Reaction coordinate of an enzymatic reaction revealed by transition path sampling

Computational Investigation of the Covalent Inhibition Mechanism of Bruton's Tyrosine Kinase by Ibrutinib

Covalent inhibitors represent a promising class of therapeutic compounds. Nonetheless, rationally designing covalent inhibitors to achieve a right balance between selectivity and reactivity remains extremely challenging. To better understand the covalent binding mechanism, a computational study is carried out using the irreversible covalent inhibitor of Bruton tyrosine kinase (BTK) ibrutinib as an example. A multi-μs classical molecular dynamics trajectory of the unlinked inhibitor is generated to explore the fluctuations of the compound associated with the kinase binding pocket. Then, the reaction pathway leading to the formation of the covalent bond with the cysteine residue at position 481 via a Michael addition is determined using the string method in collective variables on the basis of hybrid quantum mechanical-molecular mechanical (QM/MM) simulations. The reaction pathway shows a strong correlation between the covalent bond formation and the protonation/deprotonation events taking place sequentially in the covalent inhibition reaction, consistent with a 3-step reaction with transient thiolate and enolates intermediate states. Two possible atomistic mechanisms affecting deprotonation/protonation events from the thiolate to the enolate intermediate were observed: a highly correlated direct pathway involving proton transfer to the Cα of the acrylamide warhead from the cysteine involving one or a few water molecules and a more indirect pathway involving a long-lived enolate intermediate state following the escape of the proton to the bulk solution. The results are compared with experiments by simulating the long-time kinetics of the reaction using kinetic modeling.

Atomistic description of the relationship between protein dynamics and catalysis with transition path sampling

Despite initial resistance, it has been increasingly accepted that protein dynamics plays a role in enzymatic catalysis. There have been two lines of research. Some works study slow conformational motions that are not coupled to the reaction coordinate, but guide the system towards catalytically competent conformations. Understanding at the atomistic level how this is accomplished has remained elusive except for a few systems. In this review we focus on fast sub-picosecond motions that are coupled to the reaction coordinate. The use of Transition Path Sampling has allowed us an atomistic description of how these rate-promoting vibrational motions are incorporated in the reaction mechanism. We will also show how we used insights from rate-promoting motions in protein design.

Phonon-assisted electron-proton transfer in [FeFe] hydrogenases: Topological role of clusters

[FeFe] hydrogenases are enzymes that have acquired a unique capacity to synthesize or consume molecular hydrogen (H2). This function relies on a complex catalytic mechanism involving the active site and two distinct electron and proton transfer networks working in concert. By an analysis based on terahertz vibrations of [FeFe] hydrogenase structure, we are able to predict and identify the existence of rate-promoting vibrations at the catalytic site and the coupling with functional residues involved in reported electron and proton transfer networks. Our findings suggest that the positioning of the cluster is influenced by the response of the scaffold to thermal fluctuations, which in turn drives the formation of networks for electron transfer through phonon-assisted mechanisms. Thus, we address the problem of linking the molecular structure to the catalytic function through picosecond dynamics, while raising the functional gain brought by the cofactors or clusters, using the concept of fold-encoded localized vibrations.

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.

Exact reaction coordinates for flap opening in HIV-1 protease

The primary goal of protein science is to understand how proteins function, which requires understanding the functional dynamics responsible for transitions between different functional structures of a protein. A central concept is the exact reaction coordinates that can determine the value of committor for any protein configuration, which provide the optimal description of functional dynamics. Despite intensive efforts, identifying the exact reaction coordinates (RCs) in complex molecules remains a formidable challenge. Using the recently developed generalized work functional, we report the discovery of the exact RCs for an important functional process-the flap opening of HIV-1 protease. Our results show that this process has six RCs, each one is a linear combination of ~240 backbone dihedrals, providing the precise definition of collectivity and cooperativity in the functional dynamics of a protein. Applying bias potentials along each RC can accelerate flap opening by [Formula: see text] to [Formula: see text] folds. The success in identifying the RCs of a protein with 198 residues represents a significant progress beyond that of the alanine dipeptide, currently the only other complex molecule for which the exact RCs for its conformational changes are known. Our results suggest that the generalized work functional (GWF) might be the fundamental operator of mechanics that controls protein dynamics.

Perspective: Path Sampling Methods Applied to Enzymatic Catalysis

This Perspective reviews the use of Transition Path Sampling methods to study enzymatically catalyzed chemical reactions. First applied by our group to an enzymatic reaction over 15 years ago, the method has uncovered basic principles in enzymatic catalysis such as the protein promoting vibration, and it has also helped harmonize such ideas as electrostatic preorganization with dynamic views of enzyme function. It is now being used to help uncover principles of protein design necessary to artificial enzyme creation.

