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

A Thermal Gradient Approach for the Quasi-Harmonic Approximation and its Application to Improved Treatment of Anisotropic Expansion

We present a novel approach to efficiently implement thermal expansion in the quasi-harmonic approximation (QHA) for both isotropic and more importantly, anisotropic expansion. In this approach, we rapidly determine a crystal's equilibrium volume and shape at a given temperature by integrating along the gradient of expansion from zero Kelvin up to the desired temperature. We compare our approach to previous isotropic methods that rely on a brute-force grid search to determine the free energy minimum, which is infeasible to carry out for anisotropic expansion, as well as quasi-anisotropic approaches that take into account the contributions to anisotropic expansion from the lattice energy. We compare these methods for experimentally known polymorphs of piracetam and resorcinol and show that both isotropic methods agree to within error up to 300 K. Using the Gr\"{u}neisen parameter causes up to 0.04 kcal/mol deviation in the Gibbs free energy, but for polymorph free energy differences there is a cancellation in error with all isotropic methods within 0.025 kcal/mol at 300 K. Anisotropic expansion allows the crystals to relax into lattice geometries 0.01-0.23 kcal/mol lower in energy at 300 K relative to isotropic expansion. For polymorph free energy differences all QHA methods produced results within 0.02 kcal/mol of each other for resorcinol and 0.12 kcal/mol for piracetam, the two molecules tested here, demonstrating a cancellation of error for isotropic methods. We also find that when expanding in more than a single volume variable, there is a non-negligible rate of failure of the basic approximations of QHA. Specifically, while expanding into new harmonic modes as the box vectors are increased, the system often falls into alternate, structurally distinct harmonic modes unrelated by continuous deformation from the original harmonic mode.Comment: 38 pages, including 9 pages supporting informatio

Rapid Computation of Thermodynamic Properties Over Multidimensional Nonbonded Parameter Spaces using Adaptive Multistate Reweighting

We show how thermodynamic properties of molecular models can be computed over a large, multidimensional parameter space by combining multistate reweighting analysis with a linear basis function approach. This approach reduces the computational cost to estimate thermodynamic properties from molecular simulations for over 130,000 tested parameter combinations from over a thousand CPU years to tens of CPU days. This speed increase is achieved primarily by computing the potential energy as a linear combination of basis functions, computed from either modified simulation code or as the difference of energy between two reference states, which can be done without any simulation code modification. The thermodynamic properties are then estimated with the Multistate Bennett Acceptance Ratio (MBAR) as a function of multiple model parameters without the need to define a priori how the states are connected by a pathway. Instead, we adaptively sample a set of points in parameter space to create mutual configuration space overlap. The existence of regions of poor configuration space overlap are detected by analyzing the eigenvalues of the sampled states' overlap matrix. The configuration space overlap to sampled states is monitored alongside the mean and maximum uncertainty to determine convergence, as neither the uncertainty or the configuration space overlap alone is a sufficient metric of convergence. This adaptive sampling scheme is demonstrated by estimating with high precision the solvation free energies of charged particles of Lennard-Jones plus Coulomb functional form. We also compute entropy, enthalpy, and radial distribution functions of unsampled parameter combinations using only the data from these sampled states and use the free energies estimates to examine the deviation of simulations from the Born approximation to the solvation free energy