Predicting Activation Barriers With Ring and Bond Critical Points in the Kemp Elimination Reaction

Abstract

The Kemp elimination reaction can be catalyzed by computationally designed enzymes. After undergoing directed evolution, changes to the positioning of active site residues in Kemp eliminases can alter electric fields (EFs) and improve catalytic efficiency. However, optimizing EFs during the enzyme design process remains a challenge. Here we investigate how implicit solvation and external EFs reposition bond and ring critical points (CPs) in the reactant state of a truncated Kemp eliminase system using density functional theory and the quantum theory of atoms in molecules. We find that a linear correlation exists between the bond‐ring CP distances integral to the Kemp elimination ring‐opening reaction and activation barriers. The N─O stretching mode of the benzisoxazole ring that coincides with the reaction coordinate is also found to move ring CPs to varying degrees for each EF strength tested. EFs that move the ring CPs the largest magnitude and in the proper orientation toward the N─O bond CP correlate with the smallest activation barriers. Since reactant state QTAIM calculations are significantly less expensive than directly calculating activation barriers, this may provide an avenue for screening computationally designed enzymes that catalyze ring‐opening reactions.

Electric fields (EFs) alter activation barriers in the enzymatic Kemp elimination reaction and reposition bond and ring critical points (CPs). CPs that start closer together in the reactant state correlate with lower activation barriers, as it takes less energy to move the CPs together for the ring‐opening reaction.

1. Introduction

Predicting the efficiency of computationally designed enzymes is a challenging but potentially rewarding task. Much of this challenge arises from an incomplete understanding of the origin of an enzyme’s catalytic power. Slight changes to the environment within and outside of the enzyme active site have been shown to often have large impacts on reaction kinetics, which has been especially evident through laboratory‐directed evolution experiments [1, 2, 3, 4, 5, 6, 7]. Some ways these mutations have been studied include changes to the geometry of active sites [2, 3, 6, 7, 8], altered electric field (EF) environments [2, 9, 10, 11, 12, 13, 14], residue vibrations that couple to rate‐promoting vibrations (RPVs) of the reaction coordinate [15, 16, 17], and protein dynamics [1, 3, 13]. In terms of EFs, the position, polarity, and charge of each amino acid residue alter the EF in the enzyme active site. The magnitude and direction of this field along relevant bonds have been shown to correlate with activation barriers [18, 19]. Warshel has proposed that an enzyme’s catalytic ability is thus due to the electrostatic preorganization of the active site [20, 21]. Solvent choice has also been shown to affect activation barriers. For both enzymatic and nonenzymatic reactions, correlations have been found between modeled dielectric constants, ε, and rates of reactions [19, 22, 23, 24].

The quantum theory of atoms in molecules (QTAIM) enables the quantification of subtle changes to the electron charge density, ρ( r ), whether these changes are due to solvation, EFs, neighboring molecules, or atomic motion during vibrations or chemical reactions [25, 26, 27, 28, 29]. The presence of bond critical points (CPs) in ρ( r ) generally indicates stabilizing bonding interactions in chemical species. Ring CPs exist in ρ( r ) any time rings of bond CPs that are connected by bond paths form. These rings can be composed entirely of covalent bonds or other interactions such as hydrogen bonds. From a topological perspective, ring‐opening reactions occur as the ring CP and the bond CP of the bond to be broken physically move toward each other until the CPs coalesce and topologically annihilate one another. In carboxypeptidase A, it was shown that the ring CP formed between the substrate carbonyl, a water oxygen, and a zinc cation in the active site moved significantly in ρ( r ) when mutations were made to the enzyme, corresponding to changes in activation barriers [30]. When modeling a histone deacetylase, the protein environment around the active site was shown to also have a large effect on the position of the ring CP vital to reactivity [27]. These studies suggest that the positions of CPs in the reactant state can be used as an inexpensive predictor of catalytic efficiency.

