Coulomb replica‐exchange method: Handling electrostatic attractive and repulsive forces for biomolecules

We propose a new type of the Hamiltonian replica‐exchange method (REM) for molecular dynamics (MD) and Monte Carlo simulations, which we refer to as the Coulomb REM (CREM). In this method, electrostatic charge parameters in the Coulomb interactions are exchanged among replicas while temperatures are exchanged in the usual REM. By varying the atom charges, the CREM overcomes free‐energy barriers and realizes more efficient sampling in the conformational space than the REM. Furthermore, this method requires only a smaller number of replicas because only the atom charges of solute molecules are used as exchanged parameters. We performed Coulomb replica‐exchange MD simulations of an alanine dipeptide in explicit water solvent and compared the results with those of the conventional canonical, replica exchange, and van der Waals REMs. Two force fields of AMBER parm99 and AMBER parm99SB were used. As a result, the CREM sampled all local‐minimum free‐energy states more frequently than the other methods for both force fields. Moreover, the Coulomb, van der Waals, and usual REMs were applied to a fragment of an amyloid‐β peptide (Aβ) in explicit water solvent to compare the sampling efficiency of these methods for a larger system. The CREM sampled structures of the Aβ fragment more efficiently than the other methods. We obtained β‐helix, α‐helix, 3₁₀‐helix, β‐hairpin, and β‐sheet structures as stable structures and deduced pathways of conformational transitions among these structures from a free‐energy landscape. © 2012 Wiley Periodicals, Inc.     

The temperature intervals with global exchange of replicas empirical accelerated sampling method: parameter sensitivity and extension to a complex molecular system

The recently developed "temperature intervals with global exchange of replicas" (TIGER2) algorithm is an efficient replica-exchange sampling algorithm that provides the freedom to specify the number of replicas and temperature levels independently of the size of the system and temperature range to be spanned, thus making it particularly well suited for sampling molecular systems that are considered to be too large to be sampled using conventional replica exchange methods. Although the TIGER2 method is empirical in nature, when appropriately applied it is able to provide sampling that satisfies the balance condition and closely approximates a Boltzmann-weighted ensemble of states. In this work, we evaluated the influence of factors such as temperature range, temperature spacing, replica number, and sampling cycle design on the accuracy of a TIGER2 simulation based on molecular dynamics simulations of alanine dipeptide in implicit solvent. The influence of these factors is further examined by calculating the properties of a complex system composed of the B1 immunoglobulin-binding domain of streptococcal protein G (protein G) in aqueous solution. The accuracy of a TIGER2 simulation is particularly sensitive to the maximum temperature level selected for the simulation. A method to determine the appropriate maximum temperature level to be used in a TIGER2 simulation is presented.     

Walking freely in the energy and temperature space by the modified replica exchange molecular dynamics method

Replica Exchange Molecular Dynamics (REMD) method is a powerful sampling tool in molecular simulations. Recently, we made a modification to the standard REMD method. It places some inactive replicas at different temperatures as well as the active replicas. The method completely decouples the number of the active replicas and the number of the temperature levels. In this article, we make a further modification to our previous method. It uses the inactive replicas in a different way. The inactive replicas first sample in their own knowledge‐based energy databases and then participate in the replica exchange operations in the REMD simulation. In fact, this method is a hybrid between the standard REMD method and the simulated tempering method. Using different active replicas, one can freely control the calculation quantity and the convergence speed of the simulation. To illustrate the performance of the method, we apply it to some small models. The distribution functions of the replicas in the energy space and temperature space show that the modified REMD method in this work can let the replicas walk freely in both of the two spaces. With the same number of the active replicas, the free energy surface in the simulation converges faster than the standard REMD. © 2016 Wiley Periodicals, Inc.     

