The importance of ensemble averaging in enzyme kinetics

Perspectives on Computational Enzyme Modeling: From Mechanisms to Design and Drug Development

Understanding enzyme mechanisms is essential for unraveling the complex molecular machinery of life. In this review, we survey the field of computational enzymology, highlighting key principles governing enzyme mechanisms and discussing ongoing challenges and promising advances. Over the years, computer simulations have become indispensable in the study of enzyme mechanisms, with the integration of experimental and computational exploration now established as a holistic approach to gain deep insights into enzymatic catalysis. Numerous studies have demonstrated the power of computer simulations in characterizing reaction pathways, transition states, substrate selectivity, product distribution, and dynamic conformational changes for various enzymes. Nevertheless, significant challenges remain in investigating the mechanisms of complex multistep reactions, large-scale conformational changes, and allosteric regulation. Beyond mechanistic studies, computational enzyme modeling has emerged as an essential tool for computer-aided enzyme design and the rational discovery of covalent drugs for targeted therapies. Overall, enzyme design/engineering and covalent drug development can greatly benefit from our understanding of the detailed mechanisms of enzymes, such as protein dynamics, entropy contributions, and allostery, as revealed by computational studies. Such a convergence of different research approaches is expected to continue, creating synergies in enzyme research. This review, by outlining the ever-expanding field of enzyme research, aims to provide guidance for future research directions and facilitate new developments in this important and evolving field.

Catalytic Effects of Active Site Conformational Change in the Allosteric Activation of Imidazole Glycerol Phosphate Synthase

Imidazole glycerol phosphate synthase (IGPS) is a class-I glutamine amidotransferase (GAT) that hydrolyzes glutamine. Ammonia is produced and transferred to a second active site, where it reacts with N1-(5'-phosphoribosyl)-formimino-5-aminoimidazole-4-carboxamide ribonucleotide (PrFAR) to form precursors to purine and histidine biosynthesis. Binding of PrFAR over 25 Å away from the active site increases glutaminase efficiency by ∼4500-fold, primarily altering the glutamine turnover number. IGPS has been the focus of many studies on allosteric communication; however, atomic details for how the glutamine hydrolysis rate increases in the presence of PrFAR are lacking. We present a density functional theory study on 237-atom active site cluster models of IGPS based on crystallized structures representing the inactive and allosterically active conformations and investigate the multistep reaction leading to thioester formation and ammonia production. The proposed mechanism is supported by similar, well-studied enzyme mechanisms, and the corresponding energy profile is consistent with steady-state kinetic studies of PrFAR + IGPS. Additional active site models are constructed to examine the relationship between active site structural change and transition-state stabilization via energy decomposition schemes. The results reveal that the inactive IGPS conformation does not provide an adequately formed oxyanion hole structure and that repositioning of the oxyanion strand relative to the substrate is vital for a catalysis-competent oxyanion hole, with or without the hVal51 dihedral flip. These findings are valuable for future endeavors in modeling the IGPS allosteric mechanism by providing insight into the atomistic changes required for rate enhancement that can inform suitable reaction coordinates for subsequent investigations.

Protein Dynamics and Enzymatic Catalysis

This Perspective presents a review of our work and that of others in the highly controversial topic of the coupling of protein dynamics to reaction in enzymes. We have been involved in studying this topic for many years. Thus, this perspective will naturally present our own views, but it also is designed to present an overview of the variety of viewpoints of this topic, both experimental and theoretical. This is obviously a large and contentious topic.

