Dynamical Reweighting for Biased Rare Event Simulations.

Dynamical reweighting techniques aim to recover the correct molecular dynamics from a simulation at a modified potential energy surface. They are important for unbiasing enhanced sampling simulations of molecular rare events. Here, we review the theoretical frameworks of dynamical reweighting for modified potentials. Based on an overview of kinetic models with increasing level of detail, we discuss techniques to reweight two-state dynamics, multistate dynamics, and path integrals. We explore the natural link to transition path sampling and how the effect of nonequilibrium forces can be reweighted. We end by providing an outlook on how dynamical reweighting integrates with techniques for optimizing collective variables and with modern potential energy surfaces.

Data-driven dynamical coarse-graining for condensed matter systems.

Simulations of condensed matter systems often focus on the dynamics of a few distinguished components but require integrating the full system. A prime example is a molecular dynamics simulation of a (macro)molecule in a solution, where the molecule(s) and the solvent dynamics need to be integrated, rendering the simulations computationally costly and often unfeasible for physically/biologically relevant time scales. Standard coarse graining approaches can reproduce equilibrium distributions and structural features but do not properly include the dynamics. In this work, we develop a general data-driven coarse-graining methodology inspired by the Mori-Zwanzig formalism, which shows that macroscopic systems with a large number of degrees of freedom can be described by a few relevant variables and additional noise and memory terms. Our coarse-graining method consists of numerical integrators for the distinguished components, where the noise and interaction terms with other system components are substituted by a random variable sampled from a data-driven model. The model is parameterized using data from multiple short-time full-system simulations, and then, it is used to run long-time simulations. Applying our methodology to three systems-a distinguished particle under a harmonic and a bistable potential and a dimer with two metastable configurations-the resulting coarse-grained models are capable of reproducing not only the equilibrium distributions but also the dynamic behavior due to temporal correlations and memory effects. Remarkably, our method even reproduces the transition dynamics between metastable states, which is challenging to capture correctly. Our approach is not constrained to specific dynamics and can be extended to systems beyond Langevin dynamics, and, in principle, even to non-equilibrium dynamics.

Decoding dissociation of sequence-specific protein-DNA complexes with non-equilibrium simulations.

Sequence-specific protein-DNA interactions are crucial in processes such as DNA organization, gene regulation and DNA replication. Obtaining detailed insights into the recognition mechanisms of protein-DNA complexes through experiments is hampered by a lack of resolution in both space and time. Here, we present a molecular simulation approach to quantify the sequence specificity of protein-DNA complexes, that yields results fast, and is generally applicable to any protein-DNA complex. The approach is based on molecular dynamics simulations in combination with a sophisticated steering potential and results in an estimate of the free energy difference of dissociation. We provide predictions of the nucleotide specific binding affinity of the minor groove binding Histone-like Nucleoid Structuring (H-NS) protein, that are in agreement with experimental data. Furthermore, our approach offers mechanistic insight into the process of dissociation. Applying our approach to the major groove binding ETS domain in complex with three different nucleotide sequences identified the high affinity consensus sequence, quantitatively in agreement with experiments. Our protocol facilitates quantitative prediction of protein-DNA complex stability, while also providing high resolution insights into recognition mechanisms. As such, our simulation approach has the potential to yield detailed and quantitative insights into biological processes involving sequence-specific protein-DNA interactions.

Molecular Free Energies, Rates, and Mechanisms from Data-Efficient Path Sampling Simulations.

Molecular dynamics is a powerful tool for studying the thermodynamics and kinetics of complex molecular events. However, these simulations can rarely sample the required time scales in practice. Transition path sampling overcomes this limitation by collecting unbiased trajectories and capturing the relevant events. Moreover, the integration of machine learning can boost the sampling while simultaneously learning a quantitative representation of the mechanism. Still, the resulting trajectories are by construction non-Boltzmann-distributed, preventing the calculation of free energies and rates. We developed an algorithm to approximate the equilibrium path ensemble from machine-learning-guided path sampling data. At the same time, our algorithm provides efficient sampling, mechanism, free energy, and rates of rare molecular events at a very moderate computational cost. We tested the method on the folding of the mini-protein chignolin. Our algorithm is straightforward and data-efficient, opening the door to applications in many challenging molecular systems.

