Equilibrium of Homochiral Oligomerization of a Mixture of Enantiomers

J. Phys. Chem. B 2008, 112, 15361–15368

15361

Equilibrium of Homochiral Oligomerization of a Mixture of Enantiomers. Its Relevance to Nonlinear Effects in Asymmetric Catalysis Masaki Tsukamoto, Kovuru Gopalaiah, and Henri B. Kagan* Institut de Chimie Molećulaire et des Mate´riaux d’Orsay (UMR 8182, CNRS), Laboratoire de Catalyse Molećulaire, UniVersite´ Paris-Sud, 91405 Orsay, France ReceiVed: July 4, 2008; ReVised Manuscript ReceiVed: September 2, 2008

The origin of nonlinear effects (no proportionality between enantiomeric excess (ee) of chiral auxiliary and ee of product) is first summarized in general terms, underlining the importance of the presence of molecular species bearing several moieties deriving from the chiral auxiliary. The presence of a heterochiral species, produced from enantioimpure chiral auxiliaries, usually explains well the deviation to linearity, especially asymmetric amplification. In this article it is shown that the absence of a heterochiral species is not incompatible with an asymmetric amplification. The demonstration has been done on a simple model, the equilibrium of homochiral dimerization. The monomers R and S being in equilibrium with the dimers R2 and S2, it was possible to calculate eemonomer and eedimer as a function of the initial concentration and the initial ee of the monomer. The asymmetric amplification can be quite substantial for the dimer, while asymmetric depletion characterizes the residual monomers. Similar conclusions apply to homochiral tri- and tetramerizations. The extension to irreversible reactions was briefly analyzed as well as the use of these results. Introduction In asymmetric synthesis there is not always the expected proportionality between the enantiomeric excesses of the chiral auxiliary and of the product. Nonlinear effects (NLE) were experimentally demonstrated in 19861 and have been subsequently discovered in many types of asymmetric catalyzed reactions.2-11 Simplified models have been proposed in order to broadly describe the various facets of this phenomena.9,12 Some of the basic models are summarized in Scheme 1. The simple MLn model was proposed by Kagan et al.,1,12 and several variations of it were treated by Blackmond et al.9 The case of catalyzed addition of diethylzinc on benzaldehyde was mechanistically studied in great details by Noyori, Kitamura et al. and illuminated one key aspect of nonlinear effects exemplified by model C in Scheme 1.13-15 Aggregation16 is the key feature for explaining deviations from an ideal behavior in various phenomena related to physical or chemical properties of mixtures of enantiomers. The formation of diastereomeric species in the homo- and the heteroassociation of a mixture of enantiomers makes a basic difference with the autoassociation involving an enantiopure compound (only giving homochiral species). Herein we will address the question: is the formation of heterochiral species really necessary for observation of nonlinear effects? For giving insight into this question, we want to discuss it by considering first homochiral dimerization (without any heterochiral dimerization) as a model system.17,18 Then we will extend the results to homochiral trimerization19,20 and tetramerization. Results and Discussion 1. Homochiral Dimerization. Dimerization equilibrium is described in Scheme 2. R and S are the two enantiomers which will give rise to the homochiral dimers R2 and S2, respectively. The heterochiral dimer (R)(S) is assumed to be not formed. * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: +33-1-69-15-78-95. Fax: +33-1-69-15-46-80.

SCHEME 1: Some Simplified Models Providing Nonlinear Effects Involving Chiral Organometallic Catalystsa

a M and L stand for metal and ligand, respectively. In C the monomer is catalytically active, in D the dimer is the catalyst, eemax stands for the ee of the product obtained with enantiopure ligand.

Since the discussion is very general, it is not necessary to specify what the structure of the dimers is. The word dimerization means proximity of the two initial chiral molecules into a molecular

10.1021/jp8058917 CCC: $40.75  2008 American Chemical Society Published on Web 11/11/2008

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Tsukamoto et al.

