An improved replica-exchange sampling method: temperature intervals with global energy reassignment

In a molecular dynamics (MD) simulation, representative sampling over the entire phase space is desired to obtain an accurate canonical distribution at a given temperature. For large molecules, such as proteins, this is problematic because systems tend to become trapped in local energy minima. The extensively used replica-exchange molecular dynamics (REMD) simulation technique overcomes this kinetic-trapping problem by allowing Boltzmann-weighted configuration exchange processes to occur between numerous thermally adjacent and compositionally identical simulations that are thermostated at sequentially higher temperatures. While the REMD method provides much better sampling than conventional MD, there are two substantial difficulties that are inherent in its application: (1) the large number of replicas that must be used to span a designated temperature range and (2) the subsequent long time required for configurations sampled at high temperatures to exchange down for potential inclusion within the low-temperature ensemble of interest. In this work, a new method based on temperature intervals with global energy reassignment (TIGER) is presented that overcomes both of these problems. A TIGER simulation is conducted as a series of short heating-sampling-quenching cycles. At the end of each cycle, the potential energies of all replicas are simultaneously compared at the same temperature using a Metropolis sampling method and then globally reassigned to the designated temperature levels. TIGER is compared with regular MD and REMD methods for the alanine dipeptide in water. The results indicate that TIGER increases sampling efficiency while substantially reducing the number of central processing units required for a comparable conventional REMD simulation.     

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.     

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.     

A convective replica‐exchange method for sampling new energy basins

Replica‐exchange is a powerful simulation method for sampling the basins of a rugged energy landscape. The replica‐exchange method's sampling is efficient because it allows replicas to perform round trips in temperature space, thereby visiting both low and high temperatures in the same simulation. However, replicas have a diffusive walk in temperature space, and the round trip rate decreases significantly with the system size. These drawbacks make convergence of the simulation even more difficult than it already is when bigger systems are tackled. Here, we present a simple modification of the exchange method. In this method, one of the replicas steadily raises or lowers its temperature. We tested the convective replica‐exchange method on three systems of varying complexity: the alanine dipeptide in implicit solvent, the GB1 β‐hairpin in explicit solvent and the Aβ_25–35 homotrimer in a coarse grained representation. For the highly frustrated Aβ_25–35 homotrimer, the proposed “convective” replica‐exchange method is twice as fast as the standard method. It discovered 24 out of 27 free‐energy basins in less than 500 ns. It also prevented the formation of groups of replicas that usually form on either side of an exchange bottleneck, leading to a more efficient sampling of new energy basins than in the standard method. © 2012 Wiley Periodicals, Inc.     

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.     

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     

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.     

Temperature weighted histogram analysis method, replica exchange, and transition paths

We analyzed the data from a replica exchange molecular dynamics simulation using the weighted histogram analysis method to combine data from all of the temperature replicas (T-WHAM) to obtain the room-temperature potential of mean force of the G-peptide (the C-terminal beta-hairpin of the B1 domain of protein G) in regions of conformational space not sampled at room temperature. We were able to determine the potential of mean force in the transition region between a minor alpha-helical population and the major beta-hairpin population and identify a possible transition path between them along which the peptide retains a significant amount of secondary structure. This observation provides new insights into a possible mechanism of formation of beta-sheet secondary structures in proteins. We developed a novel Bayesian statistical uncertainty estimation method for any quantity derived from WHAM and used it to validate the calculated potential of mean force. The feasibility of estimating regions of the potential of mean force with unfavorable free energy at room temperature by T-WHAM analysis of replica exchange simulations was further tested on a system that can be solved analytically and presented some of the same challenges found in more complex chemical systems.     

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.     

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.     

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.     

Reversible folding simulation by hybrid Hamiltonian replica exchange

Reversible foldings of a beta-hairpin peptide, chignolin, by recently invented hybrid Hamiltonian replica exchange molecular dynamics simulations based on Poisson-Boltzmann model in explicit water are demonstrated. Initiated from extended structures the peptide folded and unfolded a couple of times in seven out of eight replica trajectories during 100 nanoseconds simulation. The folded states have the lowest all-atom root mean squared deviation of 1.3 A with respect to the NMR structures. At T=300 K the occurrence of folded states was converged to 62% during 80 ns simulation which agrees well with experimental data. Especially, a detailed structural evolution map was constructed based on 800,000 structural snapshots and from where a unique folding doorway emerges. Compared with 130 ns standard replica exchange simulation using 24 replicas on the same system, the hybrid Hamiltonian replica exchange molecular dynamics simulation presents consistent results.     

