Absence of Non-Linear Effects despite evidence for catalyst aggregation
An in-depth study of the catalytic system, consisting of the enantioselective addition of ZnEt 2 to benzaldehyde with (1R,2S)-(-)-N-Methylephedrine (NME) as chiral ligand, suggests the presence of dimeric and trimeric aggregates, as deduced from product ee vs catalyst loading and NMR investigations ( 1 H, DOSY). Formation of catalyst aggregate was excluded in earlier studies as this system displays a linear product ee vs ligand ee-correlation, which is usually taken as an indication for the absence of catalyst aggregation. A subsequent theoretical study, using the monomer-dimer competition model we recently have developed, highlights the possible parameter configurations leading to linear product ee vs ligand ee plotsdespite the presence of catalyst dimers. It shows that, while the Kagan and Noyori models allow linearity in very specific cases only, a multitude of scenarios may lead to linearity here, especially if heterochiral dimers are catalytically active.
INTRODUCTION<!>RESULTS AND DISCUSSION<!>α and β (with α = [R]+[S] and β = [R][S]<!>A case study: the enantioselective addition of ZnEt 2 to benzaldehyde catalysed by (-)-NME.<!>and ee 2 , respectively) and relative kinetic constants (k 2 /k 1 , k 2 '/k 1 ), the total catalyst concentration [Cat tot ] and the dimerization constants K Homo and K Hetero . As in the Noyori model, α and β (with α = [R]+[S] and β = [R][S]<!>Modelling the absence of NLEs<!>The contribution depends on the different catalyst concentrations ([R], [S], [RR], [SS] and [RS]<!>Figure 8. Screening of the necessary KHetero/KHomo-ratio to obtain linear eeP vs eeL-plots at different k2/k1 ratios, a) in conditions of low catalyst aggregation (low KHomo, low [Cattot]) and b) high catalyst aggregation (high KHomo, high [Cattot]).<!>Conclusion<!>Data availability<!>Table of Contents Graphic
<p>Non-linear effects (NLEs) appear in asymmetric catalytic reactions when the product ee (ee P ) does not scale linearly with the catalyst ee (ee L , as the chirality is often carried by a chiral ligand attached to a metal). [1][2][3][4] In a positive non-linear effect (or (+)-NLE) ee P can become much higher than ee L and lead to asymmetric amplification (Figure 1). On the other hand, a small reduction of the ligand's enantiopurity can also lead to a depleted ee P (negative NLE, (-)-NLE). The concepts and first experimental examples of NLEs in asymmetric catalysis have been reported by Kagan and co-workers in the mid-80's and early 90's. [1,2] Their work followed studies on the influence of a compound's enantiomeric purity on its physical properties by Horeau and Guetté [5] and on its chemical reactivity in stoichiometric reactions by Wynberg and Feringa in the mid-70's. [6] It should also be noted that already in 1936 (!) Langenbeck and Triem reported a NLE in the double addition of menthol to oxalyl chloride, along with the conceptual basics for asymmetric amplification and an outlook to catalytic reactions. [7,8] Nonlinear enantioselectivity behaviours are also widely discussed in the frame of understanding the origin of biological homochirality, in which (+)-NLEs might have played a role. [9] Figure 1. Scheme for typical product ee (ee P ) vs ligand ee (ee L )-relationships. Non-linear effects (here, (-) or (+)-NLE; hyperpositive NLEs are not shown) are indicative of catalyst aggregation, which leads commonly to the assumption that the absence of NLE goes hand in hand with the absence of catalyst aggregation.