Time-Lagged Flux in the Transition Path Ensemble: Flux Maximization and Relation to Transition Path Theory

The transition path ensemble is of special interest in reaction coordinate identification as it consists of reactive trajectories that start from the reactant state and end in the product one. As a theoretical framework for describing the transition path ensemble, the transition path theory has been introduced more than 10 years ago, and so far, its applications have only been illustrated in several low-dimensional systems. Given the transition path ensemble, expressions for calculating flux, current (a vector field), and principal curves are derived here in the space of collective variables from the transition path theory, and they are applicable to time series obtained from molecular dynamics simulations of high-dimensional systems, i.e., the position coordinates as a function of time in the transition path ensemble. The connection of the transition path theory is made to a density-weighted average flux, a quantity proposed in a previous work to appraise the relevance of a coordinate to the reaction coordinate [Li, W. J. Chem. Phys. 2022, 156, 054117]. Most importantly, as an extension of the existing quantities, time-lagged quantities such as flux and current are also proposed. The main insights and objects provided by these time-lagged quantities are illustrated in the application to the alanine peptide in vacuum.

A Rigorous Method for Identifying a One-Dimensional Reaction Coordinate in Complex Molecules

Understanding the mechanism of functional protein dynamics is critical to understanding protein functions. Reaction coordinate (RC) is a central topic in protein dynamics, and the grail is to find the one-dimensional RC (1D-RC) that can fully determine the value of a committor (i.e., the reaction probability in configuration space) for any protein configuration. We present a new method that, for the first time, uses a fundamental mechanical operator, the generalized work functional, to identify the rigorous 1D-RC in complex molecules. For a prototypical biomolecular isomerization reaction, the 1D-RC identified by the current method can determine the committor with an accuracy far exceeding what was achieved by previous methods. This method only requires modest computational cost and can be readily applied to large molecules. Most importantly, the generalized work functional is the physical determinant of the collectivity in functional protein dynamics and provides a tentative roadmap that connects the structure of a protein to its function.

Method for Identifying Common Features in Reactive Trajectories of a Transition Path Sampling Ensemble

Simulation methods like transition path sampling (TPS) generate an abundance of information buried in the collection of reactive trajectories that they generate. However, only limited use has been made of this information, mainly for the identification of the reaction coordinate. The standard TPS tools have been designed for monitoring the progress of the system from reactants to products. However, the reaction coordinate does not contain all the information regarding the mechanism. In our earlier work, we have used TPS on enzymatic systems and have identified important motions in the reactant well that prepares the system for the reaction. Since these events take place in the reactant well, they are beyond the reach of standard TPS postprocessing methods. We present a simple scheme for identifying the common trends in enzymatic trajectories. This scheme was designed for a specific class of enzymatic reactions: it can be used for identifying motions that guide the system to reaction-ready conformations. We have applied it to two enzymatic systems that we have studied in the past, formate dehydrogenase and purine nucleoside phosphorylase, and we were able to identify interactions, far from the transition state, that are important for preparing the system for the reaction but that had been overlooked in earlier work.

Explaining reaction coordinates of alanine dipeptide isomerization obtained from deep neural networks using Explainable Artificial Intelligence (XAI)

A method for obtaining appropriate reaction coordinates is required to identify transition states distinguishing product and reactant in complex molecular systems. Recently, abundant research has been devoted to obtaining reaction coordinates using artificial neural networks from deep learning literature, where many collective variables are typically utilized in the input layer. However, it is difficult to explain the details of which collective variables contribute to the predicted reaction coordinates owing to the complexity of the nonlinear functions in deep neural networks. To overcome this limitation, we used Explainable Artificial Intelligence (XAI) methods of the Local Interpretable Model-agnostic Explanation (LIME) and the game theory-based framework known as Shapley Additive exPlanations (SHAP). We demonstrated that XAI enables us to obtain the degree of contribution of each collective variable to reaction coordinates that is determined by nonlinear regressions with deep learning for the committor of the alanine dipeptide isomerization in vacuum. In particular, both LIME and SHAP provide important features to the predicted reaction coordinates, which are characterized by appropriate dihedral angles consistent with those previously reported from the committor test analysis. The present study offers an AI-aided framework to explain the appropriate reaction coordinates, which acquires considerable significance when the number of degrees of freedom increases.