In a similar vein, the eigenvectors of the electronic stress tensor at CPs have been used to show how bonds will form and break during chemical reactions [31]. For an enol‐to‐aldehyde tautomerization reaction as well as an HONS isomerization to ONSH, it was found that the least compressive eigenvector of the stress tensor at the O─H bond CPs in the reactant states showed the lowest energy pathways for the CPs to move in order to break the bonds for both reactions. At the transition states, rings were temporarily formed in both reactions, and the eigenvectors at the ring CPs indicated the most “crushable” directions of the rings to proceed to the product states. It has been suggested that the stress tensor is a meaningful value for predicting chemical reactivity as the trace of the stress tensor is proportional to the kinetic energy density and kinetic energy is known to be useful for characterizing valence electron behavior [31, 32, 33, 34, 35, 36]. However, one must keep in mind that CPs are not static, even in species that are not currently undergoing reactions, due to vibrations. Shahbazian has proposed that multiple molecular graphs should be considered in QTAIM as vibrational motions cause CPs to move and sometimes even disappear [37].

Vibrations during enzymatic reactions have recently been studied in terms of RPVs. Certain enzymes, such as liver alcohol dehydrogenase, have been shown to possess residues that undergo vibrational motions at similar frequencies as the RPVs for hydride and proton transfers [38, 39, 40, 41]. These residues vibrate in such a way as to push the donor and acceptor species closer together at the same frequency that the reaction occurs. Vibrations, whether they are correlated to RPVs or not, also physically move the positions of CPs in ρ( r ).

QTAIM descriptors have previously been employed for quantitative structure–activity relationships (QSAR). Scalar properties calculated at bond CPs (e.g., ρ( r ), ∇² ρ( r ), ellipticity) [42, 43, 44, 45], atomic properties [46], localization‐delocalization matrices [47], and vector properties at CPs calculated through next‐generation QTAIM [48, 49] have all been used for predicting properties of molecules, most often for drug design applications. This paper explores the utility of ρ( r ) values at CPs and positions of CPs in static structures and during vibrations for predicting activation barriers. Anytime predictive models are created, one should keep in mind that to predict is not to explain [50]. The focus of this work is on uncovering correlations, but these correlations do not necessarily imply causation. However, it has been argued that QSAR models based on electron density have the potential to uncover physical origins of chemistry more so than most models and thus are a promising direction if the end goal is understanding in addition to predicting [47].

Here we focus on the ring‐opening reaction catalyzed by Kemp eliminase to study how the positions of ring and bond CPs and their ρ( r ) values in reactant states relate to activation barriers of a ring‐opening reaction. Kemp eliminase is an artificial enzyme that has been computationally designed, undergone many rounds of directed evolution experiments, and for which a multitude of crystal structures exist [7, 51, 52, 53, 54, 55]. Kemp eliminase is a highly studied enzyme system, lending itself well as a test case for predictive CP models. The Kemp elimination reaction involves a base (usually Asp or Glu) abstracting a proton from a substituted benzisoxazole (typically nitrobenzisoxazole) substrate, which breaks the isoxazole N─O bond in a ring‐opening reaction. A truncated version of this enzymatic reaction modeled in this work is shown in Figure 1.

FIGURE 1.

Kemp elimination minimal model with external electric field direction. (A) Kemp elimination reaction scheme. (B) QTAIM representation of reactant state showing +EF orientation pointing from Asp oxygen atom to benzisoxazole carbon atom along the z‐axis. The purple arrow in the inset indicates the bond (red circle) to ring (green circle) CP distances measured.

We hypothesize that as the isoxazole ring CP moves closer to the N─O bond CP in the reactant state, the activation energy will decrease. We computationally alter the reaction environment to change the positions and ρ( r ) values of the CPs by varying either the dielectric constant with an implicit solvation model or the magnitude of external EFs. We then consider the direction of motion of the ring and bond CPs during vibrational modes that are relevant to the reaction coordinate by calculating the effect of individual modes on the CP positions. The CP motion during vibrations is compared with the direction the ring and bond CP must move to annihilate each other during the ring‐opening reaction. Since calculating reactant state CP positions is significantly faster than determining activation barriers, CP positions may provide an inexpensive screening method for catalyst design.

