Tunneling in enzymatic and nonenzymatic hydrogen transfer reactions

Hierarchical Equations of Motion for Quantum Chemical Dynamics: Recent Methodology Developments and Applications

ConspectusQuantum effects are critical to understanding many chemical dynamical processes in condensed phases, where interactions between molecules and their environment are usually strong and non-Markovian. In this Account, we review recent progress from our group in development and application of the hierarchical equations of motion (HEOM) method, highlighting its ability to address some challenging problems in quantum chemical dynamics.In the HEOM method, the bath degrees of freedom are represented using effective modes from exponential decomposition of the bath correlation function. Complex spectral densities and low temperature simulations often require a larger number of modes, making the simulations very expensive. Recent advances, such as the barycentric spectral decomposition (BSD) technique, can significantly reduce the number of effective modes, allowing to handle complex spectral densities and enabling simulations at very low temperatures, including near-zero temperature dynamics.Another key improvement in the computational efficiency is the use of tensor network methods like matrix product states and hierarchical tensor networks. These techniques allow for efficient HEOM propagation with thousands of effective modes, crucial for simulating large molecular systems interacting with multiple baths. This combination enables simulations of excitation energy transfer (EET) in systems like the Fenna-Matthews-Olson (FMO) complex and even larger systems with experimentally determined spectral densities.The versatility of the HEOM method is demonstrated through applications to a wide range of chemical dynamics problems. Simulations of EET and related ultrafast spectroscopy are first briefly covered. Applications of the HEOM to quantum tunneling effects in proton transfer reactions are then presented. Early works have studied the non-Kramers dependence of the rate constant as a function of bath friction due to deep tunneling and revealed vibrationally nonadiabatic dynamics within the so-called nontraditional view of proton transfer reactions. A recent work on the large kinetic isotope effects in soybean lipoxygenase also indicated that many quantum correction approximations to classical transition-state theory may fall short in describing deep tunneling effects.Charge transport and separation dynamics in organic semiconductors are another area where the HEOM method has been instrumental. We first demonstrate that the HEOM provides a unified description of both band-like and thermally assisted charge carrier transport in organic materials. The effect of non-nearest neighbor transitions is then investigated by combining generalized master equations with exact memory kernels. The HEOM method also enables simulation of charge separation in organic photovoltaics (OPVs) and reveals how factors such as external electric fields, entropy, and charge delocalization influence the charge separation barrier and dynamics.Moreover, HEOM has been applied to investigate hydrogen atom scattering on the Au(111) surface and vibrational energy relaxation at molecule-metal interfaces. These studies provide deeper insights into how electron-hole pair excitations and temporary charge transfer states influence the nuclear motion, offering a new framework for simulating nonadiabatic dynamics on metal surfaces.In summary, the HEOM method has developed into a robust tool for simulating quantum effects in condensed phases. Future developments in algorithm efficiency and computational power will likely expand its applicability to even more complex systems.

Hydrogen Tunneling Exhibiting Unexpectedly Small Primary Kinetic Isotope Effects

Probing quantum mechanical tunneling (QMT) in chemical reactions is crucial to understanding and developing new transformations. Primary H/D kinetic isotopic effects (KIEs) beyond the semiclassical maximum values of 7-10 (room temperature) are commonly used to assess substantial QMT contributions in one-step hydrogen transfer reactions, because of the much greater QMT probability of protium vs. deuterium. Nevertheless, we report here the discovery of a reaction model occurring exclusively by H-atom QMT with residual primary H/D KIEs. 2-Hydroxyphenylnitrene, generated in N2 matrix, was found to isomerize to an imino-ketone via sequential (domino) QMT involving anti to syn OH-rotamerization (rate determining step) and [1,4]-H shift reactions. These sequential QMT transformations were also observed in the OD-deuterated sample, and unexpected primary H/D KIEs between 3 and 4 were measured at 3 to 20 K. Analogous residual primary H/D KIEs were found in the anti to syn OH-rotamerization QMT of 2-cyanophenol in a N2 matrix. Evidence strongly indicates that these intriguing isotope-insensitive QMT reactivities arise due to the solvation effects of the N2 matrix medium, putatively through coupling with the moving H/D tunneling particle. Should a similar scenario be extrapolated to conventional solution conditions, then QMT may have been overlooked in many chemical reactions.

Proton-Coupled Electron Transfer and Hydrogen Tunneling in Olive Oil Phenol Reactions

Olive oil phenols are recognized as molecules with numerous positive health effects, many of which rely on their antioxidative activity, i.e., the ability to transfer hydrogen to radicals. Proton-coupled electron transfer reactions and hydrogen tunneling are ubiquitous in biological systems. Reactions of olive oil phenols, hydroxytyrosol, tyrosol, oleuropein, oleacein, oleocanthal, homovanillyl alcohol, vanillin, and a few phenolic acids with a DPPH• (2,2-diphenyl-1-picrylhydrazyl) radical in a 1,4-dioxane:water = 95:5 or 99:1 v/v solvent mixture were studied through an experimental kinetic analysis and computational chemistry calculations. The highest rate constants corresponding to the highest antioxidative activity are obtained for the ortho-diphenols hydroxytyrosol, oleuropein, and oleacein. The experimentally determined kinetic isotope effects (KIEs) for hydroxytyrosol, homovanillyl alcohol, and caffeic acid reactions are 16.0, 15.4, and 16.7, respectively. Based on these KIEs, thermodynamic activation parameters, and an intrinsic bond orbital (IBO) analysis along the IRC path calculations, we propose a proton-coupled electron transfer mechanism. The average local ionization energy and electron donor Fukui function obtained for the phenolic compounds show that the most reactive electron-donating sites are associated with π electrons above and below the aromatic ring, in support of the IBO analysis and proposed PCET reaction mechanism. Large KIEs and isotopic values of Arrhenius pre-exponential factor AH/AD determined for the hydroxytyrosol, homovanillyl alcohol, and caffeic acid reactions of 0.6, 1.3, and 0.3, respectively, reveal the involvement of hydrogen tunneling in the process.

