The incomplete beta function law for parallel tempering sampling of classical canonical systems

We show that the acceptance probability for swaps in the parallel tempering Monte Carlo method for classical canonical systems is given by a universal function that depends on the average statistical fluctuations of the potential and on the ratio of the temperatures. The law, called the incomplete beta function law, is valid in the limit that the two temperatures involved in swaps are close to one another. An empirical version of the law, which involves the heat capacity of the system, is developed and tested on a Lennard-Jones cluster. We argue that the best initial guess for the distribution of intermediate temperatures for parallel tempering is a geometric progression and we also propose a technique for the computation of optimal temperature schedules. Finally, we demonstrate that the swap efficiency of the parallel tempering method for condensed-phase systems decreases naturally to zero at least as fast as the inverse square root of the dimensionality of the physical system.     

Optimized parallel tempering simulations of proteins

We apply a recently developed adaptive algorithm that systematically improves the efficiency of parallel tempering or replica exchange methods in the numerical simulation of small proteins. Feedback iterations allow us to identify an optimal set of temperatures/replicas which are found to concentrate at the bottlenecks of the simulations. A measure of convergence for the equilibration of the parallel tempering algorithm is discussed. We test our algorithm by simulating the 36-residue villin headpiece subdomain HP-36 where we find a lowest-energy configuration with a root-mean-square deviation of less than 4 A to the experimentally determined structure.     

A constant entropy increase model for the selection of parallel tempering ensembles

The present paper explores a simple approach to the question of parallel tempering temperature selection. We argue that to optimize the performance of parallel tempering it is reasonable to require that the increase in entropy between successive temperatures be uniform over the entire ensemble. An estimate of the system's heat capacity, obtained either from experiment, a preliminary simulation, or a suitable physical model, thus provides a means for generating the desired tempering ensemble. Applications to the two-dimensional Ising problem indicate that the resulting method is effective, simple to implement, and robust with respect to its sensitivity to the quality of the underlying heat capacity model.     

Equilibrium sampling of self-associating polymer solutions: a parallel selective tempering approach

We present a novel simulation algorithm based on tempering a fraction of relaxation-limiting interactions to accelerate the process of obtaining uncorrelated equilibrium configurations of self-associating polymer solutions. This approach consists of tempering (turning off) the attractive interactions for a fraction of self-associating groups determined by a biasing field h. A number of independent configurations (replicas) with overlapping Hamiltonian distributions in the expanded (NVTh) ensemble with constant NVT but different biasing fields, forming a chain of Hamiltonians, were simulated in parallel with occasional attempts to exchange the replicas associated with adjacent fields. Each field had an associated distribution of tempered interactions, average fraction of tempered interactions, and structural decorrelation time. Tempering parameters (number of replicas, fields, and exchange frequencies) were chosen to obtain the highest efficiency in sampling equilibrium configurations of a self-association polymer solution based on short serial simulation runs and a statistical model. Depending on the strength of the relaxation-limiting interactions, system size, and thermodynamic conditions, the algorithm can be orders of magnitude more efficient than conventional canonical simulation and is superior to conventional temperature parallel tempering.     

Ingenuity in performing replica permutation: How to order the state labels for improving sampling efficiency

In the replica-permutation method, an advanced version of the replica-exchange method, all combinations of replicas and parameters are considered for parameter permutation, and a list of all the combinations is prepared. Here, we report that the temperature transition probability depends on how the list is created, especially in replica permutation with solute tempering (RPST). We found that the transition probabilities decrease at large replica indices when the combinations are sequentially assigned to the state labels as in the originally proposed list. To solve this problem, we propose to modify the list by randomly assigning the combinations to the state labels. We performed molecular dynamics simulations of amyloid-β(16–22) peptides using RPST with the “randomly assigned” list (RPST-RA) and RPST with the “sequentially assigned” list (RPST-SA). The results show the decreases in the transition probabilities in RPST-SA are eliminated, and the sampling efficiency is improved in RPST-RA.     

