Ghenia Bentabed-Ababsa

Samira Hamza-Reguig

Aïcha Derdour

Luis R Domingo
email: domingo@utopia.uv.es

José A Sáez

Thierry Roisnel

Vincent Dorcet

Ekhlass Nassar

Florence Mongin
email: florence.mongin@univ-rennes1.fr

Experimental and theoretical study of the [3+2] cycloaddition of carbonyl ylides with alkynes

The [3+2] cycloaddition reaction between carbonyl ylides generated from epoxides and alkynes (phenylacetylene, methyl propiolate, methyl but-2-ynoate and methyl 3-phenylpropiolate) to give substituted 2,5-dihydrofurans was investigated. The effect of indium(III) chloride on the outcome of the reaction was studied in the case of phenylacetylene and methyl propiolate. The thermal reaction between the carbonyl ylide coming from 2,2-dicyano-3-phenyloxirane and both methyl propiolate and methyl but-2-ynoate was theoretically investigated using DFT methods in order to explain the reactivity and regioselectivity observed.

Introduction

Cycloaddition reactions are one of the most important synthetic processes, with both synthetic and mechanistic interest in organic chemistry. Among them, [3+2] cycloaddition (32CA) reactions, also named 1,3-dipolar cycloadditions, whose general concept was introduced by Huisgen and co-workers in 1960s, 1 are versatile tools for building five-membered heterocycles, and current understanding of the underlying principles in these reactions has grown from a fruitful interplay between theory and experiment. [START_REF]1.32 g.cm -3 , μ = 0.083 mm -1 . A final refinement on F 2 with 1721 unique intensities and 190 parameters converged at ωR(F 2 ) = 0.067 (R(F) = 0.0275) for 1649 observed reflections with I > 2σ(I)[END_REF] Carbonyl ylides, generated by thermal ring opening of epoxides, are known to react with -bonds of alkynes, 3 alkenes, 3a,b,4 imines, 5 aldehydes 5c,6 and ketones, 7 and thioketones, 8 affording highly substituted dihydrofurans, tetrahydrofurans, oxazolidines, dioxolanes, and oxathiolanes, respectively. Following recent studies of the 32CA reactions between, on the one hand, carbonyl ylides and, on the other hand, aldehydes, 5c imines, 5c and ketones, 7 in which we have performed theoretical calculations using density functional theory (DFT) methods to depict the mechanism of these reactions, we here report our studies concerning the reactions involving unsymmetrical alkynes.

Since the first thermal reactions between carbonyl ylides, generated from epoxides, and activated alkynes (such as methyl acetylenedicarboxylate) reported more than 40 years ago, 3a,b few studies have been devoted to such an access to dihydrofurans. 3c-e In particular, it was of interest to study, in the light of DFT calculations, the regioselectivity results obtained by involving unsymmetrical alkynes in the reaction with carbonyl ylides.

Results and discussion

Synthetic aspects

We first considered 32CA reactions between 2,2-dicyano-3-(4-45 substituted)phenyloxiranes 1a-c 9 and terminal alkynes. Using phenylacetylene (2, 5 equiv), monitoring the conversion to the corresponding 2,5-dihydrofurans by TLC showed that the reactions carried out in refluxing toluene were complete after about 40 h. NMR analysis of the crude obtained after removal of 50 the solvent showed the presence of two regioisomers in each case, with ratios of 84:16, 84:16 and 75:25 respectively obtained using 1a (X = H), 1b (X = Cl) and 1c (X = OMe). The major products 3 were purified by column chromatography over silica gel, in yields ranging from 40 to 52% whereas 3'b was the only 55 minor product isolated (12% yield). The observed 1 H NMR coupling constants of 1.8 and 2.1 Hz were respectively attributed to H J 4-5 (major regioisomer 3) and H J 3-5 (minor regioisomer 3') on the based of a previous study. 10 The major regioisomers 3 were then identified unequivocally by X-ray structure analysis (Table 60 1, entries 1-3). † The same reactions have then been repeated, but in the presence of indium(III) chloride (0.1 equiv) which has been recognized as a suitable catalyst for cycloaddition reactions. 11 Shortened reaction times of 29 h, 27 h and 20 h, increased 3:3' ratios of 90 :10, 89:11 and 83:17, and higher yields of 3 (60 to 64%) were respectively noted using 1a (X = H), 1b (X = Cl) and 1c (X = OMe) under these conditions using 4 equiv of phenylacetylene (Table 1, entries 4-6).

We next turned to the reaction of methyl propiolate (4) with the 2,2-dicyano-3-(4-substituted)phenyloxiranes 1a,b (4 equiv) and 1c (1 equiv) under the conditions used before. Without catalyst, NMR analysis of the crudes showed the formation of two regioisomers in 24:76, 26:74 and 29:71 ratios, using 1a (48 h reaction time for complete conversion), 1b (48 h) and 1c (24 h) respectively. They were separated by column chromatography over silica gel, and isolated in yields ranging from 10 to 12% and 40 to 65% for the major and minor isomer, respectively. As before, 10 the major compounds with the higher 2,5-dihydrofuran 1 H NMR coupling constants (2.1 Hz) were assigned to the 4substituted regioisomers 5' ( H J 3-5 ), and the minor compounds with the lower 2,5-dihydrofuran 1 H NMR coupling constants (1.5-1.8 Hz) to the 3-substituted regioisomers 5 ( H J 4-5 ). This was corroborated by NMR HMQC sequences performed on 5'c and 5c, and the structures of 5'a and 5b were confirmed by X-ray diffraction data (Table 2, entries 1-3). † In the presence of indium(III) chloride (0.1 equiv), the reaction times were slightly shortened (36 h in the case of 1a,b and 20 h with 1c), as well as the regioselectivities (Table 2, entries 4-6).

We then considered 32CA reactions between 2,2-dicyano-3-(4substituted)phenyloxiranes 1a-c and internal alkynes. Using methyl but-2-ynoate (6) and methyl 3-phenylpropiolate (7) without catalyst proved completely regioselective. The cycloadducts 8 and 9 respectively formed (extended reaction times were required in the case of 7) were isolated after purification by column chromatography over silica gel in 44-66% yields (Table 3).

What is remarkable on the 1 H NMR spectra of the compounds 8 is a 2.1 Hz coupling constant between the methyl protons (2.44 ppm) and H5 (6.18 ppm) (see Table 3). The presence of a methyl group at C3, already shown on 8c by an NMR HMQC sequence, was confirmed by X-ray diffraction on crystals of 8a,b, allowing an unambiguous assignment (Table 3, entries 1-3). † 2,2-Dicyano-3-(4-chlorophenyl)oxirane (1c) was also reacted with 3-phenylpropiolonitrile (10) as before to afford the expected product 11. The reaction nevertheless proved less efficient, and 45 the cycloadduct 11 could not be separated from the dioxolanes 12 coming from a reaction with the aldehyde resulting from epoxide degradation (Scheme 1).

Calculations

1) Mechanistic study of the 32CA reactions of the carbonyl ylide 50 1'a with methyl propiolate (4) and methyl but- 2-ynoate (6). In order to explain the reactivity and regioselectivity of the 32CA reactions of the carbonyl ylide (CY) 1'a with methyl propiolate (4) and methyl but-2-ynoate (6), the mechanisms of these cycloaddition reactions were theoretically studied using DFT 55 methods at the B3LYP/6-31G* level.

The CY 1'a can adopt the E or Z configuration through the restricted rotation of the C1-O2 bond. In this way, while (E)-1'a adopts a planar rearrangement, (Z)-1'a is twisted as a consequence of the hindrance between the phenyl and one cyano 60 group. This hindrance makes (Z)-1'a 8.6 kcal.mol -1 higher in energy than (E)-1'a. In addition, the barrier height associated with the C1-O2 bond rotation is very large, 27.1 kcal.mol -1 . 5c Consequently, only the E configuration of the CY 1'a is considered in the present study.

65

Due to the asymmetry of both reagents two competitive reaction channels are feasible for each 32CA reaction. They are related to the two regioisomeric approach modes of the CY 1'a towards the alkynes 4 and 6, named m and o (the m and o acronyms, traditionally used to refer to the relative meta and 70 ortho positions in the benzene series, are here employed to distinguish the cycloadduct (CA) with distant ester and nitrile functions from that with adjacent ester and nitrile functions). Along the channel m, the C1-C5 and C3-C4 bonds are formed, while along the channel o, the C1-C4 and C3-C5 bonds are 75 formed. The two regioisomeric channels were studied. Analysis of the stationary points associated with these 32CA reactions indicates that they have a one-step mechanism. Therefore, two (10).

transition states (TSs), TSxm and TSxo, and two CAs, CAxm and CAxo, were located and characterized for each 32CA reaction (see Scheme 2).

The two 32CA reactions present very low activation barriers: 2.3 kcal.mol -1 for TS1m and 3.6 kcal.mol -1 for TS2m (see Table 4). On the other hand, the two regioisomeric TSs are 2.0 (TS1o) and 4.3 (TS2o) kcal.mol -1 higher in energy than those associated with the most favorable channels m. These results agree entirely with the experimentally observed regioselectivity. In this way, a mixture of CA1m and CA1o is expected for the 32CA reaction between 1'a and 4, while for the reaction with 6 only one CA, CA2m, is expected. Experimentally, a CA1m:CA1o 76:24 mixture is obtained. All these 32CA reactions are strongly exothermic: between -68.3 and -72.6 kcal.mol -1 .

The B3LYP/6-31G* geometries were further optimized using the 6-311G* and 6-311+G* basis sets. The energy results are given in the Supplementary Information part. A comparison of the relative energies using the three selected basis sets indicates that there are not significant differences. The activation energies increase slightly with the size of the basis set, but this fact is a consequence of the higher stabilization of CY 1'a than the TSs. Anyway, the regioselectivity found at these 32CA reactions is kept no matter which basis set is used.

Recent studies devoted to 32CA reactions have shown that solvent effects in the geometry optimization have poor effects 25 due to the low polar character of both TSs and CAs, and of toluene and dichloromethane solvents. 12 Consequently, solvent effects of toluene on energies were considered through singlepoint energy calculations over the gas-phase optimized geometries using the PCM method. Solvent effects stabilize all species between 2 and 7 kcal.mol -1 (see Table 4), with the reagents being more stabilized than the TSs. 12 Therefore, the activation barrier for the cycloadditions increases by 2.1 (TS1m) and 2.7 (TS2m) kcal.mol -1 . However, solvent effects do not change the gas phase regioselectivity. Therefore, solvent effects 35 appear to have little influence on these 32CA reactions.

Further thermodynamic calculations with regard to toluene showed a similar free energy difference between each pair of regioisomeric TSs (see Table 4). Adding thermal corrections to energies and entropies increases the activation free energies to 40 20.9 (TS1m), 21.9 (TS1o), 24.4 (TS2m) and 26.8 (TS2o) kcal.mol -1 . Several conclusions can be drawn from these energies: i) these free activation energies are not very high. However, the endergonic character of the formation of the CY 1'a, which rises the activation free energies of these 32CA reactions to 29.6 45 (TS1m) and 33.1 (TS2m) kcal.mol -1 , is responsible for the high temperature demanded in these thermal reactions. 7 Note that the free energy of the CY 1'a is 8.7 kcal.mol -1 above than the corresponding epoxide; 7 ii) the presence of an electron-releasing methyl group in methyl but-2-ynoate (6) increases the free 50 activation energy by 3.5 kcal.mol -1 relative to that using methyl propiolate (4); iii) the regioselectivity measured as the activation free energy difference between the two regioisomeric TSs remains almost unaltered after the thermodynamic calculations: G (TS1o-TS1m) = 1.0 kcal.mol -1 and G (TS2o-TS2m) = 2.4 kcal.mol -1 ; 55 and iv) these 32CA reactions are strongly exergonic: between -42.5 to -48.7 kcal.mol -1 . Consequently, these reactions are irreversible.