Origin of Catalysis by Nitroalkane Oxidase

The rate of proton abstraction of the carbon acid nitroethane by Asp402 is accelerated by a factor of 10⁸ in the enzyme nitroalkane oxidase (NAO) relative to that by the organic base acetate ion in water. The Cα proton of nitroalkanes is known to exhibit an abnormal correlation between its acidity strength and the rate of deprotonation, with an unusually slow rate of deprotonation in water. This work examines the origin of NAO catalysis, revealing that the rate enhancement by the enzyme is due to transition-state stabilization, restoring the normal behavior of the linear free energy relationship of Bronsted acids. Interestingly, NAO employs the ubiquitous cofactor flavin adenosine diphosphate (FAD) to perform the subsequent oxidation. Does the FAD cofactor also affect the catalytic rate of the initial proton transfer process of the overall nitroalkane oxidation? Classical molecular dynamics and path-integral simulations using a reaction-specific combined quantum mechanics/molecular mechanics (QM/MM) approach were carried out to obtain the free energy reaction profiles, or the potentials of mean force, for the enzymatic reaction and for a model reaction in aqueous solution, as well as for the 2'-deoxy-FAD co-factor-modified NAO. Free energy perturbation calculations suggest that transition-state stabilization of the reactive fragment is the primary cause of the catalytic effect. It is found that the FAD cofactor plays a crucial role in increasing the Cα proton acidity, via specific hydrogen bonding and π-stacking interactions, although these factors have a smaller effect on the enhancement of the rate of deprotonation. Model QM calculations of the π-stacking complexes between the FAD isoalloxazine ring and the neutral and anionic nitroethane, respectively, reveal that the anionic π-stacking complex is more stable than the neutral one by 15.7 kcal/mol, and a net π-stacking energy of 17.3 kcal/mol is obtained. Hence, the isoalloxazine ring, in addition to serving as a very potent oxidizing agent via the formation of covalent intermediate structures, is able to exert a considerable amount of catalytic effect through noncovalent π-stacking interactions.

Unravelling Enzymatic Features in a Supramolecular Iridium Catalyst by Computational Calculations

Non-biological catalysts following the governing principles of enzymes are attractive systems to disclose unprecedented reactivities. Most of those existing catalysts feature an adaptable molecular recognition site for substrate binding that are prone to undergo conformational selection pathways. Herein, we present a non-biological catalyst that is able to bind substrates via the induced fit model according to in-depth computational calculations. The system, which is constituted by an inflexible substrate-recognition site derived from a zinc-porphyrin in the second coordination sphere, features destabilization of ground states as well as stabilization of transition states for the relevant iridium-catalyzed C-H bond borylation of pyridine. In addition, this catalyst appears to be most suited to tightly bind the transition state rather than the substrate. Besides these features, which are reminiscent of the action modes of enzymes, new elementary catalytic steps (i. e. C-B bond formation and catalyst regeneration) have been disclosed owing to the unique distortions encountered in the different intermediates and transition states.

Portable Models for Entropy Effects on Kinetic Selectivity

Differences in entropies of competing transition states can direct kinetic selectivity. Understanding and modeling such entropy differences at the molecular level is complicated by the fact that entropy is statistical in nature; i.e., it depends on multiple vibrational states of transition structures, the existence of multiple dynamically accessible pathways past these transition structures, and contributions from multiple transition structures differing in conformation/configuration. The difficulties associated with modeling each of these contributors are discussed here, along with possible solutions, all with an eye toward the development of portable qualitative models of use to experimentalists aiming to design reactions that make use of entropy to control kinetic selectivity.

On the Origin of Hemoglobin Cooperativity under Non-equilibrium Conditions

Abnormal hemoglobins can have major consequences for tissue delivery of oxygen. Correct diagnosis of hemoglobinopathies with altered oxygen affinity requires a determination of hemoglobin oxygen dissociation curve, which relates the hemoglobin oxygen saturation to the partial pressure of oxygen in the blood. Determination of the oxygen dissociation curve of human hemoglobin is typically carried out under conditions in which hemoglobin is in equilibrium with O2 at each partial pressure. However, in the human body due to the fast transit of red blood cells through tissues hemoglobin oxygen exchanges occur under non-equilibrium conditions. We describe the determination of non-equilibrium oxygen dissociation curve and show that under these conditions the true nature of hemoglobin cooperativity is revealed as emerging solely from the consecutive binding of oxygen to each one of the four subunits of hemoglobin until the entire tetramer is saturated. We call this form of cooperativity the sequential cooperativity of hemoglobin and define the simplest model that includes it as the minimalist model of hemoglobin. A single instantiation of this model accounts for ~70% of hemoglobin cooperativity under non-equilibrium conditions. The total cooperativity of hemoglobin can be viewed more correctly as the summation of two instantiations of the minimalist model (each one corresponding to a tetramer of low and high affinity for O2, respectively) in equilibrium with each other, as in the Monod-Wyman-Changeux model of hemoglobin. In addition to offering new insights on the nature of hemoglobin reaction with oxygen, the methodology described here for the determination of hemoglobin non-equilibrium oxygen dissociation curve provides a simple, fast, low-cost alternative to complex spectrophotometric methods, which is expected to be particularly valuable in regions where hemoglobinopathies are a significant public health problem, but where highly specialized laboratories capable of determining a traditional oxygen dissociation curve are not easily accessible.

