Overcoming finite-sampling errors in fast-switching free-energy estimates: extrapolative analysis of a molecular system

Convergence to the Asymptotic Large Deviation Limit

Large deviation theory offers a powerful and general statistical framework to study the asymptotic dynamical properties of rare events. The application of the formalism to concrete experimental situations is, however, often restricted by finite statistics. Data might not suffice to reach the asymptotic regime or judge whether large deviation estimators converge at all. We here experimentally study the large deviation properties of the stochastic work and heat of a levitated nanoparticle subjected to nonequilibrium feedback control. This setting allows us to determine for each quantity the convergence domain of the large deviation estimators using a criterion that does not require the knowledge of the probability distribution. By extracting both the asymptotic exponential decay and the subexponential prefactors, we demonstrate that singular prefactors significantly restrict the convergence characteristics close to the singularity. Our results provide unique insight into the approach to the asymptotic large deviation limit and underscore the pivotal role of singular prefactors.

Simulation of Linear and Cyclic Alkanes with Second-Order Møller-Plesset Perturbation Theory through Adaptive Force Matching

Predicting ensemble properties, such as density and heat of vaporization, of small hydrocarbons is challenging due to the dispersion-dominated weak interactions between these molecules. With the adaptive force matching (AFM) method, the bonded and short-range nonbonded interactions are fitted to second-order Møller-Plesset perturbation theory (MP2) references computed with the def2-TZVP basis set. The dispersion is modeled using symmetry adapted perturbation theory (SAPT) at MP4 accuracy using the def2-TZVPD basis set. A new charge matrix decomposition technique is described to obtain partial charges in AFM. Although the models developed do not have any empirical parameters, several properties of the resulting models are compared with experiments as validations. The density, heat of vaporization, pressure dependence of density, diffusion constants, and surface tensions all show quantitative agreement with experiments. Although the density shows a very small systematic error, which could be due to missing three-body dispersion, the heat of vaporization agrees with experiments of within 0.5%. The paper shows that AFM can be used as a reliable tool to enable simulations at post-Hartree-Fock quality at the cost of molecular mechanics force fields.

Reweighting from Molecular Mechanics Force Fields to the ANI-2x Neural Network Potential

To achieve chemical accuracy in free energy calculations, it is necessary to accurately describe the system's potential energy surface and efficiently sample configurations from its Boltzmann distribution. While neural network potentials (NNPs) have shown significantly higher accuracy than classical molecular mechanics (MM) force fields, they have a limited range of applicability and are considerably slower than MM potentials, often by orders of magnitude. To address this challenge, Rufa et al. [Rufa et al. bioRxiv 2020, 10.1101/2020.07.29.227959.] suggested a two-stage approach that uses a fast and established MM alchemical energy protocol, followed by reweighting the results using NNPs, known as endstate correction or indirect free energy calculation. This study systematically investigates the accuracy and robustness of reweighting from an MM reference to a neural network target potential (ANI-2x) for an established data set in vacuum, using single-step free-energy perturbation (FEP) and nonequilibrium (NEQ) switching simulation. We assess the influence of longer switching lengths and the impact of slow degrees of freedom on outliers in the work distribution and compare the results to those of multistate equilibrium free energy simulations. Our results demonstrate that free energy calculations between NNPs and MM potentials should be preferably performed using NEQ switching simulations to obtain accurate free energy estimates. NEQ switching simulations between the MM potentials and NNPs are efficient, robust, and trivial to implement.

A modified Jarzynski free-energy estimator to eliminate non-conservative forces and its application in nanoparticle-membrane interactions

Computational methods to understand interactions in bio-complex systems are however limited to time-scales typically much shorter than in Nature. For example, on the nanoscale level, interactions between nanoparticles (NPs)/molecules/peptides and membranes are central in complex biomolecular processes such as membrane-coated NPs or cellular uptake. This can be remedied by the application of e.g. Jarzynski's equality where thermodynamic properties are extracted from non-equilibrium simulations. Although, the out of equilibrium work leads to non-conservative forces. We here propose a correction Pair Forces method, that removes these forces. Our proposed method is based on the calculation of pulling forces in backward and forward directions for the Jarzynski free-energy estimator using steered molecular dynamics simulation. Our results show that this leads to much improvement for NP-membrane translocation free energies. Although here we have demonstrated the application of the method in molecular dynamics simulation, it could be applied for experimental approaches.

