Linear Free Energy Relationships for Enzymatic Reactions: Fresh Insight from a Venerable Probe

Conspectus

Linear free energy relationships (LFERs) for substituent effects on reactions that proceed through similar transition states provide insight into transition state structures. A classical approach to the analysis of LFERs showed that differences in the slopes of Brønsted correlations for addition of substituted alkyl alcohols to ring-substituted 1-phenylethyl carbocations and to the β-galactopyranosyl carbocation intermediate of reactions catalyzed by β-galactosidase provide evidence that the enzyme catalyst modifies the curvature of the energy surface at the saddle point for the transition state for nucleophile addition. We have worked to generalize the use of LFERs in the determination of enzyme mechanisms. The defining property of enzyme catalysts is their specificity for binding the transition state with a much higher affinity than the substrate. Triosephosphate isomerase (TIM), orotidine 5′-monophosphate decarboxylase (OMPDC), and glycerol 3-phosphate dehydrogenase (GPDH) show effective catalysis of reactions of phosphorylated substrates and strong phosphite dianion activation of reactions of phosphodianion truncated substrates, with rate constants k_cat/K_m (M^–1 s^–1) and k_cat/K_dK_HPi (M^–2 s^–1), respectively. Good linear logarithmic correlations, with a slope of 1.1, between these kinetic parameters determined for reactions catalyzed by five or more variant forms of each catalyst are observed, where the protein substitutions are mainly at side chains which function to stabilize the cage complex between the enzyme and substrate. This shows that the enzyme-catalyzed reactions of a whole substrate and substrate pieces proceed through transition states of similar structures. It provides support for the proposal that the dianion binding energy of whole phosphodianion substrates and of phosphite dianion is used to drive the conversion of these protein catalysts from flexible and entropically rich ground states to stiff and catalytically active Michaelis complexes that show the same activity toward catalysis of the reactions of whole and phosphodianion truncated substrates. There is a good linear correlation, with a slope of 0.73, between values of the dissociation constants log K_i for release of the transition state analog phosphoglycolate (PGA) trianion and log k_cat/K_m for isomerization of GAP for wild-type and variants of TIM. This correlation shows that the substituted amino acid side chains act to stabilize the complex between TIM and the PGA trianion and that ca. 70% of this stabilization is observed at the transition state for substrate deprotonation. The correlation provides evidence that these side chains function to enhance the basicity of the E165 side chain of TIM, which deprotonates the bound carbon acid substrate. There is a good linear correlation, with a slope of 0.74, between the values of ΔG^‡ and ΔG° determined by electron valence bond (EVB) calculations to model deprotonation of dihydroxyacetone phosphate (DHAP) in water and when bound to wild-type and variant forms of TIM to form the enediolate reaction intermediate. This correlation provides evidence that the stabilizing interactions of the transition state for TIM-catalyzed deprotonation of DHAP are optimized by placement of amino acid side chains in positions that provide for the maximum stabilization of the charged reaction intermediate, relative to the neutral substrate.

KEY REFERENCES

• Zhai X.; Amyes T. L.; Richard J. P.. Role of Loop-Clamping Side Chains in Catalysis by Triosephosphate Isomerase. J. Am. Chem. Soc. 2015, 137, 15185–15197.¹The report of a linear logarithmic correlation between kinetic parameters for wild-type and variant TIM-catalyzed reactions of a whole substrate and substrate pieces. This provides evidence that the two reaction transition states are stabilized by similar interactions with the enzyme.

• Zhai X.; Reinhardt C. J.; Malabanan M. M.; Amyes T. L.; Richard J. P.. Enzyme Architecture: Amino Acid Side-Chains That Function To Optimize the Basicity of the Active Site Glutamate of Triosephosphate Isomerase. J. Am. Chem. Soc. 2018, 140, 8277–8286.²The report of a linear correlation between dissociation constants log K_ifor release of the transition state analog phosphoglycolate trianion from wild-type and variants of TIM and log k_cat/K_mfor the respective enzyme-catalyzed isomerization reactions of GAP.

• Mydy L. S.; Cristobal J.; Katigbak R.; Bauer P.; Reyes A. C.; Kamerlin S. C. L.; Richard J. P.; Gulick A. M.. Human Glycerol 3-Phosphate Dehydrogenase: X-Ray Crystal Structures that Guide the Interpretation of Mutagenesis Studies. Biochemistry 2019, 58, 1061–1073.³The report of a linear logarithmic correlation between kinetic parameters for wild-type and variant GPDH-catalyzed reactions of a whole substrate and substrate pieces. This provides evidence that the two reaction transition states are stabilized by similar interactions with the enzyme.

