Replica exchange and expanded ensemble simulations as Gibbs sampling: Simple improvements for enhanced mixing

Simulated tempering based on global balance or detailed balance conditions: Suwa-Todo, heat bath, and Metropolis algorithms

Simulated tempering (ST) is a useful method to enhance sampling of molecular simulations. When ST is used, the Metropolis algorithm, which satisfies the detailed balance condition, is usually applied to calculate the transition probability. Recently, an alternative method that satisfies the global balance condition instead of the detailed balance condition has been proposed by Suwa and Todo. In this study, ST method with the Suwa-Todo algorithm is proposed. Molecular dynamics simulations with ST are performed with three algorithms (the Metropolis, heat bath, and Suwa-Todo algorithms) to calculate the transition probability. Among the three algorithms, the Suwa-Todo algorithm yields the highest acceptance ratio and the shortest autocorrelation time. These suggest that sampling by a ST simulation with the Suwa-Todo algorithm is most efficient. In addition, because the acceptance ratio of the Suwa-Todo algorithm is higher than that of the Metropolis algorithm, the number of temperature states can be reduced by 25% for the Suwa-Todo algorithm when compared with the Metropolis algorithm.

Accelerated weight histogram method for exploring free energy landscapes

Calculating free energies is an important and notoriously difficult task for molecular simulations. The rapid increase in computational power has made it possible to probe increasingly complex systems, yet extracting accurate free energies from these simulations remains a major challenge. Fully exploring the free energy landscape of, say, a biological macromolecule typically requires sampling large conformational changes and slow transitions. Often, the only feasible way to study such a system is to simulate it using an enhanced sampling method. The accelerated weight histogram (AWH) method is a new, efficient extended ensemble sampling technique which adaptively biases the simulation to promote exploration of the free energy landscape. The AWH method uses a probability weight histogram which allows for efficient free energy updates and results in an easy discretization procedure. A major advantage of the method is its general formulation, making it a powerful platform for developing further extensions and analyzing its relation to already existing methods. Here, we demonstrate its efficiency and general applicability by calculating the potential of mean force along a reaction coordinate for both a single dimension and multiple dimensions. We make use of a non-uniform, free energy dependent target distribution in reaction coordinate space so that computational efforts are not wasted on physically irrelevant regions. We present numerical results for molecular dynamics simulations of lithium acetate in solution and chignolin, a 10-residue long peptide that folds into a β-hairpin. We further present practical guidelines for setting up and running an AWH simulation.

A Stochastic Solution to the Unbinned WHAM Equations

The weighted histogram analysis method (WHAM) and unbinned versions such as the multistate Bennett acceptance ratio (MBAR) and unbinned WHAM (UWHAM) are widely used to compute free energies and expectations from data generated by independent or coupled parallel simulations. Here we introduce a replica exchange-like algorithm (RE-SWHAM) that can be used to solve the UWHAM equations stochastically. This method is capable of analyzing large data sets generated by hundreds or even thousands of parallel simulations that are too large to be "WHAMMED" using standard methods. We illustrate the method by applying it to obtain free energy weights for each of the 240 states in a simulation of host-guest ligand binding containing ∼3.5 × 10(7) data elements collected from 16 parallel Hamiltonian replica exchange simulations, performed at 15 temperatures. In addition to using much less memory, RE-SWHAM showed a nearly 80-fold improvement in computational time compared with UWHAM.

Large-scale asynchronous and distributed multidimensional replica exchange molecular simulations and efficiency analysis

We describe methods to perform replica exchange molecular dynamics (REMD) simulations asynchronously (ASyncRE). The methods are designed to facilitate large scale REMD simulations on grid computing networks consisting of heterogeneous and distributed computing environments as well as on homogeneous high-performance clusters. We have implemented these methods on NSF (National Science Foundation) XSEDE (Extreme Science and Engineering Discovery Environment) clusters and BOINC (Berkeley Open Infrastructure for Network Computing) distributed computing networks at Temple University and Brooklyn College at CUNY (the City University of New York). They are also being implemented on the IBM World Community Grid. To illustrate the methods, we have performed extensive (more than 60 ms in aggregate) simulations for the beta-cyclodextrin-heptanoate host-guest system in the context of one- and two-dimensional ASyncRE, and we used the results to estimate absolute binding free energies using the binding energy distribution analysis method. We propose ways to improve the efficiency of REMD simulations: these include increasing the number of exchanges attempted after a specified molecular dynamics (MD) period up to the fast exchange limit and/or adjusting the MD period to allow sufficient internal relaxation within each thermodynamic state. Although ASyncRE simulations generally require long MD periods (>picoseconds) per replica exchange cycle to minimize the overhead imposed by heterogeneous computing networks, we found that it is possible to reach an efficiency similar to conventional synchronous REMD, by optimizing the combination of the MD period and the number of exchanges attempted per cycle.

