Transition Path Sampling Simulations Research Articles

Combining Transition Path Sampling with Data-Driven Collective Variables through a Reactivity-Biased Shooting Algorithm.

Rare event sampling is a central problem in modern computational chemistry research. Among the existing methods, transition path sampling (TPS) can generate unbiased representations of reaction processes. However, its efficiency depends on the ability to generate reactive trial paths, which in turn depends on the quality of the shooting algorithm used. We propose a new algorithm based on the shooting success rate, i.e., reactivity, measured as a function of a reduced set of collective variables (CVs). These variables are extracted with a machine learning approach directly from TPS simulations, using a multitask objective function. Iteratively, this workflow significantly improves the shooting efficiency without any prior knowledge of the process. In addition, the optimized CVs can be used with biased enhanced sampling methodologies to accurately reconstruct the free energy profiles. We tested the method on three different systems: a two-dimensional toy model, conformational transitions of alanine dipeptide, and hydrolysis of acetyl chloride in bulk water. In the latter, we integrated our workflow with an active learning scheme to learn a reactive machine learning-based potential, which allowed us to study the mechanism and free energy profile with an ab initio-like accuracy.

Enhanced Sampling of Configuration and Path Space in a Generalized Ensemble by Shooting Point Exchange.

The computer simulation of many molecular processes is complicated by long timescales caused by rare transitions between long-lived states. Here, we propose a new approach to simulate such rare events, which combines transition path sampling with enhanced exploration of configuration space. The method relies on exchange moves between configuration and trajectory space, carried out based on a generalized ensemble. This scheme substantially enhances the efficiency of the transition path sampling simulations, particularly for systems with multiple transition channels, and yields information on thermodynamics, kinetics and reaction coordinates of molecular processes without distorting their dynamics. The method is illustrated using the isomerization of proline in the KPTP tetrapeptide.

How far can we stretch the timescale with RETIS?

Molecular dynamics (MD) and Monte Carlo (MC) have long coexisted as two main independent branches of molecular simulation. In the late eighties, however, algorithms based on the combination of both were created such as hybrid Monte Carlo which uses large MD steps as MC moves. An entirely different kind of combination emerged a decade later via the transition path sampling (TPS) method in which MD trajectories are not just part of the MC move, but also form the state space being sampled. Algorithms like replica exchange transition interface sampling (RETIS) exploit this idea to compute reaction rates via a series of TPS simulations. RETIS yields results identical to hypothetical long MD runs, but with exponentially reduced computation time. This perspective describes the RETIS method and discusses recent and future advancements that will enable the study of even longer molecular timescales with reasonable computational resources.

Artificial intelligence for molecular mechanism discovery.

We develop a machine learning algorithm to extract the mechanism of transitions between metastable states from molecular dynamics simulations. Our algorithm combines transition path sampling (TPS), deep learning, and statistical inference to simulate the dynamics of complex molecular reorganizations. We iteratively train a deep learning model on the outcomes of the shooting moves used in TPS. By learning to predict the transition dynamics, the artificial intelligence (AI) system at the center of the algorithm gradually reveals the underlying mechanism of the transition dynamics, and at the same time increases the efficiency of the rare-event sampling. The AI system can simultaneously learn from and guide multiple TPS simulations running in parallel. In a second step, we then distill the knowledge about the transition mechanism encoded in the deep learning model into a reduced mathematical model. The reduced model concisely represents the key features of the mechanism in an explicit analytical, human-understandable form. We apply the algorithm to molecular systems ranging from ion dissociation in aqueous solution over gas hydrate formation to the oligomerization of a transmembrane alpha helix involved in membrane sensing. In all cases, the AI system accurately predicts the transition dynamics and extracts reduced mathematical models of the underlying mechanisms.

Homogeneous nucleation of crystalline methane hydrate in molecular dynamics transition paths sampled under realistic conditions.

Methane hydrates are important from a scientific and industrial perspective, and form by nucleation and growth from a supersaturated aqueous solution of methane. Molecular simulation is able to shed light on the process of homogeneous nucleation of hydrates, using straightforward molecular dynamics or rare event enhanced sampling techniques with atomistic and coarse grained force fields. In our previous work [Arjun, T. A. Berendsen, and P. G. Bolhuis, Proc. Natl. Acad. Sci. U. S. A. 116, 19305 (2019)], we performed transition path sampling (TPS) simulations using all atom force fields under moderate driving forces at high pressure, which enabled unbiased atomistic insight into the formation of methane hydrates. The supersaturation in these simulations was influenced by the Laplace pressure induced by the spherical gas reservoir. Here, we investigate the effect of removing this influence. Focusing on the supercooled, supersaturated regime to keep the system size tractable, our TPS simulations indicate that nuclei form amorphous structures below roughly 260K and crystalline sI structures above 260K. For these temperatures, the average transition path lengths are significantly longer than in our previous study, pushing the boundaries of what can be achieved with TPS. The temperature to observe a critical nucleus of certain size was roughly 20K lower compared to a spherical reservoir due to the lower concentration of methane in the solution, yielding a reduced driving force. We analyze the TPS results using a model based on classical nucleation theory. The corresponding free energy barriers are estimated and found to be consistent with previous predictions, thus adding to the overall picture of the hydrate formation process.

