Non-linear Effects in Asymmetric Catalysis: Whys and Wherefores

It is argued that the titled non-linear effects (NLE) may arise whenever the order of the reaction in the chiral catalyst in greater than 1. In a fundamental departure from previous approaches, this is mathematically elaborated for the second order case. (NLE may also be observed if the chiral catalyst forms non-reacting dimers in a competing equilibrium; practically, however, this implies the in situ resolution of the catalyst.) The amplification of enantiomeric excess by NLE implies a relative (although modest) reduction in the entropy of mixing. The consequent increase in free energy apparently indicates a non-equilibrium process. It is suggested, based on arguments involving the chemical potential, that kinetically-controlled reactions lead to a state of “quasi-equilibrium”: in this, although overall equilibrium is attained, the product-spread is far from equilibrium. Thus, both the linear and NLE cases of chiral catalysis represent departures from equilibrium (which requires that the product e.e. = 0). Interesting similarities exist with models of non-equilibrium systems, the NLE cases apparently being analogs of open systems just after the bifurcation point has been crossed. 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 INTRODUCTION The enhancement of enantiomeric excess (e.e.) is justly a cause for jubilation, but sometimes, for concern too! Whilst asymmetric syntheses clearly aim for the highest possible e.e.’s, unsolicited enhancements of e.e. raise intriguing conceptual issues. Thermodynamic stability is represented by e.e. = 0, and a departure from this theoretical comfort zone implies a higher level of order: it is then incumbent on the investigator to explain the origin of the apparent increase in free energy content, and the nature of the intervention that brought it about. The phenomenon of non-linear effects in asymmetric catalysis (NLEAC) burst upon the scene of asymmetric synthesis relatively recently. NLEAC is predicated on the reasonable assumption that in a reaction that is catalysed by a chiral catalyst, the e.e. of the product cannot be greater than the e.e. of the catalyst itself. A lower product e.e., relative to the catalyst, can be accommodated by assuming correspondingly poor enantioselectivity. (This represents a drift towards the comfort zone of e.e. = 0, so can be dismissed as another failed asymmetric synthesis!) The caveat, however, is that the observed lower e.e. should be in constant proportion to the catalyst e.e. Intriguing conceptual questions arise, however, when either the above constant proportion is not maintained, or worse still, the e.e. of the product is greater than that of the catalyst! (In the latter case too, proportionality may not be maintained.) Typical plots of the product e.e. (Pee) vs. the catalyst e.e. (Cee), representing these cases (“asymmetric depletion and amplification” respectively), are shown in Fig. 1. Fig. 1 Scheme 1 3 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 Interestingly, such examples can be mediated by complexation of two molecules of the chiral catalyst – usually with a metal ion – prior to the reaction (Scheme 1). This leads to the formation of all three diastereomeric dimeric complexes (CR and CS are the enantiomeric catalysts, and M a metal ion): CRMCR, CSMCS and CRMCS. These are, in fact, the effective catalytic species in the reaction. Current treatments are apparently successful in explaining the observed NLEAC on the basis of the pre-equilibrium formation of the above diastereomeric complexes. However, a rigorous formulation would be based on a kinetic treatment of the formation of the enantiomeric products, as discussed at length below. [This differs fundamentally from previous approaches, reviewed critically in the Supplementary Information (S.I.) section.] Also, NLEAC may manifest itself under other mechanistic conditions, particularly when the non-racemic catalyst forms a relatively unreactive meso dimer species. A variant of this is preferential diastereomer formation between chiral catalyst and chiral product (as occurs in certain cases of homogeneous amino acid catalysis). These cases, however, apparently involve a prior in