</p><p>Although the concepts of non-linear effects are also valid in stoichiometric reactions and kinetic resolution processes, they have been mostly studied in catalytic reactions. This is mainly due to their usefulness in deciphering catalytic reaction mechanisms: the presence of a NLE is generally considered to be a footprint of catalyst aggregation or catalysis by multiligand-complexes. [3,4] Building a NLE curve requires nothing more than having both enantiomers of the ligand and the analysis is allegedly simple, as one may conclude in a binary way: if there is an NLE there is also catalyst aggregation ; if not, there is no aggregation. Therefore, the use of NLEs as a probe has gained a lot of success in the study of asymmetric catalytic mechanisms, especially when other analytical tools failed. [10] Although this approach leads to correct conclusions in most cases, it may also lead to false interpretations as it consists in the simplification of a complex system. This has been recently highlighted by Fu and Kalek who reported a special case of kinetic resolution, where an intrinsic (-)-NLE was observed without any catalyst aggregation. [11] In this paper, we wish to discuss a specific case complementary to that of Fu and Kalek, [11] that is an asymmetric catalytic system in which the catalyst does aggregate although a linear ee P /ee L -relationship was observed. We will first discuss how such a situation may arise according to the Kagan and Noyori models for NLEs. Then we will discuss experimental evidence for such a case, namely in the catalytic asymmetric addition of ZnEt 2 to benzaldehyde in presence of the N-methylephedrine (NME) ligand. Since this catalytic reaction follows the model we recently exposed, [12][13][14] we will use plot simulations to explore possible scenarios leading to linearity. Overall, the present contribution shows that the study of non-linear effects alone should be taken with much more caution as it can lead to false interpretations in particular if one takes into account linearity as an indicator of nonaggregation.</p><!><p>Aggregation without NLE: known theoretical cases. In 1986, Henri Kagan and co-workers established a model for non-linear effects in asymmetric catalysis which proved helpful in analysing experimental data. [1,2] They proposed a model where the non-linear behaviour is the result of the formation of catalytic species that involve several chiral ligands (i.e. ML n model). The simplest case would be assembling two chiral ligands within a complex (called ML 2 model) or merely a dimerization of a complex, i.e. (ML) 2 . In such system, three stereoisomers are formed as shown in Figure 2. Two homochiral complexes (RR and SS, which give rise to products with ee 2 and -ee 2 , respectively) and one heterochiral complex (RS, which yields racemic product) are present, the chiral product is thus obtained from the three catalysts that operate simultaneously. Therefore, the enantiomeric excess of the product [ee P , eq. ( 1)] will depend on the ligand ee (ee L ), on the relative reactivity g [eq. ( 2)], with k 2 and k 2 ' as the corresponding kinetic constants) and on the relative catalyst concentration β (with [RR], [SS] and [RS] as the individual catalyst concentrations). β is further expressed as a function of ee L and the equilibrium constant K [eq. ( 3)]. Thus, equations ( 1)-( 3) form a function of type ee P = f(ee L ) with g and K as fixed parameters.  1), ( 2) and ( 3) that allow to simulate ee P vs ee L -curves. The overall ee P will depend on ligand ee (ee L ), the enantioselectivity of the homochiral dimer (ee 2 ), the relative amounts of the catalysts present ([RR],</p><p>[SS] and [RS], expressed through the equilibrium constant K) and their relative reactivity g. Note: the nomenclature of all constants in this paper has been unified for the sake of clarity and thus differs from Kagan's original notation.</p><p>The fraction in eq. ( 1) is the factor that introduces non-linearity into the system, as it depends itself on ee L . However, eq. ( 1) becomes a linear function if that fraction is equal to 1, which may occur in two distinct cases: (i) when no heterochiral dimers are generated (i.e. β = 0) or (ii) when hetero-and homochiral dimers catalyse at the same rate (i.e. g = 1). In these two cases, complex aggregation occurs but it does not influence the ee P vs ee Lrelationship, which thus remains linear.</p><p>While cases where only homochiral species can form are not that common, a reactivity of the same order of both the homo-and heterochiral catalysts wouldn't be surprising to occur in some systems. For example in the case of dimers of chiral metal complexes that act as Lewis acid catalyst: if the chiral environment does not significantly influence the reactivity of the metal, the Lewis acidity at the metal centre of both dimers (homo and meso species) may be fairly close.