Multidimensional generalized-ensemble algorithms for complex systems

Hysteresis Elimination for an Anisotropic Liquid-Crystal Model via Molecule Design and Replica-Exchange Optimization

The phenomenon of hysteresis in simulations, in which a system's current state is correlated to previous states and inhibits the transition to a more stable phase, may often lead to misleading results in physical chemistry. In this study, in addition to the replica exchange method (REM), a novel approach was taken by combining an evolution strategy based on the evolutionary principles of nature to predict phase transitions for the Hess-Su liquid-crystal model. In this model, an anisotropy term is added to the simple 6-12 Lennard-Jones model to intuitively reproduce the behavior of liquid crystals. We first applied the pressure-temperature REM to the Hess-Su model and optimized the replica spacing for the energy distribution to gain the maximum advantage from the REM. We then used the same approach as for the Hamiltonian REM, seeking to optimize the replica spacing in the same way. Based on both results, we attempted to predict this coarse-grained liquid-crystal model's exact phase transition point. In the Hamiltonian REM, replicas were prepared with different molecular aspect ratios corresponding to the values of the anisotropy terms in the potential function. The Hess-Su liquid-crystal model, which undergoes a direct transition from the nematic to the solid phase without going through a smectic phase, is a challenging research target for understanding phase transitions. Despite the tremendous computational difficulty in overcoming the strong hysteresis present in this system, our method could predict the phase transition point clearly and significantly reduce the extent of hysteresis. Our approach is beneficial when simulating more complex systems and, above all, shows great potential for more accurate and efficient phase transition predictions in the field of molecular simulation in the future.

Sigmoid Accelerated Molecular Dynamics: An Efficient Enhanced Sampling Method for Biosystems

Gaussian accelerated molecular dynamics (GaMD) is recognized as a popular enhanced sampling method for tackling long-standing challenges in biomolecular simulations. Inspired by GaMD, Sigmoid accelerated molecular dynamics (SaMD) is proposed in this work by adding a Sigmoid boost potential to improve the balance between the highest acceleration and accurate reweighting. Compared with GaMD, SaMD extends the accessible time scale and improves the computational efficiency as tested in three tasks. In the alanine dipeptide task, SaMD can produce the free energy landscape with better accuracy and efficiency. In the chignolin folding task, the estimated Gibbs free energy difference can converge to the experimental value ∼30% faster. In the protein-ligand binding task, the bound conformations are closer to the crystal structure with a minimal ligand root-mean-square deviation of 1.7 Å. The binding of the ligand XK263 to the HIV protease is reproduced by SaMD in ∼60% less simulation time.

Computer Simulations Aimed at Exploring Protein Aggregation and Dissociation

Protein aggregation can lead to well-defined structures that are functional, but is also the cause of the death of neuron cells in many neurodegenerative diseases. The complexity of the molecular events involved in the aggregation kinetics of amyloid proteins and the transient and heterogeneous characters of all oligomers prevent high-resolution structural experiments. As a result, computer simulations have been used to determine the atomic structures of amyloid proteins at different association stages as well as to understand fibril dissociation. In this chapter, we first review the current computer simulation methods used for aggregation with some atomistic and coarse-grained results aimed at better characterizing the early formed oligomers and amyloid fibril formation. Then we present the applications of non-equilibrium molecular dynamics simulations to comprehend the dissociation of protein assemblies.

Analysis of liquids, gases, and supercritical fluids by a two-dimensional replica-exchange Monte Carlo method in temperature and chemical potential space

We investigate the liquid, gas, and supercritical fluid phases of a Lennard-Jones 12-6 potential system by a two-dimensional replica-exchange method in which not only temperature but also chemical potential is exchanged. The method is referred to as the grand canonical replica-exchange method (GCREM). While one-dimensional replica exchange, which exchanges only temperature, cannot cross first-order phase transition points, GCREM can avoid this problem by making a detour in the two-dimensional parameter space. From only one simulation run, we can obtain probability distributions in the grand canonical ensemble for wide temperature and chemical potential values using the multiple-histogram reweighting techniques. We define a phase diagram near the critical point using thermodynamic quantities. Moreover, we discuss structures in each defined phase and at phase transition points.

Protein structure predictions by enhanced conformational sampling methods

In this Special Festschrift Issue for the celebration of Professor Nobuhiro Gō's 80th birthday, we review enhanced conformational sampling methods for protein structure predictions. We present several generalized-ensemble algorithms such as multicanonical algorithm, replica-exchange method, etc. and parallel Monte Carlo or molecular dynamics method with genetic crossover. Examples of the results of these methods applied to the predictions of protein tertiary structures are also presented.

