Hyperpositive non-linear effects: enantiodivergence and modelling

Abstract

The chiral ligand N-methylephedrine (NME) was found to catalyse the addition of dimethylzinc to benzaldehyde in an enantiodivergent way, with a monomeric and a homochiral dimeric complex both catalysing the reaction at a steady state and giving opposite product enantiomers. A change in the sign of the enantiomeric product was thus possible by simply varying the catalyst loading or the ligand ee, giving rise to an enantiodivergent non-linear effect. Simulations using a mathematical model confirmed the possibility of such behaviour and showed that this can lead to situations where a reaction gives racemic products, although the system is composed only of highly enantioselective individual catalysts. Furthermore, depending on the dimer's degree of participation in the catalytic conversion, enantiodivergence may or may not be observed experimentally, which raises questions about the possibility of enantiodivergence in other monomer/dimer-catalysed systems. Simulations of the reaction kinetics showed that the observed kinetic constant k_obs is highly dependent on user-controlled parameters, such as the catalyst concentration and the ligand ee, and may thus vary in a distinct way from one experimental setup to another. This unusual dependency of k_obs allowed us to confirm that a previously observed U-shaped catalyst order vs. catalyst loading-plot is linked to the simultaneous catalytic activity of both monomeric and dimeric complexes.

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Introduction

Non-linear effects (NLEs) in asymmetric catalysis refer to cases in which the enantiomeric excess of the product does not scale linearly with the enantiomeric excess of the catalyst.¹ The first examples and models of such behavioural differences between scalemic and enantiomerically pure catalysts were established by Kagan in 1986.² Since then NLEs are considered as ubiquitous phenomena that provide additional information regarding the aggregation state of the catalyst or the formation of multiligand species^2,3 (cf. also reviews^4,5 and some recent examples^6–9). Not only being indicative of the catalytic system, NLEs also give clues to discussions on the origin of molecular homochirality in biology which is related to the origin of life.¹⁰

Several models for NLEs have been described and discussed in the literature, all of them being the results of interactions between the enantiomers of the chiral catalyst thus generating diastereomeric perturbations of the entire system. A positive non-linear effect (i.e. asymmetric amplification, (+)-NLE) is essentially generated by the presence of a reservoir of racemic ideally catalytically inactive hetero-aggregate (meso),^11–13 although pure homochiral aggregation can also lead to (+)-NLEs in certain cases.¹⁴ Amongst these models, Kagan established a hypothetical case wherein an unprecedented phenomenon could occur – that is, the chiral catalyst [would] be much more efficient when partially resolved than when enantiomerically pure. We recently have observed such a case, known as hyperpositive NLE, in the enantioselective addition of dialkylzincs to benzaldehyde when catalysed by the chiral N-benzylephedrine (NBE) ligand.^15,16 Subsequent mechanistic investigations pointed towards a two-component catalysis where monomeric as well as homochiral dimeric catalysts are in equilibrium and in competition: both catalyse the reaction with different enantioselectivities, the dimeric catalyst being the less enantioselective one (Fig. 1). Through the precipitation of a heterochiral aggregate, variation of the ligand ee leads to a change of the overall catalyst concentration and, therefore, to a change of the monomer–dimer equilibrium. This favours the more enantioselective monomeric catalyst at low ligand ee and gives rise to the hyperpositive non-linear effect (Fig. 2a, orange crosses). These findings challenge the widely applied Noyori model for asymmetric dialkylzinc additions, where only monomers are catalytically active, and shows how complex systems with concurrent catalytic cycles can emerge from a minimum of components.^13,17

Fig. 1 Competitive reaction pathway of a monomeric and a dimeric catalyst in an enantioselective reaction that accounts for a hyperpositive non-linear effect: the case of NBE-catalysed enantioselective addition of dialkylzinc to benzaldehyde.

Fig. 2 (a) NLE-curve and (b) ee_Pvs. catalyst loading-plot of the (−)-NME- (blue dots) and (−)-NBE- (orange crosses, from ref. 11) catalyzed enantioselective addition of ZnMe₂ to benzaldehyde at 0 °C. Each point is the mean of three independent experiments; the vertical bars depict standard deviations. The second-order polynomial fits (dotted lines) serve as visual guidelines. The dashed line in (a) simulates a linear ee_P/ee_L relationship for the NME-catalysed reaction. The product ee is defined as (P_R − P_S)/(P_R + P_S).