Exploring biomolecular dynamics and interactions using advanced sampling methods

Molecular dynamics (MD) and Monte Carlo (MC) simulations have emerged as a valuable tool to investigate statistical mechanics and kinetics of biomolecules and synthetic soft matter materials. However, major limitations for routine applications are due to the accuracy of the molecular mechanics force field and due to the maximum simulation time that can be achieved in current simulations studies. For improving the sampling a number of advanced sampling approaches have been designed in recent years. In particular, variants of the parallel tempering replica-exchange methodology are widely used in many simulation studies. Recent methodological advancements and a discussion of specific aims and advantages are given. This includes improved free energy simulation approaches and conformational search applications.

Ab initio RNA folding

RNA molecules are essential cellular machines performing a wide variety of functions for which a specific three-dimensional structure is required. Over the last several years, the experimental determination of RNA structures through x-ray crystallography and NMR seems to have reached a plateau in the number of structures resolved each year, but as more and more RNA sequences are being discovered, the need for structure prediction tools to complement experimental data is strong. Theoretical approaches to RNA folding have been developed since the late nineties, when the first algorithms for secondary structure prediction appeared. Over the last 10 years a number of prediction methods for 3D structures have been developed, first based on bioinformatics and data-mining, and more recently based on a coarse-grained physical representation of the systems. In this review we are going to present the challenges of RNA structure prediction and the main ideas behind bioinformatic approaches and physics-based approaches. We will focus on the description of the more recent physics-based phenomenological models and on how they are built to include the specificity of the interactions of RNA bases, whose role is critical in folding. Through examples from different models, we will point out the strengths of physics-based approaches, which are able not only to predict equilibrium structures, but also to investigate dynamical and thermodynamical behavior, and the open challenges to include more key interactions ruling RNA folding.

Enhancing constant-pH simulation in explicit solvent with a two-dimensional replica exchange method

We present a new method for enhanced sampling for constant-pH simulations in explicit water based on a two-dimensional (2D) replica exchange scheme. The new method is a significant extension of our previously developed constant-pH simulation method, which is based on enveloping distribution sampling (EDS) coupled with a one-dimensional (1D) Hamiltonian exchange method (HREM). EDS constructs a hybrid Hamiltonian from multiple discrete end state Hamiltonians that, in this case, represent different protonation states of the system. The ruggedness and heights of the hybrid Hamiltonian's energy barriers can be tuned by the smoothness parameter. Within the context of the 1D EDS-HREM method, exchanges are performed between replicas with different smoothness parameters, allowing frequent protonation-state transitions and sampling of conformations that are favored by the end-state Hamiltonians. In this work, the 1D method is extended to 2D with an additional dimension, external pH. Within the context of the 2D method (2D EDS-HREM), exchanges are performed on a lattice of Hamiltonians with different pH conditions and smoothness parameters. We demonstrate that both the 1D and 2D methods exactly reproduce the thermodynamic properties of the semigrand canonical (SGC) ensemble of a system at a given pH. We have tested our new 2D method on aspartic acid, glutamic acid, lysine, a four residue peptide (sequence KAAE), and snake cardiotoxin. In all cases, the 2D method converges faster and without loss of precision; the only limitation is a loss of flexibility in how CPU time is employed. The results for snake cardiotoxin demonstrate that the 2D method enhances protonation-state transitions, samples a wider conformational space with the same amount of computational resources, and converges significantly faster overall than the original 1D method.

Equilibrium molecular thermodynamics from Kirkwood sampling

We present two methods for barrierless equilibrium sampling of molecular systems based on the recently proposed Kirkwood method (J. Chem. Phys.2009, 130, 134102). Kirkwood sampling employs low-order correlations among internal coordinates of a molecule for random (or non-Markovian) sampling of the high dimensional conformational space. This is a geometrical sampling method independent of the potential energy surface. The first method is a variant of biased Monte Carlo, where Kirkwood sampling is used for generating trial Monte Carlo moves. Using this method, equilibrium distributions corresponding to different temperatures and potential energy functions can be generated from a given set of low-order correlations. Since Kirkwood samples are generated independently, this method is ideally suited for massively parallel distributed computing. The second approach is a variant of reservoir replica exchange, where Kirkwood sampling is used to construct a reservoir of conformations, which exchanges conformations with the replicas performing equilibrium sampling corresponding to different thermodynamic states. Coupling with the Kirkwood reservoir enhances sampling by facilitating global jumps in the conformational space. The efficiency of both methods depends on the overlap of the Kirkwood distribution with the target equilibrium distribution. We present proof-of-concept results for a model nine-atom linear molecule and alanine dipeptide.