Optimizing molecular potential models by imposing kinetic constraints with path reweighting.

Empirical force fields employed in molecular dynamics simulations of complex systems are often optimized to reproduce experimentally determined structural and thermodynamic properties. In contrast, experimental knowledge about the interconversion rates between metastable states in such systems is hardly ever incorporated in a force field due to a lack of an efficient approach. Here, we introduce such a framework based on the relationship between dynamical observables, such as rate constants, and the underlying molecular model parameters using the statistical mechanics of trajectories. Given a prior ensemble of molecular dynamics trajectories produced with imperfect force field parameters, the approach allows for the optimal adaption of these parameters such that the imposed constraint of equally predicted and experimental rate constant is obeyed. To do so, the method combines the continuum path ensemble maximum caliber approach with path reweighting methods for stochastic dynamics. When multiple solutions are found, the method selects automatically the combination that corresponds to the smallest perturbation of the entire path ensemble, as required by the maximum entropy principle. To show the validity of the approach, we illustrate the method on simple test systems undergoing rare event dynamics. Next to simple 2D potentials, we explore particle models representing molecular isomerization reactions and protein-ligand unbinding. Besides optimal interaction parameters, the methodology gives physical insights into what parts of the model are most sensitive to the kinetics. We discuss the generality and broad implications of the methodology.

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 260 K and crystalline sI structures above 260 K. 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 20 K 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.

Machine-guided path sampling to discover mechanisms of molecular self-organization.

Molecular self-organization driven by concerted many-body interactions produces the ordered structures that define both inanimate and living matter. Here we present an autonomous path sampling algorithm that integrates deep learning and transition path theory to discover the mechanism of molecular self-organization phenomena. The algorithm uses the outcome of newly initiated trajectories to construct, validate and-if needed-update quantitative mechanistic models. Closing the learning cycle, the models guide the sampling to enhance the sampling of rare assembly events. Symbolic regression condenses the learned mechanism into a human-interpretable form in terms of relevant physical observables. Applied to ion association in solution, gas-hydrate crystal formation, polymer folding and membrane-protein assembly, we capture the many-body solvent motions governing the assembly process, identify the variables of classical nucleation theory, uncover the folding mechanism at different levels of resolution and reveal competing assembly pathways. The mechanistic descriptions are transferable across thermodynamic states and chemical space.

Path Sampling Simulations Reveal How the Q61L Mutation Alters the Dynamics of KRas.

Flexibility is essential for many proteins to function, but can be difficult to characterize. Experiments lack resolution in space and time, while the time scales involved are prohibitively long for straightforward molecular dynamics simulations. In this work, we present a multiple state transition path sampling simulation study of a protein that has been notoriously difficult to characterize in its active state. The GTPase enzyme KRas is a signal transduction protein in pathways for cell differentiation, growth, and division. When active, KRas tightly binds guanosine triphosphate (GTP) in a rigid state. The protein-GTP complex can also visit more flexible states, in which it is not active. KRas mutations can affect the conversion between these rigid and flexible states, thus prolonging the activation of signal transduction pathways, which may result in tumor formation. In this work, we apply path sampling simulations to investigate the dynamic behavior of KRas-4B (wild type, WT) and the oncogenic mutant Q61L (Q61L). Our results show that KRas visits several conformational states, which are the same for WT and Q61L. The multiple state transition path sampling (MSTPS) method samples transitions between the different states in a single calculation. Tracking which transitions occur shows large differences between WT and Q61L. The MSTPS results further reveal that for Q61L, a route to a more flexible state is inaccessible, thus shifting the equilibrium to more rigid states. The methodology presented here enables a detailed characterization of protein flexibility on time scales not accessible with brute-force molecular dynamics simulations.

Revealing viscoelastic bending relaxation dynamics of isolated semiflexible colloidal polymers.