In line with our studies on NLEs, we have explored additional ephedrine-based ligands in dialkylzinc addition reactions. The results presented in this work show that N-methylephedrine ligand (NME) follows the same model as the related NBE ligand, with monomer and dimer both catalysing the reaction, the only difference being that no meso aggregate precipitates in the case of NME. However, the homochiral dimer catalyst not only yields the product with a lower ee than the monomer catalyst, it selectively gives the opposite enantiomeric product. This enantiodivergent behaviour allowed us to switch between the preference for either enantiomer of the product, just by varying the catalyst loading. Variation of the enantiomeric excess of the ligand also allowed switching the selectivity of the product formation, leading to an enantiodivergent non-linear effect. In addition, we developed a theoretical model giving a closed mathematical expression which takes into account the concomitant catalysis by monomeric and dimeric species. From this we simulated [product ee vs. catalyst concentration] and [product ee vs. ligand ee]-plots which allowed us to gain a better understanding of the two-component catalytic system. Furthermore, the model also allowed us to analyse the reaction kinetics. Simulations showed that the observed kinetic constant of the system k_obs is indeed not unique for each catalytic system, but depends on user-controlled parameters such as the total catalyst concentration and the ligand ee. This has confirmed the hypotheses made earlier and allowed the simulation of a U-shaped catalyst order vs. catalyst-curve, which had been previously observed experimentally in the NBE-catalysed reaction.¹⁵

The paper is organised as follows: we will (1) first present the experimental results obtained from the NME ligand, showing enantiodivergence in product ee (ee_P) vs. ligand ee (ee_L)- and ee_Pvs. catalyst loading-plots, (2) then go on to a theoretical analysis where we present the mathematical model and discuss ee_Pvs. catalyst concentration- and ee_Pvs. ee_L-plots, (3) this is followed by a kinetic analysis of the system in which we investigate the relationship of k_obs with the catalyst concentration and ee_L and simulate catalyst order vs. catalyst loading-plots. (4) Finally, we will discuss the impact of the simulations in a more general way and compare them with the experimental results.

Results & discussion

Nonlinear effect in the presence of N-methylephedrine as a chiral ligand

The NME and NBE ligands only differ by the N-methyl group being changed by an N-benzyl group. Therefore, we suspected that NME might generate diastereomeric and aggregation-induced perturbations in the same way as in Fig. 1 for the same reaction. The ligand has been previously applied in asymmetric alkylations using ZnEt₂ as the reagent^18–21 but, to the best of our knowledge, ZnMe₂ has not been investigated so far. Knowing that the NBE ligand displays a more pronounced NLE with ZnMe₂ than with ZnEt₂,¹⁵ we suspected that NME might show an NLE in conjunction with ZnMe₂, even though it had been shown previously that there is no apparent NLE with ZnEt₂.²²Fig. 2a displays the correlation between the enantiomeric excess of the product (ee_P) and the enantiomeric excess of the ligand (ee_L) for the asymmetric addition of dimethylzinc to benzaldehyde using chiral (−)-NME (blue dots). Although its (1R,2S)-configuration is identical to the one of (−)-NBE, it surprisingly yielded mainly the S-product (P_S) while (−)-NBE (orange crosses) gave the R-product (P_R) under similar reaction conditions. The NME-dataset in Fig. 2a generates a negative non-linear effect [(−)-NLE], albeit in an apparently inverted manner when compared to common representations of (−)-NLEs (this is due to ee_P being defined here as the enantiomeric excess of P_R, which leads to negative ee_P values when P_S is predominant).

In parallel, we conducted a screening of the catalyst loading using enantiopure (−)-NME (Fig. 2b, blue dots). While a catalytic loading of 20 mol% (−)-NME gave the product in −11% ee (P_S being predominant), lowering the catalyst loading progressively displaced ee_P into the positive scale, giving P_R in +18% ee at 2.5 mol%. The shape of the curve was found similar to the one observed when using NBE (orange crosses), as well as the ee_P increase from 20 mol% to 2.5 mol% (NME: +31%; NBE: +33%).

Such behaviour with enantiopure NME, similar to what was observed with the NBE ligand, suggests that both systems follow an analogous catalytic scheme, where both monomeric and dimeric complexes are catalytically active. However, in contrast to NBE the catalytic runs with scalemic NME ligand were homogeneous with no apparent precipitate in the reaction mixture. This may account for the observed differences in NLE curves in Fig. 2a. While the hyperpositive NLE with NBE ligand was caused by the precipitation of the RS-dimer (leaving only R and RR complexes in solution and allowing high asymmetric amplification), the (−)-NLE with NME ligand might be the result of the presence of all catalytic species in solution (including S, SS and the meso dimer RS).