Transition Path Sampling as Markov Chain Monte Carlo of Trajectories: Recent Algorithms, Software, Applications, and Future Outlook

AbstractThe development of enhanced sampling methods to investigate rare but important events has always been a focal point in the molecular simulation field. Such methods often rely on prior knowledge of the reaction coordinate. However, the search for this reaction coordinate is at the heart of the rare event problem. Transition path sampling (TPS) circumvents this problem by generating an ensemble of dynamical trajectories undergoing the activated event. The reaction coordinate is extracted from the resulting path ensemble using variants of machine learning, making it an output of the method instead of an input. Over the last 20 years, since its inception, many extensions of TPS have been developed. Perhaps surprisingly, large‐scale TPS simulations on complex molecular systems have become possible only recently. Other important developments include the transition interface sampling (TIS) methodology to compute rate constants, the application to multiple states, and adaptive path sampling. The development of OpenPathSampling and PyRETIS has enabled easy and flexible use and implementation of these and other novel path sampling algorithms. In this progress report, a brief overview of recent developments, novel algorithms, and software is given. In addition, several application areas are discussed, and a future outlook for the next decade is given.

Dynamic Features of Transition States for β‐Scission Reactions of Alkenes over Acid Zeolites Revealed by AIMD Simulations

AbstractZeolite‐catalyzed alkene cracking is key to optimize the size of hydrocarbons. The nature and stability of intermediates and transition states (TS) are, however, still debated. We combine transition path sampling and blue moon ensemble density functional theory simulations to unravel the behavior of C7 alkenes in CHA zeolite. Free energy profiles are determined, linking π‐complexes, alkoxides and carbenium ions, for B1 (secondary to tertiary) and B2 (tertiary to secondary) β‐scissions. B1 is found to be easier than B2. The TS for B1 occurs at the breaking of the C−C bond, while for B2 it is the proton transfer from propenium to the zeolite. We highlight the dynamic behaviors of the various intermediates along both pathways, which reduce activation energies with respect to those previously evaluated by static approaches. We finally revisit the ranking of isomerization and cracking rate constants, which are crucial for future kinetic studies.

Dynamic Features of Transition States for β-Scission Reactions of Alkenes over Acid Zeolites Revealed by AIMD Simulations.

Zeolite-catalyzed alkene cracking is key to optimize the size of hydrocarbons. The nature and stability of intermediates and transition states (TS) are, however, still debated. We combine transition path sampling and blue moon ensemble density functional theory simulations to unravel the behavior of C7 alkenes in CHA zeolite. Free energy profiles are determined, linking π-complexes, alkoxides and carbenium ions, for B1 (secondary to tertiary) and B2 (tertiary to secondary) β-scissions. B1 is found to be easier than B2 . The TS for B1 occurs at the breaking of the C-C bond, while for B2 it is the proton transfer from propenium to the zeolite. We highlight the dynamic behaviors of the various intermediates along both pathways, which reduce activation energies with respect to those previously evaluated by static approaches. We finally revisit the ranking of isomerization and cracking rate constants, which are crucial for future kinetic studies.

Coordination Dynamics of Zinc Triggers the Rate Determining Proton Transfer in Human Carbonic Anhydrase II.

We present, for the first time, how transient changes in the coordination number of zinc ion affects the rate determining step in the enzyme human carbonic anhydrase (HCA) II. The latter involves an intramolecular proton transfer between a zinc-bound water and a distant histidine residue (His-64). In the absence of time-resolved experiments, results from classical and QM-MM molecular dynamics and transition path sampling simulations are presented. The catalytic zinc ion is found to be present in two possible coordination states; viz. a stable tetra-coordinated state, T and a less stable penta-coordinated state, P with tetrahedral and trigonal bipyramidal coordination geometries, respectively. A fast dynamical inter-conversion occurs between T and P due to reorganization of active site water molecules making the zinc ion more positively charged in state P. When initiated from different coordination environments, the most probable mechanism of proton transfer is found to be deprotonation of the equatorial water molecule from state P and transfer of the excess proton via a short path formed by hydrogen bonded network of active site water molecules. We estimate the rate constant of proton transfer as from P and from T. A quantitative match of estimated kP with the experimental value, ( ) suggests that dynamics of Zn coordination triggers the rate determining proton transfer step in HCA II.

Atomistic insight into the kinetic pathways for Watson-Crick to Hoogsteen transitions in DNA.