The B3LYP/6-31G* geometries of the TSs associated with the 32CA reactions between the CY 1'a and methyl propiolate (4) 60 and methyl but-2-ynoate ( 6) are given in Figure 1, while those obtained using the 6-311G* and 6-311+G* basis sets are given in the Supplementary Information part. At the TSs associated with the 32CA reaction between the CY 1'a and methyl propiolate (4), the lengths of the C-C forming bonds are 2.375 Å (C1-C5) and The electronic structure of the TSs involved in the 32CA reactions between the CY 1'a and methyl propiolate (4) and methyl but-2-ynoate (6) was analyzed using the Wiberg bond 75 order 13 (BO) and charge transfer (CT) at the TSs. The extent in Table 4. Relative energies (ΔE, in kcal.mol -1 , relative to 1'a+4 and 1'a+6) in gas phase and in toluene, and relative enthalpies (ΔH, in kcal.mol -1 ), entropies (ΔS, in eu) and free energies (ΔG in kcal.mol -1 ) at 110 ºC in toluene of the stationary points involved in the 32CA reactions of CY 1'a with methyl propiolate (4) and methyl but-2-ynoate (6). Scheme 2 32CA reactions between CY 1'a and methyl propiolate (4) or methyl but-2-ynoate (6).

E

bond formation at the TSs is provided by the BOs. At the TSs associated with the 32CA reaction between the CY 1'a and methyl propiolate (4), the BO values of the C-C forming bonds are 0.22 (C1-C5) and 0.17 (C3-C4) at TS1m, and 0.26 (C3-C5) and 0.15 (C1-C4) at TS1o, while those in the 32CA reaction involving methyl but-2-ynoate (6) are 0.22 (C1-C5) and 0.19 (C3-C4) at TS2m, and 0.22 (C3-C5) and 0.18 (C1-C4) at TS2o. Several conclusions can be drawn from these values: i) the relative low BO values found at the four TSs indicate that they have an earlier character, in clear agreement with the very low activation energies and the high exothermic character of these processes; 14 ii) the similar BO values of the two forming bonds at the TSs indicate that they do not present a marked asynchronicity in bond formation as would be expected in a polar process involving asymmetric reagents; iii) in the four TSs, the C-C bond formation at the  position of 4 and 6 is slightly more advanced than at the  position.

The polar or non-polar character of these 32CA reactions was analyzed evaluating the CT at the four TSs. The natural charges were shared between the CY and the alkyne frameworks. At the TSs, the charge at the CY framework is 0.05 at TS1m, 0.02 at TS1o, 0.00 TS2m and -0.01 at TS2o. These negligible CTs indicate that these 32CA reactions have a non-polar character.

2) Analysis based on the global reactivity indices at the ground state of the reagents.

Studies devoted to Diels-Alder and 32CA reactions have shown that the analysis of the global indices defined within the context of conceptual DFT 15 is a powerful tool to understand the behavior of polar cycloadditions. 16 In Table 5, we report the static global properties, namely, electronic chemical potential μ, chemical hardness η, global electrophilicity ω and nucleophilicity N of the CY 1'a and methyl propiolate (4) and methyl but-2-ynoate ( 6).

The electronic chemical potential μ of methyl propiolate (4), μ = -4.42 eV, and methyl but-2-ynoate (6), μ = -4.04 eV, is slightly higher than that for the CY 1'a, -4.61 eV. The gaps of electronic chemical potentials between reagents, μ, are below 0.6 eV. These low values do not permit to establish a tendency for the flux of the CT along the cycloaddition, in clear agreement with the NBO analysis at the corresponding TSs.

The CY 1'a has a high electrophilicity value,  = 4.29 eV, being classified as a strong electrophile in the electrophilicity scale. On the other hand, the CY 1'a also has a very high nucleophilicity value, N = 3.28 eV, also being classified as a strong nucleophile in the nucleophilicity scale. Consequently, the CY 1'a can act as a strong electrophile towards nucleophilic species and as a strong nucleophile towards electrophilic species.

Methyl propiolate (4) and methyl but-2-ynoate (6) present low electrophilicity values,  = 1.52 and 1.29 eV, respectively, being classified as marginal electrophiles. On the other hand, they have somewhat low nucleophilicity values, N = 1.48 and 1.92 eV, respectively, being also classified as poor nucleophiles. As expected, the methyl substitution in methyl but-2-ynoate (6) decreases the electrophilicity and increases the nucleophilicity of methyl propiolate (4).

Analysis of the reactivity indices indicates that although the 55 CY 1'a can participate in polar 32CA reactions as a consequence of its high electrophilic and nucleophilic character, the low electrophilic and nucleophilic power of methyl propiolate (4) and methyl but-2-ynoate (6) prevent their participation in polar cycloadditions. The behavior of the alkynes 4 and 6 could be 60 responsible for the low CT observed at the corresponding TSs.

3) ELF topological analysis of the 32CA reaction between the CY 1'a and methyl propiolate (4). Recent theoretical studies have shown that the topological analysis of the ELF along the reaction path associated with a 65 cycloaddition is a valuable tool for understanding the bonding changes along the reaction path. 17 Consequently, a topology analysis of the ELF of some selected points along the most favorable reaction path associated with the 32CA reaction between the CY 1'a and methyl propiolate (4) was performed in 70 order to understand the bond-formation in these 32CA reactions. The N populations of the most relevant ELF valence basins are listed in Table 6. A schematic picture of the attractor positions and atom numbering are shown in Figure 2.

ELF analysis of the CY 1'a shows two disynaptic basins 75 V(C1,O2) and V(O2,C3), whose electronic populations integrate 1.69e and 2.14e, respectively, one monosynaptic basin V(O2) integrating 3.57e, and two monosynaptic basins V(C1) and V'(C1), integrating 1.17e. Recently, we studied the non-polar 32CA reaction between the CY 13 and tetramethylethylene ( 14) 80 (see Scheme 3). 17b ELF analysis of the CY 13 showed two pairs of monosynaptic basins V(Cx) and V'(Cx) at each C2 and C2' carbons, integrating ca 1.0e each pair, and one monosynaptic basin V(O1) integrating 3.58e. The high reactivity of the CY 13 towards unactivated tetramethylethylene ( 14) was attributed to its pseudodiradical electronic structure, which allows for the C-C bond formation. Note that the activation enthalpy associated with this non-polar 32CA reaction was only of 4.7 kcal.mol -1 . 17b The absence of the two monosynaptic basins at the C3 carbon of the CY 1'a can be explained by a delocalization of the electron density of this pseudodiradical center at the adjacent phenyl group. This delocalization is supported by the changes of electron density of the disynaptic basin V(C3,C6) along the IRC, which decreases from 2.79e in CY 1'a to 2.05e in CA1m.

On the other hand, methyl propiolate (4) has two disynaptic basins V(C4,C5) and V'(C4,C5), integrating 5.40e. These disynaptic basins account for the C4-C5 triple bond drawn in the Lewis structure of methyl propiolate (4).

Along the 32CA pathway, the ELF analysis of TS1m, d1 = 2.375 Å and d2 = 2.505 Å, shows that while the two monosynaptic basins V(C1) and V'(C1) have merged into one monosynaptic basin V(C1), integrating 0.90e, a new 20 monosynaptic basin V(C4), integrating 0.35e, has been created at the C4 carbon of methyl propiolate (4). Simultaneously, the two disynaptic basins V(C4,C5) and V'(C4,C5) belonging to the C4-C5 triple bond region of 4 have been depopulated by ca. 0.40e. At this point of the IRC, the CT is very low, 0.05e. This low value indicates that this cycloaddition has no polar character. Unlike polar cycloadditions in which monosynaptic basins appear at the most electrophilic center of the electrophile as a consequence of the CT with takes place along the nucleophilic attack, in the present non-polar cycloaddition, the monosynaptic basin V(C4) appears as a consequence of the internal electron reorganization that takes place at the C4-C5 triple bond region along the reaction path.

At P1, d1 = 2.075 Å and d2 = 2.255 Å, while the monosynaptic basin V(C1) has slightly increased its population, 35 the monosynaptic basin V(C4) has reached 0.71e. Additionally, at this point of the IRC two monosynaptic basins V(C3) and V(C5), integrating 0.46e and 0.42e respectively, have been created at the C3 carbon of the CY 1'a and at C5 carbon of 4. Simultaneously, the C4-C5 triple bond region has been depopulated by 0.64e. At 40 P2, d1 = 1.767 Å and d2 = 1.922 Å, the four monosynaptic basins V(C1) and V(C5), and V(C3) and V(C4), have merged into two new disynaptic basins, V(C1,C5) and V(C3, C4), which integrate 1.79e and 1.69e, respectively. Consequently, at this point of the IRC, the two new C1-C5 and C3-C4  bonds have already been formed. At P2, while the populations of the two disynaptic basins V(C4,C5) and V'(C4,C5) have decreased to 1.94e and 1.96e, the monosynaptic basin V(O2) present in the CY 1'a has split into two monosynaptic basins V(O2) and V'(O2), integrating 2.49e and 2.26e. Finally, CA1m, d1 = 1.523 Å and d2 = 1.515 Å, 50 presents a similar bonding pattern than that at P2. As shown in Table 6, only slight changes in the basin populations take place at the end of the reaction.

The ELF bonding analysis at TS1m indicates that the main change in the electronic structure of methyl propiolate ( 4) is 55 associated with the formation of C4 pseudoradical center. Note that at the CY 1'a, only the fusion of the two monosynaptic basins V(C1) and V'(C1) into the monosynaptic basin V(C1) takes place. This change is associated with the depopulation in electron density at the C4-C5 triple bond region. This behavior 60 together with the very low CT at TS1m, 0.06e (ELF) and 0.05 (NBO), suggests a non-polar 32CA reaction with a pseudodiradical character. Recently, we established that the high reactivity of CYs in non-polar processes can be associated with their pseudodiradical character, which enables the reaction to 65 take place through an unappreciable barrier. 17b Finally, ELF bonding analysis at the C1-O2-C3 framework of the CY 1'a along the 32CA reaction shows that while the electron density of the monosynaptic basins V(O2) and V'(O2) increases from 3.57e in the CY 1'a to 5.04 e in CA1m, the 70 electron density of the disynaptic basins V(C1,O2) and V(O2,C3) decreases from 3.83e in the CY 1'a to 2.46e in CA1m. This behavior, which is similar to that found in the non-polar 32CA reaction between the CY 13 and tetramethylethylene (14) in Scheme 3,17d indicates that the O2 lone pairs of the CY 1'a do not 75 participate in the C-C bond formation in the 32CA reaction; while in the CY 1'a these O2 lone pairs are partially delocated in the pseudoradical C1 and C3 centers, in CA1m they are mainly located at the O2 oxygen as a consequence of the saturation of the C1 and C3 carbons after the C-C bond formations. This behavior 80 allows for the establishment that these 32CA reactions may be electronically classified as [2n + 2π] processes, 17c,d in which only the two electrons of the HOMO of the CY 1'a and the two electrons of HOMO-1 of methyl propiolate (4) participate in the 32CA reaction (see Figure 3).