Study of the Effects of Remote Heavy Group Vibrations on the Temperature Dependence of Hydride Kinetic Isotope Effects of the NADH/NAD+ Model Reactions

It has recently been observed that the temperature(T)-dependence of KIEs in H-tunneling reactions, characterized by isotopic activation energy difference (ΔEa = EaD - EaH), is correlated to the rigidity of the tunneling ready states (TRSs) in enzymes. A more rigid system with narrowly distributed H-donor-acceptor distances (DADs) at the TRSs gives rise to a weaker T-dependence of KIEs (i.e., a smaller ΔEa). Theoreticians have attempted to develop new H-tunneling models to explain this, but none has been universally accepted. In order to further understand the observations in enzymes and provide useful data to build new theoretical models, we have studied the electronic and solvent effects on ΔEa's for the hydride-tunneling reactions of NADH/NAD+ analogues. We found that a tighter charge-transfer (CT) complex system gives rises to a smaller ΔEa, consistent with the enzyme observations. In this paper, we use the remote heavy group (R) vibrational effects to mediate the system rigidity to study the rigidity-ΔEa relationship. The specific hypothesis is that slower vibrations of a heavier remote group would broaden the DAD distributions and increase the ΔEa value. Four NADH/NAD+ systems were studied in acetonitrile but most of such heavy group vibrations do not appear to significantly increase the ΔEa. The remote heavy group vibrations in these systems may have not affected the CT complexation rigidity to a degree that can significantly increase the DADs, and further, the ΔEa values.

Activation volume and quantum tunneling in the hydrogen transfer reaction between methyl radical and methane: A first computational study

We present a theory of the effect of quantum tunneling on the basic parameter that characterizes the effect of pressure on the rate constant of chemical reactions in a dense phase, the activation volume. This theory results in combining, on the one hand, the extreme pressure polarizable continuum model, a quantum chemical method to describe the effect of pressure on the reaction energy profile in a dense medium, and, on the other hand, the semiclassical version of the transition state theory, which includes the effect of quantum tunneling through a transmission coefficient. The theory has been applied to the study of the activation volume of the model reaction of hydrogen transfer between methyl radical and methane, including the primary isotope substitution of hydrogen with deuterium (H/D). The analysis of the numerical results offers, for the first time, a clear insight into the effect of quantum tunneling on the activation volume for this hydrogen transfer reaction: this effect results from the different influences that pressure has on the competing thermal and tunneling reaction mechanisms. Furthermore, the computed kinetic isotope effect (H/D) on the activation volume for this model hydrogen transfer correlates well with the experimental data for more complex hydrogen transfer reactions.

Correlation of temperature dependence of hydride kinetic isotope effects with donor-acceptor distances in two solvents of different polarities

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 DADTRS's at the tunneling ready states (TRSs) gives rise to a weaker T-dependence of KIEs, i.e., a smaller ΔEa (= EaD - EaH). In a former work, the solvent effects of acetonitrile versus chloroform on the ΔEa of NADH/NAD+ model reactions were determined, and the DADPRC's of the productive reactant complexes (PRCs) were computed to substitute the DADTRS for the DADTRS-ΔEa correlation study. A smaller ΔEa was found in the more polar acetonitrile where the positively charged PRC is better solvated and has a shorter DADPRC, indirectly supporting the hypothesis. In this work, the TRS structures of different DADTRS'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 DADTRS order in both solutions. It was found that the equilibrium DADTRS is shorter in acetonitrile than in chloroform. Results directly support the DADTRS-ΔEa correlation hypothesis as well as the explanation that links T-dependence of KIEs to DAD sampling catalysis in enzymes.

Quantum Tunneling in Reactions Modulated by External Electric Fields: Reactivity and Selectivity

Quantum tunneling and external electric fields (EEFs) can promote some reactions. However, the synergetic effect of an EEF on a tunneling-involving reaction and its temperature-dependence is not very clear. In this study, we extensively investigated how EEFs affect three reactions that involve hydrogen- or (ground- and excited-state) carbon-tunneling using reliable DFT, DLPNO-CCSD(T1), and variational transition-state theory methods. Our study revealed that oriented EEFs can significantly reduce the barrier and corresponding barrier width (and vice versa) through more electrostatic stabilization in transition states. These EEF effects enhance the nontunneling and tunneling-involving rates. Such EEF effects also decrease the crossover temperatures and quantum tunneling contribution, albeit with lower and thinner barriers. Moreover, EEFs can modulate and switch on/off the tunneling-driven 1,2-H migration of hydroxycarbenes under cryogenic conditions. Furthermore, our study predicts for the first time that EEF/tunneling synergy can control the chemo- or site-selectivity of one molecule bearing two similar/same reactive sites.

