Correlation of Temperature Dependence of Hydride Kinetic Isotope Effects with Donor-Acceptor Distances in Two Solvents of Different Polarities

Abstract

Recently observed nearly temperature(T)-independent kinetic isotope effects (KIEs) in wild-type enzymes and T-dependent KIEs in variants were used to suggest that H-tunneling in enzymes is assisted by the fast protein vibrations that help sample short donor-acceptor distances (DADs). This supports the recently proposed role of protein vibrations in DAD sampling catalysis. However, use of T-dependence of KIEs to suggest DAD sampling associated with protein vibrations is debated. We have formulated a hypothesis regarding the correlation and designed experiments in solution to investigate it. The hypothesis is, a more rigid system with shorter ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s at the tunneling ready states (TRSs) gives rise to a weaker T-dependence of KIEs, i.e., a smaller $\mathrm{\Delta }{E}_{\mathrm{a}}\left(=E_{\mathrm{a}\mathrm{D}}-E_{\mathrm{a}\mathrm{H}}\right)$. In a former work, the solvent effects of acetonitrile versus chloroform on the $\mathrm{\Delta }{E}_{\mathrm{a}}$ of NADH/NAD⁺ model reactions were determined, and the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{P}\mathrm{R}\mathrm{C}}$’s of the productive reactant complexes (PRCs) were computed to substitute the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ for the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}-{\mathrm{\Delta }E}_{\mathrm{a}}$ correlation study. A smaller $\mathrm{\Delta }{E}_{\mathrm{a}}$ was found in the more polar acetonitrile where the positively charged PRC is better solvated and has a shorter ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{P}\mathrm{R}\mathrm{C}}$, indirectly supporting the hypothesis. In this work, the TRS structures of different ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s for the hydride tunneling reaction from 1,3-dimethyl-2-phenylimidazoline to 10-methylacridinium were computed. The N-CH₃/CD₃ secondary KIEs on both reactants were calculated and fitted to the observed values to find the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ order in both solutions. It was found that the equilibrium ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ is shorter in acetonitrile than in chloroform. Results directly support the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}-{\mathrm{\Delta }E}_{\mathrm{a}}$ correlation hypothesis as well as the explanation that links T-dependence of KIEs to DAD sampling catalysis in enzymes.

Graphical Abstract

INTRODUCTION

The focus on the enhanced transition state binding as the origin of enzyme catalysis has recently given way to the challenging question as to whether fast dynamics in the active site are coupled to the chemistry of the enzymes helping further lower the activation energy of the reaction.^1–10 This physical origin of catalysis refers to an activation process in which protein dynamics/vibrations regulate the proper reaction-center distances for reaction to take place. One strategy to search for the origin uses enzyme-catalyzed H-tunneling reactions that are sensitive to the H-donor/acceptor distances (DADs).^11–13 Over the past 20+ years, it has been frequently observed that the hydrogen kinetic isotope effects (KIEs) are temperature(T)-independent in a variety of wild-type enzymes, whereas they become T-dependent to various extents with their variants.^2,5,14–25 Since these results cannot be explained using the traditional Bell tunneling model and sometimes cannot be explained with the modified Bell model that incorporates heavy atom motions and multi-dimensional H-tunneling,^11,26 the vibration-assisted activated H-tunneling (VA-AHT) model was proposed and claimed to be able to reconcile all of the results.^11,27,28 In the latter model, H-tunneling takes place in between the degenerate reactant and product states of the activated tunneling-ready state (TRS) that has a spectrum of ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s sampled by enzyme fast dynamics. The model links KIE to the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ and correlates heavy atom motions to the thermal sampling of the short ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s, therefore, T-dependence of KIEs reflects T-dependence of ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s.^{11,17,28–30} The T-independence of KIEs is explained in terms of the well-organized reaction coordinate in which ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ is short and narrowly distributed, implicating that the wild-type enzyme active site has strong compressive vibrations that press the two reactants close to each other. In variants, however, the naturally evolved vibrations are impaired, the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ becomes longer and its distribution becomes broader. That is, variants do not have strong compressive vibrations as in the wild-type enzymes so that sampling of the short ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s is hindered and T-dependent. As a result, the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ sampling for tunneling of a heavier isotope, due to its shorter deBroglie wavelength, requires higher energy, and hence the isotopic activation energy difference $({\mathrm{\Delta }E}_{\mathrm{a}}=E_{\mathrm{a}\mathrm{D}}-E_{\mathrm{a}\mathrm{H}})$ becomes larger. i.e., the T-dependence of KIEs/${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s becomes stronger. Note that further experiments also used heavy enzymes and viscous solvents to control the DAD distributions to investigate the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}-{\mathrm{\Delta }E}_{\mathrm{a}}$ correlation, and the correlation is sometimes supported.^31–37