Replica state exchange metadynamics for improving the convergence of free energy estimates

Metadynamics (MTD) is a powerful enhanced sampling method for systems with rugged energy landscapes. It constructs a bias potential in a predefined collective variable (CV) space to overcome barriers between metastable states. In bias‐exchange MTD (BE‐MTD), multiple replicas approximate the CV space by exchanging bias potentials (replica conditions) with the Metropolis–Hastings (MH) algorithm. We demonstrate that the replica‐exchange rates and the convergence of free energy estimates of BE‐MTD are improved by introducing the infinite swapping (IS) or the Suwa‐Todo (ST) algorithms. Conceptually, IS and ST perform transitions in a replica state space rather than exchanges in a replica condition space. To emphasize this, the proposed scheme is called the replica state exchange MTD (RSE‐MTD). Benchmarks were performed with alanine polypeptides in vacuum and water. For the systems tested in this work, there is no significant performance difference between IS and ST. © 2015 Wiley Periodicals, Inc.     

Application of replica exchange umbrella sampling to protein structure refinement of nontemplate models

We provide an assessment of a computational strategy for protein structure refinement that combines self‐guided Langevin dynamics with umbrella‐potential biasing replica exchange using the radius of gyration as a coordinate (R_g‐ReX). Eight structurally nonredundant proteins and their decoys were examined by sampling conformational space at room temperature using the CHARMM22/GBMV2 force field to generate the ensemble of structures. Two atomic statistical potentials (RWplus and DFIRE) were analyzed for structure identification and compared to the simulation force‐field potential. The results show that, while the R_g‐ReX simulations were able to sample conformational basins that were more structurally similar to the X‐ray crystallographic structures than the starting first‐order ranked decoys, the potentials failed to detect these basins from refinement. Of the three potential functions, RWplus yielded the highest accuracy for recognition of structures that refined to an average of nearly 20% increase in native contacts relative to the starting decoys. The overall performance of R_g‐ReX is compared to an earlier study of applying temperature‐based replica exchange to refine the same decoy sets and highlights the general challenge of achieving consistently the sampling and detection threshold of 70% fraction of native contacts. © 2013 Wiley Periodicals, Inc.     

How hot? Systematic convergence of the replica exchange method using multiple reservoirs

We have devised a systematic approach to converge a replica exchange molecular dynamics simulation by dividing the full temperature range into a series of higher temperature reservoirs and a finite number of lower temperature subreplicas. A defined highest temperature reservoir of equilibrium conformations is used to help converge a lower but still hot temperature subreplica, which in turn serves as the high‐temperature reservoir for the next set of lower temperature subreplicas. The process is continued until an optimal temperature reservoir is reached to converge the simulation at the target temperature. This gradual convergence of subreplicas allows for better and faster convergence at the temperature of interest and all intermediate temperatures for thermodynamic analysis, as well as optimizing the use of multiple processors. We illustrate the overall effectiveness of our multiple reservoir replica exchange strategy by comparing sampling and computational efficiency with respect to replica exchange, as well as comparing methods when converging the structural ensemble of the disordered Aβ_21‐30 peptide simulated with explicit water by comparing calculated Rotating Overhauser Effect Spectroscopy intensities to experimentally measured values. © 2009 Wiley Periodicals, Inc. J Comput Chem 31: 620–627, 2010     

Replica exchange with nonequilibrium switches: enhancing equilibrium sampling by increasing replica overlap

We describe a replica exchange strategy where trial swap configurations are generated by nonequilibrium switching simulations. By devoting simulation time to the switching simulations, one can systematically increase an effective overlap between replicas, which leads to an increased exchange acceptance rate and less correlated equilibrium samples. In this paper, we derive our method for a general class of stochastic dynamics, and discuss various strategies for enhancing replica overlap through novel dynamical schemes and prudent choices of switching protocols. We then demonstrate our method on a model system of alanine dipeptide in implicit solvent, characterizing decreases in data correlations and gains in sampling efficiency.     