Multiscale methods for computational RNA enzymology

RNA catalysis is of fundamental importance to biology and yet remains ill-understood due to its complex nature. The multidimensional "problem space" of RNA catalysis includes both local and global conformational rearrangements, changes in the ion atmosphere around nucleic acids and metal ion binding, dependence on potentially correlated protonation states of key residues, and bond breaking/forming in the chemical steps of the reaction. The goal of this chapter is to summarize and apply multiscale modeling methods in an effort to target the different parts of the RNA catalysis problem space while also addressing the limitations and pitfalls of these methods. Classical molecular dynamics simulations, reference interaction site model calculations, constant pH molecular dynamics (CpHMD) simulations, Hamiltonian replica exchange molecular dynamics, and quantum mechanical/molecular mechanical simulations will be discussed in the context of the study of RNA backbone cleavage transesterification. This reaction is catalyzed by both RNA and protein enzymes, and here we examine the different mechanistic strategies taken by the hepatitis delta virus ribozyme and RNase A.

Xenon NMR of liquid crystals confined to cylindrical nanocavities: a simulation study

Coarse-grained simulations show that the ¹²⁹Xe NMR shielding reflects the smooth changes of orientational order in liquid crystals confined to nanocavities.

Computational Recipe for Efficient Description of Large-Scale Conformational Changes in Biomolecular Systems

Characterizing large-scale structural transitions in biomolecular systems poses major technical challenges to both experimental and computational approaches. On the computational side, efficient sampling of the configuration space along the transition pathway remains the most daunting challenge. Recognizing this issue, we introduce a knowledge-based computational approach toward describing large-scale conformational transitions using (i) nonequilibrium, driven simulations combined with work measurements and (ii) free energy calculations using empirically optimized biasing protocols. The first part is based on designing mechanistically relevant, system-specific reaction coordinates whose usefulness and applicability in inducing the transition of interest are examined using knowledge-based, qualitative assessments along with nonequilirbrium work measurements which provide an empirical framework for optimizing the biasing protocol. The second part employs the optimized biasing protocol resulting from the first part to initiate free energy calculations and characterize the transition quantitatively. Using a biasing protocol fine-tuned to a particular transition not only improves the accuracy of the resulting free energies but also speeds up the convergence. The efficiency of the sampling will be assessed by employing dimensionality reduction techniques to help detect possible flaws and provide potential improvements in the design of the biasing protocol. Structural transition of a membrane transporter will be used as an example to illustrate the workings of the proposed approach.

Hamiltonian Switch Metropolis Monte Carlo Simulations for Improved Conformational Sampling of Intrinsically Disordered Regions Tethered to Ordered Domains of Proteins

There is growing interest in the topic of intrinsically disordered proteins (IDPs). Atomistic Metropolis Monte Carlo (MMC) simulations based on novel implicit solvation models have yielded useful insights regarding sequence-ensemble relationships for IDPs modeled as autonomous units. However, a majority of naturally occurring IDPs are tethered to ordered domains. Tethering introduces additional energy scales and this creates the challenge of broken ergodicity for standard MMC sampling or molecular dynamics that cannot be readily alleviated by using generalized tempering methods. We have designed, deployed, and tested our adaptation of the Nested Markov Chain Monte Carlo sampling algorithm. We refer to our adaptation as Hamiltonian Switch Metropolis Monte Carlo (HS-MMC) sampling. In this method, transitions out of energetic traps are enabled by the introduction of an auxiliary Markov chain that draws conformations for the disordered region from a Boltzmann distribution that is governed by an alternative potential function that only includes short-range steric repulsions and conformational restraints on the ordered domain. We show using multiple, independent runs that the HS-MMC method yields conformational distributions that have similar and reproducible statistical properties, which is in direct contrast to standard MMC for equivalent amounts of sampling. The method is efficient and can be deployed for simulations of a range of biologically relevant disordered regions that are tethered to ordered domains.