Efficient simulation protocol for determining the density of states: Combination of replica-exchange Wang-Landau method and multicanonical replica-exchange method

By combining two generalized-ensemble algorithms, the replica-exchange Wang-Landau (REWL) method and the multicanonical replica-exchange method (MUCAREM), we propose an effective simulation protocol to determine the density of states with high accuracy. The new protocol is referred to as REWL-MUCAREM, and REWL is first performed and then MUCAREM is performed. In order to verify the effectiveness of our protocol, we performed simulations of a square-lattice Ising model using the three methods, namely REWL, MUCAREM, and REWL-MUCAREM. The results showed that the density of states obtained by REWL-MUCAREM is more accurate than that is estimated by the two methods separately.

Enhanced sampling method in molecular simulations using genetic algorithm for biomolecular systems

We propose a molecular simulation method using genetic algorithm (GA) for biomolecular systems to obtain ensemble averages efficiently. In this method, we incorporate the genetic crossover, which is one of the operations of GA, to any simulation method such as conventional molecular dynamics (MD), Monte Carlo, and other simulation methods. The genetic crossover proposes candidate conformations by exchanging parts of conformations of a target molecule between a pair of conformations during the simulation. If the candidate conformations are accepted, the simulation resumes from the accepted ones. While conventional simulations are based on local update of conformations, the genetic crossover introduces global update of conformations. As an example of the present approach, we incorporated genetic crossover to MD simulations. We tested the validity of the method by calculating ensemble averages and the sampling efficiency by using two kinds of peptides, ALA3 and (AAQAA)₃ . The results show that for ALA3 system, the distribution probabilities of backbone dihedral angles are in good agreement with those of the conventional MD and replica-exchange MD simulations. In the case of (AAQAA)₃ system, our method showed lower structural correlation of α-helix structures than the other two methods and more flexibility in the backbone ψ angles than the conventional MD simulation. These results suggest that our method gives more efficient conformational sampling than conventional simulation methods based on local update of conformations. © 2018 Wiley Periodicals, Inc.

GPU-Accelerated Molecular Dynamics and Free Energy Methods in Amber18: Performance Enhancements and New Features

We report progress in graphics processing unit (GPU)-accelerated molecular dynamics and free energy methods in Amber18. Of particular interest is the development of alchemical free energy algorithms, including free energy perturbation and thermodynamic integration methods with support for nonlinear soft-core potential and parameter interpolation transformation pathways. These methods can be used in conjunction with enhanced sampling techniques such as replica exchange, constant-pH molecular dynamics, and new 12-6-4 potentials for metal ions. Additional performance enhancements have been made that enable appreciable speed-up on GPUs relative to the previous software release.

Locally weighted histogram analysis and stochastic solution for large-scale multi-state free energy estimation

The weighted histogram analysis method (WHAM) including its binless extension has been developed independently in several different contexts, and widely used in chemistry, physics, and statistics, for computing free energies and expectations from multiple ensembles. However, this method, while statistically efficient, is computationally costly or even infeasible when a large number, hundreds or more, of distributions are studied. We develop a locally WHAM (local WHAM) from the perspective of simulations of simulations (SOS), using generalized serial tempering (GST) to resample simulated data from multiple ensembles. The local WHAM equations based on one jump attempt per GST cycle can be solved by optimization algorithms orders of magnitude faster than standard implementations of global WHAM, but yield similarly accurate estimates of free energies to global WHAM estimates. Moreover, we propose an adaptive SOS procedure for solving local WHAM equations stochastically when multiple jump attempts are performed per GST cycle. Such a stochastic procedure can lead to more accurate estimates of equilibrium distributions than local WHAM with one jump attempt per cycle. The proposed methods are broadly applicable when the original data to be "WHAMMED" are obtained properly by any sampling algorithm including serial tempering and parallel tempering (replica exchange). To illustrate the methods, we estimated absolute binding free energies and binding energy distributions using the binding energy distribution analysis method from one and two dimensional replica exchange molecular dynamics simulations for the beta-cyclodextrin-heptanoate host-guest system. In addition to the computational advantage of handling large datasets, our two dimensional WHAM analysis also demonstrates that accurate results similar to those from well-converged data can be obtained from simulations for which sampling is limited and not fully equilibrated.

