Near-Enantiopure Trimerization of 9-Ethynylphenanthrene on a Chiral Metal Surface.

Enantioselectivity in heterogeneous catalysis strongly depends on the chirality transfer between catalyst surface and all reactants, intermediates, and the product along the reaction pathway. Herein we report the first enantioselective on-surface synthesis of molecular structures from an initial racemic mixture and without the need of enantiopure modifier molecules. The reaction consists of a trimerization via an unidentified bonding motif of prochiral 9-ethynylphenanthrene (9-EP) upon annealing to 500 K on the chiral Pd3 -terminated PdGa{111} surfaces into essentially enantiopure, homochiral 9-EP propellers. The observed behavior strongly contrasts the reaction of 9-EP on the chiral Pd1 -terminated PdGa{111} surfaces, where 9-EP monomers that are in nearly enantiopure configuration, dimerize without enantiomeric excess. Our findings demonstrate strong chiral recognition and a significant ensemble effect in the PdGa system, hence highlighting the huge potential of chiral intermetallic compounds for enantioselective synthesis and underlining the importance to control the catalytically active sites at the atomic level.

Molecular motor crossing the frontier of classical to quantum tunneling motion.

The reliability by which molecular motor proteins convert undirected energy input into directed motion or transport has inspired the design of innumerable artificial molecular motors. We have realized and investigated an artificial molecular motor applying scanning tunneling microscopy (STM), which consists of a single acetylene (C2H2) rotor anchored to a chiral atomic cluster provided by a PdGa(111) surface that acts as a stator. By breaking spatial inversion symmetry, the stator defines the unique sense of rotation. While thermally activated motion is nondirected, inelastic electron tunneling triggers rotations, where the degree of directionality depends on the magnitude of the STM bias voltage. Below 17 K and 30-mV bias voltage, a constant rotation frequency is observed which bears the fundamental characteristics of quantum tunneling. The concomitantly high directionality, exceeding 97%, implicates the combination of quantum and nonequilibrium processes in this regime, being the hallmark of macroscopic quantum tunneling. The acetylene on PdGa(111) motor therefore pushes molecular machines to their extreme limits, not just in terms of size, but also regarding structural precision, degree of directionality, and cross-over from classical motion to quantum tunneling. This ultrasmall motor thus opens the possibility to investigate in operando effects and origins of energy dissipation during tunneling events, and, ultimately, energy harvesting at the atomic scales.

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.

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.

PyEMMA 2: A Software Package for Estimation, Validation, and Analysis of Markov Models.

Markov (state) models (MSMs) and related models of molecular kinetics have recently received a surge of interest as they can systematically reconcile simulation data from either a few long or many short simulations and allow us to analyze the essential metastable structures, thermodynamics, and kinetics of the molecular system under investigation. However, the estimation, validation, and analysis of such models is far from trivial and involves sophisticated and often numerically sensitive methods. In this work we present the open-source Python package PyEMMA ( http://pyemma.org ) that provides accurate and efficient algorithms for kinetic model construction. PyEMMA can read all common molecular dynamics data formats, helps in the selection of input features, provides easy access to dimension reduction algorithms such as principal component analysis (PCA) and time-lagged independent component analysis (TICA) and clustering algorithms such as k-means, and contains estimators for MSMs, hidden Markov models, and several other models. Systematic model validation and error calculation methods are provided. PyEMMA offers a wealth of analysis functions such that the user can conveniently compute molecular observables of interest. We have derived a systematic and accurate way to coarse-grain MSMs to few states and to illustrate the structures of the metastable states of the system. Plotting functions to produce a manuscript-ready presentation of the results are available. In this work, we demonstrate the features of the software and show new methodological concepts and results produced by PyEMMA.

Projected metastable Markov processes and their estimation with observable operator models.

