360521		-0.666203	-0.552782	0.462601
	∆GDPt+1	∆GDPt+2	∆GDPt+3	∆GDPt+4
	-0.140414	-0.034988		-0.020162	0.13383
		∆ut-4	∆ut-3	∆ut-2	∆ut-1
		0.057445	0.174961	0.294040	0.243801
		∆ut+1	∆ut+2	∆ut+3	∆ut+4
		0.013995	0.296894	0.141647	0.089200

Table 3 :

 3 Cross correlations of unemployment with GDP and our graphical analyzes carried out in the data part. This means that the more GDP increases, the more unemployment decreases.

Table 5 :

 5 2 + DF C 2009 + DCC 2020 + t Tests on residuals

		coef	std err	t	P>ltl
	const	0.0586	0.018	3.279	0.001***
	∆GDPt-1	-0.1023	0.009	-11.233	0.0001***
	∆GDPt-2	-0.0306	0.015	-2.103	0.037**
	∆ut-1	0.1943	0.081	2.413	0.017**
	∆ut-2	0.3679	0.072	5.125	0.0001***
	DF C2009	0.4627	0.146	3.166	0.002***
	DCC2020	1.1253	0.288	3.910	0.0001***
		Table 4: Regression

Table 6 :

 6 Granger causality test

	F test	F=55.1666	p=0.00001
	Chi2 test	chi2=113.9866	p=0.00001
	Likelihood ratio test	chi2=85.5686	p=0.00001

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