Infrared Spectra and Phototransformations of meta-Fluorophenol Isolated in Argon and Nitrogen Matrices

Monomers of meta-fluorophenol (mFP) were trapped from the gas phase into cryogenic argon and nitrogen matrices. The estimated relative energies of the two conformers are very close, and in the gas phase they have nearly equal populations. Due to the similarity of their structures (they only differ in the orientation of the OH group), the two conformers have also similar predicted vibrational signatures, which makes the vibrational characterization of the individual rotamers challenging. In the present work, it has been established that in an argon matrix only the most stable trans conformer of mFP exists (the OH group pointing away from the fluorine atom). On the other hand, the IR spectrum of mFP in a nitrogen matrix testifies to the simultaneous presence in this matrix of both the trans conformer and of the higher-energy cis conformer (the OH group pointing toward the fluorine atom), which is stabilized by interaction with the matrix gas host. We found that the exposition of the cryogenic N₂ matrix to the Globar source of the infrared spectrometer affects the conformational populations. By collecting experimental spectra, either in the full mid-infrared range or only in the range below 2200 cm^-1, we were able to reliably distinguish two sets of experimental bands originating from individual conformers. A comparison of the two sets of experimental bands with computed infrared spectra of the conformers allowed, for the first time, the unequivocal vibrational identification of each of them. The joint implementation of computational vibrational spectroscopy and matrix-isolation infrared spectroscopy proved to be a very accurate method of structural analysis. Some mechanistic insights into conformational isomerism (the quantum tunneling of hydrogen atom and vibrationally-induced conformational transformations) have been addressed. Finally, we also subjected matrix-isolated mFP to irradiations with UV light, and the phototransformations observed in these experiments are also described.

Solvent Effects on the Temperature Dependence of Hydride Kinetic Isotope Effects: Correlation to the Donor-Acceptor Distances

Protein structural effects on the temperature (T) dependence of kinetic isotope effects (KIEs) in H-tunneling reactions have recently been used to discuss about the role of enzyme thermal motions in catalysis. Frequently observed nearly T-independent KIEs in the wild-type enzymes and T-dependent KIEs in variants suggest that H-tunneling in the former is assisted by the naturally evolved protein constructive vibrations that help sample short donor-acceptor distances (DADs) needed. This explanation that correlates the T-dependence of KIEs with DAD sampling has been highly debated as simulations following other H-tunneling models sometimes gave alternative explanations. In this paper, solvent effects on the T-dependence of KIEs of two hydride tunneling reactions of NADH/NAD⁺ analogues (represented by ΔE_a = E_aD - E_aH) were determined in attempts to replicate the observations in enzymes and test the protein vibration-assisted DAD sampling concept. Effects of selected aprotic solvents on the DAD_PRC's of the productive reactant complexes (PRCs) and the DAD_TRS's of the activated tunneling ready states (TRSs) were obtained through computations and analyses of the kinetic data, including 2° KIEs, respectively. A weaker T-dependence of KIEs (i.e., smaller ΔE_a) was found in a more polar aprotic solvent in which the system has a shorter average DAD_PRC and DAD_TRS. Further results show that a charge-transfer (CT) complexation made of a stronger donor/acceptor gives rise to a smaller ΔE_a. Overall, the shorter and less broadly distributed DADs resulting from the stronger CT complexation vibrations give rise to a smaller ΔE_a. Our results appear to support the explanation that links the T-dependence of KIEs to the donor-acceptor rigidity in enzymes.

Master Equation Study of Hydrogen Abstraction from HCHO by OH Via a Chemically Activated Intermediate

The abstraction reaction of hydrogen from formaldehyde by OH radical plays an important role in formaldehyde oxidation. The reaction involves a bimolecular association to form a chemically activated hydrogen-bonded reaction...

Direct Correlation between Donor-Acceptor Distance and Temperature Dependence of Kinetic Isotope Effects in Hydride-Tunneling Reactions of NADH/NAD+ Analogues

Recent study of structural effects on primary kinetic isotope effects (1° KIEs) of H-transfer reactions in enzymes and solution revealed that a more rigid reaction system gave rise to a weaker temperature dependence of 1° KIEs, i.e., a smaller isotopic activation energy difference (ΔE_a = E_aD - E_aH). This has been explained within the contemporary vibrationally assisted activated H-tunneling (VA-AHT) model in which rigidity is defined according to the density of donor-acceptor distance (DAD_TRS) populations at the tunneling ready state (TRS) sampled by heavy atom motions. To test the relationship between DAD_TRS and ΔE_a in the model, we developed a computational method to obtain the TRS structures for H-transfer reactions. The method was applied to three hydride transfer reactions of NADH/NAD⁺ analogues for which the ΔE_a's as well as secondary (2°) KIEs have been reported. The 2° KIEs computed from each TRS structure were fitted to the observed values to obtain the optimal TRSs/DAD_TRS's. It was found that a shorter DAD_TRS does correspond with a smaller ΔE_a. This appears to support the VA-AHT model. Moreover, an analysis of hybridizations at the bent TRS structures shows that rehybridizations at the donor-acceptor centers are much more advanced than predicted from the classical mechanism. This implies that more orbital preparations are required for the nonclassical H-tunneling to take place.

Understanding the Large Kinetic Isotope Effect of Hydrogen Tunneling in Condensed Phases by Using Double-Well Model Systems

In recent years, many experiments have shown large kinetic isotope effects (KIEs) for hydrogen transfer reactions in condensed phases as evidence of strong quantum tunneling effects. Since accurate calculation of the tunneling dynamics in such systems still present significant challenges, previous studies have employed different types of approximations to estimate the tunneling effects and KIEs. In this work, by employing model systems consisting of a double-well coupled to a harmonic bath, we calculate the tunneling effects and KIEs using the numerically exact hierarchical equations of motion (HEOM) method. It is found that hydrogen and deuterium transfer reactions in the same system may show rather different behaviors, where hydrogen transfer is dominated by tunneling between the two lowest vibrational states and deuterium transfer is controlled by excited vibrational states close to the barrier top. The simulation results are also used to test the validity of various approximate methods. It is shown that the Wolynes theory of dissipative tunneling gives a good estimation of rate constants in the over-the-barrier regime, while the nonadiabatic reaction rate theory based on the Landau-Zener formula is more suitable for deep tunneling reactions.

