Free Energy Guided Sampling

Protein Structure Validation Derives a Smart Conformational Search in a Physically Relevant Configurational Subspace

Since proteins perform biological functions through their dynamic properties, molecular dynamics (MD) simulation is a sophisticated strategy for investigating their functions. Analyses of trajectories provide statistical information about a specific protein as a free-energy landscape (FEL). However, the timescale of normal MD is shorter than that of biological functions, resulting in statistically insufficient conformational sampling, finally leading to unreliable FEL calculation. To search for a broad configurational subspace, an external bias is imposed on a target protein as biased sampling. However, its regulation is challenging because the optimal strength of the perturbation is unknown. Furthermore, a physically irrelevant configurational subspace was searched when imposing an inappropriate external bias. To address this issue, we newly proposed an external biased regulation scheme known as the G-factor external bias limiter (GERBIL). In GERBIL, protein configurations generated by external bias are structurally validated by an indicator (G-factor), enabling the search for a physically relevant subspace. In addition to biased sampling, nonbiased sampling might search for a physically irrelevant configurational subspace because repeating multiple MD simulations from several initial structures tends to search for an overly broad configurational subspace. For this issue, the structural qualities of configurations generated by nonbiased sampling have not been investigated. Therefore, we confirmed whether the G-factor screened the collapsed (low-quality) configurations generated by nonbiased sampling. To address this issue, the outlier flooding method (OFLOOD) was adopted in GERBIL as a nonbiased sampling method, which is referred to as OFLOOD-GERBIL. OFLOOD rapidly expands a configurational subspace by resampling the rarely occurring states of a given protein and tends to search an overly broad subspace. Thus, we considered that GERBIL might improve the excessive conformational search of OFLOOD for a physically irrelevant configurational subspace. As a demonstration, OFLOOD and OFLOOD-GERBIL were applied to a globular protein (T4 lysozyme) and their conformational search qualities were assessed. Based on our assessment, normal OFLOOD without the outlier validation frequently sampled low-quality configurations, whereas OFLOOD-GERBIL with the outlier validation intensively sampled high-quality configurations. In conclusion, OFLOOD-GERBIL derives a smart conformational search in a physically relevant configurational subspace, indicating that protein structure validation works in both nonbiased and biased sampling methods.

Additive eigenvectors as optimal reaction coordinates, conditioned trajectories, and time-reversible description of stochastic processes

A fundamental way to analyze complex multidimensional stochastic dynamics is to describe it as diffusion on a free energy landscape—free energy as a function of reaction coordinates (RCs). For such a description to be quantitatively accurate, the RC should be chosen in an optimal way. The committor function is a primary example of an optimal RC for the description of equilibrium reaction dynamics between two states. Here, additive eigenvectors (addevs) are considered as optimal RCs to address the limitations of the committor. An addev master equation for a Markov chain is derived. A stationary solution of the equation describes a sub-ensemble of trajectories conditioned on having the same optimal RC for the forward and time-reversed dynamics in the sub-ensemble. A collection of such sub-ensembles of trajectories, called stochastic eigenmodes, can be used to describe/approximate the stochastic dynamics. A non-stationary solution describes the evolution of the probability distribution. However, in contrast to the standard master equation, it provides a time-reversible description of stochastic dynamics. It can be integrated forward and backward in time. The developed framework is illustrated on two model systems—unidirectional random walk and diffusion.

Nonparametric Analysis of Nonequilibrium Simulations

We extend the nonparametric framework of reaction coordinate optimization to nonequilibrium ensembles of (short) trajectories. For example, we show how, starting from such an ensemble, one can obtain an equilibrium free-energy profile along the committor, which can be used to determine important properties of the dynamics exactly. A new adaptive sampling approach, the transition-state ensemble enrichment, is suggested, which samples the configuration space by "growing" committor segments toward each other starting from the boundary states. This framework is suggested as a general tool, alternative to the Markov state models, for a rigorous and accurate analysis of simulations of large biomolecular systems, as it has the following attractive properties. It is immune to the curse of dimensionality, does not require system-specific information, can approximate arbitrary reaction coordinates with high accuracy, and has sensitive and rigorous criteria to test optimality and convergence. The approaches are illustrated on a 50-dimensional model system and a realistic protein folding trajectory.

How Effectively Can Adaptive Sampling Methods Capture Spontaneous Ligand Binding?

Molecular dynamics (MD) simulations that capture the spontaneous binding of drugs and other ligands to their target proteins can reveal a great deal of useful information, but most drug-like ligands bind on time scales longer than those accessible to individual MD simulations. Adaptive sampling methods―in which one performs multiple rounds of simulation, with the initial conditions of each round based on the results of previous rounds―offer a promising potential solution to this problem. No comprehensive analysis of the performance gains from adaptive sampling is available for ligand binding, however, particularly for protein–ligand systems typical of those encountered in drug discovery. Moreover, most previous work presupposes knowledge of the ligand’s bound pose. Here we outline existing methods for adaptive sampling of the ligand-binding process and introduce several improvements, with a focus on methods that do not require prior knowledge of the binding site or bound pose. We then evaluate these methods by comparing them to traditional, long MD simulations for realistic protein–ligand systems. We find that adaptive sampling simulations typically fail to reach the bound pose more efficiently than traditional MD. However, adaptive sampling identifies multiple potential binding sites more efficiently than traditional MD and also provides better characterization of binding pathways. We explain these results by showing that protein–ligand binding is an example of an exploration–exploitation dilemma. Existing adaptive sampling methods for ligand binding in the absence of a known bound pose vastly favor the broad exploration of protein–ligand space, sometimes failing to sufficiently exploit intermediate states as they are discovered. We suggest potential avenues for future research to address this shortcoming.

