Marie Varin

Angèle Chiaroni

Jean-Yves Lallemand

Bogdan Iorga
email: iorga@icsn.cnrs-gif.fr

Catherine Guillou
email: guillou@icsn.cnrs-gif.fr

MeOH Meona

A new access to dihydrotropones through ring expansion of spirocyclohexadienones: Synthesis and mechanism §

Keywords: dienone, dihydrotropone, rearrangement, DFT calculations

In this paper we report the rearrangement of spirocyclohexadienones into dihydrotropones in basic conditions as a new method for the preparation of seven-membered ring ketones, which are key building blocks for the synthesis of tropoloalkaloids. DFT calculations and deuterium labeling studies support the mechanism we propose for this rearrangement, involving the ring opening of a spirocyclopropane intermediate followed by successive base-catalyzed 1,3-hydrogen shifts. The X-ray 2 structure of the resulting dihydrotropone shows near-perfect planarity and the conjugation gain is likely to be the driving force of the reaction.

Introduction

Substituted carbocyclic seven-membered rings are frequently found in natural and medicinally active compounds (e.g. hinokitiol, [START_REF] Miyashita | [END_REF] thapsigargins, 2 colchicine 3,4 ), and several methods for their synthesis are available, [5][6][7] among them, cycloadditions of allyl cations with 1,3-dienes, [8][9][10] metal-catalyzed cycloadditions, [11][12][13][14][15][16][17][18][19][20] electrochemical oxidations, 21 radical, [22][23][24] acidic 25,26 or thermal 27,28 ring expansion reactions.

We describe here a new method, the rearrangement of spirocyclohexadienones to one-carbon ring expansion products, dihydrotropones, under basic conditions. 29 A mechanism for this reaction is also proposed, which is fully supported by DFT calculations and deuterium labeling studies.

Results and Discussion

Reaction conditions. In basic conditions, we observed conversion of the dienone 1 30 into two products: a one carbon ring expansion product (2a) and a tricyclic ester (3) resulting from the lactone ring opening (Scheme 1). These two compounds are obtained in various ratios according to the reaction conditions. Scheme 1. Rearrangement and lactone ring opening products of spirocyclohexadienone 1 in basic conditions. Throughout this study, the following naming system is used: (a) for the keto forms; (b) for the enol forms; (c) for the enolate forms (see Table 1 for more details).

The presence of an efficient protonating agent seems to be essential for the rearrangement of 1 into 2a to occur. Several systems have been investigated at different temperatures (-78 °C to 75 °C): i) in the absence of protonating agent (NaH/THF) no reaction is observed; ii) in the presence of stoichiometric quantity of protonating agent (DBU/CH 2 Cl 2 , potassium phthalimide/THF, lithium N-methyltoluenesulfonamide/THF)

we observed the suppression of the lactone ring opening, a slow reaction rearrangement and low yields of 2a (5-30 %); iii) when the protonating agent is the solvent (MeONa/MeOH, EtONa/EtOH, i-PrONa/i-PrOH) we observed a fast reaction and the formation of two compounds: the rearrangement product 2a and the tricyclic ester 3. The best results are obtained with MeONa in MeOH at 40 °C for 30 min, the product 2a being isolated in 75 % yield in these conditions.

During the study of the reaction of 1 with MeONa in MeOH at 40 °C we evidenced two competitive reactions: i) the rearrangement described above, leading to 2a after protonation and ii) the intermediate formation of the ester 3, [START_REF]The compound 3 can be prepared in a straightforward manner in acidic medium (TFA, MeOH) from spirocyclohexadienone[END_REF] through the nucleophilic opening of the lactone ring followed by a Michael addition of the resulting phenoxide 8 (Scheme 2). The formation of compound 3 proved to be reversible and the equilibrium is shifted towards the thermodynamic product 2a (Figure S1, Supporting information). Figure 1 shows that the ester 3 has completely disappeared after 10 min, but the best yields of 2a are obtained after 30 min at 40 °C. At lower temperatures (-78 °C to 20 °C), the ester 3 is almost exclusively observed. When the reaction of 1 with CD 3 ONa in CD 3 OD at 40 °C is followed by [START_REF] Miyashita | [END_REF] H-NMR (Figure S2, Supporting information) it is noteworthy that: i) an equilibrium between three species (5, 7c and 9) is immediately established, which is completely shifted towards the delocalized anion 7c after 30 min; ii) the deuterium exchange is very fast and all the enolisable positions are completely deuterated after 3 min. The complete incorporation of deuterium into both positions α to the ketone in 5 proves the existence of a dynamic equilibrium between the ester 5 and the open bicyclic phenoxide 9. [START_REF] Highet | The compound 9 is the anionic phenoxide form rather than the neutral phenol form, as indicated by the important shift of the aromatic protons observed in its 1 H-NMR spectrum[END_REF] Moreover, the reversibility of this process is confirmed by the exclusive formation of the rearrangement product 2a when the ester 3 is treated with MeONa/MeOH at 40 °C.

The rearrangement product 2a, in the presence of CD 3 ONa in CD 3 OD at room temperature, is completely deprotonated and immediately forms the enolate 6c, which then undergoes a slow deuterium exchange, complete after 20 h, leading to the enolate 7c (Figure S3, Supporting information).

Reaction mechanism. In the light of the results presented above, we propose the following mechanism for this rearrangement (Scheme 3): the deprotonation of the lactone 1, followed by the intramolecular Michael addition of anion 10 leads to the cyclopropyl derivative 11, which is transformed into the enolate 12c by a cyclopropane ring opening. [START_REF] Kende | A single example of fused tricyclic cyclopropane intermediate similar to 11 has been previously proposed as intermediate in the transformation of phenolic nitroalkanes into tropone derivatives, but the reaction conditions and the overall mechanism are different from ours, which lead to the dihydrotropone derivative 2a[END_REF] The enolate 12c is then converted into the final protonated product In order to support the proposed mechanism, the structures of intermediates and transition states were optimized at the B3LYP level with the 6-31++G(d,p) basis set using Gaussian 03 [START_REF] Frisch | Gaussian 03 revision B.05; Gaussian[END_REF] (Scheme 3). Intrinsic reaction coordinate (IRC) path calculations are in full agreement with a spirocyclopropane intermediate 11,

which rearranges to give a seven-membered ring enolate 12c. The activation barrier energies for TS1 and TS2 (9.3 and 4.7 kcal/mol, respectively) are not very high, which is consistent with the mild reaction conditions used for this transformation. The driving force of the rearrangement is likely to be the greater extent of conjugation in 2a than in 1, confirmed by the near-perfect planarity of the X-ray structure.

Whereas the mechanism for the conversion of 1 into 12c is relatively intuitive, the second part of the mechanism presented in Scheme 3, the conversion of the enolate 12c into the rearrangement product 2a, is not so obvious. We have formulated two hypotheses: a) a sigmatropic hydrogen shift and b) a basecatalyzed 1,3-hydrogen shift.

a) Sigmatropic hydrogen shift. Three pathways can be envisaged for the conversion of 12b into 2b through a sigmatropic mechanism: i) one direct [1,3] hydrogen shift; ii) three consecutive [1,5] hydrogen shifts; iii) two consecutive [1,7] hydrogen shifts (Scheme 4). These sigmatropic shifts can theoretically take place before or after the protonation step, thus both neutral and anionic forms of the possible intermediates must be considered. In the cycloheptatrienic systems only the suprafacial hydrogen shifts are possible for steric reasons, thus the [1,3] and [1,7] hydrogen shifts are thermally forbidden, whereas the [1,5] hydrogen shift is allowed. [START_REF] Spangler | [END_REF][36][37][38][39] In order to evaluate the conditions required for the sigmatropic shift in this rearrangement, we optimized the structures and calculated the energies for all intermediates, as well as for the transition states in the first sigmatropic shift (Scheme 4, Table 1).

We succeeded in locating valid transition states only in three cases, for neutral TS4n, anionic TS4a and anionic TS5a. 40 The difference of activation barriers between the neutral and anionic forms of TS4 is almost 9 kcal/mol, so it seems that the sigmatropic rearrangement is more favored in the anionic series.

However, in spite of the lower value obtained for TS4a (24.2 kcal/mol), this activation barrier is still too high to be compatible with the mild conditions used for this reaction. The activation barrier of the anionic form TS5a is higher than that of TS4a, as expected for a forbidden sigmatropic shift.

All these calculations data suggest that there is no sigmatropic shift involved in this mechanism; this is further supported by the complete deuterium exchange observed for the two methylene groups in 4a (Scheme 2), which is not compatible with a sigmatropic shift.