The determination of kinetics of high-dimensional dynamical systems, such as macromolecules, polymers, or spin systems, is a difficult and generally unsolved problem - both in simulation, where the optimal reaction coordinate(s) are generally unknown and are difficult to compute, and in experimental measurements, where only specific coordinates are observable. Markov models, or Markov state models, are widely used but suffer from the fact that the dynamics on a coarsely discretized state spaced are no longer Markovian, even if the dynamics in the full phase space are. The recently proposed projected Markov models (PMMs) are a formulation that provides a description of the kinetics on a low-dimensional projection without making the Markovianity assumption. However, as yet no general way of estimating PMMs from data has been available. Here, we show that the observed dynamics of a PMM can be exactly described by an observable operator model (OOM) and derive a PMM estimator based on the OOM learning.

Highly enantioselective adsorption of small prochiral molecules on a chiral intermetallic compound.

Intrinsically chiral surfaces of intermetallic compounds are shown to be novel materials for enantioselective processes. Their advantage is the significantly higher thermal and chemical stability, and therefore their extended application range for catalyzed chiral reactions compared to surfaces templated with chiral molecular modifiers or auxiliaries. On the Pd1 -terminated PdGa(111) surface, room-temperature adsorption of a small prochiral molecule (9-ethynylphenanthrene) leads to exceptionally high enantiomeric excess ratios of up to 98 %. Our findings highlight the great potential of intrinsically chiral intermetallic compounds for the development of novel, enantioselective catalysts that can be operated at high temperatures and potentially also in harsh chemical environments.

Spectral Rate Theory for Two-State Kinetics.

Classical rate theories often fail in cases where the observable(s) or order parameter(s) used is a poor reaction coordinate or the observed signal is deteriorated by noise, such that no clear separation between reactants and products is possible. Here, we present a general spectral two-state rate theory for ergodic dynamical systems in thermal equilibrium that explicitly takes into account how the system is observed. The theory allows the systematic estimation errors made by standard rate theories to be understood and quantified. We also elucidate the connection of spectral rate theory with the popular Markov state modeling approach for molecular simulation studies. An optimal rate estimator is formulated that gives robust and unbiased results even for poor reaction coordinates and can be applied to both computer simulations and single-molecule experiments. No definition of a dividing surface is required. Another result of the theory is a model-free definition of the reaction coordinate quality. The reaction coordinate quality can be bounded from below by the directly computable observation quality, thus providing a measure allowing the reaction coordinate quality to be optimized by tuning the experimental setup. Additionally, the respective partial probability distributions can be obtained for the reactant and product states along the observed order parameter, even when these strongly overlap. The effects of both filtering (averaging) and uncorrelated noise are also examined. The approach is demonstrated on numerical examples and experimental single-molecule force-probe data of the p5ab RNA hairpin and the apo-myoglobin protein at low pH, focusing here on the case of two-state kinetics.

Adsorption of small hydrocarbons on the three-fold PdGa surfaces: the road to selective hydrogenation.

Intermetallic compounds are a promising class of materials as stable and selective heterogeneous catalysts. Here, the (111) and (-1-1-1) single crystal surfaces of the PdGa intermetallic compound were studied as model catalysts with regard to the selective hydrogenation of acetylene (C2H2) to ethylene (C2H4). The distinct atomic surface structures exhibit isolated active centers of single atomic and three atomic Pd ensembles, respectively. For the two prototypal model catalyst surfaces, the adsorption sites and configurations for hydrogen (H2), acetylene, and ethylene were investigated by combining scanning tunneling microscopy, temperature-programmed desorption, and ab initio modeling. The topmost Pd surface atoms provide the preferred adsorption sites for all studied molecules. The structural difference of the Pd ensembles has a significant influence on the adsorption energy and configuration of C2H2, while the influence of the ensemble structure is weak for C2H4 and H2 adsorption. To approach the question of catalytic performance, we simulated the reaction pathways for the heterogeneous catalytic hydrogenation of acetylene on the two surfaces by means of density functional theory. Due to the geometrical separation of the Pd sites on the surfaces, the steric approach of the reactants (H and C2Hx) was found to be of importance to the energetics of the reaction. The presented study gives a direct comparison of binding properties of catalytic Pd on-top sites vs three-fold Pd hollow sites and is therefore of major relevance to the knowledge-based design of highly selective hydrogenation catalysts.

Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules.