Moreover, the observed negative ee_P at high catalyst loading unambiguously indicates that the dimeric NME-catalyst is not only less enantioselective, it even yields mainly the opposite enantiomeric product compared to the monomeric catalyst. Catalysis in which both enantiomeric products may be obtained from the same catalyst enantiomer is called enantiodivergent catalysis. It has attracted a lot of attention as a means to access easily both product enantiomers. Over the last two decades, many examples have been reported in which slight changes of the catalyst (substituent, metal, counter-ion, etc.) or of the reaction conditions (solvent, temperature, additive, ligand-to-metal ratio) have inverted the stereochemistry of the product.^23–26 In our case, the switch which allows toggling between both product enantiomers is the catalyst concentration, a factor which has not been discussed in this context so far.²⁷

These results made us wonder whether it is possible to switch the product enantiomer's sign within an NLE curve, i.e. by varying ee_L instead of the catalyst loading. Such “enantiodivergent non-linear effect” has already been observed in other catalytic systems,^28–30 but their origin has never been studied; even the possible presence of two catalysts yielding opposite enantiomers has not been proposed.³¹ Therefore, we continued our studies on NME by performing a temperature screening of the catalysis with the enantiopure ligand (Fig. 3a, blue dots) or only 50% ee_L (orange triangles). At low temperature (0 °C), the product ee of the enantiopure ligand was well below the ee_P of the scalemic sample, both being negative. Increasing the temperature increased ee_P in both cases but not in the same manner: the difference between 100% and 50% ee_L decreased progressively. At 40 °C the enantiopure ligand even surpassed the 50% ee_L-sample.

Fig. 3 (a) ee_P as a function of the reaction temperature (blue dots: 100% ee_L; orange triangles: 50% ee_L) and (b) NLE at room temperature of (−)-NME (blue dots) and (+)-NME (red squares) of the NME-catalysed enantioselective addition of ZnMe₂ to benzaldehyde. Each point is the mean of three different experiments; the vertical bars depict standard deviations. The second-order polynomial fits (dotted lines) serve as visual guidelines. The product ee is defined as (P_R − P_S)/(P_R + P_S).

However, an interesting situation occurred at room temperature (20–25 °C): here, (−)-NME in 50% ee gave positive ee_P-values (i.e. R product), while the enantiopure ligand stayed negative (i.e. S product): this is nothing but the requirement for an enantiodivergent non-linear effect. The full NLE curve at room temperature (Fig. 3b, blue dots) confirmed this observation: the curve starts at 0% ee_L in the positive ee_P-range, reaches a maximum, then falls down to cross the ee_L-axis (at ee_L of ca. 80%) and ends up in the negative part of the ee_P-scale. The use of (+)-NME (red squares) gave the appropriate mirror image of this curve. Thus, going to room temperature changes the (−)-NLE to a hyperpositive NLE, which in addition is also enantiodivergent.

Model studies: product ee vs. catalyst concentration (enantiopure system)

In the past, non-linear effects have been simulated and quantified using mathematical expressions of the considered models, such as the Kagan ML_n models (catalysis by the aggregate)¹¹ or the Noyori model (catalysis by the monomer, dimer acting as inactive reservoirs).¹³ These models have been further extended (electron-rich substrates,³² product inhibition³³ for the Noyori model; monomers as reservoir species with catalytically active homochiral aggregates by Kagan¹⁴). However, to the best of our knowledge there has been no attempt to unify both approaches and to consider both monomers and dimers being simultaneously catalytically active.³⁴ In order to fill this gap and to get a better understanding of hyperpositive and, in particular, enantiodivergent NLEs, we developed mathematical models which allow us to simulate ee_Pvs. catalyst concentration- and ee_Pvs. ee_L-plots. We begin with Model I, which is based on an enantiopure system where an enantiopure ligand reacts with a metal to give monomeric and dimeric homochiral complexes (R and RR, respectively) both of which catalyse the reaction at different rates (k₁ and k₂) and with different enantioselectivities (ee₁ and ee₂), as shown in Fig. 4; [Cat_tot] represents the total catalyst concentration. For the sake of simplicity, we assume that both R and RR-catalysts follow a similar mechanism with a rate law of type −d[Sub]/dt = k_i[Cat_i][Sub][Rea] (with k_i and [Cat_i] being the respective rate constants and catalyst concentrations, [Sub] and [Rea] the substrate and reactant concentrations; all species are first-order). We also assume that the [RR]/[R]-ratio stays constant over the course of the reaction and depends only on the homochiral dimerization constant K_Homo.³⁵ This makes Model I reminiscent of Kagan's ML_n model, which also considers the ratio between different catalytic species to be constant over time. The case of a time-dependent [RR]/[R]-ratio will be discussed at the end of this study.