As suggested by one of the referees, in the light of Herndon's studies, 11 we investigated the possibility to convert 2a into 13a by two successive thermal [1,5] hydrogen shifts. However, heating 2a in p-xylene at 140 °C for 16 h led only to the partial conversion of 2a into 13a and from the mixture of these two compounds (ratio 2a / 13a = 3 / 1) 13a was isolated in 11 % yield. Longer reaction time (3 days) resulted in extensive decomposition. These results are in agreement with an extended delocalized system in our case, a high activation barrier for the thermal [1,5] hydrogen shifts and support the greater stability (1.6 kcal/mol) calculated for 2a compared to 13a. it is noteworthy that, with one exception (structures 14), the keto form is always more stable than the enol, the difference of energies being very important for 2 and 13. This is in agreement with the experimental results 41 showing that the cycloheptadienone is the major form in this keto-enol equilibrium. Moreover, the energies of compounds 12, 14 and 17, bearing ring junction hydrogen atoms, are unusually high, for both keto and enol forms. This is the consequence of the significant geometrical change induced by the presence of these hydrogen atoms, which prevents the conjugation between the 7-membered ring and the other two rings. b) Base-catalyzed 1,3-hydrogen shift. Several examples of this type of transformation are described in cycloheptatrienic systems, catalyzed by triethylamine, [42][43][44][45] 1,8-diazabicyclo [5.4.0]undec-7-ene 27,28 or potassium tert-butoxide. 46,47 The mechanism and regioselectivity have been studied for cycloheptatrienes bearing electron withdrawing groups. [48][49][50][51][52] There are also a few examples of thermal, [53][54][55][56][57] photochemical 58 or acid-catalyzed 57,58 1,3-hydrogen shifts. In most cases, the hydrogen atom shifts from a ring-junction position to the nearest unsubstituted position (path A, Scheme 5). 27,28,[42][43][44][56][57][58] In the case where cycloheptatriene 18 has R 1 ≠ H (path B, Scheme 5), the 1,3-shift is very unfavored. The system evolves towards the spirodienone 21 43,[53][54][55] if an oxygenated substituent is present at R 2 or remains in the less constrained norcaradiene form 20 59 in the other cases. To sum up the above, the pathways and the final products are determined by the stability given by the substituents present on the molecule. This equilibrium is completely shifted towards the enolate 6c, which is the only species visible in the NMR spectrum. The complete incorporation of deuterium in the positions marked with red dots in Scheme 6 supports the existence of successive base-catalyzed 1,3-hydrogen shifts leading to the enolate 6c, which is the most stable of all anionic intermediates (Table 1). The protonation of 6c leads exclusively to the isomer 2a, the most stable compound among 2a, 6a and 12a (Table 1).

R 2 R 1 R 1 R 2 R 1 O R 1 = H R 2
The deuterium labeling results indicate that the mechanism of the rearrangement involves base-catalyzed 1,3-hydrogen shifts. This conclusion is supported by the calculations data.

Conclusion

In this paper we report the rearrangement of spirocyclohexadienones into dihydrotropones in basic conditions as a new method for the preparation of seven-membered ring ketones, which are key building blocks for the synthesis of tropoloalkaloids. We gained insight into the mechanism that we proposed for this rearrangement by DFT calculations of the possible intermediates and transition states as well as by deuterium labeling. All these data allowed us to choose between two possible paths and thus conclude that the mechanism involves a base-catalyzed 1,3-hydrogen shift in the cycloheptatrienic system obtained from the opening of a fused cyclopropane intermediate.

Experimental Section

General Remarks. Starting materials and reagents were purchased from a commercial supplier and used without further purification with the exception of MeOH, which was distilled over Mg and I 2 .

Synthesis of methyl 2'-(

Computational Methods

Geometry optimization and vibrational frequency calculations were performed at the Density Functional

Theory (DFT) level by using the Gaussian 03 program suite. [START_REF] Frisch | Gaussian 03 revision B.05; Gaussian[END_REF] Becke's three-parameter exchange functional (B3) 60,61 was employed in conjunction with the Lee-Yang-Parr correlation functional (LYP), 62 as implemented in Gaussian 03 [START_REF] Frisch | Gaussian 03 revision B.05; Gaussian[END_REF] at a 6-31++G(d,p) basis set level. Geometries were optimized without constraint, and vibrational frequencies were then computed to characterize each structure as a minimum or transition structure (TS), via the number of imaginary frequencies (zero for minima and one for saddle points, respectively). After locating a TS, an intrinsic reaction coordinate (IRC) [63][64][65][66] calculation was carried out to identify its respective reactant and product. Crystallographic results. Data were obtained from a small colorless crystal plate (0.55 x 0.50 x 0.20 mm).

Empirical formula : C 15 H 12 O 4 , M w = 256.25. The compound crystallizes in the triclinic system, space group P-1, racemic. There are four molecules in the unit-cell (Z= 4) and so, two independent molecules in the asymmetric unit. Cell parameters : a = 7.510 (5), b = 12.824(5), c = 13.261(7) Å, α = 73.61(3), β = 78.80(3), γ = 87.33(3)°, V = 1202 Å 3 ; d c = 1.416 g cm -3 , F(000) = 536, λ (Mo Kα) = 0.71073 Å, µ= 0.103 mm -1 . Data were measured with a Nonius Kappa-CCD linear area-detector diffractometer, using graphite monochromated Mo Kα radiation, according to the phi and omega scan method, up to θ = 27.50°. A total of 9306 intensity data was collected reduced to 7000 triclinic reflections of which 5273

were unique (Rint = 0.020). [START_REF] Miyashita | [END_REF] Structure was solved with program SHELXS86 2 and refined by full-matrix least-squares, upon all unique F 2 with program SHELXL97. 3 All the hydrogen atoms were located in difference Fourier maps. They were fitted at theoretical positions and treated as riding, assigned of an isotropic displacement parameter equivalent to 1.12 the one of the bonded atom, 1.15 for those of the methyl groups. Refinement of 345 parameters converged to R1(F) = 0.0486 with the 3639 observed reflections having I ≥ 2 sigma (I) and wR2(F 2 ) = 0.1272 with all the 5273 unique data ; goodness-of-fit S factor of 1.024. The residual electron density was found between -0.18 and 0.19 eÅ -3 . All the crystallographic results are given in Cif file and Tables S1 to S6.

Comparison of molecular conformations. The two independent molecules of the asymmetric unit named ----------------------------------------

* O6-C6-C6a-C7 5.3(3) C11a-C6a-C7-C8 24.3(3) C6a-C7-C8-C9 6.0(4) C7-C8-C9-C10 -6.1(3) O9-C9-C10-C11 137.5(2) C8-C9-C10-C11 -43.7(3) C9-C10-C11-C11a 86.2(2) C7-C6a-C11a-C11 -4.3(3) C10-C11-C11a-C6a -56.2(2) C10-C11-C11a-C11b 124.8(2) C11-C11a-C11b-C1 0.8(3) C3-C4-O4-C12 -4.1(3) ** O6-C6-C6a-C7 1.8(3) C11a-C6a-C7-C8 32.1(3) C6a-C7-C8-C9 6.9(3) C7-C8-C9-C10 -28.1(3) O9-C9-C10-C11 160.4(2) C8-C9-C10-C11 -19.0(3) C9-C10-C11-C11a 76.9(2) C7-C6a-C11a-C11 -2.1(3) C10-C11-C11a-C6a -63.8(2) C10-C11-C11a-C11b 115.9(2) C11-C11a-C11b-C1 0.4(3) C3-C4-O4-C12 -4.7(2) _____________________________________________________________

S19

Theoretical data for compound 1 :

Final Energy in Hartrees: HF = -879.7282735 (B3LYP/6-31++G(D,P))

Standard orientation: -------------------------------------------------------------------- ------------------------------------------------------------------- ------------------------------------------------------------------- Theoretical data for compound 10 :

Final Energy in Hartrees: HF = -879.1713081 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------ ------------------------------------------------------------------- Theoretical data for transition state TS1 :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.1565668 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------- Theoretical data for compound 11 :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.1575149 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------- Theoretical data for transition state TS2 :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.1498774 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------- Theoretical data for compound 12c :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.1830086 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------- ------------------------------------------------------------------- ------------------------------------------------------------------- Theoretical data for transition state TS4n :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.662717 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------ ------------------------------------------------------------------- ------------------------------------------------------------------ Theoretical data for transition state TS5a :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.1087276 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- -------------------------------------------------------------------- Theoretical data for compound 2b :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.7303913 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- -------------------------------------------------------------------- Theoretical data for compound 2c :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.1790705 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- -------------------------------------------------------------------- Theoretical data for compound 13a :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.7447082 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------- Theoretical data for compound 13b :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.7297326 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------- Theoretical data for compound 13c :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.1767698 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------- Theoretical data for compound 12a :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.7161544 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------- Theoretical data for compound 12b :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.7154875 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------ ------------------------------------------------------------------ ------------------------------------------------------------------- Theoretical data for compound 14b : ------------------------------------------------------------------ ------------------------------------------------------------------- ------------------------------------------------------------------ ------------------------------------------------------------------- ------------------------------------------------------------------- ------------------------------------------------------------------ Theoretical data for compound 15b :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.7291446 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------- Theoretical data for compound 15c :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.2055479 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------ Theoretical data for compound 6b :

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -
Final Energy in Hartrees: HF = -879.7281245 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- -------------------------------------------------------------------

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -

S43

Theoretical data for compound 6c :

Final Energy in Hartrees: HF = -879.2136405 (B3LYP/6-31++G(D,P)) ------------------------------------------------------------------- ------------------------------------------------------------------ ------------------------------------------------------------------ ------------------------------------------------------------------ ------------------------------------------------------------------ ------------------------------------------------------------------ ------------------------------------------------------------------ ------------------------------------------------------------------

Standard orientation: --------------------------------------------------------------------- Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z -

Scheme 2 .

 2 Scheme 2. Deuterium labeling studies during the rearrangement and lactone ring opening of spirocyclohexadienone 1 in basic conditions.

Figure 1 .

 1 Figure 1. Time-dependent formation of compounds 2a and 3 (MeONa, MeOH, 40 °C)

Scheme 3 .