2. Computational Methods

All calculations were performed with the Amsterdam density functional (ADF) module of the Amsterdam modeling suite (AMS) [56, 57] using the M06‐2X functional and an all‐electron TZ2P basis set [58]. To ensure the most consistent geometries between models with applied EFs and solvation, a minimal model was employed that included only a benzisoxazole substrate molecule and a truncated Asp residue, as has been done in other work [19], resulting in an overall system charge of −1.

For systems including COSMO implicit solvation [59], all solvent radii were fixed at the van der Waals radius of water (1.93 Å) and ε values were set at 1, 3, 6, 8, 25, 45, 55, and 78. These ε values were chosen to represent some common solvents, tetrahydrofuran (8), dimethyl sulfoxide (45), and water (78), and to provide additional values for denser data. These solvation settings were applied to transition state searches, intrinsic reaction coordinates (IRCs), and single‐point calculations with frequencies enabled on resulting reactant states for Gibbs energy calculations.

For calculations where an external EF was applied, an oxygen atom of the Asp residue and the carbon atom of the benzisoxazole losing a proton were used to define the origin (Asp oxygen) and z‐axis (see Figure 1). The x‐ and y‐coordinates of these two atoms were held fixed during transition state calculations to ensure the external EF remained aligned with the reaction axis, resulting in either inhibition (+EF) or facilitation (−EF) of the flow of electrons. The EF direction is defined here such that the field points from a negative source to a positive sink, resulting in electrons generally moving opposite the EF, which matches AMS notation. EF strengths ranged between −0.25 and +0.25 V/Å at intervals of 0.05 V/Å. All EF calculations were performed in the gas phase.

All transition states were verified as having one imaginary frequency. Subsequent IRCs with all geometry constraints removed, following the vibrational mode corresponding to the imaginary frequency, were performed to ensure each transition state corresponded to the appropriate reactant and product states. For EF calculations, some rotation off the z‐axis (EF direction) occurred during IRCs. For these calculations, the molecular system was rotated to realign the Asp oxygen atom and benzisoxazole carbon atom with the z‐axis before calculating reactant state energies. This rotation was performed to better mimic the positioning of the benzisoxazole molecule if it were constrained within the active site of an enzyme.

Activation energies were calculated from the Gibbs free energy difference between the resulting reactant state and the verified transition state. Gibbs energies for stationary points include zero‐point energy, vibrational enthalpy, and entropy, which are obtained from a harmonic vibrational frequency calculation. Low vibrational frequency free rotor interpolation corrections were applied for frequencies less than 20 cm⁻¹ [60]. A self‐consistent field convergence criteria of 10⁻⁶ (error of the commutator of the Fock matrix and the density matrix in the representation of the basis functions from which the Fock matrix was obtained) was used for all calculations. Stationary points on the potential energy surface (transition and reactant states) used geometry convergence criteria of 10⁻³ Hartree/Å for gradient convergence, 10⁻⁵ Hartree per atom for energy convergence (resulting in 2.1 × 10⁻⁴ Hartree), and 10⁻² for step convergence. QTAIM analysis was performed within ADF on the resulting reactant molecule structures with default 0.5 a.u. cartesian grid spacing for starting points during the CP search [61].

For EF systems, the calculated normal modes of each reactant state were examined to identify frequencies that indicated atom displacements in directions relevant to the reaction coordinate. These modes involved either the stretching of the isoxazole proton to be abstracted toward the catalytic Asp or the stretching of the isoxazole N─O bond.

To determine how CP positions changed over the duration of a vibrational mode, the atomic displacements of each chosen normal mode in the reactant state were scaled by a factor of 0.5 from the output files. This scaling factor was chosen to ensure that atoms did not overlap and to ensure calculation convergence. The magnitude of vibrational motions was also considered from QM/MM simulations on Kemp eliminase enzymes from a directed evolution study [16]. This study showed changes in the N─O bond of approximately 0.6 Å and the C─H bond of approximately 0.1 Å before the Kemp elimination reaction occurred during transition state sampling. These values loosely correspond to scalings of 0.17 for isoxazole C─H and 0.30 for ring opening of N─O, which gave qualitatively similar results to 0.5 scaling (see Figures S1 and S2). For each normal mode selected for each system, two displaced systems were generated by addition or subtraction of scaled displacements to reactant state coordinates. Single‐point calculations were performed on the added and subtracted displaced systems with the same computational settings as above. Displaced structures for one EF strength with CPs are shown in Figure 2.

