# Non-linear effects

In enantioselective synthesis, a non-linear effect refers to a process in which the enantiopurity of the catalyst (or the chiral auxiliary) does not correlate linearly with the enantiopurity of the product produced. This deviation from linearity is described as the non-linear effect, NLE. The linearity can be expressed mathematically, as shown in Equation 1. Stereoselection (i.e. the ee$_{product}$) that is higher or lower than the enantiomeric excess of the catalyst (ee$_{catalyst}$, relative to the equation) is considered non-routine behavior.
    $ee_{\text{product}}=ee_{\max }ee_{\text{catalyst}}$
For an ideal asymmetric reaction, the ee$_{product}$ may be described as the product of ee$_{max}$ multiplied by the ee$_{catalyst}$. This is not the case for reactions exhibiting NLE's.
In 1976, Wynberg and Feringa observed different chemical behavior in the reaction of an enantiopure and racemic substrate in a phenol coupling reaction. In 1981, Kagan and collaborators described the first non-linear effects in asymmetric catalysis and gave rational explanations for these phenomena. General definitions and mathematical models are essential for understanding nonlinear effects and their application to specific chemical reactions. In recent decades, the study of nonlinear effects has helped elucidate reaction mechanism and guide synthetic applications.

## Types of non-linear effects
### Positive non-linear effect, (+)-NLE

A positive non-linear effect, (+)-NLE, is present in an asymmetric reaction which demonstrates a higher product ee (ee$_{product}$ ) than predicted by an ideal linear situation (Figure 1). It is often referred to as asymmetric amplification, a term coined by Oguni and co-workers. An example of a positive non-linear effect is observed in the case of Sharpless epoxidation with the substrate geraniol.In all cases of chemical reactivity exhibiting (+)-NLE, there is an innate tradeoff between overall reaction rate and enantioselectivity. The overall rate is slower and the enantioselectivity is higher relative to a linear behaving reaction.

### Negative non-linear effect, (−)-NLE

Referred to as asymmetric depletion, a negative non-linear effect is present when the ee$_{product}$ is lower than predicted by an ideal linear situation. In contrast to a (+)-NLE, a (−)-NLE results in a faster overall reaction rate and a decrease in enantioselectivity. Synthetically, a (−)-NLE effect could be beneficial with a reasonable assay for separating product enantiomers and a high output is necessary . An interesting example of a (−)-NLE effect has been reported in asymmetric sulfide oxidations.

## Modeling non-linear effects

1n 1986, Henri B. Kagan and coworkers observed a series of known reactions that followed a non-ideal behavior. A correction factor, f, was adapted to Equation 1 to fit the kinetic behavior of reactions with NLEs (Equation 2).
    $ee_{\text{product}}=f\,ee_{\max }ee_{\text{catalyst}}$
Equation 2: A general mathematical equation that describes non-linear behavior
Unfortunately, Equation 2 is too general to apply to specific chemical reactions. Due to this, Kagan and coworkers also developed simplified mathematical models to describe the behavior of catalysts which lead to non-linear effects. These models involve generic ML$_n$ species, based on a metal (M) bound to n number of enantiomeric ligands (L). The type of MLn model varies among asymmetric reactions, based on the goodness of fit with reaction data. With accurate modeling, NLE may elucidate mechanistic details of an enantioselective, catalytic reaction.