A new class of enhanced kinetic sampling methods for building Markov state models

Markov state models (MSMs) and other related kinetic network models are frequently used to study the long-timescale dynamical behavior of biomolecular and materials systems. MSMs are often constructed bottom-up using brute-force molecular dynamics (MD) simulations when the model contains a large number of states and kinetic pathways that are not known a priori. However, the resulting network generally encompasses only parts of the configurational space, and regardless of any additional MD performed, several states and pathways will still remain missing. This implies that the duration for which the MSM can faithfully capture the true dynamics, which we term as the validity time for the MSM, is always finite and unfortunately much shorter than the MD time invested to construct the model. A general framework that relates the kinetic uncertainty in the model to the validity time, missing states and pathways, network topology, and statistical sampling is presented. Performing additional calculations for frequently-sampled states/pathways may not alter the MSM validity time. A new class of enhanced kinetic sampling techniques is introduced that aims at targeting rare states/pathways that contribute most to the uncertainty so that the validity time is boosted in an effective manner. Examples including straightforward 1D energy landscapes, lattice models, and biomolecular systems are provided to illustrate the application of the method. Developments presented here will be of interest to the kinetic Monte Carlo community as well.

Focused conformational sampling in proteins

A detailed understanding of the conformational dynamics of biological molecules is difficult to obtain by experimental techniques due to resolution limitations in both time and space. Computer simulations avoid these in theory but are often too short to sample rare events reliably. Here we show that the progress index-guided sampling (PIGS) protocol can be used to enhance the sampling of rare events in selected parts of biomolecules without perturbing the remainder of the system. The method is very easy to use as it only requires as essential input a set of several features representing the parts of interest sufficiently. In this feature space, new states are discovered by spontaneous fluctuations alone and in unsupervised fashion. Because there are no energetic biases acting on phase space variables or projections thereof, the trajectories PIGS generates can be analyzed directly in the framework of transition networks. We demonstrate the possibility and usefulness of such focused explorations of biomolecules with two loops that are part of the binding sites of bromodomains, a family of epigenetic "reader" modules. This real-life application uncovers states that are structurally and kinetically far away from the initial crystallographic structures and are also metastable. Representative conformations are intended to be used in future high-throughput virtual screening campaigns.

Enhancing pairwise state-transition weights: A new weighting scheme in simulated tempering that can minimize transition time between a pair of conformational states

Simulated tempering (ST) is a widely used enhancing sampling method for Molecular Dynamics simulations. As one expanded ensemble method, ST is a combination of canonical ensembles at different temperatures and the acceptance probability of cross-temperature transitions is determined by both the temperature difference and the weights of each temperature. One popular way to obtain the weights is to adopt the free energy of each canonical ensemble, which achieves uniform sampling among temperature space. However, this uniform distribution in temperature space may not be optimal since high temperatures do not always speed up the conformational transitions of interest, as anti-Arrhenius kinetics are prevalent in protein and RNA folding. Here, we propose a new method: Enhancing Pairwise State-transition Weights (EPSW), to obtain the optimal weights by minimizing the round-trip time for transitions among different metastable states at the temperature of interest in ST. The novelty of the EPSW algorithm lies in explicitly considering the kinetics of conformation transitions when optimizing the weights of different temperatures. We further demonstrate the power of EPSW in three different systems: a simple two-temperature model, a two-dimensional model for protein folding with anti-Arrhenius kinetics, and the alanine dipeptide. The results from these three systems showed that the new algorithm can substantially accelerate the transitions between conformational states of interest in the ST expanded ensemble and further facilitate the convergence of thermodynamics compared to the widely used free energy weights. We anticipate that this algorithm is particularly useful for studying functional conformational changes of biological systems where the initial and final states are often known from structural biology experiments.

A molecular simulation protocol to avoid sampling redundancy and discover new states

BACKGROUND

For biomacromolecules or their assemblies, experimental knowledge is often restricted to specific states. Ambiguity pervades simulations of these complex systems because there is no prior knowledge of relevant phase space domains, and sampling recurrence is difficult to achieve. In molecular dynamics methods, ruggedness of the free energy surface exacerbates this problem by slowing down the unbiased exploration of phase space. Sampling is inefficient if dwell times in metastable states are large.