 3 Scheme 3. Proposed mechanism for the rearrangement of 1 into 2a. Calculated relative energies (kcal/mol) of neutral compounds are represented in blue and those of anions and transition states (TS) in red. In brackets are reported the activation barriers (kcal/mol).

  e (24.2) f 39.2 e (46.5) f a calculated at B3LYP/6-31++G(d,p) level ; b could not be located ; c compared to 1 ; d compared to 12b (activation barrier) ; e compared to 10 ; f compared to 12c (activation barrier).

Scheme 4 .

 4 Scheme 4. Three different possible pathways for the conversion of 12b into 2b through sigmatropic reactions.

a

  Calculated relative energies of all possible intermediates in the sigmatropic rearrangement Relative energy (kcal/mol) a Structure keto (a) b enol (b) b enolate (c) c calculated at B3LYP/6-31++G(d,p) level; b compared to 1; c compared to 10; d in this case, alcohol/alcoholate instead of enol/enolate However, from the results of the

Scheme 5 .Scheme 6 .

 56 Scheme 5. Influence of the ring substituents in the evolution of cycloheptatrienic systems

Figure S2 .

 S2 Figure S2. Superposed 1 H-NMR spectra (500 MHz, CD 3 OD) of intermediates from the reaction of 1 with CD 3 ONa in CD 3 OD at 40°C. The competitive formation of the ester 5, phenoxide 9 and 7-membered ring anion 7c is observed. The equilibrium is shifted almost completely towards the more thermodynamically stable anion 7c after 30 min.

Figure S3 .

 S3 Figure S3. Superposed 1 H-NMR spectra (500 MHz, CD 3 OD) of intermediates from the reaction of 2a with CD 3 ONa in CD 3 OD at room temperature. We observe the immediate formation of the anion 6c, which is progressively deuterated, leading to the complete formation of the anion 7c after 20 h.

  respectively, * and **, appear in Figure S5, showing the same diastereoisomer. In both molecules, bond distance and angle values are nearly the same with double bonds C6a-C11a (1.362 Å) and C7-C8 (av. 1.341Å) clearly precised. However, differences appear in torsion angle values of the seven membered rings principally along the bonds: C8-C9 (-6.1 in *, -28.1° in **), C9-C10 (-43.7 and -19.0°) and C10-C11 (76.9 and 86.2), indicating differences in the ring conformations. Fitting the two structures, atoms C7 and C11 are quitely coplanar with the bicyclic aromatic part in each molecule, while atoms C8, C9 and C10 are respectively deviated by 0.226(2), 0.885(2) and 1.183(2) Å in *, by 0.500(2), 1.372(2) and 1.284(2) Å in **, from the mean plane of the other twelve atoms. If atom C8 is included in the aromatic mean plane, deviations of atoms C9 and C10 become : 0.800(2) and 1.114(2) Å in *, 1.180(2) and 1.144 (2) Å -nearly the same -in **. So, the angles between the planes [C11, C11a, C6a, C7, C8] and [C8, C9, C10, C11] in the seven membered rings, are respectively 136.3° in *, 131.0° in **. More, if we consider only the mean plane of the double bonds involving atoms [C11a, C6a, C7, C8, C9], atoms C10 and C11 are deviated from this plane by respectively 0.327(2) and -0.620(2) Å in *, while atom C10 lies in this plane in **, with C11 situated below by -0.890(2) Å. The crystal packing study shows that only van der Waals contacts exist between the molecules. 1 Collect (Nonius BV, 1997-2000); HKL Denzo and Scalepack (Otinowski & Minor, 1997).

Theoretical data for compound 2a :Final

  Energy in Hartrees: HF = -879.7472803 (B3LYP/6-31++G(D,P)) Standard orientation: ------------------------------------------------------------------

  Theoretical data for transition state TS4a : Final Energy in Hartrees: HF = -879.1443616 (B3LYP/6-31++G(D,P)) Standard orientation:------------------------------------------------------------------

Theoretical data for compound 14a :Final

  Energy in Hartrees: HF = -879.7186584 (B3LYP/6-31++G(D,P)) Standard orientation: ------------------------------------------------------------------

Final

  Energy in Hartrees: HF = -879.7203451 (B3LYP/6-31++G(D,P)) Standard orientation: ------------------------------------------------------------------

Theoretical data for compound 14c :Final

  Energy in Hartrees: HF = -879.1933354 (B3LYP/6-31++G(D,P)) Standard orientation: ------------------------------------------------------------------

  Theoretical data for compounds 15a and 6a : Final Energy in Hartrees: HF = -879.7342275 (B3LYP/6-31++G(D,P))Standard orientation:------------------------------------------------------------------

Final

  Energy in Hartrees: HF = -879.7239323 (B3LYP/6-31++G(D,P)) Standard orientation: ------------------------------------------------------------------

Final

  Energy in Hartrees: HF = -879.1417281 (B3LYP/6-31++G(D,P)) Standard orientation: ------------------------------------------------------------------

Final

  Energy in Hartrees: HF = -879.7122644 (B3LYP/6-31++G(D,P)) Standard orientation: ------------------------------------------------------------------

Table

Table

4-methoxy-7-oxo-5a,6,7,9a-tetrahydrodibenzo[b,d]furan-9a-yl)acetate (3). To

	a solution of 8-methoxyspiro[chroman-4,1'-cyclohexa[2',5']diene]-2,4'-dione 1 30 (20 mg, 0.078
	mmol) in methanol (0.5 mL) was added trifluoroacetic acid (0.1 mL, 1.3 mmol). After stirring 16 h at room
	temperature, the mixture was concentrated in vacuo. Preparative silica gel TLC (heptane / ethyl acetate 50 /
	50) afforded 20 mg of methyl 2'-(4-methoxy-7-oxo-5a,6,7,9a-tetrahydrodibenzo[b,d]furan-9a-yl) acetate 3
	as a pale yellow oil (87 %): 1 H NMR (500 MHz, CDCl 3 ) δ 6.94 (dd, J = 7.7, 7.8 Hz, 1H), 6.86 (d,
	J = 7.7 Hz, 1H), 6.83 (d, J = 7.8 Hz, 1H), 6.57 (dd, J = 10.3, 2 Hz, 1H), 6.02 (d, J = 10.3 Hz, 1H), 5.13
	(m, 1H), 3.88 (s, 3H), 3.69 (s, 3H), 3.09 (d, J = 14.3 Hz, 1H), 3.13 (dd, J = 17.7, 2.8 Hz, 1H), 6.03 (dd,
	J = 17.7, 4.2 Hz, 1H), 2.90 (d, J = 15 Hz, 1H); 13 C NMR (75 MHz, CDCl 3 ) δ 195.0, 170.3, 146.9, 146.3,
	145.1, 131.4, 127.6, 122.6, 115.0, 112.7, 86.0, 56.2, 52.2, 47.4, 41.5, 38.5; IR (CDCl 3 ) 1199, 1280, 1492,
	1680, 1731 cm -1 ; MS (ESI) m/z : 311(M+Na) + ; HRMS (ESI) m/z calcd for C 16 H 16 O 5 Na	+ 311.0895,
	found 311.0878.

Synthesis of 4-methoxy-10,11-dihydro-cyclohepta[c]chromene-6,9-dione (2a).

		A solution of 8-
	methoxyspiro[chroman-4,1'-cyclohexa[2',5']diene]-2,4'-dione 1 30 (20 mg, 0.078 mmol) in methanol (0.6
	mL) / THF (0.3 mL) is added at 40 °C to a solution of sodium methoxide, freshly prepared from sodium
	(13 mg, 0.56 mmol) and methanol (0.6 mL). After stirring for 30 min at 40 °C, the reaction is quenched
	with 1 M hydrochloric acid (2 mL), stirred for 10 min and extracted with dichloromethane (3 x 5 mL).
	Concentration in vacuo of the combined organic layers followed by preparative silica gel TLC (heptane /
	ethyl acetate 70 / 30) afforded 15 mg of 4-methoxy-10,11-dihydro-cyclohepta[c]chromene-6,9-dione 2a
	(75 %): 1 H NMR (500 MHz, CDCl 3 ) δ 7.59 (d, J = 12.9 Hz, 1H), 7.39 (d, J = 8.35 Hz, 1H), 7.30 (dd,
	J = 8.35, 8.04 Hz, 1H), 7.18 (d, J = 8.04 Hz, 1H), 6.40 (d, J = 12.8 Hz, 1H), 4.01 (s, 3H), 3.20 (dd,
	J = 5.5, 6.0 Hz, 2H), 2.83 (dd, J = 5.6, 6.1 Hz, 2H); 13 C NMR (125 MHz, CDCl 3 ) δ 199.8, 159.7, 155.8,
	148.0, 143.4, 135.6, 132.0, 124.7, 121.5, 119.4, 116.7, 115.0, 56.6, 41.2, 22.7; IR (CDCl 3 ) 1469, 1662,
	1710 cm -1 ; MS (ESI) m/z : 279 (M+Na) + ; HRMS (ESI) m/z calcd for C 15 H 12 O 4 Na	+ 279.0633, found
	279.0616.