On the Use of Interaction Entropy and Related Methods to Estimate Binding Entropies

Molecular mechanics combined with Poisson-Boltzmann or generalized Born and solvent-accessible area solvation energies (MM/PBSA and MM/GBSA) are popular methods to estimate the free energy for the binding of small molecules to biomacromolecules. However, the estimation of the entropy has been problematic and time-consuming. Traditionally, normal-mode analysis has been used to estimate the entropy, but more recently, alternative approaches have been suggested. In particular, it has been suggested that exponential averaging of the electrostatic and Lennard-Jones interaction energies may provide much faster and more accurate entropies, the interaction entropy (IE) approach. In this study, we show that this exponential averaging is extremely poorly conditioned. Using stochastic simulations, assuming that the interaction energies follow a Gaussian distribution, we show that if the standard deviation of the interaction energies (σIE) is larger than 15 kJ/mol, it becomes practically impossible to converge the interaction entropies (more than 10 million energies are needed, and the number increases exponentially). A cumulant approximation to the second order of the exponential average shows a better convergence, but for σIE > 25 kJ/mol, it gives entropies that are unrealistically large. Moreover, in practical applications, both methods show a steady increase in the entropy with the number of energies considered.

Alchemical Binding Free Energy Calculations in AMBER20: Advances and Best Practices for Drug Discovery

Predicting protein-ligand binding affinities and the associated thermodynamics of biomolecular recognition is a primary objective of structure-based drug design. Alchemical free energy simulations offer a highly accurate and computationally efficient route to achieving this goal. While the AMBER molecular dynamics package has successfully been used for alchemical free energy simulations in academic research groups for decades, widespread impact in industrial drug discovery settings has been minimal because of the previous limitations within the AMBER alchemical code, coupled with challenges in system setup and postprocessing workflows. Through a close academia-industry collaboration we have addressed many of the previous limitations with an aim to improve accuracy, efficiency, and robustness of alchemical binding free energy simulations in industrial drug discovery applications. Here, we highlight some of the recent advances in AMBER20 with a focus on alchemical binding free energy (BFE) calculations, which are less computationally intensive than alternative binding free energy methods where full binding/unbinding paths are explored. In addition to scientific and technical advances in AMBER20, we also describe the essential practical aspects associated with running relative alchemical BFE calculations, along with recommendations for best practices, highlighting the importance not only of the alchemical simulation code but also the auxiliary functionalities and expertise required to obtain accurate and reliable results. This work is intended to provide a contemporary overview of the scientific, technical, and practical issues associated with running relative BFE simulations in AMBER20, with a focus on real-world drug discovery applications.

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.

A GPU-Accelerated Parameter Interpolation Thermodynamic Integration Free Energy Method

There has been a resurgence of interest in free energy methods motivated by the performance enhancements offered by molecular dynamics (MD) software written for specialized hardware, such as graphics processing units (GPUs). In this work, we exploit the properties of a parameter-interpolated thermodynamic integration (PI-TI) method to connect states by their molecular mechanical (MM) parameter values. This pathway is shown to be better behaved for Mg2+ → Ca2+ transformations than traditional linear alchemical pathways (with and without soft-core potentials). The PI-TI method has the practical advantage that no modification of the MD code is required to propagate the dynamics, and unlike with linear alchemical mixing, only one electrostatic evaluation is needed (e.g., single call to particle-mesh Ewald) leading to better performance. In the case of AMBER, this enables all the performance benefits of GPU-acceleration to be realized, in addition to unlocking the full spectrum of features available within the MD software, such as Hamiltonian replica exchange (HREM). The TI derivative evaluation can be accomplished efficiently in a post-processing step by reanalyzing the statistically independent trajectory frames in parallel for high throughput. We also show how one can evaluate the particle mesh Ewald contribution to the TI derivative evaluation without needing to perform two reciprocal space calculations. We apply the PI-TI method with HREM on GPUs in AMBER to predict p Ka values in double stranded RNA molecules and make comparison with experiments. Convergence to under 0.25 units for these systems required 100 ns or more of sampling per window and coupling of windows with HREM. We find that MM charges derived from ab initio QM/MM fragment calculations improve the agreement between calculation and experimental results.