Explanations that correlate ${\mathrm{\Delta }E}_{\mathrm{a}}$ to ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ sampling have, however, been debated, which makes the use of ${\mathrm{\Delta }E}_{\mathrm{a}}$’s to propose the DAD sampling catalysis role arguable. One argument is about the applicability of the VA-AHT model that includes the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}-{\mathrm{\Delta }E}_{\mathrm{a}}$ correlation to the adiabatic hydride- and proton-transfer systems, as the initial form of the model was established for the nonadiabatic H-atom transfer systems. But, quite a few latter studies seemed to indicate that the correlation exists in hydride tunneling enzymes as well.^28,38 Other arguments come from the simulation of the ${\mathrm{\Delta }E}_{\mathrm{a}}$’s by using other H-tunneling theories that do not include such an apparent correlation. For example, studies using the ensemble averaged variational transition state (TS) theory with multi-dimensional H-tunneling (EA-VTST/MT) have shown that the T-dependence of KIEs could also result from the effects of temperature on the position of the TS and the shape of the potential barrier.^39,40 In another study, simulations ascribed the different ${\mathrm{\Delta }E}_{\mathrm{a}}$’s to the higher entropic barrier in the variant than in the wild-type enzyme.⁴¹ On the other hand, it should be noted that simulations using the EA-VTST/MT model and the empirical valence bond theory sometimes did show that the weak T-dependence of hydride KIEs (small ${\mathrm{\Delta }E}_{\mathrm{a}}$) results from the insensitivity of the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s to temperature (short and narrowly distributed ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$).^{17,27,28,42,43} Therefore, whether the ${\mathrm{\Delta }E}_{\mathrm{a}}$ magnitude reflects the ease of DAD sampling for general H-transfer enzymes and whether their correlation can be used to examine the proposed contentious role of protein dynamics in enzyme catalysis are still questionable.

We regard that the correlation between ${\mathrm{\Delta }E}_{\mathrm{a}}$ and ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ sampling could be examined using the “simpler” H-transfer reactions in solution. To this end, we have formulated a testable hypothesis, which is, a more rigid system in which the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s are more densely populated gives rise to a smaller ${\mathrm{\Delta }E}_{\mathrm{a}}$.⁴⁴ We have used the electronic, steric, and solvent effects to control the system rigidity for the study and have found that the hypothesis appears to be supported.^44–47

One challenge for investigation of the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}-{\mathrm{\Delta }E}_{\mathrm{a}}$ correlation for both enzymatic and solution reactions is to acquire the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ information to estimate the TRS rigidity. Often, this type of research, including ours, uses the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{P}\mathrm{R}\mathrm{C}}$ distribution of the productive reactant complexes (PRCs) or the similar DAD information for the Michaelis complexes in enzymes to substitute the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ information for the study.^{20,48,44,45,47} For example, we have recently reported the correlation between the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{P}\mathrm{R}\mathrm{C}}$’s and ${\mathrm{\Delta }E}_{\mathrm{a}}$’s for two hydride transfer reactions of NADH/NAD⁺ analogues in aprotic acetonitrile and chloroform that have very different polarities (the dielectric constants are 37.5 and 4.8, respectively).⁴⁷ Our design was based on the expectation that a more polar solvent would stabilize the positively charged TRS, increasing the system rigidity that would lead to a smaller ${\mathrm{\Delta }E}_{\mathrm{a}}$. We did find that the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{P}\mathrm{R}\mathrm{C}}$’s are shorter in acetonitrile than in chloroform and the ${\mathrm{\Delta }E}_{\mathrm{a}}$ is smaller in the former solvent. This appeared to support the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}-{\mathrm{\Delta }E}_{\mathrm{a}}$ relationship hypothesis, but the direct ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ information was missing in that study.