Observation of helix associations for insertion of a retinal molecule and distortions of helix structures in bacteriorhodopsin

We applied a newly proposed prediction method for membrane protein structures to bacteriorhodopsin that has distorted transmembrane helices in the native structure. This method uses an implicit membrane model, which restricts sampling space during folding in a membrane region, and includes helix bending. Replica-exchange simulations were performed with seven transmembrane helices only without a retinal molecule. Obtained structures were classified into clusters of similar structures, which correspond to local-minimum free energy states. The two lowest free energy states corresponded to a native-like structure with the correct empty space for retinal and a structure with this empty space filled with a helix. Previous experiments of bacteriorhodopsin suggested that association of transmembrane helices enables them to make a room for insertion of a retinal. Our results are consistent with these results. Moreover, distortions of helices in the native-like structures were successfully reproduced. In the distortions, whereas the locations of kinks for all helices were similar to those of Protein Data Bank's data, the amount of bends was more similar for helices away from the retinal than for those close to the retinal in the native structure. This suggests a hypothesis that the amino-acid sequence specifies the location of kinks in transmembrane helices and that the amount of distortions depends on the interactions with the surrounding molecules such as neighboring helices, lipids, and retinal.

On the use of mass scaling for stable and efficient simulated tempering with molecular dynamics

Simulated tempering (ST) is a generalized-ensemble algorithm that employs trajectories exploring a range of temperatures to effectively sample rugged energy landscapes. When implemented using the molecular dynamics method, ST can require the use of short time steps for ensuring the stability of trajectories at high temperatures. To address this shortcoming, a mass-scaling ST (MSST) method is presented in which the particle mass is scaled in proportion to the temperature. Mass scaling in the MSST method leads to velocity distributions that are independent of temperature and eliminates the need for velocity scaling after the accepted temperature updates that are required in conventional ST simulations. The homogeneity in time scales with changing temperature improves the stability of simulations and allows for the use of longer time steps at high temperatures. As a result, the MSST is found to be more efficient than the standard ST method, particularly for cases in which a large temperature range is employed. © 2016 Wiley Periodicals, Inc.

Mass-scaling replica-exchange molecular dynamics optimizes computational resources with simpler algorithm

We develop a novel method of replica-exchange molecular dynamics (REMD) simulation, mass-scaling REMD (MSREMD) method, which improves numerical stability of simulations. In addition, the MSREMD method can also simplify a replica-exchange routine by eliminating velocity scaling. As a pilot system, a Lennard-Jones fluid is simulated with the new method. The results suggest that the MSREMD method improves the numerical stability at high temperatures compared with the conventional REMD method. For the Nosé-Hoover thermostats, we analytically demonstrate that the MSREMD simulations can reproduce completely the same trajectories of the conventional REMD ones with shorter time steps at high temperatures. Accordingly, we can easily compare the computational costs of the REMD and MSREMD simulations. We conclude that the MSREMD method decreases the instability and optimizes the computational resources with simpler algorithm.

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.

Combine umbrella sampling with integrated tempering method for efficient and accurate calculation of free energy changes of complex energy surface

Umbrella sampling is an efficient method for the calculation of free energy changes of a system along well-defined reaction coordinates. However, when there exist multiple parallel channels along the reaction coordinate or hidden barriers in directions perpendicular to the reaction coordinate, it is difficult for conventional umbrella sampling to reach convergent sampling within limited simulation time. Here, we propose an approach to combine umbrella sampling with the integrated tempering sampling method. The umbrella sampling method is applied to chemically more relevant degrees of freedom that possess significant barriers. The integrated tempering sampling method is used to facilitate the sampling of other degrees of freedom which may possess statistically non-negligible barriers. The combined method is applied to two model systems, butane and ACE-NME molecules, and shows significantly improved sampling efficiencies as compared to standalone conventional umbrella sampling or integrated tempering sampling approaches. Further analyses suggest that the enhanced performance of the new method come from the complemented advantages of umbrella sampling with a well-defined reaction coordinate and integrated tempering sampling in orthogonal space. Therefore, the combined approach could be useful in the simulation of biomolecular processes, which often involves sampling of complex rugged energy landscapes.

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.

Protein folding pathways extracted by OFLOOD: Outlier FLOODing method

The Outlier FLOODing method (OFLOOD) is proposed as an efficient conformational sampling method to extract biologically rare events such as protein folding. In OFLOOD, sparse distributions (outliers in the conformational space) were regarded as relevant states for the transitions. Then, the transitions were enhanced through conformational resampling from the outliers. This evidence indicates that the conformational resampling of the sparse distributions might increase chances for promoting the transitions from the outliers to other meta-stable states, which resembles a conformational flooding from the outliers to the neighboring clusters. OFLOOD consists of (i) detections of outliers from conformational distributions and (ii) conformational resampling from the outliers by molecular dynamics (MD) simulations. Cycles of (i) and (ii) are simply repeated. As demonstrations, OFLOOD was applied to folding of Chignolin and HP35. In both cases, OFLOOD automatically extracted folding pathways from unfolded structures with ns-order computational costs, although µs-order canonical MD failed to extract them.

