Statistically optimal analysis of samples from multiple equilibrium states

We present a new estimator for computing free energy differences and thermodynamic expectations as well as their uncertainties from samples obtained from multiple equilibrium states via either simulation or experiment. The estimator, which we term the multistate Bennett acceptance ratio (MBAR) estimator because it reduces to the Bennett acceptance ratio when only two states are considered, has significant advantages over multiple histogram reweighting methods for combining data from multiple states. It does not require the sampled energy range to be discretized to produce histograms, eliminating bias due to energy binning and significantly reducing the time complexity of computing a solution to the estimating equations in many cases. Additionally, an estimate of the statistical uncertainty is provided for all estimated quantities. In the large sample limit, MBAR is unbiased and has the lowest variance of any known estimator for making use of equilibrium data collected from multiple states. We illustrate this method by producing a highly precise estimate of the potential of mean force for a DNA hairpin system, combining data from multiple optical tweezer measurements under constant force bias.Comment: 13 pages (including appendices), 1 figure, LaTe

Spectral rate theory for projected two-state kinetics

Classical rate theories often fail in cases where the observable(s) or order parameter(s) used are poor reaction coordinates or the observed signal is deteriorated by noise, such that no clear separation between reactants and products is possible. Here, we present a general spectral two-state rate theory for ergodic dynamical systems in thermal equilibrium that explicitly takes into account how the system is observed. The theory allows the systematic estimation errors made by standard rate theories to be understood and quantified. We also elucidate the connection of spectral rate theory with the popular Markov state modeling (MSM) approach for molecular simulation studies. An optimal rate estimator is formulated that gives robust and unbiased results even for poor reaction coordinates and can be applied to both computer simulations and single-molecule experiments. No definition of a dividing surface is required. Another result of the theory is a model-free definition of the reaction coordinate quality (RCQ). The RCQ can be bounded from below by the directly computable observation quality (OQ), thus providing a measure allowing the RCQ to be optimized by tuning the experimental setup. Additionally, the respective partial probability distributions can be obtained for the reactant and product states along the observed order parameter, even when these strongly overlap. The effects of both filtering (averaging) and uncorrelated noise are also examined. The approach is demonstrated on numerical examples and experimental single-molecule force probe data of the p5ab RNA hairpin and the apo-myoglobin protein at low pH, here focusing on the case of two-state kinetics

Time step rescaling recovers continuous-time dynamical properties for discrete-time Langevin integration of nonequilibrium systems

When simulating molecular systems using deterministic equations of motion (e.g., Newtonian dynamics), such equations are generally numerically integrated according to a well-developed set of algorithms that share commonly agreed-upon desirable properties. However, for stochastic equations of motion (e.g., Langevin dynamics), there is still broad disagreement over which integration algorithms are most appropriate. While multiple desiderata have been proposed throughout the literature, consensus on which criteria are important is absent, and no published integration scheme satisfies all desiderata simultaneously. Additional nontrivial complications stem from simulating systems driven out of equilibrium using existing stochastic integration schemes in conjunction with recently-developed nonequilibrium fluctuation theorems. Here, we examine a family of discrete time integration schemes for Langevin dynamics, assessing how each member satisfies a variety of desiderata that have been enumerated in prior efforts to construct suitable Langevin integrators. We show that the incorporation of a novel time step rescaling in the deterministic updates of position and velocity can correct a number of dynamical defects in these integrators. Finally, we identify a particular splitting that has essentially universally appropriate properties for the simulation of Langevin dynamics for molecular systems in equilibrium, nonequilibrium, and path sampling contexts.Comment: 15 pages, 2 figures, and 2 table

Splitting probabilities as a test of reaction coordinate choice in single-molecule experiments

To explain the observed dynamics in equilibrium single-molecule measurements of biomolecules, the experimental observable is often chosen as a putative reaction coordinate along which kinetic behavior is presumed to be governed by diffusive dynamics. Here, we invoke the splitting probability as a test of the suitability of such a proposed reaction coordinate. Comparison of the observed splitting probability with that computed from the kinetic model provides a simple test to reject poor reaction coordinates. We demonstrate this test for a force spectroscopy measurement of a DNA hairpin

Towards Automated Benchmarking of Atomistic Forcefields: Neat Liquid Densities and Static Dielectric Constants from the ThermoML Data Archive