In order to directly investigate the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}-{\mathrm{\Delta }E}_{\mathrm{a}}$ correlation in the aforementioned solvent polarity effect study, in this work, we computed the TRS structures of various ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s and fitted the calculated $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ to the observed ones to find the equilibrium (“average”) TRS structures and the corresponding ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ information for one such hydride transfer reaction in the two solvents. We chose the reaction between 1,3-dimethyl-2-phenylimidzoline (DMPBIH, hydride donor) and 10-methylacridinum ion (MA⁺, acceptor) (eqn. 1) for this study. This is because the N-CH₃/CD₃ $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ at the two $\mathrm{\gamma }$-positions of DMPBIH and one ε-position of MA⁺ can be determined and simultaneous simulation of both would give rise to a more reasonable TRS structure. We found that the equilibrium ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ is indeed shorter in acetonitrile than in chloroform, supporting the hypothesized correlation between TRS rigidity and T-dependence of KIEs as well as the corresponding “compressive vibrations” explanations for the enzyme observations.

(1)

Computation Methods

TRS Structure Computations

A TRS contains a spectrum of activated degenerate reactant and product structures of different DADs and different energies. A TRS of certain DAD for a H-tunneling reaction from a donor (D-H) to an acceptor (A) can be written as [D-H---H-A]^≠. In this TRS, H is in a quantum state being at both sides of the degenerate reactant ([D-H A]^≠) and product ([D H-A]^≠) states simultaneously. It is thus a hybrid (linear combination) of the two degenerate structures. We have recently developed a method to compute the degenerate [D-H A]^≠ and [D H-A]^≠ structures for the TRS structures.⁴⁶ In our method, we use the classical transition state (TS) structure to construct the initial guess of the TRS structure. The dihedral angles with respect to the reaction centers at D and A in [D-H A]^≠ at a selected ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ are scanned to form the 2-Dimentional potential energy surface (2D-PES) for reactant state and the same is done for the [D H-A]^≠ product state structure. The minimum energy point of the intersecting parabola curve of the two PESs is taken as the TRS structure. More specifically, the dihedrals were initially scanned within the range of ±2.5 degrees with 0.5 degrees interval. If the expected lowest energy intersecting area is not found on the 3D-PES plot, the scanning ranges will be extended. The preliminary intersecting parabola is refined by repeating the scan with a smaller dihedral interval to achieve a more accurate TRS structure.

In this work, we found four classical TSs for reaction (1), but only found two TRSs for each DAD that we selected for computation, which are 2.8, 3.0, and 3.2 Å, respectively. The selection of this range of ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s is because these equilibrium ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ values have been found appropriate for some H-tunneling reactions to use in enzymes and solution.^49–53 It should be stressed that a TRS has a spectrum of ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s and the selected ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ only refers to a typical one, rather than that the tunneling only takes place at the particular DAD. As a matter of fact, the most efficient ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ for H-tunneling to occur is about 2.7–2.8 Å.⁴³

Both the gas-phase and solution phase TRSs were computed. For the latter, the universal solvation model (SMD) was used. Figure 1 delineates the process for searching for a TRS structure at 2.8 Å for the reaction between DMPBIH and MA⁺ in gas-phase. The dihedrals chosen to scan are defined therein. It can be seen that hydride-transfer takes place in a $\mathrm{\pi }-\mathrm{\pi }$ stacking charge-transfer (CT) complex, which is known for the NADH/NAD⁺ model reactions.^44,54–56

Figure 1.