The role of intramolecular nonbonded interaction and angle sampling in single-step free energy perturbation

Single-step free energy perturbation (sFEP) has often been proposed as an efficient tool for a quick free energy scan due to its straightforward protocol and the ability to recycle an existing molecular dynamics trajectory for free energy calculations. Although sFEP is expected to fail when the sampling of a system is inefficient, it is often expected to hold for an alchemical transformation between ligands with a moderate difference in their sizes, e.g., transforming a benzene into an ethylbenzene. Yet, exceptions were observed in calculations for anisole and methylaniline, which have similar physical sizes as ethylbenzene. In this study, we show that such exceptions arise from the sampling inefficiency on an unexpected rigid degree of freedom, namely, the bond angle θ. The distributions of θ differ dramatically between two end states of a sFEP calculation, i.e., the conformation of the ligand changes significantly during the alchemical transformation process. Our investigation also reveals the interrelation between the ligand conformation and the intramolecular nonbonded interactions. This knowledge suggests a best combination of the ghost ligand potential and the dual topology setting, which improves the accuracy in a single reference sFEP calculation by bringing down its error from around 5k_BT to k_BT.

Jarzynski matrix equality: Calculating the free-energy difference by nonequilibrium simulations with an arbitrary initial distribution

The Jarzynski equality (JE) method, which relates the work of a nonequilibrium process to the free-energy difference between its initial and final states, provides an efficient way to calculate free energies of thermodynamic systems in simulations or experiments. However, more extensive applications of the JE are hindered by the requirement that the initial state must be in equilibrium. In this work we extend the JE method to be the Jarzynski matrix equality (JME) method, which relates the work of trajectories connecting metastable conformational regions to their local free energies, and thus we can estimate the free energy from the nonequilibrium trajectories starting from an almost arbitrary initial distribution. We then apply the JME to toy models, Lennard-Jones fluids, and polymer chain models, demonstrating its efficiency in free-energy calculations with satisfactory accuracy. The JME extends the applicability of the nonequilibrium methods to complex systems whose initial equilibrium states are difficult to reach.

Convergence of large-deviation estimators

We study the convergence of statistical estimators used in the estimation of large-deviation functions describing the fluctuations of equilibrium, nonequilibrium, and manmade stochastic systems. We give conditions for the convergence of these estimators with sample size, based on the boundedness or unboundedness of the quantity sampled, and discuss how statistical errors should be defined in different parts of the convergence region. Our results shed light on previous reports of "phase transitions" in the statistics of free energy estimators and establish a general framework for reliably estimating large-deviation functions from simulation and experimental data and identifying parameter regions where this estimation converges.

Prior-predictive value from fast-growth simulations: error analysis and bias estimation

Variants of fluctuation theorems recently discovered in the statistical mechanics of nonequilibrium processes may be used for the efficient determination of high-dimensional integrals as typically occurring in Bayesian data analysis. In particular for multimodal distributions, Monte Carlo procedures not relying on perfect equilibration are advantageous. We provide a comprehensive statistical error analysis for the determination of the prior-predictive value (the evidence) in a Bayes problem, building on a variant of the Jarzynski equation. Special care is devoted to the characterization of the bias intrinsic to the method and statistical errors arising from exponential averages. We also discuss the determination of averages over multimodal posterior distributions with the help of a consequence of the Crooks relation. All our findings are verified by extensive numerical simulations of two model systems with bimodal likelihoods.

The adaptive biasing force method: everything you always wanted to know but were afraid to ask