situ resolution of the catalyst, and are hence considered only briefly further below. RESULTS AND DISCUSSION The rates of formation (vR and vS) of the two enantiomeric products, PR and PS (respectively) in the reaction in Scheme 1, would be given by eqs. 1 and 2. (We note that two molecules of catalyst are involved in the reaction, hence the second order dependence, the k’s being rate constants; the treatment extends previous ones as explained in the S.I. section.) For the sake of simplicity, at this stage, it is assumed that 4 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 the reaction of (achiral) substrate (S) with CR yields only PR, and with CS only PS, i.e. 100% enantioselectivity; also, CR and CS do not act in concert. (All this implies that only CRMCR and CSMCS react with total selectivity, and that CRMCS is completely unreactive.) v v P ( C P R = d[PR]/dt = kR[S][M][CR] (1) S = d[PS]/dt = kS[S][M][CS] (2) The e.e. of the product (Pee) would then be given by eq. 3, that of the catalyst (Cee) by eq. 4 ([CR] > [CS]). It is also assumed that the yield of each product, at a given time (t), bears the same proportion to its rate of formation, i.e. vR = [PR] and vS = [PS],  being a proportionality constant. ee = ([PR]-[PS])/([PR]+[PS]) = (vR-vS)/(vR+vS) = [CR]-[CS])/([CR]+[CS]) (3) ee = ([CR]-[CS])/([CR]+[CS]) (4) ee = ([CR]-[CS])/{([CR]+[CS])-2[CR][CS]/([CR]+[CS])} (5) This is valid as the catalyst is regenerated, so [CR] and [CS] remain constant during the course of the reaction; also, S and M are achiral precursors common to the two pathways. (In an achiral medium, kR = kS.) It is particularly noteworthy that this mechanism thus gives rise to the “real” NLEAC, and differs fundamentally from cases involving a prior in situ resolution of catalyst (discussed further below). Furthermore, eq. 3 can be reduced to eq. 5, which can be compared with eq. 4: it is seen that since {[CR][CS]/([CR] +[CS])} > 0, the denominator of eq. 5 would be less than that of eq. 4. Clearly, therefore, Pee would be enhanced relative to Cee. This enhancement, 5 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 of course, is a direct consequence of the fact that the reaction is second order in the catalyst. [Note: 10 x Pee = Pee(%);10 x Cee = Cee(%).] v v Pe P R = d[PR]/dt = kR[S][M][CR] + k 2 RS[S][M][CR][CS] (6) S = d[PS]/dt = kS[S][M][CS] + k 2 RS[S][M][CR][CS] (7) e = ([CR] -[C 2 S] )/([C 2 R] +[C 2 S] +2(k 2 RS/kR)[CR][CS]) (8) ee = ([CR]-[CS])/{([CR]+[CS])+2(-1)[CR][CS]/([CR]+[CS])} (9) In contrast to this amplification, depletion would be observed when the meso complex, CRMCS, is more reactive than its chiral diastereomers. (The meso diastereomer would produce equal amounts of PR and PS). For then, the rates of formation of PR and PS would be given by eqs. 6 and 7 (kRS being the rate constant corresponding to the meso catalyst, noting that kRS > kR, kS). Pee is now given by eq. 8; reduction of this to eq. 9 (where  = kRS/kR) sets it up for analogous arguments as with eq. 5 above. In this case, the denominator would be greater than that of eq. 4, as  > 1 and (2(kRS/kR)[CR][CS]/([CR]+[CS]) > 0, so Pee < Cee. (Note that [CR]/([CR]+[CS]) and [CS]/([CR]+[CS]) are the mole fractions of CR and CS respectively.) The above simplified treatment can evolve to represent real-life scenarios by the inclusion of a factor for the enantioselectivity (es) and the (very real!) possibility that all three diastereomeric complexes (CRMCR, CSMCS and CRMCS) react concurrently (vide infra). This was considered above for the case of depletion of e.e., with kRS >> kR, kS  0; however, for the case of amplification, it was assumed that kRS = 0. (Note that for the case kRS >> kR, kS = 0, Pee = 0.) In fact, it is quite likely that kR, kS >> kRS  0: this 6 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 would moderate the observed amplification of e.e. Thus, CRMCR and CSMCS would have identical reactivity (via analogous transition states), but CRMCS may be either more or less reactive than them. (These two cases would lead to depletion or moderated amplification of e.e. respectively.) Pe Pe e = ([CR]-[CS])/{([CR]+[CS])-2(1-)[CR][CS]/([CR]+[CS])} (10) e = (es)([CR]-[CS])/{([CR]+[CS])-2(1-)[CR][CS]/([CR]+[CS])} (11) Fig. 2 The case