SCHEME 2: Homochiral Dimerization Where x1, x2, y1, and y2 are the Concentrations at Equilibrium

or supramolecular system. This can be the consequence of hydrogen bonds, of a chemical reaction, or of bonding to a template like a metal center. In the mixture of R and S enantiomers arising from homochiral dimerization, the two dimerization equilibria are independent, since they have no common components. If the initial concentrations of R and S are defined as x0 and y0, these quantities are expressed as eq 1 by using the total initial concentration (a) of R and S (a ) x0 + y0) as well as the initial enantiomeric excess (ee0).

x0 )

a(1 + ee0) , 2

y0 )

a(1 - ee0) 2

(1)

where ee0 ) (x0 - y0)/(x0 + y0). The quantities x0 and y0 will decrease with time because of the establishment of the two dimerization equilibria. At the equilibrium, the concentrations of the four species, R, S, R2, and S2 (in amounts, x1, y1, x2, and y2) can be calculated by using the equilibrium constant K2. The relations, x2 ) K2x12 and y2 ) K2y12 are derived from the definition of K2. From the material balance of R and S, one finds the relationships

x0)x1+2x2)x1 + 2K2x21,

y0)y1+2y2)y1 + 2K2y21 (2)

Figure 1. Nonlinear behavior of enantiomeric excesses of dimers (eedimer) and monomers (eemonomer) when the reversible homochiral dimerization is obtained from a scalemic monomers (ee0). Curves computed from eq 4a and 4b; K2 ) equilibrium constant (Scheme 2); a ) 10-1 mol L-1.

The two above equations are easily solved:

x1 )

√1 + 8K2x0 - 1 , 4K2

y1 )

√1 + 8K2y0 - 1 4K2

(3)

Enantiomeric excess of monomer (eemonomer) and that of the dimer (eedimer) are defined as

eemonomer ) eedimer )

x1 - y1 x1 + y1

(4a)

x2 - y2 x21 - y21 ) x2 + y2 x2 + y2 1

(4b)

1

Thus from eq 1, 3, and 4, eemonomer and eedimer are calculated as a function of ee0 with given a and K2. The unit of concentration is taken in mol L-1, giving K in L mol-1. The relationships between ee0 and eemonomer or eedimer are depicted in Figure 1 under a ) 10-1 with K2 ranging from 10-1 to 103.21 Positive nonlinear effects are calculated in the dimer formation (Figure 1a). When the K2 value is less than 10-1, the asymmetric amplification is reaching the maximum. With the increased values of K2, the degree of amplification is decreased and at a K2 value of 103, and eedimer is almost equal to ee0. In contrast, negative nonlinear effects are calculated for the remaining monomer (Figure 1b). K2 values inferior to 10-1 give linearity between ee0 and eemonomer, and as K2 values are increased, the degree of asymmetric depletion is stronger. For the evaluation of the material balance of R, S, R2, and S2, the dimer amount defined as eq 5 (amount of initial monomer included in the dimer at equilibrium) was plotted in Figure 2

Figure 2. Homochiral dimerization: Amount of dimer (proportion of initial monomer in the dimer) as a function of equilibrium constant K2 (Scheme 2); a ) 10-1 mol L-1, initial enantiopurity of ee0.

as a function of ee0 with various K2 values. A significant amplification of ee of the dimer is occurring when the dimer is produced in a small amount (K2 e 10-1) (Figure 1a). As K2 value is larger (K2 ) 103), the dimer is dominant (Figure 2), while there is a remarkable asymmetric depletion for the monomer (Figure 1b).

2(x2 + y2) 2K2(x12 + y12) dimer amount ) ) a a

(5)

The degree of asymmetric amplification in the dimer formation is an important point to discuss. The ratio eedimer/ee0, taken

Homochiral Oligomerization of a Mixture of Monomers

J. Phys. Chem. B, Vol. 112, No. 48, 2008 15363

Figure 3. Asymmetric amplification in dimerization equilibria (Scheme 2) as a function of equilibrium constant K2 and initial monomer ee (ee0); a ) 10-1 mol L-1.