Synthesis of 4-methoxy-7,8-dihydro-cyclohepta[c]chromene-6,9-dione (13a). A

	heated at 140 °C for 16 h, concentrated in vacuo and the resulting mixture (ratio 2a / 13a = 3 / 1) is
	purified by preparative silica gel TLC (eluting with heptane / ethyl acetate 40 / 60) to afford 2.5 mg of 4-
	methoxy-7,8-dihydro-cyclohepta[c]chromene-6,9-dione 13a (11 %) : 1 H NMR (500 MHz, CDCl 3 ) δ 7.42
	(d, J = 12.8 Hz, 1H), 7.28 (m, 2H), 7.14 (dd, J = 7.0, 2.1 Hz, 1H), 6.65 (d, J = 12.8 Hz, 1H), 4.01 (s, 3H),
	3.17 (m, 2H), 2.84 (m, 2H); 13 C NMR (125 MHz, CDCl 3 ) 200.9, 160.6, 147.7, 142.5, 142.1, 136.3, 132.5,
	129.3, 124.4, 119.1, 115.0, 113.3, 56.3, 44.1, 20.9; IR (CDCl 3 ) 1473, 1671, 1707 cm -1 ; MS (ESI) m/z :
	279 (M+Na) + ; HRMS (ESI) m/z calcd for C 15 H 12 O 4 Na	+ 279.0633, found 279.0631.
		solution of 4-
	methoxy-10,11-dihydro-cyclohepta[c]chromene-6,9-dione 2a (22 mg, 0.086 mmol) in p-xylene (1 mL) is

Table S1 .

 S1 Crystal data and structure refinement for compound 2a.

	______________________________________________________________________
	Identification code	'VARIN051'
	Empirical formula	C15 H12 O4
	Formula weight	256.25
	Temperature	293(2) K
	Wavelength	0.71073 A
	Crystal system, space group	Triclinic, P -1
	Unit cell dimensions	a = 7.510(5) A	alpha = 73.61(3) deg.
		b = 12.824(5) A	beta = 78.80(3) deg.
		c = 13.261(7) A	gamma = 87.33(3) deg.
	Volume	1201.9(11) A^3
	Z, Z', Calculated density	4, 2, 1.416 Mg/m^3
	Absorption coefficient	0.103 mm^-1
	F(000)	536
	Crystal size	0.55 x 0.50 x 0.20 mm
	Theta range for data collection	2.76 to 27.50 deg.
	Limiting indices	-9<=h<=9, -16<=k<=16, -16<=l<=17
	Reflections collected / unique	9306 -7000 / 5273 [R(int) = 0.0199]
	Completeness to theta = 27.50	95.3 %
	Absorption correction	Not measured
	Refinement method	Full-matrix least-squares on F^2
	Data / restraints / parameters	5273 / 0 / 345
	Goodness-of-fit on F^2	1.024
	Final R indices [I>2sigma(I)]	R1 = 0.0486, wR2 = 0.1101 (3639 Fo)
	R indices (all data)	R1 = 0.0779, wR2 = 0.1272 (5273 data)
	Largest diff. peak and hole	0.187 and -0.177 e.A^-3
	_______________________________________________________________________

2 Sheldrick, G.M. (1990). SHELXS86. Acta Cryst. A 46, 467-473.

3

Sheldrick, G.M. (1997). SHELX97. Program for the Refinement of Crystal Structures, Univ. of Göttingen, Germany. S13

Table S2 .

 S2 Atomic coordinates (x 10^4) and equivalent isotropic displacement parameters (A^2 x 10^3) for compound 2a. U(eq) is defined as one third of the trace of the orthogonalized Uij tensor.

		(* Molecule 1, ** Molecule 2)
	________________________________________________________________
		x	y	z	U(eq)
	________________________________________________________________
	* C1	5205(3)	3658(2)	2140(2)	54(1)
	C2	4806(3)	2587(2)	2304(2)	62(1)
	C3	5442(3)	2054(2)	1534(2)	57(1)
	C4	6506(3)	2600(1)	571(1)	48(1)
	C5	6902(2)	3696(1)	402(1)	41(1)
	O5	7928(2)	4211(1)	-580(1)	46(1)
	C6	8399(2)	5287(1)	-854(1)	41(1)
	C6a	7777(2)	5888(1)	-64(1)	40(1)
	C7	8222(3)	7045(1)	-455(1)	49(1)
	C8	8325(3)	7798(2)	62(2)	58(1)
	C9	8167(3)	7658(2)	1207(2)	58(1)
	C10	7663(3)	6568(2)	1970(2)	59(1)
	C11	6121(3)	6017(2)	1708(1)	54(1)
	C11a	6766(2)	5385(1)	910(1)	41(1)
	C11b	6285(2)	4246(1)	1170(1)	42(1)
	C12	6884(4)	1034(2)	-73(2)	85(1)
	O4	7196(2)	2168(1)	-249(1)	62(1)
	O6	9313(2)	5647(1)	-1733(1)	56(1)
	O9	8478(3)	8426(1)	1518(1)	90(1)
	** C1	9034(3)	1683(2)	3263(1)	53(1)
	C2	8869(3)	922(2)	2744(2)	60(1)
	C3	8165(3)	-106(2)	3296(2)	56(1)
	C4	7624(3)	-378(1)	4389(1)	47(1)
	C5	7810(2)	394(1)	4922(1)	42(1)
	O5	7255(2)	74(1)	6015(1)	50(1)
	C6	7373(3)	757(1)	6629(1)	48(1)
	C6a	8072(2)	1858(1)	6080(1)	42(1)
	C7	8174(3)	2531(1)	6792(1)	49(1)
	C8	8020(3)	3609(2)	6606(2)	53(1)
	C9	7519(3)	4385(1)	5663(2)	53(1)
	C10	7904(3)	4170(1)	4592(2)	58(1)
	C11	9339(3)	3317(1)	4460(1)	50(1)
	C11a	8623(2)	2182(1)	4997(1)	40(1)
	C11b	8504(2)	1432(1)	4377(1)	41(1)
	C12	6606(3)	-2132(2)	4486(2)	66(1)
	O4	6921(2)	-1359(1)	5015(1)	64(1)
	O6	6862(2)	398(1)	7580(1)	72(1)
	O9	6781(2)	5233(1)	5757(1)	83(1)
	________________________________________________________________

Table S3 .

 S3 Hydrogen coordinates (x 10^4) and isotropic displacement parameters (A^2 x 10^3) for compound 2a.

	_______________________________________________________________
		x	y	z	U(eq)
	_______________________________________________________________
	* H1	4763	3999	2670	61
	H2	4091	2208	2947	69
	H3	5153	1324	1664	63
	H7	8482	7311	-1197	55
	H8	8525	8508	-371	65
	H10A	7304	6646	2687	66
	H10B	8721	6104	1961	66
	H11A	5471	5529	2363	60
	H11B	5276	6567	1427	60
	H12A	5603	897	49	97
	H12B	7362	627	542	97
	H12C	7477	817	-690	97
	** H1	9498	2370	2875	59
	H2	9238	1096	2005	68
	H3	8059	-610	2926	63
	H7	8373	2156	7471	55
	H8	8259	3898	7135	60
	H10A	8286	4846	4052	65
	H10B	6782	3941	4455	65
	H11A	9768	3397	3703	56
	H11B	10364	3438	4761	56
	H12A	7735	-2305	4089	76
	H12B	5788	-1834	4006	76
	H12C	6081	-2779	5006	76
	_____________________________________________________________

Table S4 .

 S4 Comparison of selected torsion angles [deg] for compound 2a. _____________________________________________________________

Table S5 .

 S5 Bond lengths [A] and angles [deg] for compound 2a.

	______________________________________________
	* C1-C2	1.367(3)
	C1-C11b	1.409(3)
	C2-C3	1.383(3)
	C3-C4	1.382(3)
	C4-O4	1.361(2)
	C4-C5	1.397(2)
	C5-O5	1.378(2)
	C5-C11b	1.393(2)
	O5-C6	1.369(2)
	C6-O6	1.207(2)
	C6-C6a	1.464(2)
	C6a-C11a	1.362(2)
	C6a-C7	1.458(2)
	C7-C8	1.345(2)
	C8-C9	1.460(3)
	C9-O9	1.218(2)
	C9-C10	1.491(3)
	C10-C11	1.526(3)
	C11-C11a	1.504(2)
	C11a-C11b	1.449(2)
	C12-O4	1.429(2)
	* C2-C1-C11b	120.08(18)
	C1-C2-C3	121.49(18)
	C4-C3-C2	120.31(17)
	O4-C4-C3	125.72(16)
	O4-C4-C5	116.13(16)
	C3-C4-C51	118.15(16)
	O5-C5-C11b	121.67(15)
	O5-C5-C4	115.90(15)
	C11b-C5-C4	122.42(16)
	C6-O5-C5	121.93(13)
	O6-C6-O5	116.09(14)
	O6-C6-C6a	126.08(15)
	O5-C6-C6a	117.83(14)
	C11a-C6a-C7	125.30(15)
	C11a-C6a-C6	120.60(15)
	C7-C6a-C6	113.93(15)
	C8-C7-C6a	131.71(17)
	C7-C8-C9	129.12(18)
	O9-C9-C8	119.4(2)
	O9-C9-C10	121.46(19)
	C8-C9-C10	119.08(16)
	C9-C10-C11	113.17(17)
	C11a-C11-C10	113.11(16)
	C6a-C11a-C11b	120.06(15)
	C6a-C11a-C11	119.97(15)
	C11b-C11a-C11	119.96(15)
	C5-C11b-C1	117.54(16)
	C5-C11b-C11a	117.90(15)
	C1-C11b-C11a	124.55(16)
	C4-O4-C12	117.01(16)
	-

Acknowledgment. We thank the Institut de Chimie des Substances Naturelles (ICSN) and the Centre National de la Recherche Scientifique (CNRS) for the financial support.

Supporting Information Available. Spectral characterization of compounds 3, 2a and 13a, equilibrium between different intermediates and deuterium labeling followed by NMR, detailed data for the X-ray structure of compound 2a, theoretical data for calculated compounds, intermediates and transition states, a movie generated from the IRC calculations showing the first part of the mechanism. This material is available free of charge via the Internet at http://pubs.acs.org.