of moderated amplification (kR, kS >> kRS  0) may be dealt with analogously as the derivation of eq. 9, but now  < 1, so eq. 9 may be rewritten as eq. 10: comparison with eq. 4 clearly shows that Pee would be enhanced but not to the same extent as in eq. 5. Plots of Pee (eqs. 5 and 10) vs. Cee (eq. 4) can thus be generated and are shown in Fig. 2. The inclusion of the enantioselectivity term, es, leads to eq. 11 (cf. S.I. for details). Thus, the above kinetic treatment succeeds in reproducing the observed NLEAC for the case of the intermediate formation of dimeric complexes. Case of Non-reactive Dimers in Equilibrium Scheme 2 An interesting variant of the above phenomena, which also apparently displays NLEAC, is shown in Scheme 2. In this, the reactive form of the chiral catalyst is a monomer, which, however, is in equilibrium with non-reacting dimeric species. (This variant was originally observed in the addition of diethylzinc to aldehydes, catalysed by chiral amino alcohols. An analogous case was discovered in certain proline-catalysed 7 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 aldol condensations.) In fact, these systems can also be described by equations analogous to eqs. 5-11 above. (However, there is no second order dependence with the catalyst concentration in these cases. Eqs. 12-17 specifically refer to the diethylzinc addition case; the proline-catalysis case can be treated analogously.) v v [ [C C Pee 2 . R = d[PR]/dt = kR[S][CR] (12) S = d[PS]/dt = kS[S][CS] (13) CR] = [CR]o – (2[CR]/KR) – ([CR][CS]/KRS) (14) S] = [CS]o – (2[CS]/KS) – ([CR][CS]/KRS) (15) ee = ([CR]o – [CS]o)/([CR]o + [CS]o) (16) = {([CR]o – [CS]o) – 2([CR] – [CS])/KR}/{([CR]o + [CS]o) – (2([CR] + [CS])/KR) – ([CR][CS]/KRS)} (17) Thus, the rates of formation of enantiomeric products follow the rate laws in eqs. 12 and 13. (For simplicity, total stereoselectivity is assumed.) Now, however, the concentrations of the reactive monomeric species are governed by the equilibrium constants (KR, KS and KSR, cf. Scheme 2) for the formation of the three diastereomeric dimer species, viz. CR-CR, CS-CS and CS-CR. Eqs. 14-17 follow, analogously to eqs. 3-8 above. ([CR]o and [CS]o refer to total catalyst concentrations; note that kR = kS and KR = KS; an enantioselectivity term, es, can be included in eq. 17, as in eq. 11 above.) Comparison of eqs. 16 and 17 indicates that amplification is to be expected when KR, KS >> KRS. This is because the terms ([CR] – [CS])/KR and ([CR] + [CS])/KR << ([CR][CS]/KRS) for large KR, KS, so may be neglected. Also then, since ([CR][CS]/KRS) > 0, the denominator of eq. 17 would be less than that of eq. 16 8 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 The case for depletion is less straightforward, but obtains for KR, KS << KRS as follows. Firstly, ([CR][CS]/KRS) may be neglected. Furthermore, as [CR] > [CS], [CR] >> [CS], so ([CR] – [CS]) and ([CR] + [CS]) ~ [CR]. Thus, eq. 17 now reduces to Pee ~ {([CR]o – [CS]o) – 2([CR]/KR) }/{([CR]o + [CS]o) – 2([CR]/KR)}. Comparing this with Cee (eq. 16) indicates that both its numerator and denominator have been reduced by an equal quantity (2([CR]/KR). However, since Cee < 1, the numerator has been reduced by a greater fraction of itself than has the denominator. Hence, Pee < Cee, leading to deple defin tion. For both large KR, KS and KRS, i.e. insignificant formation of the dimers, linearity would be observed between Cee and Pee: Pee = esCee (es being the enantioselectivity as ed above). This, then, represents the transition between amplification and depletion. Interestingly, these systems are analogs of a precedented phenomenon. Thus, it is known that, when a racemic compound is crystallized from its enantiomerically-enriched solution, the supernatant solution displays an enhanced e.e. The preferred formation of the meso dimer (CR-CS) above is the analog of crystalline racemic-compound formation. Also, for amplification of e.e. to be observed, the analog of conglomerate formation, i.e. the formation of CS-CS and CR-CR, needs to be avoided or suppressed: this condition obtains when KR, KS >> KRS. Thus, the condition KR, KS << KRS (leading to depletion) is analogous to the conglomerate form being more stable than the racemic compound