DNA predominantly contains Watson–Crick (WC) base pairs, but a non-negligible fraction of base pairs are in the Hoogsteen (HG) hydrogen bonding motif at any time. In HG, the purine is rotated ∼180° relative to the WC motif. The transitions between WC and HG may play a role in recognition and replication, but are difficult to investigate experimentally because they occur quickly, but only rarely. To gain insight into the mechanisms for this process, we performed transition path sampling simulations on a model nucleotide sequence in which an AT pair changes from WC to HG. This transition can occur in two ways, both starting with loss of hydrogen bonds in the base pair, followed by rotation around the glycosidic bond. In one route the adenine base converts from WC to HG geometry while remaining entirely within the double helix. The other route involves the adenine leaving the confines of the double helix and interacting with water. Our results indicate that this outside route is more probable. We used transition interface sampling to compute rate constants and relative free energies for the transitions between WC and HG. Our results agree with experiments, and provide highly detailed insights into the mechanisms of this important process.

Advantages of a distant cellulase catalytic base

The inverting glycoside hydrolase Trichoderma reesei (Hypocrea jecorina) Cel6A is a promising candidate for protein engineering for more economical production of biofuels. Until recently, its catalytic mechanism had been uncertain: The best candidate residue to serve as a catalytic base, Asp-175, is farther from the glycosidic cleavage site than in other glycoside hydrolase enzymes. Recent unbiased transition path sampling simulations revealed the hydrolytic mechanism for this more distant base, employing a water wire; however, it is not clear why the enzyme employs a more distant catalytic base, a highly conserved feature among homologs across different kingdoms. In this work, we describe molecular dynamics simulations designed to uncover how a base with a longer side chain, as in a D175E mutant, affects procession and active site alignment in the Michaelis complex. We show that the hydrogen bond network is tuned to the shorter aspartate side chain, and that a longer glutamate side chain inhibits procession as well as being less likely to adopt a catalytically productive conformation. Furthermore, we draw comparisons between the active site in Trichoderma reesei Cel6A and another inverting, processive cellulase to deduce the contribution of the water wire to the overall enzyme function, revealing that the more distant catalytic base enhances product release. Our results can inform efforts in the study and design of enzymes by demonstrating how counterintuitive sacrifices in chemical reactivity can have worthwhile benefits for other steps in the catalytic cycle.

Reaction Coordinate, Free Energy, and Rate of Intramolecular Proton Transfer in Human Carbonic Anhydrase II.

The role of structure and dynamics of an enzyme has been investigated at three different stages of its function including the chemical event it catalyzes. A one-pot computational method has been designed for each of these stages on the basis of classical and/or quantum mechanical-molecular mechanical molecular dynamics and transition path sampling simulations. For a pair of initial and final states A and B separated by a high free-energy barrier, using a two-stage selection process, several collective variables (CVs) are identified that can delineate A and B. However, these CVs are found to exhibit strong cross-coupling over the transition paths. A set of mutually orthogonal order parameters is then derived from these CVs and an optimal reaction coordinate, r, determined applying half-trajectory likelihood maximization along with a Bayesian information criterion. The transition paths are also used to project the multidimensional free energy surface and barrier crossing dynamics along r. The proposed scheme has been applied to the rate-determining intramolecular proton transfer reaction of the well-known enzyme human carbonic anhydrase II. The potential of mean force, F( r), in the absence of the chemical step is found to reproduce earlier results on the equilibrium population of two side-chain orientations of key residue His-64. Estimation of rate constants, k, from mean first passage times for the three different stages of catalysis shows that the rate-determining step of intramolecular proton transfer occurs with k ≃ 1.0 × 106 s-1, in close agreement with known experimental results.

Atomistic insight into the non-classical nucleation mechanism during solidification in Ni.

Nucleation is a key step during crystallization, but a complete understanding of the fundamental atomistic processes remains elusive. We investigate the mechanism of nucleation during solidification in nickel for various undercoolings using transition path sampling simulations. The temperature dependence of the free energy barriers and rate constants that we obtain is consistent with the predictions of classical nucleation theory and experiments. However, our analysis of the transition path ensemble reveals a mechanism that deviates from the classical picture of nucleation: the growing solid clusters have predominantly non-spherical shapes and consist of face-centered-cubic and random hexagonal-close-packed coordinated atoms surrounded by a cloud of prestructured liquid. The nucleation initiates in regions of supercooled liquid that are characterized by a high orientational order with structural features that predetermine the polymorph selection. These results provide atomistic insight not only into the nucleation mechanism of nickel but also into the role of the preordered liquid regions as precursors for crystallization.

Thermodynamic and kinetic solid-liquid interface properties from transition path sampling.