Supporting Information

A new access to dihydrotropones through ring expansion of spirocyclohexadienones: Synthesis and mechanism

Marie Varin, Angèle Chiaroni, Jean-Yves Lallemand, Bogdan Iorga, Catherine Guillou Institut de Chimie des Substances Naturelles, CNRS UPR 2301, F-91198 Gif sur Yvette, France

Table of contents :

1 H and 13 C-NMR spectra of compound 3 S2

1 H and 13 C-NMR spectra of compound 2a S4 Crystals suitable for an X-ray study have been obtained for the 6,7-dihydrotropone 2a and the tridimensional structure is shown in Figure S4. All atoms are coplanar except the atoms C8, C9, C10 and O9. There are two independent molecules in the asymmetric unit, the main difference between them being the degree of deviation of the atoms C8, C9, C10 and O9 from the molecule plane (Figure S5).


Bogdan Iorga
email: bogdan.iorga@icsn.cnrs-gif.fr

Denyse Herlem

Elvina Barré

Catherine Guillou
email: catherine.guillou@icsn.cnrs-gif.fr

Acetylcholine Nicotinic Receptors: Finding the Putative Binding Site of Allosteric Modulators Using the "Blind Docking" Approach

Keywords:

come

Introduction

Alzheimer's disease (AD) [1,2] is the most common form of degenerative dementia of the human central nervous system characterized by progressive memory loss, disorientation, and pathological markers (senile plaques and neurofibrillary tangles). [3] The main existing treatments for AD are the inhibition of acetylcholinesterase (AChE) [4,5] and the blocking of NMDA receptors. [6] Presently, three acetylcholinesterase inhibitors (AChEIs) are available on the market: donepezil (Aricept ® ), rivastigmine (Exelon ® ) and galanthamine (Reminyl ® ). The memantine (Ebixa ® ), a NMDA receptors antagonist, is the latest drug released for the treatment of AD.

An interesting alternative for the treatment of AD has emerged in recent years. Indeed, the allosteric modulation of nicotinic acetylcholine receptors (nAChRs) appears as a new, promising approach for the treatment of this disease. [7][8][9][10][11][12][13][14][15][16][17] As the allosteric modulators do not interact with the agonist binding site, the secondary effects due to receptor desensitization observed for the nAChR agonists are expected to be suppressed. [13] Nicotinic acetylcholine receptors are pentameric membrane proteins traditionally classified into "muscle" and "neuronal" types. The "muscle" nAChRs are heteropentamers with the stoichiometry (a1)2(b1)gd prior to innervation and (a1)2(b1)de after innervation. The "neuronal" nAChRs can be classified, based on their subunit composition, in two subtypes: homopentamers consisting of a7-a10 subunits and heteropentamers consisting of various combinations of a2-a6 and b2-b4 subunits. These subunit combinations confer specific pharmacologies to the nAChRs. [START_REF] Maelicke | Neuronal nicotinic receptors[END_REF] The generally accepted mechanism for the cholinergic neurotransmission implies the binding of ACh to the amino-terminal domains of nAChRs which undergo conformational changes that trigger the opening of the ion channel (gating mechanism). The fixation of an allosteric modulator induces an apparent enhancement of receptor affinity and increases the probability of channel opening.

Three different conformations of the nAChRs have been postulated, based on the Monod-Wyman-Changeux model of allosteric transitions: the open, closed and desensitized states. [START_REF] Edelstein | [END_REF] A number of hypotheses concerning the channel opening mechanism have been proposed [20][21][START_REF] Colquhoun | ) Burger's Medicinal Chemistry[END_REF][START_REF] Grutter | [END_REF][24][25][26][27] on the basis of the recently published cryo-electron microscopy structures of nAChRs. [28,29] However, the lack of more precise information on the nAChRs three-dimensional structure has limited the attempts to gain molecular insights into the ligand-receptor interaction. The publication of the crystal structure of the Acetylcholine Binding Protein (AChBP), [30][31][32][33] a soluble protein homologue to the extracellular domain of nAChRs, bound to HEPES, [30,33] nicotine [33] and carbamylcholine, [33] changed significantly the situation. In functional terms, AChBP shares virtually all of the ligand binding characteristics with the nicotinic receptor family, and reveals a structure largely consistent with the electron microscopy image, chemical modification, mutagenesis, and spectroscopic data. [34] Considering its sequence identity with the amino-terminal part of a7 nAChR (26%) and the similar behaviour towards the "classical" nAChR ligands, AChBP is considered as a reliable structure for nAChR homology modelling and docking simulations. [35][36][37] A number of modelled nAChRs structures have been published in recent years. [38][39][40][41][42][43][44][45][46][47] Galanthamine, codeine and eserine (physostigmine) are three known allosteric modulators (allosteric potentiating ligands, APLs) of nAChRs (Figure 1). [9,[48][49][50]] Some of them (galanthamine and eserine) show a dual mode of action, being also good AChEIs. This duality confers to galanthamine its effectiveness in the treatment of AD. [5,12,51] In addition, APLs can act as noncompetitive agonists of very low efficacy, and as direct blockers of ACh-activated channels. These actions are observed with nAChRs from brain, muscle and electric tissue. They depend on the structure of the APL and the concentration range applied. [50,52,53] Place Figure 1 Here

The exact position of the binding site of nAChR APLs is not very clear. [54] Photoaffinity labelling experiments and epitope mapping for the eserinecompetitive antibody FK1 have identified the aLys125 residue of Torpedo nAChR in the allosteric binding site, or in its immediate vicinity. [55][56][57] These results have been exploited recently [45] for the docking of eserine in a region defined within 14Å from aLys122 of murine nAChR (corresponding to aLys125 of Torpedo nAChR) using the DOCK program. [58] In this paper we present the results of a more general investigation intending to elucidate the location of the allosteric binding site of nAChRs using the "blind docking" approach, [59] a powerful feature of the AUTODOCK program. [60] For this purpose, we created models of human a7, a3b4 and a4b2 nAChRs on which we docked three allosteric modulators, without imposing a binding site. Three binding sites were identified in the channel pore, one of them being located opposite to the agonist binding site. These findings are discussed and a possible interpretation, considering the experimental results previously published, is proposed.

Materials and methods

Sequence Alignment

The multiple alignment between AChBP and the amino-terminal domains of human a3, a4, a7, b2 and b4 nAChRs subunits was obtained by means of the CLUSTALX package applying the default parameters. [61] The initial alignment was further manually refined.

Model building

The program MODELLER (version 6v2) [62,63] was used to build the threedimensional models of the amino-terminal domains of human a7, a4b2 and a3b4 nAChRs according to the comparative protein modeling method. The template used was the X-ray structure of AChBP (2.7Å resolution, PDB entry code 1I9B).

To maintain the complementarity between subunits at their interfaces, all five units were modeled simultaneously. As expected, the backbone atoms of the predicted models and AChBP overlapped well (RMSD values computed using CHIMERA [64] are 0.71Å for a7, 0.60Å for a3b4 and 0.63Å for a4b2, respectively), due to the algorithm used by MODELLER and to the low number of gaps in the alignment. Additionally, the structures were checked with PROCHECK. [65] The comparison of the Ramachandran plots shows that the models have good quality, with 85-86 % of the residues in the most favourable regions (see Supplementary material). For AChBP and a7 model, the subunit at the clockwise side of each interface as viewed from the N-terminus is called "plus" and the other "minus".

Ligand structure

For galanthamine, the coordinates extracted from the X-ray structure of the complex with AChE (PDB entry 1DX6) [66] were used. For codeine and eserine, the coordinates were taken from the Cambridge Structural Database (CSD), entries ZZZTSE01 [67] and ESERIN10, [68] respectively.

Molecular docking

The docking of galanthamine, codeine and eserine into AChBP and human a7, a4b2 and a3b4 models was performed with the program AUTODOCK (version 3.0.5). [60] Its graphical front-end, AUTODOCKTOOLS, [69] was used to add polar hydrogens and partial charges for proteins and ligands using the Kollman United Atom and Gasteiger charges, respectively. Atomic solvation parameters and fragmental volumes for the proteins were assigned using the ADDSOL tool (included in the program package). Flexible torsions in the ligands were assigned with AUTOTORS module and all dihedral angles were allowed to rotate freely. In general, these were all acyclic, non-terminal single bonds (excluding amide bonds) in a given ligand molecule. Affinity grid fields were generated using the auxiliary program AUTOGRID.

The genetic algorithm-local search (GA-LS) hybrid was used to perform an automated molecular docking. Default parameters were used, except for the number of generations, energy evaluations, and docking runs were set to 1,000, 25,000,000 and 256, respectively. The docking process was performed in two steps. In the first one, the docking procedure was realised on the whole protein target, without imposing the binding site ("blind docking"). [59] The grid field was a 60Å cube with grid points separated by 1Å centred at the middle of the protein. In the second step, we docked the ligands in each of the three binding sites found in the first step ("refined docking"). This time, the grid field was a 60Å cube with grid points separated by 0.3Å centred on the best scored conformation obtained in the first step. Lennard-Jones parameters 12-10 and 12-6 (supplied with the program package) were used for modelling H-bonds and Van der Waals interactions, respectively. The resulting docked conformations were clustered into families of similar binding modes, with a root mean squares deviation (RMSD) clustering tolerance of 2Å. In almost all cases the lowest docking energy conformations were included in the largest cluster found (which usually contains 80-100% of total conformations). Otherwise, the lowest docking energy conformations were considered as the most stable orientations. The docking energy represents the sum of the intermolecular energy and the internal energy of the ligand while the free binding energy is the sum of the intermolecular energy and the torsional free energy. [58] The polar and apolar surface areas of the binding sites were calculated using the GETAREA 1.1 program. [70] The volumes were obtained with the module "SiteID Find Pockets", part of the SYBYL molecular modeling environment, [71] with a 1 Å grid resolution and 3 Å protein film depth, after addition of all hydrogen atoms to the protein. The depth of the binding site was considered as the distance between the most distant spheres generated by the above mentioned module.