• Kulkarni Y. S.; Amyes T. L.; Richard J. P.; Kamerlin S. C. L.. Uncovering the Role of Key Active-Site Side Chains in Catalysis: An Extended Brønsted Relationship for Substrate Deprotonation Catalyzed by Wild-Type and Variants of Triosephosphate Isomerase J. Am. Chem. Soc. 2019, 141, 16139–16150.⁴The report of a linear correlation between ΔG^‡and ΔG°, determined by electron valence bond calculations, for deprotonation of DHAP in water and when bound to wild-type or variants of TIM, to form the free or enzyme-stabilized enediolate reaction intermediate.

1. Introduction

The Hammett equation was used to establish linear free energy relationships (LFERs) between the effect of aromatic ring substituents X on ΔG° for the reference ionization of substituted benzoic acids in water and on ΔG° or ΔG^‡ for reactions at second aromatic frameworks (Figure 1).⁵ This equation spurred the characterization of other LFERs, including those correlated by equations developed by Taft, Brønsted, and Grunwald and Winstein.⁶ These LFERs are a cornerstone of physical organic chemistry.⁷

Figure 1.

LFER between the effect of changing X on (ΔG°)_ref for the reference ionization of ring-substituted benzoic acids and on (ΔG°)_X or (ΔG^‡)_X for a second reaction at a similar rigid organic framework.

Figure 2A shows a pair of reactions for which a LFER is observed. The correlation between ΔG° for ionization of substituted phenylacetic and benzoic acids in water, with a slope of 0.5,⁸ is consistent with a 2-fold falloff in electrostatic interactions between the ring substituent X and the reacting −COOH that arises from insertion of the methylene group. The slopes determined for Brønsted correlations of log k against the pK_a of varied reactants, such as Brønsted acid (slope of α) or base (slope of β) catalysts, or reacting alkyl alcohol nucleophiles (slope of β_nuc) or alkoxy leaving groups (slope of β_lg) generally lie between 0 and 1.0. These slopes provide a metric for the change in the effective charge “seen” by the varied substituent on proceeding to the reaction transition state, relative to the 1.0 unit change in the charge for the reference ionization reaction.⁹

Figure 2.

(A) The ionization of ring-substituted benzoic acids ((ΔG°)_ref, n = 0) and phenylacetic acids ((ΔG°)_X, n = 1). (B) Addition of substituted ethanols to ring-substituted 1-phenylethyl carbocations, with rate constants k_X (M^–1 s^–1), to form protonated alkyl 1-phenylethyl ethers and general-base catalysis by alkane carboxylate anions of nucleophile addition, with rate constants k_YAc (M^–2 s^–1), to form ring-substituted alkyl 1-phenylethyl ethers.

Brønsted coefficients provide insight into changes in bonding interactions during steps that are not easily probed by kinetic experiments. For example, the value of (β_nuc)_obs = −0.1 determined for phosphoryl transfer from phosphorylated pyridine to 3-substituted quinuclidines in water requires a surprising decrease in the effective positive charge at the quinuclidine nitrogen at the transition state for the phosphoryl transfer reaction, so that the reacting amine nitrogen must carry a partial positive charge at the ground state in water.¹⁰ These results were rationalized by Scheme 1, where β_des = −0.2 for removal of a hydrogen-bonded water from the tertiary amine nucleophile to form the desolvated encounter complex with the phosphorylated pyridine has a larger absolute value than (β_nuc)_chem = 0.1 for conversion of this encounter complex to the reaction transition state. There is a similar decrease in amine reactivity, with increasing amine basicity, for nucleophilic addition of amines to the 1-(methylthiophenyl)-2,2,2-trifluoroethyl carbocation.¹¹

Scheme 1. Transfer of a Phosphoryl Group from Phosphorylated Pyridine to 3-Substituted Quinuclidines in Water.

2. Beginnings

J.P.R. first encountered the Brønsted equation in studies on addition of alkyl alcohols to ring-substituted 1-phenylethyl carbocations. The values of β_nuc for these reactions (Figure 2B, no general base catalysis), determined as the slopes of correlations of log k_X against alcohol pK_a, were found to increase from β_nuc = 0.1 to β_nuc = 0.5 as the carbocation is stabilized by a change in the aromatic ring substituent from Z = 4-CH₃ to 4-NMe₂ (Figure 2B).¹² This increase in the effective transition state positive charge at the oxygen nucleophile, with the decreasing thermodynamic driving force for carbocation addition, arises from a Hammond-type shift from a reactant to product-like transition state, where there is an increase in bonding between the oxygen nucleophile and carbon electrophile.¹³

These addition reactions of alcohols are catalyzed by alkane carboxylate anions, with third-order rate constants k_YAc (Figure 2B).^14,15 The slopes of Brønsted-type correlations of log k_YAc against the pK_a of the conjugate acid of the catalyst increase from β = 0.23 to β = 0.33 for addition of trifluoroethanol, as the aromatic ring substituent is changed from Z = 4-OCH₃ to 4-NMe₂.¹⁵ This is consistent with an increase in transition state bonding between the alkane carboxylate base and the transferred alcohol proton (Figure 3A).¹⁶

Figure 3.

