Transition Path Sampling Research Articles

Mechanism of oligosaccharide synthesis via a mutant GH29 fucosidase

First unbiased transition path sampling study of a glycosynthase enzyme reveals single-step mechanism with oxocarbenium-like transition state.

Studying rare nonadiabatic dynamics with transition path sampling quantum jump trajectories.

We present a method to study rare nonadiabatic dynamics in open quantum systems using transition path sampling and quantum jump trajectories. As with applications of transition path sampling to classical dynamics, the method does not rely on prior knowledge of transition states or reactive pathways and thus can provide mechanistic insight into ultrafast relaxation processes in addition to their associated rates. In particular, we formulate a quantum path ensemble using the stochastic realizations of an unravelled quantum master equation, which results in trajectories that can be conditioned on starting and ending in particular quantum states. Because the dynamics rigorously obeys detailed balance, rate constants can be evaluated from reversible work calculations in this conditioned ensemble, allowing for branching ratios and yields to be computed in an unbiased manner. We illustrate the utility of this method with three examples: energy transfer in a donor-bridge-acceptor model, and models of photo-induced proton-coupled electron transfer and thermally activated electron transfer. These examples demonstrate the efficacy of path ensemble methods and pave the way for their use in studying complex reactive quantum dynamics.

Maximum Likelihood Analysis of Reaction Coordinates during Solidification in Ni.

Understanding the underlying mechanism of crystal nucleation is a fundamental aspect in the prediction and control of materials properties. Classical nucleation theory (CNT) assumes that homogeneous nucleation occurs via random fluctuations within the supercooled liquid, that the structure of the growing clusters resembles the most stable bulk phase, and that the nucleus size is the sole reaction coordinate (RC) of the process. Many materials are, however, known to exhibit multiple steps during crystallization, forming different polymorphs. As a consequence, more complex RCs are often required to capture all relevant information about the process. Here, we employ transition path sampling together with a maximum likelihood analysis of candidate order parameters to identify suitable RCs for the nucleation mechanism during solidification in Ni. In contrast to CNT, the analysis of the reweighted path ensemble shows that a prestructured liquid region that surrounds the crystal cluster is a relevant order parameter that enhances the RC and therefore plays a key role in the description of the nucleus and the interfacial free energy. We demonstrate that prestructured liquid clusters that emerge within the liquid act as precursors of the crystallization in a nonclassical two-step mechanism, which predetermines the coordination of the selected polymorphs.

OpenPathSampling: A Python Framework for Path Sampling Simulations. 2. Building and Customizing Path Ensembles and Sample Schemes.

The OpenPathSampling (OPS) package provides an easy-to-use framework to apply transition path sampling methodologies to complex molecular systems with a minimum of effort. Yet, the extensibility of OPS allows for the exploration of new path sampling algorithms by building on a variety of basic operations. In a companion paper [Swenson et al. J. Chem. Theory Comput.2018, 10.1021/acs.jctc.8b00626] we introduced the basic concepts and the structure of the OPS package, and how it can be employed to perform standard transition path sampling and (replica exchange) transition interface sampling. In this paper, we elaborate on two theoretical developments that went into the design of OPS. The first development relates to the construction of path ensembles, the what is being sampled. We introduce a novel set-based notation for the path ensemble, which provides an alternative paradigm for constructing path ensembles and allows building arbitrarily complex path ensembles from fundamental ones. The second fundamental development is the structure for the customization of Monte Carlo procedures; how path ensembles are being sampled. We describe in detail the OPS objects that implement this approach to customization, the MoveScheme and the PathMover, and provide tools to create and manipulate these objects. We illustrate both the path ensemble building and sampling scheme customization with several examples. OPS thus facilitates both standard path sampling application in complex systems as well as the development of new path sampling methodology, beyond the default.

OpenPathSampling: A Python Framework for Path Sampling Simulations. 1. Basics.