Converging free energies of binding in cucurbit[7]uril and octa-acid host-guest systems from SAMPL4 using expanded ensemble simulations

Molecular containers such as cucurbit[7]uril (CB7) and the octa-acid (OA) host are ideal simplified model test systems for optimizing and analyzing methods for computing free energies of binding intended for use with biologically relevant protein-ligand complexes. To this end, we have performed initially blind free energy calculations to determine the free energies of binding for ligands of both the CB7 and OA hosts. A subset of the selected guest molecules were those included in the SAMPL4 prediction challenge. Using expanded ensemble simulations in the dimension of coupling host-guest intermolecular interactions, we are able to show that our estimates in most cases can be demonstrated to fully converge and that the errors in our estimates are due almost entirely to the assigned force field parameters and the choice of environmental conditions used to model experiment. We confirm the convergence through the use of alternative simulation methodologies and thermodynamic pathways, analyzing sampled conformations, and directly observing changes of the free energy with respect to simulation time. Our results demonstrate the benefits of enhanced sampling of multiple local free energy minima made possible by the use of expanded ensemble molecular dynamics and may indicate the presence of significant problems with current transferable force fields for organic molecules when used for calculating binding affinities, especially in non-protein chemistries.

Roadmaps through free energy landscapes calculated using the multi-dimensional vFEP approach

The variational free energy profile (vFEP) method is extended to two dimensions and tested with molecular simulation applications. The proposed 2D-vFEP approach effectively addresses the two major obstacles to constructing free energy profiles from simulation data using traditional methods: the need for overlap in the re-weighting procedure and the problem of data representation. This is especially evident as these problems are shown to be more severe in two dimensions. The vFEP method is demonstrated to be highly robust and able to provide stable, analytic free energy profiles with only a paucity of sampled data. The analytic profiles can be analyzed with conventional search methods to easily identify stationary points (e.g. minima and first-order saddle points) as well as the pathways that connect these points. These "roadmaps" through the free energy surface are useful not only as a post-processing tool to characterize mechanisms, but can also serve as a basis from which to direct more focused "on-the-fly" sampling or adaptive force biasing. Test cases demonstrate that 2D-vFEP outperforms other methods in terms of the amount and sparsity of the data needed to construct stable, converged analytic free energy profiles. In a classic test case, the two dimensional free energy profile of the backbone torsion angles of alanine dipeptide, 2D-vFEP needs less than 1% of the original data set to reach a sampling accuracy of 0.5 kcal/mol in free energy shifts between windows. A new software tool for performing one and two dimensional vFEP calculations is herein described and made publicly available.

Identifying ligand binding sites and poses using GPU-accelerated Hamiltonian replica exchange molecular dynamics

We present a method to identify small molecule ligand binding sites and poses within a given protein crystal structure using GPU-accelerated Hamiltonian replica exchange molecular dynamics simulations. The Hamiltonians used vary from the physical end state of protein interacting with the ligand to an unphysical end state where the ligand does not interact with the protein. As replicas explore the space of Hamiltonians interpolating between these states, the ligand can rapidly escape local minima and explore potential binding sites. Geometric restraints keep the ligands from leaving the vicinity of the protein and an alchemical pathway designed to increase phase space overlap between intermediates ensures good mixing. Because of the rigorous statistical mechanical nature of the Hamiltonian exchange framework, we can also extract binding free energy estimates for all putative binding sites. We present results of this methodology applied to the T4 lysozyme L99A model system for three known ligands and one non-binder as a control, using an implicit solvent. We find that our methodology identifies known crystallographic binding sites consistently and accurately for the small number of ligands considered here and gives free energies consistent with experiment. We are also able to analyze the contribution of individual binding sites to the overall binding affinity. Our methodology points to near term potential applications in early-stage structure-guided drug discovery.

Optimization of Umbrella Sampling Replica Exchange Molecular Dynamics by Replica Positioning

The positioning of sampling windows in an umbrella sampling simulation has an effect on the rate of convergence and computational efficiency. When such simulation is coupled with a Hamiltonian replica exchange setup, we show that such positioning can be optimized for maximal convergence of the results. We present a method for estimating the exchange acceptance ratio (EAR) between two arbitrary positions on a reaction coordinate in umbrella sampling replica exchange (USRE) molecular dynamics (MD). We designed a scoring function to optimize the position of the set of replicas (windows). By maximizing the scoring function, we make EAR the same for all neighbor replica pairs, increasing the efficiency of the method. We tested our algorithm by sampling a torsion for butane in implicit solvent and by studying a salt bridge in explicit solvent. We found that the optimized set of replicas recovers the correct free energy profile much faster than for equally spaced umbrellas.

