Directionally negative friction: a method for enhanced sampling of rare event kinetics

A method exploiting the properties of an artificial (nonphysical) Langevin dynamics with a negative frictional coefficient along a suitable manifold and positive friction in the perpendicular directions is presented for the enhanced calculation of time-correlation functions for rare event problems. Exact time-correlation functions that describe the kinetics of the transitions for the all-positive, physical system can be calculated by reweighting the generated trajectories according to stochastic path integral treatment involving a functional weight based on an Onsager-Machlup action functional. The method is tested on a prototypical multidimensional model system featuring the main elements of conformational space characteristic of complex condensed matter systems. Using the present method, accurate estimates of rate constants require at least three order of magnitudes fewer trajectories than regular Langevin dynamics. The method is particularly useful in calculating kinetic properties in the context of multidimensional energy landscapes that are characteristic of complex systems such as proteins and nucleic acids.     

Force-momentum-based self-guided Langevin dynamics: a rapid sampling method that approaches the canonical ensemble

The self-guided Langevin dynamics (SGLD) is a method to accelerate conformational searching. This method is unique in the way that it selectively enhances and suppresses molecular motions based on their frequency to accelerate conformational searching without modifying energy surfaces or raising temperatures. It has been applied to studies of many long time scale events, such as protein folding. Recent progress in the understanding of the conformational distribution in SGLD simulations makes SGLD also an accurate method for quantitative studies. The SGLD partition function provides a way to convert the SGLD conformational distribution to the canonical ensemble distribution and to calculate ensemble average properties through reweighting. Based on the SGLD partition function, this work presents a force-momentum-based self-guided Langevin dynamics (SGLDfp) simulation method to directly sample the canonical ensemble. This method includes interaction forces in its guiding force to compensate the perturbation caused by the momentum-based guiding force so that it can approximately sample the canonical ensemble. Using several example systems, we demonstrate that SGLDfp simulations can approximately maintain the canonical ensemble distribution and significantly accelerate conformational searching. With optimal parameters, SGLDfp and SGLD simulations can cross energy barriers of more than 15 kT and 20 kT, respectively, at similar rates for LD simulations to cross energy barriers of 10 kT. The SGLDfp method is size extensive and works well for large systems. For studies where preserving accessible conformational space is critical, such as free energy calculations and protein folding studies, SGLDfp is an efficient approach to search and sample the conformational space.     

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.     

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.     

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.     

Optimal allocation of replicas in parallel tempering simulations

We have studied the efficiency of parallel tempering simulations for a variety of systems including a coarse-grained protein, an atomistic model polypeptide, and the Lennard-Jones fluid. A scheme is proposed for the optimal allocation of temperatures in these simulations. The method is compared to the existing empirical approaches used for this purpose. Accuracy associated with the computed thermodynamic quantities such as specific heat is also computed and their dependence on the trial-exchange acceptance rate is reported.     

Mean field kinetic theory for a lattice gas model of fluids confined in porous materials

We consider the mean field kinetic equations describing the relaxation dynamics of a lattice model of a fluid confined in a porous material. The dynamical theory embodied in these equations can be viewed as a mean field approximation to a Kawasaki dynamics Monte Carlo simulation of the system, as a theory of diffusion, or as a dynamical density functional theory. The solutions of the kinetic equations for long times coincide with the solutions of the static mean field equations for the inhomogeneous lattice gas. The approach is applied to a lattice gas model of a fluid confined in a finite length slit pore open at both ends and is in contact with the bulk fluid at a temperature where capillary condensation and hysteresis occur. The states emerging dynamically during irreversible changes in the chemical potential are compared with those obtained from the static mean field equations for states associated with a quasistatic progression up and down the adsorption/desorption isotherm. In the capillary transition region, the dynamics involves the appearance of undulates (adsorption) and liquid bridges (adsorption and desorption) which are unstable in the static mean field theory in the grand ensemble for the open pore but which are stable in the static mean field theory in the canonical ensemble for an infinite pore.     