Serial replica exchange

Parallel tempering (or the replica exchange method (REM)) is a powerful method for speeding up the sampling of conformational states of systems with rough energy landscapes, like proteins, where stable conformational states can be separated by large energy barriers. The usual implementation of the REM is performed on local computer clusters (or parallel processors) where the different replicas must be run synchronously. Here, we present serial replica exchange (SREM), a method that is equivalent to the standard REM in terms of efficiency yet runs asynchronously on a distributed network of computers. A second advantage is the method's greatly enhanced fault tolerance, which enables the study of biological systems on worldwide distributed computing environments, such as Folding@Home. For proof of concept, we apply the SREM to a single alanine dipeptide molecule in explicit water. We show that the SREM reproduces the thermodynamic and structural properties determined by the REM.     

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

The widespread popularity of replica exchange and expanded ensemble algorithms for simulating complex molecular systems in chemistry and biophysics has generated much interest in discovering new ways to enhance the phase space mixing of these protocols in order to improve sampling of uncorrelated configurations. Here, we demonstrate how both of these classes of algorithms can be considered as special cases of Gibbs sampling within a Markov chain Monte Carlo framework. Gibbs sampling is a well-studied scheme in the field of statistical inference in which different random variables are alternately updated from conditional distributions. While the update of the conformational degrees of freedom by Metropolis Monte Carlo or molecular dynamics unavoidably generates correlated samples, we show how judicious updating of the thermodynamic state indices--corresponding to thermodynamic parameters such as temperature or alchemical coupling variables--can substantially increase mixing while still sampling from the desired distributions. We show how state update methods in common use can lead to suboptimal mixing, and present some simple, inexpensive alternatives that can increase mixing of the overall Markov chain, reducing simulation times necessary to obtain estimates of the desired precision. These improved schemes are demonstrated for several common applications, including an alchemical expanded ensemble simulation, parallel tempering, and multidimensional replica exchange umbrella sampling.     

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.     

Monte Carlo methods for small molecule high-throughput experimentation

By analogy with Monte Carlo algorithms, we propose new strategies for design and redesign of small molecule libraries in high-throughput experimentation, or combinatorial chemistry. Several Monte Carlo methods are examined, including Metropolis, three types of biased schemes, and composite moves that include swapping or parallel tempering. Among them, the biased Monte Carlo schemes exhibit particularly high efficiency in locating optimal compounds. The Monte Carlo strategies are compared to a genetic algorithm approach. Although the best compounds identified by the genetic algorithm are comparable to those from the better Monte Carlo schemes, the diversity of favorable compounds identified is reduced by roughly 60%.     

All-exchanges parallel tempering

An alternative exchange strategy for parallel tempering simulations is introduced. Instead of attempting to swap configurations between two randomly chosen but adjacent replicas, the acceptance probabilities of all possible swap moves are calculated a priori. One specific swap move is then selected according to its probability and enforced. The efficiency of the method is illustrated first on the case of two Lennard-Jones (LJ) clusters containing 13 and 31 atoms, respectively. The convergence of the caloric curve is seen to be at least twice as fast as in conventional parallel tempering simulations, especially for the difficult case of LJ31. Further evidence for an improved efficiency is reported on the ergodic measure introduced by Mountain and Thirumalai [J. Phys. Chem. 93, 6975 (1989)], calculated here for LJ13 close to the melting point. Finally, tests on two simple spin systems indicate that the method should be particularly useful when a limited number of replicas are available.     

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.     

Hierarchical parallelization of divide-and-conquer density functional tight-binding molecular dynamics and metadynamics simulations

Massively parallel divide-and-conquer density functional tight-binding (DC-DFTB) molecular dynamics and metadynamics simulations are efficient approaches for describing various chemical reactions and dynamic processes of large complex systems via quantum mechanics. In this study, DC-DFTB simulations were combined with multi-replica techniques. Specifically, multiple walkers metadynamics, replica exchange molecular dynamics, and parallel tempering metadynamics methods were implemented hierarchically into the in-house Dcdftbmd program. Test simulations in an aqueous phase of the internal rotation of formamide and conformational changes of dialanine showed that the newly developed extensions increase the sampling efficiency and the exploration capabilities in DC-DFTB configuration space.     