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...

Multidimensional Free Energy and Accelerated Quantum Library Methods Provide a Gateway to Glycoenzyme Conformational, Electronic, and Reaction Mechanisms

Enzyme reactions are complex to simulate accurately, and none more so than glycoenzymes (glycosyltransferase and glycosidases). A rigorous sampling of the protein frame and the conformationally plural carbohydrate substrate coupled with an unbiased treatment of the electron dynamics is needed to discover the true reaction landscapes. Here, we demonstrate the effectiveness of two computational methods ported in libraries that we have developed. The first is a flat histogram free energy method called FEARCF capable of multidimensional sampling and rapidly converging to a complete coverage of phase space. The second, the Quantum Supercharger Library (QSL), is a method that accelerates the computation of the ab initio electronic wave function as well as the integral derivatives on graphical processing units (GPUs). These QSL accelerated computations form the core components needed for direct quantum dynamics and QM/MM dynamics when coupled with legacy codes such as GAMESS and NWCHEM, making state of the art hyper-parallel electronic computations in chemistry and chemical biology possible. The combination of QSL (acceleration of ab initio QM computation) and FEARCF (multidimensional hyper-parallel reaction dynamics) makes the simulation of ab initio QM/MM reaction dynamics of enzyme catalysis feasible. Enzymes that process carbohydrates pose an added challenge as their pyranose ring substrates span multidimensional conformational space whose sampling is an intimate function of the catalytic mechanism. Here, we use the pairing of FEARCF and QSL to simulate the catalytic effect of the glycoenzyme β-N-acetylglucosamine transferase (OGT). The reaction mechanism is discovered from a variable three bond reaction surface using SCCDFTB. The role of the OGT in distorting the pyranose ring of β-N-acetylglucosamine (GlcNAc) away from the equilibrium ⁴C₁ chair conformation toward the E₃ envelope needed for the transition state is discovered from its pucker free energy hypersurfaces (or free energy volume, FEV). A complete GlcNAc ring pucker HF 6-31g FEV is constructed from ab initio QM dynamics in vacuum and ab initio QM/MM dynamics in the OGT catalytic domain. The OGT is shown to clearly lower the pathway toward the transition state E₃ ring conformer as well as stabilize it by 1.63 kcal/mol. Illustrated here is the use of QSL accelerated ab initio QM/MM dynamics that thoroughly explores carbohydrate catalyzed reactions through a FEARCF multidimensional sampling of the interdependence between reaction and conformational space. This demonstrates how experimentally inaccessible molecular and electronic mechanisms that underpin enzyme catalysis can be discovered by directly modeling the dynamics of these complex reactions.

Machine-Learning-Assisted Free Energy Simulation of Solution-Phase and Enzyme Reactions

Despite recent advances in the development of machine learning potentials (MLPs) for biomolecular simulations, there has been limited effort on developing stable and accurate MLPs for enzymatic reactions. Here we report a protocol for performing machine-learning-assisted free energy simulation of solution-phase and enzyme reactions at the ab initio quantum-mechanical/molecular-mechanical (ai-QM/MM) level of accuracy. Within our protocol, the MLP is built to reproduce the ai-QM/MM energy and forces on both QM (reactive) and MM (solvent/enzyme) atoms. As an alternative strategy, a delta machine learning potential (ΔMLP) is trained to reproduce the differences between the ai-QM/MM and semiempirical (se) QM/MM energies and forces. To account for the effect of the condensed-phase environment in both MLP and ΔMLP, the DeePMD representation of a molecular system is extended to incorporate the external electrostatic potential and field on each QM atom. Using the Menshutkin and chorismate mutase reactions as examples, we show that the developed MLP and ΔMLP reproduce the ai-QM/MM energy and forces with errors that on average are less than 1.0 kcal/mol and 1.0 kcal mol^-1 Å^-1, respectively, for representative configurations along the reaction pathway. For both reactions, MLP/ΔMLP-based simulations yielded free energy profiles that differed by less than 1.0 kcal/mol from the reference ai-QM/MM results at only a fraction of the computational cost.