Multiple scaling replica exchange for the conformational sampling of biomolecules in explicit water

A multiple scaling replica exchange method for the efficient conformational sampling of biomolecular systems in explicit solvent is presented. The method is a combination of the replica exchange with solute tempering (REST) technique and a Tsallis biasing potential. The Tsallis biasing increases the sampling efficiency, while the REST minimizes the number of replicas needed. Unbiased statistics can be obtained by reweighting of the data using a weighted histogram analysis technique. The method is illustrated by its application to a ten residue peptide in explicit water.     

Virtual‐system‐coupled adaptive umbrella sampling to compute free‐energy landscape for flexible molecular docking

A novel enhanced conformational sampling method, virtual‐system‐coupled adaptive umbrella sampling (V‐AUS), was proposed to compute 300‐K free‐energy landscape for flexible molecular docking, where a virtual degrees of freedom was introduced to control the sampling. This degree of freedom interacts with the biomolecular system. V‐AUS was applied to complex formation of two disordered amyloid‐β (Aβ_30–35) peptides in a periodic box filled by an explicit solvent. An interpeptide distance was defined as the reaction coordinate, along which sampling was enhanced. A uniform conformational distribution was obtained covering a wide interpeptide distance ranging from the bound to unbound states. The 300‐K free‐energy landscape was characterized by thermodynamically stable basins of antiparallel and parallel β‐sheet complexes and some other complex forms. Helices were frequently observed, when the two peptides contacted loosely or fluctuated freely without interpeptide contacts. We observed that V‐AUS converged to uniform distribution more effectively than conventional AUS sampling did. © 2015 Wiley Periodicals, Inc.     

Sampling enhancement for the quantum mechanical potential based molecular dynamics simulations: a general algorithm and its extension for free energy calculation on rugged energy surface

An approach is developed in the replica exchange framework to enhance conformational sampling for the quantum mechanical (QM) potential based molecular dynamics simulations. Importantly, with our enhanced sampling treatment, a decent convergence for electronic structure self-consistent-field calculation is robustly guaranteed, which is made possible in our replica exchange design by avoiding direct structure exchanges between the QM-related replicas and the activated (scaled by low scaling parameters or treated with high "effective temperatures") molecular mechanical (MM) replicas. Although the present approach represents one of the early efforts in the enhanced sampling developments specifically for quantum mechanical potentials, the QM-based simulations treated with the present technique can possess the similar sampling efficiency to the MM based simulations treated with the Hamiltonian replica exchange method (HREM). In the present paper, by combining this sampling method with one of our recent developments (the dual-topology alchemical HREM approach), we also introduce a method for the sampling enhanced QM-based free energy calculations.     

Velocity-scaling optimized replica exchange molecular dynamics of proteins in a hybrid explicit/implicit solvent

We propose a scheme for replica exchange molecular dynamics of proteins in explicit solvent that minimizes the number of required replicas using velocity rescaling. Our approach relies on a hybrid method where the protein evolves at each temperature in an explicit solvent, but replica exchange moves utilize an implicit solvent term. The two terms are coupled through the velocity rescaling. We test the efficiency of this approach for a common test case, the trp-cage protein.     

Hamiltonian replica‐exchange simulations with adaptive biasing of peptide backbone and side chain dihedral angles

A Hamiltonian Replica‐Exchange Molecular Dynamics (REMD) simulation method has been developed that employs a two‐dimensional backbone and one‐dimensional side chain biasing potential specifically to promote conformational transitions in peptides. To exploit the replica framework optimally, the level of the biasing potential in each replica was appropriately adapted during the simulations. This resulted in both high exchange rates between neighboring replicas and improved occupancy/flow of all conformers in each replica. The performance of the approach was tested on several peptide and protein systems and compared with regular MD simulations and previous REMD studies. Improved sampling of relevant conformational states was observed for unrestrained protein and peptide folding simulations as well as for refinement of a loop structure with restricted mobility of loop flanking protein regions. © 2013 Wiley Periodicals, Inc.     