Inverse heavy enzyme isotope effects in methylthioadenosine nucleosidases

Heavy enzyme isotope effects occur in proteins substituted with 2H-, 13C-, and 15N-enriched amino acids. Mass alterations perturb femtosecond protein motions and have been used to study the linkage between fast motions and transition-state barrier crossing. Heavy enzymes typically show slower rates for their chemical steps. Heavy bacterial methylthioadenosine nucleosidases (MTANs from Helicobactor pylori and Escherichia coli) gave normal isotope effects in steady-state kinetics, with slower rates for the heavy enzymes. However, both enzymes revealed rare inverse isotope effects on their chemical steps, with faster chemical steps in the heavy enzymes. Computational transition-path sampling studies of H. pylori and E. coli MTANs indicated closer enzyme-reactant interactions in the heavy MTANs at times near the transition state, resulting in an improved reaction coordinate geometry. Specific catalytic interactions more favorable for heavy MTANs include improved contacts to the catalytic water nucleophile and to the adenine leaving group. Heavy bacterial MTANs depart from other heavy enzymes as slowed vibrational modes from the heavy isotope substitution caused improved barrier-crossing efficiency. Improved sampling frequency and reactant coordinate distances are highlighted as key factors in MTAN transition-state stabilization.

Reaction Coordinate and Thermodynamics of Base Flipping in RNA

Base flipping is a key biophysical event involved in recognition of various ligands by ribonucleic acid (RNA) molecules. However, the mechanism of base flipping in RNA remains poorly understood, in part due to the lack of atomistic details on complex rearrangements in neighboring bases. In this work, we applied transition path sampling (TPS) methods to study base flipping in a double-stranded RNA (dsRNA) molecule that is known to interact with RNA-editing enzymes through this mechanism. We obtained an ensemble of 1000 transition trajectories to describe the base-flipping process. We used the likelihood maximization method to determine the refined reaction coordinate (RC) consisting of two collective variables (CVs), a distance and a dihedral angle between nucleotides that form stacking interactions with the flipping base. The free energy profile projected along the refined RC revealed three minima, two corresponding to the initial and final states and one for a metastable state. We suggest that the metastable state likely represents a wobbled conformation of nucleobases observed in NMR studies that is often characterized as the flipped state. The analyses of reactive trajectories further revealed that the base flipping is coupled to a global conformational change in a stem-loop of dsRNA.

Sequential Water and Headgroup Merger: Membrane Poration Paths and Energetics from MD Simulations

Membrane topology changes such as poration, stalk formation, and hemifusion rupture are essential to cellular function, but their molecular details, energetics, and kinetics are still not fully understood. Here, we present a unified energetic and mechanistic picture of metastable pore defects in tensionless lipid membranes. We used an exhaustive committor analysis to test and select optimal reaction coordinates and also to determine the nucleation mechanism. These reaction coordinates were used to calculate free-energy landscapes that capture the full process and end states. The identified barriers agree with the committor analysis. To enable sufficient sampling of the complete transition path for our molecular dynamics simulations, we developed a “gizmo” potential biasing scheme. The simulations suggest that the essential step in the nucleation is the initial merger of lipid headgroups at the nascent pore center. To facilitate this event, an indentation pathway is energetically preferred to a hydrophobic defect. Continuous water columns that span the indentation were determined to be on-path transients that precede the nucleation barrier. This study gives a quantitative description of the nucleation mechanism and energetics of small metastable pores and illustrates a systematic approach to uncover the mechanisms of diverse cellular membrane remodeling processes.

Role of Protein Motions in Catalysis by Formate Dehydrogenase

We have analyzed the reaction catalyzed by formate dehydrogenase using transition path sampling. This system has recently received experimental attention using infrared spectroscopy and heavy-enzyme studies. Some of the experimental results point to the possible importance of protein motions that are coupled to the chemical step. We found that the residue Val123 that lies behind the nicotinamide ring occasionally comes into van der Waals contact with the acceptor and that in all reactive trajectories, the barrier-crossing event is preceded by this contact, meaning that the motion of Val123 is part of the reaction coordinate. Experimental results have been interpreted with a two-dimensional formula for the chemical rate, which cannot capture effects such as the one we describe.

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.

Nonstatistical Reaction Dynamics

Nonstatistical dynamics is important for many chemical reactions. The Rice-Ramsperger-Kassel-Marcus (RRKM) theory of unimolecular kinetics assumes a reactant molecule maintains a statistical microcanonical ensemble of vibrational states during its dissociation so that its unimolecular dynamics are time independent. Such dynamics results when the reactant's atomic motion is chaotic or irregular. Intrinsic non-RRKM dynamics occurs when part of the reactant's phase space consists of quasiperiodic/regular motion and a bottleneck exists, so that the unimolecular rate constant is time dependent. Nonrandom excitation of a molecule may result in short-time apparent non-RRKM dynamics. For rotational activation, the 2J + 1 K levels for a particular J may be highly mixed, making K an active degree of freedom, or K may be a good quantum number and an adiabatic degree of freedom. Nonstatistical dynamics is often important for bimolecular reactions and their intermediates and for product-energy partitioning of bimolecular and unimolecular reactions. Post–transition state dynamics is often highly complex and nonstatistical.