Effective stochastic dynamics on a protein folding energy landscape

We present an approach to protein folding kinetics using stochastic reaction-coordinate dynamics, in which the effective drift velocities and diffusion coefficients are determined from microscopic simulation data. The resultant Langevin equation can then be used to directly simulate the folding process. Here, we test this approach by applying it to a toy two-state dynamical system and to a funnellike structure-based (Go-type) model. The folding time predictions agree very well with full simulation results. Therefore, we have in hand a fast numerical tool for calculating the folding kinetic properties, even when full simulations are not feasible. In addition, the local drift and diffusion coefficients provide an alternative way to compute the free energy profile in cases where only local sampling can be achieved.     

Prediction of nitroxide hyperfine coupling constants in solution from combined nanosecond scale simulations and quantum computations

We present a combined theoretical approach based on analyzing molecular dynamics trajectories (at the nanosecond scale) generated by use of classical polarizable force fields and on quantum calculations to compute averaged hyperfine coupling constants. That method is used to estimate the constant of a prototypical nitroxide: the dimethylnitroxide. The molecule is embedded during the simulations in a cubic box containing about 500 water molecules and the molecular dynamics is generated using periodic conditions. Once the trajectories are achieved, the nitroxide and its first hydration shell molecules are extracted, and the coupling constants are computed by considering the latter aggregates by means of quantum computations. However, all the water molecules of the bulk are also accounted for during those computations by means of the electrostatic potential fitted method. Our results exhibit that in order to predict accurate and reliable coupling constants, one needs to describe carefully the out-of-plane motion of the nitroxide nitrogen and to sample trajectories with a time interval of 400 fs at least to generate an uncorrelated large set of nitroxide structures. Compared to Car-Parrinello molecular dynamics techniques, our approach can be used readily to compute hyperfine coupling constants of large systems, such as nitroxides of great size interacting with macromolecules such as proteins or polymers.     

Molecular latent space simulators

Small integration time steps limit molecular dynamics (MD) simulations to millisecond time scales. Markov state models (MSMs) and equation-free approaches learn low-dimensional kinetic models from MD simulation data by performing configurational or dynamical coarse-graining of the state space. The learned kinetic models enable the efficient generation of dynamical trajectories over vastly longer time scales than are accessible by MD, but the discretization of configurational space and/or absence of a means to reconstruct molecular configurations precludes the generation of continuous atomistic molecular trajectories. We propose latent space simulators (LSS) to learn kinetic models for continuous atomistic simulation trajectories by training three deep learning networks to (i) learn the slow collective variables of the molecular system, (ii) propagate the system dynamics within this slow latent space, and (iii) generatively reconstruct molecular configurations. We demonstrate the approach in an application to Trp-cage miniprotein to produce novel ultra-long synthetic folding trajectories that accurately reproduce atomistic molecular structure, thermodynamics, and kinetics at six orders of magnitude lower cost than MD. The dramatically lower cost of trajectory generation enables greatly improved sampling and greatly reduced statistical uncertainties in estimated thermodynamic averages and kinetic rates.

Latent space simulators learn kinetic models for atomistic simulations and generate novel trajectories at six orders of magnitude lower cost.     

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.     

Configurational temperature in membrane simulations using dissipative particle dynamics

The use of excessively long time steps in dissipative particle dynamics simulations may produce simulation artifacts due to the generation of configurations which are not representative of the desired canonical ensemble. The configurational temperature, among other quantities, may be used to assess the extent of the deviation from equilibrium. This paper presents results for simulations of models of water and lipid bilayer membranes to illustrate the nature of the problems.     