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.

Dissecting the Dynamics during Enzyme Catalysis: A Case Study of Pin1 Peptidyl-Prolyl Isomerase

Free energy surfaces have played a central role in studying protein conformational changes and enzymatic reactions over decades. Yet, free energy barriers and kinetics are highly dependent on the coordinates chosen to define the surface, and furthermore, the dynamics during the reactions are often overlooked. Our recent study on the Pin1-catalyzed isomerization reaction has indicated that the isomerization transition events remarkably deviate from the free energy path, highlighting the need to understand the reaction dynamics in more detail. To this end, here we investigate the reaction coordinates that describe the transition states of the free energy and transition pathways by minimizing the cross-entropy function. We show that the isomerization transition events can be expressed by the concerted changes in the improper torsion angle ζ and nearby backbone torsional angles of the ligand, whereas the transition state of the free energy surface involves changes in a broad range of coordinates including multiple protein-ligand interactions. The current result supports the previous finding that the isomerization transitions occur quickly from the conformational excited states, which is in sharp contrast to the slow and collective changes suggested from the free energy path. Our results further indicate that the coordinates derived from the transition trajectories are not sufficient for finding the transition states on the free energy surfaces due to the lack of information from conformational excited states.

Temperature, Dynamics, and Enzyme-Catalyzed Reaction Rates

We review the adaptations of enzyme activity to different temperatures. Psychrophilic (cold-adapted) enzymes show significantly different activation parameters (lower activation enthalpies and entropies) from their mesophilic counterparts. Furthermore, there is increasing evidence that the temperature dependence of many enzyme-catalyzed reactions is more complex than is widely believed. Many enzymes show curvature in plots of activity versus temperature that is not accounted for by denaturation or unfolding. This is explained by macromolecular rate theory: A negative activation heat capacity for the rate-limiting chemical step leads directly to predictions of temperature optima; both entropy and enthalpy are temperature dependent. Fluctuations in the transition state ensemble are reduced compared to the ground state. We show how investigations combining experiment with molecular simulation are revealing fundamental details of enzyme thermoadaptation that are relevant for understanding aspects of enzyme evolution. Simulations can calculate relevant thermodynamic properties (such as activation enthalpies, entropies, and heat capacities) and reveal the molecular mechanisms underlying experimentally observed behavior.

Real-time nanoscale organization of amyloid precursor protein

Despite an intuitive understanding of the role of APP in health and disease, there exist few attempts to dissect its molecular localization at excitatory synapses. Though the biochemistry involved in the enzymatic processing of APP is well understood, there is a void in understanding the nonuniformity of the product formation in vivo. Here, we employed multiple paradigms of single molecules and ensemble based nanoscopic imaging to reveal that APP molecules are organized into regulatory nanodomains that are differentially compartmentalized in the functional zones of an excitatory synapse. Furthermore, with the aid of high density single particle tracking, we show that the lateral diffusion of APP in live cells dictates an equilibrium between these nanodomains and their nano-environment, which is affected in a detrimental variant of APP. Additionally, we incorporate this spatio-temporal detail 'in silico' to generate a realistic nanoscale topography of APP in dendrites and synapses. This approach uncovers a nanoscale heterogeneity in the molecular organization of APP, depicting a locus for differential APP processing. This holistic paradigm, to decipher the real-time heterogeneity of the substrate molecules on the nanoscale, could enable us to better evaluate the molecular constraints overcoming the ensemble approaches used traditionally to understand the kinetics of product formation.

Grafting metal complexes onto amorphous supports: from elementary steps to catalyst site populationsviakernel regression

We use transition state theory, kernel regression, and population balance modeling techniques to model the grafting of metal complexes onto amorphous catalyst supports.