Fig. 4 Schematic representation of Model I, which consists of a monomeric (R) and a dimeric (RR) enantiopure catalyst both of which operate at a steady state and are linked through the equilibrium constant K_Homo. The catalysts are issued from the reaction of a metal salt (M) with a chiral, enantiopure ligand (L_R) and promote the reaction of a substrate (Sub) and a reactant (Rea) to form a chiral product with the overall enantiomeric excess ee_P. The R- and RR-catalysts yield a product with a rate constant of k₁ and k₂, respectively, and with an enantioselectivity of ee₁ and ee₂.

By combining the set of equations displayed in Fig. 4, it was possible to obtain eqn (1) and (2) which relate ee_P to the parameters k₂/k₁, ee₁, ee₂, K_Homo and [Cat_tot], and allowed us to compute ee_Pvs. [Cat_tot]-curves. Fig. 5a displays the evolution of ee_P for selected values of ee₂ with fixed values of K_Homo, k₁ and k₂ (k₂/k₁ was set to 1, ee₁ to 100% and K_Homo to 33, which corresponds to the association constant of DAIB-ZnMe).^13,36 The graphs show that an enantiodivergent behaviour can indeed be observed by varying [Cat_tot], as long as ee₂ has a sufficiently low and negative value. This is further favoured by high values of K_Homo (Fig. 5b) and k₂/k₁ (Fig. 5c): the more the dimeric complex prevails over the monomer and the higher its relative activity, the more the curve becomes hyperbolic, pushing the point at which it crosses the [Cat_tot]-axis (i.e. ee_P = 0, labelled [Cat_tot]⁰) to lower [Cat_tot]. [Cat_tot]⁰ corresponds to an overall catalytic system where R and RR catalysts compensate each other to yield an overall racemic product – even if both give independently enantiopure products.

Fig. 5 Simulation of the relationship between ee_P and [Cat_tot] according to eqn (1) and (2). The basic set of parameters is ee₁ = 100, ee₂ = −100, K_Homo = 33 and k₂/k₁ = 1. Each panel shows curves where one of the parameters has been varied: (a) ee₂, (b) K_Homo, (c) k₂/k₁. The product ee is defined as (P_R − P_S)/(P_R + P_S).

Model studies: product ee vs. ligand ee (scalemic system)

In order to also simulate NLE curves, we expanded Model I to non-enantiopure ligands, as shown in Fig. 6. The resulting system (Model II) now includes the catalytic species S and SS (which, like their enantiomeric counterparts, are linked through K_Homo and catalyse with the kinetic constants k₁ and k₂) and also the heterochiral dimer RS, which is related to R and S through the dimerization constant K_Hetero and may generate racemic products with a rate constant k₂′. Thus, a single scalemic ligand gives rise to 5 different catalytic species. To derive closed mathematical expressions, we followed the approach used by Noyori for the DAIB-model which consists in introducing α = [R] + [S] and β = [R][S] to simplify the equations. ee_P and ee_L are then given by eqn (3) and (4) as functions of α and β. Since β is itself a function of α [cf.eqn (5)], ee_P and ee_L are linked through α and depend only on the parameters K_Homo, K_Hetero, k₁, k₂, k₂′, ee₁, ee₂ and [Cat_tot]. After defining these parameters, ee_Pvs. ee_L-datasets could be obtained by choosing appropriate values for α (cf. ESI Methods† for the details of the calculations and the general expressions for the upper and lower limits of α). For this study, we will focus on the cases where the NLE is hyperpositive and potentially enantiodivergent – that is with K_Hetero > 2K_Homo which is, as in the Noyori model, a necessary condition to obtain (+)-NLEs – and ee₁ > ee₂.¹³Fig. 7 shows several cases computed from Model II. To simplify the discussion we have set k₂′ = 0 in all simulations except in Fig. 7f.