Markov state models (MSMs) have been successful in computing metastable states, slow relaxation timescales and associated structural changes, and stationary or kinetic experimental observables of complex molecules from large amounts of molecular dynamics simulation data. However, MSMs approximate the true dynamics by assuming a Markov chain on a clusters discretization of the state space. This approximation is difficult to make for high-dimensional biomolecular systems, and the quality and reproducibility of MSMs has, therefore, been limited. Here, we discard the assumption that dynamics are Markovian on the discrete clusters. Instead, we only assume that the full phase-space molecular dynamics is Markovian, and a projection of this full dynamics is observed on the discrete states, leading to the concept of Projected Markov Models (PMMs). Robust estimation methods for PMMs are not yet available, but we derive a practically feasible approximation via Hidden Markov Models (HMMs). It is shown how various molecular observables of interest that are often computed from MSMs can be computed from HMMs/PMMs. The new framework is applicable to both, simulation and single-molecule experimental data. We demonstrate its versatility by applications to educative model systems, a 1 ms Anton MD simulation of the bovine pancreatic trypsin inhibitor protein, and an optical tweezer force probe trajectory of an RNA hairpin.

Dynamic neutron scattering from conformational dynamics. I. Theory and Markov models.

The dynamics of complex molecules can be directly probed by inelastic neutron scattering experiments. However, many of the underlying dynamical processes may exist on similar timescales, which makes it difficult to assign processes seen experimentally to specific structural rearrangements. Here, we show how Markov models can be used to connect structural changes observed in molecular dynamics simulation directly to the relaxation processes probed by scattering experiments. For this, a conformational dynamics theory of dynamical neutron and X-ray scattering is developed, following our previous approach for computing dynamical fingerprints of time-correlation functions [F. Noé, S. Doose, I. Daidone, M. Löllmann, J. Chodera, M. Sauer, and J. Smith, Proc. Natl. Acad. Sci. U.S.A. 108, 4822 (2011)]. Markov modeling is used to approximate the relaxation processes and timescales of the molecule via the eigenvectors and eigenvalues of a transition matrix between conformational substates. This procedure allows the establishment of a complete set of exponential decay functions and a full decomposition into the individual contributions, i.e., the contribution of every atom and dynamical process to each experimental relaxation process.

Dynamic neutron scattering from conformational dynamics. II. Application using molecular dynamics simulation and Markov modeling.

Neutron scattering experiments directly probe the dynamics of complex molecules on the sub pico- to microsecond time scales. However, the assignment of the relaxations seen experimentally to specific structural rearrangements is difficult, since many of the underlying dynamical processes may exist on similar timescales. In an accompanying article, we present a theoretical approach to the analysis of molecular dynamics simulations with a Markov State Model (MSM) that permits the direct identification of structural transitions leading to each contributing relaxation process. Here, we demonstrate the use of the method by applying it to the configurational dynamics of the well-characterized alanine dipeptide. A practical procedure for deriving the MSM from an MD is introduced. The result is a 9-state MSM in the space of the backbone dihedral angles and the side-chain methyl group. The agreement between the quasielastic spectrum calculated directly from the atomic trajectories and that derived from the Markov state model is excellent. The dependence on the wavevector of the individual Markov processes is described. The procedure means that it is now practicable to interpret quasielastic scattering spectra in terms of well-defined intramolecular transitions with minimal a priori assumptions as to the nature of the dynamics taking place.

Probing molecular kinetics with Markov models: metastable states, transition pathways and spectroscopic observables.

Markov (state) models (MSMs) have attracted a lot of interest recently as they (1) can probe long-term molecular kinetics based on short-time simulations, (2) offer a way to analyze great amounts of simulation data with relatively little subjectivity of the analyst, (3) provide insight into microscopic quantities such as the ensemble of transition pathways, and (4) allow simulation data to be reconciled with measurement data in a rigorous and explicit way. Here we sketch our current perspective of Markov models and explain in short their theoretical basis and assumptions. We describe transition path theory which allows the entire ensemble of protein folding pathways to be investigated and that combines naturally with Markov models. Experimental observations can be naturally linked to Markov models with the dynamical fingerprint theory, by which experimentally observable timescales can be equipped with an understanding of the structural rearrangement processes that take place at these timescales. The concepts of this paper are illustrated by a simple kinetic model of protein folding.