The gas-phase PES scans as a function of dihedrals for the activated complexes of the reactants state (in blue color) and products state (in orange color) at DAD = 2.8 Å for the hydride-transfer reaction from DMPBIH to MA⁺ (top 3D plot). The dihedral φ is defined by N11-N12-C13-C17 in DMPBIH, and θ by C36-C39-C44-H60 in MA⁺. The lowest intersecting energy point of the two PES’s is taken as the TRS structure (circled point). It is the hybrid of the reactant state ([DMPBI-H MA]^≠, bottom left) and product state ([DMPBI H-MA]^≠, bottom right). This is one of the two TRS’s found with ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}=2.8\mathrm{\AA }$ and it has the lower electronic energy. In [DMPBI-H MA]^≠, the transferring H16 is at C13; in [DMPBIH-MA]^≠, it is at C39. The ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ refers to the C13-C39 distance.

$2^{\circ }KIE$ Calculations

We use the following steps to calculate the $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ on the basis of the TRS structures found.⁴⁶ (i) Calculate free energies ${(G}^{\circ }=E_{\text{electronic}}+E_{\text{thermal\_correction}})$ of the ground state (GS) reactants D-H, A⁺, and two degenerate states of the TRS ([D-H A]^≠ and [D H-A]^≠), here the $E_{\text{electronic}}$ is the electronic energy and $E_{\text{thermal\_correction}}$ is the thermal correction to the Gibbs free energy; (ii) Calculate two rate constants ($k\left[\mathrm{T}\mathrm{R}\mathrm{S}\right]$’s) for system to reach the two TRSs ($k[\text{D-H}\mathrm{A}{]}^{\ne }$ and $k[\mathrm{D}\text{H-A}{]}^{\ne }$), respectively, using the Eyring equation (free energy of activation $\left.\Delta G^{\ne }=G^{°}{}_{\text{TRS}}-G^{°}{}_{\text{GS}}(\text{D}-\text{H})-G^{°}{}_{\text{GS}}\left({\text{A}}^{+}\right)\right)$; (iii) Take composite rate constant ($k\left[\mathrm{T}\mathrm{R}\mathrm{S}\right]$) as the geometric mean of the $k[\text{D-H}\mathrm{A}{]}^{\ne }$ and $k[\mathrm{D}\text{H}-\text{A}{]}^{\ne }$; (iv) Calculate $k\left[\mathrm{T}\mathrm{R}\mathrm{S}\right]$ for reaction involving isotopes, here the $\mathrm{\gamma },\mathrm{\gamma }-2{\mathrm{C}\mathrm{D}}_3$ substitutions in DMPBIH and the $\mathrm{\epsilon }-{\mathrm{C}\mathrm{D}}_3$ substitution in MA⁺; and (v) Calculate $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}=k\left[\mathrm{T}\mathrm{R}\mathrm{S}\left({\mathrm{C}\mathrm{H}}_3\right)\right]/k\left[\mathrm{T}\mathrm{R}\mathrm{S}\left({\mathrm{C}\mathrm{D}}_3\right)\right]$. Since each reaction has two TRSs for a DAD leading to two $k\left[\mathrm{T}\mathrm{R}\mathrm{S}\right]$ values ($k[\text{TRS}{]}_i,i$ represents different TRSs), the overall KIE is calculated using eqn. 2 where $\mathrm{n}=2$ (TRSs).

$$2^{°}\mathrm{K}\mathrm{I}\mathrm{E}=\sum _n{k\left[\mathrm{T}\mathrm{R}\mathrm{S}\right(2{\mathrm{C}\mathrm{H}}_3\left)\right]}_i/\sum _n{k\left[\mathrm{T}\mathrm{R}\mathrm{S}\left(2{\mathrm{C}\mathrm{D}}_3\right)\right]}_i$$ (2)

It should be noted that the $2^{°}\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ calculated here are the same as the ones calculated using the Isoeff software⁵⁷ that requires vibrational analysis of the TRSs and GSs. In the latter method, the $E_{\text{electronic}}$ is not used as it is treated as the same for two isotopic structures and is cancelled out in the $2^{°}\mathrm{K}\mathrm{I}\mathrm{E}$ calculations.