[71]

Results and discussion

Sequence alignment

Although the sequence identity between the AChBP and the human nAChR extracellular domains is relatively low (18-26 %, Table 1), the presence of highly conserved ACh binding residues in the AChBP [30] and the nicotinic pharmacology of the AChBP [72] suggest that homology modelling of nAChR extracellular domains using the AChBP structure is appropriate.

Place Table 1 Here

The most important requirement in homology modelling is the correct alignment of the sequence to be modelled with that of the template structure. Regions containing insertions relative to AChBP are the greatest sources of uncertainty in modelling. [38] However, only a few insertions are present in our alignment (Figure 2), situated in non-conserved regions, the modelling program used being designed to accommodate them. [START_REF] Fiser | Computational Biochemistry and Biophysics[END_REF] Several alignments of nAChRs subunits have been previously published, [41][42][43]45,46] our alignment being very similar to those described by Le Novère et al. [41] The main differences are the residue numbering (our reference is AChBP) and the position of the residues Thr13 and Ala91-Val106, which are shifted one residue lower in our alignment. Henchman et al. [46] used an alignment based on lysine scanning mutagenesis results, involving one supplementary residue of a7 nAChRs at the beginning of the alignment. These differences are observed in regions with low sequence identity, so it is difficult to say which alignment is better.

Place Figure 2 Here

Model building

We used MODELLER [62,63] to generate homology models of extracellular domains of human a7, a3b4 and a4b2 nAChRs using spatial constraints provided by AChBP (see Materials and Methods section). As expected from the model generation algorithm, the output structures do not show major differences with the AChBP structure, as confirmed by the RMSD values for the backbone atoms (0.71Å, 0.60Å and 0.63Å, respectively) and the Ramachandran plots (see Supplementary material). These results are in good accord with those previously reported for the chick a7 model. [41]

Molecular docking

The AUTODOCK program [60] is one of the most reliable docking tools available today, owing its efficiency to the use of a genetic algorithm and to a scoring function comprising several terms (dispersion/repulsion energy, directional hydrogen bonding, screened Coulomb potential electrostatics, a volume-based solvation term, and a weighted sum of torsional degrees of freedom to estimate the entropic cost of binding). [41] Furthermore, it allows the docking of ligands on the entire protein surface, without prior specification of the binding site ("blind docking"). A parameter set based on the AMBER force field [START_REF] Cornell | [END_REF] and the possibility of using flexible as well as fixed torsions for the ligands during the docking procedure make AUTODOCK an appropriate tool for this purpose. [59] The docking of three known allosteric modulators of nAChRs (galanthamine, codeine and eserine) with AChBP and the models of human a7, a3b4 and a4b2 nAChRs generated before was performed in two steps.

"Blind docking"

In the first step, the "blind docking" approach was used in order to identify the potential fixation sites of nAChRs. The resulting conformations of ligands were clustered (RMSD 2Å) and most of them were found to be located in the channel pore, distributed over three main sites (Figure 3). The first one is located between the L1 and L4 loops (a/+ subunit) and the b3 and b5 sheets (b/-subunit) and the most important residues are those corresponding to Pro16, Asp17, Leu81, Trp82, Val83, Pro84 and Asp85 (a/+ subunit) and Ser75, Pro77, Pro100 and Leu102 (b/subunit) of AChBP. The second site is situated between the loops L4 (a/+ subunit) and L5 (b/-subunit), the conserved residues being those corresponding to Val83, Leu86, Ala87, Ala88, Lys94 and Pro95 (a/+ subunit) and Leu98, Thr99 and Pro100 (b/-subunit) of AChBP. The third one is positioned close to the L5 loop and b7 and b2 sheets (b/-subunit) and comprises mainly the residues corresponding to Asp49, Ser93, Lys94, Pro95, Glu96, Arg118, Gln119 and Arg120 (b/-subunit) of AChBP. For more details about the residues composing these three binding sites see Figure 2 and Tables S1-S3.

Place Figure 3 Here An important finding is the location of the first binding site, situated exactly opposite to the agonist binding site, at less than 12Å through the protein wall. The proximity of these two sites, as well as the high affinity of the ligands for this allosteric site suggest that the allosteric site 1 plays an important role in the potentiation of the nAChR. Indeed, the most direct manner to transmit an information between two sites is to place them nearby. It must be pointed out that for the first crystallographic structure of AChBP [30] all the seven structural water molecules present in the channel pore observed are located in this allosteric site 1.

Additionally, it is worth noting that this high affinity binding site was evidenced without imposing any constraint and in the absence of water molecules in the docking process. Some differences in the population of the three sites were observed (Table S4 in Supplementary material), depending on the nature of protein and ligand, but two remarks can be made: a) in all cases, the great majority of conformers were found in the sites 1 and 2 and b) the docking energy and the free binding energy of the conformations found in the third site were always higher than those found in the sites 1 and 2 (Tables S5 andS6 in Supplementary material). As the lack of flexibility in the protein may influence the binding modes of the ligands and the affinities and orientations may importantly vary from one site to other one, these results should be considered with care. However, the agonist binding site of acetylcholine nicotinic receptors has been correctly identified previously using AUTODOCK on homology models constructed in a similar manner. [41] The reproducibility observed for all these dockings may suggest that the third site shows an apparent low affinity for the allosteric modulators, in contrast with the sites 1 and 2. Considering the position of this third site at the bottom of the ligand binding domain, it might be also implicated in the modulation of the motions needed for opening/closing the narrow ion gate of the transmembrane domain.

Additionally, the third site is situated not far from the location previously identified as the allosteric site by photoaffinity labelling, [55,56] and used later for docking studies. [45] However, the experimental protocol used for the photoaffinity labelling (irradiation of nAChR-enriched Torpedo marmorata membrane fragments incubated with 8-azidoATP and [ 3 H]physostigmine) [55,56] allows only the identification of allosteric sites located in close proximity to an ATP binding site. [75][76][77] Several parameters of these three allosteric sites identified by "blind docking" have been calculated (polar and apolar surface areas, volume and depth) and compared with those of the agonist binding site (Table S7 in Supplementary material). The sites are mostly hydrophobic, with some polar regions. The site 1 is rather narrow and deep, similar to the agonist site, whereas the sites 2 and 3 are more open and less deep. The third site is smaller than the other ones, and this may explain the weaker affinity of the allosteric ligands observed for this site.

"Refined docking"

In the second step, the ligands were docked in each of the three binding sites previously found ("refined docking"). The use of an improved grid resolution allows a better evaluation of the protein-ligand interactions, and consequently lower docked energies are obtained with respect to the "blind docking" (see Table S5 in Supplementary material). The Figure 4 shows some representative binding modes of the best docked conformations in the three allosteric sites. The hydrogen bonds were automatically identified using the CHIMERA's "FindHBond" module [64] and the other interactions were visually evidenced, with a cut-off distance of 4Å. In most cases, a salt bridge was observed between the positive nitrogen atoms of the ligands and Asp or Glu residues of the protein. Additionally, the oxygen atoms of the ligands were found to be implicated in hydrogen bonds and electrostatic interactions with the protein residues.

Place Figure 4 Here On the basis of the results presented above, a possible mechanism for the allosteric activation of the nAChRs can be outlined. We propose the existence of three allosteric binding sites showing different affinities for the allosteric modulators. Since the reversibility of the allosteric modulators binding is generally accepted [53] we can presume an exchange of the ligands between the three allosteric sites. The allosteric binding in the first site probably induces an increase of the agonist binding affinity as a consequence of the proximity of these two sites, which might be connected through a hydrogen bond network in which protein residues and structural water molecules are involved. On the other hand, despite its apparent low affinity for the ligands, the third site may be implicated the transmission of the allosteric deformation from the agonist site to the transmembrane domain, due to its key position at the interface between the transmembrane and ligand binding domains. All these processes will result in an increase of gate opening frequency and the overall potentiation of nAChRs.

Additional work is in progress in our laboratory to obtain more information about the exact nature of the interaction between the two adjacent sites, especially the role of the structural water molecules in the activation process. The validation of these results by means of photoaffinity labelling and mutagenesis studies is under way and will be presented elsewhere.

Conclusions

In the present work we generated models of human a7, a4b2 and a3b4 nAChRs by homology modelling, which were docked with three known allosteric modulators of nAChRs using the "blind docking" approach. Three binding sites were identified in the channel pore, one of them being situated in the immediate proximity of the agonist binding site. We propose a possible mechanism for the nAChRs activation, which is the first attempt to rationalise this process at the molecular level. Further studies are in progress to investigate the role of crystallographic water molecules in the activation, probably through a hydrogen bond network. The identification of the allosteric binding site(s) of the nAChRs will subsequently allow the design of new molecules for the treatment of Alzheimer's disease. represent the fully conserved residues, while the dots (:) and (.) represent "strongly" and "weakly" conserved residues, respectively. The boxes represent the most important residues of the three allosteric sites: 1 (blue), 2 (green) and 3 (red). See Tables S1-S3 (Supplementary material) for more detailed information

Representative binding modes of the most stable docked orientations of galanthamine with human α3β4 model (sites 1, 2 and 3) and codeine with human α4β2 model (site 1). The α/(+) subunits are coloured in yellow and the β/(-) subunits are coloured in blue. The H-bonds are represented in green, the salt bridges in magenta, the ionic interactions in orange and the cation-π or π-π interactions in mauve.