In the host of numerical schemes devised to calculate free energy differences by way of geometric transformations, the adaptive biasing force algorithm has emerged as a promising route to map complex free-energy landscapes. It relies upon the simple concept that as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it yields a Hamiltonian devoid of an average force acting along the transition coordinate of interest. This means that sampling proceeds uniformly on a flat free-energy surface, thus providing reliable free-energy estimates. Much of the appeal of the algorithm to the practitioner is in its physically intuitive underlying ideas and the absence of any requirements for prior knowledge about free-energy landscapes. Since its inception in 2001, the adaptive biasing force scheme has been the subject of considerable attention, from in-depth mathematical analysis of convergence properties to novel developments and extensions. The method has also been successfully applied to many challenging problems in chemistry and biology. In this contribution, the method is presented in a comprehensive, self-contained fashion, discussing with a critical eye its properties, applicability, and inherent limitations, as well as introducing novel extensions. Through free-energy calculations of prototypical molecular systems, many methodological aspects are examined, from stratification strategies to overcoming the so-called hidden barriers in orthogonal space, relevant not only to the adaptive biasing force algorithm but also to other importance-sampling schemes. On the basis of the discussions in this paper, a number of good practices for improving the efficiency and reliability of the computed free-energy differences are proposed.

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.

Parallel-pulling protocol for free-energy evaluation

Jarzynski's equality (JE) allows us to compute free-energy differences from distributions of work. In molecular dynamics simulations, the traditional way of constructing work distributions is to perform as many pulling simulations as possible. But reliable work distributions are not always produced in a finite number of simulations. The computational cost of using JE is not less than other commonly used methods such as thermodynamic integration and umbrella sampling methods. Here we first show a different proof of JE based on the idea of stepwise pulling procedures that is efficient in computing free energies by using JE. The key point in our proof is that the processes of turning on or off a harmonic potential to perform work are described by double Heaviside functions of time. We then show that the distributions of work performed by the potential can be easily generated from the distributions of a reaction coordinate along a pathway. Based on the proof, we propose sequential and parallel stepwise pulling protocols for generating work distributions that require suitable relaxation time at each pulling step. The criterion for reliable work distributions is that there must be sufficient mutual overlaps between the adjacent distributions of the reaction coordinate along the pathway. We arrive at an alternative formula (besides JE) to compute free-energy differences from the averaged values of the reaction coordinate. The combination of JE and the alternative formula provides a viable way to determine the accuracy of computed free-energy differences. For the stretching of a deca-alanine molecule, our approach requires 21 parallel simulations and relaxation time as small as 0.4 ns for each simulation to estimate free-energy differences with an uncertainty of about 13%.

Good practices in free-energy calculations

As access to computational resources continues to increase, free-energy calculations have emerged as a powerful tool that can play a predictive role in a wide range of research areas. Yet, the reliability of these calculations can often be improved significantly if a number of precepts, or good practices, are followed. Although the theory upon which these good practices rely has largely been known for many years, it is often overlooked or simply ignored. In other cases, the theoretical developments are too recent for their potential to be fully grasped and merged into popular platforms for the computation of free-energy differences. In this contribution, the current best practices for carrying out free-energy calculations using free energy perturbation and nonequilibrium work methods are discussed, demonstrating that at little to no additional cost, free-energy estimates could be markedly improved and bounded by meaningful error estimates. Monitoring the probability distributions that underlie the transformation between the states of interest, performing the calculation bidirectionally, stratifying the reaction pathway, and choosing the most appropriate paradigms and algorithms for transforming between states offer significant gains in both accuracy and precision.

Assessment of nonequilibrium free energy methods

One of the factors preventing the general application of free energy methods in rational drug design remains the lack of sufficient computational resources. Many nonequilibrium (NE) free energy methods, however, are easily made embarrassingly parallel in comparison to equilibrium methods and may be conveniently run on desktop computers using distributed computing software. In recent years, there has been a proliferation of NE methods, but the general applicability of these approaches has not been determined. In this study, a subset including only those NE methods which are easily parallelised were considered for examination, with a view to their application to the prediction of protein-ligand binding affinities. A number of test systems were examined, including harmonic oscillator (HO) systems and the calculation of relative free energies of hydration of water-methane. The latter system uses identical potentials to the protein ligand case and is therefore an appropriate model system on which methods may be tested. As well as investigating existing protocols, a replica exchange NE approach was developed, which was found to offer advantages over conventional methods. It was found that Rosenbluth-based approaches to optimizing the NE work values used in NE free energy estimates were not consistent in the improvements in accuracy achieved and that, given their computational cost, the simple approach of taking each work value in an unbiased way is to be preferred. Of the two free energy estimators examined, Bennett's acceptance ratio was the most consistent and is, therefore, to be preferred over the Jarzynski estimator. The recommended protocols may be run very efficiently within a distributed computing environment and are of similar accuracy and precision to equilibrium free energy methods.