Ionic melts with waterlike anomalies: thermodynamic properties of liquid BeF2

Thermodynamic properties of liquid beryllium difluoride (BeF(2)) are studied using canonical ensemble molecular dynamics simulations of the transferable rigid ion model potential. The negative slope of the locus of points of maximum density in the temperature-pressure plane is mapped out. The excess entropy, computed within the pair correlation approximation, is found to show an anomalous increase with isothermal compression at low temperatures which will lead to diffusional as well as structural anomalies resembling those in water. The anomalous behavior of the entropy is largely connected with the behavior of the Be-F pair correlation function. The internal energy shows a T(35) temperature dependence. The pair correlation entropy shows a T(-25) temperature dependence only at high densities and temperatures. The correlation plots between internal energy and the pair correlation entropy for isothermal compression show the characteristic features expected of network-forming liquids with waterlike anomalies. The tagged particle potential energy distributions are shown to have a multimodal form at low temperatures and densities similar to those seen in other liquids with three-dimensional tetrahedral networks, such as water and silica.     

Static and dynamical properties of heavy water at ambient conditions from first-principles molecular dynamics

The static and dynamical properties of heavy water have been studied at ambient conditions with extensive Car-Parrinello molecular-dynamics simulations in the canonical ensemble, with temperatures ranging between 325 and 400 K. Density-functional theory, paired with a modern exchange-correlation functional (Perdew-Burke-Ernzerhof), provides an excellent agreement for the structural properties and binding energy of the water monomer and dimer. On the other hand, the structural and dynamical properties of the bulk liquid show a clear enhancement of the local structure compared to experimental results; a distinctive transition to liquidlike diffusion occurs in the simulations only at the elevated temperature of 400 K. Extensive runs of up to 50 ps are needed to obtain well-converged thermal averages; the use of ultrasoft or norm-conserving pseudopotentials and the larger plane-wave sets associated with the latter choice had, as expected, only negligible effects on the final result. Finite-size effects in the liquid state are found to be mostly negligible for systems as small as 32 molecules per unit cell.     

Configurational constant pressure molecular dynamics

We propose two new algorithms for generating isothermal-isobaric molecular dynamics. The algorithms are based on an extended phase space dynamics where two extra degrees of freedom, representing the thermostat and the barostat, are included. These new methods adopt a totally different approach towards molecular dynamics simulation in the isothermal-isobaric ensemble. They are fully configurational in the sense that only the particle positions are required in the control of the system temperature and pressure. Following on from the works of Delhommelle and Evans [Mol. Phys., 99, 1825 (2001)] and of Braga and Travis [J. Chem. Phys., 123, 134101 (2005)] concerning configurational canonical dynamics, these new algorithms can be seen as a natural extension to the isothermal-isobaric ensemble. We have validated both of our new configurational isothermal-isobaric schemes by conducting molecular dynamics simulations of a Lennard-Jones fluid and comparing the static and dynamic properties for a single state point. We find that both schemes generate similar results compared with schemes which use kinetic temperature and pressure control. We have also monitored the response of the system to a series of isothermal compressions and isobaric quenches. We find that the configurational schemes performed at least as well as the kinetic based scheme in bringing the system temperature and pressure into line with the set point values of these variables. These new methods will potentially play a significant role in simulations where the calculation of the kinetic temperature and pressure can be problematic. A well known example resides in the field of nonequilibrium simulations where the kinetic temperature and pressure require a knowledge of the streaming velocity of the fluid in order to calculate the true peculiar velocities (or momenta) that enter into their definitions. These are completely avoided by using our configurational thermostats and barostats, since these are independent of momenta. By extending the analysis of Kusnezov et al. [Ann. Phys., 204, 155 (1990)] in order to derive a set of generalized Nose-Hoover equations of motion which can generate isothermal-isobaric dynamics in a number of different ways, we are able to show that both of our new configurational barostats and Hoover's kinetic isothermal-isobaric scheme are special cases of this more general set of equations. This generalization can be very powerful in generating constant pressure dynamics for a variety of systems.     

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 .