Universality of fold-encoded localized vibrations in enzymes

Enzymes speed up biochemical reactions at the core of life by as much as 15 orders of magnitude. Yet, despite considerable advances, the fine dynamical determinants at the microscopic level of their catalytic proficiency are still elusive. In this work, we use a powerful mathematical approach to show that rate-promoting vibrations in the picosecond range, specifically encoded in the 3D protein structure, are localized vibrations optimally coupled to the chemical reaction coordinates at the active site. Remarkably, our theory also exposes an hithertho unknown deep connection between the unique localization fingerprint and a distinct partition of the 3D fold into independent, foldspanning subdomains that govern long-range communication. The universality of these features is demonstrated on a pool of more than 900 enzyme structures, comprising a total of more than 10,000 experimentally annotated catalytic sites. Our theory provides a unified microscopic rationale for the subtle structure-dynamics-function link in proteins.

Molecular Modeling Investigations of Sorption and Diffusion of Small Molecules in Glassy Polymers

With a wide range of applications, from energy and environmental engineering, such as in gas separations and water purification, to biomedical engineering and packaging, glassy polymeric materials remain in the core of novel membrane and state-of the art barrier technologies. This review focuses on molecular simulation methodologies implemented for the study of sorption and diffusion of small molecules in dense glassy polymeric systems. Basic concepts are introduced and systematic methods for the generation of realistic polymer configurations are briefly presented. Challenges related to the long length and time scale phenomena that govern the permeation process in the glassy polymer matrix are described and molecular simulation approaches developed to address the multiscale problem at hand are discussed.

Optimization of the Turnover in Artificial Enzymes via Directed Evolution Results in the Coupling of Protein Dynamics to Chemistry

The design of artificial enzymes is an emerging field of research. Although progress has been made, the catalytic proficiency of many designed enzymes is low compared to natural enzymes. Nevertheless, recently Hilvert et al. ( Nat. Chem. 2017, 9, 50-56) created a series of five artificial retro-aldolase enzymes via directed evolution, with the final variant exhibiting a rate comparable to the naturally occurring enzyme fructose 1,6 bisphosphate aldolase. We present a study of this system in atomistic detail that elucidates the effects of mutational changes on the chemical step. Transition path sampling is used to create ensembles of reactive trajectories, and committor analysis is used to identify the stochastic separatrix of each ensemble. The application of committor distribution analysis to constrained trajectories allows the identification of changes in important protein motions coupled to reaction across the generated series of the artificial retro-aldolases. We observed two different reaction mechanisms and analyzed the role of the residues participating in the reaction coordinate of each enzyme. However, only in the most evolved variant we identified a fast motion that promotes catalysis, suggesting that this rate promoting vibration was introduced during directed evolution. This study provides further evidence that protein dynamics must be taken into account in designing efficient artificial enzymes.

Roles of Closed- and Open-Loop Conformations in Large-Scale Structural Transitions of l-Lactate Dehydrogenase

The mechanism of l-lactate generation from pyruvate by l-lactate dehydrogenase (LDH) from the rabbit muscle was studied theoretically by the multistructural microiteration (MSM) method combined with the quantum mechanics/molecular mechanics (QM/MM)-ONIOM method, where the MSM method describes the MM environment as a weighted average of multiple different structures that are fully relaxed during geometry optimization or a reaction path calculation for the QM part. The results showed that the substrate binding and product states were stabilized only in the open-loop conformation of LDH and the reaction occurred in the closed-loop conformation. In other words, before and after the chemical reaction, a large-scale structural transition from the open-loop conformation to the closed-loop conformation and vice versa occurred. The closed-loop conformation stabilized the transition state of the reaction. In contrast, the open-loop conformation stabilized the substrate binding and final states. In other words, the closed- to open-loop transition at the substrate binding state urges capture of the substrate molecule, the subsequent open- to closed-loop transition promotes the product generation, and the final closed- to open-loop transition at the final state prevents the reverse reaction going back to the substrate binding state. It is thus suggested that the exchange of stability between the closed- and open-loop conformations at different states promotes the catalytic cycle.

A Biophysical Perspective on Enzyme Catalysis

Even after a century of investigation, our understanding of how enzymes work remains far from complete. In particular, several factors that enable enzymes to achieve high catalytic efficiencies remain only poorly understood. A number of theories have been developed, which propose or reaffirm that enzymes work as structural scaffolds, serving to bring together and properly orient the participants so that the reaction can proceed; therefore, leading to enzymes being viewed as only passive participants in the catalyzed reaction. A growing body of evidence shows that enzymes are not rigid structures but are constantly undergoing a wide range of internal motions and conformational fluctuations. In this Perspective, on the basis of studies from our group, we discuss the emerging biophysical model of enzyme catalysis that provides a detailed understanding of the interconnection among internal protein motions, conformational substates, enzyme mechanisms, and the catalytic efficiency of enzymes. For a number of enzymes, networks of conserved residues that extend from the surface of the enzyme all the way to the active site have been discovered. These networks are hypothesized to serve as pathways of energy transfer that enables thermodynamical coupling of the surrounding solvent with enzyme catalysis and play a role in promoting enzyme function. Additionally, the role of enzyme structure and electrostatic effects has been well acknowledged for quite some time. Collectively, the recent knowledge gained about enzyme mechanisms suggests that the conventional paradigm of enzyme structure encoding function is incomplete and needs to be extended to structure encodes dynamics, and together these enzyme features encode function including catalytic rate acceleration.