Catalytic Hydrolysis Mechanism of Cocaine by Human Carboxylesterase 1: An Orthoester Intermediate Slows Down the Reaction

Human carboxylesterase 1 (hCES1) is a major carboxylesterase in the human body and plays important roles in the metabolism of a wide variety of substances, including lipids and drugs, and therefore is attracting more and more attention from areas including lipid metabolism, pharmacokinetics, drug–drug interactions, and prodrug activation. In this work, we studied the catalytic hydrolysis mechanism of hCES1 by the quantum mechanics computation method, using cocaine as a model substrate. Our results support the four-step theory of the esterase catalytic hydrolysis mechanism, in which both the acylation stage and the deacylation stage include two transition states and a tetrahedral intermediate. The roles and cooperation of the catalytic triad, S221, H468, and E354, were also analyzed in this study. Moreover, orthoester intermediates were found in hCES1-catalyzed cocaine hydrolysis reaction, which significantly elevate the free energy barrier and slow down the reaction. Based on this finding, we propose that hCES1 substrates with β-aminocarboxylester structure might form orthoester intermediates in hCES1-catalyzed hydrolysis, and therefore prolong their in vivo half-life. Thus, this study helps to clarify the catalytic mechanism of hCES1 and elucidates important details of its catalytic process, and furthermore, provides important insights into the metabolism of hCES1 substrates and drug designing.

Linking inhibitor motions to proteolytic stability of sunflower trypsin inhibitor-1

The remarkable capability of an enzyme isn't only determined by its active site but also controlled by the environment. To unravel the environment role in catalysis, the dynamic motions as well as the static mechanism need to be studied. In this work, QM/MM MD simulations were employed to study the proteolysis process of SFTI-1 and BiKF, which revealed that a combination of static non-bonded interactions and dynamic motions along the reaction coordinate can account for the different hydrolysis rates between them. A comparison among SFTI-1 and three analogs with similar non-bonded interactions further revealed a positive correlation between the mobility of inhibitors and the hydrolysis rates. Apart from the cyclic backbone and disulfide bond, intramolecular hydrogen bonds also increase the rigidity of the backbone of inhibitors, and therefore hinder inhibitor motions to resist proteolysis. These new detailed mechanistic insights suggest the need to consider inhibitor motions in the rational design of peptide inhibitors.

Besides the non-bonded interactions, inhibitor motions especially rotation of the scissile bond also influence proteolytic stability.

Prebiotic synthesis of nucleic acids and their building blocks at the atomic level - merging models and mechanisms from advanced computations and experiments

The origin of life on Earth is one of the most fascinating questions of contemporary science. Extensive research in the past decades furnished diverse experimental proposals for the emergence of first informational polymers that could form the basis of the early terrestrial life. Side by side with the experiments, the fast development of modern computational chemistry methods during the last 20 years facilitated the use of in silico modelling tools to complement the experiments. Modern computations can provide unique atomic-level insights into the structural and electronic aspects as well as the energetics of key prebiotic chemical reactions. Many of these insights are not directly obtainable from the experimental techniques and the computations are thus becoming indispensable for proper interpretation of many experiments and for qualified predictions. This review illustrates the synergy between experiment and theory in the origin of life research focusing on the prebiotic synthesis of various nucleic acid building blocks and on the self-assembly of nucleotides leading to the first functional oligonucleotides.

Influence of water and enzyme SpnF on the dynamics and energetics of the ambimodal [6+4]/[4+2] cycloaddition

SpnF is the first monofunctional Diels-Alder/[6+4]-ase that catalyzes a reaction leading to both Diels-Alder and [6+4] adducts through a single transition state. The environment-perturbed transition-state sampling method has been developed to calculate free energies, kinetic isotope effects, and quasi-classical reaction trajectories of enzyme-catalyzed reactions and the uncatalyzed reaction in water. Energetics calculated in this way reproduce the experiment and show that the normal Diels-Alder transition state is stabilized by H bonds with water molecules, while the ambimodal transition state is favored in the enzyme SpnF by both intramolecular hydrogen bonding and hydrophobic binding. Molecular dynamics simulations show that trajectories passing through the ambimodal transition state bifurcate to the [6+4] adduct and the Diels-Alder adduct with a ratio of 1:1 in the gas phase, 1:1.6 in water, and 1:11 in the enzyme. This example shows how an enzyme acts on a vibrational time scale to steer post-transition state trajectories toward the Diels-Alder adduct.

Spiers Memorial Lecture : Introductory lecture: quantum dynamics of chemical reactions

This Spiers Memorial Lecture discusses quantum effects that can be calculated and observed in the chemical reactions of small molecules.