(A) Transition state for addition of trifluoroethanol to the 1-(4-dimethylaminophenyl)ethyl carbocation catalyzed by substituted alkane carboxylate anions (β = 0.33). (B) Transition state for the addition of substituted alkyl alcohols XCH₂CH₂OH to the 1-(4-dimethylaminophenyl)ethyl carbocation catalyzed by acetate anion. The value of β_nuc = 0.11 is consistent with the buildup of the positive charge at the nucleophilic oxygen. (C) Transition state for addition of XCH₂CH₂OH to the β-d-galactopyranosyl cation intermediate of reactions catalyzed by β-galactosidase. The values of β_nuc = −0.19 and β_nuc = −0.07 for Z = OH and Z = H, respectively, are consistent with a transition state where proton transfer from the alcohol to the enzyme is further advanced than bond formation between the alcohol and the sugar, leaving a net negative charge at the oxygen nucleophile.

Proton transfer from XCH₂CH₂OH to the carboxylate anion reduces the effective negative charge at the base catalyst and the effective positive charge at the oxygen nucleophile.¹⁶ The latter change is reflected in the decrease in β_nuc = 0.5 for the uncatalyzed addition of alkyl alcohols to the 1-(4-dimethylaminophenyl)ethyl carbocation to β_nuc = 0.11 for the addition reaction catalyzed by acetate anion (Figure 3B).¹⁵ These changes in first-derivative effects describe changes, with the changing reaction driving force, in the transition state structure: they define second-derivative effects. The results were modeled on a reaction coordinate profile, which assigns separate axes for bond formation between the alcohol oxygen and benzylic carbon, and for proton transfer from the alcohol oxygen to the carboxylate anion,¹⁷ using a protocol developed by William Jencks that expands upon a model from Bell, Marcus, Hammond, Polanyi, Thornton, and Leffler.¹⁶

3. Early Studies at Buffalo

Enzymes achieve rate accelerations through stabilization of their bound transition states by interactions with protein catalysts.¹⁸⁻²⁰ The structures of these transition states are of intellectual interest and may guide the development of tight-binding transition state analog enzyme inhibitors, with the potential to function as therapeutic agents.²¹ The first step in β-galactosidase-catalyzed hydrolysis of alkyl β-d-galactopyranosides is transfer of the β-d-galactopyranosyl group from the substrate to the carboxylate side chain of E537 (Figure 4),^22,23 by a stepwise mechanism where the acidic E461 side chain provides Brønsted catalysis of glycoside bond cleavage to form the β-d-galactopyranosyl cation intermediate,²⁴ which is then trapped by the E537 side chain (Figure 4). The similarity between the transition states for the general base-catalyzed addition of alkyl alcohols to (i) ring-substituted 1-phenylethyl carbocations (Figures 3A and 3B) and (ii) the cation formed at β-galactosidase (Figure 3C) prompted studies to compare the structure–reactivity parameters determined for these enzymatic reactions and the nonenzymatic model.²⁵⁻²⁷

Figure 4.

Stepwise β-galactosidase-catalyzed sugar transfer from alkyl β-d-galactopyranosides to the carboxylate side chain of E537, with Brønsted acid catalysis by the E461 side chain.

Figures 5A and 5B (●) show Brønsted plots of log k_cat/K_m and log k_ROH against alcohol pK_a, with slopes of β_lg = −0.75 and β_nuc = −0.19, respectively, for reversible β-galactosidase-catalyzed cleavage of alkyl β-galactopyranosides to form the alkyl alcohol and covalent reaction intermediate (Figure 4).^25,26Figure 5C shows the plot of equilibrium constants log K_eq = log[(k_cat/K_m)/(k_ROH)], with slope β_eq = −0.75 – (−0.19) = −0.56 for the β-d-galactopyranosyl group transfer from alkyl β-galactopyranosides to the enzyme.²⁵ The large negative β_eq of −0.56 shows that interactions between electron-withdrawing alkoxy substituents −X and the electron-deficient β-d-galactopyranosyl group or the reactant are destabilizing, compared to the interactions between these −X and the proton of the alcohol product. It is consistent with a sugar substituent that imparts an effective positive charge of +0.56 to the alkoxy oxygen relative to the normalized charge of 0.0 at the alcohol product.⁹ The value (β_nuc)_obs = −0.19 for alcohol addition is consistent with an effective transition state negative charge of −0.19 at the alkoxy oxygen. This is equal to the sum of the contribution of the effective positive charge from the β-d-galactopyranosyl group plus the true formal negative charge at oxygen. The value of (β_nuc)_obs underestimates the buildup of the formal negative charge at this oxygen, because stabilizing charge interactions at oxygen are masked [canceled] by destabilizing dipole–dipole interactions between the electron-deficient sugar and electron-deficient substituents at XCH₂CH₂OH.