Calculation of the Donor/Acceptor Hybridizations at the TRS

The hybridization states $\left({\mathrm{s}\mathrm{p}}^H\right)$ of the donor and acceptor carbons in the TRS structures were calculated using eqn. 3.^49–51

$$H=2+(180-\theta )/\left(180-{\theta }_0\right)$$ (3)

In this equation, $\theta$ is the out-of-plane bending angle at the TRS, ${\theta }_0$ is the angle of the reduced form of each reactant.

Weight averaged hybridizations ($H\left(W\right)$’s) are calculated using eqn. 4 ($\mathrm{n}=2$ (TRSs)).

$$H\left(W\right)=\sum _n{p}_i{H}_i$$ (4)

In this equation, $p_i$ is the percentage of the individual TRS calculated using Boltzmann distribution of the average free energies (${\epsilon }_{\mathrm{i}}$’s) of the reactant- and product-state of the TRS (eqn. 5, ${\epsilon }_i=(G^{\circ }([\text{D-H}\mathrm{A}{]}^{\ne })+G^{\circ }([\mathrm{D}\text{H-A}{]}^{\ne })/2),k(B)$ is the Boltzmann constant),

$$p_i=\frac{e^{-{\epsilon }_i/k(B)T}}{\sum _n{e}^{-{\epsilon }_i/k(B)T}}$$ (5)

and $H_{\mathrm{i}}$ is the hybridization state of the donor or acceptor carbon of the corresponding TRS, and $T=298.15\mathrm{K}$.

The hybridization states $\left({\mathrm{s}\mathrm{p}}^H\right)$ of the donor and acceptor carbons in the classical gas-phase TS structure were calculated using the same method ($\mathrm{n}=4$ (TSs) in eqn. 4) for comparison with the same for the TRS structures.

Results and Discussion

The N-CH/CD $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}$ originates from the isotopic difference in negative hyperconjugation between the lone-pair of electrons on N and the empty ${\mathrm{\sigma }}^{*}$ orbital of the C-H/D bond.^44,58 It is resulted from the loss/gain of electron density on N during the reaction. The electron density loss tightens the C-H/D bonds, leading to an inverse $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}$, whereas the electron density gain loosens the C-H/D bonds, leading to a normal $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}$.^58,59 According to this analysis, for the reaction (1), the 1,3-N,N-2CH₃/2CD₃ $\mathrm{\gamma }-2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ on DMPBIH should be inverse while the 10-N-CH₃/CD₃ $\mathrm{\epsilon }-2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ on MA⁺ be normal. Table 1 lists the $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ computed in both the gas-phase and solution phase. The observed $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ that we reported are also listed in the table for comparison⁴⁷. Results show that the signs of the $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ (normal or inverse) are consistent with the expectations.

Table 1.

The observed vs. computed nonclassical gas-phase N-CH₃/CD₃ $2^{°}\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ for the reaction of DMPBIH with MA⁺ at 25 °C

	γ-2CH3/2CD3 2º KIE on DMPBIH	(ε-CH3/CD3) 2º KIE on MA+
	Computed from TRSs in gas-phase
${\text{DAD}}_{\mathrm{T}\mathrm{R}\mathrm{S}}=2.8$ Å	0.86	1.04
${\text{DAD}}_{\mathrm{T}\mathrm{R}\mathrm{S}}=3.0$ Å	0.87	1.02
${\text{DAD}}_{\mathrm{T}\mathrm{R}\mathrm{S}}=3.2$ Å	0.88	1.03
	Computed from TRSs in solution a
${\text{DAD}}_{\mathrm{T}\mathrm{R}\mathrm{S}}=2.8$ Å
In acetonitrile	0.93	1.13
In chloroform	0.88	1.09
${\text{DAD}}_{\mathrm{T}\mathrm{R}\mathrm{S}}=3.2$ Å
In acetonitrile	0.92	1.16
In chloroform	0.89	1.13
	Observed b
In acetonitrile	0.88 (0.01)	1.03 (0.01)
In 95.8%chlorform and 4.2% acetonitrile (v/v)	0.92 (0.01)	1.03 (0.01)

^aUsing the universal solvation model (SMD)

^bFrom ref.⁴⁷, data in paratheses are standard deviations.