Ab Initio QM/MM Modeling of the Rate-Limiting Proton Transfer Step in the Deamination of Tryptamine by Aromatic Amine Dehydrogenase

Aromatic amine dehydrogenase (AADH) and related enzymes are at the heart of debates on the roles of quantum tunneling and protein dynamics in catalysis. The reaction of tryptamine in AADH involves significant quantum tunneling in the rate-limiting proton transfer step, shown by large H/D primary kinetic isotope effects (KIEs), with unusual temperature dependence. We apply correlated ab initio combined quantum mechanics/molecular mechanics (QM/MM) methods, at levels up to local coupled cluster theory (LCCSD(T)/(aug)-cc-pVTZ), to calculate accurate potential energy surfaces for this reaction, which are necessary for quantitative analysis of tunneling contributions and reaction dynamics. Different levels of QM/MM treatment are tested. Multiple pathways are calculated with fully flexible transition state optimization by the climbing-image nudged elastic band method at the density functional QM/MM level. The average LCCSD(T) potential energy barriers to proton transfer are 16.7 and 14.0 kcal/mol for proton transfer to the two carboxylate atoms of the catalytic base, Asp128β. The results show that two similar, but distinct pathways are energetically accessible. These two pathways have different barriers, exothermicity and curvature, and should be considered in analyses of the temperature dependence of reaction and KIEs in AADH and other enzymes. These results provide a benchmark for this prototypical enzyme reaction and will be useful for developing empirical models, and analyzing experimental data, to distinguish between different conceptual models of enzyme catalysis.

Coupled Valence-Bond State Molecular Dynamics Description of an Enzyme-Catalyzed Reaction in a Non-Aqueous Organic Solvent

Enzymes are widely used in nonaqueous solvents to catalyze non-natural reactions. While experimental measurements showed that the solvent nature has a strong effect on the reaction kinetics, the molecular details of the catalytic mechanism in nonaqueous solvents have remained largely elusive. Here we study the transesterification reaction catalyzed by the paradigm subtilisin Carlsberg serine protease in an organic apolar solvent. The rate-limiting acylation step involves a proton transfer between active-site residues and the nucleophilic attack of the substrate to form a tetrahedral intermediate. We design the first coupled valence-bond state model that simultaneously describes both reactions in the enzymatic active site. We develop a new systematic procedure to parametrize this model on high-level ab initio QM/MM free energy calculations that account for the molecular details of the active site and for both substrate and protein conformational fluctuations. Our calculations show that the reaction energy barrier changes dramatically with the solvent and protein conformational fluctuations. We find that the mechanism of the tetrahedral intermediate formation during the acylation step is similar to that determined under aqueous conditions, and that the proton transfer and nucleophilic attack reactions occur concertedly. We identify the reaction coordinate to be mostly due to the rearrangement of some residual water molecules close to the active site.

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.

Protein Ensembles: How Does Nature Harness Thermodynamic Fluctuations for Life? The Diverse Functional Roles of Conformational Ensembles in the Cell

All soluble proteins populate conformational ensembles that together constitute the native state. Their fluctuations in water are intrinsic thermodynamic phenomena, and the distributions of the states on the energy landscape are determined by statistical thermodynamics; however, they are optimized to perform their biological functions. In this review we briefly describe advances in free energy landscape studies of protein conformational ensembles. Experimental (nuclear magnetic resonance, small-angle X-ray scattering, single-molecule spectroscopy, and cryo-electron microscopy) and computational (replica-exchange molecular dynamics, metadynamics, and Markov state models) approaches have made great progress in recent years. These address the challenging characterization of the highly flexible and heterogeneous protein ensembles. We focus on structural aspects of protein conformational distributions, from collective motions of single- and multi-domain proteins, intrinsically disordered proteins, to multiprotein complexes. Importantly, we highlight recent studies that illustrate functional adjustment of protein conformational ensembles in the crowded cellular environment. We center on the role of the ensemble in recognition of small- and macro-molecules (protein and RNA/DNA) and emphasize emerging concepts of protein dynamics in enzyme catalysis. Overall, protein ensembles link fundamental physicochemical principles and protein behavior and the cellular network and its regulation.