Bayesian estimates of free energies from nonequilibrium work data in the presence of instrument noise

The Jarzynski equality and the fluctuation theorem relate equilibrium free energy differences to nonequilibrium measurements of the work. These relations extend to single-molecule experiments that have probed the finite-time thermodynamics of proteins and nucleic acids. The effects of experimental error and instrument noise have not been considered previously. Here, we present a Bayesian formalism for estimating free energy changes from nonequilibrium work measurements that compensates for instrument noise and combines data from multiple driving protocols. We reanalyze a recent set of experiments in which a single RNA hairpin is unfolded and refolded using optical tweezers at three different rates. Interestingly, the fastest and farthest-from-equilibrium measurements contain the least instrumental noise and, therefore, provide a more accurate estimate of the free energies than a few slow, more noisy, near-equilibrium measurements. The methods we propose here will extend the scope of single-molecule experiments; they can be used in the analysis of data from measurements with atomic force microscopy, optical, and magnetic tweezers.

Gibbs free-energy estimates from direct path-sampling computations

We have implemented a path-sampling scheme enabling a direct estimation of Gibbs free energy. This scheme consists of a Monte Carlo sampling of constant-pressure Langevin paths, followed by an ensemble averaging carried out over the Markov chain of paths. In practice, we sample an umbrella path ensemble, which requires to rigorously define a statistical weight for the paths, equivalent of the Boltzmann weight. This statistical weight is a function of an effective work related to the path. The umbrella ensemble is chosen so that its work histogram overlaps with the histograms corresponding to the ensembles of forward and backward paths. We have finally investigated the relations between numerical efficiency and overlapping properties of the various work histograms. This analysis yields a built-in criterion for diagnosing the convergence during a single-run simulation.

Peptide conformational equilibria computed via a single-stage shifting protocol

We study the conformational equilibria of two peptides using a novel statistical mechanics approach designed for calculating free energy differences between highly dissimilar conformational states. Our results elucidate the contrasting roles of entropy in implicitly solvated leucine dipeptide and decaglycine. The method extends earlier work by Voter and overcomes the notorious "overlap" problem in free energy computations by constructing a mathematically equivalent calculation with high conformational similarity. The approach requires only equilibrium simulations of the two states of interest, without the need for sampling transition states. We discuss possible extensions and optimizations of the approach.

Single molecule pulling with large time steps

Recently, we presented a generalization of the Jarzynski nonequilibrium work theorem for phase space mappings. The formalism shows that one can determine free energy differences from approximate trajectories obtained from molecular dynamics simulations in which very large time steps are used. In this work we test the method by simulating the force-induced unfolding of a deca-alanine helix in vacuum. The excellent agreement between results obtained with a small, conservative time step of 0.5 fs and results obtained with a time step of 3.2 fs (i.e., close to the stability limit) indicates that the large-time-step approach is practical for such complex biomolecules. We further adapt the method of Hummer and Szabo for the simulation of single molecule force spectroscopy experiments to the large-time-step method. While trajectories generated with large steps are approximate and may be unphysical--in the simulations presented here we observe a violation of the equipartition theorem--the computed free energies are exact in principle. In terms of efficiency, the optimum time step for the unfolding simulations lies in the range 1-3fs .

Single-ensemble nonequilibrium path-sampling estimates of free energy differences

We introduce a straightforward, single-ensemble, path sampling approach to calculate free energy differences based on Jarzynski's relation. For a two-dimensional "toy" test system, the new (minimally optimized) method performs roughly one hundred times faster than either optimized "traditional" Jarzynski calculations or conventional thermodynamic integration. The simplicity of the underlying formalism suggests the approach will find broad applicability in molecular systems.

