Local Sampling in Steered Monte Carlo Simulations Decreases Dissipation and Enhances Free Energy Estimates via Nonequilibrium Work Theorems

Local Resampling Trick for Focused Molecular Dynamics

We describe a method that focuses sampling effort on a user-defined selection of a large system, which can lead to substantial decreases in computational effort by speeding up the calculation of nonbonded interactions. A naive approach can lead to incorrect sampling if the selection depends on the configuration in a way that is not accounted for. We avoid this pitfall by introducing appropriate auxiliary variables. This results in an implementation that is closely related to "configurational freezing" and "elastic barrier dynamical freezing." We implement the method and validate that it can be used to supplement conventional molecular dynamics in free energy calculations (absolute hydration and relative binding).

Lessons learned about steered molecular dynamics simulations and free energy calculations

The calculation of free energy profiles is central in understanding differential enzymatic activity, for instance, involving chemical reactions that require QM-MM tools, ligand migration, and conformational rearrangements that can be modeled using classical potentials. The use of steered molecular dynamics (sMD) together with the Jarzynski equality is a popular approach in calculating free energy profiles. Here, we first briefly review the application of the Jarzynski equality to sMD simulations, then revisit the so-called stiff-spring approximation and the consequent expectation of Gaussian work distributions and, finally, reiterate the practical utility of the second-order cumulant expansion, as it coincides with the parametric maximum-likelihood estimator in this scenario. We illustrate this procedure using simulations of CO, both in aqueous solution and in a carbon nanotube as a model system for biologically relevant nanoheterogeneous environments. We conclude the use of the second-order cumulant expansion permits the use of faster pulling velocities in sMD simulations, without introducing bias due to large dispersion in the non-equilibrium work distribution.

Correspondence between light-absorption spectrum and nonequilibrium work distribution as a mean to access free energy differences between electronic states

The problem of recovering the free energy difference between two electronic states has been investigated by Frezzato [Chem. Phys. Lett. 533, 106 (2012)], exploring the equivalence between light-absorption spectra and work distribution, hence opening to the application of a spectroscopic version of the Jarzynski equality (JE) [Phys. Rev. Lett. 78, 2690 (1997)]. Here, assuming the validity of the time-dependent perturbation theory, we demonstrate that such equivalence does not lead to the known form of the JE. This is ascribed to the fact that light-absorption processes cannot be described as stochastic processes. To emphasize such an aspect, we devise a stochastic model for the UV-vis (ultraviolet and visible) absorption, suitable for determining the free energy difference between two generic quantum manifolds in a JE-like fashion. However, the model would require explicit knowledge of the transition dipole moments, which are in general not available. Nonetheless, we derive a spectroscopic version of the JE that allows us to recover the free energy difference between the ground and an excited electronic state when the latter state is the only one observed in the spectrum.

Driving Structural Transitions in Molecular Simulations Using the Nonequilibrium Candidate Monte Carlo

Hybrid simulation procedures which combine molecular dynamics with Monte Carlo are attracting increasing attention as tools for improving the sampling efficiency in molecular simulations. In particular, encouraging results have been reported for nonequilibrium candidate protocols, in which a Monte Carlo move is applied gradually, and interleaved with a process that equilibrates the remaining degrees of freedom. Although initial studies have uncovered a substantial potential of the method, its practical applicability for sampling structural transitions in macromolecules remains incompletely understood. Here, we address this issue by systematically investigating the efficiency of the nonequilibrium candidate Monte Carlo on the sampling of rotameric distributions of two peptide systems at atomistic resolution both in vacuum and explicit solvent. The studied systems allow us to directly probe the efficiency with which a single or a few slow degrees of freedom can be driven between well-separated free-energy minima and to explore the sensitivity of the method toward the involved free parameters. In line with results on other systems, our study suggests that order-of-magnitude gains can be obtained in certain scenarios but also identifies challenges that arise when applying the procedure in explicit solvent.

Elastic Barrier Dynamical Freezing in Free Energy Calculations: A Way To Speed Up Nonequilibrium Molecular Dynamics Simulations by Orders of Magnitude

An important issue concerning computer simulations addressed to free energy estimates via nonequilibrium work theorems, such as the Jarzynski equality [Phys. Rev. Lett. 1997, 78, 2690], is the computational effort required to achieve results with acceptable accuracy. In this respect, the dynamical freezing approach [Phys. Rev. E 2009, 80, 041124] has been shown to improve the efficiency of this kind of simulations, by blocking the dynamics of particles located outside an established mobility region. In this report, we show that dynamical freezing produces a systematic spurious decrease of the particle density inside the mobility region. As a consequence, the requirements to apply nonequilibrium work theorems are only approximately met. Starting from these considerations, we have developed a simulation scheme, called "elastic barrier dynamical freezing", according to which a stiff potential-energy barrier is enforced at the boundaries of the mobility region, preventing the particles from leaving this region of space during the nonequilibrium trajectories. The method, tested on the calculation of the distance-dependent free energy of a dimer immersed into a Lennard-Jones fluid, provides an accuracy comparable to the conventional steered molecular dynamics, with a computational speedup exceeding a few orders of magnitude.