Figure 5.

Brønsted correlations of kinetic parameters for β-galactosidase-catalyzed transfer of the β-d-galactopyranosyl group from alkyl β-d-galactopyranosides to the carboxylate side chain of E537, with rate-determining substrate ionization to form the glycosyl carbocation intermediate (Figure 4). (A) The correlation of rate constants (k_cat/K_m) for the reaction in the cleavage direction.²⁶ (B) Correlations of rate constants k_ROH for the reaction in the reverse, synthesis direction: (●) addition of alkyl alcohols to the β-d-galactopyranoside intermediate²⁵ and (▲) addition of alkyl alcohols to the 2-deoxy-β-d-galactopyranoside intermediate.²⁷ (c) The correlation of equilibrium constants K_eq for transfer of the β-d-galactopyranosyl group from the substrate to the enzyme.²⁵

The value of β_nuc = −0.19 for alcohol addition to the glycosyl cation intermediate reflects the balance between bond development between the alkoxy oxygen and the glycosyl carbon and proton transfer from oxygen to the carboxylate catalyst. It is consistent with a strong coupling of changes in C–O and H–O bonding at the reaction transition state,¹⁶ as was documented for catalysis of the reversible addition alkyl alcohols to formaldehyde and acetaldehyde.^28,29 A weaker coupling between these changes in bonding was determined for general base catalysis of alcohol addition to ring-substituted 1-phenylethyl carbocations (β_nuc > 0, Figure 3B), where motion on the reaction coordinate at the transition state is dominated by formation of the C–O bond, with smaller movement of the proton toward the Brønsted base catalyst.¹⁵

The C-2 hydroxyl of the β-d-galactopyranosyl cation reaction intermediate stabilizes the transition state for β-galactosidase-catalyzed addition of trifluoroethanol by 7.6 kcal/mol.³⁰ This is accompanied by a decrease in β_nuc for alcohol addition from β_nuc = −0.07 (▲, Figure 5B) to β_nuc = −0.19 (●).²⁷ The decrease in β_nuc, with increasing stability of the glycosyl cation intermediate, corresponds to an anti-Hammond shift of the transition state toward reactants,¹⁶ in contrast to the Hammond-type in the shift in the transition state observed for alcohol addition to ring-substituted-1-phenylethyl carbocations.^15,16 Anti-Hammond shifts in transition state structure have also been reported for general base catalysis of addition of alcohols to the carbonyl groups of formaldehyde and to acetaldehyde.^28,29 They were rationalized by a saddle point that shows a strongly coupled motion of the nucleophilic oxygen toward the carbon electrophile and of proton transfer from XCH₂CH₂OH to the base catalyst.¹⁶ These differences in coupling observed for general base catalysis of alcohol addition to ring-substituted 1-phenylethyl carbocations, compared to formaldehyde and acetaldehyde, were proposed to arise because of changes in the curvature of the energy surface in the region of the saddle point on a two-dimensional reaction coordinate profile.^16,31 These results are consistent with a similar tight coupling of changes in bonding for addition of XCH₂CH₂OH to the protein-stabilized sugar carbocation intermediates of reactions catalyzed by β-galactosidase.^24,27

4. Enzyme Catalysis at Protein Cages

The insight gained from studies on β-galactosidase has helped us to uncover LFERs of kinetic data for other enzymatic reactions. The transition states for decarboxylation catalyzed by orotidine 5′-monophosphate decarboxylase (OMPDC),^32,33 isomerization catalyzed by triosephosphate isomerase (TIM),³³⁻³⁵ and hydride transfer catalyzed by glycerol 3-phosphate dehydrogenase (GPDH)^33,36 show a similar 12 kcal/mol stabilization by interactions with the phosphodianion of the whole substrate (Figure 6). This transition state stabilization is calculated from the ratio of second-order rate constants for the enzyme-catalyzed reactions of whole and phosphodianion truncated substrates: [−RT ln[(k_cat/K_m)_W/(k_cat/K_d)_P] = 12 ± 1 kcal/mol). In all three cases, the slow enzymatic reaction of the truncated substrate is strongly activated by the substrate piece phosphite dianion. This activation corresponds to a 6–8 kcal/mol stabilization of the transition states for reactions of truncated substrates by enzyme-bound phosphite dianion.