Single-molecule experiments in biological physics: methods and applications

I review single-molecule experiments (SMEs) in biological physics. Recent technological developments have provided the tools to design and build scientific instruments of high enough sensitivity and precision to manipulate and visualize individual molecules and measure microscopic forces. Using SMEs it is possible to manipulate molecules one at a time and measure distributions describing molecular properties, characterize the kinetics of biomolecular reactions and detect molecular intermediates. SMEs provide additional information about thermodynamics and kinetics of biomolecular processes. This complements information obtained in traditional bulk assays. In SMEs it is also possible to measure small energies and detect large Brownian deviations in biomolecular reactions, thereby offering new methods and systems to scrutinize the basic foundations of statistical mechanics. This review is written at a very introductory level, emphasizing the importance of SMEs to scientists interested in knowing the common playground of ideas and the interdisciplinary topics accessible by these techniques. The review discusses SMEs from an experimental perspective, first exposing the most common experimental methodologies and later presenting various molecular systems where such techniques have been applied. I briefly discuss experimental techniques such as atomic-force microscopy (AFM), laser optical tweezers (LOTs), magnetic tweezers (MTs), biomembrane force probes (BFPs) and single-molecule fluorescence (SMF). I then present several applications of SME to the study of nucleic acids (DNA, RNA and DNA condensation) and proteins (protein-protein interactions, protein folding and molecular motors). Finally, I discuss applications of SMEs to the study of the nonequilibrium thermodynamics of small systems and the experimental verification of fluctuation theorems. I conclude with a discussion of open questions and future perspectives.

Calculating potentials of mean force from steered molecular dynamics simulations

Steered molecular dynamics (SMD) permits efficient investigations of molecular processes by focusing on selected degrees of freedom. We explain how one can, in the framework of SMD, employ Jarzynski's equality (also known as the nonequilibrium work relation) to calculate potentials of mean force (PMF). We outline the theory that serves this purpose and connects nonequilibrium processes (such as SMD simulations) with equilibrium properties (such as the PMF). We review the derivation of Jarzynski's equality, generalize it to isobaric--isothermal processes, and discuss its implications in relation to the second law of thermodynamics and computer simulations. In the relevant regime of steering by means of stiff springs, we demonstrate that the work on the system is Gaussian-distributed regardless of the speed of the process simulated. In this case, the cumulant expansion of Jarzynski's equality can be safely terminated at second order. We illustrate the PMF calculation method for an exemplary simulation and demonstrate the Gaussian nature of the resulting work distribution.

Comparison of efficiency and bias of free energies computed by exponential averaging, the Bennett acceptance ratio, and thermodynamic integration

Recent work has demonstrated the Bennett acceptance ratio method is the best asymptotically unbiased method for determining the equilibrium free energy between two end states given work distributions collected from either equilibrium and nonequilibrium data. However, it is still not clear what the practical advantage of this acceptance ratio method is over other common methods in atomistic simulations. In this study, we first review theoretical estimates of the bias and variance of exponential averaging (EXP), thermodynamic integration (TI), and the Bennett acceptance ratio (BAR). In the process, we present a new simple scheme for computing the variance and bias of many estimators, and demonstrate the connections between BAR and the weighted histogram analysis method. Next, a series of analytically solvable toy problems is examined to shed more light on the relative performance in terms of the bias and efficiency of these three methods. Interestingly, it is impossible to conclusively identify a "best" method for calculating the free energy, as each of the three methods performs more efficiently than the others in at least one situation examined in these toy problems. Finally, sample problems of the insertion/deletion of both a Lennard-Jones particle and a much larger molecule in TIP3P water are examined by these three methods. In all tests of atomistic systems, free energies obtained with BAR have significantly lower bias and smaller variance than when using EXP or TI, especially when the overlap in phase space between end states is small. For example, BAR can extract as much information from multiple fast, far-from-equilibrium simulations as from fewer simulations near equilibrium, which EXP cannot. Although TI and sometimes even EXP can be somewhat more efficient in idealized toy problems, in the realistic atomistic situations tested in this paper, BAR is significantly more efficient than all other methods.

Simple estimation of absolute free energies for biomolecules

One reason that free energy difference calculations are notoriously difficult in molecular systems is due to insufficient conformational overlap, or similarity, between the two states or systems of interest. The degree of overlap is irrelevant, however, if the absolute free energy of each state can be computed. We present a method for calculating the absolute free energy that employs a simple construction of an exactly computable reference system which possesses high overlap with the state of interest. The approach requires only a physical ensemble of conformations generated via simulation and an auxiliary calculation of approximately equal central-processing-unit cost. Moreover, the calculations can converge to the correct free energy value even when the physical ensemble is incomplete or improperly distributed. As a "proof of principle," we use the approach to correctly predict free energies for test systems where the absolute values can be calculated exactly and also to predict the conformational equilibrium for leucine dipeptide in implicit solvent.