</p><p>In 1989, Ryoji Noyori and co-workers reported the enantioselective addition of dialkylzincs to aldehydes promoted by the chiral DAIB ligand and observed unprecedented strong (+)-NLEs. [15] A few years later, they published a complete mathematical treatment and quantitative analysis of the phenomena, where they demonstrated that the chiral amplification is the consequence of several structural, thermodynamic and kinetic parameters. [16] Here, the (+)-NLEs originate from the reversible formation of homochiral and heterochiral dimers of the enantiomeric catalysts (Figure 3) and can be modelled with equations ( 4)- (6). In contrast to Kagan's ML 2 model, the reaction is catalysed by monomeric complexes while dimers (both homochiral and heterochiral) are inactive and in a dynamic equilibrium with monomers. Here, the NLE is governed only by the relative stabilities of the dimeric species (i.e. dimerisation constants, K Homo and K Hetero ), which in turn alter the R/S ratio of the catalytically active monomers. A (+)-NLE occurs when the heterochiral dimer is more stable than the homochiral one (i.e. K Hetero > 2K Homo ). On the other hand, a (-)-NLE occurs when the homochiral dimers are favoured (i.e. K Hetero < 2K Homo ). In between, the particular case of K Hetero = 2K Homo (what Henri Kagan called "a statistical distribution of ligands" in the ML 2 model, [2] see also section I.2.4 in the Supporting Information) was highlighted as the "branching point" of the system: the ee P vs ee L -relationship is strictly linear here. Dimerization takes place, but in a way that it doesn't influence the original ratio of the monomeric R-and S-catalysts. It should be noted that the Noyori model takes also into account the association of the catalyst with the reagent and the substrate (with the association constant K assoc ) prior to the actual enantioselective step. Thus, a system may exhibit linearity -or weak nonlinearity -if the catalyst concentration is very low or if the substrate and/or reagent concentration is very high: the latter consume the free monomeric complexes and thus drive dimer dissociation. The absence of NLEs in these particular experimental conditions indeed concords with the absence of dimers.  4), ( 5) and ( 6) that allow to simulate ee P vs ee L -curves. The overall ee P is a consequence of the R/S-ratio of monomeric catalyststhe dimers are inactive -which depends on the ligand ee (ee L ), the catalyst's enantioselectivity (ee 1 ), the dimerization constants K Homo and K Hetero , the total catalyst concentration [Cat tot ] and the catalystsubstrate-reactant association constant K assoc . The kinetic constant k 1 does not influence ee P .</p><!><p>) are working variables introduced to simplify the expressions. Furthermore, α links ee P and ee L through equations ( 4) and ( 5): a valid range of α-values must be defined first so that paired ee P -and ee L -values can be calculated then. The nomenclature of all constants in this paper has been unified for the sake of clarity and thus differs from Noyori's original notation; equations ( 4) -( 6) have also been modified in order to contain association instead of dissociation constants. Details are found in section I.2.5 of the Supporting Information.</p><!><p>We currently study the addition of dialkylzinc reagents to benzaldehyde catalysed by Nsubstituted ephedrine ligands which are, as Noyori's DAIB, chiral aminoalcohols. One could expect that both give rise to similar catalytic systems, but an unexpected finding proved otherwise: N-benzylephedrine (NBE) induces a hyperpositive nonlinear effect -that is when a chiral catalyst, when not enantiopure, is more enantioselective than its enantiopure counterpart. [12,13] Mechanistic investigations established that the hyperpositive NLE originates from a two-component catalysis where both monomeric and dimeric complexes catalyse the reaction with different enantioselectivities, thus being in equilibrium and in competition. Figure 4 displays our proposed reaction network (with NBE, the RS-dimer precipitates and can be considered as inactive, i.e. k 2 ' = 0). This model stands in striking contrast to the Noyori model, which excludes dimer catalysis, and thus opens new possibilities in non-linear phenomena. The parent ligand N-methylephedrine (NME) proved to follow the same system than its benzylated counterpart when using ZnMe 2 as the dialkylzinc reagent. [14] The emergence of an enantiodivergent NLE additionally showed that the monomeric and homochiral dimeric complexes from NME give the product not only in different ee, they even yield opposite enantiomers.  