form. (In th not, the suppression would not be “kinetically effective” even if the equilibrium constant is case the crystalline phase would be enantiomerically enriched.) It is also noteworthy that, for NLEAC to be observed in these cases, the reverse of the “favored” equilibrium needs to be much slower than the rate of the overall reaction. If 9 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 favors the formation of the dimer. Ideally, therefore, the meso dimer should be formed irreversibly, for maximal amplification to be achieved. Furthermore, this practically implies the in situ resolution of the chiral catalyst. In this sense, perhaps, these cases (that are mediated by competing dimer formation) cannot strictly be considered as NLEAC. Thermodynamic Considerations The above kinetic treatment provides a model that explains the manner in which the observed NLEAC arise. However, the fundamental origin of the effects remains intriguing, as a singular thermodynamic stumbling block still needs to be addressed. This is because the phenomenon of NLEAC, in proposing that a product of higher e.e. can be obtained from a catalyst of lower e.e., implies the creation of a higher level of order from a lower one, apparently without additional input of energy. A racemate is known to possess an entropy of mixing (Smix) by virtue of the presence of both the enantiomers (eq. 18, n being the number of moles of the racemate present and R the gas constant). Smix is modest in magnitude, contributing ~ 0.4 kcals/mol in the Gibbs free energy, towards stabilizing the racemate relative to an enantiomer (whose Smix = 0). A “partial racemate”, i.e. an enantiomerically enriched mixture, would thus possess Smix in between 0 and Rln2, which would but contribute marginally in terms of the Gibbs free energy of stabilization (relative to e.e. of 100%). The relation between Smix and e.e. can be derived in a straightforward manner (eqs. 19-20, N being the total number of moles of the sample). Eq. 20 – which shows that higher e.e. implies a lower Smix – follows (cf. S.I. for details). Smix = nRln2 (18) 10 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 n = N[1-(e.e./100)] (19) Smix = N[1-(e.e./100)]Rln2 (20) Smix = NRln2(e.e./100) (21) Gmix = -TSmix (22) In the case of NLEAC, the entropy of mixing (Smix) of the reaction product would be different from the (hypothetical) case in which the relationship between the catalyst and product e.e.’s is linear. The corresponding change in the entropy of mixing (Smix) in terms of the change in the e.e. of the product relative to the catalyst (e.e.) is given by eq. 21. This leads to eq. 22, in which Gmix is the change in the standard Gibbs free energy corresponding to the change in the standard entropy of mixing of Smix (T is the absolute temperature). In the case of a positive NLEAC, the Smix would be lower relative to the linear case, so Smix would be negative and Gmix would be positive. However, the marginal magnitude of Gmix (at least at normal temperatures) belies the overwhelming stability of a racemate (e.e. = 0) relative to an enantiomer (e.e. = 100%), as is generally observed. Thus Gmix = RT ln2 (from eqs. 18 and 22, per mole of racemate), which implies that the free energy difference associated with Smix is even less than the average thermal energy (RT). However, it is noteworthy that a racemate is never known to spontaneously convert to an enantiomerically enriched form (in an achiral environment in free solution). The reason for this apparent paradox, of course, is that there are two enantiomeric species of equal energy (ignoring parity violation, deemed undetectable generally). Thus, the probability of disequilibration of a racemate to either enantiomer is identical, so 11 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 the racemate form prevails. Conversely, an observed disequilibration – defined to include any enhancement of e.e. – would represent a departure from the above norm, clearly challenging cherished assumptions in stereochemical practice! It is also noteworthy that the above thermodynamic quantities, viz. free energy, enthalpy and entropy, are scalar quantities (i.e. invariant to reflection). Hence, they cannot be used to distinguish between chiral