Figure 1 Figure 2

 12 Figure 1 Chemical structures of three allosteric modulators of nicotinic acetylcholine receptors

Figure 3 .Figure 4

 34 Figure 3. Representation of two subunits of AChBP with carbamoylcholine (CCE) in the agonist binding site (PDB entry code 1UV6, [33] back) and codeine (COD), galanthamine (GAL) and eserine (ESE) in the putative allosteric binding sites (1, 2 and 3, respectively) found using the "blind docking" approach (front)

  Parameters of the allosteric sites identified by "blind docking" compared with those of the agonist site

Table 1

 1 Percentage of sequence identities computed using MODELLER software[62,63] from the alignment shown in Figure2. The number of identical residues is shown in brackets.

		AChBP	a3hs	a4hs	a7hs	b2hs
	a3hs	21.0 (43)
	a4hs	20.5 (42)	61.1 (127)
	a7hs	25.9 (53)	43.2 (89)	44.7 (92)
	b2hs	18.0 (37)	47.8 (99)	52.2 (108) 39.8 (82)
	b4hs	19.5 (40)	47.3 (98)	50.2 (104) 41.7 (86)	70.0 (145)

Acknowledgements

We thank to the Centre National de la Recherche Scientifique (CNRS) for the financial support.

Supplementary material available Ramachandran plots of the models created by homology, representative binding modes of the most stable docked orientations found (Figures S1-S2), list of residues of the three binding sites (Tables S1-S3), distribution of populations on the three binding sites and estimated docking energies in "blind docking" (Table S4), estimated docking energies (Table S5) and free binding energies (Table S6) in "refined docking", parameters of the allosteric sites identified by "blind docking" compared with those of the agonist site (Table S7).

Supporting Information

Acetylcholine nicotinic receptors: Finding the putative binding site of allosteric modulators using the "blind docking" approach  Representative binding modes of the most stable docked orientations found (Figures S1-S2)

 List of residues of the three binding sites (Tables S1-S3)

 Distribution of populations on the three binding sites and estimated docking energies in "blind docking" (Table S4)

 Estimated docking energies (Table S5) and free binding energies (Table S6) in "refined docking"

 Parameters of the allosteric sites identified by "blind docking" compared with those of the agonist site (Table S7)

Representative binding modes of the most stable docked orientations of eserine with AChBP (site 1), codeine with AChBP (site 3) and galanthamine with human α7 model (sites 1 and 2). The α/(+) subunits are coloured in yellow and the β/(-) subunits are coloured in blue. The H-bonds are represented in green, the salt bridges in magenta, the ionic interactions in orange and the cation-π or π-π interactions in mauve.


Philippe Blanchard

Daniel Gandolfo

Jean Ruiz

Marc Wouts

Thermodynamic versus Topological Phase Transitions: Cusp in the Kertész Line

Keywords: 05.50.+q -Lattice theory and statistics (Ising, Potts, etc.) 05.70.Fh -Phase transitions: general studies 64.60.ah -Percolation

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Introduction . -In [1], Kertész pointed out that a very interesting phenomenon arises in the Ising model subject to an external field. The so-called Coniglio-Klein droplets [2] associated to Fortuin-Kasteleyn clusters [3] have a whole percolation transition line extending from the Curie point to infinite fields. This seems, at first sight in contradiction with the fact that on the other hand, thermodynamics quantities do not have any singularities in any of their derivatives (with respect to the temperature or the field) as soon as the field is non-zero. But, as Kertész already mentioned: However, we emphasize that the suggested picture is not in contradiction with the non existence of singularities in the thermodynamic quantities because the total free energy remains analytic.

Indeed in such a model, the criticality can be specified in two different ways: the thermodynamic criticality associated to the thermodynamic limit of the bulk free energy and the geometric criticality associated in the same limit to another lower order free energy.

Since then, the Kertész line has remained the subject of interests over the years for various models of statistical mechanics. In particular, it has been considered recently in [4] within the Potts' model on the regular lattice Z d . There it was found, that, in dimension d = 2, the whole thermodynamic first order transition line (when such first order behavior is present in the system) coincides with the Kertész line. The latter is also first order in the corresponding range of values of temperature and field.

The aim of this letter is to present a study of such a model when the underlying lattice is the complete graph with n vertices. In this case the model is called mean-field or Curie-Weiss Potts model.

The model. -To introduce the Potts model on the complete graph we attach to the sites i = 1, . . . , n of the graph, spin variables σ i that take values in the set {1, . . . , q}. The model at temperature T = 1/β and subject to an external field H = h/β is then defined by the Gibbs measure

µ Potts (σ) = 1 Z Potts i<j e (β/n)(δσ i ,σ j -1) i e hδσ i ,1 (1)
over spins configurations σ. Here, Z Potts denotes the partition function, the indices i, j runs over the set {1, . . . , n}, and δ is the Kronecker symbol. The critical (thermody-namic) behaviour of this model is well known [5], [6], [7], and mainly governed by the mean field equation

h = -βs + ln 1 + (q -1)s 1 -s (2)
Namely, when h = 0, there exists a threshold value

β (q) c = 2 for q = 2 (3)
β (q) c = 2
q -1 q -2 ln(q -1) for q 3 (4) such that the system exhibits at β (q) c a continuous transition when q = 2, and a first order transition when q 3. When h > 0, no transition occurs if q = 2, while as soon as q 3 a first order transition line appears for which the microcanonical free energy of the model has two minima associated with two different solutions of the mean field equation [5], [7].

To study the behaviour of clusters previously mentioned, we turn to the Edwards-Sokal joint measure [8] given by

µ ES (σ, η) = 1 Z ES i<j [e -β n (1 -η ij ) + (1 -e -β n )η ij δ σi,σj ] × i e hδσ i ,1 (5)
where the edges variables η ij belong to {0, 1}. Notice that when h = ∞, all σ i = 1. This means that we open edges (η ij = 1) with probability (w.p.) p = 1-e -β/n and close edges (η ij = 0) w.p. e -β/n . This is nothing else than the well known Erdös-Rényi random graph G(n, p) [9]. This random graph is known to exhibit a (topological) transition at β = 1 such that with probability tending to 1 as n → ∞: i. for β < 1 all components of open edges are at most of order ln n.

ii. at β = 1 a giant component of order n 2/3 appears.

iii. for β > 1 this giant component becomes of order s * n where s * is the largest root of the mean field equation ( 2) with q = 1 and h = 0.

We refer the reader to [START_REF] Bollobás | Random Graphs[END_REF], [START_REF] Janson | Random Graphs[END_REF] for detailed discussions and proofs (see also [START_REF] Biskup | [END_REF] for a new approach of this transition). Notice also that, on the other hand, when h = 0, the marginal of the ES measure over the edges variables is the random cluster model:

µ RC (η) = 1 Z RC i<j e -(β/n)(1-ηij) (1 -e -β/n ) ηij q C(η) (6)
where C(η) denotes the number of connected components (including isolated sites) of open edges of the configuration η. For q = 1, this model again reduces to G(n, p) with p as before. A refined study of the random cluster model ( 6) is given in [13] (see also [START_REF] Grimmett | The Random-Cluster Model[END_REF], [START_REF] Luczak | [END_REF]). There, it is shown that with the threshold value β (q) c , given by (3) for 0 < q 2 and by (4) for q > 2, then with probability tending to 1 as n → ∞:

a) if β < β (q) c , the largest component (of open edges) of µ RC is of order ln n. b) if β > β (q)
c , the largest component of µ RC is of order s 0 n, where s 0 > 0 is the largest root of the mean field equation ( 2) with h = 0.

c) if β = β (q) c and 0 < q 2, µ RC has largest component of order n 2/3 . d) if β = β (q) c
and q > 2, µ RC is either as in a) or as in b).

According to these results, it is natural to expect for model µ ES (and its marginal over the edges variables), a Kertész line h K (β), interpolating betwen h K (1) = ∞ and h K (β (q) c ) = 0, that signs the emergence of a giant component in the open edges of the configuration η.

To study this line, it is convenient to consider a colored version of the Edwards-Sokal formulation. Namely, whenever the endpoints ij of a given edge are occupied by a spin of color a, we label the variable η ij with a superscript indicating the color. For a spin configuration σ, let n a (σ) be the number of sites occupied by spins of color a = 1, . . . , q. We then relabel the variables η a ij by η a kl : k < l ∈ {1, . . . , n a (σ)}. This means that, for any pairs i < j ∈ {1, . . . , n a (σ)}, we open an edge η a ij = 1 w.p. 1 -e -β/n , and close this edge η a ij = 0 w.p. e -β/n . All the other edges between two sites of different colors are closed. The resulting measure becomes

µ CES (σ, η 1 , . . . , η q ) = e hn1(σ) Z CES q a=1 e βna(σ)(na(σ)-1)/2n × i<j∈{1,...,na(σ)} [e -β/n (1 -η a ij ) + (1 -e -β/n )η a ij ] (7)
By summing over the spins variables, we get the following Colored-Random-Cluster model

µ CRC (η 1 , . . . , η q ) = 1 Z CRC n1+•••+nq=n Z(n 1 , . . . , n q ) × q a=1 i<j∈{1,...,na} [e -βxa/na (1 -η a ij ) + (1 -e -βxa/na )η a ij ] (8)
where x a = n a /n are the densities of the colors and

Z(n 1 , . . . , n q ) = n! n 1 ! • • • n q ! e hn1 q a=1
e βna(na-1)/2n (9) This colored random cluster model can be thought as follows. Given a partition n 1 , . . . , n q of n, we have q classical Erdös-Rényi random graphs where the edges are open w.p.