7), ( 8) and ( 9) that allow to simulate ee P vs ee L -curves. Both monomers and dimers are catalytically active, thus ee P depends on the ligand ee (ee L ), the catalysts' enantioselectivities (ee 1</p><!><p>) are working variables introduced to simplify the expressions. Furthermore, α links ee P and ee L through equations ( 7) and ( 8): a valid range of α-values must be defined first so that paired ee P -and ee Lvalues can be calculated then.</p><p>In the light of these observations, it is worth to reconsider an earlier report from Fitzpatrick and co-workers. [17] They used NME in conjunction with ZnEt 2 and found an "essentially linear" relationship between ee P and ee L . Cryoscopic molecular weight (MW) measurements showed that the NME-ZnEt complex has a MW of 770 g/mol, which the authors interpreted to be a 1:1-mix of dimers and tetramers -trimers were excluded without further explanation -which, however, break down to monomers upon addition of one equivalent of ZnEt 2 . This led them conclude that NME-ZnEt doesn't form any aggregate under catalytic conditions, which would concord with the absence of an NLE. This is rather surprising: why would switching from ZnMe 2 to ZnEt 2 result in a total loss of aggregates? With NBE as ligand, the homochiral dimeric complex is present as well as catalytically active with both ZnMe 2 and ZnEt 2 , even though the dimer's contribution to the overall ee P is probably lower with ZnEt 2 (as seen in a hyperpositive NLE with a weaker amplitude). [13] This made us conduct our own investigations on the NME/ZnEt 2 -system. Using our usual reaction conditions (0.83 M benzaldehyde in toluene, 1.2 equiv. ZnEt 2 , 20 mol% ligand) we first could reproduce the linear ee P vs ee L -relationship (Figure 5a, blue dots) as previously noted by Fitzpatrick; for comparison, we have added the data for the ZnMe 2 addition in similar reaction conditions (Figure 5a, grey dots) which shows an upside-down (-)-NLE. [14] When using only enantiopure NME, a reduction of the catalyst loading led to a significant increase of ee P (Figure 5b, blue dots). While the alcohol product was obtained in 68% ee when using 20% mol% of (-)-NME, 80% ee was measured in presence of only 2.5 mol% ligand. Such behaviour is typical of a monomer-dimer-competition system, as shown in Figure 4, where a dimeric catalyst is present and gives a lower ee P than the monomer: a decrease in catalyst concentration tips the equilibrium from the dimer to the more enantioselective monomer, which leads to an increase of the overall ee P . A similar ee P increase is obtained with NBE as ligand using the same reaction conditions (Figure 5b, grey crosses) [12,13] and the NME/ZnMe 2 -system also shows this behaviour (Figure 5b, grey dots). [14] This points to the presence, and also to the catalytic activity, of homochiral NME-ZnEt dimers -even though no NLE is observed for this system in Figure 5a.  [14] ) or (-)-NBE (ZnEt 2 , grey crosses). [12,13] Each point is the mean of three independent experiments; the vertical bars depict standard deviations. The linear (a, blue dotted lines) and 2 nd -order polynomial fits (a and b, grey dotted or dashed lines) serve as visual guidelines.