possibilities. In the above case of enantiomers possessing identical free energy content, the entropic difference between one enantiomer and the racemate should perhaps be double that were both the enantiomers taken into account. This is because, in considering only one of the enantiomers, only half of the available energy states is being taken into account. Based on this argument, the Smix between the racemate and an enantiomer would be 2Rln2, and the corresponding Gmix = 2RT ln2 (~ 0.8 kcal/mol at normal temperatures). This is still a modest difference in free energy, although perhaps not insubstantial (being discernibly > RT). However, it does serve to bolster that argument that any “spontaneous” enhancement of e.e. needs to be justified in thermodynamic terms. It is also noteworthy that the fact that a chiral catalyst is regenerated does not confer an exemption from the above thermodynamic strictures. (The catalyst is thus analogous to a template.) The e.e. of the product would reflect the e.e. of the catalyst (in a characteristic way). As the rate constants for the formation of the enantiomeric products are identical, the Gibbs free energy contents of the enantiomeric transition states would also be identical. Thus, the enhanced e.e.’s observed in NLEAC represent a purely concentration effect. An 12 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 amplified product e.e., relative to the catalyst e.e., then implies – in effect – a correspondingly selective reaction of one of the catalyst enantiomers. However the play of catalyst e.e., i.e. relative enantiomer concentrations, on the product e.e. is explained in kinetic terms, it remains a thermodynamic enigma. The following arguments are noteworthy in this context. Kinetic Control and “Quasi-equilibrium” Even apart from the NLEAC cases, processes involving the enhancement of e.e., including the spontaneous generation of optical activity, have been well-documented (but remain controversial in some cases). Thus, several examples of second-order asymmetric transformation are known, which are crystallization-driven processes resulting in the spontaneous generation of optical activity. The enhancement of e.e. merely by recrystallization is also known; this, however, does not imply an absolute increase in the e.e., as the enhancement results from the preferential crystallization (hence does not apply to the whole sample, i.e. there is no change in the total Smix). An analog of this process, based on compound formation with an achiral but “bis-functionalized” auxiliary, is also well-exemplified. Perhaps the most intriguing and spectacular examples of enhancement of e.e., are those resulting from autocatalytic amplification reported by Soai and coworkers. These represent the only known reaction analogs of the crystallization-driven processes mentioned above. Also, the Soai reactions are capable of spontaneous generation of optical activity, i.e. without any initial enrichment. Theoretically, however, any spontaneous increase in e.e. – crystallization-driven or not – remains enigmatic (certainly so when there is a change in the total Smix). Current 13 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 approaches view these phenomena in the light of non-equilibrium theory and irreversibility. These approaches essentially make a distinction between conventional “static” stability, on one hand, and a dynamic stability achieved as a steady state in an open system, on the other. The racemate represents stability in the static sense, but can be countervailed under suitable conditions of matter flow, particularly involving the removal or recycling of product. Under these conditions, two enantiomerically-related steady states develop, the system then evolving irreversibly to one of them. Under the inexorable influence of autocatalytic loops and processes effecting mutual destruction of the enantiomers, the system careens into a state that is both dynamically stable and far from thermodynamic equilibrium. Thus, enhancement of e.e. is the result of the interplay of two seemingly incompatible conditions – stability and disequilibrium. It is interesting to note that an analogous combination is to be found in the case of kinetically-controlled