4) What is the origin of the regioselectivity in these non-polar

32CA reactions? While Diels-Alder reactions present a high regioselectivity, which increases with the polar character of the reaction, 18 32CA reactions present a low regioselectivity, which also depends on polarity. Thus, it is expected that these non-polar 32CA reactions will have a poor regioselectivity. However, both experimental and theoretical calculations show that both reactions are regioselective. Analyses of the local reactivity indices indicates that the C1 carbon of the CY 1'a is the most nucleophilic center, 7 and that the  conjugated C5 position of methyl propiolate (4) and methyl but-2-ynoate ( 6) is the most electrophilic center of these molecules. Therefore, although the polar analysis explains the observed regioselectivity, the bonding pattern involved in polar cycloadditions can not rule in these non-polar 32CA reactions. 19 ELF analysis of the electronic structures of the four TSs involved in these non-polar 32CA reactions allows to obtain some interesting information. For simplicity, only the ELF attractors of the four TSs are given in Figure 4. The four TSs show two monosynaptic basins, the V(C1) and the V(C4) at the regioisomeric TS1m and TS2m, and the V(C1) and the V(C5) at the regioisomeric TS1o and TS2o. Therefore, the four TSs share the presence of the monosynaptic basin V(C1), which is already present in the CY 1'a with similar population, and the mosynaptic basins V(C4) in channels m and the V(C5) in channels o, which are created along the reaction. Interestingly, in the four TSs, the first monosynaptic basin at the acetylenic system appears at the opposite carbon of the CY, regardless of the carbonyl group position. Note that in a polar cycloaddition, these basins appear always at the  conjugated position, which correspond with the most electrophilic center of these acetylene derivatives. 19 Consequently, we can associate the relative stability of the two pairs of regioisomeric TSs with the relative stability of the corresponding pseudodiradical species formed on going to the corresponding TS. Unlike the polar cycloadditions in which the formation of the monosynaptic basins takes place firstly at the conjugated position, in these non-polar 32CA reactions the formation of the monosynaptic basin at the  position appears to be favored over the  one. This finding could explain the larger regioselectivity found in the 32CA reaction using methyl but-2ynoate (6) than that with methyl propiolate (4). The presence of the electron-releasing methyl group on C5 carbon of but-2-ynoate (6) favors the formation of monosynaptic basin V(C4) in TS2m respect to TS1m. This behavior is supported by the larger population of the monosynaptic basin V(C4) in TS2m, 0.58e, than that in TS1m, 0.35e. Note that in both TSs, the monosynaptic basin V(C1) present the same population, 0.90e. 7). All the compounds tested showed moderate bactericidal activities against Staphylococcus Aureus, Escherichia Coli and Pseudomonas aeroginosa compared to that of ciprofloxacin as a reference drug. The 60 compounds tested showed moderate fungicidal activities towards Fusarium oxysporium and Aspergillus niger compared to that of nystin as a reference, except the compound 5'c that showed significant activities, in particular against Fusarium oxysporium. Only the compound 5'c showed a moderate activity against 65 Candida albicans.

The compounds 3a,b, 5'c and 9c were also tested against a human liver carcinoma cell line (HEPG2), and 3b against human breast (MCF7), and cervix (HELA) carcinoma cell lines (Table 8). Moderate cytotoxic activities were observed for all the 70 compounds tested, compared to a reference drug (doxorubicin), except in the case of 3b, for which activities close to those of doxorubicin were observed. The difference in the value of IC 50 between the two compounds 3a and 3b may be due to the presence of the chloro group in 3b, a substituent that can affect 75 the biological activity of the compounds. 21

Conclusions

Thus, substituted 2,5-dihydrofurans were synthesized by 32CA reaction between carbonyl ylides generated from epoxides and alkynes. While 3-phenyl-5-(4-substituted phenyl)-(2H,5H)dihydrofuran-2,2-dicarbonitrile 3 and methyl 2,2-dicyano-5-(4substituted phenyl)-(2H,5H)-dihydrofuran-4-carboxylate 5' only dominate using phenylacetylene ( 2) and methyl propiolate (4), respectively, the use of methyl but-2-ynoate ( 6) and methyl 3phenylpropiolate ( 7) led to only one regioisomer, the methyl 5-(4substituted phenyl)-2,2-dicyano-(2H,5H)-dihydrofuran-4carboxylates 8 and 9, respectively methylated or phenylated at C3. The different observed regioisomeric ratios using methyl propiolate ( 4) and methyl but-2-ynoate ( 6) in the reactions with the CY 1'a were explained by theoretical calculations. Despite the high electrophilic and nucleophilic character of the CY 1'a, the low electrophilic and nucleophilic character of the alkynes 4 and 6 causes that these 32CA reactions take place through a non-polar mechanism via TSs with a pseudodiradical character. In spite of the low activation energy associated with these non-polar 32CA reactions, the endothermic character associated with the formation of the CY 1'a was found responsible for the high temperature required to carry out these thermal reactions. Finally, analysis of the bonding changes along these 32CA reactions allows for the classification of these nonpolar cycloadditions as [2n + 2π] processes.

Experimental Syntheses: general methods

Liquid chromatography separations were achieved on silica gel Merck-Geduran Si 60 (40-63 m). Petrol refers to petroleum ether (bp 40-60 °C). Melting points were measured on a Kofler 30 apparatus. Nuclear Magnetic Resonance spectra were acquired using a Bruker AC-300 spectrometer (300 MHz and 75 MHz for 1 H and 13 C respectively). 1 H chemical shifts () are given in ppm relative to the solvent residual peak, and 13 C chemical shifts relative to the central peak of the solvent signal. NOESY, HMBC

and HMQC experiments were performed on an Avance 500 spectrometer (500 MHz and 125 MHz for 1 H and 13 C, respectively). High resolution mass spectra measurements were recorded at the Centre Régional de Mesures Physiques de l'Ouest (CRMPO) in Rennes. Oxiranes were prepared according to 40 described procedures. 9 Toluene was dried before use. Reactions were performed under dry argon.

General procedure 1.

A mixture of epoxide (2 mmol) and phenylacetylene (1.0 g, 10 mmol) in anhydrous toluene (30 mL)

was heated at reflux under Ar for 40 h. The mixture was then 45 evaporated to dryness and the product isolated by column chromatography over silica gel (eluent: 25:75 CH 2 Cl 2 -petrol).

General procedure 2.

A mixture of epoxide (2 mmol), phenylacetylene (0.80 g, 8 mmol) and InCl 3 (40 mg, 0.2 mmol) in anhydrous toluene (30 mL) was heated at reflux under Ar for 29 h 50 (1a), 27 h (1b) and 20 h (1c). The mixture was then evaporated to dryness and the product isolated by column chromatography over silica gel (eluent: 25:75 CH 2 Cl 2 -petrol). (1c). The mixture was then evaporated to dryness and the product isolated by column chromatography over silica gel (eluent: 10:90 AcOEt-petrol in the case of 1a,b; 20:80 with 1c).

Methyl

2,2-dicyano-5-phenyl-(2H,5H)-dihydrofuran-3carboxylate (5a). Yield: 12% (general procedure 3). Yellow oil. 1 H NMR (300 MHz, CDCl 3 ): 3.95 (s, 3H), 6.27 (d, 1H, J = 1.8 Hz), 7.26 (d, 1H, J = 1.8 Hz), 7.29-7.32 (m, 3H), 7.42-7.45 (m, 2H). 13 C NMR (75 MHz, CDCl 3 ): 53.4, 72.4, 91.7, 112.0, 112.3, 126.9 (2C), 128.1, 130.2 (2C), 130.2, 134.6, 146.7, 159.1. HRMS (ESI): calcd for C 14 H 10 N 2 O 3 (M +• ) 254.0589, found 254.0592. (general procedure 4). White powder, mp 102 °C. 1 H NMR (300 MHz, CDCl 3 ): 3.73 (s, 3H), 6.24 (d, 1H, J = 2.1 Hz), 6.78 (d, 1H, J = 2.1 Hz), 7.32-7.35 (m, 2H), 7.41-7.43 (m, 3H). 13 C NMR ( 75MHz, CDCl 3 ): 53. 0, 72.3, 91.1, 111.6, 111.8, 127.6 (2C), 128.7, 129.1 (2C), 130.1, 135.1, 142.9, 160.2 (d, 1H, J = 1.8 Hz), 7.23 (d, 1H, J = 1.8 Hz), 7.24-7.27 (m, 2H), 7.40-7.43 (m, 2H). 13 C NMR (75 MHz, CDCl 3 ): 54. 5, 72.4, 90.8, 111.9, 112.2, 128.3 (2C), 128.6, 129.8 (2C), 133.1, 136.3, 146.0, 159.0 (5'b). Yield: 54% (general procedure 3), 57% (general procedure 4). White powder, mp 105 °C. 1 H NMR (300 MHz, CDCl 3 ): 3.75 (s, 3H), 6.21 (d, 1H, J = 30 2.1 Hz), 6.78 (d, 1H, J = 2.1 Hz), 7.26-7.29 (m, 2H), 7.39-7.41 (m, 2H). 13 C NMR (75 MHz, CDCl 3 ): 53. 1, 72.3, 90.3, 111.4, 111.7, 128.9, 129.0 (2C), 129.5 (2C), 133.7, 136.2, 142.6, 160.0 )dihydrofuran-3-carboxylate (5c). Yield: 11% (general procedure 3), 8% (general procedure 4). Yellow oil. 1 H NMR (300 MHz, CDCl 3 ): 3.82 (s, 3H), 3.96 (s, 3H), 6.22 (d, 1H, J = 1.5 Hz), 6.93-6.96 (m, 2H), 7.20-7.23 (m, 2H), 7.24 (d, 1H, J = 3H, J = 2.10 Hz), 3.67 (s, 3H), 3.79 (s, 3H), 6.16 (q, 1H, J = 2.10 Hz), 6. 89-6.92 (m, 2H), 7.21-7.24 (m, 2H). 13 C NMR (75 MHz, CDCl 3 ): 11.1, 52.4, 55.3, 76.8, 91.8, 111.9, 112.1, 114.3 (2C), 128.1, 129.0 (2C), 134.0, 140.0, 160.7, 161.4. An NMR HMQC sequence performed on 8c showed a relationship between the 90 methyl group protons connected to the dihydrofuran ring and the nitrile functions, in accordance with a methyl at C3 (see Table 3). HRMS (ESI): calcd for C 16 H 14 N 2 O 4 (M +• ) 298.0846, found 298.0848. General procedure 6. A mixture of epoxide (2 mmol) and 95 methyl 3-phenylpropiolate (0.32 g, 2 mmol) in anhydrous toluene (30 mL) was heated at reflux under Ar for 96 h (1a,b) or 55 h (1c). The mixture was then evaporated to dryness and the product isolated by column chromatography over silica gel (eluent: 5:95 AcOEt-petrol in the case of 1a,b; 10:90 with 1c) followed by 100 recrystallization in CH 2 Cl 2 .