Fig. 6 Schematic representation of Model II, which consists of monomeric (R, S) and both homo- (RR, SS) and heterochiral (RS) dimeric catalysts that all operate at a steady state and are linked through the equilibrium constants K_Homo and K_Hetero. The catalysts are issued from the reaction of a metal salt (M) with a mixture of both ligand enantiomers (L_R and L_S) with an enantiomeric excess of ee_L. They promote the reaction of a substrate (Sub) and a reactant (Rea) to form a chiral product with the overall enantiomeric excess ee_P. The monomeric, the homochiral dimeric and heterochiral dimeric catalysts yield a product with a rate constant of k₁, k₂, and k₂′, respectively, and with an enantioselectivity of ee₁ and ee₂ (R and RR) or -ee₁ and -ee₂ (S and SS). The RS dimer yields a racemic product (ee = 0).

Fig. 7 Simulation of NLEs with Model II, varying parameters (a) K_Homo, (b) [Cat_tot], (c) k₂/k₁, (d) ee₁ and ee₂, (e) K_Hetero and (f) k₂′/k₁. Fixed parameters: [Cat_tot] = 0.11, k₂/k₁ = 1, ee₁ = 100, ee₂ = −100 and k₂′/k₁ = 0 in all curves except where the corresponding parameter is varied; K_Homo = 33 (b, c, e), 100 (d) and 130 (f); K_Hetero = 330 000 (a, d, f), 100 000 (b) and 33 000 (c). The product ee is defined as (P_R − P_S)/(P_R + P_S).

Influence of K_Homo, [Cat_tot], k₂/k₁ and ee_1/2. A non-linear effect is hyperpositive as long as the highest product ee (labelled ee^max_P) is different from the ee_P for the enantiopure ligand (ee_P¹⁰⁰). In Model II, ee_P¹⁰⁰ will be strongly dependent on K_Homo, [Cat_tot] and k₂/k₁ (Fig. 7a–c): higher the K_Homo, [Cat_tot] or k₂/k₁, lower the ee_P¹⁰⁰. This is consistent with a higher proportion and a higher activity of the low ee_P-yielding RR-catalyst over its monomeric counterpart.

In all representations in Fig. 7, we selected conditions in which ee_P¹⁰⁰ could be negative and where the NLE curve crosses the ee_L-axis, making it an enantiodivergent NLE. Lower the ee_P¹⁰⁰, lower the crossing point at which ee_P = 0 (ee⁰_L). At this point, the outcomes of all catalysts compensate each other to yield a racemic product. ee^max_P diminishes as ee_P¹⁰⁰ decreases; the maximum's ee_L-value (ee^max_L) is only slightly affected by K_Homo and, to a somewhat greater extent, by [Cat_tot] and k₂/k₁. This is seen nicely if, for a given set of parameters, K_Homo is multiplied and [Cat_tot] divided by the same value: ee_P¹⁰⁰ remains unchanged but ee^max_P and ee^max_L do not (cf. ESI Fig. 1†). Fig. 7d shows the impact of ee₁ and ee₂ on ee_P¹⁰⁰: lower the ee₂, lower the ee_P¹⁰⁰. This is also true for ee₁; however, if both ee₁ and ee₂ are negative, the enantiodivergent NLE curve becomes a classic (+)-NLE for the S-product (P_S, Fig. 7d, dashed curves); the same holds for the R-product (P_R) if ee₁ ≤ ee₂ and if both are positive. In a similar way, very high K_Homo-, [Cat_tot]- or k₂/k₁-values lead to apparent (+)-NLEs as ee^max_P and ee^max_L become exceedingly close to 0. Lowering the absolute amount of both a positive ee₁ and a negative ee₂ leads to a compression of the spectra (cf. ESI Fig. 2†).

Influence of K_Hetero and k₂′. In contrast to the previously discussed parameters, an increase in K_Hetero (Fig. 7e) does not affect ee_P¹⁰⁰ but has a great impact on the hyperpositive maximum, which is shifted to higher ee^max_P and lower ee^max_L values. Consequently, the ee⁰_L is shifted to higher ee_L values under the same conditions. However, this is only true if the meso dimer is catalytically inactive: RS performing racemic catalysis (k₂′/k₁ ≠ 0, Fig. 6f) leads to the inverse effect, namely a compacting of the curve. The values of ee_P¹⁰⁰ and ee⁰_L remain unchanged, the latter being an isobestic point. The value of ee^max_P decreases significantly even at low k₂′/k₁ as the concentration of RS at low ee_L is particularly high. k₂′/k₁-values higher than k₂/k₁ (k₂′/k₁ > 1 in Fig. 7f) additionally lead to a contraction of the curve between ee⁰_L and ee_L = 100 and push its appearance towards a classical (−)-NLE. As in Kagan's ML₂-model, a high activity of the meso catalyst leads to asymmetric depletion.

