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.     

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

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

Competing through-space and through-bond, intramolecular triplet-energy transfer in a supposedly rigid ruthenium(II) tris(2,2'-bipyridine)--fullerene molecular dyad

A ditopic ruthenium(II) tris(2,2'-bipyridyl)-based fullerene conjugate has been synthesized so as to separate the photoactive terminals by way of a short ethynylene spacer group that is expected to act as a rigid rod. Intramolecular triplet-energy transfer from the metal complex to the fullerene is quantitative at all temperatures and there is no indication for competing electron transfer. Temperature dependence studies indicate two pathways for triplet-energy transfer. An activationless route dominates at low temperature and is attributed to through-bond electron exchange that takes place via super-exchange interactions. The triplet energy of the bridging unit lies well above that of the metal complex. An activated process is switched-on at high temperatures and is believed to involve through-space electron exchange within closed conformations. Molecular dynamics simulations predict that, in addition to an extended conformation, the linker can distort in such a way that the terminals come into orbital contact. In fact, the resultant closed conformation possesses an idealised geometry for fast electron exchange.     

Simple continuous and discrete models for simulating replica exchange simulations of protein folding

The efficiency of temperature replica exchange (RE) simulations hinge on their ability to enhance conformational sampling at physiological temperatures by taking advantage of more rapid conformational interconversions at higher temperatures. While temperature RE is a parallel simulation technique that is relatively straightforward to implement, kinetics in the RE ensemble is complicated, and there is much to learn about how best to employ RE simulations in computational biophysics. Protein folding rates often slow down above a certain temperature due to entropic bottlenecks. This "anti-Arrhenius" behavior represents a challenge for RE. However, it is far from straightforward to systematically explore the impact of this on RE by brute force molecular simulations, since RE simulations of protein folding are very difficult to converge. To understand some of the basic mechanisms that determine the efficiency of RE, it is useful to study simplified low dimensionality systems that share some of the key characteristics of molecular systems. Results are presented concerning the efficiency of temperature RE on a continuous two-dimensional potential that contains an entropic bottleneck. Optimal efficiency was obtained when the temperatures of the replicas did not exceed the temperature at which the harmonic mean of the folding and unfolding rates is maximized. This confirms a result we previously obtained using a discrete network model of RE. Comparison of the efficiencies obtained using the continuous and discrete models makes it possible to identify non-Markovian effects, which slow down equilibration of the RE ensemble on the more complex continuous potential. In particular, the rate of temperature diffusion and also the efficiency of RE is limited by the time scale of conformational rearrangements within free energy basins.     

Boltzmann bias grand canonical Monte Carlo

We derive an efficient method for the insertion of structured particles in grand canonical Monte Carlo simulations of adsorption in very confining geometries. We extend this method to path integral simulations and use it to calculate the isotherm of adsorption of hydrogen isotopes in narrow carbon nanotubes (two-dimensional confinement) and slit pores (one-dimensional confinement) at the temperatures of 20 and 77 K, discussing its efficiency by comparison to the standard path integral grand canonical Monte Carlo algorithm. We use this algorithm to perform multicomponent simulations in order to calculate the hydrogen isotope selectivity for adsorption in narrow carbon nanotubes and slit pores at finite pressures. The algorithm described here can be applied to the study of adsorption of real oligomers and polymers in narrow pores and channels.     

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.     

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.     

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.     

Comparison of two adaptive temperature-based replica exchange methods applied to a sharp phase transition of protein unfolding-folding

Temperature-based replica exchange (T-ReX) enhances sampling of molecular dynamics simulations by autonomously heating and cooling simulation clients via a Metropolis exchange criterion. A pathological case for T-ReX can occur when a change in state (e.g., folding to unfolding of a protein) has a large energetic difference over a short temperature interval leading to insufficient exchanges amongst replica clients near the transition temperature. One solution is to allow the temperature set to dynamically adapt in the temperature space, thereby enriching the population of clients near the transition temperature. In this work, we evaluated two approaches for adapting the temperature set: a method that equalizes exchange rates over all neighbor temperature pairs and a method that attempts to induce clients to visit all temperatures (dubbed "current maximization") by positioning many clients at or near the transition temperature. As a test case, we simulated the 57-residue SH3 domain of alpha-spectrin. Exchange rate equalization yielded the same unfolding-folding transition temperature as fixed-temperature ReX with much smoother convergence of this value. Surprisingly, the current maximization method yielded a significantly lower transition temperature, in close agreement with experimental observation, likely due to more extensive sampling of the transition state.     