FIGURE 2.

Molecular geometries and CP positions for  −0.25 V/Å reactant state due to N─O bond stretch, with a blue arrow indicating the direction of +EF. (A) Subtracted displacement geometry, (B) nondisplaced geometry, (C) added displacement geometry.

3. Results and Discussion

During the Kemp elimination reaction, the N─O bond CP and the isoxazole ring CP physically move toward one another until they coalesce. At this point, the two saddle points annihilate one another and the topology of the charge density changes to indicate that the N─O bond is broken and the isoxazole ring no longer exists. We thus hypothesized that the closer the bond and ring CP are to one another in the reactant state, the less distance they will have to travel for the ring to open, and the lower the activation barrier will be. However, there are two ways that one can measure the distance between the ring and bond CP. One can consider the Euclidean distance in space, which is shown in Figure 1, or the difference in ρ( r ) between the points. For the CPs to coalesce, they must come to the same point in 3D space and have the same value of ρ( r ). Here we calculate both these distance metrics and compare them to ΔG ^‡ when varying either implicit solvation ε values or external EF strength. We then consider how vibrational motions relevant to the reaction coordinate affect the positions of reactant state CPs for each EF strength.

3.1. Solvation

Other works have shown that an increase in dielectric constant resulted in decreased reaction barriers when there were high concentrations of charge in the transition state [22, 24, 62, 63]. The stabilization of the transition state can be attributed to the partially charged molecules interacting with a polarizable solvent. In cases where the transition state was less polar than the corresponding reactant state, an increase in the dielectric constant resulted in an increased activation barrier [19, 22, 23] due to the polarizable solvent stabilizing the reactant state more than the transition state. In both cases, altering the dielectric constant has been found to have diminishing returns, resulting in a logarithmic relationship between ΔG ^‡ and ε. For the Kemp elimination reaction shown in Figure 1, the reactant state has a  −1 charge on the Asp and a neutral charge on the benzisoxazole. The transition state (which has partial charges during the proton transfer) has less charge separation than the reactant state, and it has previously been demonstrated that a logarithmic relationship between ΔG ^‡ and ε exists for this reaction [19]. Our truncated system showed similar results with an R ² value of 0.7171 (Figure 3A). Due to the limited number of data points (eight), the R ² values should only be considered as a rough estimate of the fit. Furthermore, residual plots for solvation data indicate small residuals for low ε values and large residuals (positive and negative) for larger ε values, in line with the large spread of data for more polar solvents (Figure S3).

FIGURE 3.

Effect of dielectric constant on Kemp elimination activation barrier (A), difference in ρ( r ) values between relevant bond and ring CPs (B), and Euclidean distance between bond and ring CPs (C).

Figure 3 also shows the relationship between either the Δρ( r ) between the bond and ring CPs (Panel B) or the distance between the bond and ring CPs (Panel C) in the reactant state for each ε value tested. Both metrics show a logarithmic relationship with a slightly better R ² value for ρ( r ). Figure 4 displays the relationship between activation barrier and either Δρ( r ) (Panel A) or CP distance (Panel B) in the reactant state. Both metrics were found to have a linear relationship with the activation barrier, with R ² values of 0.7189 and 0.6743, respectively. For low ε values, there is a larger spread in the data with a clear linear relationship. For larger ε values, CP positions and ρ( r ) values are no longer able to differentiate between high and low activation barriers. This is consistent with an increase in the dielectric constant having diminishing returns with altering activation barriers.

FIGURE 4.

Relationships between Kemp elimination activation barrier and difference in ρ( r ) values between relevant bond and ring CPs (A) and Euclidean distance between bond and ring CPs (B) when varying dielectric constants.