Methyl

2,2-dicyano-3,5-diphenyl-(2H,5H)-dihydrofuran-4carboxylate (9a). Yield: 60% (general procedure 6). Yellow powder, mp 85 °C. 1 H NMR (300 MHz, CDCl 3 ): 3.53 (s, 3H), 6. 40 (s, 1H), 7.44-7.46 (m, 5H), 7.52-7.56 (m, 3H), 7.60-7.63 (m, 105 2H). 13 C NMR (75 MHz, CDCl 3 ): 52.6, 76.9, 92.2, 112.0, 112.2, 127.0, 127.7 (2C), 128.8 (2C), 129.1 (2C), 129.2 (2C), 130.1, 131.1, 135.7, 136.0, 140.2, 161.1. HRMS 3.53 (s, 3H), 6.36 (s, 1H), 7.35-7.38 (m, 2H), 7.42-7.45 (m, 2H), 7.50-7.60 (m, 5H). 13 C NMR (75 MHz, CDCl 3 ): 52.7, dihydrofuran-4-carboxylate (9c). Yield: 66% (general procedure 6). Yellow oil. 1 H NMR (300 MHz, CDCl 3 ): 3.53 (s, 3H), 3.83 (s, 3H), 6.35 (s, 1H), 6.95-6.98 (m, 2H), 7.34-7.37 (m, 2H), 7.51-7.55 (m, 3H), 7.60-7.63 (m, 2H). 13 C NMR (75 MHz, CDCl 3 ): 52.6, 55.4, 76.5, 91.9, 112.1, 112.2, 114.5 (2C), 127.0, 127.6, 128.7 (2C)

Crystallography

Single crystals suitable for X-ray diffraction were grown after 20 slow evaporation (several days at room temperature) of solutions of 3a, 3b and 3c in CH 2 Cl 2 . The samples were studied with graphite monochromatized Mo K radiation ( = 0.71073 Å). Xray diffraction data were collected at T = 100( 2

Computational methods

DFT calculations were carried out using the B3LYP 26 exchangecorrelation functionals, together with the standar 6-31G* basis set. 27 All stationary points were further optimized using the 6-80 311G* and 6-311+G* basis sets. The corresponding results are given in the Supplementary Information part. The optimisations were carried out using the Berny analytical gradient optimisation method. 28 The stationary points were characterized by frequency calculations in order to verify that TSs have one and only one 85 imaginary frequency. The IRC 29 paths were traced in order to check the energy profiles connecting each TS to the two associated minima of the proposed mechanism using the second order González-Schlegel integration method. 30 Solvent effects were considered at the same level of theory by single-point 90 energy calculations of the gas-phase structures using a selfconsistent reaction field (SCRF) 31 based on the polarizable continuum model (PCM) of the Tomasi's group. 32 Since this cycloaddition was carried out in toluene, we selected its dielectric constant at 298.0 K, ε=2.38. Values of enthalpies, entropies and 95 free energies in toluene were calculated with standard statistical thermodynamics at 110 ºC and 1 atm. 27 The electronic structures of stationary points were analyzed by the natural bond orbital (NBO) method 33 and by the topological analysis of the ELF, (r). 34 The ELF study was performed with the TopMod 100 program 35 using the corresponding monodeterminantal wavefunctions of the selected structures of the IRC. All calculations were carried out with the Gaussian 09 suite of programs. 36 The global electrophilicity index, 37 , is given by the nucleophilicity index, 39 , based on the HOMO energies obtained within the Kohn-Sham scheme, [START_REF] Frisch | Revision A.02[END_REF] and defined as  = E HOMO (Nu) -E HOMO (TCE). The nucleophilicity is referred to TCE, because it presents the lowest HOMO energy in a large series of molecules already investigated in the context of polar cycloadditions. This choice allows us conveniently to handle a nucleophilicity scale of positive values. 39

Pharmacology

Applying the agar plate diffusion technique, 20 the compounds were screened in vitro for their bactericidal activity against Gram 10 positive bacteria (Staphylococcus aureus) and Gram negative bacteria (Escherichia Coli and Pseudomonas aeroginosa), and for their fungicidal activity against Fusarium oxysporium, Aspergillus niger and Candida albicans. In this method, a standard 5 mm diameter sterilized filter paper disc impregnated with the compound (0.3 mg/0.1 ml of DMF) was placed on an agar plate seeded with the test organism. The plates were incubated for 24 h at 37 °C for bacteria and 28 °C for fungi. The zone of inhibition of bacterial and fungi growth around the disc was observed.

20

Compounds were tested against human liver (HEPG2), human breast (MCF7), and cervix carcinoma cell lines. The method applied is similar to that reported by Skehan et al. 41 using 20 Sulfo-Rhodamine-B stain (SRB). Cells were plated in 96multiwell plate (104 cells/well) for 24 h before treatment with the test compound to allow attachment of cell to the wall of the plate. Different concentrations of the compound under test (0, 1.0, 2.5, 5.0, and 10 μg/ml) were added to the cell monolayer in triplicate wells individual dose, and monolayer cells were incubated with the compounds for 48 h at 37 °C and in atmosphere of 5% CO 2 .

After 48 h, cells were fixed, washed and stained with SRB stain, excess stain was washed with acetic acid and attached stain was recovered with Tris-EDTA buffer. Color intensity was measured in an ELISA reader, and the relation between surviving fraction and drug concentration is plotted to get the survival curve of each 35 tumor cell line after the specified compound and the IC50 was calculated.

grateful to the Spanish Government (project CTQ2009-11027/BQU).

Notes and references

a Laboratoire de Synthèse Organique Appliquée, Faculté des Sciences, Université d'Oran, BP 1524 El M'Naouer, 31000 Oran, Algeria. Fax: Substituted 2,5-dihydrofurans were synthesized by [3+2] cycloaddition reaction between carbonyl ylides, generated from epoxides, and alkynes. The mechanism and regioselectivity were theoretically studied using DFT methods at the B3LYP/6-31G* level.

65 2 .

 2 505 Å (C3-C4) at TS1m, and 2.283 Å (C3-C5) and 2.626 Å (C1-C4) at TS1o, while those in the 32CA reaction involving methyl but-2-ynoate (6) are 2.487 Å (C1-C5) and 2.374 Å (C3-C4) at TS2m, and 2.415 Å (C3-C5) and 2.484 Å (C1-C4) at TS2o. A comparison of these lengths with those obtained using 70 the 6-311G* and 6-311+G* basis sets indicate that there are not significant geometrical differences.

Figure 1

 1 Figure 1 Transition structures involved in the reactions between CY 1'a and methyl propiolate (4) and methyl but-2-ynoate (6) (distances are given in Å).

Figure 2

 2 Scheme 3

Figure 3

 3 Figure 3 HOMO of CY 1'a and the two electrons of HOMO-1 of methyl propiolate (4).

  plate diffusion technique, 20 the newly synthesized compounds 3a, 5'c and 9c were screened in vitro for their bactericidal activity against Gram positive bacteria (Staphylococcus aureus) and Gram negative bacteria (Escherichia Coli and Pseudomonas aeroginosa), and for their 55 fungicidal activity towards Fusarium oxysporium, Aspergillus niger and Candida albicans (Table

Figure 4

 4 Figure4ELF attractors at the TSs involved in the non-polar 32CA reactions between CY 1'a and methyl propiolate (4) and methyl but-2ynoate (6).

  ) K for 3a,b, at T = 150(2) K for 3c, and at T = 200(2) K for 5'a, 5b, 8a,b, using 25 APEXII Bruker-AXS diffractometer(3a-c, 5'a, 5b, 8b) or KappaCCD diffractometer (8a). The structure was solved by direct methods using the SIR97 program, 22 and then refined with full-matrix least-square methods based on F 2 (SHELX-97)23 with the aid of the WINGX program.24 All non-hydrogen atoms were refined with anisotropic atomic displacement parameters. H atoms were finally included in their calculated positions. Molecular diagrams were generated by ORTEP-3 (version 1.08) 25 and furnished as ESI. † Crystal data for 3a: C 18 H 12 N 2 O, M = 272.30, monoclinic, P = 0.0449) for 6009 observed reflections with I > 2σ(I). Crystal data for 3c: C 19 H 14 N 2 O 2 ; M = 302.32, triclinic, P -1, a = 7.446(4), b = 7.670(3), c = 15.675(11) Å, α = 97.60(4), β = 91.12(3),  = 118.99(2) °, V = 772.5(7) Å 3 , Z = 2, d = 1.30 g.cm -3 , μ = 0.086 mm -1 . A final refinement on F 2 with 3395 unique intensities and 211 parameters converged at ωR(F 2 ) = 0.2758 (R(F) = 0.0897) for 2690 observed reflections with I > 2σ(I). Crystal data for 5'a: C 14 H 10 N 2 O 3 , M = 254.24, orthorhombic, P 2 1 2 1 2 1 , a = 6.355(1), b = 8.010(1), c = 24.798(3) Å, V = 1262.4(3) Å 3 , Z = 4, d = 1.34 g.cm -3 , μ = 0.096 mm -1 . A final 55 refinement on F 2 with 2876 unique intensities and 173 parameters converged at ωR(F 2 ) = 0.0921 (R(F) = 0.0391) for 2380 observed reflections with I > 2σ(I). Crystal data for 5b: C 14 H 9 ClN 2 O 3 , M = 288.68, monoclinic, P 2 1 /c, a = 11.0681(3), b = 9.2126(3), c = 13.3847(3) Å, β = 60 90.0100(10) °, V = 1364.78(7) Å 3 , Z = 4, d = 1.40 g.cm -3 , μ = 0.288 mm -1 . A final refinement on F 2 with 3101 unique intensities and 182 parameters converged at ωR(F 2 ) = 0.0906 (R(F) = 0.0372) for 2394 observed reflections with I > 2σ(I). Crystal data for 8a: C 15 H 12 N 2 O 3 , M = 268.27, monoclinic, P 65 2 1 , a = 8.5077(4), b = 9.6154(5), c = 9.3240(4) Å, β = 116.932(7) °, V = 680.02(6) Å 3 , Z = 2, d = 1.31 g.cm -3 , μ = 0.093 mm -1 . A final refinement on F 2 with 1659 unique intensities and 183 parameters converged at ωR(F 2 ) = 0.0784 (R(F) = 0.0346) for 1509 observed reflections with I > 2σ(I). 70 Crystal data for 8b: C 15 H 11 ClN 2 O 3 , M = 302.71, triclinic, P-1, a = 6.545(1), b = 10.072(2), c = 11.676(2) Å, α = 107.83(5), β = 101.23(4),  = 92.80(3) °, V = 713.8(4) Å 3 , Z = 2, d = 1.41 g.cm -3 , μ = 0.279 mm -1 . A final refinement on F 2 with 3257 unique intensities and 192 parameters converged at ωR(F 2 ) = 0.0972 75 (R(F) = 0.047) for 2050 observed reflections with I > 2σ(I).