Temperature dependence of coarse-grained potentials for liquid hexane

The present molecular dynamics study is an investigation of the temperature (T) dependence of liquid hexane coarse-grained potentials optimized with the Iterative Boltzmann Inversion method. An approach for the derivation of coarse-grained potentials at temperatures T different from the optimization temperature T(0) has recently been proposed for ethylbenzene. This method is based on the use of a T-dependent scaling factor f(T) to generate ethylbenzene potentials at T≠T(0). The approach is here extended to hexane, considering different reference temperatures T(0) and functional forms for f(T). From our simulations, it appears that the accuracy of the temperature transferability depends simultaneously on the T(0) chosen and the analytic form of f(T). Such a behavior is suppressed by the use of a new 2-point interpolation formula to generate coarse-grained potentials as a function of T. This scheme employs a linear interpolation based on the optimization of coarse-grained potentials at two reference temperatures, T(L) and T(U), with T(L)≤T≤T(U). Accurate coarse-grained simulations of liquid hexane can be performed using the new interpolation scheme. The results are encouraging for the use of potential interpolations as a practical means for devising coarse-grained potentials within a wider temperature range.     

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.     

Dynamical reweighting: improved estimates of dynamical properties from simulations at multiple temperatures

Dynamical averages based on functionals of dynamical trajectories, such as time-correlation functions, play an important role in determining kinetic or transport properties of matter. At temperatures of interest, the expectations of these quantities are often dominated by contributions from rare events, making the precise calculation of these quantities by molecular dynamics simulation difficult. Here, we present a reweighting method for combining simulations from multiple temperatures (or from simulated or parallel tempering simulations) to compute an optimal estimate of the dynamical properties at the temperature of interest without the need to invoke an approximate kinetic model (such as the Arrhenius law). Continuous and differentiable estimates of these expectations at any temperature in the sampled range can also be computed, along with an assessment of the associated statistical uncertainty. For rare events, aggregating data from multiple temperatures can produce an estimate with the desired precision at greatly reduced computational cost compared with simulations conducted at a single temperature. Here, we describe use of the method for the canonical (NVT) ensemble using four common models of dynamics (canonical distribution of Hamiltonian trajectories, Andersen thermostatting, Langevin, and overdamped Langevin or Brownian dynamics), but it can be applied to any thermodynamic ensemble provided the ratio of path probabilities at different temperatures can be computed. To illustrate the method, we compute a time-correlation function for solvated terminally-blocked alanine peptide across a range of temperatures using trajectories harvested using a modified parallel tempering protocol.     

Diffusion processes and growth on aluminum cluster surfaces

Diffusion processes of adatoms on icosahedral and Wulff polyhedral aluminum cluster surfaces have been studied by molecular dynamics simulations using the effective medium theory. Activation energies of diffusion mechanisms along {111} and {100} facets and from one facet to another, including different hopping and exchange processes as well as more exotic events, have been calculated. Exchange diffusion of an adatom by a chain mechanism through a {100} facet between two {111} facets and hopping diffusion across the edge between two {111} facets via a pull of another adatom on the neighbour facet are shown to play an important role. Adatoms on {111} facets are mobile already at very low temperatures, but on {100} facets diffusion starts above the room temperature as well as diffusion from {111} facets to {100} facets. Diffusion from {100} facet to other facets was not observed until at temperatures close to the melting temperatures of clusters. Dynamical simulations at different temperatures confirmed the appearance of diffusion mechanisms predicted by the activation energies.     

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.     

Conformational equilibrium of cytochrome P450 BM-3 complexed with N-palmitoylglycine: a replica exchange molecular dynamics study

UV-vis absorbance measurements and associated studies of cytochrome P450 BM-3 in complex with N-palmitoylglycine (NPG) indicate that a conformational change occurs in the active site of the complex where the terminal atoms of the ligand move from a site distant from the heme iron, as seen in the low temperature crystal structure to a site proximal to the heme iron at biological temperatures. We employ replica exchange molecular dynamics simulations to study this conformational change. The population of the proximal state is found to increase with temperature in agreement with UV-vis absorbance and NMR measurements. In addition to the conformations characterized by X-ray crystallography and computer modeling, this study shows that a new conformational state is significantly populated at room temperature. The observed increase in the population of conformations where the terminal atoms of NPG are proximal to the heme iron with increasing temperature indicates that the proximal state is stabilized by conformational entropy. A proposal for the origin of this entropic stabilization is provided on the basis of the structure of the newly identified state. We use the temperature weighted histogram (T-WHAM) method to characterize the transition state regions of the conformational ensemble and propose a mechanism of interconversion between these low free energy conformational states.     