To further investigate this relationship, ρ( r ) values at bond and ring CPs were considered individually and compared to the dielectric constant. As the value of ε increased, both bond and ring CPs increased in ρ( r ) values following similar logarithmic trends as CP distance and Δρ( r ) (Figure S4). Though the bond and ring CPs both followed logarithmic trends, the bond CP ρ( r ) values increased by 0.011 e/ų as ε increased from 1 to 78, while the ring CP ρ( r ) values only increased by 0.00031 e/ų, leading to an increasing difference between the two values as ε increased. Furthermore, individual bond and ring CP ρ( r ) values also displayed a linear relationship with calculated activation barriers as shown in Figure S5. Interpretation of individual bond and ring CP ρ( r ) values will be discussed in more detail in the EFs section.

3.2. EFs

Figure 5A shows how different strengths of external EFs along the reaction axis affect the activation barriers of the Kemp elimination reaction. As has been found in other works [19], a linear relationship exists with −EFs catalyzing the reaction and +EFs inhibiting the Kemp elimination reaction. Similarly, Boxer et al. showed that a linear relationship exists between the magnitude of EF in the direction of the transition state dipole and the activation barrier for other enzymatic reactions [18, 64]. A linear relationship was also found between the difference in ρ( r ) values at the bond and ring CPs and EF strengths (Figure 5B) and when comparing the Euclidean distance between CPs with EF strengths (Figure S6). However, similar to solvation results, considering Δρ( r ) insead of Euclidean distance resulted in a slightly higher R ² value. Figure 5C directly compares the difference in ρ( r ) at the CPs to activation energies. The strong linear relationship found here suggests that the values of ρ( r ) in the reactant state CPs may be useful as a predictor of the activation barrier. Unlike solvation results, the correlation between activation barrier and Δρ( r ) from applied EFs is relatively consistent across the entire range of EF strengths tested (see residual plots in Figure S3). The relationship between activation barrier and Euclidean distance between CPs showed a similar relationship but with a slightly lower R ² value of 0.8213 (Figure S6).

FIGURE 5.

Effect of EF strength on Kemp elimination activation barrier (A) and difference in ρ( r ) values between relevant bond and ring CPs (B). Also shown is the relationship between reactant state CP ρ( r ) values and activation barriers for various EF strengths (C).

It was found that both bond and ring CPs increased individually in charge density as the EF magnitude increased, similar to the results for dielectric constants (Figure S7). The values of ρ( r ) again increased at a more rapid rate for the bond CP than the ring CP with +EFs, resulting in larger differences in ρ( r ) values between the ring and bond CPs overall. Separate plots were also created for bond and ring CP ρ( r ) values against activation barrier, and linear correlations were again found with R ² values of 0.8762 and 0.8943, respectively (Figure S8).

To understand these results, consider that the bond force constants from a set of diatomic molecules have previously been shown to linearly correlate with EF strength [65]. For these same systems, a linear correlation was also found between EF strength and values of ρ( r ) at bond CPs [66]. These relationships in diatomic molecules provide insight into the observed relationships between activation barriers and changes in ρ( r ) from external EFs in the Kemp elimination reaction. The −EF direction that catalyzed the reaction also lowered the bond CP ρ( r ) value and thus lowered the bond force constant, weakening the N─O bond [67, 68]. Activation energies are fundamentally related to the underlying potential energy surface that also defines bond force constants. Stronger bonds with larger force constants generally exhibit higher activation barriers for bond cleavage, as demonstrated through correlations with bond and lattice energies [69, 70].

Interpretation of values of ρ( r ) at ring CPs is less straightforward. Other work has shown that increasing values of ρ( r ) at ring CPs corresponded to increasing ring strain energy in some four‐membered rings [71] and to higher aromatic properties within benzenoids [72]. However, there is no known direct correlation between ring CP properties and aromaticity for heterocycles in general. In this work, −EFs that catalyze the Kemp elimination reaction decreased the values of ρ( r ) at isoxazole ring CPs, suggesting that the aromaticity index may also decrease, resulting in weaker bonds.

