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 temperature predictor for parallel tempering simulations

An algorithm is proposed that generates a set of temperatures for use in parallel tempering simulations (also known as temperature-replica exchange molecular dynamics simulations) of proteins to obtain a desired exchange probability Pdes. The input consists of the number of protein atoms and water molecules in the system, information about the use of constraints and virtual sites and the lower temperature limits. The temperatures generated yield probabilities which are very close to Pdes (correlation 97%), independent of force field and over a wide temperature range. To facilitate its use, the algorithm has been implemented as a web server at .     

Optimal use of data in parallel tempering simulations for the construction of discrete-state Markov models of biomolecular dynamics

Parallel tempering (PT) molecular dynamics simulations have been extensively investigated as a means of efficient sampling of the configurations of biomolecular systems. Recent work has demonstrated how the short physical trajectories generated in PT simulations of biomolecules can be used to construct the Markov models describing biomolecular dynamics at each simulated temperature. While this approach describes the temperature-dependent kinetics, it does not make optimal use of all available PT data, instead estimating the rates at a given temperature using only data from that temperature. This can be problematic, as some relevant transitions or states may not be sufficiently sampled at the temperature of interest, but might be readily sampled at nearby temperatures. Further, the comparison of temperature-dependent properties can suffer from the false assumption that data collected from different temperatures are uncorrelated. We propose here a strategy in which, by a simple modification of the PT protocol, the harvested trajectories can be reweighted, permitting data from all temperatures to contribute to the estimated kinetic model. The method reduces the statistical uncertainty in the kinetic model relative to the single temperature approach and provides estimates of transition probabilities even for transitions not observed at the temperature of interest. Further, the method allows the kinetics to be estimated at temperatures other than those at which simulations were run. We illustrate this method by applying it to the generation of a Markov model of the conformational dynamics of the solvated terminally blocked alanine peptide.     

On the efficiency of exchange in parallel tempering monte carlo simulations

We introduce the concept of effective fraction, defined as the expected probability that a configuration from the lowest index replica successfully reaches the highest index replica during a replica exchange Monte Carlo simulation. We then argue that the effective fraction represents an adequate measure of the quality of the sampling technique, as far as swapping is concerned. Under the hypothesis that the correlation between successive exchanges is negligible, we propose a technique for the computation of the effective fraction, a technique that relies solely on the values of the acceptance probabilities obtained at the end of the simulation. The effective fraction is then utilized for the study of the efficiency of a popular swapping scheme in the context of parallel tempering in the canonical ensemble. For large dimensional oscillators, we show that the swapping probability that minimizes the computational effort is 38.74%. By studying the parallel tempering swapping efficiency for a 13-atom Lennard-Jones cluster, we argue that the value of 38.74% remains roughly the optimal probability for most systems with continuous distributions that are likely to be encountered in practice.     

On the convergence of parallel tempering Monte Carlo simulations of LJ38

The convergence of parallel tempering Monte Carlo simulations of the 38-atom Lennard-Jones cluster starting from the Oh global minimum and from the C(5v) second-lowest-energy minimum is investigated. It is found that achieving convergence is appreciably more difficult, particularly at temperatures in the vicinity of the Oh --> C(5v) transformation when starting from the C(5v) structure. A strategy combining the Tsallis generalized ensemble and the parallel tempering algorithm is implemented and used to improve the convergence of the simulations in the vicinity of the Oh --> C(5v) transformation.     

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.     

Comparing different protocols of temperature selection in the parallel tempering method

Parallel tempering Monte Carlo simulations have been applied to a variety of systems presenting rugged free-energy landscapes. Despite this, its efficiency depends strongly on the temperature set. With this query in mind, we present a comparative study among different temperature selection schemes in three lattice-gas models. We focus our attention in the constant entropy method (CEM), proposed by Sabo et al. In the CEM, the temperature is chosen by the fixed difference of entropy between adjacent replicas. We consider a method to determine the entropy which avoids numerical integrations of the specific heat and other thermodynamic quantities. Different analyses for first- and second-order phase transitions have been undertaken, revealing that the CEM may be an useful criterion for selecting the temperatures in the parallel tempering.     

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.     

Structure of aqueous proline via parallel tempering molecular dynamics and neutron diffraction

The structure of aqueous L-proline amino acid has been the subject of much debate centering on the validity of various proposed models, differing widely in the extent to which local and long-range correlations are present. Here, aqueous proline is investigated by atomistic, replica exchange molecular dynamics simulations, and the results are compared to neutron diffraction and small angle neutron scattering (SANS) data, which have been reported recently (McLain, S.; Soper, A.; Terry, A.; Watts, A. J. Phys. Chem. B 2007, 111, 4568). Comparisons between neutron experiments and simulation are made via the static structure factor S(Q) which is measured and computed from several systems with different H/D isotopic compositions at a concentration of 1:20 molar ratio. Several different empirical water models (TIP3P, TIP4P, and SPC/E) in conjunction with the CHARMM22 force field are investigated. Agreement between experiment and simulation is reasonably good across the entire Q range although there are significant model-dependent variations in some cases. In general, agreement is improved slightly upon application of approximate quantum corrections obtained from gas-phase path integral simulations. Dimers and short oligomeric chains formed by hydrogen bonds (frequently bifurcated) coexist with apolar (hydrophobic) contacts. These emerge as the dominant local motifs in the mixture. Evidence for long-range association is more equivocal: No long-range structures form spontaneously in the MD simulations, and no obvious low-Q signature is seen in the SANS data. Moreover, associations introduced artificially to replicate a long-standing proposed mesoscale structure for proline correlations as an initial condition are annealed out by parallel tempering MD simulations. However, some small residual aggregates do remain, implying a greater degree of long-range order than is apparent in the SANS data.     