We perform transition path sampling simulations to determine two of the key quantities in solidification, the solid-liquid interface energy and velocity, in a Lennard-Jones system. Our approach is applicable to a wide range of temperature and pressure conditions, at the melting temperature and out-of-equilibrium. We show that small system sizes are sufficient for good values of interface energies and velocities. The transition path sampling method thus offers an attractive and robust alternative for the evaluation of solid-liquid interface properties.

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.

Using Bayes formula to estimate rates of rare events in transition path sampling simulations.

Transition path sampling is a method for estimating the rates of rare events in molecular systems based on the gradual transformation of a path distribution containing a small fraction of reactive trajectories into a biased distribution in which these rare trajectories have become frequent. Then, a multistate reweighting scheme is implemented to postprocess data collected from the staged simulations. Herein, we show how Bayes formula allows to directly construct a biased sample containing an enhanced fraction of reactive trajectories and to concomitantly estimate the transition rate from this sample. The approach can remediate the convergence issues encountered in free energy perturbation or umbrella sampling simulations when the transformed distribution insufficiently overlaps with the reference distribution.

Modeling Diffusion of Linear Hydrocarbons in Silica Zeolite LTA Using Transition Path Sampling

The diffusivities of linear hydrocarbons (CH4, C2H6, C2H4, C3H8, C3H6, and C4H10) in pure silica zeolite LTA (ITQ-29) are computed at 300 K and infinite dilution. To overcome the time scale problem arising from the slow diffusion process at room temperature, we used transition path sampling (TPS). The influence of framework flexibility on diffusion is investigated by combining TPS simulations with fully flexible molecular dynamics performed in the NpT ensemble. The ensemble of the collected reactive trajectories was used to characterize sets of transition states, and the corresponding configurations were analyzed to construct window size distributions during the molecular hopping events. The diffusion process is affected by framework flexibility, and the influence of framework flexibility on diffusion of propane and butane is much larger than for methane and ethane.

Enzyme homologues have distinct reaction paths through their transition states.

Recent studies of the bacterial enzymes EcMTAN and VcMTAN showed that they have different binding affinities for the same transition state analogue. This was surprising given the similarity of their active sites. We performed transition path sampling simulations of both enzymes to reveal the atomic details of the catalytic chemical step, which may be the key for explaining the inhibitor affinity differences. Even though all experimental data would suggest the two enzymes are almost identical, subtle dynamic differences manifest in differences of reaction coordinate, transition state structure, and eventually significant differences in inhibitor binding. Unlike EcMTAN, VcMTAN has multiple distinct transition states, which is an indication that multiple sets of coordinated protein motions can reach a transition state. Reaction coordinate information is only accessible from transition path sampling approaches, since all experimental approaches report averages. Detailed knowledge could have a significant impact on pharmaceutical design.

The water-catalyzed mechanism of the ring-opening reaction of glucose.

The hexopyranose mutarotation is an important focus for carbohydrate chemistry for more than 150 years. The paper describes the results of advanced computational studies aimed at elucidating the ring-opening reaction of glucose. Molecular simulations based on the combination of the DFT method with the molecular dynamics formalism allowed for a detailed insight into the mechanism of the process accompanied by the information of the kinetic and dynamic nature. The results indicate that the process is initiated by deprotonation of the anomeric hydroxyl group by water molecules and the subsequent proton transfer to the ring oxygen atom. The latter event has been identified as a 'bottleneck' of the process triggering the ring cleavage. The most time-consuming steps of the ring-opening reaction are the orientational rearrangements of water molecule(s) participating in the proton transfer(s) and the final extension of the newly-formed aldehyde chain. The orientational preferences of the aldehyde group present in the acyclic form of D-hexopyranoses are responsible for the anomeric equilibrium characteristics.

Structure, Kinetics, and Thermodynamics of the Aqueous Uranyl(VI) Cation

In this work, molecular simulation techniques were employed to gain insight into the structural, kinetic, and thermodynamic properties of the uranyl(VI) cation (UO2(2+)) in aqueous solution. The simulations made use of an atomistic potential model (force field) derived in this work and based on the model of Guilbaud and Wipff [J. Mol. Struct. (THEOCHEM) 1996 , 366 , 55 - 63]. Reactive flux and thermodynamic integration calculations show that the derived potential model yields predictions for the water exchange rate and free energy of hydration, respectively, that are in agreement with experimental data. The water binding energies, hydration shell structure, and self-diffusion coefficient were also calculated and analyzed. Finally, a combination of metadynamics and transition path sampling simulations was employed to probe the mechanisms of water exchange reactions in the first hydration shell of the uranyl ion. These atomistic simulations indicate, based on two-dimensional free energy surfaces, that water exchanges follow an associative interchange mechanism. The nature and structure of the water exchange transition states were also determined. The improved potential model is expected to lead to more accurate predictions of uranyl adsorption energies at mineral surfaces using potential-based molecular dynamics simulations.