  105following simple expression,  = ( 2 /2), in terms of the electronic chemical potential  and the chemical hardness . Both quantities may be approached in terms of the one electron energies of the frontier molecular orbital HOMO and LUMO,  H and  L , as   ( H +  L )/2 and   ( L - H ), respectively. 38 110 Recently, we have introduced an empirical (relative)

  , Cl, OMe, R = Me, Ph: 44-66% yield

Table 1

 1 Cycloaddition reaction between carbonyl ylides generated from the epoxides 1 and phenylacetylene(2).

										H 5	H 4
	X	H O	CN CN	+	Ph	C CH	toluene reflux 40 h	X	3	O NC H 5	Ph CN Ph
		1				2		X		O	H 3
									3'		CN
										NC
	entry X (1)	catalyst			3:3' ratio product(s), yield(s) (%)
	1	H (1a)	-				84:16	3a, 40
	2	Cl (1b)	-				84:16	3b, 52; 3'b, 12
	3	MeO (1c) -				75:25	3c, 50
	4	H (1a)	InCl3 (0.1 equiv) 90:10	3a, 60
	5	Cl (1b)	InCl3 (0.1 equiv) 89:11	3b, 64
	6	MeO (1c) InCl3 (0.1 equiv) 83:17	3c, 60

Table 2

 2 Cycloaddition reaction between carbonyl ylides generated from the epoxides 1 and methyl propiolate (4).

										H 5	H 4
	X	H O	CN CN	+	MeO 2 C	C CH	toluene reflux 40 h	X	5	O NC H 5	CO 2 Me CN CO 2 Me
		1				4		X		O	H 3
									5'		CN
										NC
	entry X (1)	catalyst		5:5' ratio product(s), yield(s) (%)
	1	H (1a)	-				24:76	5a, 12; 5'a, 40
	2	Cl (1b)	-				26:74	5b, 10; 5'b, 54
	3	MeO (1c) -				29:71	5c, 11; 5'c, 65
	4	H (1a)	InCl3 (0.1 equiv) 33:67	5'a, 58
	5	Cl (1b)	InCl3 (0.1 equiv) 31:69	5'b, 57
	6	MeO (1c) InCl3 (0.1 equiv) 18:82	5c, 8; 5'c, 67

Table 3

 3 Cycloaddition reaction between carbonyl ylides generated from the epoxides 1 and methyl but-2-ynoate and3-phenylpropiolate (6,7).

	X	H O	CN CN	+	MeO 2 C	C C	R	toluene reflux 40 h	X		H 5 O	CO 2 Me CN R 3
		1			6: R = Me				8: R = Me	NC
					7: R = Ph				9: R = Ph
	entry X (1)		R		reaction time (h)	product, yield (%)
	1	H (1a)		Me (6)	48				8a, 66
	2	Cl (1b)		Me (6)	48				8b, 62
	3	MeO (1c)	Me (6)	24				8c, 44
	4	H (1a)		Ph (7)	96				9a, 60
	5	Cl (1b)		Ph (7)	96				9b, 47
	6	MeO (1c)	Ph (7)	55				9c, 66
						Cl
			H 5	CN
		X		O	Ph CN			O	O		Cl
			11	NC				12	CN
								NC
	Scheme 1 The different products obtained by reaction of 2,2-dicyano-3-
	(4-chlorophenyl)oxirane (1c) with 3-phenylpropiolonitrile

Table 5

 5 Electronic chemical potential (μ, in au), chemical hardness (η, in au), global electrophilicity (ω, in eV) and global nucleophilicity (N, in eV)

	values of CY 1'a and methyl propiolate (4) and methyl but-2-ynoate (6).
					N
	1'a	-4.61	2.47	4.29	3.28
	4	-4.42	6.43	1.52	1.48
	6	-4.04	6.32	1.29	1.92

Table 6

 6

	MeO 2 C 6 4 3 4	O 2 +	CN 1 5	CN	TS1m	CA1m O 1 3 MeO 2 C Ph 5 4	CN CN
		1'a
		1'a+4 TS1m P1	P2	CA1m
	d1(C1-C5)				2.375 2.075 1.767 1.523
	d2(C3-C4)				2.505 2.255 1.922 1.515
	CT(NBO)				0.05	0.07	0.07
	CT(ELF)				0.06	0.07	0.08
	V(C1,O2)	1.69		1.54	1.44	1.34	1.24
	V(O2)	3.57		3.95	4.38	2.49	2.59
	V'(O2)						2.26	2.45
	V(O2,C3)	2.14		1.91	1.62	1.41	1.22
	V(C3,C6)	2.79		2.69	2.35	2.17	2.05
	V(C4,C5)	2.70		2.51	2.17	1.94	1.84
	V'(C4,C5)	2.70		2.47	2.17	1.96	1.74
	V(C1)	0.59		0.90	1.05
	V'(C1)	0.58
	V(C3)					0.46
	V(C4)				0.35	0.71
	V(C5)					0.42
	V(C1,C5)						1.79	1.97
	V(C3,C4)						1.69	2.04

Valence basin populations N calculated from the ELF of some selected points of the IRC associated with the most favorable regioisomeric channel of the 32CA reaction between CY 1'a and methyl propiolate (4). The CT is also included.

Table 7

 7 Bactericidal and fungicidal activity a of the compounds 3a, 5'c, 9c, and ciprofloxacin and nystin. The diameters of zones of inhibition are given in mm. Stock solution: 5 g in 1 mL of DMF. 0.1 mL of stock solution in each hole of each paper disk.

	entry compound	Staphylococcus aureus Escherichia coli Pseudomonas aeroginosa Fusarium oxysporium Aspergillus niger Candida albicans
	1 2	3a 5'c	19 (++) 27 (+++)	17 (++) 15 (++)	17 (++) 22 (++)	19 (++) 43 (++++)	20 (++) 27 (+++)	-18 (++)
	3	9c	18 (++)	17 (++)	16 (++)	16 (++)	18 (++)	-
	4	ciprofloxacin ++++	++++	++++	-	-	-
	5	nystin	-	-	-	++++	++++	++++
	a

+: < 15 mm; ++: 15-24 mm; +++: 25-34 mm; ++++: 35-44 mm, etc.

4-Methoxyphenyl)-3-phenyl-(2H,5H)-dihydrofuran-2,2-

		3,5-Diphenyl-(2H,5H)-dihydrofuran-2,2-dicarbonitrile (3a).
	55	Yield: 40% (general procedure 1), 60% (general procedure 2).
		White powder, mp 133 °C. 1 H NMR (300 MHz, CDCl 3 ): 6.25 (d,
		1H, J = 1.8 Hz), 6.65 (d, 1H, J = 1.8 Hz), 7.36-7.39 (m, 2H),
		7.43-7.48 (m, 3H), 7.48-7.50 (m, 3H), 7.65-7.68 (m, 2H). 13 C
		NMR (75 MHz, CDCl 3 ): 73.8, 91.6, 113.1, 113.2, 126.8 (2C),
	60	127.0 (2C), 127.5, 129.3 (2C), 129.5 (2C), 129.8, 130.5, 131.0, 134.6, 136.5. HRMS (ESI): calcd for C 18 H 12 N 2 O (M +• ) 272.0842,
		found 272.0842.
		5-(4-Chlorophenyl)-3-phenyl-(2H,5H)-dihydrofuran-2,2-
		dicarbonitrile (3b). Yield: 52% (general procedure 1), 64%
	65	(general procedure 2). Yellow powder, mp 85 °C. 1 H NMR (300
		MHz, CDCl 3 ): 6.22 (d, 1H, J = 1.8 Hz), 6.62 (d, 1H, J = 1.8 Hz),
		7.30 (d, 2H, J = 8.4 Hz), 7.42 (d, 2H, J = 8.4 Hz), 7.46-7.50 (m,
		3H), 7.63-7.67 (m, 2H). 13 C NMR (75 MHz, CDCl 3 ): 73.7, 90.7,
		112.8, 113.1, 126.8 (2C), 127.2, 128.4 (2C), 129.5 (2C), 129.6
	70	(2C), 130.5, 130.7, 134.9, 135.0, 135.7. HRMS (ESI): calcd for C 18 H 11 35 ClN 2 O (M +• ) 306.0576, found 306.0455.
		5-(4-Chlorophenyl)-4-phenyl-(2H,5H)-dihydrofuran-2,2-
		dicarbonitrile (3'b). Yield: 12% (general procedure 1). Pale
		yellow oil. 1 H NMR (300 MHz, CDCl 3 ): 6.33 (d, 1H, J = 2.1 Hz),
	75	6.41 (d, 1H, J = 2.1 Hz), 7.26-7.29 (m, 2H), 7.29-7.32 (m, 3H),
		7.34-7.37 (m, 4H). 13 C NMR (75 MHz, CDCl 3 ): 72.6, 91.1,
		112.7, 113.0, 127.3, 127.5 (2C), 129.1, 129.2 (2C), 129.6 (2C),
		129.6, 129.7 (2C), 130.9, 134.6, 136.1. HRMS (ESI): calcd for C 18 H 11 35 ClN 2 O (M +• ) 306.0576, found 306.0459.
	5-(80
		dicarbonitrile (3c). Yield: 50% (general procedure 1), 60%
		(general procedure 2). Yellow powder, mp 118 °C. 1 H NMR (300
		MHz, CDCl 3 ): 3.83 (s, 3H), 6.20 (d, 1H, J = 1.8 Hz), 6.62 (d, 1H,
		J = 1.8 Hz), 6.94-6.97 (m, 2H), 7.27-7.31 (m, 2H), 7.47-7.50 (m,
	85	3H), 7.64-7.67 (m, 2H). 13 C NMR (75 MHz, CDCl 3 ): 55.5, 73.5,
		91.4, 113.1, 113.3, 114.6 (2C), 126.8 (2C), 127.6, 128.5, 128.8
		(2C), 129.5 (2C), 130.5, 131.0, 134.6, 160.8. HRMS (ESI): calcd for C 19 H 14 N 2 O 2 [(M+Na) +• ] 302.0953, found 302.0953.
		General procedure 3. A mixture of epoxide (2 mmol) and
	90	methyl propiolate (0.67 g, 8 mmol in the case of 1a,b; 0.16 g, 2
		mmol in the case of 1c) in anhydrous toluene (30 mL) was heated
		at reflux under Ar for 48 h (1a,b) or 24 h (1c). The mixture was
		then evaporated to dryness and the product isolated by column

chromatography over silica gel (eluent: 10:90 AcOEt-petrol in the case of 1a,b; 20:80 with 1c). 95 General procedure 4. A mixture of epoxide (2 mmol), methyl propiolate (0.67 g, 8 mmol in the case of 1a,b; 0.16 g, 2 mmol in the case of 1c) and InCl 3 (40 mg, 0.2 mmol) in anhydrous toluene (30 mL) was heated at reflux under Ar for 36 h (1a,b) or 20 h

Table 8

 8 In vitro cytotoxic activity (IC50) a of the compounds 3a,b, 5'c, 9c, and doxorubicin against carcinoma cell lines.

	entry compound HEPG2 (g.mL -1 ) MCF7 (g.mL -1 ) HELA (g.mL -1 )
	1	3a	2.88	-	-
	2 3	3b 5'c	0.67 2.98	0.74 -	0.89 -
	4	9c	3.34	-	-
	5	doxorubicin 0.60	0.70	0.85
	a IC50 is defined as the concentration which results in a 50% decrease in
	cell number as compared with that of the control structures in the absence
	of an inhibitor.

  . HRMS (ESI): calcd for C 14 H 10 N 2 O 3 (M +• ) 254.0589, found 254.0593.

		Methyl	5-(4-chlorophenyl)-2,2-dicyano-(2H,5H)-
	20	dihydrofuran-3-carboxylate (5b). Yield: 10% (general
		procedure 3). Yellow oil. 1 H NMR (300 MHz, CDCl 3 ): 3.96 (s,
		3H), 6.24

-dicyano-5-(4-methoxyphenyl)-(2H,5H

	. 35 ClN 2 O 3 [(M-H) +• ] HRMS (APCI/ASAP): calcd for C 14 H 8
	287.0218, found 287.0219.
	Methyl	2,2

  (APCI/ASAP): calcd for C 20 H 13 N 2 O 3 [(M-H) +• ] 329.0920, found 329.0924.