Optimal position of solute for simulations

It is shown that, by optimizing the placement of a solute in a solvent droplet or in a periodic simulation cell, the number of solvent molecules can be reduced without affecting the quality of the simulation. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 812–815, 1997     

Free-energy landscape for beta hairpin folding from combined parallel tempering and metadynamics

We develop a new free-energy method, based on the combination of parallel tempering and metadynamics, and apply this method to the calculation of the free-energy landscape of the folding beta hairpin in explicit water. We show that the combined method greatly improves the performance of both parallel tempering and metadynamics. In particular, we are able to sample the high free-energy regions, which are not accessible with conventional parallel tempering. We use our results to calculate the difference in entropy and enthalpy between the folded and the unfolded state and to characterize the most populated configurations in the relevant free-energy basins.     

Langevin dynamics simulations of macromolecules on parallel computers

A parallel algorithm is developed that allows efficient Langevin-dynamics simulations of macromolecular coils, which is the usual structure of synthetic polymers in solution and in bulk. Contrary to usual so-called spatial decomposition algorithms, we map the one-dimensional topology of the chain molecule on the parallel computer. The speedup of the algorithm is measured on different multi-processor systems. The reliability of the parallel calculations is shown by comparison with sequential simulations.     

Multicanonical parallel simulations of proteins with continuous potentials

The determination of the three‐dimensional (3D) structure of a protein or peptide is a very important research problem in biological and medical sciences. Anfinsen's experiments (Science 1973, 181, 223) on renaturation of denatured proteins have shown that the native 3D structure of a (small) protein at low (room) temperatures is uniquely determined by its amino acid sequence, which suggests that it might be possible to determine the 3D structure of a protein from its amino acid sequence by pure computations. As a step toward that goal, in this article we present a simple approach for parallelization of multicanonical Monte Carlo simulations of proteins with continuous potentials. Our method is based on the parallel calculation of the protein energy function. The algorithm is tested by simulated annealing and multicanonical simulations of two small peptides, and known results are reproduced accurately. An acceptable degree of parallelization can be achieved in the simulation of Protein L using up to 30 PCs. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1287–1296, 2001     

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.     

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.     

Symplectic molecular dynamics simulations on specially designed parallel computers

We have developed a computer program for molecular dynamics (MD) simulation that implements the Split Integration Symplectic Method (SISM) and is designed to run on specialized parallel computers. The MD integration is performed by the SISM, which analytically treats high-frequency vibrational motion and thus enables the use of longer simulation time steps. The low-frequency motion is treated numerically on specially designed parallel computers, which decreases the computational time of each simulation time step. The combination of these approaches means that less time is required and fewer steps are needed and so enables fast MD simulations. We study the computational performance of MD simulation of molecular systems on specialized computers and provide a comparison to standard personal computers. The combination of the SISM with two specialized parallel computers is an effective way to increase the speed of MD simulations up to 16-fold over a single PC processor.     

Improving the performance of molecular dynamics simulations on parallel clusters

In this article a procedure is derived to obtain a performance gain for molecular dynamics (MD) simulations on existing parallel clusters. Parallel clusters use a wide array of interconnection technologies to connect multiple processors together, often at different speeds, such as multiple processor computers and networking. It is demonstrated how to configure existing programs for MD simulations to efficiently handle collective communication on parallel clusters with processor interconnections of different speeds.     

Searching for globally optimal functional forms for interatomic potentials using genetic programming with parallel tempering

Molecular dynamics and other molecular simulation methods rely on a potential energy function, based only on the relative coordinates of the atomic nuclei. Such a function, called a force field, approximately represents the electronic structure interactions of a condensed matter system. Developing such approximate functions and fitting their parameters remains an arduous, time-consuming process, relying on expert physical intuition. To address this problem, a functional programming methodology was developed that may enable automated discovery of entirely new force-field functional forms, while simultaneously fitting parameter values. The method uses a combination of genetic programming, Metropolis Monte Carlo importance sampling and parallel tempering, to efficiently search a large space of candidate functional forms and parameters. The methodology was tested using a nontrivial problem with a well-defined globally optimal solution: a small set of atomic configurations was generated and the energy of each configuration was calculated using the Lennard-Jones pair potential. Starting with a population of random functions, our fully automated, massively parallel implementation of the method reproducibly discovered the original Lennard-Jones pair potential by searching for several hours on 100 processors, sampling only a minuscule portion of the total search space. This result indicates that, with further improvement, the method may be suitable for unsupervised development of more accurate force fields with completely new functional forms.