</p><p>Next, we investigated the behaviour of the complexes in solution using 1 H and DOSY NMR spectroscopy. The isolated enantiopure (-)-NME-ZnEt complex in toluene-d 8 (Figure 6a) shows a clear trimodal pattern which is well seen in the -CH-O-region of the spectra ( = 4.8 to 5.2 ppm). The integratals of the three doublets are identical, even if the concentration is decreased from 185 to 6 mM (cf. Supporting Information). This indicates that the three sets of signals do not arise from different species in equilibrium (i.e. monomer and dimer) but from a single molecule containing several non-equivalent NME-ZnEt-subunits -i.e. a trimeric aggregate, for which ample precedent exists with other aminoalcohols and organozincs. [18][19][20][21][22][23] This confirms Fitzpatrick's MW measurements, even though they excluded the possibility of trimers. Our own measurements using 1 H DOSY NMR and the ECC-methodology developed by Stalke and co-workers [24][25][26] gave a mean molecular weight (mMW) of 732 g/mol at a concentration of 185 mM, which corresponds to a degree of aggregation (A d ) of 2.7. This is consistent with Fitzpatrick's cryoscopic molecular weight measurements, even though they didn't interpret the data as resulting from the trimerization of the complex. At 6 mM, 619 g/mol and A d = 2.3 were obtained, indicating that the trimer is in equilibrium with smaller aggregates. However, addition of 5 equiv. of ZnEt 2 didn't disrupt the trimer: the trimodal pattern persists in 1 H NMR at 185 mM (Figure 6b) and the mean MW decreased only slightly (701 g/mol, A d = 2.6)). This stands in contrast to  We then performed a catalytic run (i. e. NME-ZnEt in presence of both ZnEt 2 and benzaldehyde) monitored by 1 H NMR. Figure 7 shows the spectra (a) before, (b) 5 min after and (c) 6 h after the beginning of the reaction. Upon addition of the benzaldehyde substrate the CH-O region dramatically changes: the trimodal pattern broadens and almost totally disappears. At the end of the catalytic reaction the trimer's signal becomes more clearly defined (ca. 20% of its original intensity). This is a strong indication for the trimer being only a resting species which dissociates to the actual catalysts in presence of substrate and reagent, and which partly reforms once the latter are consumed. At this stage we could not assign the various broad peaks to possible catalytic species yet, but the complexity of the spectrum renders unlikely the hypothesis that a single monomeric complex is present. A DOSY analysis after completion of the reaction (the reaction is too fast to be analysed during the conversion) resulted in a complex spectrum, leading to average MW values between 679 and 805 g/mol (A d = 2.5 to 3.0, cf. Figure 7c).</p><!><p>Altogether, the catalyst loading screening and the NMR data show that NME-ZnEt complex does aggregate under reaction conditions, despite the ee P vs ee L -relationship being linear and in contrast to conclusions drawn in earlier reports. Therefore, the latter must be the result of a "lucky" coincidence: the parameters governing the reaction are configured in a way so that the curve doesn't deviate from linearity although the catalyst does aggregate - much like in the examples shown for the Kagan and Noyori models vide supra. This drove us to investigate such particular parameter configurations using our own model for monomerdimer competition reactions, which has allowed us to simulate hyperpositive NLE curves, ee P vs [Cat tot ] curves and kinetic profiles. [14] The following simulations have been calculated using [equations ( 7)-( 9), cf. Figure 4]. It should be noted this set of equations doesn't allow to set K Hetero = 2K Homo because of the denominator in eq. ( 9) would become equal to 0. This issue can be easily circumvented but, for the sake of completeness, we have developed a new set of expressions which integrates K Hetero = 2K Homo ("Model II.b", cf. section I.2.3 of the Supplementary Information).</p><p>The present theoretical study shows that there are various scenarios which allow the emergence of a linear ee P vs ee L -relationship, which depend on the contribution of the different catalysts to the overall reaction outcome.</p><!