Promoting Vibrations and the Function of Enzymes. Emerging Theoretical and Experimental Convergence

A complete understanding of enzyme catalysis requires knowledge of both transition state features and the detailed motions of atoms that cause reactant molecules to form and traverse the transition state. The seeming intractability of the problem arises from the femtosecond lifetime of chemical transition states, preventing most experimental access. Computational chemistry is admirably suited to short time scale analysis but can be misled by inappropriate starting points or by biased assumptions. Kinetic isotope effects provide an experimental approach to transition state structure and a method for obtaining transition state analogues but, alone, do not inform how that transition state is reached. Enzyme structures with transition state analogues provide computational starting points near the transition state geometry. These well-conditioned starting points, combined with the unbiased computational method of transition path sampling, provide realistic atomistic motions involved in transition state formation and passage. In many, but not all, enzymatic systems, femtosecond local protein motions near the catalytic site are linked to transition state formation. These motions are not inherently revealed by most approaches of transition state theory, because transition state theory replaces dynamics with the statistics of the transition state. Experimental and theoretical convergence of the link between local catalytic site vibrational modes and catalysis comes from heavy atom ("Born-Oppenheimer") enzymes. Fully labeled and catalytic site local heavy atom labels perturb the probability of finding enzymatic transition states in ways that can be analyzed and predicted by transition path sampling. Recent applications of these experimental and computational approaches reveal how subpicosecond local catalytic site protein modes play important roles in creating the transition state.

Directed Evolution as a Probe of Rate Promoting Vibrations Introduced via Mutational Change

In this article, we study with transition path sampling and reaction coordinate analysis how directed evolution in the Kemp eliminase family of artificial enzymes makes differential use of rapid rate promoting vibrations as a component of their chemical mechanism. Even though this family was initially created by placing the expected active site in a fixed protein matrix, we find a shift from largely static to more dynamic active sites that make use of donor-acceptor compression as the evolutionary process proceeds. We see that this introduction of dynamics significantly shifts the order of processes in the reaction. We also suggest that the lack of "design for dynamics" may help explain the relatively low proficiency of such designed enzymes.

Equipartition terms in transition path ensemble: Insights from molecular dynamics simulations of alanine dipeptide

Transition path ensemble consists of reactive trajectories and possesses all the information necessary for the understanding of the mechanism and dynamics of important condensed phase processes. However, quantitative description of the properties of the transition path ensemble is far from being established. Here, with numerical calculations on a model system, the equipartition terms defined in thermal equilibrium were for the first time estimated in the transition path ensemble. It was not surprising to observe that the energy was not equally distributed among all the coordinates. However, the energies distributed on a pair of conjugated coordinates remained equal. Higher energies were observed to be distributed on several coordinates, which are highly coupled to the reaction coordinate, while the rest were almost equally distributed. In addition, the ensemble-averaged energy on each coordinate as a function of time was also quantified. These quantitative analyses on energy distributions provided new insights into the transition path ensemble.

Mechanistic Insights on Human Phosphoglucomutase Revealed by Transition Path Sampling and Molecular Dynamics Calculations

Human α-phosphoglucomutase 1 (α-PGM) catalyzes the isomerization of glucose-1-phosphate into glucose-6-phosphate (G6P) through two sequential phosphoryl transfer steps with a glucose-1,6-bisphosphate (G16P) intermediate. Given that the release of G6P in the gluconeogenesis raises the glucose output levels, α-PGM represents a tempting pharmacological target for type 2 diabetes. Here, we provide the first theoretical study of the catalytic mechanism of human α-PGM. We performed transition-path sampling simulations to unveil the atomic details of the two catalytic chemical steps, which could be key for developing transition state (TS) analogue molecules with inhibitory properties. Our calculations revealed that both steps proceed through a concerted S_N 2-like mechanism, with a loose metaphosphate-like TS. Even though experimental data suggests that the two steps are identical, we observed noticeable differences: 1) the transition state ensemble has a well-defined TS region and a late TS for the second step, and 2) larger coordinated protein motions are required to reach the TS of the second step. We have identified key residues (Arg23, Ser117, His118, Lys389), and the Mg²⁺ ion that contribute in different ways to the reaction coordinate. Accelerated molecular dynamics simulations suggest that the G16P intermediate may reorient without leaving the enzymatic binding pocket, through significant conformational rearrangements of the G16P and of specific loop regions of the human α-PGM.