Calculating potentials of mean force and diffusion coefficients from nonequilibrium processes without Jarzynski’s equality

In general, the direct application of the Jarzynski equality (JE) to reconstruct potentials of mean force (PMFs) from a small number of nonequilibrium unidirectional steered molecular-dynamics (SMD) paths is hindered by the lack of sampling of extremely rare paths with negative dissipative work. Such trajectories that transiently violate the second law of thermodynamics are crucial for the validity of JE. As a solution to this daunting problem, we propose a simple and efficient method, referred to as the FR method, for calculating simultaneously both the PMF U(z) and the corresponding diffusion coefficient D(z) along a reaction coordinate z for a classical many-particle system by employing a small number of fast SMD pullings in both forward (F) and time reverse (R) directions, without invoking JE. By employing Crooks [Phys. Rev. E 61, 2361 (2000)] transient fluctuation theorem (that is more general than JE) and the stiff-spring approximation, we show that (i) the mean dissipative work W(d) in the F and R pullings is the same, (ii) both U(z) and W(d) can be expressed in terms of the easily calculable mean work of the F and R processes, and (iii) D(z) can be expressed in terms of the slope of W(d). To test its viability, the FR method is applied to determine U(z) and D(z) of single-file water molecules in single-walled carbon nanotubes (SWNTs). The obtained U(z) is found to be in very good agreement with the results from other PMF calculation methods, e.g., umbrella sampling. Finally, U(z) and D(z) are used as input in a stochastic model, based on the Fokker-Planck equation, for describing water transport through SWNTs on a mesoscopic time scale that in general is inaccessible to MD simulations.

Using non-Gaussian density functional fits to improve relative free energy calculations

The accurate and reliable computation of relative free energy differences remains an important long-term goal. Major stumbling blocks for achieving this goal reflect the difficulty of sampling in a known fashion along the reaction coordinate and of maximally combining information that has been collected from the simulation along the reaction coordinate. In this paper we examine the utility of a probability density functional type fit to the distribution of work events collected during a nonequilibrium sample along the reaction coordinate. This approach can readily be generalized to equilibrium sampling and has the potential to estimate the quality of a relative free energy estimate as data are being collected. The method may have the greatest utility for nonequilibrium sampling where non-Gaussian work distributions are generally present that are strongly dominated by rare event sampling in the tail region. We believe that the approach can be used to augment the design and the error analysis of relative free energy computations thus improving the ability to reliably and with known accuracy compute a relative free energy.

Enhancing the accuracy, the efficiency and the scope of free energy simulations

Many different methods exist for computing free energy changes from molecular simulations. Recent advances have led to improvements in the theoretical framework underlying these calculations, as well as in the accuracy and sampling efficiency of the algorithms. Novel methods combining the advantages afforded by various existing approaches offer promising strategies and open up new perspectives to help elucidate the physical basis of important biological processes.

Tools for channels: moving towards molecular calculations of gating and permeation in ion channel biophysics

Recent X-ray structures of voltage gated potassium channels provide an exciting opportunity to connect molecular structures with measured biological function. Two of the most important connections for these channels are: first, to the molecular basis behind selectivity and the associated free energy profile underlying ionic current flow and, second, to a true molecular understanding of the large-scale conformational transitions that underlie voltage dependent gating. But, existing computational tools need to be further developed to reach these goals. In this contribution to the symposia on sampling methods we outline our dynamic importance sampling method for sampling large-scale conformational transitions as well as our studies with non-equilibrium work events and equilibrium overlap sampling (OS) methods for sampling events related to the calculation of relative free energies.