><p>) and their respective kinetic constants (k 1 , k 2 and k 2 '). The catalyst concentrations are considered to be constant over time and depend on the interplay of the dimerization constants (K Homo , K Hetero ) and the overall catalyst concentration [Cat tot ]. The monomer's and homochiral dimer's enantioselectivities (ee 1 and ee 2 ) have been set to 100% and -100%, respectively. Globally, we can distinguish between two different extreme cases: one where only the monomers and one where only the dimers catalyse. In those extreme cases the system behaves either like the Noyori or the ML 2 model, respectively, and the ee P vs ee L -plots become linear within the parameter configurations discussed vide supra.</p><p>For a negligible activity of both homo-and heterochiral dimers (k 2 /k 1 = 0.01 and k 2 '/ k 1 = 0.01), the plot becomes linear if K Hetero /K Homo = 2 -exactly as in the Noyori modelregardless of the absolute value of K Homo , of K Hetero or of the kinetic constants. This changes when k 2 /k 1 is increased: the necessary K Hetero /K Homo ratio deviates from 2 and becomes dependent on the absolute values of K Homo , K Hetero and [Cat tot ]. Figure 8a shows a screening of k 1 /k 2 in a system with only weak dimerization (K Homo = 10, [Cat tot ] = 0.01). The K Hetero /K Homo ratio necessary to keep the plot linear decreases progressively to 0 as k 2 /k 1 is increased, the plot's slope also decreases. Interestingly, once it has crossed the ee L axis and gets a negative slope, the plot cannot become linear anymore; the dashed lines in Figure 8a show plots which get closest to linear as possible (cf. Supporting Information section I.2.1 for the fitting procedure used here). For k 2 /k 1 = 10 one obtains a seemingly linear plot when setting K Hetero /K Homo to 1.03 (a close zoom shows that it is slightly S-shaped) but for higher k 2 /k 1values the plots become distorted.</p><!><p>The non-full lines (dashed and coloured in a), dotted in b)) represent cases were linearity cannot be achieved, they show instead the curve closest to linearity which could be obtained. All curves have been obtained by setting the indicated parameters (k1 = 100, k2' = 1, ee1 = 100%, ee2 = -100%) and by fitting the curve, through variation of KHetero, to a simulated linear plot. The full procedure and data are found in the Supporting Information.</p><p>A similar behaviour is found when working in conditions with high catalyst aggregation ( i.e. K Homo = 1000 and [Cat tot ] = 0.16, Figure 8b). For k 2 /k 1 = 0.01 the optimal K Hetero /K Homo to get a linear plot is again 2, although the lower ee P at ee L = 100% (ee P 100 ) indicates that the homochiral dimer participates more to the reaction outcome as in Figure 8a. However, the system is much more sensitive to changes in k 2 /k 1 : an increase by factor 10 is almost sufficient to make the plot coincide with the ee L -axis; K Hetero /K Homo lowers to 1.63 for k 2 /k 1 = 0.06 but then increases back to 1.8. At k 2 /k 1 = 0.2 the plot's slope is negative and, again, only seemingly linear. However, at lower values the system's behaviour differs from Figure 8a: close to linear plots (dotted lines) are obtained by setting K Hetero = 0. This is observed only in Figure 8b; setting K Hetero = 0 for the dashed curves in Figure 8b doesn't lead to more linear plots.</p><p>The reason why the ee L -axis represents a crossing point in these k 2 /k 1 -screenings is because it sets the limit between monomer-dominated (upper half of the graph) and dimerdominated catalysis (lower half). Since ee 1 = 100% and ee 2 = -100% here, negative ee P -values can only arise if the homochiral dimer is the main contributor to the overall catalytic outcome. Therefore, the graph's lower half is also the region in which the system resembles more the ML 2 model than the Noyori model. This explains the deviations observed in Figure 8: to obtain linear plots the ML 2 model requires k 2 ' to be nonnegligible (g, i. e. k 2 '/k 2 , should be equal to 1) which is not the case here as it has been set to 0.01. The other possibility is β = 0, i. e. RS-dimers are completely absent from the reaction mixture. This is given if K Hetero = 0 and thus explains the linearity of the dotted lines in Figure 8b. The reason why this doesn't work in Figure 8a is probably that in low aggregation conditions, the monomer's contribution perturbs too much the curve, even in the lower half of the graph.