Electric Fields and Fast Protein Dynamics in Enzymes

In recent years, there has been much discussion regarding the origin of enzymatic catalysis and whether including protein dynamics is necessary for understanding catalytic enhancement. An important contribution in this debate was made with the application of the vibrational Stark effect spectroscopy to measure electric fields in the active site. This provided a window on electric fields at the transition state in enzymatic reactions. We performed computational studies on two enzymes where we have shown that fast dynamics is part of the reaction mechanism and calculated the electric field near the bond-breaking event. We found that the fast motions that we had identified lead to an increase of the electric field, thus preparing an enzymatic configuration that is electrostatically favorable for the catalytic chemical step. We also studied the enzyme that has been the subject of Stark spectroscopy, ketosteroid isomerase, and found electric fields of a similar magnitude to the two previous examples.

High-resolution studies of hydride transfer in the ferredoxin:NADP+ reductase superfamily

Ferredoxin: NADP⁺ reductase (FNR) is an FAD-containing enzyme best known for catalysing the transfer of electrons from ferredoxin (Fd) to NADP⁺ to make NADPH during photosynthesis. It is also the prototype for a broad enzyme superfamily, including the NADPH oxidases (NOXs) that all catalyse similar FAD-enabled electron transfers between NAD(P)H and one-electron carriers. Here, we define further mechanistic details of the NAD(P)H ⇌ FAD hydride-transfer step of the reaction based on spectroscopic studies and high-resolution (~ 1.5 Å) crystallographic views of the nicotinamide-flavin interaction in crystals of corn root FNR Tyr316Ser and Tyr316Ala variants soaked with either nicotinamide, NADP⁺ , or NADPH. The spectra obtained from FNR crystal complexes match those seen in solution and the complexes reveal active site packing interactions and patterns of covalent distortion of the FAD that imply significant active site compression that would favour catalysis. Furthermore, anisotropic B-factors show that the mobility of the C4 atom of the nicotinamide in the FNR:NADP⁺ complex has a directionality matching that expected for boat-like excursions of the nicotinamide ring thought to enhance hydride transfer. Arguments are made for the relevance of this binding mode to catalysis, and specific consideration is given to how the results extrapolate to provide insight to structure-function relations for the membrane-bound NOX enzymes for which little structural information has been available.

DATABASES

Structural data are available in the PDB database under the accession numbers 3LO8 (wild-type), 5VW4 [Y316S:nicotinamide (P3₂ 21)], 5VW9 [Y316S:nicotinamide (P3₁ 21)], 5VW3 [Y316S:NADP⁺ (P3₂ 21)], 5VW8 [Y316S:NADP⁺ (P3₁ 21)], 5VW2 [Y316S:NADPH (P3₂ 21)], 5VW5 [Y316A:nicotinamide (P3₂ 21)], 5VW6 [Y316A:NADP⁺ (P3₂ 21)], 5VW7 [Y316A:NADPH (P3₂ 21)], 5VWA [Y316F (P3₂ 21)], and 5VWB [Y316F:NADP⁺ (P3₁ 21)]. Enzyme Commission number: ferredoxin:NADP⁺ reductase - E C1.18.1.2.

Incorporating Fast Protein Dynamics into Enzyme Design: A Proposed Mutant Aromatic Amine Dehydrogenase

In recent years, there has been encouraging progress in the engineering of enzymes that are designed to catalyze reactions not accelerated by natural enzymes. We tested the possibility of reengineering an existing enzyme by introducing a fast protein motion that couples to the reaction. Aromatic amine dehydrogenase is a system that has been shown to use a fast substrate motion as part of the reaction mechanism. We identified a mutation that preserves this fast motion but also introduces a favorable fast motion near the active site that did not exist in the native enzyme. Transition path sampling was used for the analysis of the atomic details of the mechanism.

Perspective: chemical dynamics simulations of non-statistical reaction dynamics

Non-statistical chemical dynamics are exemplified by disagreements with the transition state (TS), RRKM and phase space theories of chemical kinetics and dynamics. The intrinsic reaction coordinate (IRC) is often used for the former two theories, and non-statistical dynamics arising from non-IRC dynamics are often important. In this perspective, non-statistical dynamics are discussed for chemical reactions, with results primarily obtained from chemical dynamics simulations and to a lesser extent from experiment. The non-statistical dynamical properties discussed are: post-TS dynamics, including potential energy surface bifurcations, product energy partitioning in unimolecular dissociation and avoiding exit-channel potential energy minima; non-RRKM unimolecular decomposition; non-IRC dynamics; direct mechanisms for bimolecular reactions with pre- and/or post-reaction potential energy minima; non-TS theory barrier recrossings; and roaming dynamics.This article is part of the themed issue 'Theoretical and computational studies of non-equilibrium and non-statistical dynamics in the gas phase, in the condensed phase and at interfaces'.