Experimental test of Hatano and Sasa's nonequilibrium steady-state equality

Most natural processes occur far from equilibrium and cannot be treated within the framework of classical thermodynamics. In 1998, Oono and Paniconi [Oono, Y. & Paniconi, M. (1998) Prog. Theor. Phys. Suppl. 130, 29-44] proposed a general phenomenological framework, steady-state thermodynamics, encompassing nonequilibrium steady states and transitions between such states. In 2001, Hatano and Sasa [Hatano, T. & Sasa, S. (2001) Phys. Rev. Lett. 86, 3463-3466] derived a testable prediction of this theory. Specifically, they were able to show that the exponential average of Y, a quantity similar to a dissipated work, should be equal to zero for arbitrary transitions between nonequilibrium steady states, -ln = 0. We have tested this strong prediction by measuring the dissipation and fluctuations of microspheres optically driven through water. We have found that -ln approximately 0 for three different nonequilibrium systems, supporting Hatano and Sasa's proposed extension of thermodynamics to arbitrary steady states and irreversible transitions.

Using overlap and funnel sampling to obtain accurate free energies from nonequilibrium work measurements

Two concepts are presented for accurate nonequilibrium work free-energy measurements, realized both in molecular simulation and experiment. First, the need for an intermediate important to both the reference and the target systems (overlap) is indicated. Second, the use of a soft path from each end point to the intermediate (funnel) is demonstrated. Schemes implementing these concepts dramatically improve efficiency and accuracy of free energy calculations, as shown by calculation of the free energy of ion charging in water, and the free energy change in mutation of an adenosine molecule.

Efficient use of nonequilibrium measurement to estimate free energy differences for molecular systems

A promising method for calculating free energy differences DeltaF is to generate nonequilibrium data via "fast-growth" simulations or by experiments--and then use Jarzynski's equality. However, a difficulty with using Jarzynski's equality is that DeltaF estimates converge very slowly and unreliably due to the nonlinear nature of the calculation--thus requiring large, costly data sets. The purpose of the work presented here is to determine the best estimate for DeltaF given a (finite) set of work values previously generated by simulation or experiment. Exploiting statistical properties of Jarzynski's equality, we present two fully automated analyses of nonequilibrium data from a toy model, and various simulated molecular systems. Both schemes remove at least several k(B)T of bias from DeltaF estimates, compared to direct application of Jarzynski's equality, for modest sized data sets (100 work values), in all tested systems. Results from one of the new methods suggest that good estimates of DeltaF can be obtained using 5-40-fold less data than was previously possible. Extending previous work, the new results exploit the systematic behavior of bias due to finite sample size. A key innovation is better use of the more statistically reliable information available from the raw data.

Equilibrium free energies from nonequilibrium measurements using maximum-likelihood methods

We present a maximum likelihood argument for the Bennett acceptance ratio method, and derive a simple formula for the variance of free energy estimates generated using this method. This derivation of the acceptance ratio method, using a form of logistic regression, a common statistical technique, allows us to shed additional light on the underlying physical and statistical properties of the method. For example, we demonstrate that the acceptance ratio method yields the lowest variance for any estimator of the free energy which is unbiased in the limit of large numbers of measurements.

Bias and error in estimates of equilibrium free-energy differences from nonequilibrium measurements

In 1997, Jarzynski proved a remarkable equality that allows one to compute the equilibrium free-energy difference DeltaF between two states from the probability distribution of the nonequilibrium work W done on the system to switch between the states, e-DeltaF/kappaT =e-W/kappaT [Jarzynski, C. (1997) Phys. Rev. Lett. 87, 2690-2693]. The Jarzynski equality provides a powerful free-energy difference estimator from a set of N irreversible experiments and is closely related to free-energy perturbation, a common computational technique for estimating free-energy differences. Despite the many applications of the Jarzynski estimator, its behavior is only poorly understood. In this article we derive the large N limit for the Jarzynski estimator bias, variance, and mean square error that is correct for arbitrary perturbations. We then analyze the properties of the Jarzynski estimator for all N when the probability distribution of work values is Gaussian, as occurs, for example, in the near-equilibrium regime. This allows us to quantitatively compare it to two other free-energy difference estimators: the mean work estimator and the fluctuation-dissipation theorem estimator. We show that, for near-equilibrium switching, the Jarzynski estimator is always superior to the mean work estimator and is even superior to the fluctuation-dissipation estimator for small N. The Jarzynski-estimator bias is shown to be the dominant source of error in many cases. Our expression for the bias is used to develop a bias-corrected Jarzynski free-energy difference estimator in the near-equilibrium regime.