reactions. These reactions are driven to completion by a highly favorable equilibrium constant, although the ratios of the various products that may be formed need not reflect their thermodynamic stability. In other words, despite the large equilibrium constant in favor of products, equilibrium between the various products themselves is not attained. This appears to represent a state of “quasi-equilibrium”, which is, of course, the result of the fact that the reverse of the overall reaction is extremely (often immeasurably) slow. Given sufficient time, however, the reaction – in principle – could be reversed, thus leading to the various products attaining mutual thermodynamic equilibrium. In the case of a reaction occurring under chiral catalysis, this would correspond to racemic products. (This may appear intriguing, but it should be noted that a catalyst – 14 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 chiral or achiral – cannot alter the equilibrium constant between enantiomers, which perforce equals 1.) Therefore, the obtention of non-racemic products under chiral catalysis implies a non-equilibrium state. This is true even if the relationship between catalyst and product e.e.’s is linear. The NLEAC case, then, would represent an extension of the non-equilbrium state (beyond the linear case), the origins of which can be addressed as below. To reiterate, the abnormally high e.e. observed in the case of NLEAC (asymmetric amplification) represents a state of apparent stability (reaction being driven to completion), although not of overall equilibrium (e.e. being non-zero). According to current theory, apparently, two conditions are necessary for the manifestation of the nonracemic state: non-equilibrium and autocatalysis. For the emergence of chiral states from a totally racemic one, autocatalysis is apparently de rigueur. However, for the transformation of a previously chiral state to another chiral state, experience indicates that autocatalysis need not be mechanistically involved. (This applies to any of the myriad examples of asymmetric synthesis, whether involving chiral auxiliaries or catalysts.) In the case of NLEAC, autocatalysis is also not a necessity. These arguments seem to indicate that the NLEAC cases represent states that are farther from equilibrium than the cases in which the catalyst and product e.e.’s are linearly related. The origins of this “extended non-equilibrium” lie both in the kinetic characteristics of NLEAC phenomena and in the nature of the non-equilibrium state. Thus, as has been discussed at length above, NLEAC arise when the overall kinetic order is greater than one. This also has an interesting consequence on the manner in which the non-equilibrium state is attained. 15 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 Thus, it is believed that a non-equilibrium state can be attained and maintained in an open system, in which there is a continuous inflow of reactants and outflow of products. This ensures that the chemical potential () of reactants remains high relative to the products, thus avoiding the state of conventional thermodynamic equilibrium (characterized by  = 0). These conditions are also conducive to the maintenance of non-racemic states. Also, a relatively high  between products and reactants, apparently, may lead to a non-equilibrium state that is correspondingly farther from equilibrium, as discussed below.   R = o(R) + RT ln([S][M][CR]) (23) S = o(S) + RT ln([S][M][CS]) (24) Now, the chemical potentials of the reactants (R and S) relating to eqs. 1 and 2 (Scheme 1) are given by eqs. 23 and 24. The o’s are the standard potentials, noting that o(R) = o(S). The fact that the reactions are second order in catalyst (CR and CS) means that R and S are also correspondingly greater, relative to the (hypothetical) first order case. Interestingly, this is analogous to the high chemical potential of the reactants in the case of open systems, which prevents a reversal of the reaction and the attainment of equilibrium. In the NLEAC cases, the reactions are essentially driven by a large equilibrium constant in favor of products, arising from a correspondingly large difference in the standard potential between products and reactants (o). The second