	Methyl	5-(4-chlorophenyl)-2,2-dicyano-3-phenyl-(2H,5H)-
	dihydrofuran-4-carboxylate (9b). Yield: 47% (general
	procedure 6). White powder, mp 142 °C. 1 H NMR (300 MHz,
	CDCl 3 ):

  , 129.0 (2C), 129.2 (2C), 131.0, 135.9, 139.9, 160.9, 161.2. HRMS (ESI): calcd for C 12 H 16 N 2 O 4 (M +• ) 360.1002, found 360.1002.

Acknowledgment

The authors thank Rennes Métropole and the Institut Universitaire de France for financial support to F.M. We are also

Methyl

2,2-dicyano-5-(4-methoxyphenyl)-(2H,5H)dihydrofuran-4-carboxylate (5'c). Yield: 65% (general procedure 3), 67% (general procedure 4). Yellow oil. 1 H NMR (300 MHz, CDCl 3 ): 3.71 (s, 3H), 3.79 (s, 3H), 6.19 (d, 1H, J = 2.1 Hz), 6.77 (d, 1H, J = 2.1 Hz), 6.91-6.94 (m, 2H), 7.23-7.26 50 (m, 2H). 13 C NMR (75 MHz, CDCl 3 ): 52. 9, 55.4, 71.9, 90.8, 111.6, 111.9, 114.5 (2C), 127.1, 128.5, 129.1 (2C), 142.7, 160.2, 160.8. NMR HMQC sequences performed on 5'c showed relationship between the phenyl protons and the carbonyl group. (d, 3H, J = 2.10 Hz), 3.70 (s, 3H), 6.18 (q, 1H, J = 2.10 Hz), 7.23-7.26 (m, 2H), 7.37-7.40 (m, 2H). 13 C NMR (75 MHz, CDCl 3 ): 11.2, 52.6, 77.1, 91.3, 111.7, 111.9, 129.0 (2C), 129.3 (2C), 133.7, 134.6, 135.9, 140.8, 161. (d,


Technical Appendix on Sparse Bayesian Regression

Loïc Le Folgoc 1 , Hervé Delingette 1 , Antonio Criminisi 2 , and Nicholas Ayache 1 1 Asclepios Research Project, INRIA Sophia Antipolis, France 2 Machine Learning and Perception Group, Microsoft Research Cambridge, UK Abstract. We report the technical details for a sparse bayesian approach to regression. It can be seen as an extension of the Relevance Vector Machine of Tipping et al [START_REF] Tipping | Fast marginal likelihood maximisation for sparse bayesian models[END_REF] to a more general setting that can handle vector-valued regression and generic quadratic priors.

Quadratic Energies & Marginal Likelihood of the Data

We want to minimize in w the following cost:

E(w) = (t -Φw) ⊺ βH(t -Φw) + w ⊺ (A + λR)w (1)
and jointly optimize in A = diag(A i ), β, λ. w|t, A, λ, β follows a Gaussian distribution N (µ, Σ), where

µ = ΣΦ ⊺ βHt, Σ = (A + λR + Φ ⊺ βHΦ) -1 (2)
A key element is that the distribution of t|A is also Gaussian, t|A ∼ N (0, C), with

C = (βH) -1 + Φ(A + λR) -1 Φ ⊺ (3)
Indeed, we see that:

p(t|A) = p(t|w)p(w|A)dw ∝ exp - 1 2 (t -Φw) ⊺ βH(t -Φw) • exp - 1 2 w ⊺ (A + λR)w • dw ∝ exp - 1 2 µ ⊺ Σ -1 µ • exp - 1 2 t ⊺ βHt • exp - 1 2 (w -µ) ⊺ Σ -1 (w -µ)dw
where the integral sums to 1. Thus using (2),

p(t|A) ∝ exp - 1 2 t ⊺ (βH)t - 1 2 t ⊺ (βH)ΦΣΦ ⊺ (βH))t
t|A is Gaussian since the distribution is proportional to a Gaussian, and by identification it ensues that C -1 = βH-(βH)ΦΣΦ ⊺ (βH). We then get the desired result using a matrix inversion identity. The fast RVM algorithm proceeds by iteratively implementing a single change to one of the A i 's on the block-diagonal matrix A; specifically the one that maximizes the increase of a quantity known as the evidence, log p(t|A), then re-estimating the parameters of the conditional posterior w|t, A, λ, β using (2). The algorithm starts with all A i set to ∞. The computations involve rank-one "block" updates; it also turns out that the optimal A i 's are rank one matrices (so we actually have rank-one updates).

Computation of the gain in evidence for a given action

We want to evaluate the change in log p(t|A -i , A i ) when a single prior weight

A i is changed. Recall that t|A -i , A i ∼ N (0, C), thus log p(t|A -i , A i ) = -1/2 log |C| + t ⊺ C -1 t (4)
Now let's single out the contribution of A i , for each of the two terms above. Noting the form of eq. ( 3), let L = (A + λR) -1 . We can single out the contribution of the ith basis to L first, as a rank one update:

L = L -i 0 0 0 + U i l i U ⊺ i (5)
with

U ⊺ i = (λL -i R i ) ⊺ I ⊺ and l i = {A i + λR ii -(λR i ) ⊺ L -i (λR i )} -1 {A i + κ i } -1
. Note also that any basis for which A j = ∞ can be disregarded, since its corresponding line and column in L and L -i will be null. We can interpret L as a square matrix of dimension the number of active bases (including the basis under consideration), and the algorithmic complexity of matrix operations involving L or L -i will indeed be related to the reduced sized of these matrices. This is also true for U i , as a direct consequence, and for all of the other quantities involved.

Injecting (5) into (3) gives a decomposition of C into the sum of a term that does not depend on the ith basis and of a rank-one term:

C = C -i + Φ i U i l i U ⊺ i Φ i ⊺ (6)
Φ is superscripted with i to recall that along with all the other active bases, the ith basis φ i is present in this matrix. Using rank-one updates for, respectively, the determinant and the inverse, and letting C -1 -i

(C -i ) -1 , we get the two following expressions:

|C| = |C -i | |l i | |l -1 i + U ⊺ i Φ i ⊺ C -1 -i Φ i U i | (7) t ⊺ C -1 t = t ⊺ C -1 -i -C -1 -i Φ i U i l -1 i + U ⊺ i Φ i ⊺ C -1 -i Φ i U i -1 U ⊺ i Φ i ⊺ C -1 -i t (8)
These quantities rewrite more compactly if we introduce appropriate notations. Namely, let q j (i)

φ ⊺ j C -1 -i t ∈ R d and s jk (i) φ ⊺ j C -1 -i φ k ∈ M d,d .
The concatenation of these q j 's for all active bases plus the basis under scrutiny (total of m bases), a.k.a q i ∈ R d×m , will come in helpful. Similarly s i ∈ M d×m,d×m will denote the matrix with s kl (i) as (k, l)th coefficient, where indices span the set of active bases plus the ith basis. Now, let

q i U ⊺ i Φ i ⊺ C -1 -i t = U ⊺ i q i ∈ R d ,

and

s i U ⊺ i Φ i ⊺ C -1 -i Φ i U i = U ⊺ i s i U i ∈ M d,d
. With these notations in hand and recalling that l -1 i = A i + κ i , we can rewrite eq. ( 7) and eq. ( 8) as:

log |C| = log |C -i | -log |A i + κ i | + log |A i + κ i + s i | (9) t ⊺ C -1 t = t ⊺ C -1 -i t -q i ⊺ A i + κ i + s i -1 q i (10)
Ignoring the terms that do not depend on the ith basis, we see that the contribution to the evidence of the model for a given value of A i is directly related to:

l(A i ) = log |A i + κ i | -log |A i + κ i + s i | + q i ⊺ A i + κ i + s i -1 q i (11)
Naturally if λ = 0 (no additional regularization) this comes down to the regular RVM, with q i = q i , s i = s ii and κ i = 0.

Maximization of the gain in evidence

If q i q i ⊺s i has no positive eigenvalue, the maximum A i lies at infinity and the basis should remain inactive, or be removed. Otherwise the gradient of Eq. ( 11) provides ground to look for rank-one maximizers

A i = α i η i η ⊺ i .
To that aim we compute n i = s i -1 q i and

a i = (n ⊺ i s i n i ) 2 (n ⊺ i q i ) 2 -n ⊺ i s i n i -n ⊺ i κ i n i (12)
If a i ≥ 0 the maximizer is given by α i = a i and η i = n i . Otherwise (a i < 0) we set α i = 0 and numerically solve over the optimal orientation η i . This latter case arises when the regularization level alone is sufficient to make additional "shrinkage" unnecessary.

Update of λ

We derive an update rule via an expectation-maximization procedure. Knowing w, it would be straightforward to derive an estimate of λ by maximizing the log-likelihood of w or the posterior of λ given w. However w is hidden in our model. Instead, we try to maximize the log-likelihood on average (i.e. to minimize the average loss): max λ E w∼N (µ,Σ) [log p(t, w|A, λ, β)|A * , β * ], where A * and β * are our current estimates of the respective quantities. Discarding terms which are constants of λ, we obtain the following:

λ * = arg max λ - λ 2 tr(ΣR) + 1 2 log |A + λR| - λ 2 µ ⊺ Rµ (13)
Deriving leads to:

∂ ∂λ f (λ) ∝ tr({A + λR} -1 R) -tr(ΣR) -µ ⊺ Rµ (14)
This is a decreasing function of λ and thus has at most one zero. If ∂ ∂λ f (λ) is negative at the origin, λ * = 0. Otherwise, we optimize by using the Newton method on a log scale. This is motivated by the fact that the function of interest is not only decreasing, but also smooth and convex. Lastly note that {A + λR} -1 R = {R -1 A + λI} -1 , so we can compute the eigenvalues δ k of A 1/2 R -1 A 1/2 once and rely on the fact that tr({A + λR} -1 R) = k 1/(δ k + λ) to avoid repeated matrix inversions.


Kai Siedenburg

Matthieu Kowalski

Monika Dörfler

AUDIO DECLIPPING WITH SOCIAL SPARSITY

Keywords: Structured sparsity, Audio declipping, Iterative Shrinkage/Thresholding Algorithm

We consider the audio declipping problem by using iterative thresholding algorithms and the principle of social sparsity. This recently introduced approach features thresholding/shrinkage operators which allow to model dependencies between neighboring coefficients in expansions with time-frequency dictionaries. A new unconstrained convex formulation of the audio declipping problem is introduced. The chosen structured thresholding operators are the so called windowed group-Lasso and the persistent empirical Wiener. The usage of these operators significantly improves the quality of the reconstruction, compared to simple soft-thresholding. The resulting algorithm is fast, simple to implement, and it outperforms the state of the art in terms of signal to noise ratio.

INTRODUCTION 1.Problem Statement

An important task in digital audio restoration is the recovery of missing or corrupted samples of a signal. Two important cases of this problem concern a) missing samples (or even full intervals of samples) and b) clipped audio. While the former mostly arises through errors of signal transmission, clipping denotes a situation in which a signal's amplitude exceeds a certain threshold and is truncated. For clipped signals, as opposed to the data loss case, at least the lost samples' correct sign values are known. Clipping is a common problem in digital audio systems whose maximum gain can be exceeded for many reasons. The resulting signal truncation leads to very unpleasant digital distortion.