Figure 6.

Reactions of whole substrates [(k_cat/K_m)_W], the phosphodianion truncated substrate [(k_cat/K_m)_P], and phosphite dianion activation of the truncated substrate [(k_cat/K_dK_HPi)] catalyzed by OMPDC, TIM, and GPDH.

The 12 kcal/mol stabilization of these enzymatic transition states by interactions with the substrate phosphodianion shows the power of this small fragment to promote efficient enzymatic catalysis. The large fraction of these interactions specifically expressed at these transition states shows that TIM, OMPDC, and GPDH have evolved mechanisms that avoid the full expression of the strong ligand binding at the Michaelis complex, in order to avoid tight and irreversible binding of the substrate.^20,37,38 The observation of dianion activation of enzyme-catalyzed proton and hydride transfer and decarboxylation reactions is significant because such activation had not previously been considered for these three intensively studied enzymes. This suggests that [many] other exciting results await discovery by mechanistic enzymologists. At the very least, these results are the tip of the iceberg of enzymatic reactions for which there is a large specificity of nonreacting substrate fragments in the stabilization of the enzymatic transition state.³⁹

The similarity in the kinetic results for activation of TIM, OMPDC, and GPDH by phosphite suggested a common mechanism for obtaining dianion specificity for transition state binding. The enzyme X-ray crystal structures provide the critical clue that reveals the mechanism of dianion activation. These structures show that TIM, OMPDC, and GPDH each utilizes phosphodianion binding energy to drive extensive conformational changes that are dominated by the closure of flexible phosphodianion gripper loops over the substrate.^3,40−44 Loop closure converts flexible open enzymes to closed Michaelis complexes, with the substrate trapped in a protein cage and sequestered from interaction with the solvent water.⁴⁵

Scheme 2 shows a compelling model for dianion activation. The catalytic activity of the ground-state open form (E_O) of these three enzymes is low, compared to the active closed form E_C, because the side chains are poorly positioned for catalysis. The enzyme activity toward catalysis of reactions of phosphodianion truncated substrates is likewise low, because only a small fraction of total enzyme is present in the active closed conformation (K_C ≪ 1.0), and there is insufficient substrate binding energy to drive the enzyme-activating conformational change. A high activity is observed for phosphorylated substrates and for the reactions of truncated substrates in the presence of phosphite dianion, because of utilization of dianion binding energy to convert these enzymes from inactive E_O to active E_C•S.^20,38,46Scheme 2 is the induced fit mechanism that was proposed more than 60 years ago,⁴⁷ in another context, and whose significance has since been a subject of debate.⁴⁸ It provides a general mechanism for avoiding the full expression of binding interactions at the ground state Michaelis complex, through the utilization of substrate binding energy to drive formation of closed, high-energy, and catalytically active protein cages, where the number of stabilizing protein–ligand contacts has been optimized.^38,49 For example, OMPDC-catalyzed decarboxylation of OMP proceeds through a transition state that is stabilized by ca. 31 kcal/mol by interactions with the protein catalyst but with only a modest 8 kcal/mol stabilization of the Michaelis complex to OMP.⁵⁰ Much or all of this 23 kcal/mol difference represents the ligand binding energy utilized to drive the enzyme-activating conformational change from E_O to E_C.⁴¹

Scheme 2. Enzyme Activation by a Ligand Driven Conformational Change that Converts an Inactive Open Enzyme (E_O) to the High Energy Catalytically Active Substrate Cage (E_C).

We have examined the mechanism for dianion activation in studies of variant enzymes prepared by site-directed mutagenesis. The positions of the active site side chains examined for yeast TIM are shown in Figure 7A for the complex to the phosphoglycolate (PGA) trianion.⁵¹ Most of these side chains are buried beneath the phosphodianion gripper loop 6 (shaded green, Figure 7A) and play structural roles in optimizing the reactivity of the enzyme-bound substrate toward deprotonation.^1,52−54

Figure 7.