Reaction mechanism and reaction coordinates from the viewpoint of energy flow

Reaction coordinates are of central importance for correct understanding of reaction dynamics in complex systems, but their counter-intuitive nature made it a daunting challenge to identify them. Starting from an energetic view of a reaction process as stochastic energy flows biased towards preferred channels, which we deemed the reaction coordinates, we developed a rigorous scheme for decomposing energy changes of a system, both potential and kinetic, into pairwise components. The pairwise energy flows between different coordinates provide a concrete statistical mechanical language for depicting reaction mechanisms. Application of this scheme to the C7eq → C7ax transition of the alanine dipeptide in vacuum revealed novel and intriguing mechanisms that eluded previous investigations of this well studied prototype system for biomolecular conformational dynamics. Using a cost function developed from the energy decomposition components by proper averaging over the transition path ensemble, we were able to identify signatures of the reaction coordinates of this system without requiring any input from human intuition.

Triple Isotope Effects Support Concerted Hydride and Proton Transfer and Promoting Vibrations in Human Heart Lactate Dehydrogenase

Transition path sampling simulations have proposed that human heart lactate dehydrogenase (LDH) employs protein promoting vibrations (PPVs) on the femtosecond (fs) to picosecond (ps) time scale to promote crossing of the chemical barrier. This chemical barrier involves both hydride and proton transfers to pyruvate to form l-lactate, using reduced nicotinamide adenine dinucleotide (NADH) as the cofactor. Here we report experimental evidence from three types of isotope effect experiments that support coupling of the promoting vibrations to barrier crossing and the coincidence of hydride and proton transfer. We prepared the native (light) LDH and a heavy LDH labeled with ¹³C, ¹⁵N, and nonexchangeable ²H (D) to perturb the predicted PPVs. Heavy LDH has slowed chemistry in single turnover experiments, supporting a contribution of PPVs to transition state formation. Both the [4-²H]NADH (NADD) kinetic isotope effect and the D₂O solvent isotope effect were increased in dual-label experiments combining both NADD and D₂O, a pattern maintained with both light and heavy LDHs. These isotope effects support concerted hydride and proton transfer for both light and heavy LDHs. Although the transition state barrier-crossing probability is reduced in heavy LDH, the concerted mechanism of the hydride-proton transfer reaction is not altered. This study takes advantage of triple isotope effects to resolve the chemical mechanism of LDH and establish the coupling of fs-ps protein dynamics to barrier crossing.

Coupling Protein Dynamics with Proton Transport in Human Carbonic Anhydrase II

The role of protein dynamics in enzyme catalysis is one of the most highly debated topics in enzymology. The main controversy centers around what may be defined as functionally significant conformational fluctuations and how, if at all, these fluctuations couple to enzyme catalyzed events. To shed light on this debate, the conformational dynamics along the transition path surmounting the highest free energy barrier have been herein investigated for the rate limiting proton transport event in human carbonic anhydrase (HCA) II. Special attention has been placed on whether the motion of an excess proton is correlated with fluctuations in the surrounding protein and solvent matrix, which may be rare on the picosecond and subpicosecond time scales of molecular motions. It is found that several active site residues, which do not directly participate in the proton transport event, have a significant impact on the dynamics of the excess proton. These secondary participants are shown to strongly influence the active site environment, resulting in the creation of water clusters that are conducive to fast, moderately slow, or slow proton transport events. The identification and characterization of these secondary participants illuminates the role of protein dynamics in the catalytic efficiency of HCA II.

Targeting a Rate-Promoting Vibration with an Allosteric Mediator in Lactate Dehydrogenase

We present a new type of allosteric modulation in which a molecule bound outside the active site modifies the chemistry of an enzymatic reaction through rapid protein dynamics. As a test case for this type of allostery, we chose an enzyme with a well-characterized rate-promoting vibration, lactate dehydrogenase; identified a suitable small molecule for binding; and used transition path sampling to obtain ensembles of reactive trajectories. We found that the small molecule significantly affected the reaction by changing the position of the transition state and, through applying committor distribution analysis, showed that it removed the protein component from the reaction coordinate. The ability of a small-molecule to disrupt enzymatic reactions through alteration of subpicosecond protein motion opens the door for new experimental studies on protein motion coupled to enzymatic reactions and possibly the design of drugs to target these enzymes.

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.

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.