SCHEME 3: Homochiral Trimerization Where x1, x3, y1, and y3 Are the Concentrations at Equilibrium

as the amplification factor because of simplicity in the mathematical treatment, was plotted in Figure 3 as a function of ee0. For a fixed K2 value, the amplification factor reaches its maximum as ee0 approaches to 0. The maximum amplification factor of 2 is reached when K2 is small enough (e10-2). The general mathematical treatment concerning the maximum amplification factor is detailed in Supporting Information. 2. Homochiral Trimerization. The homochiral trimerization can be treated as the homochiral dimerization, considering the equilibrium between monomeric R, S species and the trimeric R3 and S3 species, where the equilibrium constant K3 is defined as shown in Scheme 3. For simplicity, the trimeric species are supposed to be produced from the monomeric species without any accumulations of the homochiral dimeric species, R2 or S2, and of the heterochiral trimers, (R)2(S) and (R)(S)2. If the concentrations of R, S, R3, and S3 are expressed by x1, y1, x3 () K3x13), and y3 () K3y13), respectively, the following equations are derived at the equilibrium. The x0 and y0 are initial concentrations of monomers.

x0)x1+3x3)x1 + 3K3x31,

y0)y1+3y3)y1 + 3K3 y31 (6)

Analytical expressions of x1 and y1 can be obtained by solving Eq 6 (see 2 in Supporting Information). The enantiomeric excess of the trimer (eetrimer) is defined as (x3 - y3)/(x3 + y3) ) (x13 y13)/(x13 + y13). Since eq 1 and 4a are also applicable, the enantiomeric excess of monomer or that of trimer can be analyzed as a function of K3 and of ee0 with initial concentration (a). One example is shown in Figure 4 with a ) 10-1 mol L-1 and K3 ranging from 10-1 to 103 L2 mol-2. The corresponding trimer amount as defined by eq 7 is plotted in Figure 5as a function of ee0.

trimer amount )

3(x3 + y3) 3K3(x13 + y13) ) a a

(7)

The same trends as in dimerization were also observed here. The amplification of enantiomeric excess of the trimer occurs

Figure 4. Nonlinear behavior of enantiomeric excesses of trimers (eetrimer) (a) and monomers (eemonomer) (b) when the reversible homochiral trimerization is obtained from a scalemic monomer (ee0). Curves are computed as explained in the text; K3 is the equilibrium constant (Scheme 3); a ) 10-1 mol L-1.

Figure 5. Homochiral trimerization: Amount of trimer (proportion of initial monomer in the trimer) as a function of equilibrium constant K3 (Scheme 3); a ) 10-1 mol L-1.

when it is present in small quantities, while an asymmetric depletion of the remaining monomer is produced when the monomer amount is decreased. The maximum amplification factor, eetrimer/ee0, is calculated to be equal to 3 when K3 is inferior to 10-1 and when ee0 is close to 0, as shown in Figure 6. 3. Homochiral Tetramerization. Asymmetric amplification in tetramer formation, and depletion in monomer formation are also occurring in the homochiral tetramerization shown in Scheme 4. In this system, monomeric R, S species and the tetrameric R4, S4 species coexist, with the concentrations expressed by x1, y1, x4 () K4x14), and y4 () K4y14), respectively


Tsukamoto et al. relationship between the enantiomeric excess of the tetramer (eetetramer) or that of monomer (eemonomer) and ee0 can be obtained (see 3.1. in Supporting Information) by solving eq 8, where eetetramer ) (x4 - y4)/(x4 + y4) ) (x14 - y14)/(x14 + y14). One example is shown in Figure 7 with a ) 10-1 mol L-1 and K4 ) 10-1 and 103 L3 mol-3. The corresponding tetramer amount () 4(x4 + y4)/a) as a function of ee0 is calculated in Tables S3.1 and S3.2 of Supporting Information.

x0)x1+4x4)x1 + 4K4x41,

Figure 6. Asymmetric amplification in trimerization equilibria (Scheme 3) as a function of equilibrium constant K3 and initial monomer ee (ee0); a ) 10-1 mol L-1.

Figure 7. Nonlinear behavior of enantiomeric excesses of tetramers (eetetramer) and monomers (eemonomer) when the reversible homochiral tetramerization is obtained from a scalemic monomer (ee0). Curves are computed as explained in the text; K4 is the equilibrium constant (Scheme 4); a ) 10-1 mol L-1.