The nucleus tunneling mechanisms have been proposed for the hydride transfer reactions of many NADH/NAD⁺ models, including the reaction (1).^{50–52,56,57,65−70} Our observed ${\mathrm{\Delta }E}_{\mathrm{a}}$ of 0.43 kcal/mol for reaction (1) in acetonitrile and 0.64 kcal/mol in chloroform (containing 4.2% (v/v) acetonitrile) are well outside of the semiclassical limit of 1.0 – 1.2 kcal/mol.^44,47 The results also support the nonclassical hydride tunneling mechanism.

Simulation of the observed $2^{\circ }KIEs$ to evaluate the ${DAD}_{TRS}$ order in the two solvents

The main purpose of the work is to examine whether the observed ${\mathrm{\Delta }E}_{\mathrm{a}}$’s are linked to the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s in terms of our hypothesis that a smaller ${\mathrm{\Delta }E}_{\mathrm{a}}$ for a H-tunneling reaction corresponds with the narrowly distributed short ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s, i.e., a more rigid TRS. Since ${\mathrm{\Delta }E}_{\mathrm{a}}$ of the reaction (1) in acetonitrile is smaller than that in chloroform,^44,47 we expect to see a shorter ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ in acetonitrile than in chloroform. While simulation of a $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}$ gives rise to an equilibrium (or average) TRS structure from a spectrum of TRS structures sampled, fit of the observed $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ in the two solutions to the computed TRS structures could give rise to the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ difference for correlation analysis with the ${\mathrm{\Delta }E}_{\mathrm{a}}$’s.

Ideally, $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ computed involving the two solvents should be used for the simulation study. We find, however, that the resultant values at selected ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s are not only deviate significantly from the observed ones (Table 1, computations in solution vs. observed), but the $\mathrm{\gamma }-2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ on DMPBIH are also significantly higher in acetonitrile than in chloroform being opposite to the observed order! We regard that there are two major systematic errors from our computations that are responsible for the inconsistency between the experiments and theory. The first error originates from the approximation in our method to find a TRS. We assume that the two dihedrals in the donor and acceptor C’s can represent the major characteristics of a TRS structure, so in our computation they are adjusted to be equal at between the reactant and product states to locate a TRS (c.f. Figure 1). The drawback is that all other geometric differences are ignored. For example, the heavy atom framework of the two geometries of the TRS in Figure 1 are not completely superimposed. In our previous simulations of $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ using the gas-phase results to differentiate the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s for two systems in the same solvent, however, this procedure did not make significant systematic errors and the method appeared working well.^46,50,60 The second error originates from the use of the SMD solvation model, such as its implicit solvation approximation without considering the microscopic effect from the individual solvent molecules. When the two approximations work together, the errors could be additive to or canceling each other. In this case, the errors appear to be additive. In fact, the computed gas-phase $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ reasonably agree with the observed $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ (Table 1 and see the subsequent detailed discussions). In this case, the systematic errors may have been cancelled to some extent in the comparison of $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ at different DADs. It should be noted that others have used the fit of $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ in wild-type vs. variant to the gas-phase $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ to differentiate the TRS structures in the two enzymes.⁴⁹ In that computation, the basic principle is the same as ours. Therefore, fit of the observed $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ to the computed values in the same gas-phase to find the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ order in the two solvents would be acceptable.