### ML$_2$ model
#### General description

The simplest model to describe a non-linear effect, the ML$_2$ model involves a metal system (M) with two chiral ligands, L$_R$ and L$_S$. In addition to the catalyzed reaction of interest, the model accounts for a steady state equilibrium between the unbound and bound catalyst complexes. There are three possible catalytic complexes at equilibrium (ML$_S$L$_R$, ML$_S$L$_S$, ML$_R$L$_R$). The two enantiomerically pure complexes ( ML$_S$L$_S$, ML$_R$L$_R$) are referred to as homochiral complexes. The possible heterochiral complex, ML$_R$L$_S$, is often referred to as a meso-complex.
${\begin{aligned}&{\begin{matrix}\mathrm {M} +\mathrm {L} _{R}+\mathrm {L} _{S}\quad \rightleftharpoons \quad &\mathrm {ML} _{R}\mathrm {L} _{R}&+&\mathrm {ML} _{S}\mathrm {L} _{S}&+&\mathrm {ML} _{R}\mathrm {L} _{S}&\\&x&&y&&z&\left\}\quad {\overset{\text{concentrations}}}\right.\\&\\&\left\downarrow {\begin{matrix}\\\mathrm {r} _{RR}\\\\\end{matrix}}\right.&&\left\downarrow {\begin{matrix}\\\mathrm {r} _{SS}\\\\\end{matrix}}\right.&&\left\downarrow {\begin{matrix}\\\mathrm {r} _{RS}\\\\\end{matrix}}\right.&\left\}\quad {\overset{\text{rates}}}\right.\qquad \qquad \\\end{matrix}$
The equilibrium constant that describes this equilibrium, K, is presumably independent on the catalytic chemical reaction. In Kagan's model, K is determined by the amount of aggregation present in the chemical environment. A K=4 is considered to be the state at which there is a statistical distribution of ligands to each metal complex. In other words, there is no thermodynamic disadvantage or advantage to the formation of heterochiral complexes at K=4.
Obeying the same kinetic rate law, each of the three catalytic complexes catalyze the desired reaction to form product. As enantiomers of each other, the homochiral complexes catalyze the reaction at the same rate, although opposite absolute configuration of the product is induced (i.e. r$_{RR}$=r$_{SS}$). The heterochiral complex, however, forms a racemic product at a different rate constant (i.e. r$_{RS}$).

#### Mathematical model for the ML$_2$ Model

In order to describe the ML$_2$ model in quantitative parameters, Kagan and coworkers described the following formula:
    $ee_{\text{product}}=ee_{\text{max}}ee_{\text{auxiliary}}(1+\beta )/(1+g\,\beta )$
In the correction factor, Kagan and co-workers introduced two new parameters absent in Equation 1, β and g. In general, these parameters represent the concentration and activity of three catalytic complexes relative to each other. β represents the relative amount of the heterochiral complex (ML$_R$L$_S$) as shown in Equation 3. It is important to recognize that the equilibrium constant K is independent on both β and g. As described by Donna Blackmond at Scripps Research Institute, "the parameter K is an inherent property of the catalyst mixture, independent of the ee$_{catalyst}$. K is also independent of the catalytic reaction itself, and therefore independent of the parameter g."
    $\beta =z/(x+y)$
Equation 3: The correction factor, β, may be described as z, the heterochiral complex concentration, divided by x and y, the respective concentrations of the complex concentration divided by x and y, the respective concentrations of the homochiral complexes
The parameter g represents the reactivity of the heterochiral complex relative to the homochiral complexes. As shown in Equation 5, this may be described in terms of rate constants. Since the homochiral complexes react at identical rates, g can then be described as the rate constant corresponding to the heterochiral complex divided by the rate constant corresponding to either homochiral complex.
    $g=r_{rs}/r_{rr}$
Equation 4: The correction parameter, g, can be described as the rate of product formation with the heterochiral catalyst ML$_R$L$_S$ divided by the rate of product formation of the homochiral complex (ML$_R$L$_R$ or ML$_S$L$_S$).

#### Interpretation of the mathematical results of the ML$_2$ Model

   * If β=0 or g=1, the ML$_2$ equation simplifies to the Equation 1. No meso catalyst complex is present or active. Therefore, the simple additive properties should apply to such a scenario to establish a linear relationship between product enantioselectivity and the enantiopurity of the chiral catalyst.
   * If the correction factor is greater than one, the reaction displays an asymmetric amplification, also known as a positive non-linear effect. Under the ML$_2$ model, a (+)-NLE infers a less reactive heterochiral catalyst. In this case, the equilibrium constant K also increases as the correction factor increases. Although the product enantioselectivity is relatively high compared to the enantiopurity of the chiral catalyst, this comes at a cost of the overall reaction rate. In order to achieve an asymmetric amplification, there must be a relatively large concentration of the heterochiral complex. In addition, this heterochiral complex must have a substantially slower rate of reactivity, r$_{RS}$. Therefore, the reactive catalytic species should decrease in concentration, leading to an overall slower reaction rate.
   * If the correction factor is less than one, the reaction displays an asymmetric depletion, also known as a negative non-linear effect. In this scenario, the heterochiral catalyst is relatively more reactive than the homochiral catalyst complexes. In this case, the (−)-NLE may result in an overall faster although less selective product formation.
iv. Reaction Kinetics with the ML$_2$ Model: Following H.B. Kagan's publication of the ML$_2$ model, Professor Donna Blackmond at Scripps demonstrated how this model could be used to also calculate the overall reaction rates. With these relative reaction rates, Blackmond showed how the ML$_2$ model could be used to formulate kinetic predictions which could then be compared to experimental data. The overall rate equation, Equation 6, is shown below.
    $r=(x+y+gz)r_{RR}$
In addition to the goodness of fit to the model, kinetic information about the overall reaction may further validate the proposed reaction mechanism. For instance, a positive NLE in the ML$_2$ should result in an overall lower reaction rate. By solving the reaction rate from Equation 6, one can confirm if that is the case.