It should be noted that when modeling the Kemp elimination reaction, the acetate moiety underwent significant rotation from the transition state to the reactant state in the +0.15 and +0.25 V/Å EF systems (Figure S9). This rotation was nearly identical in both systems and would not be physically possible during the enzymatic reaction due to the protein backbone holding the pertinent residues in place and the limited space for rotation in the active site. Exclusion of these two data points significantly improved the value of R ² for the correlation between activation barrier and the difference in ρ( r ) from 0.8740 to 0.9489 (Figure S10). Note that Figure 5 includes the data from the +0.15 and +0.25 V/Å EF systems with acetate rotation.

While a strong correlation was found between the ring and bond CP Δρ( r ) values and activation barriers, the magnitudes of Δρ( r ) were not large, in the range of 0.02 e/ų for the EF magnitudes tested here (see Figure 5). Similarly, the change in distances between ring and bond CPs was only around 0.005 Å (see Figure S6). Since small changes in ρ( r ) can have significant effects on energetics [66, 73, 74, 75, 76], this suggests that prediction of activation barriers may be highly sensitive to environmental factors that can alter ρ( r ). We thus also studied the motion of CPs during vibrational modes in an attempt to substantiate this predictive method and get a more realistic view of CP motion in nonstatic systems.

3.3. Vibrational Modes

Kemp eliminase is a strong candidate for the presence of RPVs as it catalyzes a proton transfer reaction [16]. Dynamic motions of residues as well as substrate molecules have been linked to lowering reaction barriers and influencing transition states [77]. Chen and Schwartz found that directed evolution on one Kemp eliminase, KE59, favored the development of RPVs as residues were repositioned in the active site [16]. The RPVs found through QM/MM transition state sampling and subsequent committor analysis indicated that certain residues were positioned to compress the donor–acceptor distance between the benzisoxazole carbon atom and the Asp oxygen atom.

We investigated the effect of vibrational motions on CP positions for each EF strength for both the C─H (proton transfer) and N─O (bond breaking) stretching modes of the reactant state relevant to the Kemp elimination reaction coordinate. For EF strengths of  +0.05,  +0.1, and  +0.2 V/Å, two modes described the expected N─O stretch. Out of these two modes, one moved the atoms a larger distance in the expected direction along the reaction coordinate (alignment‐prioritized), while the other had a higher intensity (intensity‐prioritized). Considering the intensity‐prioritized modes over alignment‐prioritized modes improved all correlations and removed contextual outliers. Thus, intensity‐prioritized modes are considered here (see Figure S11 for alignment‐prioritized N─O data).

CP motion during vibrations was analyzed using three metrics. (1) Magnitude of motion was calculated based on the change in bond‐ring CP position during displacements. This was calculated by taking the difference of the bond‐ring CP Euclidean distances between the added and subtracted displacement geometries. (2) Direction of motion was calculated based on the dot products of the reference unit vector (pointing from bond to ring CP in nondisplaced geometries) and the ring CP unit vector defined as the unit vector pointing from the added to the subtracted displacement ring CP positions, as shown in Figure 6D. (3) Combined motion was calculated by taking the same dot product as described for the direction of motion, except that unit vectors were not used. Instead, the reference vector and ring CP displacement vectors were used directly to capture both direction and magnitude of motion.

FIGURE 6.

Relationship between critical point motion during N─O normal modes and calculated activation barriers at varying applied EF strengths. (A) Magnitude of motion, (B) direction of motion (unit vector dot products), and (C) combined magnitude and direction (dot products). (D) Overlay of the isoxazole ring geometry for no displacement, added displacement, and subtracted displacement of −0.25 V/Å EF system. The black circles show the CP positions with no displacements, solid red and green circles show bond and ring CP positions from added displacements, and dashed red and green circles show CP positions from subtracted displacements. The reference vector (black) and ring CP vector (green) used to calculate dot products are also shown.

Observing only the magnitude of change in bond‐ring CP distances during the benzisoxazole N─O vibration did not produce a notable correlation with calculated activation barriers (R ² = 0.2810, Figure 6A). The CP distance values appeared to be more dependent on the magnitude of the EF applied, with systems that had EFs of larger magnitude resulting in larger CP distance changes regardless of whether the EF was applied in the positive or negative direction.