An integrate-over-temperature approach for enhanced sampling

A simple method is introduced to achieve efficient random walking in the energy space in molecular dynamics simulations which thus enhances the sampling over a large energy range. The approach is closely related to multicanonical and replica exchange simulation methods in that it allows configurations of the system to be sampled in a wide energy range by making use of Boltzmann distribution functions at multiple temperatures. A biased potential is quickly generated using this method and is then used in accelerated molecular dynamics simulations.     

Temperature-dependent H/D exchange of compact and elongated cytochrome c ions in the gas phase

Isotopic exchange reactions of compact and elongated conformations of gaseous cytochrome c ions (+5 and +9 states) with D2O have been measured as a function of temperature (from 300 to approximately 440 K) using ion mobility techniques. Rate constants for those sites that exchange at high temperatures (>400 K) are about an order of magnitude smaller than rate constants for sites that exchange at 300 K. Although the exchange rates decrease, the maximum exchange levels for rapidly exchanging sites increase with temperature. At 300 K, exchange levels of 53 +/- 3 and 63 +/- 3 are measured for the compact and elongated states, respectively. From 300 to 335 K, the exchange levels increase slightly to approximately 60 to 70 hydrogens. Above 335 K, the levels increase to a value of approximately 200 for the +5 state and approximately 190 for the +9 state, near the maximum possible levels, 200 and 204 for these respective charge states. Molecular dynamics simulations have been carried out on structures having calculated cross sections that are near the experimental values in order to explore the exchange process. Overall, it appears that charge site and exchange site proximities are important factors in the exchange profiles for the elongated +9 ion and the compact +5 ion.     

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.     

Insights into the dynamics of HIV-1 protease: a kinetic network model constructed from atomistic simulations

The conformational dynamics in the flaps of HIV-1 protease plays a crucial role in the mechanism of substrate binding. We develop a kinetic network model, constructed from detailed atomistic simulations, to determine the kinetic mechanisms of the conformational transitions in HIV-1 PR. To overcome the time scale limitation of conventional molecular dynamics (MD) simulations, our method combines replica exchange MD with transition path theory (TPT) to study the diversity and temperature dependence of the pathways connecting functionally important states of the protease. At low temperatures the large-scale flap opening is dominated by a small number of paths; at elevated temperatures the transition occurs through many structurally heterogeneous routes. The expanded conformation in the crystal structure 1TW7 is found to closely mimic a key intermediate in the flap-opening pathways at low temperature. We investigated the different transition mechanisms between the semi-open and closed forms. The calculated relaxation times reveal fast semi-open ? closed transitions, and infrequently the flaps fully open. The ligand binding rate predicted from this kinetic model increases by 38-fold from 285 to 309 K, which is in general agreement with experiments. To our knowledge, this is the first application of a network model constructed from atomistic simulations together with TPT to analyze conformational changes between different functional states of a natively folded protein.     

Dynamically biased statistical model for the ortho/para conversion in the H(2)+H(3) (+) → H(3) (+)+ H(2) reaction

In this work we present a dynamically biased statistical model to describe the evolution of the title reaction from statistical to a more direct mechanism, using quasi-classical trajectories (QCT). The method is based on the one previously proposed by Park and Light [J. Chem. Phys. 126, 044305 (2007)]. A recent global potential energy surface is used here to calculate the capture probabilities, instead of the long-range ion-induced dipole interactions. The dynamical constraints are introduced by considering a scrambling matrix which depends on energy and determine the probability of the identity/hop/exchange mechanisms. These probabilities are calculated using QCT. It is found that the high zero-point energy of the fragments is transferred to the rest of the degrees of freedom, what shortens the lifetime of H(5) (+) complexes and, as a consequence, the exchange mechanism is produced with lower proportion. The zero-point energy (ZPE) is not properly described in quasi-classical trajectory calculations and an approximation is done in which the initial ZPE of the reactants is reduced in QCT calculations to obtain a new ZPE-biased scrambling matrix. This reduction of the ZPE is explained by the need of correcting the pure classical level number of the H(5) (+) complex, as done in classical simulations of unimolecular processes and to get equivalent quantum and classical rate constants using Rice-Ramsperger-Kassel-Marcus theory. This matrix allows to obtain a ratio of hop/exchange mechanisms, α(T), in rather good agreement with recent experimental results by Crabtree et al. [J. Chem. Phys. 134, 194311 (2011)] at room temperature. At lower temperatures, however, the present simulations predict too high ratios because the biased scrambling matrix is not statistical enough. This demonstrates the importance of applying quantum methods to simulate this reaction at the low temperatures of astrophysical interest.