Parallel replica dynamics with a heterogeneous distribution of barriers: application to n-hexadecane pyrolysis

Parallel replica dynamics simulation methods appropriate for the simulation of chemical reactions in molecular systems with many conformational degrees of freedom have been developed and applied to study the microsecond-scale pyrolysis of n-hexadecane in the temperature range of 2100-2500 K. The algorithm uses a transition detection scheme that is based on molecular topology, rather than energetic basins. This algorithm allows efficient parallelization of small systems even when using more processors than particles (in contrast to more traditional parallelization algorithms), and even when there are frequent conformational transitions (in contrast to previous implementations of the parallel replica algorithm). The parallel efficiency for pyrolysis initiation reactions was over 90% on 61 processors for this 50-atom system. The parallel replica dynamics technique results in reaction probabilities that are statistically indistinguishable from those obtained from direct molecular dynamics, under conditions where both are feasible, but allows simulations at temperatures as much as 1000 K lower than direct molecular dynamics simulations. The rate of initiation displayed Arrhenius behavior over the entire temperature range, with an activation energy and frequency factor of E(a) = 79.7 kcal/mol and log A/s(-1) = 14.8, respectively, in reasonable agreement with experiment and empirical kinetic models. Several interesting unimolecular reaction mechanisms were observed in simulations of the chain propagation reactions above 2000 K, which are not included in most coarse-grained kinetic models. More studies are needed in order to determine whether these mechanisms are experimentally relevant, or specific to the potential energy surface used.     

Replica exchanging self-guided Langevin dynamics for efficient and accurate conformational sampling

This work presents a replica exchanging self-guided Langevin dynamics (RXSGLD) simulation method for efficient conformational searching and sampling. Unlike temperature-based replica exchanging simulations, which use high temperatures to accelerate conformational motion, this method uses self-guided Langevin dynamics (SGLD) to enhance conformational searching without the need to elevate temperatures. A RXSGLD simulation includes a series of SGLD simulations, with simulation conditions differing in the guiding effect and∕or temperature. These simulation conditions are called stages and the base stage is one with no guiding effect. Replicas of a simulation system are simulated at the stages and are exchanged according to the replica exchanging probability derived from the SGLD partition function. Because SGLD causes less perturbation on conformational distribution than high temperatures, exchanges between SGLD stages have much higher probabilities than those between different temperatures. Therefore, RXSGLD simulations have higher conformational searching ability than temperature based replica exchange simulations. Through three example systems, we demonstrate that RXSGLD can generate target canonical ensemble distribution at the base stage and achieve accelerated conformational searching. Especially for large systems, RXSGLD has remarkable advantages in terms of replica exchange efficiency, conformational searching ability, and system size extensiveness.     

Replica exchange with solute tempering: efficiency in large scale systems

We apply the recently developed replica exchange with solute tempering (REST) to three large solvated peptide systems: an alpha-helix, a beta-hairpin, and a TrpCage, with these peptides defined as the "central group". We find that our original implementation of REST is not always more efficient than the replica exchange method (REM). Specifically, we find that exchanges between folded (F) and unfolded (U) conformations with vastly different structural energies are greatly reduced by the nonappearance of the water self-interaction energy in the replica exchange acceptance probabilities. REST, however, is expected to remain useful for a large class of systems for which the energy gap between the two states is not large, such as weakly bound protein-ligand complexes. Alternatively, a shell of water molecules can be incorporated into the central group, as discussed in the original paper.     