Considering only directional contributions improved the correlation slightly to R ² = 0.3389 (Figure 6B). While strong positive and negative EFs increased the magnitude of the ring CP motion during the N─O stretch, the negative EFs resulted in ring CP motions that aligned with the reference vector more often. The +0.20 V/Å EF data point was a clear outlier, with a higher‐than‐expected unit vector dot product of 0.9997. Figure 6C shows that combining contributions from the magnitude and direction of ring CP motion during the N─O vibration resulted in the strongest linear correlation with the activation barrier, with R ² = 0.5918. As the change in CP distance and the alignment of expected CP motion both increased, activation barriers decreased.

The motion of the bond CP could be considered in addition to the ring CP motion. However, it was found that the ring CP moved a larger magnitude than the bond CP and that the bond CP generally moved perpendicular to the reference vector, making the bond CP motion less relevant to the reaction coordinate. Unlike the ring CP motion, which resulted in unit vector dot products close to 1, bond CP motion yielded values less than 0.5 (see Figure S12 for bond CP motion plots).

N─O vibrations also altered donor–acceptor distances (isoxazole C‐Asp O), which has previously been found to influence proton transfer catalysis, with shorter distances suggesting a higher degree of active site preorganization [77, 78, 79, 80]. Altering EF strengths from  +0.25 to −0.25 V/Å generally decreased donor–acceptor distances for reactant state geometries, as expected. However, considering the change in donor–acceptor distances from N─O vibrational modes did not correlate with EF strength as well as CP motion, especially for the geometry outliers from +0.15 and +0.25 V/Å (see Table S1). Given that direct analysis of solid‐state structures has been unable to distinguish between active and inactive variants using atom distances, CPs may be a more robust method than donor–acceptor distances for investigating vibrational motions [80].

The C─H stretching modes resulted in average bond and ring CP movements of 0.007 ± 0.002 Å and 0.044 ± 0.007 Å, respectively. These CP movements were one to two orders of magnitude smaller than the average bond CP and ring CP movements for the N─O stretching modes of 0.158 ± 0.042 Å and 0.179 ± 0.058 Å, respectively. Furthermore, the effect of EFs on the C─H vibrational mode did not correlate with calculated activation barriers (see Figures S13 and S14). This may be due to the lack of ring CP directional alignment with the reaction coordinate, evidenced by unit vector dot products less than 0.2 for the C─H stretch compared to N─O values closer to 1.

4. Conclusions

This paper studied the ring‐opening Kemp elimination reaction between a benzisoxazole substrate molecule and the side chain of an Asp residue as a minimal model of a Kemp eliminase enzyme. Previous work showed that a logarithmic relationship exists for this reaction between the activation barrier and the dielectric constant of an implicit solvation model, while a linear relationship exists between the activation barrier and external EF strength [19]. We expanded on this previous work by considering the relationships between reactant‐state ring and bond CP positions and ρ( r ) values with activation barriers for the Kemp elimination reaction. Since a ring and bond CP must coalesce for a ring‐opening reaction to occur, bond and ring CPs that are closer together in the reactant state should require less energy to move together and open the ring. Both the Euclidean distance between ring and bond CPs and the difference in charge density between these CPs were calculated and compared to activation barriers.

When varying solvation dielectric constants, a linear relationship was found between the activation barrier and the bond‐ring CP difference. The correlation was slightly stronger when considering Δρ( r ) (R ² = 0.7189) rather than CP distance (R ² = 0.6743). At high ε values, the activation barrier became less dependent on CP differences, in agreement with the logarithmic trend between activation barrier and dielectric constant, which emphasizes more significant changes for smaller ε values. Varying external EF strength along the reaction axis, from the Asp oxygen to the benzisoxazole carbon atom that transfers a proton, also resulted in linear relationships between activation barrier and the bond and ring CPs. The activation barrier and CP distance were again found to have a higher correlation with Δρ( r ) (R ² = 0.8743) than with Euclidean distance (R ² = 0.8213). While applied EFs were found to increase the ρ( r ) values at both the ring and bond CPs, the bond CPs ρ( r ) values increased to a larger degree than the ring CPs as the external EF became more positive. This is consistent with positive EFs inhibiting the reaction while strengthening the reactant state N─O bond.