Dynamic properties of models I and II

Apart from simulating ee_Pvs. ee_L-curves, NLE models have also been used to study the kinetic properties of the systems. This can be useful as some kinetic features may be characteristic for one or the other model, and thus be used as an additional probe to support the validity of a model for a specific catalytic reaction. Blackmond showed this in conjunction with the Kagan and Noyori models;^37,38 Micheau and co-workers even built a toolset, using the different kinetic properties of both systems, which allows an easy distinction between monomer- and dimer-catalysed enantioselective systems (which is not necessarily possible on the basis of ee_Pvs. ee_L-curves alone)³⁹ and verified the origin of the non-linearity in the Noyori model by a kinetic system based on differential equations.⁴⁰ This prompted us to extend our work on Models I and II to a kinetic study.

The mathematical expressions of the rate laws based on Models I and II turned out to correspond to a standard second-order rate law (eqn (6) and (7)), albeit with a more complex term for k_obs (cf. ESI Methods† for the calculation details). For aggregate-free catalysed reactions, with 1^st order in catalyst, k_obs is the product of [Cat_tot] and the rate constant k₁ (eqn (8)). According to Model I, where the monomer and dimer coexist and both of which are catalytically active, k_obs depends on the parameters previously discussed: the (now absolute instead of relative) rate constants k₁ and k₂, K_Homo and [Cat_tot] (eqn (9)). This holds for chiral, enantiopure catalysts as well as for achiral ones, since we consider only the system's kinetic behaviour and not a possible product ee. Non-enantiopure catalytic systems following Model II additionally depend on ee_L, k₂′ and K_Hetero (eqn (10)). For the sake of simplicity, we will consider here only the second-order rate law where the substrate and reactant concentrations are equal ([Sub] = [Rea]), however the following discussion applies also to its more general form where both differ from each other.

General second-order rate law (if [Sub] = [Rea]):

Integrated form of the second-order rate law:

If only monomers exist:

k_obs = k₁[Cat_tot] (8)

If monomers and dimers catalyse (Model I, achiral or enantiopure catalyst):

If monomers and dimers catalyse (Model II, chiral catalyst with any ee_L):

with and

The interesting point here is that [Cat_tot] and ee_L are user-controlled parameters, whose variation leads to singular changes in k_obs. This can be seen in simulated k_obs/[Cat_tot] vs. [Cat_tot]-plots: in the case of aggregate-free catalysed reactions (cf.eqn (8)) such a plot results in a flat line, whose y-intercept is equal to k₁ (Fig. 8a, blue line). The same is observed for Model I (eqn (9)) in a special case that is when k₂/k₁ = 2: the loss of monomeric catalyst upon increase of [Cat_tot], because of dimeric aggregation, is then perfectly compensated by the higher activity of the dimer catalyst. Otherwise, an increase in [Cat_tot] leads to a significant change of k_obs/[Cat_tot]: it increases if k₂/k₁ > 2 (grey dashed line) or decreases if k₂/k₁ < 2 (orange dotted line), which is symptomatic of the changing [RR]/[R]-ratio. At very low [Cat_tot] the amount of RR-catalyst becomes negligible and k_obs/[Cat_tot] becomes equal to k₁; on the other hand, k_obs/[Cat_tot] = 0.5k₂ at very high [Cat_tot] because of the prevalence of the dimeric catalyst. Thus, k_obs/[Cat_tot] varies over changing [Cat_tot] even if R and RR catalyse with the same rate (k₁ = k₂, green dashed/dotted line). With non-enantiopure ligands k_obs/[Cat_tot] varies in a similar way to that in Fig. 7a and depends, in addition, also on k₂′ (cf. ESI Fig. 3† for a commented example with ee_L = 0).

Fig. 8 (a) Simulated k_obs/[Cat_tot] vs. [Cat_tot]-plots computed from eqn (9) (K_Homo = 1000, k₁ and k₂ as indicated) and (b) simulated k_obsvs. ee_L plots computed from eqn (10), (4) and (5) (K_Homo = 30, K_Hetero = 3000, [Cat_tot] = 0.16, k₁ = 2.5, k₂ and k₂′ as indicated). The full blue lines in a) and b) can also be obtained from eqn (8) (i.e. no aggregates are present in the system, the monomer is the only catalyst) using the indicated k₁ value.