SCHEME 4: Homochiral Tetramerization Where x1, x4, y1, and y4 Are the Concentrations at Equilibrium

(Scheme 4). The heterochiral tetramers, (R)3(S), (R)(S)3, and (R)2(S)2 are assumed to be absent in the equilibrium. The

y0)y1+4 y4)y1 + 4K4 y41 (8)

The same general trends as in dimerization and trimerization are observed here. The amplification of enantiomeric excess of the tetramer is important when it is present in small quantities, while the asymmetric depletion of the monomer is produced when the monomer amount is low. The maximum amplification factor, eetetramer/ee0, is calculated to be equal to 4 when K4 is smaller than 10-1 and ee0 is close to 0 (see Figure S1 of Supporting Information). 4. General Trends and Perspectives. The calculations showed that the exclusive homochiral oligomerization is able to induce an asymmetric amplification in the oligomer (di-, tri-, and tetramer) while an asymmetric depletion may be observed in the residual monomer. Clearly, nonlinearity can occur without the formation of heterochiral species. To experimentally detect this nonlinearity, the equilibrium position must be shifted strongly either toward monomer or dimer. In the first case the large amount of monomer keeps an ee close to the initial value ee0, but eedimer will be amplified with respect to ee0. In the second case, the predominent dimer has eedimer close to ee0, while a significant asymmetric depletion (eemonomer < ee0) characterizes the remaining monomer. These trends are well evidenced, for example, in curves of Figure 1 (homochiral dimerization), where a good asymmetric amplification for the dimer is achieved when K2 ) 10-2, while the monomer gives the linearity. A strong asymmetric depletion is obtained for the monomer for K2 ) 103, simultaneously a nonlinear effect is absent for the dimer. In Table S1.1-S1.6 (Supporting Information) are selected some values of ee0 and the corresponding values of the equilibrium composition, eemonomer and eedimer. Tables S1.1-S3.2 (Supporting Information) involve various combinations of equilibrium constants Ki with fixed initial concentration a () 10-1 mol L-1) for homochiral di-, tri-, and tetramerization. Of course, the ultimate monomer and oligomer concentrations depend not only on the equilibrium constant but also on the total initial concentration of monomers. Another general trend is the greater efficiency of the homochiral tetramerization, compared to di- and trimerization for a set of given Ki and a values. This is apparent in Figures 8 and 9. For example, under the condition of Ki ) 10-1 and a ) 10-1 mol L-1, the highest amplification of oligomer is obtained in tetramerization (Figures 8 and 9). Amplification factors (een/ee0) in oligomerization were plotted in Figure 9 under Ki ) 10-1 and a ) 10-1 mol L-1. Here, een is the enantiomeric excess of oligomer. In each case, the maximum amplification factor is n. The general conclusions given in the study of reversible oligomerization are generalizable to some extent to irreVersible oligomerization. Thus a partial transformation of a mixture of monomers (with ee0) will provide by an irreversible homochiral dimerization a residual monomer with eemonomer ee0. In Scheme 5 are indicated the scheme and the basic equations, k being the rate constant of the pseudofirst order dimer formation. The details of the calculation are shown in Supporting Information. In Figure 10 is given a



Figure 8. Comparison of asymmetric amplifications in homochiral di-, tri-, and tetramerizations.

Figure 10. Irreversible dimerization (Scheme 5). Evaluation with time of eedimer (a) and eemonomer (b) for various ee0 values of initial scalemic monomer; K ) 1 L mol-1 s-1, and a ) R0 + S0 ) 0.1 mol L-1. Figure 9. Amplification factors (een/ee0) in homochiral oligomerization of a mixture of monomers under Ki ) 10-1 and a ) 10-1 mol L-1 with an initial enantiopurity of ee0.

SCHEME 6: Equilibrium of Homochiral Oligomerization Involving a Metal M at Initial Concentration m0a

SCHEME 5: Homochiral Irreversible Dimerization When R0 and S0 Are the Initial Monomer Concentrations

graphical representation of irreversible homochiral dimerization for a set of parameters. There is an asymmetric amplification, which is a maximum at low conversion, where a small amount of dimer is formed. This is similar to results in the reversible dimerization. For example for ee0 ) 50%, the dimer has an 80% ee at initial conversion. To come closer to the models of Scheme 1 where heterochiral species are absent, it is interesting to consider the reversible homochiral oligomerization with introduction of a metal M at an initial concentration m0 (Scheme 6). The mathematical treatment where n ) 2 is detailed in Supporting Information. The general trend is the same as in the reversible homochiral oligomerization in the absence of M: when the homochiral ML2 complex is produced in small amount, its enantiomeric excess can be greatly amplified with respect to the initial enantiomeric excess of the ligand (ee0) (Figure 11). 5. Examples of Homochiral Dimerization. A simple way to detect homochiral dimerization is to run the reaction on a racemic monomer. If a racemic dimer is the only product it is an indication that a stereoselective coupling (homo- or heterochiral) occurred. When the racemic homochiral dimer is formed, this can be easily confirmed by the comparison with the product

a x, y, X, and Y are the concentrations of LR, LS, M(LR)n and M(LS)n respectively, at the equilibrium.