Combining the biased and unbiased sampling strategy into one convenient free energy calculation method

Constructing a free energy landscape for a large molecule is difficult. One has to use either a high temperature or a strong driving force to enhance the sampling on the free energy barriers. In this work, we propose a mixed method that combines these two kinds of acceleration strategies into one simulation. First, it applies an adaptive biasing potential to some replicas of the molecule. These replicas are particularly accelerated in a collective variable space. Second, it places some unbiased and exchangeable replicas at various temperature levels. These replicas generate unbiased sampling data in the canonical ensemble. To improve the sampling efficiency, biased replicas transfer their state variables to the unbiased replicas after equilibrium by Monte Carlo trial moves. In comparison to previous integrated methods, it is more convenient for users. It does not need an initial reference biasing potential to guide the sampling of the molecule. And it is also unnecessary to insert many replicas for the requirement of passing the free energy barriers. The free energy calculation is accomplished in a single stage. It samples the data as fast as a biased simulation and it processes the data as simple as an unbiased simulation. The method provides a minimalist approach to the construction of the free energy landscape. © 2019 Wiley Periodicals, Inc.     

Modified replica exchange simulation methods for local structure refinement

Parallel tempering, also known as replica exchange molecular dynamics (REMD), has recently been successfully used to study the structure and thermodynamic properties of biomolecules such as peptides and small proteins. For large systems, however, applying REMD can be costly since the number of replicas needed increases as the square root of the number of degrees of freedom in the system. Often, enhanced sampling is only needed for a subset of atoms, such as a loop region of a large protein or a small ligand binding to a receptor. In such applications, it is often reasonable to assume a weak dependence of the structure of the larger region on the instantaneous conformation of the smaller region of interest. For these cases, we derived two variant replica exchange methods, partial replica exchange molecular dynamics (PREMD) and local replica exchange molecular dynamics (LREMD). The Hamiltonian for the system is separated, with replica exchange carried out only for terms involving the subsystem of interest while the remainder of the system is maintained at a single temperature. The number of replicas required for efficient exchange thus depends on the number of degrees of freedom in the fragment needing refinement rather than on the size of the full system. The method can be applied to much larger systems than was previously practical. This also provides a means to preserve the integrity of the structure outside the refinement region without introduction of restraints. LREMD takes this weak coupling approximation a step further, employing only a single representation of the large fragment that simultaneously interacts with all of the replicas of the subsystem of interest. This is obtained by combining replica exchange with the locally enhanced sampling approximation (LES), reducing the computational expense of replica exchange simulations to near that of a single standard molecular dynamics (MD) simulation. Use of LREMD also permits the use of LES without requiring the specification of a single temperature, a known difficulty for standard LES simulations. We tested these two methods on the loop region of an RNA hairpin model system and find significant advantages over standard MD and REMD simulations.     

Coupling of replica exchange simulations to a non-Boltzmann structure reservoir

Computing converged ensemble properties remains challenging for large biomolecules. Replica exchange molecular dynamics (REMD) can significantly increase the efficiency of conformational sampling by using high temperatures to escape kinetic traps. Several groups, including ours, introduced the idea of coupling replica exchange to a pre-converged, Boltzmann-populated reservoir, usually at a temperature higher than that of the highest temperature replica. This procedure reduces computational cost because the long simulation times needed for extensive sampling are only carried out for a single temperature. However, a weakness of the approach is that the Boltzmann-weighted reservoir can still be difficult to generate. We now present the idea of employing a non-Boltzmann reservoir, whose structures can be generated through more efficient conformational sampling methods. We demonstrate that the approach is rigorous and derive a correct statistical mechanical exchange criterion between the reservoir and the replicas that drives Boltzmann-weighted probabilities for the replicas. We test this approach on the trpzip2 peptide and demonstrate that the resulting thermal stability profile is essentially indistinguishable from that obtained using very long (>100 ns) standard REMD simulations. The convergence of this reservoir-aided REMD is significantly faster than for regular REMD. Furthermore, we demonstrate that modification of the exchange criterion is essential; REMD simulations using a standard exchange function with the non-Boltzmann reservoir produced incorrect results.     