### M*L$_2$ Model
#### General description

Similar to the ML$_2$ model, this modified system involves chiral ligands binding to a metal center (M) to create a new center of chirality. There are four pairs of enantiomeric chiral complexes in the M*L$_2$ model, as shown in Figure 5.
${\begin{matrix}{\begin{matrix}\mathrm {M} +\mathrm {L} _{R}+\mathrm {L} _{S}\quad \rightleftharpoons \rightleftharpoons \quad \\\\\\\\\\&\end{matrix}}{\begin{aligned}&{\begin{matrix}\mathrm {M} _{R}\mathrm {L} _{R}\mathrm {L} _{R}&+&\mathrm {M} _{R}\mathrm {L} _{R}\mathrm {L} _{S}&+&\mathrm {M} _{R}\mathrm {L} _{S}\mathrm {L} _{R}&+&\mathrm {M} _{R}\mathrm {L} _{S}\mathrm {L} _{S}\\\mathrm {M} _{S}\mathrm {L} _{S}\mathrm {L} _{S}&+&\mathrm {M} _{S}\mathrm {L} _{S}\mathrm {L} _{R}&+&\mathrm {M} _{S}\mathrm {L} _{R}\mathrm {L} _{S}&+&\mathrm {M} _{S}\mathrm {L} _{R}\mathrm {L} _{R}\\&\\\left\downarrow {\begin{matrix}k_{0}\\\pm EE_{0}\end{matrix}}\right.&&\left\downarrow {\begin{matrix}k_{1}\\\pm EE_{1}\end{matrix}}\right.&&\left\downarrow {\begin{matrix}k_{2}\\\pm EE_{2}\end{matrix}}\right.&&\left\downarrow {\begin{matrix}k_{3}\\\pm EE_{3}\end{matrix}}\right.\end{matrix}}\\\\&\qquad \qquad \qquad \qquad \qquad \,\,\,\,{\text{products}}\end{aligned}}\end{matrix}}$
In this model, one can make the approximation that the dimeric complexes dissociate irreversibly to the monomeric species. In this case, the same mathematical equations apply to the ML*$_2$ model that applied to the ML$_2$ model.

### ML$_3$ model
#### General description

A higher level of modeling, the ML$_3$ model involves four active catalytic complexes: ML$_R$L$_R$L$_R$, ML$_S$L$_S$L$_S$, ML$_R$L$_R$L$_S$, ML$_S$L$_S$L$_R$. Unlike the ML$_2$ model, where only the two homochiral complexes reacted to form enantiomerically enriched product, all four of the catalytic complexes react enantioselectively. However, the same steady state assumption applies to the equilibrium between unbound and bound catalytic complexes as in the more simple ML$_2$ model. This relationship is shown below in Figure 7.
${\ce {{M}+{L_{R}}+L_{S}<=>}}\ \underbrace {\ce {ML_{R}L_{R}L_{R}}} _{x}+\underbrace {\ce {ML_{S}L_{S}L_{S}}} _{y}+\underbrace {\ce {ML_{R}L_{R}L_{S}}} _{z}+\underbrace {\ce {ML_{S}L_{S}L_{R}}} _{w}$

#### Mathematical modeling

Calculating the ee$_{product}$ is considerably more challenging than in the simple ML$_2$ model. Each of the two heterochiral catalytic complexes should react at the same rate. The homochiral catalytic complexes, similar to the ML$_2$ case, should also react at the same rate. As such, the correction parameter g is still calculated as the rate of the heterochiral catalytic complex divided by the rate of the homochiral catalytic complex. However, since the heterochiral complexes lead to enantiomerically enriched product, the overall equation for calculating the ee$_{product}$ becomes more difficult. In Figure 8., the mathematical formula for calculating enantioselectivity is shown.
    $EE_{\text{prod}}=EE_{0}ee_{\text{aux}}{\frac {3+3\,g\,EE'_{0}/EE_{0}+(1-3\,g\,EE'_{0}/EE_{0})ee_{\text{aux}}^{2}}{1+3\,g+3(1-g)ee_{\text{aux}}^{2}}}$
Figure 8: The mathematical formula describing an ML$_3$ system. The ee$_{product}$ is calculated by multiplying the ee$_{max}$ by the correction factor developed by Kagan and co-workers.

