Kinetic Rationalization of Nonlinear Effects in Asymmetric Catalytic Cascade Reactions under Curtin–Hammett Conditions

Observations of nonlinear effects of catalyst enantiopurity on product enantiomeric excess in asymmetric catalysis are often used to infer that more than one catalyst species is involved in one or more reaction steps. We demonstrate here, however, that in the case of asymmetric catalytic cascade reactions, a nonlinear effect may be observed in the absence of any higher order catalyst species or any reaction step involving two catalyst species. We illustrate this concept with an example from a recent report of an organocatalytic enantioselective [10 + 2] stepwise cyclization reaction. The disruption of pre-equilibria (Curtin–Hammett equilibrium) in reversible steps occurring prior to the final irreversible product formation step can result in an alteration of the final product ee from what would be expected based on a linear relationship with the enantiopure catalyst. The treatment accounts for either positive or negative nonlinear effects in systems over a wide range of conditions including “major-minor” kinetics or the more conventional “lock-and-key” kinetics. The mechanistic scenario proposed here may apply generally to other cascade reaction systems exhibiting similar kinetic features and should be considered as a viable alternative model whenever a nonlinear effect is observed in a cascade sequence of reactions.


General remarks
All kinetic modelling was conducted using Complex Pathway Simulator (COPASI) software. 1 The values used in each kinetic model are provided, as well as the corresponding COPASI files. The data of the nonlinear effects used in the simulations have been extracted from the experimental data reported by Jørgensen in ACS catalysis 2020 for the reaction in Scheme S1. 2 The reaction involves a catalyzed reaction between 1 and 2, producing intermediate II, which cyclises in a catalyzed reaction to from the final product 4. All the conditions used for the COPASI modelling were extracted from page S64 in the supporting information of Ref. 2 (SI) unless stated otherwise. Within our COPASI models, we refer to 1, 2, and 3 as A, B, and cat respectively.

Kinetic analysis of the nonlinear effects of the reaction network in Scheme 2
Scheme S2. Reaction scheme used for the mathematical analysis of nonlinear effects observed in cascade reactions.
We derived an equation for ee4 by applying the steady-state approximation to the reaction intermediates II(SS) and II(RR). We will use the following equations and definitions: Enantiomeric ratio of product 4: Rate of formation of 4(SSSS): Rate of formation of 4(RRRR):

(6)
Enantioselectivity of the first step: Selectivity factor of the second step (analogous to the selectivity factor of kinetic resolutions): Enantiomeric ratio and enantiomeric excess of catalyst 3: From equation (13) we derive the er4 when using enantiomerically pure catalyst 3 S (er4 ep = ): From equation (14), and definitions (7) and (8): Equation (18) allows us to rationalize the appearance of NLE and its sense (summarized in Figure S1).
1. If αβ = 1: In this scenario, the selectivity of the first step (α ) is the inverse of the selectivity of the second step (β ) and therefore they cancel each other. The result is a linear dependency of the ee of the product (ee4) with respect to the ee of the catalyst (ee3) ( Figure S1).

S-4
2. If α = γ : In this scenario, the result is a linear dependency of the ee of the product (ee4) with respect to the ee of the catalyst (ee3) (Figure S1), which may be due to different reasons. To understand the different reasons, we have used equation (14) and definitions (7) and (8) to derive the following equation: Which means that either the first step is practically irreversible (k-1 /k3 ≈ 0), or that both steps have identical selectivity (α = β ) ( Figure S1).
3. β = γ : In this scenario, the result is a linear dependency of the ee of the product (ee4) with respect to the ee of the catalyst (ee3) (Figure S1), which may be due to different reasons. To understand the different reasons, we have used equation (14) and definitions (7) and (8) to derive the following equation: Which means that either the first step is in quasi-equilibrium (because k3 /k-1 ≈ 0) or that both steps have identical selectivity (α = β ) ( Figure S1).
• If αβ = 1 , (19) is zero and the system has a linear behaviour (see point 1 in page S-3 and Figure S1).
• If αβ = 1 , (19) is zero and the system has a linear behaviour (see point 1 in page S-3 and Figure S1).