Phase Space Bottlenecks in Enzymatic Reactions

The definition of a transition state on an individual reactive trajectory is made via a committor analysis. In the past, the bottleneck definition has often been applied in configuration space. This is an approximation, and in order to expand this definition, we are revisiting an enzyme in which we had identified a fast subpicosecond motion that makes the reaction possible. First we used a time-series analysis method to identify the exact time when this motion initiates donor-acceptor compression. Then we modified the standard committor analysis of transition path sampling to identify events in phase space and found that there is a dividing surface in phase space significantly earlier than the configurationally defined transition-state crossing.

Tunneling effect in vitamin E recycling by green tea

A tunneling effect was found to play an important role in vitamin E recycling reactions by catechins contained in green tea.

Role of Protein Dynamics in Allosteric Control of the Catalytic Phosphoryl Transfer of Insulin Receptor Kinase

The catalytic and allosteric mechanisms of insulin receptor kinase (IRK) are investigated by a combination of ab initio and semiempirical quantum mechanical and molecular mechanical (QM/MM) methods and classical molecular dynamics (MD) simulations. The simulations reveal that the catalytic reaction proceeds in two steps, starting with the transfer of a proton from substrate Tyr to the catalytic Asp1132, followed by the phosphoryl transfer from ATP to substrate Tyr. The enhancement of the catalytic rate of IRK upon phosphorylations in the enzyme's activation loop is found to occur mainly via changes to the free energy landscape of the proton transfer step, favoring the proton transfer in the fully phosphorylated enzyme. In contrast, the effects of the phosphorylations on the phosphoryl transfer are smaller. Equilibrium MD simulations show that IRK phosphorylations affect the protein dynamics of the enzyme before the proton transfer to Asp1132 with only a minor effect after the proton transfer. This finding is consistent with the large change in the proton transfer free energy and the smaller change in the free energy barrier of phosphoryl transfer found by QM/MM simulations. Taken together, the present results provide details on how IRK phosphorylation exerts allosteric control of the catalytic activity via modifications of protein dynamics and free energy landscape of catalytic reaction. The results also highlight the importance of protein dynamics in connecting protein allostery and catalysis to control catalytic activity of enzymes.

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.

Review of computer simulations of isotope effects on biochemical reactions: From the Bigeleisen equation to Feynman's path integral

Enzymatic reactions are integral components in many biological functions and malfunctions. The iconic structure of each reaction path for elucidating the reaction mechanism in details is the molecular structure of the rate-limiting transition state (RLTS). But RLTS is very hard to get caught or to get visualized by experimentalists. In spite of the lack of explicit molecular structure of the RLTS in experiment, we still can trace out the RLTS unique "fingerprints" by measuring the isotope effects on the reaction rate. This set of "fingerprints" is considered as a most direct probe of RLTS. By contrast, for computer simulations, oftentimes molecular structures of a number of TS can be precisely visualized on computer screen, however, theoreticians are not sure which TS is the actual rate-limiting one. As a result, this is an excellent stage setting for a perfect "marriage" between experiment and theory for determining the structure of RLTS, along with the reaction mechanism, i.e., experimentalists are responsible for "fingerprinting", whereas theoreticians are responsible for providing candidates that match the "fingerprints". In this Review, the origin of isotope effects on a chemical reaction is discussed from the perspectives of classical and quantum worlds, respectively (e.g., the origins of the inverse kinetic isotope effects and all the equilibrium isotope effects are purely from quantum). The conventional Bigeleisen equation for isotope effect calculations, as well as its refined version in the framework of Feynman's path integral and Kleinert's variational perturbation (KP) theory for systematically incorporating anharmonicity and (non-parabolic) quantum tunneling, are also presented. In addition, the outstanding interplay between theory and experiment for successfully deducing the RLTS structures and the reaction mechanisms is demonstrated by applications on biochemical reactions, namely models of bacterial squalene-to-hopene polycyclization and RNA 2'-O-transphosphorylation. For all these applications, we used our recently-developed path-integral method based on the KP theory, called automated integration-free path-integral (AIF-PI) method, to perform ab initio path-integral calculations of isotope effects. As opposed to the conventional path-integral molecular dynamics (PIMD) and Monte Carlo (PIMC) simulations, values calculated from our AIF-PI path-integral method can be as precise as (not as accurate as) the numerical precision of the computing machine. Lastly, comments are made on the general challenges in theoretical modeling of candidates matching the experimental "fingerprints" of RLTS. This article is part of a Special Issue entitled: Enzyme Transition States from Theory and Experiment.