Dynamical reweighting: improved estimates of dynamical properties from simulations at multiple temperatures.

Dynamical averages based on functionals of dynamical trajectories, such as time-correlation functions, play an important role in determining kinetic or transport properties of matter. At temperatures of interest, the expectations of these quantities are often dominated by contributions from rare events, making the precise calculation of these quantities by molecular dynamics simulation difficult. Here, we present a reweighting method for combining simulations from multiple temperatures (or from simulated or parallel tempering simulations) to compute an optimal estimate of the dynamical properties at the temperature of interest without the need to invoke an approximate kinetic model (such as the Arrhenius law). Continuous and differentiable estimates of these expectations at any temperature in the sampled range can also be computed, along with an assessment of the associated statistical uncertainty. For rare events, aggregating data from multiple temperatures can produce an estimate with the desired precision at greatly reduced computational cost compared with simulations conducted at a single temperature. Here, we describe use of the method for the canonical (NVT) ensemble using four common models of dynamics (canonical distribution of Hamiltonian trajectories, Andersen thermostatting, Langevin, and overdamped Langevin or Brownian dynamics), but it can be applied to any thermodynamic ensemble provided the ratio of path probabilities at different temperatures can be computed. To illustrate the method, we compute a time-correlation function for solvated terminally-blocked alanine peptide across a range of temperatures using trajectories harvested using a modified parallel tempering protocol.

Optimal use of data in parallel tempering simulations for the construction of discrete-state Markov models of biomolecular dynamics.

Parallel tempering (PT) molecular dynamics simulations have been extensively investigated as a means of efficient sampling of the configurations of biomolecular systems. Recent work has demonstrated how the short physical trajectories generated in PT simulations of biomolecules can be used to construct the Markov models describing biomolecular dynamics at each simulated temperature. While this approach describes the temperature-dependent kinetics, it does not make optimal use of all available PT data, instead estimating the rates at a given temperature using only data from that temperature. This can be problematic, as some relevant transitions or states may not be sufficiently sampled at the temperature of interest, but might be readily sampled at nearby temperatures. Further, the comparison of temperature-dependent properties can suffer from the false assumption that data collected from different temperatures are uncorrelated. We propose here a strategy in which, by a simple modification of the PT protocol, the harvested trajectories can be reweighted, permitting data from all temperatures to contribute to the estimated kinetic model. The method reduces the statistical uncertainty in the kinetic model relative to the single temperature approach and provides estimates of transition probabilities even for transitions not observed at the temperature of interest. Further, the method allows the kinetics to be estimated at temperatures other than those at which simulations were run. We illustrate this method by applying it to the generation of a Markov model of the conformational dynamics of the solvated terminally blocked alanine peptide.

Markov models of molecular kinetics: generation and validation.

Markov state models of molecular kinetics (MSMs), in which the long-time statistical dynamics of a molecule is approximated by a Markov chain on a discrete partition of configuration space, have seen widespread use in recent years. This approach has many appealing characteristics compared to straightforward molecular dynamics simulation and analysis, including the potential to mitigate the sampling problem by extracting long-time kinetic information from short trajectories and the ability to straightforwardly calculate expectation values and statistical uncertainties of various stationary and dynamical molecular observables. In this paper, we summarize the current state of the art in generation and validation of MSMs and give some important new results. We describe an upper bound for the approximation error made by modeling molecular dynamics with a MSM and we show that this error can be made arbitrarily small with surprisingly little effort. In contrast to previous practice, it becomes clear that the best MSM is not obtained by the most metastable discretization, but the MSM can be much improved if non-metastable states are introduced near the transition states. Moreover, we show that it is not necessary to resolve all slow processes by the state space partitioning, but individual dynamical processes of interest can be resolved separately. We also present an efficient estimator for reversible transition matrices and a robust test to validate that a MSM reproduces the kinetics of the molecular dynamics data.