The Thermodynamics of Folding of a β Hairpin Peptide Probed Through Replica Exchange Molecular Dynamics Simulations

Peptides that possess a well defined native state are ideal model systems to study the folding of proteins. They possess many of the complexities of larger proteins, yet their small size renders their study computationally tractable. Recent advances in sampling techniques, including replica exchange molecular dynamics, now permit a full characterization of the thermodynamics of folding of small peptides. These simulations not only yield insight into the folding of larger proteins, but equally importantly, they allow, through comparison with experiment, an objective test of the accuracy of force fields, water models and of different numerical schemes for dealing with electrostatic interactions. In this account, we present a molecular dynamics simulation of a small β-hairpin peptide using the replica exchange algorithm and illustrate how this enhanced sampling scheme enables a thorough characterization of the native and unfolded states, and sheds new light into its folding mechanism.     

Replica sub-permutation method for molecular dynamics and monte carlo simulations

We propose an improvement of the replica-exchange and replica-permutation methods, which we call the replica sub-permutation method (RSPM). Instead of considering all permutations, this method uses a new algorithm referred to as sub-permutation to perform parameter transition. The RSPM succeeds in reducing the number of combinations between replicas and parameters without the loss of sampling efficiency. For comparison, we applied the replica sub-permutation, replica-permutation, and replica-exchange methods to a β-hairpin mini protein, chignolin, in explicit water. We calculated the transition ratio and number of tunneling events in the parameter space, the number of folding–unfolding events, the autocorrelation function, and the autocorrelation time as measures of sampling efficiency. The results indicate that among the three methods, the proposed RSPM is the most efficient in both parameter and conformational spaces. © 2019 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.     

Hamiltonian replica‐permutation method and its applications to an alanine dipeptide and amyloid‐β(29–42) peptides

We propose the Hamiltonian replica‐permutation method (RPM) (or multidimensional RPM) for molecular dynamics and Monte Carlo simulations, in which parameters in the Hamiltonian are permuted among more than two replicas with the Suwa‐Todo algorithm. We apply the Coulomb RPM, which is one of realization of the Hamiltonian RPM, to an alanine dipeptide and to two amyloid‐β(29–42) molecules. The Hamiltonian RPM realizes more efficient sampling than the Hamiltonian replica‐exchange method. We illustrate the protein misfolding funnel of amyloid‐β(29–42) and reveal its dimerization pathways. © 2013 Wiley Periodicals, Inc.     

Adaptively biased molecular dynamics for free energy calculations

We present an adaptively biased molecular dynamics (ABMD) method for the computation of the free energy surface of a reaction coordinate using nonequilibrium dynamics. The ABMD method belongs to the general category of umbrella sampling methods with an evolving biasing potential and is inspired by the metadynamics method. The ABMD method has several useful features, including a small number of control parameters and an O(t) numerical cost with molecular dynamics time t. The ABMD method naturally allows for extensions based on multiple walkers and replica exchange, where different replicas can have different temperatures and/or collective variables. This is beneficial not only in terms of the speed and accuracy of a calculation, but also in terms of the amount of useful information that may be obtained from a given simulation. The workings of the ABMD method are illustrated via a study of the folding of the Ace-GGPGGG-Nme peptide in a gaseous and solvated environment.     

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.     

Hamiltonian replica exchange molecular dynamics using soft-core interactions

To overcome the problem of insufficient conformational sampling within biomolecular simulations, we have developed a novel Hamiltonian replica exchange molecular dynamics (H-REMD) scheme that uses soft-core interactions between those parts of the system that contribute most to high energy barriers. The advantage of this approach over other H-REMD schemes is the possibility to use a relatively small number of replicas with locally larger differences between the individual Hamiltonians. Because soft-core potentials are almost the same as regular ones at longer distances, most of the interactions between atoms of perturbed parts will only be slightly changed. Rather, the strong repulsion between atoms that are close in space, which in many cases results in high energy barriers, is weakened within higher replicas of our proposed scheme. In addition to the soft-core interactions, we proposed to include multiple replicas using the same Hamiltonian/level of softness. We have tested the new protocol on the GTP and 8-Br-GTP molecules, which are known to have high energy barriers between the anti and syn conformation of the base with respect to the sugar moiety. During two 25 ns MD simulations of both systems the transition from the more stable to the less stable (but still experimentally observed) conformation is not seen at all. Also temperature REMD over 50 replicas for 1 ns did not show any transition at room temperature. On the other hand, more than 20 of such transitions are observed in H-REMD using six replicas (at three different Hamiltonians) during 6.8 ns per replica for GTP and 12 replicas (at six different Hamiltonians) during 8.7 ns per replica for 8-Br-GTP. The large increase in sampling efficiency was obtained from an optimized H-REMD scheme involving soft-core potentials, with multiple simulations using the same level of softness. The optimization of the scheme was performed by fast mimicking [J. Hritz and C. Oostenbrink, J. Chem. Phys. 127, 204104 (2007)].     

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.