Distinguishing Concerted versus Stepwise Mechanisms Using Isotope Effects on Isotope Effects

Double isotope fractionation was discovered independently by two groups during the early 1980s when it was first published in the pages of Biochemistry and the Journal of the American Chemical Society. The principle offers a method for testing whether two bonds are changed stepwise or concertedly in the midst of an enzyme-catalyzed reaction. Initially demonstrated with malic enzyme and proline racemase, it has since been applied to many systems over the years, becoming a standard approach in enzymology. This Viewpoint aims to highlight the initial work on double isotope fractionation and provide an overview of how it works.

Malic enzyme catalyzes the oxidative decarboxylation of L-malate (1) to pyruvate (2), and this reaction could proceed via a stepwise (1 → 3 → 4) or concerted (1 → 4) mechanism as shown in Figure 1. Hermes, Roeske, O’Leary, and Cleland addressed this question by measuring the isotope effect induced on V/K when C4 of malate is labeled with ¹³C.¹ They found that this ¹³C isotope effect was significantly decreased from 1.031 to 1.025 when [¹³C]malate was simultaneously labeled with a deuterium at C2. As described below, this allowed the authors to conclude that dehydrogenation of the C2 hydroxyl takes place prior to the elimination of CO₂ in a stepwise mechanism.

Figure 1.

Mechanisms proposed for malic enzyme and proline racemase. The positions of the fractionating and reweighting labels are colored red and blue, respectively.

Proline racemase catalyzes the racemization of proline (5 ⇌ 6) via two proton transfers at opposite faces of the proline α- carbon (see Figure 1). Belasco, Albery, and Knowles tested whether these Hydron transfers take place in one versus two steps by essentially measuring the solvent deuterium isotope effect on the net rate constant for the Michaelis complex.² They found no significant difference in this isotope effect regardless of whether the substrate was initially deuterated at the α-carbon. As explained below, the authors were thus able to infer that the Hydron transfers are concerted or take place stepwise with cysteines as the catalytic residues.

Double isotope fraction thus involves looking for an isotope effect on an isotope effect. Such effects are not expected due to the rule of the geometric mean when the reaction is elementary and quantum corrections are small or the relevant molecular vibrations are at least uncoupled.³ However, the net rate constants investigated by Hermes et al.¹ and Belasco et al.² do not describe elementary reactions but rather a sequence of reactions at the steady state. Thus, an isotope effect on a net rate constant can be altered by the presence of a second label that adjusts how each step limits the rate.

To see this more concretely, consider a serial mechanism with n intermediates such as

$${\text{E}}_1+\text{S}\rightleftharpoons {\text{E}}_2\rightleftharpoons \dots \rightleftharpoons {\text{E}}_{n-1}\rightleftharpoons {\text{E}}_n\to {\text{E}}_1+\text{P}$$

If e_i is the steady state concentration of intermediate E_i for some steady state velocity v, then the net rate constant for E_i is given by the ratio k_i′ = v/e_i.⁴ Note that V/K is just k₁′ multiplied by the ratio of the total enzyme and substrate concentrations.⁴ An isotope effect *k_i′ = k_i′/k_i*′, where the subscript asterisk denotes the presence of the fractionating isotopic label, can be expressed as a weighted average^4b

$$*k_i^{\prime }=\sum _{j=i}^n{w}_{ij}*{\kappa }_{ij}$$ (1)

where the weights w_ij are independent of the fractionating label and the isotope effects *κ_ij represent fractionation of this label between E_i and the jth transition state downstream (see Figure 2A). If the rule of the geometric mean holds for each individual step, then the *κ_ij terms will remain unchanged in the presence of a second reweighting isotopic label. Therefore, adding such a label in addition to the fractionating label will affect only the weighting terms w_ij.

Figure 2.

Example free energy diagrams illustrating how the double isotope fractionation experiment essentially works for k₁′. (A) Free energy profile in the presence of no labels (black) vs the fractionating label alone (red). Bond changes associated with the fractionating label take place in the E₃ → E₄ elementary reaction. (B) Reweighting label alone (blue) vs that with the fractionating label (brown) for concerted bond changes. (C) Reweighting label alone (blue) vs that with the fractionating label (brown) for a stepwise mechanism where the reweighting bond change takes place first (E₂ → E₃). (D) Same as panel C with a normal equilibrium iosotopic effect associated with the reweighting bond change (all other equilibrium isotope effects are treated as unity for the sake of clarity). The isotope effects are exaggerated, and the plots have been adjusted such that the free energy of E₁ is the same in each diagram. Thus, isotopic fractionation of the jth transition state vs E₁ (i.e., *κ_1j) is exponentially related to the free energy difference between the transition states in the two plots of each panel (*κ₁₁ = *κ₁₂ = *κ₁₄ = 1; *κ₁₃ > 1). Likewise, the weights (w_1j) are exponentially related to the relative transition state free energies in the black and blue plots alone.

When Hermes et al.¹ deuterated C2 of malate as the reweighting label, they were slowing the C2–H bond cleavage (1 → 3 or 1 → 4), thereby increasing its weight in eq 1. Had decarboxylation (e.g., E₃ → E₄ in Figure 2) taken place in the same step as C2 oxidation, then the isotope effect induced by the ¹³C fractionating label would have been more strongly expressed in the isotope effect on k₁′ (i.e., V/K), making it larger or at least leaving it unchanged (Figure 2A vs Figure 2B). The observed decrease in the isotope effect, however, meant that a step (e.g., E₂ → E₃) different from decarboxylation (E₃ → E₄) had been made more rate limiting by slowing C2–H bond cleavage (Figure 2A vs Figure 2C). Hence, oxidation and decarboxylation must be on different steps.

The same interpretation holds for the results observed by Belasco et al.,² leading them to propose that the two Hydron transfers catalyzed by proline racemase might be concerted (Figure 2A vs Figure 2B, with the Michaelis complex of 5 indexed as E₁). However, these authors also had to consider the possibility that the catalytic acid and base were thiols, which later turned out to be the case.⁵ This would decrease the size of the kinetic isotope effect induced by the fractionating label on protonation of the α-carbon (7 → 6 as E₃ → E₄ in Figure 2), because the S–H bond is already relatively “loose” in the reactant state (5 as E₁).^3b Likewise, a relatively large, normal equilibrium isotope effect would also be induced by the αCD reweighting label upon the deprotonation of proline (5 → 7 as E₂ → E₃) by a sulfur as opposed to an oxygen or nitrogen base. This would tend to offset the reweighting caused by the kinetic isotope effect on deprotonation of the α- carbon (Figure 2A vs Figure 2D). The combined effect would make it difficult to discern any difference in the observed isotope effects even if the mechanism were stepwise.

The elementary reactions that make up the catalytic cycles of enzymes often represent a black box requiring considerable ingenuity and creativity to investigate. The discovery of double isotope fractionation represents just one example to have been published in Biochemistry over the years. These papers have provided not only a new experimental methodology but also a deeper understanding of enzymatic steady state kinetics. Therefore, revisiting these classic works offers a valuable source of insight and inspiration for the modern study of biochemical systems.