The other user-controlled parameter, ee_L, also influences k_obs (at constant [Cat_tot]) as seen in k_obsvs. ee_L plots (Fig. 8b, computed from eqn (10)). k_obs is constant if k₂/k₁ = k₂′/k₁ = 2 (blue full line). The case of k₂/k₁-values higher than 2 leads to an increase in k_obs especially at high ee_L, where the concentration of the homochiral dimers is also higher, and results in a positive slope (orange dotted line). On the other hand, increasing k₂′/k₁ gives a negative slope as it affects k_obs mostly at low ee_L, where the proportion of RS-dimers is highest (grey dashed line). A simultaneous increase of k₂′/k₁ and k₂/k₁ by the same amount also yields a negative slope if K_Hetero > 2K_Homo (green dashed/dotted line). If the K_Hetero/K_Homo-relationship is inverted then a positive slope is obtained, cf. ESI Fig. 4.†

In our previous study, we also determined the catalyst order c of the NBE-catalysed reaction when considering the system to follow the rate law −d[Sub]/dt = k[Cat_tot]^c[Sub][Rea].¹⁵ For this, we had determined the catalyst order c using Variable Time-Normalised Analysis (VTNA)^41–44 of rate profiles obtained from enantiopure NBE at different catalyst loadings. Plotting catalyst order c vs. catalyst loading gave an unusual U-shaped plot which we postulated to originate in a [Cat_tot]-induced change of k_obs. The present kinetic model now allows us to verify this assumption, since Fig. 8a shows that k_obs does indeed change with varying [Cat_tot]. To this end, we generated sets of rate profiles from eqn (7) and (9) by varying [Cat_tot] and leaving all other parameters unchanged. Then, c was determined from two different rate profiles at a time using VTNA; the results are shown in Fig. 9. In concordance with the previous discussion, k₂/k₁ = 2 gives a constant c value while varying the catalyst loading, since k_obs is constant in that special case. k₂/k₁ < 2 gives a U-shaped curve, here with a minimum at c ≈ 0.7 (blue dots) and similar to the experimental one obtained from NBE. On the other hand k₂/k₁ > 2 gives a bell-shaped curve with a maximum at c ≈ 1.5 (grey triangles). Increasing K_Homo displaces the curve's extremum to lower catalyst loading values (green crosses).

Fig. 9 Simulated c vs. catalyst loading plot of a catalytic system following the kinetics of Model I and treated as if following the rate law −d[Sub]/dt = k[Cat_tot]^c[Sub]^a[Rea]^b, with c as a variable partial order in catalyst, with undetermined rate constant k and partial substrate/reactant order a and b. Each datapoint relates the c-value obtained from two Model I-rate profiles via VTNA (from eqn (7) and (9), with [Sub]₀ = 0.833, k₁ = 2.5, K_Homo and k₂ as indicated, var. [Cat_tot]) with the mean of their catalyst loading values; the procedure is described in the ESI Methods†. The full lines are free-hand drawings which serve as visual guidelines.

Discussion

Models I and II, although kept conceptually simple, allow us to gain a better understanding of several aspects of the NME- and NBE-catalysed reactions, which we will discuss in this section. Globally, the shape of the experimental curves (ee_Pvs. catalyst loading as well as ee_Pvs. ee_L, Fig. 2 and 3) corresponds well with the computed curves in Fig. 5 and 7. Previously published graphs from reactions with NBE and ZnEt₂ as the dialkylzinc reagent,¹⁶ giving a much weaker hyperpositive NLE than ZnMe₂, also concord with the simulated ones in this study. The simulations have also given insights into the differences between the NME- and NBE-based catalysts: the overall lower ee_P-values with NME may be due to lower ee₁ or ee₂ values, as well as to higher K_Homo and/or k₂/k₁. The slight hyperpositive (and enantiodivergent) NLE with NME at room temperature (Fig. 3b) may arise from K_Hetero being only slightly higher than 2K_Homo, since pronounced hyperpositive NLEs were simulated only when K_Hetero and K_Homo are highly different (cf.Fig. 7e). At 0 °C (Fig. 2a), NME even generates a (−)-NLE which precludes that upon cooling, K_Hetero falls below the value of 2K_Homo.