Free-energy analyses of a proton transfer reaction by simulated-tempering umbrella sampling and first-principles molecular dynamics simulations

A simulated tempering method, which is referred to as simulated-tempering umbrella sampling, for calculating the free energy of chemical reactions is proposed. First principles molecular dynamics simulations with this simulated tempering were performed to study the intramolecular proton transfer reaction of malonaldehyde in an aqueous solution. Conformational sampling in reaction coordinate space can be easily enhanced with this method, and the free energy along a reaction coordinate can be calculated accurately. Moreover, the simulated-tempering umbrella sampling provides trajectory data more efficiently than the conventional umbrella sampling method.

Enhanced sampling algorithms

In biomolecular systems (especially all-atom models) with many degrees of freedom such as proteins and nucleic acids, there exist an astronomically large number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. Enhanced conformational sampling techniques are thus in great demand. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present various extensions of these three generalized-ensemble algorithms. The effectiveness of the methods is tested with short peptide and protein systems.

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.

A preformed binding interface in the unbound ensemble of an intrinsically disordered protein: evidence from molecular simulations

Intrinsically disordered proteins play an important role in cellular signalling, mediated by their interactions with other biomolecules. A key question concerns the nature of their binding mechanism, and whether the bound structure is induced only by proximity to the binding partner. This is difficult to answer through experiment alone because of the very heterogeneous nature of the unbound ensemble, and the probable rapid interconversion of the various unbound structures. Here we report the most extensive set of simulations on NCBD to date: we use large-scale replica exchange molecular dynamics to explore the unbound state. An important feature of the study is the use of an atomistic force field that has been parametrised against experimental data for weakly structured peptides, together with an accurate explicit water model. Neither the force field nor the starting conformations are biased towards a particular structure. The regions of NCBD that have high helical propensity in the simulations correspond closely to helices in the 'core' unbound conformation determined by NMR, although no single member of the simulated unbound ensemble closely resembles the core conformation, or either of the two known bound conformations. We have validated the results against NMR spectroscopy and SAXS measurements, obtaining reasonable agreement. The two helices which most stabilise the binding of NCBD with ACTR are formed readily; the third helix, which is less important for binding but is involved in most of the intraprotein contacts of NCBD in the bound conformation, is formed more rarely, and tends not to coexist with the other helices. These results support a mechanism by which NCBD gains the advantages of disorder, while forming binding-competent structures in the unbound state. We obtain support for this mechanism from coarse-grained simulations of NCBD with, and without, its binding partner.

Generalized simulated tempering for exploring strong phase transitions

An extension of the simulation tempering algorithm is proposed. It is shown to be particularly suited to the exploration of first-order phase transition systems characterized by the backbending or S-loop in the statistical temperature or a microcanonical caloric curve. A guided Markov process in an auxiliary parameter space systematically combines a set of parametrized Tsallis-weight ensemble simulations, which are targeted to transform unstable or metastable energy states of canonical ensembles into stable ones and smoothly join ordered and disordered phases across phase transition regions via a succession of unimodal energy distributions. The inverse mapping between the sampling weight and the effective temperature enables an optimal selection of relevant Tsallis-weight parameters. A semianalytic expression for the biasing weight in parameter space is adaptively updated "on the fly" during the simulation to achieve rapid convergence. Accelerated tunneling transitions with a comprehensive sampling for phase-coexistent states are explicitly demonstrated in systems subject to strong hysteresis including Potts and Ising spin models and a 147 atom Lennard-Jones cluster.

Serial tempering without exchange

Serial tempering is a computational method that turns the temperature T (or more generally any independent λ parameter) into a dynamical variable. It is shown that, under conditions for which this variable is fast, serial tempering is equivalent to the umbrella sampling method with a single effective potential. This equivalence is demonstrated using both a small one-dimensional system and a small solvated peptide. The suggestion is then made to replace the serial tempering protocol with the equivalent umbrella sampling calculation. This approach, serial tempering without exchange (STeWiE), has the same performance as serial tempering in the limit that exchanges are frequent, is simpler to implement, and has fewer adjustable parameters than conventional serial tempering. The equivalence of serial tempering and STeWiE also provides a convenient route for estimating and optimizing the performance of serial tempering simulations and other generalized-ensemble methods.