Two‐dimensional replica‐exchange method for predicting protein–ligand binding structures

We have developed a two‐dimensional replica‐exchange method for the prediction of protein–ligand binding structures. The first dimension is the umbrella sampling along the reaction coordinate, which is the distance between a protein binding pocket and a ligand. The second dimension is the solute tempering, in which the interaction between a ligand and a protein and water is weakened. The second dimension is introduced to make a ligand follow the umbrella potential more easily and enhance the binding events, which should improve the sampling efficiency. As test cases, we applied our method to two protein‐ligand complex systems (MDM2 and HSP 90‐alpha). Starting from the configuration in which the protein and the ligand are far away from each other in each system, our method predicted the ligand binding structures in excellent agreement with the experimental data from Protein Data Bank much faster with the improved sampling efficiency than the replica‐exchange umbrella sampling method that we have previously developed. © 2013 Wiley Periodicals, Inc.     

Combining smart darting with parallel tempering using Eckart space: application to Lennard-Jones clusters

The smart-darting algorithm is a Monte Carlo based simulation method used to overcome quasiergodicity problems associated with disconnected regions of configurations space separated by high energy barriers. As originally implemented, the smart-darting method works well for clusters at low temperatures with the angular momentum restricted to zero and where there are no transitions to permutational isomers. If the rotational motion of the clusters is unrestricted or if permutational isomerization becomes important, the acceptance probability of darting moves in the original implementation of the method becomes vanishingly small. In this work the smart-darting algorithm is combined with the parallel tempering method in a manner where both rotational motion and permutational isomerization events are important. To enable the combination of parallel tempering with smart darting so that the smart-darting moves have a reasonable acceptance probability, the original algorithm is modified by using a restricted space for the smart-darting moves. The restricted space uses a body-fixed coordinate system first introduced by Eckart, and moves in this Eckart space are coupled with local moves in the full 3N-dimensional space. The modified smart-darting method is applied to the calculation of the heat capacity of a seven-atom Lennard-Jones cluster. The smart-darting moves yield significant improvement in the statistical fluctuations of the calculated heat capacity in the region of temperatures where the system isomerizes. When the modified smart-darting algorithm is combined with parallel tempering, the statistical fluctuations of the heat capacity of a seven-atom Lennard-Jones cluster using the combined method are smaller than parallel tempering when used alone.     

On easy implementation of a variant of the replica exchange with solute tempering in GROMACS

To reduce the number of replicas required in the conventional replica exchange method for huge systems, recently the replica exchange with solute tempering (REST) method was proposed. Here we showed that a variant of REST realized by rescaling the force-field parameters can be performed with GROMACS 4 without changing the code. We tested the variant REST for alanine dipeptide and an N-terminal peptide from p53 confirming its performance nearly equal to the original REST.     

Replica-exchange extensions of simulated tempering method

In this paper we consider combinations of two well-known generalized-ensemble algorithms, namely, simulated tempering and replica-exchange method. We discuss two examples of such combinations. One is the replica-exchange simulated tempering and the other is the simulated tempering replica-exchange method. In the former method, a short replica-exchange simulation is first performed and the simulated tempering weight factor is obtained by the multiple-histogram reweighting techniques. This process of simulated tempering weight factor determination is faster and simpler than that in the usual iterative process. A long simulated tempering production run is then performed with this weight factor. The latter method is a further extension of the former in which a simulated tempering replica-exchange simulation is performed with a small number of replicas. These algorithms are particularly useful for studying frustrated systems with rough energy landscape. We give the formulations of these two methods in detail and demonstrate their effectiveness taking the example of the system of a 17-residue helical peptide.     

Protein structure prediction by tempering spatial constraints

Summary The probability to predict correctly a protein structure can be enhanced through introduction of spatial constraints – either from NMR experiments or from homologous structures. However, the additional constraints lead often to new local energy minima and worse sampling efficiency in simulations. In this work, we present a new parallel tempering variant that alleviates the energy barriers resulting from spatial constraints and therefore yields to an enhanced sampling in structure prediction simulations.