#### Interpretation of the ML$_3$ Model

In general, interpreting the correction parameter values of g to predict positive and negative non-linear effects is considerably more difficult. In the case where the heterochiral complexes ML$_R$L$_R$L$_S$ and ML$_S$L$_S$L$_R$ are less reactive than the homochiral complexes ML$_S$L$_S$L$_S$ and ML$_R$L$_R$L$_R$, a kinetic behavior similar to the ML$_2$ model is observed (Figure 9). However, a substantially different behavior is observed in the case where the heterochiral complexes are more reactive than the homochiral complexes. In such case, Kagan and collaborators showed that it is possible to have a case “where the enantiomeric excess could take on much larger values for a partially resolved ligand than for an enantiomerically pure ligand”. The authors proposed the term “hyperpositive nonlinear effect” to characterize this situation.

### Reservoir Effect
#### General Description

Often described adjacent or in collaboration with the ML$_2$ model, the reservoir effect describes the scenario in which part of the chiral ligand is allocated to a pool of inactive heterochiral catalytic complexes outside the catalytic cycle. A pool of unreactive heterochiral catalysts, described with an ee$_{pool}$, develops an equilibrium with the catalytically active homochiral complexes, described with an ee$_{effective}$. Depending on the concentration of the inactive pool of catalysts, one can calculate the enantiopurity of the active catalyst complexes. The general result of the reservoir effect is an asymmetric amplification, also known as a (+)-NLE.

#### Origin of the Reservoir Effect

The pool of unreactive catalytic complexes, as described in the reservoir effect, can be the result of several factors. One of these could potentially be an aggregation effect amongst the heterochiral catalytic complexes that takes place prior to the steady state equilibrium.

## Early Examples of the Non-Linear Effect
### Sharpless Epoxidation of Geraniol

In 1986, Kagan and co-workers were able to demonstrate NLE with the Sharpless epoxidation of (E)-Geraniol (Figure 11). Under Sharpless oxidizing conditions with Ti(O-i-Pr)$_4$/(+)-DET/t-BuOOH, Kagan and coworkers were able to demonstrate that there was a non-linear correlation between the ee$_{product}$ and the ee of the chiral catalyst, diethyl tartrate (DET). As one can see from Figure 11, a greater ee$_{product}$ than expected was observed. According to the ML$_2$ model, Kagan and coworkers were able to conclude that a less reactive heterochiral DET complex was present. This would therefore explain the asymmetric amplification observed. The NLE data is also consistent with the Sharpless mechanism of asymmetric epoxidation.

### Asymmetric Sulfide Oxidation

In 1994, Kagan and co-workers reported a NLE in asymmetric sulfide oxidation. The goodness of fit for the reaction data matched the ML$_4$ model. This implied that a dimeric Titanium complexed with 4 DET ligands was the active catalytic species. In this case, the reaction rate would be significantly faster relative to ideal reaction kinetics. The downfall, as is the case in all (−)-NLE scenarios, is that the enantioselectivity was lower than expected. Below, in Figure 12, one can see the concavity of the data points is highly indicative of a (−)-NLE.

## Prebiotic Catalysis and the Non-linear Effect

In pre-biotic chemistry, autocatalytic systems play a significant rule in understanding the origin of chirality in life. An autocatalytic reaction, a reaction in which the product acts as a catalyst for itself, serves as a model for homochirality. The asymmetric Soai reaction is commonly referred to as chemical plausibility for this pre-biotic hypothesis. In this system, an asymmetric amplification is observed during the process of autocatalytic catalysis. Professor Donna Blackmond has studied the NLE of this reaction extensively using Kagan's ML$_2$ model. From this mathematical analysis, Blackmond was able to conclude that a dimeric, homochiral complex was the active catalyst in promoting homochirality for the Soai reaction.