Overcoming the Rare Event Sampling Problem in Biological Systems with Infinite Swapping

Infinite swapping (INS) is a recently developed method to address the rare event sampling problem. For INS, an expanded computational ensemble composed of a number of replicas at different temperatures is used, similar to the widely used parallel tempering (PT) method. While the basic concept of PT is to sample various replicas of the system at different temperatures and exchange information between the replicas occasionally, INS uses the symmetrized distribution of configurations in temperature space, which corresponds to the infinite swapping limit of PT. The effect of this symmetrization and the enhanced information exchange between replicas is evaluated for three different biological systems representing different sampling problems in biology: (1) blocked alanine dipeptide, which is a small system and therefore optimal to evaluate sampling efficiency quantitatively, (2) Villin headpiece, which is used as a test case for the protein folding process, and (3) neuroglobin, which is used to evaluate the effects of enhanced information exchange between replicas for sampling the substate space of a folded protein. For these three test systems, PINS is compared to PT, and it is found that in all cases the sampling with PINS is substantially more efficient.

The native GCN4 leucine-zipper domain does not uniquely specify a dimeric oligomerization state

The dimerization domain of the yeast transcription factor GCN4, one of the first coiled-coil proteins to be structurally characterized at high resolution, has served as the basis for numerous fundamental studies on α-helical folding. Mutations in the GCN4 leucine zipper are known to change its preferred oligomerization state from dimeric to trimeric or tetrameric; however, the wild-type sequence has been assumed to encode a two-chain assembly exclusively. Here we demonstrate that the GCN4 coiled-coil domain can populate either a dimer or trimer fold, depending on environment. We report high-resolution crystal structures of the wild-type sequence in dimeric and trimeric assemblies. Biophysical measurements suggest populations of both oligomerization states under certain experimental conditions in solution. We use parallel tempering molecular dynamics simulations on the microsecond time scale to compare the stability of the dimer and trimer folded states in isolation. In total, our results suggest that the folding behavior of the well-studied GCN4 leucine-zipper domain is more complex than was previously appreciated. Our results have implications in ongoing efforts to establish predictive algorithms for coiled-coil folds and the selection of coiled-coil model systems for design and mutational studies where oligomerization state specificity is an important consideration.

Simulated tempering and magnetizing: application of two-dimensional simulated tempering to the two-dimensional Ising model and its crossover

We have performed two-dimensional simulated tempering (ST) simulations of the two-dimensional Ising model with different lattice sizes in order to investigate the two-dimensional ST's applicability to dealing with phase transitions and study the crossover of critical scaling behavior. The external field, as well as the temperature, was treated as a dynamical variable updated during the simulations. Thus this simulation can be referred to as simulated tempering and magnetizing (STM). We also performed simulated magnetizing (SM) simulations, in which the external field was considered as a dynamical variable and temperature was not. As discussed in previous studies, the ST method is not always compatible with first-order phase transitions. This is also true in the magnetizing process. Flipping of the entire magnetization did not occur in the SM simulations under the critical temperature T{c} in large-lattice-size simulations; however, the phase changed through the high-temperature region in the STM simulations. Thus the dimensional extension let us eliminate the difficulty of the first-order phase transitions and study a wide area of the phase space. We discuss how frequently parameter-updating attempts should be made for optimal convergence. The results favor frequent attempts. We finally study the crossover behavior of the phase transitions with respect to the temperature and external field. The crossover behavior is clearly observed in the simulations, in agreement with the theoretical implications.

Enhancing Conformation and Protonation State Sampling of Hen Egg White Lysozyme Using pH Replica Exchange Molecular Dynamics

We evaluate the efficiency of the pH replica exchange molecular dynamics (pH-REMD) method proposed by Itoh et al. (Proteins2011, 79, 3420-3436) by using it to predict the pKa values of the titratable residues in hen egg white lysozyme (HEWL). pKa values predicted using pH-REMD converge significantly faster than those calculated using constant pH molecular dynamics (CpHMD). Furthermore, increasing the frequency between exchange attempts in pH-REMD simulations improves protonation and conformational state sampling. By enabling the simulation to sample both conformational and protonation states more rapidly, pH-REMD simulations provide valuable insight into the pH-dependence of HEWL that the CpHMD simulations failed to capture. We present an efficient and highly scalable implementation of pH-REMD as an attractive enhancement to traditional CpHMD methods.