The viscoelastic properties of filaments and biopolymers play a crucial role in soft and biological materials from biopolymer networks to novel synthetic metamaterials. Colloidal particles with specific valency allow mimicking polymers and more complex molecular structures at the colloidal scale, offering direct observation of their internal degrees of freedom. Here, we elucidate the time-dependent viscoelastic response in the bending of isolated semi-flexible colloidal polymers, assembled from dipatch colloidal particles by reversible critical Casimir forces. By tuning the patch-patch interaction strength, we adjust the polymers' viscoelastic properties, and follow spontaneous bending modes and their relaxation directly on the particle level. We find that the elastic response is well described by that of a semiflexible rod with persistence length of order 1000 µm, tunable by the critical Casimir interaction strength. We identify the viscous relaxation on longer timescales to be due to internal friction, leading to a wavelength-independent relaxation time similar to single biopolymers, but in the colloidal case arising from the contact mechanics of the bonded patches. These tunable mechanical properties of assembled colloidal filaments open the door to "colloidal architectures", rationally designed (network) structures with desired topology and mechanical properties.

Revealing Polymerization Kinetics with Colloidal Dipatch Particles.

Limited-valency colloidal particles can self-assemble into polymeric structures analogous to molecules. While their structural equilibrium properties have attracted wide attention, insight into their dynamics has proven challenging. Here, we investigate the polymerization dynamics of semiflexible polymers in 2D by direct observation of assembling divalent particles, bonded by critical Casimir forces. The reversible critical Casimir force creates living polymerization conditions with tunable chain dissociation, association, and bending rigidity. We find that unlike dilute polymers that show exponential size distributions in excellent agreement with Flory theory, concentrated samples exhibit arrest of rotational and translational diffusion due to a continuous isotropic-to-nematic transition in 2D, slowing down the growth kinetics. These effects are circumvented by the addition of higher-valency particles, cross linking the polymers into networks. Our results connecting polymer flexibility, polymer interactions, and the peculiar isotropic-nematic transition in 2D offer insight into the polymerization processes of synthetic two-dimensional polymers and biopolymers at membranes and interfaces.

Homogenous nucleation rate of CO2 hydrates using transition interface sampling.

Carbon dioxide and water can form solid clathrate structures in which water cages encapsulate the gas molecules. Such hydrates have sparked much interest due to their possible application in CO2 sequestration. How the solid structure forms exactly from the liquid phase via a homogenous nucleation process is still poorly understood. This nucleation event is rare on the molecular timescale even under moderate undercooling or supersaturation conditions because of the large free energy barrier toward crystallization, rendering a brute force simulation of hydrate nucleation unfeasible for moderate undercooling or supersaturation. Here, we perform transition interface sampling simulations to quantify the homogenous nucleation rate for CO2 hydrate formation using accurate atomistic force fields at 500 bars for three different temperatures between 260 and 273 K. Collecting more than 100 000 pathways comprising roughly two milliseconds of simulation time, we computed a nucleation rate in the amorphous phase of ∼1021 nuclei s-1 cm-3 for a temperature of 260 K and a rate of ∼1012 nuclei s-1 cm-3 for a temperature of 265 K. For a temperature of 273 K, we find that the hydrate forms an sI crystalline phase with a rate of order of ∼101 nuclei s-1 cm-3. We compare these rates to classical nucleation theory estimates as well as experiments, and to nucleation rate estimates for methane hydrates and discuss possible causes of the observed differences. Our findings shed light on the kinetics of this important clathrate and should assist in future hydrate formation investigation.

A method of incorporating rate constants as kinetic constraints in molecular dynamics simulations.

From the point of view of statistical mechanics, a full characterization of a molecular system requires an accurate determination of its possible states, their populations, and the respective interconversion rates. Toward this goal, well-established methods increase the accuracy of molecular dynamics simulations by incorporating experimental information about states using structural restraints and about populations using thermodynamics restraints. However, it is still unclear how to include experimental knowledge about interconversion rates. Here, we introduce a method of imposing known rate constants as constraints in molecular dynamics simulations, which is based on a combination of the maximum-entropy and maximum-caliber principles. Starting from an existing ensemble of trajectories, obtained from either molecular dynamics or enhanced trajectory sampling, this method provides a minimally perturbed path distribution consistent with the kinetic constraints, as well as modified free energy and committor landscapes. We illustrate the application of the method to a series of model systems, including all-atom molecular simulations of protein folding. Our results show that by combining experimental rate constants and molecular dynamics simulations, this approach enables the determination of transition states, reaction mechanisms, and free energies. We anticipate that this method will extend the applicability of molecular simulations to kinetic studies in structural biology and that it will assist the development of force fields to reproduce kinetic and thermodynamic observables. Furthermore, this approach is generally applicable to a wide range of systems in biology, physics, chemistry, and material science.