of the dimerization from the enantiopure monomer. Below are presented selected examples of homochiral dimerization occurring in solution. At the beginning we mention autoassociation of metal complexes (such as M2L2, M2L3, and M3L2 systems) and then organic compounds autoassociated by hydrogen bonds. Hayashi et al. have reported a kinetic study of Rh/BINAP catalyzed asymmetric 1,4-addition of phenylboronic acid to R,βunsaturated ketones.24 An inactive homochiral dimeric hydroxorhodium complex is a dominant species in the catalytic cycle, which was confirmed by quantitative investigation of the nonlinear effect. A negative NLE was obtained and the dimerization constant has been estimated. The homochiral recognition has been observed for various classes of organometallic complexes. For example, ML2 type complexes 1 and 2 (Figure 12) are proved to be obtained dominantly from a mixture of enantiomers.25,26 Reaction of either enantiopure or racemic 3,3′-disubstituted1,1′-bi-2-naphthol (H2Me2BINO) with Ti(O-i-Pr)4 in diethyl ether afforded [(Me2BINO)Ti(O-i-Pr)2]2 complex.27 Similarly, either the racemic or the enantiopure bis(trimethylsilyl)ether of hydrogenated binaphthol reacted with TiCl4 to produce the


Tsukamoto et al.

Figure 14. Some ligands used in homochiral association with Ag(I) salt.

Figure 15. Ligand 6 used for the homochiral self-assembly with Cu(I) salt.

Figure 16. Ligand 7 used for the homochiral self-assembly with Ga(acac)3.

SCHEME 7: Ligands 8 Were Used in the Homochiral Self-Assembly with Ag(I) Salt Figure 11. Nonlinear behavior of metal mediated reversible homochiral association of mixture of LR and LS (Scheme 6, n ) 2) of initial enantiomeric excess ee0; K is the equilibrium constant; a ) 10-2 mol L-1.

Figure 12. Homochiral association of BINOL and BINAP in Ti and Pd complexes, respectively.

Figure 13. Titanium(IV) octahydrobinaphtholate dimer.

complex 3 (Figure 13).28 These complexes exist as homochiral dimers both in solution and in the solid-state, as established by 1H and 13C NMR spectra, solution molecular weight, and X-ray crystallographic studies. Complexes of , Ti(diolate)Cl2 . where diolate is derived from BINOL or 3,3′-disubstituted BINOL are well-known for their catalytic activities in enantioselective Diels-Alder reactions and carbonyl-ene reactions.29,30 A detailed structural study of titanium naphtholate deriving from 3,3′-dimethyl-BINOL showed formation of exclusive homochiral dimerization in solution when starting from either racemic or enantiopure dimethyl-BINOL.31 The dimeric Ti2(Me2BINO)2Cl4 have a catalytic activity for the condensation between cyclopentadiene

and methyl acrylate. However the low enantioselectivity (26% ee) with the enantiopure ligand precludes the use of this system to study the asymmetric amplification predicted in the present article. The synthesis of enantiopure helicates using chiral bis(oxazolyl) pyridine ligands (4 and 5) with silver(I) tetrafluoroborate in solution have been reported by Williams and co-workers (Figure 14).32 A racemic mixture of R,R-4 and S,S-4 produced [Ag2(R,R-4)2](BF4)2 and [Ag2(S,S-4)2](BF4)2 by homochiral selfassembly. The solution studies on a racemic ligand 5 gave the similar results as with 4. Mascharak et al. have established the formation of homochiral dimeric copper(II) complexes [Cu2((R)-PEA)2](ClO4)2 and [Cu2((S)-PEA)2](ClO4)2 in equal amounts from a racemic mixture of a chiral ligand N-(1,2-bis(2-pyridyl)ethyl)pyridine-2-carboxamide (PEAH) and Cu(ClO4)2 · 6H2O in DMF.33 An X-ray crystallographic study revealed that homochiral dimers formed in the solid-state. The racemic mixture of monomeric complex [Cu((R/S)-PEA)(Cl)(H2O)] reacted with AgClO4 in acetonitrile afforded the homochiral dimers [Cu2((R)-PEA)2](ClO4)2 and [Cu2((S)-PEA)2](ClO4)2 in equal amounts. The formation of only homochiral dimeric complexes from racemic mixtures of PEAH has been rationalized.