</p><p>The precedent conclusions prompted us to study how variation of k 2 ' (i.e. racemic contribution) may drive system towards linearity. Figure 9a shows the exact same simulated curves as in Figure 8a (dashed lines in both figures) with k 2 ' = 0.01. Their parameter sets were taken to calculate another curve set (Figure 9a, full lines) with increased k 2 '-values in order to get more linear plots. Increasing k 2 '/k 1 indeed flattens the curves and reduces the difference (i. e. the sum of squares, cf. Supporting Information I.2.1) from linearity, but only to a certain extent. This is due to the high K Hetero /K Homo -ratio which twists the curve into an "S" shape (and makes it hyperpositive [14] ) and which cannot be fully compensated by tweaking k 2 '/k 1 . However, for K Hetero /K Homo < 100 the S-shape is negligible and linear plots are obtained for any k 2 /k 1 -ratio if the correct k 2 ' is chosen (Figure 9b), regardless of the plot being in the upper or lower half of the graph. The system becomes equivalent to the ML 2 model (with g = k 2 '/k 2 = 1 to achieve linearity) only in very high aggregation conditions (very high K Homo , K Hetero , [Cat tot ]) as the ML 2 model assumes that monomers do not exist in the catalytic system. If the presence of monomers is not negligible, then k 2 '/k 2 = 1 gives linear plots only if K Hetero /K Homo = 2. Higher K Hetero /K Homo -ratios afford lower k 2 '/k 2 (and vice-versa) to keep the plot linear. This is seen in Figure 9b: here, K Hetero /K Homo = 4 and thus k 2 '/k 2 is slightly below 1 if the monomers have a negligible activity (k 2 /k 1 = 100). Increasing the monomer's contribution (lowering of k 2 /k 1 ) lifts the plot up and decreases k 2 '/k 2 until it crosses the ee Laxis, then k 2 '/k 2 increases again. Globally, we can say that especially when the meso dimer is active, a multitude of parameter configurations exists where the ee L vs ee P -plot becomes linear although catalyst dimers are present (and possibly catalytically active) in the reaction mixture.</p><!><p>The present study shows that probing for non-linear effects alone can lead to false interpretations considering the aggregation state of the catalyst in a catalytic reaction: although the absence of a NLE is commonly interpreted as the catalyst not aggregating, we show here an example where this is not the case. The NME-catalysed addition of ZnEt 2 to benzaldehyde shows a linear ee P vs ee L correlation although catalyst aggregates are present and, moreover, catalytically active as seen in the ee P vs catalyst loading screening and the 1 H and DOSY NMR studies. Kagan and Noyori already had pinpointed such exceptions to the rule in their respective models when applying a particular configuration of parameters. Our own model confirms these exceptions and shows that, in presence and with the catalytic  What we can take from this study is that probing for non-linear effects is still a good tool to find evidence for catalyst aggregation, but it should be complemented with other ones in order to avoid short-cut conclusions. In this study, 1 H and DOSY NMR proved to be such additional and valuable tools, and ee P vs catalyst loading-plots have now become good indicators for the possible catalytic activity of both monomeric and aggregated catalyststhus, they also give evidence about the catalyst's aggregation state. Kinetic studies have also been used to complement NLE probing: relationships such as rate vs ee L [27] and catalyst order vs catalyst concentration [12,14,28] may show features that are reminiscent of certain models.</p><!><p>Synthetic and mathematical procedures, as well as additional data, chromatograms and NMR spectra ( 1 H and DOSY) can be found in the SI.docx. 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_Simulation_Data.xlsx).</p><!><p>One might think that the absence of a non-linear effect (NLE) might necessarily mean that only monomeric complexes catalyse a studied enantioselective reaction. Here we show a counter-example using an ephedrine-derived ligand in asymmetric alkylzinc additions. The system is catalysed by both monomers and dimers, although no NLE is apparent.</p>