Transition path sampling techniques allow molecular dynamics simulations of complex systems to focus on rare dynamical events, providing insight into mechanisms and the ability to calculate rates inaccessible by ordinary dynamics simulations. While path sampling algorithms are conceptually as simple as importance sampling Monte Carlo, the technical complexity of their implementation has kept these techniques out of reach of the broad community. Here, we introduce an easy-to-use Python framework called OpenPathSampling (OPS) that facilitates path sampling for (bio)molecular systems with minimal effort and yet is still extensible. Interfaces to OpenMM and an internal dynamics engine for simple models are provided in the initial release, but new molecular simulation packages can easily be added. Multiple ready-to-use transition path sampling methodologies are implemented, including standard transition path sampling (TPS) between reactant and product states and transition interface sampling (TIS) and its replica exchange variant (RETIS), as well as recent multistate and multiset extensions of transition interface sampling (MSTIS, MISTIS). In addition, tools are provided to facilitate the implementation of new path sampling schemes built on basic path sampling components. In this paper, we give an overview of the design of this framework and illustrate the simplicity of applying the available path sampling algorithms to a variety of benchmark problems.

Phases of quantum dimers from ensembles of classical stochastic trajectories

We study the connection between the phase behaviour of quantum dimers and the dynamics of classical stochastic dimers. At the so-called Rokhsar-Kivelson (RK) point a quantum dimer Hamiltonian is equivalent to the Markov generator of the dynamics of classical dimers. A less well understood fact is that away from the RK point the quantum-classical connection persists: in this case the Hamiltonian corresponds to a non-stochastic "tilted" operator that encodes the statistics of time-integrated observables of the classical stochastic problem. This implies a direct relation between the phase behaviour of quantum dimers and properties of ensembles of stochastic trajectories of classical dimers. We make these ideas concrete by studying fully packed dimers on the square lattice. Using transition path sampling - supplemented by trajectory umbrella sampling - we obtain the large deviation statistics of dynamical activity in the classical problem, and show the correspondence between the phase behaviour of the classical and quantum systems. The transition at the RK point between quantum phases of distinct order corresponds, in the classical case, to a trajectory phase transition between active and inactive dynamical phases. Furthermore, from the structure of stochastic trajectories in the active dynamical phase we infer that the ground state of quantum dimers has columnar order to one side of the RK point. We discuss how these results relate to those from quantum Monte Carlo, and how our approach may generalise to other problems.

Nested Transition Path Sampling.

We introduce a novel transition path (TPS) sampling scheme employing nested sampling. Analogous to how nested sampling explores the entire configurational phase space for atomistic systems, nested TPS samples the entire available trajectory space in one simulation. Thermodynamic and path observables can be constructed aposteriori for all temperatures simultaneously. We exploit this to compute the rate of rare processes at arbitrarily low temperature through the coupling to easily accessible rates at high temperature. We illustrate the method on several model systems.

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.

Importance sampling large deviations in nonequilibrium steady states. I.

Large deviation functions contain information on the stability and response of systems driven into nonequilibrium steady states and in such a way are similar to free energies for systems at equilibrium. As with equilibrium free energies, evaluating large deviation functions numerically for all but the simplest systems is difficult because by construction they depend on exponentially rare events. In this first paper of a series, we evaluate different trajectory-based sampling methods capable of computing large deviation functions of time integrated observables within nonequilibrium steady states. We illustrate some convergence criteria and best practices using a number of different models, including a biased Brownian walker, a driven lattice gas, and a model of self-assembly. We show how two popular methods for sampling trajectory ensembles, transition path sampling and diffusion Monte Carlo, suffer from exponentially diverging correlations in trajectory space as a function of the bias parameter when estimating large deviation functions. Improving the efficiencies of these algorithms requires introducing guiding functions for the trajectories.

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.

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.