Approximating free energy and committor landscapes in standard transition path sampling using virtual interface exchange.

Transition path sampling is a powerful technique for investigating rare transitions, especially when the mechanism is unknown and one does not have access to the reaction coordinate. Straightforward application of transition path sampling does not directly provide the free energy landscape nor the kinetics. This drawback has motivated the development of path sampling extensions able to simultaneously access both kinetics and thermodynamics, such as transition interface sampling, and the reweighted path ensemble. However, performing transition interface sampling is more involved than standard two-state transition path sampling and still requires (some) insight into the reaction to define interfaces. While packages that can efficiently compute path ensembles for transition interface sampling are now available, it would be useful to directly compute the free energy from a single standard transition path sampling simulation. To achieve this, we present here an approximate method, denoted virtual interface exchange transition path sampling, that makes use of the rejected pathways in a form of waste recycling. The method yields an approximate reweighted path ensemble that allows an immediate view of the free energy landscape from a standard single transition path sampling simulation, as well as enables a committor analysis.

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.

Unbiased atomistic insight in the competing nucleation mechanisms of methane hydrates.

Methane hydrates have important industrial and climate implications, yet their formation via homogeneous nucleation under natural, moderate conditions is poorly understood. Obtaining such understanding could lead to improved control of crystallization, as well as insight into polymorph selection in general, but is hampered by limited experimental resolution. Direct molecular dynamics simulations using atomistic force fields could provide such insight, but are not feasible for moderate undercooling, due to the rare event nature of nucleation. Instead, we harvest ensembles of the rare unbiased nucleation trajectories by employing transition path sampling. We find that with decreasing undercooling the mechanism shifts from amorphous to crystalline polymorph formation. At intermediate temperature the 2 mechanisms compete. Reaction coordinate analysis reveals the amount of a specific methane cage type is crucial for crystallization, while irrelevant for amorphous solids. Polymorph selection is thus governed by kinetic accessibility of the correct cage type and, moreover, occurs at precritical nucleus sizes, apparently against Ostwald's step rule. We argue that these results are still in line with classical nucleation theory. Our findings illuminate how selection between competing methane hydrate polymorphs occurs and might generalize to other hydrates and molecular crystal formation.

Allostery in Its Many Disguises: From Theory to Applications.

Allosteric regulation plays an important role in many biological processes, such as signal transduction, transcriptional regulation, and metabolism. Allostery is rooted in the fundamental physical properties of macromolecular systems, but its underlying mechanisms are still poorly understood. A collection of contributions to a recent interdisciplinary CECAM (Center Européen de Calcul Atomique et Moléculaire) workshop is used here to provide an overview of the progress and remaining limitations in the understanding of the mechanistic foundations of allostery gained from computational and experimental analyses of real protein systems and model systems. The main conceptual frameworks instrumental in driving the field are discussed. We illustrate the role of these frameworks in illuminating molecular mechanisms and explaining cellular processes, and describe some of their promising practical applications in engineering molecular sensors and informing drug design efforts.

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.

Ratchet-induced variations in bulk states of an active ideal gas.

We study the distribution of active, noninteracting particles over two bulk states separated by a ratchet potential. By solving the steady-state Smoluchowski equations in a flux-free setting, we show that the ratchet potential affects the distribution of particles over the bulks and thus exerts an influence of infinitely long range. As we show, an external potential that is nonlinear is crucial for having such a long-range influence. We characterize how the difference in bulk densities depends on activity and on the ratchet potential, and we identify power law dependencies on system parameters in several limiting cases. While weakly active systems are often understood in terms of an effective temperature, we present an analytical solution that explicitly shows that this is not possible in the current setting. Instead, we rationalize our results by a simple transition state model that presumes particles to cross the potential barrier by Arrhenius rates modified for activity. While this model does not quantitatively describe the difference in bulk densities for feasible parameter values, it does reproduce-in its regime of applicability-the complete power law behavior correctly.

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 a posteriori 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.