Free‐energy calculations of residue mutations in a tripeptide using various methods to overcome inefficient sampling

Previous free‐energy calculations have shown that the seemingly simple transformation of the tripeptide KXK to KGK in water holds some unobvious challenges concerning the convergence of the forward and backward thermodynamic integration processes (i.e., hysteresis). In the current study, the central residue X was either alanine, serine, glutamic acid, lysine, phenylalanine, or tyrosine. Interestingly, the transformation from alanine to glycine yielded the highest hysteresis in relation to the extent of the chemical change of the side chain. The reason for that could be attributed to poor sampling of φ₂/ψ₂ dihedral angles along the transformation. Altering the nature of alanine's C_β atom drastically improved the sampling and at the same time led to the identification of high energy barriers as cause for it. Consequently, simple strategies to overcome these barriers are to increase simulation time (computationally expensive) or to use enhanced sampling techniques such as Hamiltonian replica exchange molecular dynamics and one‐step perturbation. © 2016 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Fast,metadynamics‐based method for prediction of the stereochemistry‐dependent relative free energies of ligand–receptor interactions

The computational approach applicable for the molecular dynamics (MD)‐based techniques is proposed to predict the ligand–protein binding affinities dependent on the ligand stereochemistry. All possible stereoconfigurations are expressed in terms of one set of force‐field parameters [stereoconfiguration‐independent potential (SIP)], which allows for calculating all relative free energies by only single simulation. SIP can be used for studying diverse, stereoconfiguration‐dependent phenomena by means of various computational techniques of enhanced sampling. The method has been successfully tested on the β₂‐adrenergic receptor (β₂‐AR) binding the four fenoterol stereoisomers by both metadynamics simulations and replica‐exchange MD. Both the methods gave very similar results, fully confirming the presence of stereoselective effects in the fenoterol‐β₂‐AR interactions. However, the metadynamics‐based approach offered much better efficiency of sampling which allows for significant reduction of the unphysical region in SIP. © 2014 Wiley Periodicals, Inc.     

Hydrophobic aided replica exchange: an efficient algorithm for protein folding in explicit solvent

A hydrophobic aided replica exchange method (HAREM) is introduced to accelerate the simulation of all-atom protein folding in explicit solvent. This method is based on exaggerating the hydrophobic effect of various protein amino acids in water by attenuating the protein-water attractive interactions (mimicking the Chaperon effect) while leaving other interactions among protein atoms and water molecules unchanged. The method is applied to a small representative protein, the alpha-helix 3K(I), and it is found that the HAREM method successfully folds the protein within 4 ns, while the regular replica exchange method does not fold the same protein within 5 ns, even with many more replicas.     

Scalable free energy calculation of proteins via multiscale essential sampling

A multiscale simulation method, "multiscale essential sampling (MSES)," is proposed for calculating free energy surface of proteins in a sizable dimensional space with good scalability. In MSES, the configurational sampling of a full-dimensional model is enhanced by coupling with the accelerated dynamics of the essential degrees of freedom. Applying the Hamiltonian exchange method to MSES can remove the biasing potential from the coupling term, deriving the free energy surface of the essential degrees of freedom. The form of the coupling term ensures good scalability in the Hamiltonian exchange. As a test application, the free energy surface of the folding process of a miniprotein, chignolin, was calculated in the continuum solvent model. Results agreed with the free energy surface derived from the multicanonical simulation. Significantly improved scalability with the MSES method was clearly shown in the free energy calculation of chignolin in explicit solvent, which was achieved without increasing the number of replicas in the Hamiltonian exchange.