Representations from X-ray crystal structures of enzyme active sites, where the bound ligands are buried beneath the shaded protein surfaces: (A) The complex between TIM and PGA (PDB ID 2YPI). The phosphodianion gripper loop is shaded green. Figure 8 shows a correlation of kinetic parameters for wild-type TIMs and for the following enzyme variants: I170 V, I170A, Y208T, Y208A, Y208S, Y208F, Y208T/S211G, P166A, S211A, and S211G.^1,53 (B) The complex between OMPDC and 6-hydroxyuridine 5′-monophosphate (BMP, PDB ID 1DQX). The phosphodianion and pyrimidine gripper loops are shaded blue and pink, respectively. Figure 8 shows a correlation of kinetic parameters for wild-type OMPDC and the following variants: Q215A, Y217F, Q215A/R235A, R235A, S154A, and S154A/Q215A.⁵⁵ (C) The nonproductive complex between GPDH, NAD, and DHAP (PDB ID 6E90), with the flexible capping loop shaded red and additional surface residues shaded blue. Figure 8 shows a correlation of kinetic parameters for wild-type hlGPDH and for the following enzyme variants: Q295G, Q295S Q295A, Q295N, and K120A.^3,56

The kinetic parameters k_cat/K_m for catalysis of the reaction of the whole substrate and k_cat/K_dK_HPi for phosphite dianion activation of the catalyzed reactions of the substrate piece glycolaldehyde (Scheme 3) were determined for catalysis by TIM. We first focused on rationalizing the effects of individual mutations on these rate constants^57,58 but then noted an excellent linear free energy relationship, with a slope of 1.06 ± 0.04,¹ between the effect these substitutions have on log k_cat/K_m and log k_cat/K_dK_HPi (Figure 8 (●)). This correlation shows that the individual substitutions cause nearly the same change in the stability of the transition states for the two reactions. In other words, these transition states are nearly identical over the regions affected by these substitutions. There is a roughly constant difference of ΔΔG^‡ ≈ 6.6 kcal/mol in the activation barriers for TIM-catalyzed reactions of the whole substrate compared with the substrate pieces. This represents the entropic advantage to binding and reaction of the single whole substrate, compared with the two pieces.^1,53,59

Scheme 3. Enzyme-Catalyzed Reactions of the Whole Substrate and Phosphite Dianion Activated Reactions of Truncated Substrates with Rate Constants k_cat/K_m (M^–1 s^–1) and k_cat/K_dK_HPi (M^–2 s^–1), Respectively.

Figure 8.

Linear free energy relationships between the effect of site-directed substitutions at TIM,¹ OMPDC,⁵⁵ and GPDH³ on the kinetic parameters k_cat/K_m and k_cat/K_dK_HPi for enzyme-catalyzed reactions of whole and truncated substrates, respectively. The positions of the substituted side chains at the enzyme active sites are shown in Figure 7. The values of k_cat/K_m are for TIM-catalyzed isomerization of GAP, OMPDC-catalyzed decarboxylation of OMP, and GPDH-catalyzed reduction of dihydroxyacetone phosphate by NADH. The truncated substrates are glycolaldehyde (TIM and GPDH) and 1-(β-d-erythrofuranosyl)orotic acid (OMPDC).

The correlation from Figure 8 was rationalized by Scheme 3. The interactions between TIM and the substrate phosphodianion, or the phosphite dianion piece, are utilized to drive the common enzyme conformational change from E_O to E_C. This moves active-site side chains into positions that provide the same stabilization of the transition states for deprotonation of whole or truncated carbon acid substrates. The results of a high-level electron valence bond (EVB) computational study to model the reactivity of E_C, using the X-ray structure of the Michaelis complex to DHAP,⁴⁴ gave similar activation barriers for deprotonation of enzyme-bound whole and truncated substrates to form the respective enediolate reaction intermediates. This is consistent with Scheme 2, where dianion binding interactions are utilized to stabilize a caged Michaelis complex, which shows the full catalytic activity toward deprotonation of both the truncated substrate glycolaldehyde and the whole substrate GAP.⁶⁰

Figure 8 shows two additional LFERs between the values of log k_cat/K_m and log k_cat/K_dK_HPi (Scheme 3), with slopes of 1.065 ± 0.08 and 1.09 ± 0.03, respectively, for reactions catalyzed by wild-type and variant forms of OMPDC⁵⁵ and GPDH³ (Figure 6), with the positions of the substituted side chains shown in Figures 7B and 7C, respectively. None of the side chains replaced act to directly stabilize the transition states for the respective wild-type enzyme-catalyzed reactions, except for the cationic side chain of K120 from GPDH, which provides electrostatic stabilization of the negative charge that develops at the transition state for enzyme-catalyzed hydride transfer from NADH to DHAP.^3,61 In the other cases, the side chains function to lock these enzymes into their closed conformations (E_C), mainly though interactions with the substrate dianion. The near unit slopes for these correlations show that the amino-acid substitutions have nearly the same effect on the stability of the transition states for the enzyme-catalyzed reactions of whole substrates and the substrate pieces and are consistent with the conclusion that E_C shows nearly the same activity toward catalysis of reactions of whole substrates and the substrate pieces.⁶²⁻⁶⁴ As noted above, this is predicted by the model shown in Scheme 2.