order concentration terms, therefore, also contribute to a raised ground state, i.e. high (R + S). (This would not affect the overall equilibrium as the reverse reaction would also be second order in 16 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 the catalyst; however, the effect would be important under kinetic control, i.e. away from equilibrium.) Also, the second order dependence of the concentration terms in eqs. 23 and 24 leads to a correspondingly greater enantioselectivity, by virtue of a greater difference in the ground state energy (in terms of R and S) for the two parallel routes. The enhanced enantioselectivity in NLEAC (amplification) indicates a state of non-equilibrium, and the relatively high chemical potentials of the reactants apparently enable the phenomenon. Furthermore, the equilibrium condition of  = 0 is compatible with both a “quasiequilibrium” or total equilibrium. “Quasi-equilibrium” is achieved under kinetic control, with the product-spread being unrelated to relative stability. Total equilibrium, however, implies that the products are present in proportion to their thermodynamic stability. Thus, if there are two products P1 and P2 that are formed, the chemical potential of the product (P) is given by eq. 25, wherein P(1) and P(2) are the chemical potentials of P1 and P2 respectively. The P(1) and P(2) can be represented in terms of their standard potentials (oP(1) and oP(2)) by eqs. 26 and 27, respectively. If the sum total of the reactant (Rt) chemical potentials be Rt, this is given by eq. 28 (oRt being the corresponding standard potential). The equilibrium condition is given by eq. 29. Involving eqs. 25-27 in eq. 29 leads to eq. 30, thence by rearrangement to eq. 31.   P = P(1) + P(2) (25) P(1) = oP(1) + RT ln[P1] (26) P(2) = oP(2) + RT ln[P2] (27) Rt = oRt + RT ln[Rt] (28) 17 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2   = (P Rt) = (P(1) + P(2) Rt) = 0 (29)  = (oP(1) + oP(2) + RT ln([P1][P2])) – (oRt + RT ln[Rt]) = 0 (30) (oP(1) + oP(2)) – (oRt + RT ln[Rt]) = -RT ln([P1][P2]) (31) K = ([P1][P2])/[Rt] (32) oP = (oP(1) oP(2)) = -RT ln([P1]/[P2]) (33) The left hand side of eq. 31 is a constant at given values of [Rt] and T, noting that the o’s are constants at given standard states. The product ([P1][P2]) on the right hand side would then also be a constant, although [P1] and [P2] themselves may vary. (A similar conclusion may be reached, perhaps more simply, by involving the equilibrium constant K, as defined in eq. 32; however, the chemical potentials apparently relate better to non-equilibrium theory, as discussed above.) Eqs. 31 and 32 thus show that the fundamental equilibrium condition ( = 0, or K constant) may be satisfied for either the case of “quasi-equilibrium” or that of total equilibrium: “quasi-equilibrium” represents kinetic control, the ratio ([P1]/[P2]) reflecting only the relative rates of formation; in total equilibrium, however, ([P1]/[P2]) would reflect relative stabilities. linear cases represent departures from the equilibrium case, which must lead to racemic Also, total equilibrium implies the condition P(1) = P(2) which, along with eqs. 26 and 27, leads to eq. 33. (Thus, ([P1]/[P2]) is now a constant.) In the context of asymmetric catalysis, this implies that the product is a racemate, i.e. ([P1]/[P2]) = 1. Indeed, it is against this absolute standard that the e.e.’s observed in asymmetric catalysis, whether linear or not, have to be judged. To reiterate, both NLEAC and the 18 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 product. NLEAC, however, represents an extended departure from reversibility and equilibrium. NLEAC and Non-equilibrium It is interesting to compare NLEAC with current models of non-equilibrium processes. These models, of course, are based on open systems (with autocatalysis); the normally observed NLEAC cases are closed ones (generally without autocatalysis). However (to reiterate), the latter are driven by the irreversibility of the reactions, which apparently ensures a measure of non-equilibrium. Interestingly, the existence of three possible transition states in the NLEAC case (2 chiral and 1 meso) finds a resonance in the open system models of non-equilibrium. These, too, postulate the formation