Faster protein folding using enhanced conformational sampling of molecular dynamics simulation

In this study, we applied swarm particle-like molecular dynamics (SPMD) approach to enhance conformational sampling of replica exchange simulations. In particular, the approach showed significant improvement in sampling efficiency of conformational phase space when combined with replica exchange method (REM) in computer simulation of peptide/protein folding. First we introduce the augmented dynamical system of equations, and demonstrate the stability of the algorithm. Then, we illustrate the approach by using different fully atomistic and coarse-grained model systems, comparing them with the standard replica exchange method. In addition, we applied SPMD simulation to calculate the time correlation functions of the transitions in a two dimensional surface to demonstrate the enhancement of transition path sampling. Our results showed that folded structure can be obtained in a shorter simulation time using the new method when compared with non-augmented dynamical system. Typically, in less than 0.5 ns using replica exchange runs assuming that native folded structure is known and within simulation time scale of 40 ns in the case of blind structure prediction. Furthermore, the root mean square deviations from the reference structures were less than 2Å. To demonstrate the performance of new method, we also implemented three simulation protocols using CHARMM software. Comparisons are also performed with standard targeted molecular dynamics simulation method.

Cis-to- Trans Isomerization of Azobenzene Derivatives Studied with Transition Path Sampling and Quantum Mechanical/Molecular Mechanical Molecular Dynamics.

Azobenzene-based molecular photoswitches are becoming increasingly important for the development of photoresponsive, functional soft-matter material systems. Upon illumination with light, fast interconversion between a more stable trans and a metastable cis configuration can be established resulting in pronounced changes in conformation, dipole moment or hydrophobicity. A rational design of functional photosensitive molecules with embedded azo moieties requires a thorough understanding of isomerization mechanisms and rates, especially the thermally activated relaxation. For small azo derivatives considered in the gas phase or simple solvents, Eyring's classical transition state theory (TST) approach yields useful predictions for trends in activation energies or corresponding half-life times of the cis isomer. However, TST or improved theories cannot easily be applied when the azo moiety is part of a larger molecular complex or embedded into a heterogeneous environment, where a multitude of possible reaction pathways may exist. In these cases, only the sampling of an ensemble of dynamic reactive trajectories (transition path sampling, TPS) with explicit models of the environment may reveal the nature of the processes involved. In the present work we show how a TPS approach can conveniently be implemented for the phenomenon of relaxation-isomerization of azobenzenes starting with the simple examples of pure azobenzene and a push-pull derivative immersed in a polar (DMSO) and apolar (toluene) solvent. The latter are represented explicitly at a molecular mechanical (MM) and the azo moiety at a quantum mechanical (QM) level. We demonstrate for the push-pull azobenzene that path sampling in combination with the chosen QM/MM scheme produces the expected change in isomerization pathway from inversion to rotation in going from a low to a high permittivity (explicit) solvent model. We discuss the potential of the simulation procedure presented for comparative calculation of reaction rates and an improved understanding of activated states.

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.

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 SN 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.

Detecting reactive islands using Lagrangian descriptors and the relevance to transition path sampling.

It has been known for sometime now that isomerization reactions, classically, are mediated by phase space structures called reactive islands (RI). RIs provide one possible route to correct for the nonstatistical effects in the reaction dynamics. In this work, we map out the reactive islands for the two dimensional Müller-Brown model potential and show that the reactive islands are intimately linked to the issue of rare event sampling. In particular, we establish the sensitivity of the so called committor probabilities, useful quantities in the transition path sampling technique, to the hierarchical RI structures. Mapping out the RI structure for high dimensional systems, however, is a challenging task. Here, we show that the technique of Lagrangian descriptors is able to effectively identify the RI hierarchy in the model system. Based on our results, we suggest that the Lagrangian descriptors can be useful for detecting RIs in high dimensional systems.

Foundations and latest advances in replica exchange transition interface sampling.