We have observed other linear free energy relationships for reactions catalyzed by TIM, OMPDC, and GPDH. For example, third-order rate constants k_cat/K_dK_X (M^–2 s^–1) were determined for the activation of GPDH-catalyzed reactions of the truncated substrate glycolaldehyde by fluorophosphate, phosphite and sulfate dianion, and fourth-order rate constants k_cat/K_dK_XK_Gua(M^–3 s^–1) were determined for the activation of the R269A variant-catalyzed reaction by the combined effects of guanidine cation and each of the above three anions. The excellent LFER between values of log k_cat/K_dK_XK_Gua and log k_cat/K_dK_X (not shown) for the reaction of these three dianions, with a slope of 1.0, shows that the R269A substitution does not significantly alter the interaction of these dianions with the enzyme-bound transition state complex, so that the structural integrity of the caged complex is maintained at the R269A variant.⁶³Figure 9 shows a plot, with a slope of 0.95 and r² = 0.26, for activation of TIM- and GPDH-catalyzed proton and hydride transfer reactions, respectively, by several different tetrahedral dianions. The poor quality of this correlation shows that there are significant differences in the stabilizing interactions of these different dianions at the active sites of TIM and GPDH.³³

Figure 9.

Correlation between third-order rate constants k_cat/K_dK_X (M^–2 s^–1) for activation of GPDH and TIM by several tetrahedral dianions.³³

5. Transition State Analogs

Enzyme catalysts show specificity in binding their transition states with a higher affinity than the substrate.¹⁸ The tight binding observed for analogs of the transition states for enzymatic reactions provides evidence for this proposal.⁶⁵ The good linear correlation between values of log K_i for binding of phosphonoamidate peptide analogs (1) to thermolysin and log (k_cat/K_m) for thermolysin-catalyzed hydrolysis of the corresponding peptide substrates (2)⁶⁶ provides strong evidence that thermolysin stabilizes the covalent tetrahedral reaction intermediate formed by addition of water to the peptide substrate.

PGA binds to triosephosphate isomerase (K_i, Scheme 4) with a substantially higher affinity than substrates GAP and DHAP⁶⁵ and is a putative analog of the enediolate reaction intermediate (Scheme 4).⁶⁷Figure 10 shows the correlation, with a slope of 0.73 (r² = 0.96), between values of log K_i and log k_cat/K_m for the isomerization of GAP catalyzed by wild-type and variants of TIM.² This correlation shows that the amino acid side chains substituted for at TIM (Figure 7) act to stabilize the complex between the enzyme and PGA trianion and that ca. 70% of this stabilization is observed at the transition state for substrate deprotonation.²

Scheme 4. Inhibition of TIM by PGA.

Figure 10.

Linear logarithmic correlation between values of K_i for the breakdown of PGA-TIM complexes (Scheme 4) and k_cat/K_m for isomerization of GAP catalyzed by wild-type TIM from yeast (yTIM) and chicken muscle (cTIM) Trypanosoma brucei (TbbTIM) and by variant forms of these enzymes.²

The binding of the PGA trianion to TIM results in an a > 6 unit increase in the pK_a of the E165 side chain, from pK_a = 4 at free TIM to pK_a > 10 at the complex to PGA (Figure 11).^2,68 This is due to the combined effects of stabilization of the inhibitor complex by a hydrogen bond between the PGA trianion and the protonated E165 side chain and destabilizing electrostatic interactions between the inhibitor trianion and the ionized side chain anion. A similar increase in the pK_a for this side chain should occur upon proton transfer from the carbon acid substrate to the E165 carboxylate, to form the enediolate anion reaction intermediate (Scheme 5 and Figure 11).^2,68 The high basicity of the E165 side chain at the complex to the enediolate reaction intermediate analog PGA [(pK_a)_COOH, Figure 11] reflects the tight packing of PGA into the solvent-sequestered enzyme active site.^45,69,70 This promotes formation of the enediolate intermediate, provided that the enhanced side-chain basicity is expressed at the transition state for substrate deprotonation.⁷¹ This model is supported by the observation that several side-chain substitutions at TIM, which result in decreases in log K_i for inhibition by PGA (Figure 10), also result in decreases in (pK_a)_COOH (Figure 11) for the E165 side chain at the complex to PGA.^2,68

Figure 11.