of three states, two chiral and one racemic, which coexist at the steady state. The further evolution of such systems away from the racemic state towards one of the chiral ones is believed to occur when there is a continuous flow of matter, which ensures a large difference in chemical potential between reactants and products. This, in turn, ensures irreversibility, thus avoiding the racemic equilibrium state. In the NLEAC cases, however, a naturally large difference in standard potentials ensures irreversibility (as discussed above). Also, NLEAC requires a selective traversing of the three possible transition states. This is per se a condition of non-equilibrium, as equal traverse along the three transition states would lead to racemic products: indeed, this is the analog of the equilibrium state achieved in an open system. 19 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 It is also noteworthy that the maximal departure from linearity observed in NLEAC corresponds to a catalyst e.e. of ~ 25%. This indicates a relatively enhanced sensitivity of the product e.e. towards changes in the catalyst e.e., in the low catalyst e.e. regime. This is, of course, explicable with reference to eqs. 5 and 9, the term {[CR][CS]/([CR]+[CS])} being greater at lower e.e.’s (as may be gleaned by assigning values to [CR] and [CS]). Interestingly, however, this is also reminiscent of the hypersensitivity of open systems at the bifurcation point, which is also a state with e.e. ~ 0, and marks the beginning of the non-equilibrium regime. An important difference between NLEAC and open systems is that, in NLEAC the chiral outcome is predetermined by the e.e. of the catalyst, whereas the open systems are totally achiral to begin with. Thus, NLEAC apparently corresponds to the regime beyond the bifurcation point in open systems, after chirality has evolved and a choice made between the two possible chiral states. The low e.e. regime in NLEAC would thus be analogous to an open system which has just crossed the bifurcation point, when the system is still at its most sensitive: indeed, the curve for the evolution of chirality [in terms of , corresponding to ([PR]-[PS] in Scheme 1)] is not unlike that seen for NLE es appear that NLEAC resonates to at least some of the ideas of non-equilibrium theory! AC! In fact, the former represents the evolution of  in relation to reactant chemical potential μ. Interestingly,  relates to Pee, and the μ to Cee, in the NLEAC representation (Fig. 2), noting that a higher Cee relates to a higher μ. Therefore, although the analogy between an open system and NLEAC should not be belabored, it do 20 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 CONCLUSIONS The phenomenon of non-linear effects in asymmetric catalysis (NLEAC) has been rigorously addressed by a kinetic treatment. When the chiral catalyst forms dimeric reactive intermediates, NLEAC apparently arises from the second-order dependence of the overall rates on the concentration of the catalyst. When the chiral catalyst forms nonreactive dimers reversibly, NLEAC arises from the competition between this equilibrium and the overall rates. The thermodynamic origins of NLEAC are intriguing, although they are apparently a manifestation of the “quasi-equilibrium” status of kinetic control. (“Quasi-equilibrium” possesses some of the characteristics of both equilibrium and nonequilibrium.) Analogies possibly exist between NLEAC and irreversible processes in open systems (in particular, app point has been crossed). ng the plots in Fig. 2, and to Prof. N. S ch Centre) for interest. linear effects in asymmetric Nonlinear effects in asymmetric catalysis. J Am Chem Soc 1994;116:9430-9439. arently, after the bifurcation ACKNOWLEDGMENT The Council of Scientific and Industrial Research (CSIR), New Delhi, is thanked for generous financial support. I am indebted to Mr. Ravula Thirupathi for generati uryaprakash (NMR Resear LITERATURE CITED 1. Satayanarayana T, Abraham S, Kagan HB. 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Oxford: Oxford University Press; 1995. p 230, 271-274. 23 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2 N at ur e P re ce di ng s : d oi :1 0. 10 38 /n pr e. 20 12 .6 94 7. 1 : P os te d 27 F eb 2 01 2