Transition-State Theory, Dynamics, and Narrow Time Scale Separation in the Rate-Promoting Vibrations Model of Enzyme Catalysis

The power of transition-state theory (TST) for understanding enzymes is evidenced by its recent use in the design and synthesis of highly active de novo enzymes. However, dynamics can influence reaction kinetics, and some studies of rate-promoting vibrations even claim that dynamical theories instead of TST are needed to understand enzymatic reaction mechanisms. For the rate-promoting vibration (RPV) model of enzyme catalysis [Antoniou et al., J. Chem. Phys. 2004, 121, 6442], a reactive flux correlation function analysis shows that dynamical effects do slow the kinetics. However, the RPV model also shows extremely long-lived correlations because the RPV and the bath are not directly coupled. Additionally, earlier studies of the RPV model show a narrow time scale separation due to a small 5kT barrier. Thus earlier findings based on the RPV model may have little bearing on the properties of real enzymes. The intrinsic reaction coordinate (IRC) reveals that the RPV is an important component of the reaction coordinate at early and late stages of the pathway, but the RPV is not an important component of the reaction coordinate direction at the transition state. The unstable eigenmode from harmonic TST (which coincides with the IRC at the saddle point) gives a larger transmission coefficient than the coordinate used in the correlation functions of Antoniou et al. Thus while TST cannot predict the transmission coefficient, the RPV model suggests that TST can provide mechanistic insights on elementary steps in enzyme catalysis. Finally, we propose a method for using the transition-state ensemble as predicted from harmonic TST to distinguish promoting vibrations from other more mundane bath variables.

Reducing the cost of evaluating the committor by a fitting procedure

Correct identification of reaction coordinates in complex systems is essential for understanding the mechanisms of their reaction dynamics. Existing methods for identifying reaction coordinates typically require knowledge of the committor--the probability of a given configuration to reach the product basin. The high computational cost of evaluating committors has limited applications of methods for identifying reaction coordinates. We proposed a fitting procedure that can reduce the cost of evaluating committors by an order of magnitude or more. The method only requires evaluating the committors of a few configurations in a transition path by the standard and costly shooting procedure. The committors of the other configurations are then estimated with great accuracy by a sigmoid function derived from fitting the few numerically evaluated committors. The method has been systematically tested on a model system of a Brownian particle moving in a one-dimensional double-well potential, and a small biomolecular system--the isomerization of alanine dipeptide in vacuum and in explicit water.

Isotope-specific and amino acid-specific heavy atom substitutions alter barrier crossing in human purine nucleoside phosphorylase

Computational chemistry predicts that atomic motions on the femtosecond timescale are coupled to transition-state formation (barrier-crossing) in human purine nucleoside phosphorylase (PNP). The prediction is experimentally supported by slowed catalytic site chemistry in isotopically labeled PNP (13C, 15N, and 2H). However, other explanations are possible, including altered volume or bond polarization from carbon-deuterium bonds or propagation of the femtosecond bond motions into slower (nanoseconds to milliseconds) motions of the larger protein architecture to alter catalytic site chemistry. We address these possibilities by analysis of chemistry rates in isotope-specific labeled PNPs. Catalytic site chemistry was slowed for both [2H]PNP and [13C, 15N]PNP in proportion to their altered protein masses. Secondary effects emanating from carbon-deuterium bond properties can therefore be eliminated. Heavy-enzyme mass effects were probed for local or global contributions to catalytic site chemistry by generating [15N, 2H]His8-PNP. Of the eight His per subunit, three participate in contacts to the bound reactants and five are remote from the catalytic sites. [15N, 2H]His8-PNP had reduced catalytic site chemistry larger than proportional to the enzymatic mass difference. Altered barrier crossing when only His are heavy supports local catalytic site femtosecond perturbations coupled to transition-state formation. Isotope-specific and amino acid specific labels extend the use of heavy enzyme methods to distinguish global from local isotope effects.

Transition path sampling with quantum/classical mechanics for reaction rates

Predicting rates of biochemical reactions through molecular simulations poses a particular challenge for two reasons. First, the process involves bond formation and/or cleavage and thus requires a quantum mechanical (QM) treatment of the reaction center, which can be combined with a more efficient molecular mechanical (MM) description for the remainder of the system, resulting in a QM/MM approach. Second, reaction time scales are typically many orders of magnitude larger than the (sub-)nanosecond scale accessible by QM/MM simulations. Transition path sampling (TPS) allows to efficiently sample the space of dynamic trajectories from the reactant to the product state without an additional biasing potential. We outline here the application of TPS and QM/MM to calculate rates for biochemical reactions, by means of a simple toy system. In a step-by-step protocol, we specifically refer to our implementation within the MD suite Gromacs, which we have made available to the research community, and include practical advice on the choice of parameters.