In the case of NBE (Fig. 2a, orange crosses), the strong hyperpositive curve arises from the precipitation of the heterochiral dimer, which has an effect similar to an increase of K_Hetero. The simulations in Fig. 7 also reveal that, even if the dimeric catalyst gives the opposite product enantiomer in its pure form (ee₂ = −100%), the NLE does not need to be enantiodivergent as long as the dimer's participation in the catalytic process is sufficiently low (due to low K_Homo, [Cat_tot] or k₂/k₁, as seen in Fig. 7a–d). Therefore, dimeric NBE-ZnMe might yield the opposite enantiomer as well, even though its NLE and ee_Pvs. catalyst loading-plot (Fig. 2, orange crosses) don't show direct evidence for that. This possibility is particularly interesting as it raises questions whether enantiodivergence between monomeric and dimeric catalysts might be a systematic phenomenon. There are various cases of 2 : 1 ligand-to-metal complexes which have been reported to yield the other product enantiomer than their 1 : 1 counterparts,^45–53 Seebach's TADDOL-Ti-complexes being the first (but largely unnoticed) ones.^46–48 2 : 1 complexes can be considered as analogues of dimeric complexes, both are even equivalent in Kagan's ML₂ model.¹¹

Also, the models suggest that it is possible to obtain racemic products when working at a certain catalyst concentration ([Cat_tot]⁰, cf.Fig. 5) or ligand ee (ee⁰_L, cf.Fig. 7), although the system consists of only highly enantioselective catalytic species. This is well exemplified with the results from NME-catalysed reactions at room temperature (Fig. 3b): the only 2% ee_P yielded by the enantiopure catalyst may let one think that NME is a totally non-enantioselective catalyst in this reaction, which is definitely not the case.

Furthermore, the kinetic models show that the evolution in the ratios between the different catalytic species, when changing reaction parameters like [Cat_tot] or ee_L, results in a variation of the rate constant k_obs. This allowed us to confirm our previous assumption that a U-shaped curve for the catalyst order c vs. catalyst loading plot is consistent with Model I, assuming the monomer has more than half the catalytic activity of the dimer (k₂/k₁ < 2). However, this feature also shows the limitations of Model I and II: the simulated U-shaped c vs. catalyst loading-curve (Fig. 9) spans over several orders of magnitude of catalyst loading, whereas the experimental one obtained from NBE spans only from 20 to 2.5 mol%.¹⁵ For second order kinetic laws, plotting 1/[Sub]_tvs. time gives a linear graph, however the corresponding experimental plots from NBE- and NME-catalysed reactions are mostly curved (cf. ESI Fig. 5†). This shows that k_obs is not constant over time, most probably because of a change of the catalysts' concentrations (and their respective ratios) as the reaction goes on. It is known for DAIB-catalysed reactions that the reactants and, in certain cases, the reaction products have an influence on the catalyst composition, which results in altered reaction rates and non-constant ee_P over time.^13,33 Indeed, an increase of ee_P over the course of the reaction was observed in the NBE-catalysed reaction.¹⁵ Therefore, further studies (and quantitative analyses in particular) of NME- and NBE-catalysed dialkylzinc additions will require more complex models, which include the concentrations of the starting materials and of the products and their possible influence on the catalyst distribution.

Conclusion

In summary, we have found an enantiodivergent behaviour in NME-catalysed reactions which may be explained by the monomer–dimer competition model, known for the parent ligand NBE. Simulation allowed us to confirm this and to get further insights into the model, such as the dependence of the observed kinetic constant k_obs on the catalyst concentration and on the enantiomeric excess of the ligand. We expect this may serve as a basis for further research to understand past and future cases of enantiodivergent non-linear effects, as well as of asymmetric catalytic systems in general. Further work in our group will aim at the elucidation and the quantification of the different parameters governing reactions catalysed by ephedrine-derivatives, at the extension of the models in order to include the influence of reactants and products and at the investigation of aggregation-induced complex catalytic systems in a general way.

Data availability

Synthetic and mathematical procedures, as well as additional data and chromatograms for every experimental catalytic run can be found in the ESI.† Detailed experimental data (reactant quantities, reaction conditions, raw and treated results for all catalytic runs, including reactions with NBE) and calculated data for all simulated graphs in this paper can be found in the respective excel files (SI_experimental_Data.xlsx, SI_Simulated_NLE.xlsx, SI_kinetic_model.xlsx).

Author contributions

Y. G. performed the synthetic experiments and developed the catalytic models, along with their mathematical expressions. A. M.-F. and T. A. participated in data analyses. S. B.-L. conceptualized and supervised the study and wrote the manuscript with Y. G.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was funded by the French National Research Agency (ANR) through the Programme d'Investissement d'Avenir under contract ANR-11-LABX-0058_NIE within the Investissement d'Avenir program ANR-10-IDEX-0002-02.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0sc04724d

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