SCHEME 8: Homochiral Dimeric Zirconium Containing Macrocycle 10 from Racemic 9

Stack et al. have studied the ligand self-recognition in the assembly of chiral ligands with metal ions leading to homochiral complexes both in solution and solid-state.34 The equimolar reaction of enantiopure (R,R)-6 (Figure 15) with [Cu(MeCN)4]CF3SO3 in 1:1 CH2Cl2:CH3CN generated stereospecifically binuclear species Λ,Λ-[{Cu(R,R-6)}2]2+, while racemic 6 led to two discrete homochiral metal complexes Λ,Λ-[{Cu(R,R6)}2]2+ and ∆,∆-[{Cu(S,S-6)}2]2+. Authors have also explored the metal-assisted self-assembly in the generation of a chiral molecular tetrahedral complex with bis(2,3-dihydroxybenzamide) ligand 7 in solution (Figure 16).35 Racemic ligand (rac-7) with [Ga(acac)3] in the presence base produced only the homochiral isomers Λ,Λ,Λ,Λ-[Ga4(S,S7)6]12- and ∆,∆,∆,∆-[Ga4(R,R-7)6]12-. It was substantiated by a 1H NMR spectrum of the rac-7 metal complex that is identical to the spectrum obtained for enantiopure [Ga4(S,S-7)6]12-. Hong et al. have investigated the production of enantiopure propeller-shaped supramolecular capsules induced by stereospecific self-assembly of chiral tris(oxazoline) ligands (8) around Ag(I) ions both in solution and solid-state (Scheme 7).36 The ligands 8 act as a trismonodentate unit gave rise to D3symmetric, dimeric, trinuclear complexes with silver ions. Reaction of an equimolar mixture of Me(S)-8 and Me(R)-8 in the presence of 3 equiv of AgNO3 afforded a mixture of the homochiral complexes (M)-[Ag3(Me(S)-8)2]3+ and (P)-[Ag3(Me(R)-8)2]3+ by the ligand self-recognition. No trace of heterochiral products was noticed. This high homoleptic diastereoselectivity was also accomplished in [Ag3(Ph(S and R)-8)2]3+. Authors surmise that these dimeric chiral capsules could function as chiral catalysts. Tilley et al. have reported the diastereoselective synthesis of homochiral macrocycles from racemic BINOL derivatives via the controlled di- and tricyclizations of chiral diynes.37 The homochiral dimeric macrocycle 10 was prepared from the diyne 9 using Negishi-type zirconocene coupling (Scheme 8) and further transformed to the metal-free macrocycle by reaction with benzoic acid. Similarly, homochiral trimeric macrocycle was synthesized from a racemic tethered diyne precursor via zirconocene coupling. It is known that chiral macrocycles may act as ligands in asymmetric catalysis.38-41 Costero et al. have established the homochiral aggregation of ammonium picrates of simple chiral amines from their racemic solutions by using NMR techniques.42 The formation of hydrogen-bonded homodimers has been noticed and the dimerization equilibrium constants (Keq/M-1 ) 10-240) measured for these ammonium salts. Meijer and co-workers have reported the dimerization of some racemic bifunctional 2-ureido-4-[1H]-pyrimidinone derivatives in solution.18 These selectively produced homochiral assemblies