Based on the assumption of sparse synthesis coefficients of the original signal, the declipping problem can be modeled by

α = argmin α α 0 s.t. y r -M r Φα 2 2 ≤ ǫ (1)
Here, α ∈ C N denotes the synthesis coefficients and Φ ∈ C T ×N the synthesis operator corresponding to the employed timefrequency dictionary. The vector y r = M r y ∈ R M denotes the reliable samples of the observed signal y ∈ R T , that is, the unclipped samples in the clipping case or just the available samples in the case of data packet loss. Then, M r ∈ R M ×T is a matrix comprised of those rows of the identity matrix that choose the entries KS is supported by a Harman Scholarship from the AES Educational Foundation. MK benefited from the support of the "FMJH Program Gaspard Monge in optimization and operation research", and from the support to this program from EDF. MD is funded by the Vienna Science and Technology Fund (WWTF) through project VRG12-009.

of the reliable samples. Regarding the synthesis operator, Gabor frames (a.k.a. Short-Time Fourier-Transform, cf. [START_REF] Feichtinger | Introduction[END_REF][START_REF] Dörfler | Time-frequency Analysis for Music Signals. A Mathematical Approach[END_REF]) have proven to be well suited for the representation of audio signals, especially in the context of sparse decomposition. Therefore, Φ will denote the matrix associated to a Gabor dictionary in the following.

In this paper, we specifically address the important problem of audio declipping. Here, an additional constraint can be added to (1): reconstructed samples must be greater (in absolute value) than the clipping threshold. In analogy to the definition of M r , let M c ∈ R (T -M )×T denote the matrix picking the clipped samples. Also, let θ clip ∈ R (T -M ) be the vector of clipped samples, taking only the values ±θ clip , in dependence on the sign of the true values in y. Then, for declipping, the problem becomes

α = argmin α α 0 (2)
s.t. y r -M r Φα 2 2 ≤ ǫ and |M c Φα| ≥ |θ clip |

Previous Work

Classical approaches to audio interpolation and declipping include autoregressive (AR) modeling [START_REF] Janssen | Adaptive interpolation of discrete-time signals that can be modeled as autoregressive processes[END_REF], signal matching with bandwidth constraints [START_REF] Abel | Restoring a clipped signal[END_REF], and Bayesian estimation [START_REF] Godsill | A Bayesian approach to the restoration of degraded audio signals[END_REF]. A sparsity based formulation has been provided in [START_REF] Adler | Audio inpainting[END_REF], with (1) serving as the basic problem which was solved using orthogonal matching pursuit (OMP). The authors dubbed the method audio inpainting in reference to sparsity constrained image inpainting [START_REF] Elad | Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)[END_REF].

It is well known that the convex relaxation of (1) leads to the Lasso [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] or Basis Pursuit Denoising problem [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF], yielding the following minimization problem:

argmin α 1 2 y r -M r Φα 2 2 + λ α 1 . (3)
This convex non-smooth functional can be minimized by the popular iterative shrinkage/thresholding algorithm (ISTA) [START_REF] Daubechies | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[END_REF], also called forward-backward algorithm [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF], or its accelerated version FISTA [START_REF] Beck | A fast iterative shrinkagethresholding algorithm for linear inverse problems[END_REF]. The most straight-forward approach for obtaining a convex relaxation of (2) with linear constraints is to consider [START_REF] Janssen | Adaptive interpolation of discrete-time signals that can be modeled as autoregressive processes[END_REF] with the additional constraint to choose α, such that

|M c Φα| ≥ |θ clip | (4)
However, non-smooth convex problems with such constraints cannot be minimized directly by a forward-backward strategy. Indeed, the proximity operator of the ℓ1 penalty with the linear constraint cannot be computed in closed-form, and one has to use an inner iteration inside the forward-backward algorithm to approximate it. Thus, a common approach is to use a Douglas-Rachford algorithm as inner loop, see e.g. [START_REF] Chaux | Nested iterative algorithms for convex constrained image recovery problems[END_REF], but this is usually accompanied by a very high computational burden.

In the OMP based approach [START_REF] Adler | Audio inpainting[END_REF], the clipping constraint in (2), as well as an additional maximum constraint forcing the declipped signal below a maximum value, are satisfied by using a two-step strategy. First, OMP recovers the time-frequency support of the solution. Second, standard convex optimization solvers are applied on the obtained support. In comparison to traditional approaches, this approach leads to improvements in terms of signal to noise ratio, but has the shortcoming of being computationally very expensive, as it eventually employs high dimensional convex optimization.

An alternative iterative hard-thresholding formulation for declipping was recently described in [START_REF] Kitić | Consistent iterative hard thresholding for signal declipping[END_REF]. Here, the iteration consists of a gradient descent step followed by hard-thresholding and thus bears similarity to iterative approaches as (F)ISTA. The authors reported improvements over the constrained OMP method [START_REF] Adler | Audio inpainting[END_REF], although their evaluation was based on a rather small set of audio examples. While the algorithm to be presented in the following also makes use of an ISTA-type iteration and shares some of the properties of the hard-thresholding approach, it was developed independently, cf. [START_REF] Siedenburg | Audio inpainting with social sparsity[END_REF].

Besides the sparsity principle used in (1), many authors have investigated various structured sparsity approaches. This includes perceptually-informed compressed sensing [START_REF] Defraene | Declipping of audio signals using perceptual compressed sensing[END_REF] and Group-Lasso techniques [START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF] which allow to define grouping of coefficients to be jointly processed. Various extension to groups including overlaps have been proposed, such as the Latent-Group-Lasso [START_REF] Jacob | Group lasso with overlap and graph lasso[END_REF][START_REF] Obozinski | Group lasso with overlaps: the latent group lasso approach[END_REF]. More specifically in the context of audio processing, Group-Lasso with overlap has been studied in [START_REF] Bayram | Mixed norms with overlapping groups as signal priors[END_REF]. The main drawback of these approaches is the high computational load required to solve the suitable functionals. The methods of social sparsity proposed in the current contribution avoid this practical drawback, cf. [START_REF] Kowalski | Social sparsity! Neighborhood systems enrich structured shrinkage operators[END_REF] for more detailed theoretical background.

This article features two main contributions. First, we propose an unconstrained convex relaxation of (2) which yields a solution with desired declipping behavior: the reconstructed samples' absolute values are above the clipping threshold. This unconstrained formulation allows to employ ISTA-type algorithms. Second, we explore the benefits of the recently introduced concept of social sparsity [START_REF] Kowalski | Social sparsity! Neighborhood systems enrich structured shrinkage operators[END_REF], in order to take into account temporal dependencies between Gabor synthesis coefficients. The remainder of the paper is organized as follows. Section 2 introduces the unconstrained convex formulation for the declipping problem and derives the associated ISTA algorithm. Section 3 is a brief recap of social sparsity and associated structured shrinkage operators to be embedded in ISTA in practice. Section 4 presents numerical experiments on the declipping problem and compares the approach with the algorithms presented in [START_REF] Adler | Audio inpainting[END_REF] and [START_REF] Kitić | Consistent iterative hard thresholding for signal declipping[END_REF].

AN UNCONSTRAINED CONVEX FORMULATION

In this contribution we propose to relax the constraints (4) by means of a squared hinge function. This is a well-known function in classification, see e.g. [START_REF] Kowalski | Multiple indefinite kernel learning with mixed norm regularization[END_REF], defined as follows:

h 2 : R -→ R+ z → h 2 (z) = z 2 if z < 0 0 if z ≥ 0
By application to z = x-θ clip , for known clipping values θ clip > 0, the squared hinge sets x "free" if |x| ≥ θ clip , and penalizes otherwise. Using the notation

[θ clip -x] 2 + = k:θ clip k >0 h 2 (x k -θ clip k ) + k:θ clip k <0 h 2 (θ clip k -x k )
we thus introduce the following unconstrained convex optimization problem

argmin α 1 2 y r -M r Φα 2 2 + 1 2 [θ clip -M c Φα] 2 + + λ α 1 (5)
Since the squared hinge is differentiable with Lipschitz-continuous gradient, cf. [START_REF] Kowalski | Multiple indefinite kernel learning with mixed norm regularization[END_REF], so is

α → 1 2 y r -M r Φα 2 2 + 1 2 [θ clip -M c Φα] 2 + ( 6
)
and any algorithm from the ISTA family can be applied to solve [START_REF] Godsill | A Bayesian approach to the restoration of degraded audio signals[END_REF], cf. [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF][START_REF] Beck | A fast iterative shrinkagethresholding algorithm for linear inverse problems[END_REF].

SOCIAL SPARSITY AND THE PROPOSED ALGORITHM

For approximating a solution to the relaxed problem (5), our work explores the application of social sparsity operators [START_REF] Kowalski | Social sparsity! Neighborhood systems enrich structured shrinkage operators[END_REF]. Social sparsity allows to incorporate a priori knowledge about signal classes and artifacts. Given the structure of the declipping problem it is natural to take temporal correlation into account: signal components such as harmonics which extend over time induce temporally persistent coefficients. On the other hand, isolated high-energy coefficients or temporally localized spread of energy over frequency may be attributed to the corruption of the signal and should therefore be discarded in the reconstruction process. By extending the usual soft thresholding, corresponding to the ℓ 1 constraint as used in ( 5), it becomes possible to exploit the persistence-properties of signal components through time-frequency neighborhood systems [START_REF] Kowalski | Social sparsity! Neighborhood systems enrich structured shrinkage operators[END_REF]. Note that these generalized operators do not directly correspond to the minimization of a convex functional any more. Denote by N (t) the set of indices forming the neighborhood of the index t for time-frequency coefficients α = {α tf } and set (x) + = max(x, 0). We can then restate the classic Lasso and its persistent variation, the so-called Windowed Group-Lasso (WGL) [START_REF] Kowalski | Sparsity and persistence: mixed norms provide simple signals models with dependent coefficients[END_REF]:

• Lasso : αtf = S L λ (α tf ) = α tf 1 -λ |α tf | + • WGL : αtf = S W GL λ (α tf ) = α tf 1 - λ t ′ ∈N (t) |α t ′ f | 2 +
The application of the shrinkage operators associated to Lasso and WGL, respectively, typically leads to a loss of energy in the estimated signal. Thus, in practice, it is common to first use the Lasso in order to select the relevant time-frequency atoms, and to perform a least-square estimation of the signal w.r.t. the selected atoms [START_REF] Figueiredo | Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems[END_REF] of the employed dictionary in a second step. Another strategy, not requiring the least-squares step, is to design thresholding operators which preserve the energy in the big coefficients, and which can also be used inside ISTA [START_REF] Antoniadis | Wavelet methods in statistics: Some recent developments and their applications[END_REF]. The most well known is the Empirical Wiener (EW) operator [START_REF] Ghael | Improved wavelet denoising via empirical wiener filtering[END_REF], also known as nonnegative garrote shrinkage [START_REF] Antoniadis | Wavelet methods in statistics: Some recent developments and their applications[END_REF]. The EW operator features an altered exponentiation of the coefficient energy while having the same support as the Lasso. Such an exponentiation can also be used on the WGL operator, yielding the persistent EW (PEW) [START_REF] Siedenburg | Persistent time-frequency shrinkage for audio denoising[END_REF]. These operators read

• EW : αtf = S EW λ (α tf ) = α tf 1 -λ 2 |α tf | 2 + • PEW : αtf = S P EW λ (α tf ) = α tf 1 - λ 2 t ′ ∈N (t) |α t ′ f | 2 +
These generalized thresholding operators, denoted by S λ for a threshold λ, are subsequently used in the ISTA-framework. Algorithm 1 lays out the details and is called relaxed forward backward.