Table 1 shows that the magnitudes of the computed gas-phase ${\mathrm{\epsilon }-2}^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ on MA⁺ (1.04 – 1.03) with selected ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ are consistent with the observed one (1.03), whereas the gas-phase $\mathrm{\gamma }-2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ on DMPBIH are close to but somewhat more inverse than the observed ones (0.86 – 0.88 vs. observed 0.88 – 0.92). We find that the computed $\mathrm{\gamma }-2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}$ on DMPBIH becomes more and more inverse as the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ decreases. This suggests that the observed decrease in the $\mathrm{\gamma }-2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}$ value from the reactions in chloroform (0.92) to acetonitrile (0.88) is due to the longer equilibrium ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ in chloroform than in acetonitrile. On the other hand, the computed ${\mathrm{\epsilon }-2}^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ on MA⁺ appear to slightly decrease with increasing ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ (1.04 at 2.8 Å and 1.03 at 3.2 Å), but the observed ones are 1.03 in both solvents. It is safe to say that the two match well with each other within the experimental and computational errors. One reason that we observe the same $\mathrm{\epsilon }-2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ but very different $\mathrm{\gamma }-2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ in the two solvents might be that the $\mathrm{\epsilon }-2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}$ on 3 remote H/D’s is less sensitive to the change in DAD than the $\mathrm{\gamma }-2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}$ on DMPBIH that involves 6 H/D’s being closer to the reaction center. It should be noted that the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{P}\mathrm{R}\mathrm{C}}$ order in the two solvents that we reported⁴⁷ are consistent with the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ order obtained here. Overall, the results are consistent with our expectations that the positively charged TRS is stabilized and thus more rigid with shorter DAD in the more polar solvent of acetonitrile.

In our former paper, we have reported the charge distributions at the TRS structures of the reaction (1) in the two solvents.⁴⁷ The donor and acceptor moieties carry partial positive charge and the transferring nucleus carries partial negative charge. In the meantime, there should be partial negative charge in between the donor and acceptor for CT complexation/bonding. We found that the negative (−) charge density in between the donor and acceptor in the TRS, which is borne by the transferring H as well as is used for the CT complexation, is greater in acetonitrile (0.32-) than in chloroform (0.16-). This suggests a stronger CT-complexation and thus more rigid TRS in the former solvent. It is consistent with the results from this work − the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ is shorter in acetonitrile than in chloroform.

Correlation of the ${\Delta E}_a$’s to the ${DAD}_{TRS}$ order in the two solvents

In the activation process from PRCs to TRSs, the CT complexation is thermally compressed and the shorter ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s are sampled. From above analysis of the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ order as well as our reported ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{P}\mathrm{R}\mathrm{C}}$ order in the two solvents, both the weight averaged ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{P}\mathrm{R}\mathrm{C}}$⁴⁷ and the equilibrium ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ are shorter in acetonitrile than in chloroform (Figure 2 left⁴⁷). This suggests that the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ distribution is narrower in acetonitrile than in chloroform (Figure 2 right). That is, the TRS in the former solvent is more rigid than that in the latter solvent. Overall, our results show that the smaller ${\mathrm{\Delta }E}_{\mathrm{a}}$ in acetonitrile corresponds with a shorter (equilibrium) ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ thus a tighter or more rigid TRS (having stronger CT complexation vibrations). This supports our hypothesis concerning the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}-{\mathrm{\Delta }E}_{\mathrm{a}}$ correlation within the VA-AHT model.

Figure 2.

Schematic description of the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ sampling activation by heavy atom motions (thermal molecular vibrations) from PRCs in acetonitrile and chloroform.⁴⁷ Both the weight averaged ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{P}\mathrm{R}\mathrm{C}}$ and the equilibrium ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ are shorter in acetonitrile than in chloroform. The positively charged PRC and TRS are more stable in acetonitrile than in chloroform. It is well known that the most efficient ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ for H-tunneling to occur is about 2.7–2.8 Å.⁴³ The left figure is copied from the reference 47 (Copyright: American Chemical Society).

TRS structural comparison in the two solvents

Table 2 lists the hybridizations of both the donor and acceptor C’s at the gas-phase TRSs. Hybridizations at the assumed gas-phase classical TS are also listed for comparison. In the TRS, the hybridization does not change with ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ (from 2.8 to 3.2 Å). This suggests that the TRSs in the two solvents have similar geometries at the reaction centers. On the other hand, a comparison of these hybridizations with the same in the classical TS show that the rehybridizations in the activation coordinates of both reactions are smaller in the tunneling mechanism (from sp^~3.00 to sp^~2.90 at DMPBIH and from sp^~2.00 to sp^~2.37 at MA⁺) than in the classical mechanism (from sp^~3.00 to sp^~2.85 at DMPBIH and from sp^~2.00 to sp^~2.49 at MA⁺). This appears reasonable as the tunneling mechanism uses longer distance between the reaction centers so that the TRS arrives earlier (than the assumed classical TS) in the reaction coordinate. In the reaction between DMPBIH with a more reactive and sterically hindered hydride acceptor of 9-phenylxanthylium ion, however, we found that the rehybridization is more advanced in the formation of the TRS than the classical TS, indicating that the particular reaction requires extra orbital preparations for hydride tunneling to take place.⁴⁶

Table 2.