To put these external EFs into the context of the heterogeneous EFs created by an enzyme, we consider work from the Head‐Gordon group that calculated the contributions of mutations to the KE15 Kemp eliminase enzyme on both transition state stabilization and EF strength [10]. Four rounds of mutations altered the electrostatic free energy of stabilization of the transition state from −5.4 kcal/mol in KE15 to −7.6 kcal/mol. When projecting the EF contributions from the entire system along the C─H bond of the nitrobenzisoxazole substrate, the magnitude of the EF changed from 0.4 to 1.0 V/Å. Noting that this EF direction is nearly opposite to the z‐axis defined in this work, these results show that creating a more negative EF along the z‐axis should stabilize the transition state and catalyze the Kemp elimination reaction.

When varying either dielectric constant or EF strength, the changes in position and ρ( r ) of the bond and ring CPs were not large. For example, the change in Δρ( r ) values between +0.25 and −0.25 V/Å EFs was around 0.02 e/ų. However, subtle changes in the charge density are known to often have significant effects on energies [66, 73, 74, 75, 76]. Since atoms and CPs are not static, one should consider that the ρ( r ) is constantly changing and the actual Δρ( r ) values may be much larger at any point in time during a molecular vibration. This led us to investigate how much the CPs moved during vibrational modes relevant to the reaction coordinate for the external EFs.

We hypothesized that if the distance between the ring and bond CPs changed by a larger amount during a vibrational mode, this would correlate with a lower activation barrier, as the vibrational motion facilitates the ring opening. Instead, we found that stronger EFs (whether in the positive or negative direction) always moved the CPs to a larger degree during N─O stretching modes. However, when calculating dot products between the nondisplaced bond and ring CPs (the reference vector) and the vector representing the motion of the ring CPs during N─O stretching modes, we found that the EFs that moved the ring CPs toward the bond CPs corresponded to lower activation barriers. Thus, EFs that alter the N─O stretching mode to move the ring CP in the direction toward the bond CP with the greatest magnitude were found to correspond to the highest catalytic rate. No correlations were found between the ring and bond CP positions during a C─H stretching mode and activation barrier, suggesting that the N─O stretch is more relevant to ring CP position during this reaction.

These results provide a systematic study of how ring and bond CP positions directly correlate with activation barriers of the Kemp eliminase ring‐opening reaction. However, we recognize several limitations of this initial study. First, the minimal model used to represent the Kemp eliminase active site was small and did not include water molecules, oxyanion–stabilizing interactions, or any full active site residues. The model only included a benzisoxazole substrate molecule and the side chain of the catalytic Asp residue. Furthermore, all linear correlations were made with between eight and eleven data points. While the R ² values for the EF CP data were encouraging, a more extended study is required to validate the predictive capabilities of ring and bond CPs for ring‐opening reaction barriers.

We plan to expand this work to studying the full active sites of enzymes and also consider how multiple geometries from molecular dynamics (MD) simulations affect CP positions and activation energies. This is necessary for any application to enzymatic reactions, as protein dynamics have been found to often have a large influence on catalytic ability, and multiple snapshots along MD trajectories can help to incorporate this information [81]. If the preliminary correlations from this study are maintained for full enzyme active sites, there is potential to use bond and ring CP data in the reactant states as predictors for enzyme reaction barriers. When applying similar methods as described in this work to small enzyme active site cluster models (∼50 atoms), reactant state geometry optimizations with QTAIM took around six node‐hours compared to 100 node‐hours to calculate activation barriers (transition state search and verification, IRC, single point on reactant state). Since reactant state ρ( r ) calculations are significantly less expensive than activation barrier calculations (94% in this case), ring and bond CP ρ( r ) values may be a useful parameter for screening potential designed enzymes and other catalysts.

Funding

This work was supported by the National Science Foundation (2017917) and University of Colorado Colorado Springs.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Supplementary Material

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.