METHODS

We suggest a heuristic algorithm to terminate and reseed trajectories run in multiple copies in parallel. It uses a recent method to order snapshots, which provides notions of "interesting" and "unique" for individual simulations. We define criteria to guide the reseeding of runs from more "interesting" points if they sample overlapping regions of phase space.

RESULTS

Using a pedagogical example and an α-helical peptide, the approach is demonstrated to amplify the rate of exploration of phase space and to discover metastable states not found by conventional sampling schemes. Evidence is provided that accurate kinetics and pathways can be extracted from the simulations.

CONCLUSIONS

The method, termed PIGS for Progress Index Guided Sampling, proceeds in unsupervised fashion, is scalable, and benefits synergistically from larger numbers of replicas. Results confirm that the underlying ideas are appropriate and sufficient to enhance sampling.

GENERAL SIGNIFICANCE

In molecular simulations, errors caused by not exploring relevant domains in phase space are always unquantifiable and can be arbitrarily large. Our protocol adds to the toolkit available to researchers in reducing these types of errors. This article is part of a Special Issue entitled "Recent developments of molecular dynamics".

Rapid exploration of configuration space with diffusion-map-directed molecular dynamics

The gap between the time scale of interesting behavior in macromolecular systems and that which our computational resources can afford often limits molecular dynamics (MD) from understanding experimental results and predicting what is inaccessible in experiments. In this paper, we introduce a new sampling scheme, named diffusion-map-directed MD (DM-d-MD), to rapidly explore molecular configuration space. The method uses a diffusion map to guide MD on the fly. DM-d-MD can be combined with other methods to reconstruct the equilibrium free energy, and here, we used umbrella sampling as an example. We present results from two systems: alanine dipeptide and alanine-12. In both systems, we gain tremendous speedup with respect to standard MD both in exploring the configuration space and reconstructing the equilibrium distribution. In particular, we obtain 3 orders of magnitude of speedup over standard MD in the exploration of the configurational space of alanine-12 at 300 K with DM-d-MD. The method is reaction coordinate free and minimally dependent on a priori knowledge of the system. We expect wide applications of DM-d-MD to other macromolecular systems in which equilibrium sampling is not affordable by standard MD.

New insights into the folding of a β-sheet miniprotein in a reduced space of collective hydrogen bond variables: application to a hydrodynamic analysis of the folding flow

A new analysis of the 20 μs equilibrium folding/unfolding molecular dynamics simulations of the three-stranded antiparallel β-sheet miniprotein (beta3s) in implicit solvent is presented. The conformation space is reduced in dimensionality by introduction of linear combinations of hydrogen bond distances as the collective variables making use of a specially adapted principal component analysis (PCA); i.e., to make structured conformations more pronounced, only the formed bonds are included in determining the principal components. It is shown that a three-dimensional (3D) subspace gives a meaningful representation of the folding behavior. The first component, to which eight native hydrogen bonds make the major contribution (four in each beta hairpin), is found to play the role of the reaction coordinate for the overall folding process, while the second and third components distinguish the structured conformations. The representative points of the trajectory in the 3D space are grouped into conformational clusters that correspond to locally stable conformations of beta3s identified in earlier work. A simplified kinetic network based on the three components is constructed, and it is complemented by a hydrodynamic analysis. The latter, making use of "passive tracers" in 3D space, indicates that the folding flow is much more complex than suggested by the kinetic network. A 2D representation of streamlines shows there are vortices which correspond to repeated local rearrangement, not only around minima of the free energy surface but also in flat regions between minima. The vortices revealed by the hydrodynamic analysis are apparently not evident in folding pathways generated by transition-path sampling. Making use of the fact that the values of the collective hydrogen bond variables are linearly related to the Cartesian coordinate space, the RMSD between clusters is determined. Interestingly, the transition rates show an approximate exponential correlation with distance in the hydrogen bond subspace. Comparison with the many published studies shows good agreement with the present analysis for the parts that can be compared, supporting the robust character of our understanding of this "hydrogen atom" of protein folding.

Distribution of Reciprocal of Interatomic Distances: A Fast Structural Metric

We present a structural metric based on the Distribution of Reciprocal of Interatomic Distances (DRID) for evaluating geometrical similarity between two conformations of a molecule. A molecular conformation is described by a vector of 3N orientation-independent DRID descriptors where N is the number of molecular centroids, for example, the non-hydrogen atoms in all nonsymmetric groups of a peptide. For two real-world applications (pairwise comparison of snapshots from an explicit solvent simulation of a protease/peptide substrate complex and implicit solvent simulations of reversible folding of a 20-residue β-sheet peptide), the DRID-based metric is shown to be about 5 and 15 times faster than coordinate root-mean-square deviation (cRMSD) and distance root-mean-square deviation (dRMSD), respectively. A single core of a mainstream processor can perform about 10(8) DRID comparisons in one-half a minute. Importantly, the DRID metric has closer similarity to kinetic distance than does either cRMSD or dRMSD. Therefore, DRID is suitable for clustering molecular dynamics trajectories for kinetic analysis, for example, by Markov state models. Moreover, conformational space networks and free energy profiles derived by DRID-based clustering preserve the kinetic information.