by means of hydrogen bonding. The dimerization was determined by 1H NMR spectra as well a single-crystal X-ray analysis. Conclusion Our study shows that even in absence of heterochiral species a significative asymmetric amplification (in the oligomer) may occur. This should be useful in asymmetric catalysis if the reaction under investigation obeys a mechanism of type D (Scheme 1, with no meso complex), in which the dimer is the catalyst. It is clear that the asymmetric amplification can occur in aggregation processes, even in the absence of formation of heterochiral oligomers, the asymmetric amplification always characterizing the oligomer and not the remaining monomer.43 Homochiral dimerization is not uncommon as shown by a literature survey. It should be possible to find a catalytic system where the equations developed in this article will apply. The concept of asymmetric amplification through homochiral oligomerization finds implications for the evolution of biological homochirality. Peptides of biological interest may be formed initially in high stereochemical purity from aminoacids monomers of low ee’s if homochiral association is predominant.47 Calculation Procedures. Computation for this study was conducted either by Mathematica or Kaleida Graph and the graphics was obtained by Kaleida Graph. Acknowledgment. We thank CNRS and Universite´ ParisSud for a financial support. Two of us acknowledge JSPS (M.T.) and Servier Co (K.G.) for postdoctoral fellowship. Supporting Information Available: Calculations, mathematical treatments, homochiral oligomerization data. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Puchot, C.; Samuel, O.; Dunach, E.; Zhao, S.; Agami, C.; Kagan, H. B. J. Am. Chem. Soc. 1986, 108, 2353. (2) For some reviews see refs 3-12. (3) Girard, C.; Kagan, H. B. Angew. Chem., Int. Ed. 1998, 37, 2923. (4) Satyanarayana, T.;Abraham, S.;Kagan, H. B.;Angew. Chem., Int. Ed. 2008, in press. (5) Avalos, M.; Babiano, R.; Cintas, P.; Jimenez, J. L.; Palacios, J. C. Tetrahedron: Asymmetry 1997, 8, 2997. (6) Bolm, C. In AdVanced Asymmetric Synthesis; Stephenson, G. R., Ed.; Blackie Academic and Professional: London, 1996; pp 9. (7) Kagan, H. B.; Girard, C.; Guillaneux, D.; Rainford, D.; Samuel, O.; Zhao, S. H.; Zhang, S. Y. Acta Chem. Scand. 1996, 50, 345. (8) Kagan, H. B.; Luukas, T. O. In ComprehensiVe Asymmetric Catalysis; Jacobsen, E. N.; Pfaltz, A.; Yamamoto, H., Eds; Springer: Berlin, 1999; Vol. I, pp 101. (9) Blackmond, D. G. Acc. Chem. Res. 2000, 33, 402.

15368 J. Phys. Chem. B, Vol. 112, No. 48, 2008 (10) Kagan, H. B. Synlett 2001, 888. (11) Kagan, H. B. AdV. Synth. Cat. 2001, 343, 227. (12) Guillaneux, D.; Zhao, S. H.; Samuel, O.; Rainford, D.; Kagan, H. B. J. Am. Chem. Soc. 1994, 116, 9430. (13) Kitamura, M.; Okada, S.; Suga, S.; Noyori, R. J. Am. Chem. Soc. 1989, 111, 4028. (14) Noyori, R.; Kitamura, M. Angew. Chem., Int. Ed. 1991, 30, 49. (15) Noyori, R.; Suga, S.; Oka, H.; Kitamura, M. Chem. Record 2001, 1, 85. (16) Auto-association is taken in a broad sense here. It can be the association between two molecules by hydrogen bonds (carboxylic acids, peptides) or by dipole-dipole interactions (sulfoxides) as well as by coordination (e.g. lithium derivatives). (17) Dimerization of chiral bifunctional 2-ureido-4(1H)-pyrimidinone derivatives selectively produces homochiral assemblies. See ref 18. (18) ten Cate, A. T.; Dankers, P. Y. W.; Kooijman, H.; Spek, A. L.; Sijbesma, R. P.; Meijer, E. W. J. Am. Chem. Soc. 2003, 125, 6860. (19) For homochiral and/or heterochiral trimerization of assemblies produced by the combination of calix(4)arene bismelamines with 5,5-diethyl barbiturate, see ref 20. (20) Calama, M. C.; Hulst, R.; Fokkens, R.; Nibbering, N. M. M.; Timmerman, P.; Reinhoudt, D. N. Chem. Commun. 1998, 1021. (21) It is also possible to discuss the equilibrium of homochiral dimerization by using as parameters the enantiomeric ratios of the various species.22 Thus the initial enantiomeric ratio of the monomer is defined by er0 ) x0/y0, after equilibration it becomes ermonomer ) x1/y1, while the dimer is characterized by erdimer ) x2/y2. One immediately sees that ermonomer and erdimer are related by erdimer ) (ermonomer)2. Then, ermonomer is easily calculated: ermonomer ) ((1 + 8K2x0)1/2-1)/((1 + 8K2y0)1/2-1). (22) For a discussion on the relative advantages of using enantiomeric ratio (er) or enantiomeric excess (ee), see ref. 23. (23) Kagan, H. B. Rec. TraV. Chim. Pays-Bas 1995, 114, 203. (24) Kina, A.; Iwamura, H.; Hayashi, T. J. Am. Chem. Soc. 2006, 128, 3904. (25) Mikami, K.; Matsukawa, S. Nature 1997, 385, 613. (26) Alcazar-Roman, L. M.; Hartwig, J. F.; Rheingold, A. L.; LiableSands, L. M.; Guzei, I. A. J. Am. Chem. Soc. 2000, 122, 4618. (27) Boyle, T. J.; Barnes, D. L.; Heppert, J. A.; Morales, L.; Takusagawa, F.; Connolly, J. W. Organometallics 1992, 11, 1112.