Nonequilibrium Candidate Monte Carlo Simulations with Configurational Freezing Schemes

Nonequilibrium Candidate Monte Carlo simulation [Nilmeier et al., Proc. Natl. Acad. Sci. U.S.A. 2011, 108, E1009-E1018] is a tool devised to design Monte Carlo moves with high acceptance probabilities that connect uncorrelated configurations. Such moves are generated through nonequilibrium driven dynamics, producing candidate configurations accepted with a Monte Carlo-like criterion that preserves the equilibrium distribution. The probability of accepting a candidate configuration as the next sample in the Markov chain basically depends on the work performed on the system during the nonequilibrium trajectory and increases with decreasing such a work. It is thus strategically relevant to find ways of producing nonequilibrium moves with low work, namely moves where dissipation is as low as possible. This is the goal of our methodology, in which we combine Nonequilibrium Candidate Monte Carlo with Configurational Freezing schemes developed by Nicolini et al. (J. Chem. Theory Comput. 2011, 7, 582-593). The idea is to limit the configurational sampling to particles of a well-established region of the simulation sample, namely the region where dissipation occurs, while leaving fixed the other particles. This allows to make the system relaxation faster around the region perturbed by the finite-time switching move and hence to reduce the dissipated work, eventually enhancing the probability of accepting the generated move. Our combined approach enhances significantly configurational sampling, as shown by the case of a bistable dimer immersed in a dense fluid.

Annealed importance sampling with constant cooling rate

Annealed importance sampling is a simulation method devised by Neal [Stat. Comput. 11, 125 (2001)] to assign weights to configurations generated by simulated annealing trajectories. In particular, the equilibrium average of a generic physical quantity can be computed by a weighted average exploiting weights and estimates of this quantity associated to the final configurations of the annealed trajectories. Here, we review annealed importance sampling from the perspective of nonequilibrium path-ensemble averages [G. E. Crooks, Phys. Rev. E 61, 2361 (2000)]. The equivalence of Neal's and Crooks' treatments highlights the generality of the method, which goes beyond the mere thermal-based protocols. Furthermore, we show that a temperature schedule based on a constant cooling rate outperforms stepwise cooling schedules and that, for a given elapsed computer time, performances of annealed importance sampling are, in general, improved by increasing the number of intermediate temperatures.

Towards bulk thermodynamics via non-equilibrium methods: gaseous methane as a case study

The equation of state of bulk materials is achieved via thermodynamic derivatives of the free energy yielded by nonequilibrium transformations and Jarzynski equality.

High-precision work distributions for extreme nonequilibrium processes in large systems

The distributions of work for strongly nonequilibrium processes are studied using a very general form of a large-deviation approach, which allows one to study distributions down to extremely small probabilities of almost arbitrary quantities of interest for equilibrium, nonequilibrium stationary, and even nonstationary processes. The method is applied to quickly vary the external field in a wide range B = 3 ↔ 0 for a critical (T = 2.269) two-dimensional Ising system of size L × L = 128 × 128. To obtain free-energy differences from the work distributions, they must be studied in ranges where the probabilities are as small as 10^{-240}, which is not possible using direct simulation approaches. By comparison with the exact free energies, which are available for this model for the zero-field case, one sees that the present approach allows one to obtain the free energy with a very high relative precision of 10^{-4}. This works well also for a nonzero field, i.e., for a case where standard umbrella-sampling methods are not efficient to calculate free energies. Furthermore, for the present case it is verified that the resulting distributions of work for forward and backward processes fulfill Crooks theorem with high precision. Finally, the free energy for the Ising magnet as a function of the field strength is obtained.

Path-breaking schemes for nonequilibrium free energy calculations

We propose a path-breaking route to the enhancement of unidirectional nonequilibrium simulations for the calculation of free energy differences via Jarzynski's equality [C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997)]. One of the most important limitations of unidirectional nonequilibrium simulations is the amount of realizations necessary to reach suitable convergence of the work exponential average featuring the Jarzynski's relationship. In this respect, a significant improvement of the performances could be obtained by finding a way of stopping trajectories with negligible contribution to the work exponential average, before their normal end. This is achieved using path-breaking schemes which are essentially based on periodic checks of the work dissipated during the pulling trajectories. Such schemes can be based either on breaking trajectories whose dissipated work exceeds a given threshold or on breaking trajectories with a probability increasing with the dissipated work. In both cases, the computer time needed to carry out a series of nonequilibrium trajectories is reduced up to a factor ranging from 2 to more than 10, at least for the processes under consideration in the present study. The efficiency depends on several aspects, such as the type of process, the number of check-points along the pathway and the pulling rate as well. The method is illustrated through radically different processes, i.e., the helix-coil transition of deca-alanine and the pulling of the distance between two methane molecules in water solution.

Toward quantitative estimates of binding affinities for protein-ligand systems involving large inhibitor compounds: a steered molecular dynamics simulation route

Understanding binding mechanisms between enzymes and potential inhibitors and quantifying protein-ligand affinities in terms of binding free energy is of primary importance in drug design studies. In this respect, several approaches based on molecular dynamics simulations, often combined with docking techniques, have been exploited to investigate the physicochemical properties of complexes of pharmaceutical interest. Even if the geometric properties of a modeled protein-ligand complex can be well predicted by computational methods, it is still challenging to rank with chemical accuracy a series of ligand analogues in a consistent way. In this article, we face this issue calculating relative binding free energies of a focal adhesion kinase, an important target for the development of anticancer drugs, with pyrrolopyrimidine-based ligands having different inhibitory power. To this aim, we employ steered molecular dynamics simulations combined with nonequilibrium work theorems for free energy calculations. This technique proves very powerful when a series of ligand analogues is considered, allowing one to tackle estimation of protein-ligand relative binding free energies in a reasonable time. In our cases, the calculated binding affinities are comparable with those recovered from experiments by exploiting the Michaelis-Menten mechanism with a competitive inhibitor.