Deprotonation of TIM-bound GAP by the E165 active site side chain with equilibrium constant K_eq, where K_eq = [(K_a)_CH/(K_a)_COOH] for proton transfer between the enzyme-reactant or enzyme-intermediate complex and water.

Scheme 5. Deprotonation of DHAP by Propionate in Water (Top Reaction) and by the Carboxylate Side Chain of E165 at the Active Site of Triosephosphate Isomerase.

6. Imperatives for Catalysis of Deprotonation of Carbon

The activation barrier ΔG^‡ for nonenzymatic deprotonation of the relatively weak carbon acid DHAP (pK_CH ≈ 18) by the weak base propionate anion (pK_COOH ≈ 5, Scheme 5) is the sum of the thermodynamic reaction barrier ΔG° for proton transfer plus the contribution from the intrinsic kinetic reaction barrier Λ.⁷² It is not clear that Λ may be reduced for reactions at enzyme active sites compared to water, and attention has been focused on quantifying the reduction in ΔG°. EVB calculations to determine ΔG^‡ and ΔG° for deprotonation of DHAP bound to wild-type and variant forms of TIM to form the enediolate reaction intermediate give the linear free energy correlation shown in Figure 12, with a slope of 0.74 (r² = 0.99).⁴ The agreement between the computed activation barriers ΔG^‡ and the experimental barriers estimated from k_cat for TIM-catalyzed isomerization of GAP is within ±1 kcal/mol.

Figure 12.

Linear free energy relationship between the calculated activation barrier ΔG^† and the reaction free energy ΔG° for proton transfer from DHAP to propionate anion, wild-type TIM, and variants of TIM.⁴ Key: ●, reactions catalyzed by wild-type TIM and propionate anion; ■, I170A and L230A variants; ▲, K12G and E97 variants; ⧫, P166A variant.⁴

The substituted amino acid side chains at TIM (Figure 12) play a role in either maintaining the strong basicity of E165 (I170A, P166A, and L230A)^2,68 or in the direct stabilization of the enediolate intermediate (K12 and E97).⁷³ The Brønsted-type correlation shows that these active site side chains function to reduce ΔG° for deprotonation of the weakly acidic α-carbonyl carbon acid substrate to form the enediolate phosphate trianion intermediate (Scheme 5) and that ca. 70% of this stabilization is expressed at the transition state for formation of this intermediate and as a decrease in ΔG^‡. These results provide evidence that proton transfer from the substrate to TIM is promoted by placement of active site side chains at positions which impart an optimal basicity to the E165 side chain and optimal stabilization of the negative charge at the reaction intermediate.⁴

7. Concluding Remarks

The transformation of tables of numerical rate data into graphical LFERs often brings clarity to kinetic analyses. At the very least, these LFERs provide a framework for discussion that may reveal differences in the interpretation and conclusions of studies on reaction mechanisms. These conversations sometimes motivate additional experiments that provide a spark for progress.

The early work on β-galactosidase-catalyzed glycosyl transfer reactions represented an attempt to extend classical applications of LFERs to studies on enzyme catalysis. Our interest in LFERs then decreased, as the focus of our research shifted to more traditional studies on enzymatic catalysis. However, the opportunities discussed in this Account have resulted in the seamless incorporation of LFERs into analyses of kinetic and computational data pertinent to the effect of active-site directed substitutions on enzyme kinetic parameters. We now understand that LFERs may be observed for kinetic data that correlate the effect of protein substitutions on the rate and/or equilibrium constants for enzymatic reactions, with the caveat that the active-site substitutions are not accompanied by nonspecific changes in protein structure; and note that similar relationships have featured strongly in the work of Fersht⁷⁴ and Auerbach.⁷⁵ The few examples presented here illustrate the potential of LFERs as tools for the analysis and interpretation of kinetic data for enzymatic reactions. We encourage others to search for additional relationships in their studies on enzymatic reaction mechanisms.

Biographies

John P. Richard received a B.S. degree (1974) in biochemistry at Ohio State and a Ph.D. degree (1979) in chemistry at the same University under the direction of Perry A. Frey. He was introduced to linear free energy relationships by his postdoctoral advisor William P. Jencks. He is currently a SUNY Distinguished Professor of Chemistry at the University at Buffalo, SUNY.

Judith R. Cristobal received her B.S. degree in chemistry (2009) from the University of the Philippines Los Baños and taught chemistry at this University from 2009–2016. She is currently a Ph.D. candidate at the University at Buffalo, SUNY.

Tina L. Amyes received B.A. and M.A. degrees in natural sciences and her Ph.D. degree in chemistry (1986) from the University of Cambridge (England). She currently works as a Scientific Program Manager at the Roswell Park Cancer Institute.

The authors declare no competing financial interest.