The weight averaged hybridizations of the donor and acceptor carbons at the classical TSs and TRSs of the reactions of DMPBIH with MA⁺ in gas-phase

	Donor	Acceptor
Classical TS	Sp2.85	Sp2.49
TRSs
${\text{DAD}}_{\mathrm{T}\mathrm{R}\mathrm{S}}=2.8$ Å	Sp2.90	Sp2.37
${\text{DAD}}_{\mathrm{T}\mathrm{R}\mathrm{S}}=3.0$ Å	Sp2.88	Sp2.36
${\text{DAD}}_{\mathrm{T}\mathrm{R}\mathrm{S}}=3.2$ Å	Sp2.89	Sp2.39

Conclusions

We have recently reported the solvent effects on the ${\mathrm{\Delta }E}_{\mathrm{a}}$’s and ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{P}\mathrm{R}\mathrm{C}}$’s for two hydride transfer reactions of NADH/NAD⁺ analogues that include the reaction (1) studied in this paper. We also reported the solvent effects on the strength of the CT complexation in the TRS’s of the reaction. We found that the ${\mathrm{\Delta }E}_{\mathrm{a}}$ is smaller in a more polar aprotic solvent where the system has a shorter average ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{P}\mathrm{R}\mathrm{C}}$ and a more tightly bound TRS. In this paper, we computed the TRS structures with different DADs for the same reaction and calculated the N-CH₃/CD₃ $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ on both reactants to investigate the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}-{\mathrm{\Delta }E}_{\mathrm{a}}$ relationship that has been used by enzymologists to discuss about the role of protein dynamics in enzyme catalysis. The observed $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ in acetonitrile and chloroform were fitted to the calculated ones to give rise to the equilibrium (“average”) TRS structures in the two solvents. It was found that the equilibrium ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$ is shorter in the more polar acetonitrile than in chloroform. We have reported the smaller ${\mathrm{\Delta }E}_{\mathrm{a}}$ in acetonitrile than in chloroform. Therefore, the narrowly distributed shorter ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s in a TRS gives rise to a smaller ${\mathrm{\Delta }E}_{\mathrm{a}}$. This directly supports our hypothesis about the ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}-{\mathrm{\Delta }E}_{\mathrm{a}}$ relationship within the VA-AHT model, which is, a more rigid reaction system gives rise to a smaller ${\mathrm{\Delta }E}_{\mathrm{a}}$. The results, joining with the results from our other works, suggest that the T-independence of KIEs frequently observed in the wild-type enzymes would be resulted from the strong compressive vibrations in the enzymes that lead to the narrowly distributed short ${\mathrm{D}\mathrm{A}\mathrm{D}}_{\mathrm{T}\mathrm{R}\mathrm{S}}$’s.

At last, this paper leaves a question as to why the observed KIEs are not quite fitted to the KIEs computed from the most commonly used, if not the best, SMD solvation model. This suggests that the solvation model does not work well with our method to compute the $2^{\circ }\mathrm{K}\mathrm{I}\mathrm{E}\mathrm{s}$ from the TRS structures we find, at least for the system we study.

COMPUTATIONAL METHODS

The calculations were performed under the M06–2X⁸⁷/Def2-SVP⁸⁸ level of theory with ultrafine DFT integration grid. A fitted frequency scaling factor of 0.9695 is used to minimize the overestimation error of the harmonic model. The PES intersecting diagram is created by using the Ploty Online Chart Studio [https://chart-studio.plotly.com].

Gaussian 09 was used for all of the calculations.