Nearly 20 years ago, transition path sampling (TPS) emerged as an alternative method to free energy based approaches for the study of rare events such as nucleation, protein folding, chemical reactions, and phase transitions. TPS effectively performs Monte Carlo simulations with relatively short molecular dynamics trajectories, with the advantage of not having to alter the actual potential energy surface nor the underlying physical dynamics. Although the TPS approach also introduced a methodology to compute reaction rates, this approach was for a long time considered theoretically attractive, providing the exact same results as extensively long molecular dynamics simulations, but still expensive for most relevant applications. With the increase of computer power and improvements in the algorithmic methodology, quantitative path sampling is finding applications in more and more areas of research. In particular, the transition interface sampling (TIS) and the replica exchange TIS (RETIS) algorithms have, in turn, improved the efficiency of quantitative path sampling significantly, while maintaining the exact nature of the approach. Also, open-source software packages are making these methods, for which implementation is not straightforward, now available for a wider group of users. In addition, a blooming development takes place regarding both applications and algorithmic refinements. Therefore, it is timely to explore the wide panorama of the new developments in this field. This is the aim of this article, which focuses on the most efficient exact path sampling approach, RETIS, as well as its recent applications, extensions, and variations.

Reaction coordinates and transition states in enzymatic catalysis

Enzymatic reactions are complex chemical processes taking place in complex dynamic environments. Theoretical characterization of these reactions requires the determination of the reaction coordinate and the transition state ensemble. This is not an easy task because many degrees of freedom may be involved in principle. We present recent efforts to find good enzymatic reaction coordinates and the implications of these findings in the interpretation of enzymatic efficiency. In particular, we analyze different strategies based on the use of minimum free energy paths and direct localization of the dividing surface on multidimensional free energy surfaces. Another strategy is based on the generation of reactive trajectories, using the transition path sampling method, from which transition state configurations can be harvested. Most of the applications carried out until now coincide to stress the change in the nature of the reaction coordinate, in terms of the participation of the chemical and environmental degrees of freedom, as the reaction advances. The degrees of freedom of the chemical system are dominant at the transition state while environmental participation can be more important at early or late stages of the process. WIREs Comput Mol Sci 2018, 8:e1329. doi: 10.1002/wcms.1329This article is categorized under: Structure and Mechanism > Computational Biochemistry and Biophysics Molecular and Statistical Mechanics > Free Energy Methods Software > Simulation Methods

Understanding rare safety and reliability events using transition path sampling

In the chemical and process industries, processes and their control systems are typically well-designed to mitigate abnormal events having potential adverse consequences to human health, environment, and/or property. Strong motivation exists to understand how these events develop and propagate. These events occur so rarely that statistical analyses of their occurrences alone are incapable of describing and characterizing them − especially when they have not yet occurred. Moreover, the use of process models to understand such rare events is hampered by the orders of magnitude separating the frequencies with which reliability and safety events (years to decades) occur and the duration over which they occur (minutes to hours). To address these challenges, we adapt a Monte-Carlo based, rare-event sampling technique, Transition Path Sampling (TPS), which was developed by the molecular simulation community. Important modifications to the TPS technique are needed to apply it to process dynamics, and are discussed herein.

Transition path sampling of rare events by shooting from the top

Transition path sampling is a powerful tool in the study of rare events. Shooting trial trajectories from configurations along existing transition paths proved particularly efficient in the sampling of reactive trajectories. However, most shooting attempts tend not to result in transition paths, in particular in cases where the transition dynamics has diffusive character. To overcome the resulting efficiency problem, we developed an algorithm for "shooting from the top." We first define a shooting range through which all paths have to pass and then shoot off trial trajectories only from within this range. For a well chosen shooting range, nearly every shot is successful, resulting in an accepted transition path. To deal with multiple mechanisms, weighted shooting ranges can be used. To cope with the problem of unsuitably placed shooting ranges, we developed an algorithm that iteratively improves the location of the shooting range. The transition path sampling procedure is illustrated for models of diffusive and Langevin dynamics. The method should be particularly useful in cases where the transition paths are long so that only relatively few shots are possible, yet reasonable order parameters are known.