1 -e -βxa/na . The asymptotic behavior of the partition as n → ∞ will be determined by minimizing the free energy

f (x 1 , . . . , x q ) = -lim n→∞ 1 n ln Z(n 1 , . . . , n q ) (10)
where the limit is taken in such a way that n a /n → x a . Notice that, up to a normalizing factor, Z(n 1 , . . . , n q ) is the microcanonical partition function of the Potts model (1) restricted to configurations such that the number of spins of color a is fixed to n a . This will give the thermodynamic behavior of the system and will determine the densities x a . Once this is done the topological properties can be analysed by using the known properties of classical random graphs. Due to the presence of the field or using the symmetry of colors for vanishing field, we will pick up the first color. Then, the threshold value of the (topological) transition will be given by

β K x 1 = 1 (11)
For λ = βx 1 > 1 a giant component will appears, with size of order Θ x1 n ≡ θ βx1 x 1 n where θ λ is the largest root of the mean field equation

λθ = -ln(1 -θ) (12)
Let us briefly recall the minimization procedure for [START_REF] Bollobás | Random Graphs[END_REF]. By Stirling's formula, we have n!/n 1 ! • • • n q ! ∼ a a=1 e xa ln xa at the leading order in n, giving f (x 1 , . . . , x q ) = a a=1

x a ln x a -β 2 a a=1

x 2 a -hx 1

The minima of this function can be parametrised by a real number s, 0 s 1, such that the minimizing vector (x 1 , . . . , x q ) have the components

x 1 = 1 + (q -1)s q (14)
x a = 1 -s q a = 2, . . . , q

(with arbitrary order for h = 0). The free energy (13) then takes the form

f (s) = 1 + (q -1)s q ln(1 + (q -1)s) + q -1 q (1 -s) ln(1 -s) -β (q -1) 2q s 2 -h 1 + (q -1)s q - β 2q -ln q (16)
The minimizers have to be found among the solutions of the mean field equation (2) obtained by differentiating f with respect to s. Note that by taking into account [START_REF] Grimmett | The Random-Cluster Model[END_REF] the condition (11) becomes

β K 1 + (q -1)s q = 1 (17)
Results. -Let us first consider the case q = 2. When h = 0, the mean field equation ( 2) has a unique solution s 0 = 0 for β 2, and a unique solution s 0 > 0 for β > 2. Inserting s 0 into (17), we obtain that the giant component appears at β

(2) K = 2, getting thus β (2) K = β (2)
c . There, this component is of order n 2/3 . For β > 2, βx 1 (s 0 ) > 1, and we are in the supercritical regime with a giant component of order Θ x1(s0) n. For β < 2, βx 1 (s 0 ) < 1, and we are in the subcritical regime with components of order at most ln n.

When h > 0, the mean field equation has a unique solution s 0 > 0 for any β (no thermodynamic transition occurs). The transition line β

(2)

K (h), or equivalently h (2) K (β),
is obtained by eliminating s between ( 2) and ( 17). This Kertész line is given by

h (2) K (β) = β -2 -ln(β -1) (18)
On this line, the largest component is of order n 2/3 . For h < h K , the largest component is of order ln n, while for h > h K , the largest component is of order Θ x1(s0) n.

Let us then consider the case q 3. We have now a thermodynamic transition line

h T (β) = -β q -2 2(q -1) + ln(q -1), β 0 β β (q) c ( 19
)
with endpoints

(β 0 = 4 q -1 q , h 0 = -2 q -2 q + ln(q -1)), (β (q) c , 0) (20)
As previously mentionned, the thermodynamic transition has been already established in [7], for a curve with endpoints (20). To show that the thermodynamic curve is the straight line (19) we observe, on h T the free energy (16) has the symmetry f (s) = f ( q-2 q-1 -s) with s ∈ [0, q-2 q-1 ], see Fig. 1.

q-2 2(q-1) q-2 q-1 0 f s β = β c β > β > β 0 c
Fig. 1: The free energy on hT .

On h T the free energy has two minima s -and s + for which f (s -) = f (s + ). They satisfy the mean field equation (2) and thus are given by the parametric equations (β, h) = (β s± , -q -2 2(q -1)

β s± + ln(q -1)) (21)
where

β s = s - q -2 2(q -1) -1 ln 1 + (q -1)s (q -1)(1 -s) (22)
and s -∈ [0, q-2 2(q-1) ], s + ∈ [ q-2 2(q-1) , q-2 (q-1) ]. For β 0 < β β (q) c (or equivalently 0 h < h 0 ) these two minima are distinct: s -< s + . At (β 0 , h 0 ), s -= s + = q-2 2(q-1) , and at (β (q) c , 0), s -= 0 and s + = q-2 q-1 . Outside of the segment (19), there is only one minimum s 0 . This minimum is an analytic function of β and h, and s 0 → s -as h ↑ h T , s 0 → s + as h ↓ h T . In addition, on h T , as β increases from β 0 to β (q) c (or equivalently as h decreases from h 0 to 0), βx 1 (s -) strictly decreases from β 0 /2 > 1 to β (q) c /q < 1, while βx 1 (s + ) strictly increases from β 0 /2 to β (q) c (q -1)/q. See below. We now turn to the topological behavior. Let (β cp , h cp ) the point of h T for which βx 1 (s -) = 1. This point is distinct from the two endpoints (20). It is given by the solution of the equation (β cp -2)q = 2(q-1) ln(β cp -1) which can be solved by using the Lambert W function.

As a consequence of the above remarks, one gets the following topological behaviour for the system. On the thermodynamic line h T :

1. at (β cp , h cp ), the largest component of µ CRC is either of order n 2/3 or of order Θ x1(s+) n.

2. if h cp < h h 0 , the largest component of µ CRC is either of order Θ x1(s-) n or of order Θ x1(s+) n.

3. if 0 h < h cp , the largest component of µ CRC is either of order ln n or of order Θ x1(s+) n.

The item 3 extends to the values 0 h < h cp what happens in item d) of the random cluster model for vanishing field. It implies that the thermodynamic and topological lines coincide there. This holds also at (β cp , h cp ), with a new behaviour in the sense that the giant component is of order n 2/3 or of order Θ x1(s+) n.

The Kertész line is given in this range of temperature (and field) by

h (q) K (β) = h T (β) for β cp β β (q) c or 0 h h cp (23)
The item 2 shows that the giant component exhibits a jump, however, for given h > h cp , this giant component already appeared for lower values of β. The topological transition line is determined, as in case q = 2, by eliminating s between the mean field equation and the condition (17). This gives

h (q) K (β) = β -q q -1 -ln β -1 q -1 for β β cp or h h cp (24)
see Fig. 2. Below the Kertész line h (q)

K the largest component is of order ln n, while above h (q) K and outside of the segment (19), the largest component is of order Θ x1(s0) n. The symmetry of the free energy is better seen if we parametrize the magnetization s by z ∈ (±1) as follows:

s = q -2 2(q -1) + q 2(q -1) z. (25)
The corresponding densities are given by x 1 = 1+z 2 , x 2 = • • • = x q = 1-z 2(q-1) , and the free energy decomposes immediately into its even and odd parts:

f (z) = 1 2 [(1 + z) ln(1 + z) + (1 -z) ln(1 -z) -β q(1 + z 2 ) 4(q -1) -h -ln (4(q -1))] + z 2 ln(q -1) -β q -2 2(q -1) -h . (26)
The first derivative of f yields the mean field equation ( 2) while the second derivative is

f ′′ (z) = 1 1 -z 2 - βq 4 (q -1) . ( 27
)
Thus f is convex when β β 0 and has exactly one minima. When β > β 0 it has at most two local minimas. The analyticity of s 0 outside of h T is a consequence of f ′′ (z 0 ) > 0. The monotonicity of βx 1 (s + ) is trivial, that of βx 1 (s -) can be infered from the computation of the contour lines for βx 1 (s) below h T .

Conclusion. -To summarize, we have given the equation of the Kertész line of the mean-field Potts model and shown that when q 3 this line exhibits a cusp at some (β cp , h cp ). For low fields (h h cp ) the Kertész line coincides with the thermodynamic transition line, and for large fields (h h 0 ) only the topological transition remains while the thermodynamic transition disappears. In addition, at intermediate fields (h cp < h < h 0 ) the Kertész line separates from the thermodynamic line. This means that decreasing the temperature one sees first the appeareance of a giant component (on h K ) and, then on h T , this component exhibits a jump. This behavior is new compared to what happens for the model on the 2-dimensional regular lattice. There, no intermediate regime appears [4]. We expect for sufficiently high lattice dimensions, a behaviour similar to the model on the complete graph.

Fig. 2 :

 2 Fig. 2: The thermodynamic transition curve: thick (solid, dashed) line; the topological transition curve: solid (thick, thin) line.

Acknowledgments.

Warm hospitality and Financial support from the BiBoS Research Center, University of Bielefeld and Centre de Physique Théorique, CNRS Marseille are gratefully acknowledged.