Biological concepts for catalysis and reactivity: empowering bioinspiration

Biological systems provide attractive reactivity blueprints for the design of challenging chemical transformations. Emulating the operating mode of natural systems may however not be so easy and direct translation of structural observations does not always afford the anticipated efficiency. Metalloenzymes rely on earth-abundant metals to perform an incredibly wide range of chemical transformations. To do so, enzymes in general have evolved tools and tricks to enable control of such reactivity. The underlying concepts related to these tools are usually well-known to enzymologists and bio(inorganic) chemists but may be a little less familiar to organometallic chemists. So far, the field of bioinspired catalysis has greatly focused on the coordination sphere and electronic effects for the design of functional enzyme models but might benefit from a paradigm shift related to recent findings in biological systems. The goal of this review is to bring these fields closer together as this could likely result in the development of a new generation of highly efficient bioinspired systems. This contribution covers the fields of redox-active ligands, entatic state reactivity, energy conservation through electron bifurcation, and quantum tunneling for C-H activation.

Heavy-Atom Tunneling in Semibullvalenes: How Driving Force, Substituents, and Environment Influence the Tunneling Rates

The Cope rearrangement of selectively deuterated isotopomers of 1,5-dimethylsemibullvalene 2 a and 3,7-dicyano-1,5-dimethylsemibullvalene 2 b were studied in cryogenic matrices. In both semibullvalenes the Cope rearrangement is governed by heavy-atom tunneling. The driving force for the rearrangements is the small difference in the zero-point vibrational energies of the isotopomers. To evaluate the effect of the driving force on the tunneling probability in 2 a and 2 b, two different pairs of isotopomers were studied for each of the semibullvalenes. The reaction rates for the rearrangement of 2 b in cryogenic matrices were found to be smaller than the ones of 2 a under similar conditions, whereas differences in the driving force do not influence the rates. Small curvature tunneling (SCT) calculations suggest that the reduced tunneling rate of 2 b compared to that of 2 a results from a change in the shape of the potential energy barrier. The tunneling probability of the semibullvalenes strongly depends on the matrix environment; however, for 2 a in a qualitatively different way than for 2 b.

Substituent Effects on Temperature Dependence of Kinetic Isotope Effects in Hydride-Transfer Reactions of NADH/NAD+ Analogues in Solution: Reaction Center Rigidity Is the Key

Substituent effects on the temperature dependence of primary kinetic isotope effects, characterized by ΔE_a = E_aD - E_aH, for two series of the title reactions in acetonitrile were studied. The change from ΔE_a ≈ 0 for a highly rigid system to ΔE_a > 0 for systems with reduced rigidities was observed. The rigidities were controlled by the electronic and steric effects. This work replicates the observations in enzymes and opens a new research direction that studies structure-ΔE_a relationship.

Kinetic Isotope Effects and Hydrogen Tunnelling in PCET Oxidations of Ascorbate: New Insights into Aqueous Chemistry?

Recent experimental studies of kinetic isotope effects (KIE-s) and hydrogen tunnelling comprising three proton-coupled electron transfer (PCET) oxidations of ascorbate monoanion, (a) in aqueous reaction solutions, (b) in the mixed water-organic cosolvent systems, (c) in aqueous solutions of various salts and (d) in fairly diluted aqueous solutions of the various partial hydrophobes are reviewed. A number of new insights into the wealth of the kinetic isotope phenomena in the PCET reactions have been obtained. The modulation of KIE-s and hydrogen tunnelling observed when partially hydrophobic solutes are added into water reaction solution, in the case of fairly diluted solutions is revealed as the strong linear correlation of the isotopic ratios of the Arrhenius prefactors Ah/Ad and the isotopic differences in activation energies ΔE_a (D,H). The observation has been proposed to be a signature of the involvement of the collective intermolecular excitonic vibrational dynamics of water in activation processes and aqueous chemistry.

Substitutions of Amino Acid Residues in the Substrate Binding Site of Horse Liver Alcohol Dehydrogenase Have Small Effects on the Structures but Significantly Affect Catalysis of Hydrogen Transfer

Previous studies showed that the L57F and F93W alcohol dehydrogenases catalyze the oxidation of benzyl alcohol with some quantum mechanical hydrogen tunneling, whereas the V203A enzyme has diminished tunneling. Here, steady-state kinetics for the L57F and F93W enzymes were studied, and microscopic rate constants for the ordered bi-bi mechanism were estimated from simulations of transient kinetics for the S48T, F93A, S48T/F93A, F93W, and L57F enzymes. Catalytic efficiencies for benzyl alcohol oxidation (V₁/E_tK_b) vary over a range of ∼100-fold for the less active enzymes up to the L57F enzyme and are mostly associated with the binding of alcohol rather than the rate constants for hydride transfer. In contrast, catalytic efficiencies for benzaldehyde reduction (V₂/E_tK_p) are ∼500-fold higher for the L57F enzyme than for the less active enzymes and are mostly associated with the rate constants for hydride transfer. Atomic-resolution structures (1.1 Å) for the F93W and L57F enzymes complexed with NAD⁺ and 2,3,4,5,6-pentafluorobenzyl alcohol or 2,2,2-trifluoroethanol are almost identical to previous structures for the wild-type, S48T, and V203A enzymes. Least-squares refinement with SHELXL shows that the nicotinamide ring is slightly strained in all complexes and that the apparent donor-acceptor distances from C4N of NAD to C7 of pentafluorobenzyl alcohol range from 3.28 to 3.49 Å (±0.02 Å) and are not correlated with the rate constants for hydride transfer or hydrogen tunneling. How the substitutions affect the dynamics of reorganization during hydrogen transfer and the extent of tunneling remain to be determined.

Calculations Predict That Heavy-Atom Tunneling Dominates Möbius Bond Shifting in [12]- and [16]Annulene

The contribution of heavy-atom tunneling to reactions of [12]- and [16]annulene was probed using small-curvature tunneling rate calculations. At the CCSD(T)/cc-pVDZ//M06-2X/cc-pVDZ level, tunneling is predicted to account for more than 50% of the rate for Möbius bond shifting and ca. 35% of the rate for electrocyclization in [12]annulene, and over 80% of the rate for Möbius bond shifting in [16]annulene, at temperatures at which these reactions have been observed experimentally.

Replication of the Enzymatic Temperature Dependency of the Primary Hydride Kinetic Isotope Effects in Solution: Caused by the Protein-Controlled Rigidity of the Donor-Acceptor Centers?

The change from the temperature independence of the primary (1°) H/D kinetic isotope effects (KIEs) in wild-type enzyme-catalyzed H-transfer reactions (ΔE_a = E_aD - E_aH ∼ 0) to a strong temperature dependence with the mutated enzymes (ΔE_a ≫ 0) has recently been frequently observed. This has prompted some enzymologists to develop new H-tunneling models to correlate ΔE_a with the donor-acceptor distance (DAD) at the tunneling-ready state (TRS) as well as the protein thermal motions/dynamics that sample the short DAD_TRS's for H-tunneling to occur. While extensive evidence supporting or disproving the thermally activated DAD sampling concept has emerged, a comparable study of the simpler bimolecular H-tunneling reactions in solution has not been carried out. In particular, small ΔE_a's (∼0) have not been found. In this paper, we report a study of the hydride-transfer reactions from four NADH models to the same hydride acceptor in acetonitrile. The ΔE_a's were determined: 0.37 (small), 0.60, 0.99, and 1.53 kcal/mol (large). The α-secondary (2°) KIEs on the acceptor that serve as a ruler for the rigidity of reaction centers were previously reported or determined. All possible productive reactant complex (PRC) configurations were computed to provide insight into the structures of the TRS's. Relationships among structures, 2° KIEs, DAD_PRC's, and ΔE_a's were discussed. The more rigid system with more suppressed 2° C-H vibrations at the TRS and more narrowly distributed DAD_PRC's in PRCs gave a smaller ΔE_a. The results replicated the trend observed in enzymes versus mutated enzymes and appeared to support the concepts of different thermally activated DAD_TRS sampling processes in response to the rigid versus flexible donor-acceptor centers.

Tunneling by 16 Carbons: Planar Bond Shifting in [16]Annulene

Midsized annulenes are known to undergo rapid π-bond shifting. Given that heavy-atom tunneling plays a role in planar bond shifting of cyclobutadiene, we computationally explored the contribution of heavy-atom tunneling to planar π-bond shifting in the major (CTCTCTCT, 5a) and minor (CTCTTCTT, 6a) known isomers of [16]annulene. UM06-2X/cc-pVDZ calculations yield bond-shifting barriers of ca. 10 kcal/mol. The results also reveal extremely narrow barrier widths, suggesting a high probability of tunneling for these bond-shifting reactions. Rate constants were calculated using canonical variational transition state theory (CVT) as well as with small curvature tunneling (SCT) contributions, via direct dynamics. For the major isomer 5a, the computed SCT rate constant for bond shifting at 80 K is 0.16 s^-1, corresponding to a half-life of 4.3 s, and indicating that bond shifting is rapid at cryogenic temperatures despite a 10 kcal/mol barrier. This contrasts with the CVT rate constant of 8.0 × 10^-15 s^-1 at 80 K. The minor isomer 6a is predicted to undergo rapid bond shifting via tunneling even at 10 K. For both isomers, bond shifting is predicted to be much faster than competing conformation change despite lower barriers for the latter process. The preference for bond shifting represents cases of tunneling control in which the preferred reaction is dominated by heavy-atom motions. At all temperatures below -50 °C, tunneling is predicted to dominate the bond shifting process for both 5a and 6a. Thus, [16]annulene is predicted to be an example of tunneling by 16 carbons. Bond shifting in both isomers is predicted to be rapid at temperatures accessible by solution-phase NMR spectroscopy, and an experiment is proposed to verify these predictions.

Proton-coupled electron transfer reactions: analytical rate constants and case study of kinetic isotope effects in lipoxygenase

A general theory has been developed for proton-coupled electron transfer (PCET), which is vital to a wide range of chemical and biological processes. This theory describes PCET reactions in terms of nonadiabatic transitions between reactant and product electron-proton vibronic states and includes the effects of thermal fluctuations of the solvent or protein environment, as well as the proton donor-acceptor motion. Within the framework of this general PCET theory, a series of analytical rate constant expressions has been derived for PCET reactions in well-defined regimes. Herein, the application of this theory to PCET in the enzyme soybean lipoxygenase illustrates the regimes of validity for the various rate constant expressions and elucidates the fundamental physical principles dictating PCET reactions. Such theoretical studies provide significant physical insights that guide the interpretation of experimental data and lead to experimentally testable predictions. A combination of theoretical treatments with atomic-level simulations is essential to understanding PCET.

Primary Deuterium Kinetic Isotope Effects: A Probe for the Origin of the Rate Acceleration for Hydride Transfer Catalyzed by Glycerol-3-Phosphate Dehydrogenase

Large primary deuterium kinetic isotope effects (1° DKIEs) on enzyme-catalyzed hydride transfer may be observed when the transferred hydride tunnels through the energy barrier. The following 1° DKIEs on k_cat/K_m and relative reaction driving force are reported for wild-type and mutant glycerol-3-phosphate dehydrogenase (GPDH)-catalyzed reactions of NADL (L = H, D): wild-type GPDH, ΔΔG^⧧ = 0 kcal/mol, 1° DKIE = 1.5; N270A, 5.6 kcal/mol, 3.1; R269A, 9.1 kcal/mol, 2.8; R269A + 1.0 M guanidine, 2.4 kcal/mol, 2.7; R269A/N270A, 11.5 kcal/mol, 2.4. Similar 1° DKIEs were observed on k_cat. The narrow range of 1° DKIEs (2.4–3.1) observed for a 9.1 kcal/mol change in reaction driving force provides strong evidence that these are intrinsic 1° DKIEs on rate-determining hydride transfer. Evidence is presented that the intrinsic DKIE on wild-type GPDH-catalyzed reduction of DHAP lies in this range. A similar range of 1° DKIEs (2.4–2.9) on (k_cat/K_GA, M^–1 s^–1) was reported for dianion-activated hydride transfer from NADL to glycolaldehyde (GA) [Reyes, A. C.; Amyes, T. L.; Richard, J. P. J. Am. Chem. Soc.2016, 138, 14526–14529]. These 1° DKIEs are much smaller than those observed for enzyme-catalyzed hydrogen transfer that occurs mainly by quantum mechanical tunneling. These results support the conclusion that the rate acceleration for GPDH-catalyzed reactions is due to the stabilization of the transition state for hydride transfer by interactions with the protein catalyst. The small 1° DKIEs reported for mutant GPDH-catalyzed and for wild-type dianion-activated reactions are inconsistent with a model where the dianion binding energy is utilized in the stabilization of a tunneling ready state.

A reevaluation of the origin of the rate acceleration for enzyme-catalyzed hydride transfer

There is no consensus of opinion on the origin of the large rate accelerations observed for enzyme-catalyzed hydride transfer. The interpretation of recent results from studies on hydride transfer reactions catalyzed by alcohol dehydrogenase (ADH) focus on the proposal that the effective barrier height is reduced by quantum-mechanical tunneling through the energy barrier. This interpretation contrasts sharply with the notion that enzymatic rate accelerations are obtained through direct stabilization of the transition state for the nonenzymatic reaction in water. The binding energy of the dianion of substrate DHAP provides 11 kcal mol^-1 stabilization of the transition state for the hydride transfer reaction catalyzed by glycerol-3-phosphate dehydrogenase (GPDH). We summarize evidence that the binding interactions between (GPDH) and dianion activators are utilized directly for stabilization of the transition state for enzyme-catalyzed hydride transfer. The possibility is considered, and then discounted, that these dianion binding interactions are utilized for the stabilization of a tunnel ready state (TRS) that enables efficient tunneling of the transferred hydride through the energy barrier, and underneath the energy maximum for the transition state. It is noted that the evidence to support the existence of a tunnel-ready state for the hydride transfer reactions catalyzed by ADH is ambiguous. We propose that the rate acceleration for ADH is due to the utilization of the binding energy of the cofactor NAD+/NADH in the stabilization of the transition state for enzyme-catalyzed hydride transfer.

Fundamental Insights into Proton-Coupled Electron Transfer in Soybean Lipoxygenase from Quantum Mechanical/Molecular Mechanical Free Energy Simulations

The proton-coupled electron transfer (PCET) reaction catalyzed by soybean lipoxygenase has served as a prototype for understanding hydrogen tunneling in enzymes. Herein this PCET reaction is studied with mixed quantum mechanical/molecular mechanical (QM/MM) free energy simulations. The free energy surfaces are computed as functions of the proton donor-acceptor (C-O) distance and the proton coordinate, and the potential of mean force is computed as a function of the C-O distance, inherently including anharmonicity. The simulation results are used to calculate the kinetic isotope effects for the wild-type enzyme (WT) and the L546A/L754A double mutant (DM), which have been measured experimentally to be ∼80 and ∼700, respectively. The PCET reaction is found to be exoergic for WT and slightly endoergic for the DM, and the equilibrium C-O distance for the reactant is found to be ∼0.2 Å greater for the DM than for WT. The larger equilibrium distance for the DM, which is due mainly to less optimal substrate binding in the expanded binding cavity, is primarily responsible for its higher kinetic isotope effect. The calculated potentials of mean force are anharmonic and relatively soft at shorter C-O distances, allowing efficient thermal sampling of the shorter distances required for effective hydrogen tunneling. The primarily local electrostatic field at the transferring hydrogen is ∼100 MV/cm in the direction to facilitate proton transfer and increases dramatically as the C-O distance decreases. These simulations suggest that the overall protein environment is important for conformational sampling of active substrate configurations aligned for proton transfer, but the PCET reaction is influenced primarily by local electrostatic effects that facilitate conformational sampling of shorter proton donor-acceptor distances required for effective hydrogen tunneling.

Synthesis meets theory: Past, present and future of rational chemistry

Chemical synthesis has its roots in the empirical approach of alchemy. Nonetheless, the birth of the scientific method, the technical and technological advances (exploiting revolutionary discoveries in physics) and the improved management and sharing of growing databases greatly contributed to the evolution of chemistry from an esoteric ground into a mature scientific discipline during these last 400 years. Furthermore, thanks to the evolution of computational resources, platforms and media in the last 40 years, theoretical chemistry has added to the puzzle the final missing tile in the process of “rationalizing” chemistry. The use of mathematical models of chemical properties, behaviors and reactivities is nowadays ubiquitous in literature. Theoretical chemistry has been successful in the difficult task of complementing and explaining synthetic results and providing rigorous insights when these are otherwise unattainable by experiment. The first part of this review walks the reader through a concise historical overview on the evolution of the “model” in chemistry. Salient milestones have been highlighted and briefly discussed. The second part focuses more on the general description of recent state-of-the-art computational techniques currently used worldwide by chemists to produce synergistic models between theory and experiment. Each section is complemented by key-examples taken from the literature that illustrate the application of the technique discussed therein.

Privileged Role of Thiolate as the Axial Ligand in Hydrogen Atom Transfer Reactions by Oxoiron(IV) Complexes in Shaping the Potential Energy Surface and Inducing Significant H-Atom Tunneling

An H/D kinetic isotope effect (KIE) of 80 is found at -20 °C for the oxidation of 9,10-dihydroanthracene by [Fe^IV(O)(TMCS)]⁺, a complex supported by the tetramethylcyclam (TMC) macrocycle with a tethered thiolate. This KIE value approaches that previously predicted by DFT calculations. Other [Fe^IV(O)(TMC)(anion)] complexes exhibit values of 20, suggesting that the thiolate ligand of [Fe^IV(O)(TMCS)]⁺ plays a unique role in facilitating tunneling. Calculations show that tunneling is most enhanced (a) when the bond asymmetry between C-H bond breaking and O-H bond formation in the transition state is minimized, and (b) when the electrostatic interactions in the O---H---C moiety are maximal. These two factors-which peak for the best electron donor, the thiolate ligand-afford a slim and narrow barrier through which the H-atom can tunnel most effectively.

Heavy-Atom Tunneling Calculations in Thirteen Organic Reactions: Tunneling Contributions are Substantial, and Bell's Formula Closely Approximates Multidimensional Tunneling at ≥250 K

Multidimensional tunneling calculations are carried out for 13 reactions, to test the scope of heavy-atom tunneling in organic chemistry, and to check the accuracy of one-dimensional tunneling models. The reactions include pericyclic, cycloaromatization, radical cyclization and ring opening, and S_N 2. When compared at the temperatures that give the same effective rate constant of 3×10^-5  s^-1 , tunneling accounts for 25-95 % of the rate in 8 of the 13 reactions. Values of transmission coefficients predicted by Bell's formula, κ_Bell  , agree well with multidimensional tunneling (canonical variational transition state theory with small curvature tunneling), κ_SCT . Mean unsigned deviations of κ_Bell vs. κ_SCT are 0.08, 0.04, 0.02 at 250, 300 and 400 K. This suggests that κ_Bell is a useful first choice for predicting transmission coefficients in heavy-atom tunnelling.

Evidence for proton tunneling and a transient covalent flavin-substrate adduct in choline oxidase S101A

The effect of temperature on the reaction of alcohol oxidation catalyzed by choline oxidase was investigated with the S101A variant of choline oxidase. Anaerobic enzyme reduction in a stopped-flow spectrophotometer was biphasic using either choline or 1,2-[²H₄]-choline as a substrate. The limiting rate constants k_lim1 and k_lim2 at saturating substrate were well separated (k_lim1/k_lim2>9), and were >15-fold slower than for wild-type choline oxidase. Solvent deuterium kinetic isotope effects (KIEs) ~4 established that k_lim1 probes the proton transfer from the substrate hydroxyl to a catalytic base. Primary substrate deuterium KIEs ≥7 demonstrated that k_lim2 reports on hydride transfer from the choline alkoxide to the flavin. Between 15°C and 39°C the k_lim1 and k_lim2 values increased with increasing temperature, allowing for the analyses of H⁺ and H^- transfers using Eyring and Arrhenius formalisms. Temperature-independent KIE on the k_lim1 value (^H2Ok_lim1/^D2Ok_lim1) suggests that proton transfer occurs within a highly reorganized tunneling-ready-state with a narrow distribution of donor-acceptor distances. Eyring analysis of the k_lim2 value gave lines with the slope_(choline)>slope_(D-choline), suggesting kinetic complexity. Spectral evidence for the transient occurrence of a covalent flavin-substrate adduct during the first phase of the anaerobic reaction of S101A CHO with choline is presented, supporting the notion that an important role of amino acid residues in the active site of flavin-dependent enzymes is to eliminate alternative reactions of the versatile enzyme-bound flavin for the reaction that needs to be catalyzed.

The Cope Rearrangement of 1,5-Dimethylsemibullvalene-2(4)-d1 : Experimental Evidence for Heavy-Atom Tunneling

As an experimental test of the theoretical prediction that heavy-atom tunneling is involved in the degenerate Cope rearrangement of semibullvalenes at cryogenic temperatures, monodeuterated 1,5-dimethylsemibullvalene isotopomers were prepared and investigated by IR spectroscopy using the matrix isolation technique. As predicted, the less thermodynamically stable isotopomer rearranges at cryogenic temperatures in the dark to the more stable one, while broadband IR irradiation above 2000 cm^-1 results in an equilibration of the isotopomeric ratio. Since this reaction proceeds with a rate constant in the order of 10^-4  s^-1 despite an experimental barrier of E_a =4.8 kcal mol^-1 and with only a shallow temperature dependence, the results are interpreted in terms of heavy-atom tunneling.

Enhanced Rigidification within a Double Mutant of Soybean Lipoxygenase Provides Experimental Support for Vibronically Nonadiabatic Proton-Coupled Electron Transfer Models

Soybean lipoxygenase (SLO) is a prototype for nonadiabatic hydrogen tunneling reactions and, as such, has served as the subject of numerous theoretical studies. In this work, we report a nearly temperature-independent kinetic isotope effect (KIE) with an average KIE value of 661 ± 27 for a double mutant (DM) of SLO at six temperatures. The data are well-reproduced within a vibronically nonadiabatic proton-coupled electron transfer model in which the active site has become rigidified compared to wild-type enzyme and single-site mutants. A combined temperature-pressure perturbation further shows that temperature-dependent global motions within DM-SLO are more resistant to perturbation by elevated pressure. These findings provide strong experimental support for the model of hydrogen tunneling in SLO, where optimization of both local protein and ligand motions and distal conformational rearrangements is a prerequisite for effective proton vibrational wave function overlap between the substrate and the active-site iron cofactor.

VTST/MT studies of the catalytic mechanism of C-H activation by transition metal complexes with [Cu2(μ-O2)], [Fe2(μ-O2)] and Fe(IV)-O cores based on DFT potential energy surfaces

High-valent Cu and Fe species, which are generated from dioxygen activation in metalloenzymes, carry out the functionalization of strong C-H bonds. Understanding the atomic details of the catalytic mechanism has long been one of the main objectives of bioinorganic chemistry. Large H/D kinetic isotope effects (KIEs) were observed in the C-H activation by high-valent non-heme Cu or Fe complexes in enzymes and their synthetic models. The H/D KIE depends significantly on the transition state properties, such as structure, energies, frequencies, and shape of the potential energy surface, when the tunneling effect is large. Therefore, theoretical predictions of kinetic parameters such as rate constants and KIEs can provide a reliable link between atomic-level quantum mechanical mechanisms and experiments. The accurate prediction of the tunneling effect is essential to reproduce the kinetic parameters. The rate constants and HD/KIE have been calculated using the variational transition-state theory including multidimensional tunneling based on DFT potential energy surfaces along the reaction coordinate. Excellent agreement was observed between the predicted and experimental results, which assures the validity of the DFT potential energy surfaces and, therefore, the proposed atomic-level mechanisms. The [Cu₂(μ-O)₂], [Fe₂(μ-O)₂], and Fe(IV)-oxo species were employed for C-H activation, and their role as catalysts was discussed at an atomic level.

Variational transition state theory: theoretical framework and recent developments

This article reviews the fundamentals of variational transition state theory (VTST), its recent theoretical development, and some modern applications.

Path Sampling Methods for Enzymatic Quantum Particle Transfer Reactions

The mechanisms of enzymatic reactions are studied via a host of computational techniques. While previous methods have been used successfully, many fail to incorporate the full dynamical properties of enzymatic systems. This can lead to misleading results in cases where enzyme motion plays a significant role in the reaction coordinate, which is especially relevant in particle transfer reactions where nuclear tunneling may occur. In this chapter, we outline previous methods, as well as discuss newly developed dynamical methods to interrogate mechanisms of enzymatic particle transfer reactions. These new methods allow for the calculation of free energy barriers and kinetic isotope effects (KIEs) with the incorporation of quantum effects through centroid molecular dynamics (CMD) and the full complement of enzyme dynamics through transition path sampling (TPS). Recent work, summarized in this chapter, applied the method for calculation of free energy barriers to reaction in lactate dehydrogenase (LDH) and yeast alcohol dehydrogenase (YADH). We found that tunneling plays an insignificant role in YADH but plays a more significant role in LDH, though not dominant over classical transfer. Additionally, we summarize the application of a TPS algorithm for the calculation of reaction rates in tandem with CMD to calculate the primary H/D KIE of YADH from first principles. We found that the computationally obtained KIE is within the margin of error of experimentally determined KIEs and corresponds to the KIE of particle transfer in the enzyme. These methods provide new ways to investigate enzyme mechanism with the inclusion of protein and quantum dynamics.

Wide-dynamic-range kinetic investigations of deep proton tunnelling in proteins

Directional proton transport along 'wires' that feed biochemical reactions in proteins is poorly understood. Amino-acid residues with high pKa are seldom considered as active transport elements in such wires because of their large classical barrier for proton dissociation. Here, we use the light-triggered proton wire of the green fluorescent protein to study its ground-electronic-state proton-transport kinetics, revealing a large temperature-dependent kinetic isotope effect. We show that 'deep' proton tunnelling between hydrogen-bonded oxygen atoms with a typical donor-acceptor distance of 2.7-2.8 Å fully accounts for the rates at all temperatures, including the unexpectedly large value (2.5 × 10(9) s(-1)) found at room temperature. The rate-limiting step in green fluorescent protein is assigned to tunnelling of the ionization-resistant serine hydroxyl proton. This suggests how high-pKa residues within a proton wire can act as a 'tunnel diode' to kinetically trap protons and control the direction of proton flow.

Are there dynamical effects in enzyme catalysis? Some thoughts concerning the enzymatic chemical step

We offer some thoughts on the much debated issue of dynamical effects in enzyme catalysis, and more specifically on their potential role in the acceleration of the chemical step. Since the term 'dynamics' has been used with different meanings, we find it useful to first return to the Transition State Theory rate constant, its assumptions and the choices it involves, and detail the various sources of deviations from it due to dynamics (or not). We suggest that much can be learned about the key current questions for enzyme catalysis from prior extensive studies of dynamical and other effects in the case of reactions in solution. We analyze dynamical effects both in the neighborhood of the transition state and far from it, together with the situation when quantum nuclear motion is central to the reaction, and we illustrate our discussion with various examples of enzymatic reactions.

Transition state theory for enzyme kinetics

This article is an essay that discusses the concepts underlying the application of modern transition state theory to reactions in enzymes. Issues covered include the potential of mean force, the quantization of vibrations, the free energy of activation, and transmission coefficients to account for nonequilibrium effect, recrossing, and tunneling.

Another Look at the Mechanisms of Hydride Transfer Enzymes with Quantum and Classical Transition Path Sampling

The mechanisms involved in enzymatic hydride transfer have been studied for years, but questions remain due, in part, to the difficulty of probing the effects of protein motion and hydrogen tunneling. In this study, we use transition path sampling (TPS) with normal mode centroid molecular dynamics (CMD) to calculate the barrier to hydride transfer in yeast alcohol dehydrogenase (YADH) and human heart lactate dehydrogenase (LDH). Calculation of the work applied to the hydride allowed for observation of the change in barrier height upon inclusion of quantum dynamics. Similar calculations were performed using deuterium as the transferring particle in order to approximate kinetic isotope effects (KIEs). The change in barrier height in YADH is indicative of a zero-point energy (ZPE) contribution and is evidence that catalysis occurs via a protein compression that mediates a near-barrierless hydride transfer. Calculation of the KIE using the difference in barrier height between the hydride and deuteride agreed well with experimental results.