Tsukamoto et al. (28) Eilerts, N. W.; Heppert, J. A.; Kennedy, M. L.; Takusagawa, F. Inorg. Chem. 1994, 33, 4813. (29) Mikami, K.; Motoyama, Y.; Terada, M. J. Am. Chem. Soc. 1994, 116, 2812. (30) Mikami, K. Pure Appl. Chem. 1996, 68, 639. (31) Boyle, T. J.; Eilerts, N. W.; Heppert, J. A.; Takusagawa, F. Organometallics 1994, 13, 2218. (32) Provent, C.; Rivara-Minten, E.; Hewage, S.; Brunner, G.; Williams, A. F. Chem.sEur. J. 1999, 5, 3487. (33) Rowland, J. M.; Olmstead, M. M.; Mascharak, P. K. Inorg. Chem. 2002, 41, 1545. (34) Masood, M. A.; Enemark, E. J.; Stack, T. D. P. Angew. Chem., Int. Ed. 1998, 37, 928. (35) Enemark, E. J.; Stack, T. D. P. Angew. Chem., Int. Ed. 1998, 37, 932. (36) Kim, H.-J.; Moon, D.; Lah, M. S.; Hong, J.-I. Angew. Chem., Int. Ed. 2002, 41, 3174. (37) Schafer, L. L.; Tilley, T. D. J. Am. Chem. Soc. 2001, 123, 2683. (38) Tejeda, A.; Oliva, A.; Simon, L.; Grande, M.; Caballero, M. C.; Moran, J. R. Tetrahedron Lett. 2000, 41, 4563. (39) Zhang, X. X.; Bradshaw, J. S.; Izatt, R. M. Chem. ReV. 1997, 97, 3313. (40) Gross, Z.; Ini, S. Org. Lett. 1999, 1, 2077. (41) Collman, J. P.; Wang, Z.; Strausmanis, A.; Quelquejeu, M. J. Am. Chem. Soc. 1999, 121, 460. (42) Costero, A. M.; Colera, M.; Gavinã, P.; Gil, S.; Ochando, L. E. New. J. Chem. 2006, 30, 1263. (43) This could be an additional scenario in prebiotic chemistry for the amplification of a small enantiomeric excess. However it will relieve only a very limited amount of enantioenriched dimer as in the Yamagata accumulation principle,44-46 but a subsequent reaction, where the dimer acts as an enantioselective catalyst, will allow the generation of a large amount of a desired compound. (44) Yamagata, Y. A. J. Theor. Biol. 1966, 11, 495. (45) Bonner, W. A. Origins Life EVol. Biosphere 1999, 29, 615. (46) Chandrasekhar, S. Chirality 2008, 20, 84, and references cited therein. (47) Weissbuch, I.; Zepik, H.; Bolbach, G.; Shavit, E.; Tang, M.; Jensen, T.; Kjaer, K.; Leiserowitz, L.; Lahav, M. Chem. Eur. J. 2003, 9, 1782.

JP8058917

Equilibrium of Homochiral Oligomerization of a Mixture of Enantiomers

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