Here, γ denotes the coefficient of relaxation. For the Lasso, the convergence of the iterates α (k) towards a minimizer of (5) can be proven for -1 < γ < 1/2 and the convergence of the value of the minimization functional towards its minimum for 1/2 ≤ γ < 1, see [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF].

Algorithm 1: relaxed version of ISTA Initialization: α (0) ∈ C N , z 0 = α (0) , k = 1, δ = ΦΦ * repeat g1 = -Φ * M r T (y r -M r Φz (k-1) ); g2 = -Φ * M c T [θ clip -M c Φz (k-1) ]+; α (k) = S λ/δ z (k-1) -1 δ (g1 + g2) ; z (k) = α (k) + γ(α (k) -α (k-1) ); k = k + 1; until convergence; 4. EXPERIMENTS 4.1. Setup
We choose a tight Gabor frame 1 as time-frequency dictionary Φ in Algorithm 1. The frame is based on a Hann window of 1024 samples length (about 64 ms at 16 kHz audio sampling frequency) and a time-shift of 256 samples. Note that the corresponding analysis operator Φ * is also known as the Sliding Window or Short-Time Fourier Transform. Concerning the relaxation coefficient γ in Algorithm 1, we observed empirically that the choice of γ = 0.9 leads to an algorithm which is faster than FISTA and less prone to numerical errors. The parameter γ will thus be held constant in the following. We evaluate declipping performance using the measure of SNRm which measures estimation quality on the clipped, i.e. missing values only. For a clipped signal y and its estimation ŷ, it is computed as SNRm(y, ŷ) = 20 log

M c y M c (y-ŷ) .
For the subsequently described experiments, we used audio data provided by http://small-project.eu/ and employed for the evaluation of the respective audio inpainting toolbox [START_REF] Adler | Audio inpainting[END_REF]. Specifically, our evaluations are based on the toolbox's speech and music data sets, sampled at 16kHz, containing 10 different signals, each of 5 seconds duration. All signals were range-normalized, in order to have sample values lower than 1, and consecutively clipped at levels 0.1, 0.2, . . . , 0.9.

The operator abbreviated by OMP in the following refers to the min-max-constrained orthogonal matching pursuit, the best performing operator in [START_REF] Adler | Audio inpainting[END_REF]. We also include results from [START_REF] Kitić | Consistent iterative hard thresholding for signal declipping[END_REF]; however, in order to avoid the introduction of bias due to a different transform, in our evaluation we use a Gabor transform with the above mentioned settings instead of the discrete Cosine transform employed within the consistent iterative hard-thresholding algorithm (HT) proposed in [START_REF] Kitić | Consistent iterative hard thresholding for signal declipping[END_REF]. Here, the corresponding algorithm is run on the entire signal instead of a windowed version, and we use the strategy exposed in section 4.2 to decrease the thresholds λ. This allows for the same basic setup for all the algorithms. Sound examples featuring all mentioned algorithms can be found under http://homepage.univie. ac.at/monika.doerfler/StrucAudio.html.

1 A frame is tight, if ΦΦ * = c • I, for some positive constant c and I the identity operator [START_REF] Feichtinger | Introduction[END_REF].

Basic Properties

Regarding the choice of hyperparameter λ, we here use the classical "warm start" strategy [START_REF] Loris | On the performance of algorithms for the minimization of 1-penalized functionals[END_REF], starting with a relatively large λ which decreases in every K th step of the iteration (here K = 500). This method essentially simulates the choice of a small λ but circumvents the slow convergence of ISTA that such a choice usually implies. Note that since we do not have to deal with additive noise in the declipping scenario, low levels of the hyperparameter λ are fully appropriate. Fig. 1 exemplifies the corresponding evolution of the algorithm's SNR versus the hyperparameter λ, which assumes 10 values logarithmically spaced from λ = 10 -1 to λ = 10 -4 . This example is based on the first music signal of the evaluation set, clipped at θ clip = 0.2. The figure clearly shows the benefit of the warm start strategy. Especially PEW gains SNR rapidly during the first 500 seconds of runtime (executed on a standard consumer laptop). During that time, each decrease of λ seems to boost convergence significantly. Note that PEW achieves good performance in a computation time far below that of OMP. A qualitative example of the declipping results is displayed in Fig. 2. Again, music signal no. 1 with clipping level θ clip = 0.2 is shown, displaying estimations from all aforementioned operators. Neighborhoods for WGL and PEW extend 7 coefficients in time. On this time scale, it is obvious that (P)EW , HT and OMP operators yield much better estimations than the (WG)L: they reliably respect the clipping constraint (as do the (WG)L), and often resemble the original signal profile beyond the clip level, although OMP seems to yield too much high frequency oscillation, while HT sometimes overshoots the original amplitude values. The main differences between unclipped original and (P)EW-declipped version seem to be due to the shape of the largest amplitude values far beyond the clipping level. The (WG)L family, on the contrary, far more resembles the clipped signal than the original. In this case, the inclusion of neighborhood persistence does not even seem to yield a different solution, as both estimates behave almost identically.

Choice of the neighborhood

The choice of the neighborhoods used in the persistent thresholding operators WGL and PEW is both significant and delicate. A good choice of both size and, to a lesser extent, shape depends on the signal characteristics and the severity of corruption. Here, we evaluate neighborhoods which symmetrically extend in time and encompass Obviously, all operators improve SNRm. However, Lasso and WGL seem to be least successful overall. This confirms the qualitative insights from Fig. 2 estimates than the Lasso/WGL operators. No matter whether we consider WGL or PEW, however, the usage of neighborhoods still seems to improve performance for most clipping levels. Iterative hard-thresholding (HT) [START_REF] Kitić | Consistent iterative hard thresholding for signal declipping[END_REF] has similar performance compared to L and WGL for music signals, but yields consistently better results on speech, where it is close to OMP [START_REF] Adler | Audio inpainting[END_REF]. Interestingly, for both speech and music signals, EW and PEW lead to improvements of up to 5 dB SNRm compared to OMP. For low clipping values (i.e. massive signal deterioration) of music signals, the inclusion of neighborhood persistence seems to be particularly beneficial, yielding another 1 dB SNRm improvement.

Except for the Lasso, the algorithms could in principle be sensitive to their initialization. We observed in practice that all operators, with HT being an exception, are very robust indeed and can be initialized by the clipped signal as a default value. However, HT is known to be very sensitive to initialization [START_REF] Blumensath | Iterative thresholding for sparse approximations[END_REF], which might partly explain the disappointing results obtained here.

Let us finally note that although no formal listening experiments were conducted for the lack of resources, we noticed that PEWestimates feature remarkably few audible artifacts, even in cases of massive signal deterioration by low clipping thresholds. The approach thus not only performs well for improvement of signal to noise ratio, but seems to present a valuable tool for perceptual audio enhancement.

CONCLUSION

This paper studied the audio declipping problem by equipping the novel approach of social sparsity with a clipping constraint. We presented an algorithm which converges to a solution in the classical Lasso case and demonstrated empirically that thresholding operations beyond simple soft-thresholding lead to significant gain in declipping quality. Relevant improvements were achieved by taking into account alternative coefficient exponentiation (leading from L to EW) and neighborhood persistence (leading from Lasso to WGL and EW to PEW). In particular, the PEW operator yields significant improvements of SNRm compared to two state-of-the-art methods, while the corresponding algorithm is still less time-consuming.

Future theoretical work will focus on the characterization of PEW operators in conjunctions with ISTA-type algorithms. Furthermore, we will apply the approach to the general audio interpolation (inpainting) problem, where exploiting neighborhood persistence in the time-frequency domain might be a valuable strategy for dealing with massive data loss such as time intervals of 10ms and more.

Fig. 1 .

 1 Fig. 1. Improvement of SNR as a function of time for a music signal at clip level θ clip = 0.2. Algorithm 1 "warm-starts" with a large value of λ which is decreased every 500 iterations. Black lines indicates updates of λ. The horizontal red line indicates the SNR reached by OMP. The vertical red line indicates the cpu time taken by OMP.

Fig. 3 .

 3 Fig.3. Influence of the neighborhood on declipping performance. The 10 different speech (left) and music signals (right) are displayed on the x-axis. The average gain (over clipping levels 0.1, 0.2, . . . , 0.9) of SNRmiss with regard to the baseline set by the EW operator is displayed on the y-axis.

4. 4 .

 4 Evaluation on Speech and Music Signals

Fig. 4

 4 Fig. 4 presents systematic declipping results, depicting improvement of SNRm (with respect to the baseline of the clipped signal) as a function of clipping level. Here, each point represents the mean over the ten different signals of the evaluation set.Obviously, all operators improve SNRm. However, Lasso and WGL seem to be least successful overall. This confirms the qualitative insights from Fig.2: the better preservation of signal energy of the family of empirical Wiener-based operators yields more reliable

Fig. 4 .

 4 Fig. 4. Average SNRmiss for 10 speech (left) and music (right) signals over different clipping levels and operators. Neighborhoods extend 3 and 7 coefficients in time for speech and music signals, respectively.

  Declipped music signal using different operators for clip level θ clip = 0.2 using the Lasso, WGL, EW, PEW, HT, and OMP operators. Neighborhood size for WGL and PEW was 7.3, 5, and 7 coefficients. Neighborhoods with 3 coefficients, for instance, would encompass the centre-coefficient itself plus one coefficient preceeding and one following in time. Note that the WGL and PEW with unit-neighborhood (with only one coefficient) coincide with the Lasso and EW operator, respectively. Fig.3depicts the average gain (over all clipping levels) of SNRm obtained by using neighborhoods in conjunction with PEW and Algorithm 1, compared to the EW operator as a baseline, i.e. graphing SNRm(PEW) -SNRm(EW). Here, applying neighborhoods improves reconstruction in about 50% of the cases, detoriation occurs otherwise. It is particularly visible that shorter temporal persistence of 3 coefficients brings more benefits than larger neighborhoods. For music signals, the usage of neighborhoods turns out to be favorable in almost all cases with best results for neighborhoods of size 5 or 7. We thus use neighborhoods of length 3 for speech signals and length 7 for music signals in the following comparisons.

		0.8
		0.6
		0.4
	Amplitude	-0.2 0 0.2	Original PEW
			EW
		-0.4	L
		-0.6	WGL HT
		-0.8	OMP
		4230	4235	4240	4245	4250	4255
				Time in ms
	Fig. 2.

  : the better preservation of signal energy of the family of empirical Wiener-based operators yields more reliable

				Speech @ 16kHz					Music @ 16kHz
		18							11
	Average SNR m Improvement	2 4 6 8 10 12 14 16		L EW WGL PEW HT OMP				Average SNR m Improvement	1 2 3 4 5 6 7 8 9 10
		0	0	0.2	0.4	0.6	0.8	1	0	0	0.2	0.4	0.6	0.8	1
				Clipping Level						Clipping Level