Procedure used to obtain rate constants for the kinetic modelling of Scheme 2
To find the rate constants which produced a nonlinear effect comparable to the nonlinear effect reported in Ref.
2, we used the "solver" function in Microsoft Excel to find the values of α, β and γ. The values we found minimized the square of the differences between the value of the steady-state equation for ee4 (formula (18)) and the experimental data (page S64 of Ref. 2; we considered that the result for 98% ee3 reported by Jørgensen et al. was 100%, therefore the ee4 ep was 0.9 and γ was 19 for all our modelling of this reaction).
For Case 1, we used the following constraint in our optimization:

< <
For Case 2, we used the following constraint in our optimization:

> >
We created the COPASI model representing the reaction network in Scheme 2. We assigned all the kinetic constants to global quantities defined as follows: Since no numerical data nor reaction profiles were reported for the kinetic studies in Ref. 2, we had to estimate the progress of the reaction from the images of the stacked NMR spectra in pages S66-S68 in Ref 2. We used COPASI to do a "parameter estimation" of the kinetic constants that better fitted the estimated data.

Kinetic modelling of Scheme 2
Using the reaction scheme proposed in Scheme 2, we found two sets of rate constants which reproduced the nonlinear effect reported in Ref.
2. The first set of rate constants produce II(RR) as the major intermediate (Case 1, Figure S2, Table S1) where the reaction is under the major-minor kinetic scenario. The other set of rate constants produce II(SS) as the major intermediate (Case 2, Figure S2, Table S1) which put the reaction under the lock-and-key kinetic scenario. In both cases, the major enantiomer of product was 4(SSSS). Figure S2. Nonlinear effect observed from the reaction presented in Scheme 2. Provided COPASI models: "Scheme 2 Major-minor (case 1).cps" and "Scheme 2 Lock and key (case 2).cps" The nonlinear effects shown in Figure S2 is possible with any set of kinetic constants that maintain the values of α, β, ee4 ep . To exemplify this fact, we have reproduced the same nonlinear effect with rate constants differing by a magnitude of 14 ( Figure S3, Table S2).    S-10

Kinetic modelling of the extended reaction with unproductive diastereomeric intermediates
When the reaction in Scheme 2 is extended to include the formation of both unproductive diastereomeric intermediates (II(SR), II(RS)), and the catalyst bound intermediates (e.g. Int(SS)3 S ) we can also produce a nonlinear effect for both the kinetic scenarios in Case 1 and 2 ( Figure S5, Table S3). Figure S5. Nonlinear effect observed in the extended reaction. Provided COPASI models: "All diastereomers (case 1).cps" and "All diastereomers (case 2).cps" S-11

Kinetic modelling of Scheme 2 with a completely enantiospecific catalyst in the final step
Even with a catalyst completely enantiospecific in the second step, we can produce a nonlinear effect, but only under the major-minor kinetic scenario where II(RR) is the major intermediate and 4(SSSS) is the major product ( Figure S6, Table S4).

Kinetic modelling of Scheme 2 with an uncatalyzed final step
Even with an uncatalyzed second step, we can produce a nonlinear effect, but only under the lock-and-key kinetic scenario where II(SS) is the major intermediate and 4(SSSS) is the major product ( Figure S7, Table S5). Provided COPASI models: "Scheme 2 Uncatalyzed second step.cps"

Measurement of the order in catalyst for Scheme 2
To measure the order in catalyst for the reaction network shown in Scheme 2 we used time normalised analysis 3 on reaction profiles generated in COPASI. We used the reaction conditions used in the experimental kinetic studies reported in the supporting information of Ref. 2 (page S65). Using these conditions, we measured an order of 1 in catalyst ( Figure S8).

S-13
We also derived the order in catalyst expected from the reaction pathway shown in Scheme 2: Rate of formation of 4: