Tunneling in enzymatic and nonenzymatic hydrogen transfer 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.

The importance of ensemble averaging in enzyme kinetics

The active site of an enzyme is surrounded by a fluctuating environment of protein and solvent conformational states, and a realistic calculation of chemical reaction rates and kinetic isotope effects of enzyme-catalyzed reactions must take account of this environmental diversity. Ensemble-averaged variational transition state theory with multidimensional tunneling (EA-VTST/MT) was developed as a way to carry out such calculations. This theory incorporates ensemble averaging, quantized vibrational energies, energy, tunneling, and recrossing of transition state dividing surfaces in a systematic way. It has been applied successfully to a number of hydrogen-, proton-, and hydride-transfer reactions. The theory also exposes the set of effects that should be considered in reliable rate constants calculations.

We first review the basic theory and the steps in the calculation. A key role is played by the generalized free energy of activation profile, which is obtained by quantizing the classical potential of mean force as a function of a reaction coordinate because the one-way flux through the transition state dividing surface can be written in terms of the generalized free energy of activation. A recrossing transmission coefficient accounts for the difference between the one-way flux through the chosen transition state dividing surface and the net flux, and a tunneling transmission coefficient converts classical motion along the reaction coordinate to quantum mechanical motion. The tunneling calculation is multidimensional, accounting for the change in vibrational frequencies along the tunneling path and shortening of the tunneling path with respect to the minimum energy path (MEP), as promoted by reaction-path curvature. The generalized free energy of activation and the transmission coefficients both involve averaging over an ensemble of reaction paths and conformations, and this includes the coupling of protein motions to the rearrangement of chemical bonds in a statistical mechanically correct way. The standard deviations of the transmissions coefficients provide information on the diversity of the distribution of reaction paths, barriers, and protein conformations along the members of an ensemble of reaction paths passing through the transition state.

We first illustrate the theory by discussing the application to both wild-type and mutant Escherichia coli dihydrofolate reductase and hyperthermophilic Thermotoga maritima dihydrofolate reductase (DHFR); DHFR is of special interest because the protein conformational changes have been widely studied. Then we present shorter discussions of several other applications of EA-VTST/MT to transfer of protons, hydrogen atoms, and hydride ions and their deuterated analogs. Systems discussed include hydride transfer in alcohol dehydrogenase, xylose isomerase, and thymidylate synthase, proton transfer in methylamine dehydrogenase, hydrogen atom transfer in methylmalonyl-CoA mutase, and nucleophilic substitution in haloalkane dehalogenase and two-dimensional potentials of mean force for potentially coupled proton and hydride transfer in the β-oxidation of butyryl-coenzyme A catalyzed by short-chain acyl-CoA dehydrogenase and in the pyruvate to lactate transformation catalyzed by lactate dehydrogenase.

Applications of quantum mechanical/molecular mechanical methods to the chemical insertion step of DNA and RNA polymerization

We review theoretical attempts to model the chemical insertion reactions of nucleoside triphosphates catalyzed by the nucleic acid polymerases using combined quantum mechanical/molecular mechanical methodology. Due to an existing excellent database of high-resolution X-ray crystal structures, the DNA polymerase β system serves as a useful template for discussion and comparison. The convergence of structures of high-quality complexes and continued developments of theoretical techniques suggest a bright future for understanding the global features of nucleic acid polymerization.

Factors affecting hydrogen-tunneling contribution in hydroxylation reactions promoted by oxoiron(IV) porphyrin π-cation radical complexes

Hydrogen atom transfer with a tunneling effect (H-tunneling) has been proposed to be involved in aliphatic hydroxylation reactions catalyzed by cytochrome P450 and synthetic heme complexes as a result of the observation of large hydrogen/deuterium kinetic isotope effects (KIEs). In the present work, we investigate the factors controlling the H-tunneling contribution to the H-transfer process in hydroxylation reaction by examining the kinetics of hydroxylation reactions at the benzylic positions of xanthene and 1,2,3,4-tetrahydronaphthalene by oxoiron(IV) 5,10,15,20-tetramesitylporphyrin π-cation radical complexes ((TMP(+•))Fe(IV)O(L)) under single-turnover conditions. The Arrhenius plots for these hydroxylation reactions of H-isotopomers have upwardly concave profiles. The Arrhenius plots of D-isotopomers, clear isosbestic points, and product analysis rule out the participation of thermally dependent other reaction processes in the concave profiles. These results provide evidence for the involvement of H-tunneling in the rate-limiting H-transfer process. These profiles are simulated using an equation derived from Bell's tunneling model. The temperature dependence of the KIE values (k(H)/k(D)) determined for these reactions indicates that the KIE value increases as the reaction temperature becomes lower, the bond dissociation energy (BDE) of the C-H bond of a substrate becomes higher, and the reactivity of (TMP(+•))Fe(IV)O(L) decreases. In addition, we found correlation of the slope of the ln(k(H)/k(D)) - 1/T plot and the bond strengths of the Fe═O bond of (TMP(+•))Fe(IV)O(L) estimated from resonance Raman spectroscopy. These observations indicate that these factors modulate the extent of the H-tunneling contribution by modulating the ratio of the height and thickness of the reaction barrier.

Proton transfer pathways, energy landscape, and kinetics in creatine-water systems

We study the exchange processes of the metabolite creatine, which is present in both tumorous and normal tissues and has NH2 and NH groups that can transfer protons to water. Creatine produces chemical exchange saturation transfer (CEST) contrast in magnetic resonance imaging (MRI). The proton transfer pathway from zwitterionic creatine to water is examined using a kinetic transition network constructed from the discrete path sampling approach and an approximate quantum-chemical energy function, employing the self-consistent-charge density-functional tight-binding (SCC-DFTB) method. The resulting potential energy surface is visualized by constructing disconnectivity graphs. The energy landscape consists of two distinct regions corresponding to the zwitterionic creatine structures and deprotonated creatine. The activation energy that characterizes the proton transfer from the creatine NH2 group to water was determined from an Arrhenius fit of rate constants as a function of temperature, obtained from harmonic transition state theory. The result is in reasonable agreement with values obtained in water exchange spectroscopy (WEX) experiments.

Kinetic isotope effects as a probe of hydrogen transfers to and from common enzymatic cofactors

Enzymes use a number of common cofactors as sources of hydrogen to drive biological processes, but the physics of the hydrogen transfers to and from these cofactors is not fully understood. Researchers study the mechanistically important contributions from quantum tunneling and enzyme dynamics and connect those processes to the catalytic power of enzymes that use these cofactors. Here we describe some progress that has been made in studying these reactions, particularly through the use of kinetic isotope effects (KIEs). We first discuss the general theoretical framework necessary to interpret experimental KIEs, and then describe practical uses for KIEs in the context of two case studies. The first example is alcohol dehydrogenase, which uses a nicotinamide cofactor to catalyze a hydride transfer, and the second example is thymidylate synthase, which uses a folate cofactor to catalyze both a hydride and a proton transfer.

Synergy between experimental and theoretical methods in the exploration of homogeneous transition metal catalysis

In this Perspective, we will focus on the use of both experimental and theoretical methods in the exploration of reaction mechanisms in homogeneous transition metal catalysis. The current state-of-the-art is highlighted using examples from the literature with particular focus on the synergy between experiment and theory.

The dilemma of conformational dynamics in enzyme catalysis: perspectives from theory and experiment

The role of protein dynamics in catalysis is a contemporary issue that has stirred intense debate in the field. This chapter provides a brief overview of the approaches and findings of a wide range of experimental, computational and theoretical studies that have addressed this issue. We summarize the results of our recent atomistic molecular dynamic studies on cis-trans isomerase. Our results help to reconcile the disparate perspectives regarding the complex role of enzyme dynamics in the catalytic step and emphasize the major contribution of transition state stabilization in rate enhancement.

Molecular mechanisms for the reaction between (˙)OH radicals and proline: insights on the role as reactive oxygen species scavenger in plant stress

The accumulation of proline (Pro) and overproduction of reactive oxygen species (ROS) by plants exposed to stress is well-documented. In vitro assays show that enzyme inactivation by hydroxyl radicals ((•)OH) can be avoided in the presence of Pro, suggesting this amino acid might act as a (•)OH scavenger. Although production of hydroxyproline (Hyp) has been hypothesized in connection with such antioxidant activity, no evidence on the detailed mechanism of scavenging has been reported. To elucidate whether and how Hyp might be produced, we used density functional theory calculations coupled to a polarizable continuum model to explore 27 reaction channels including H-abstraction by (•)OH and (•)OH/H2O addition. The structure and energetics of stable species and transition states for each reaction channel were characterized at the PCM-(U)M06/6-31G(d,p) level in aqueous solution. Evidence is found for a main pathway in which Pro scavenges (•)OH by successive H-abstractions (ΔG(‡,298) = 4.1 and 7.5 kcal mol(-1)) to yield 3,4-Δ-Pro. A companion pathway with low barriers yielding Δ(1)-pyrroline-5-carboxylate (P5C) is also supported, linking with 5-Hyp through hydration. However, this connection remains unlikely in stressed plants because P5C would be efficiently recycled to Pro (contributing to its accumulation) by P5C reductase, hypothesis coined here as the "Pro-Pro cycle".

A critical test of the "tunneling and coupled motion" concept in enzymatic alcohol oxidation

The physical mechanism of C-H bond activation by enzymes is the subject of intense study, and we have tested the predictions of two competing models for C-H activation in the context of alcohol dehydrogenase. The kinetic isotope effects (KIEs) in this enzyme have previously suggested a model of quantum mechanical tunneling and coupled motion of primary (1°) and secondary (2°) hydrogens. Here we measure the 2° H/T KIEs with both H and D at the 1° position and find that the 2° KIE is significantly deflated with D-transfer, consistent with the predictions of recent Marcus-like models of H-transfer. The results suggest that the fast dynamics of H-tunneling result in a 1° isotope effect on the structure of the tunneling ready state: the trajectory of D-transfer goes through a shorter donor-acceptor distance than that of H-transfer.

Hydrogen transfer reactions in viscous media — Potential and free energy surfaces in solvent–solute coordinates and their kinetic implications

Two-dimensional potential energy and free energy surfaces are obtained using quantum mechanical and molecular dynamics calculations for four hydrogen transfer reactions in n-hexane solvent: the methyl–methane, n-propyl–n-propane, n-pentyl–n-pentane, and t-butyl–isobutane systems. The resultant surfaces have similar landscapes despite the fact the equilibrated solvent cavities for these systems are notably different in size and shape. The kinetic implications of these landscapes are discussed. The Arrhenius and tunneling kinetics of hydrogen transfer in nonpolar hydrocarbon solute–solvent systems are not expected to show any significant viscosity dependence.

Chloroform as a hydrogen atom donor in Barton reductive decarboxylation reactions

The utility of chloroform as both a solvent and a hydrogen atom donor in Barton reductive decarboxylation of a range of carboxylic acids was recently demonstrated (Ko, E. J. et al. Org. Lett. 2011, 13, 1944). In the present work, a combination of electronic structure calculations, direct dynamics calculations, and experimental studies was carried out to investigate how chloroform acts as a hydrogen atom donor in Barton reductive decarboxylations and to determine the scope of this process. The results from this study show that hydrogen atom transfer from chloroform occurs directly under kinetic control and is aided by a combination of polar effects and quantum mechanical tunneling. Chloroform acts as an effective hydrogen atom donor for primary, secondary, and tertiary alkyl radicals, although significant chlorination was also observed with unstrained tertiary carboxylic acids.

Isotope effects as probes for enzyme catalyzed hydrogen-transfer reactions

Kinetic Isotope effects (KIEs) have long served as a probe for the mechanisms of both enzymatic and solution reactions. Here, we discuss various models for the physical sources of KIEs, how experimentalists can use those models to interpret their data, and how the focus of traditional models has grown to a model that includes motion of the enzyme and quantum mechanical nuclear tunneling. We then present two case studies of enzymes, thymidylate synthase and alcohol dehydrogenase, and discuss how KIEs have shed light on the C-H bond cleavages those enzymes catalyze. We will show how the combination of both experimental and computational studies has changed our notion of how these enzymes exert their catalytic powers.

Fast protein motions are coupled to enzyme H-transfer reactions

Coupling of fast protein dynamics to enzyme chemistry is controversial and has ignited considerable debate, especially over the past 15 years in relation to enzyme-catalyzed H-transfer. H-transfer can occur by quantum tunneling, and the temperature dependence of kinetic isotope effects (KIEs) has emerged as the "gold standard" descriptor of these reactions. The anomalous temperature dependence of KIEs is often rationalized by invoking fast motions to facilitate H-transfer, yet crucially, direct evidence for coupled motions is lacking. The fast motions hypothesis underpinning the temperature dependence of KIEs is based on inference. Here, we have perturbed vibrational motions in pentaerythritol tetranitrate reductase (PETNR) by isotopic substitution where all non-exchangeable atoms were replaced with the corresponding heavy isotope ((13)C, (15)N, and (2)H). The KIE temperature dependence is perturbed by heavy isotope labeling, demonstrating a direct link between (promoting) vibrations in the protein and the observed KIE. Further we show that temperature-independent KIEs do not necessarily rule out a role for fast dynamics coupled to reaction chemistry. We show causality between fast motions and enzyme chemistry and demonstrate how this impacts on experimental KIEs for enzyme reactions.

Experimental and theoretical studies of enzyme-catalyzed hydrogen-transfer reactions

The mechanisms of enzyme-catalyzed reactions are medicinally important and present a fascinating intellectual challenge. Many experimental and theoretical techniques can shed light on these mechanisms, and here, we shall focus on the utility of kinetic isotope effects (KIEs) to study enzymatic reactions that involve hydrogen transfers. We will provide a short background on the prevailing models to interpret KIEs and then present more detailed reviews of two model enzymes: alcohol dehydrogenase and thymidylate synthase. These two examples provide a context to describe the types of experiments and theoretical calculations that drive this field forward and the kind of information each can furnish. We emphasize the importance of cooperation between experimentalists and theoreticians to continue the progress toward a comprehensive theory of enzyme catalysis.

A remote mutation affects the hydride transfer by disrupting concerted protein motions in thymidylate synthase

The role of protein flexibility in enzyme-catalyzed activation of chemical bonds is an evolving perspective in enzymology. Here we examine the role of protein motions in the hydride transfer reaction catalyzed by thymidylate synthase (TSase). Being remote from the chemical reaction site, the Y209W mutation of Escherichia coli TSase significantly reduces the protein activity, despite the remarkable similarity between the crystal structures of the wild-type and mutant enzymes with ligands representing their Michaelis complexes. The most conspicuous difference between these two crystal structures is in the anisotropic B-factors, which indicate disruption of the correlated atomic vibrations of protein residues in the mutant. This dynamically altered mutant allows a variety of small thiols to compete for the reaction intermediate that precedes the hydride transfer, indicating disruption of motions that preorganize the protein environment for this chemical step. Although the mutation causes higher enthalpy of activation of the hydride transfer, it only shows a small effect on the temperature dependence of the intrinsic KIE, suggesting marginal changes in the geometry and dynamics of the H-donor and -acceptor at the tunneling ready state. These observations suggest that the mutation disrupts the concerted motions that bring the H-donor and -acceptor together during the pre- and re-organization of the protein environment. The integrated structural and kinetic data allow us to probe the impact of protein motions on different time scales of the hydride transfer reaction within a complex enzymatic mechanism.

Protein dynamics and enzyme catalysis: the ghost in the machine?

One of the most controversial questions in enzymology today is whether protein dynamics are significant in enzyme catalysis. A particular issue in these debates is the unusual temperature-dependence of some kinetic isotope effects for enzyme-catalysed reactions. In the present paper, we review our recent model [Glowacki, Harvey and Mulholland (2012) Nat. Chem. 4, 169-176] that is capable of reproducing intriguing temperature-dependences of enzyme reactions involving significant quantum tunnelling. This model relies on treating multiple conformations of the enzyme-substrate complex. The results show that direct 'driving' motions of proteins are not necessary to explain experimental observations, and show that enzyme reactivity can be understood and accounted for in the framework of transition state theory.

Hydrogen donor-acceptor fluctuations from kinetic isotope effects: a phenomenological model

Kinetic isotope effects (KIEs) and their temperature dependence can probe the structural and dynamic nature of enzyme-catalyzed proton or hydride transfers. The molecular interpretation of their temperature dependence requires expensive and specialized quantum mechanics/molecular mechanics (QM/MM) calculations to provide a quantitative molecular understanding. Currently available phenomenological models use a nonadiabatic assumption that is not appropriate for most hydride and proton-transfer reactions, while others require more parameters than the experimental data justify. Here we propose a phenomenological interpretation of KIEs based on a simple method to quantitatively link the size and temperature dependence of KIEs to a conformational distribution of the catalyzed reaction. This model assumes adiabatic hydrogen tunneling, and by fitting experimental KIE data, the model yields a population distribution for fluctuations of the distance between donor and acceptor atoms. Fits to data from a variety of proton and hydride transfers catalyzed by enzymes and their mutants, as well as nonenzymatic reactions, reveal that steeply temperature-dependent KIEs indicate the presence of at least two distinct conformational populations, each with different kinetic behaviors. We present the results of these calculations for several published cases and discuss how the predictions of the calculations might be experimentally tested. This analysis does not replace molecular QM/MM investigations, but it provides a fast and accessible way to quantitatively interpret KIEs in the context of a Marcus-like model.

Taking Ockham's razor to enzyme dynamics and catalysis

The role of protein dynamics in enzyme catalysis is a matter of intense current debate. Enzyme-catalysed reactions that involve significant quantum tunnelling can give rise to experimental kinetic isotope effects with complex temperature dependences, and it has been suggested that standard statistical rate theories, such as transition-state theory, are inadequate for their explanation. Here we introduce aspects of transition-state theory relevant to the study of enzyme reactivity, taking cues from chemical kinetics and dynamics studies of small molecules in the gas phase and in solution--where breakdowns of statistical theories have received significant attention and their origins are relatively better understood. We discuss recent theoretical approaches to understanding enzyme activity and then show how experimental observations for a number of enzymes may be reproduced using a transition-state-theory framework with physically reasonable parameters. Essential to this simple model is the inclusion of multiple conformations with different reactivity.

Good vibrations in enzyme-catalysed reactions

Fast motions (femtosecond to picosecond) and their potential involvement during enzyme-catalysed reactions have ignited considerable interest in recent years. Their influence on reaction chemistry has been inferred indirectly from studies of the anomalous temperature dependence of kinetic isotope effects and computational simulations. But can such motion reduce the width and height of energy barriers along the reaction coordinate, and contribute to quantum mechanical and/or classical nuclear-transfer chemistry? Here we discuss contemporary ideas for enzymatic reactions invoking a role for fast 'promoting' (or 'compressive') motions that, in principle, can aid hydrogen-transfer reactions. Of key importance is the direct demonstration of a role for compressive motions and the ability to understand in atomic detail the structural origin of these fast motions, but so far this has not been achieved. Here we discuss both indirect experimental evidence that supports a role for compressive motion and the additional insight gained from computational simulations.

Dynamics and dissipation in enzyme catalysis

We use quantized molecular dynamics simulations to characterize the role of enzyme vibrations in facilitating dihydrofolate reductase hydride transfer. By sampling the full ensemble of reactive trajectories, we are able to quantify and distinguish between statistical and dynamical correlations in the enzyme motion. We demonstrate the existence of nonequilibrium dynamical coupling between protein residues and the hydride tunneling reaction, and we characterize the spatial and temporal extent of these dynamical effects. Unlike statistical correlations, which give rise to nanometer-scale coupling between distal protein residues and the intrinsic reaction, dynamical correlations vanish at distances beyond 4-6 Å from the transferring hydride. This work finds a minimal role for nonlocal vibrational dynamics in enzyme catalysis, and it supports a model in which nanometer-scale protein fluctuations statistically modulate--or gate--the barrier for the intrinsic reaction.

Triple isotopic labeling and kinetic isotope effects: exposing H-transfer steps in enzymatic systems

Kinetic isotope effect (KIE) studies can provide insight into the mechanism and kinetics of specific chemical steps in complex catalytic cascades. Recent results from hydrogen KIE measurements have examined correlations between enzyme dynamics and catalytic function, leading to a surge of studies in this area. Unfortunately, most enzymatic H-transfer reactions are not rate limiting, and the observed KIEs do not reliably reflect the intrinsic KIEs on the chemical step of interest. Given their importance to understanding the chemical step under study, accurate determination of the intrinsic KIE from the observed data is essential. In 1975, Northrop developed an elegant method to assess intrinsic KIEs from their observed values [Northrop, D. B. (1975) Steady-state analysis of kinetic isotope effects in enzymic reactions. Biochemistry 14, 2644-2651]. The Northrop method involves KIE measurements using all three hydrogen isotopes, where one of them serves as the reference isotope. This method has been successfully used with different combinations of observed KIEs over the years, but criteria for a rational choice of reference isotope have never before been experimentally determined. Here we compare different reference isotopes (and hence distinct experimental designs) using the reduction of dihydrofolate and dihydrobiopterin by two dissimilar enzymes as model reactions. A number of isotopic labeling patterns have been applied to facilitate the comparative study of reference isotopes. The results demonstrate the versatility of the Northrop method and that such experiments are limited only by synthetic techniques, availability of starting materials, and the experimental error associated with the use of distinct combinations of isotopologues.

Quantum nature of the hydrogen bond

Hydrogen bonds are weak, generally intermolecular bonds, which hold much of soft matter together as well as the condensed phases of water, network liquids, and many ferroelectric crystals. The small mass of hydrogen means that they are inherently quantum mechanical in nature, and effects such as zero-point motion and tunneling must be considered, though all too often these effects are not considered. As a prominent example, a clear picture for the impact of quantum nuclear effects on the strength of hydrogen bonds and consequently the structure of hydrogen bonded systems is still absent. Here, we report ab initio path integral molecular dynamics studies on the quantum nature of the hydrogen bond. Through a systematic examination of a wide range of hydrogen bonded systems we show that quantum nuclear effects weaken weak hydrogen bonds but strengthen relatively strong ones. This simple correlation arises from a competition between anharmonic intermolecular bond bending and intramolecular bond stretching. A simple rule of thumb is provided that enables predictions to be made for hydrogen bonded materials in general with merely classical knowledge (such as hydrogen bond strength or hydrogen bond length). Our work rationalizes the influence of quantum nuclear effects, which can result in either weakening or strengthening of the hydrogen bonds, and the corresponding structures, across a broad range of hydrogen bonded materials. Furthermore, it highlights the need to allow flexible molecules when anharmonic potentials are used in force field-based studies of quantum nuclear effects.

Transmission coefficients for chemical reactions with multiple states: role of quantum decoherence

Transition-state theory (TST) is a widely accepted paradigm for rationalizing the kinetics of chemical reactions involving one potential energy surface (PES). Multiple PES reaction rate constants can also be estimated within semiclassical approaches provided the hopping probability between the quantum states is taken into account when determining the transmission coefficient. In the Marcus theory of electron transfer, this hopping probability was historically calculated with models such as Landau-Zener theory. Although the hopping probability is intimately related to the question of the transition from the fully quantum to the semiclassical description, this issue is not adequately handled in physicochemical models commonly in use. In particular, quantum nuclear effects such as decoherence or dephasing are not present in the rate constant expressions. Retaining the convenient semiclassical picture, we include these effects through the introduction of a phenomenological quantum decoherence function. A simple modification to the usual TST rate constant expression is proposed: in addition to the electronic coupling, a characteristic decoherence time τ(dec) now also appears as a key parameter of the rate constant. This new parameter captures the idea that molecular systems, although intrinsically obeying quantum mechanical laws, behave semiclassically after a finite but nonzero amount of time (τ(dec)). This new degree of freedom allows a fresh look at the underlying physics of chemical reactions involving more than one quantum state. The ability of the proposed formula to describe the main physical lines of the phenomenon is confirmed by comparison with results obtained from density functional theory molecular dynamics simulations for a triplet to singlet transition within a copper dioxygen adduct relevant to the question of dioxygen activation by copper monooxygenases.