Atomistic molecular simulations are a powerful way to make quantitative predictions, but the accuracy of these predictions depends entirely on the quality of the forcefield employed. While experimental measurements of fundamental physical properties offer a straightforward approach for evaluating forcefield quality, the bulk of this information has been tied up in formats that are not machine-readable. Compiling benchmark datasets of physical properties from non-machine-readable sources require substantial human effort and is prone to accumulation of human errors, hindering the development of reproducible benchmarks of forcefield accuracy. Here, we examine the feasibility of benchmarking atomistic forcefields against the NIST ThermoML data archive of physicochemical measurements, which aggregates thousands of experimental measurements in a portable, machine-readable, self-annotating format. As a proof of concept, we present a detailed benchmark of the generalized Amber small molecule forcefield (GAFF) using the AM1-BCC charge model against measurements (specifically bulk liquid densities and static dielectric constants at ambient pressure) automatically extracted from the archive, and discuss the extent of available data. The results of this benchmark highlight a general problem with fixed-charge forcefields in the representation low dielectric environments such as those seen in binding cavities or biological membranes

Generation and validation

Markov state models of molecular kinetics (MSMs), in which the long-time statistical dynamics of a molecule is approximated by a Markov chain on a discrete partition of configuration space, have seen widespread use in recent years. This approach has many appealing characteristics compared to straightforward molecular dynamics simulation and analysis, including the potential to mitigate the sampling problem by extracting long-time kinetic information from short trajectories and the ability to straightforwardly calculate expectation values and statistical uncertainties of various stationary and dynamical molecular observables. In this paper, we summarize the current state of the art in generation and validation of MSMs and give some important new results. We describe an upper bound for the approximation error made by modelingmolecular dynamics with a MSM and we show that this error can be made arbitrarily small with surprisingly little effort. In contrast to previous practice, it becomes clear that the best MSM is not obtained by the most metastable discretization, but the MSM can be much improved if non-metastable states are introduced near the transition states. Moreover, we show that it is not necessary to resolve all slow processes by the state space partitioning, but individual dynamical processes of interest can be resolved separately. We also present an efficient estimator for reversible transition matrices and a robust test to validate that a MSM reproduces the kinetics of the molecular dynamics data

Replica exchange and expanded ensemble simulations as Gibbs sampling: Simple improvements for enhanced mixing

The widespread popularity of replica exchange and expanded ensemble algorithms for simulating complex molecular systems in chemistry and biophysics has generated much interest in enhancing phase space mixing of these protocols, thus improving their efficiency. Here, we demonstrate how both of these classes of algorithms can be considered a form of Gibbs sampling within a Markov chain Monte Carlo (MCMC) framework. While the update of the conformational degrees of freedom by Metropolis Monte Carlo or molecular dynamics unavoidably generates correlated samples, we show how judicious updating of the thermodynamic state indices---corresponding to thermodynamic parameters such as temperature or alchemical coupling variables---associated with these configurations can substantially increase mixing while still sampling from the desired distributions. We show how state update methods in common use lead to suboptimal mixing, and present some simple, inexpensive alternatives that can increase mixing of the overall Markov chain, reducing simulation times necessary to obtain estimates of the desired precision. These improved schemes are demonstrated for several common applications, including an alchemical expanded ensemble simulation, parallel tempering, and multidimensional replica exchange umbrella sampling.Comment: 17 pages, 2 figure

End-to-End Differentiable Molecular Mechanics Force Field Construction

Molecular mechanics (MM) potentials have long been a workhorse of computational chemistry. Leveraging accuracy and speed, these functional forms find use in a wide variety of applications from rapid virtual screening to detailed free energy calculations. Traditionally, MM potentials have relied on human-curated, inflexible, and poorly extensible discrete chemical perception rules (atom types) for applying parameters to molecules or biopolymers, making them difficult to optimize to fit quantum chemical or physical property data. Here, we propose an alternative approach that uses graph nets to perceive chemical environments, producing continuous atom embeddings from which valence and nonbonded parameters can be predicted using a feed-forward neural network. Since all stages are built using smooth functions, the entire process of chemical perception and parameter assignment is differentiable end-to-end with respect to model parameters, allowing new force fields to be easily constructed, extended, and applied to arbitrary molecules. We show that this approach has the capacity to reproduce legacy atom types and can be fit to MM and QM energies and forces, among other targets