Weighted Histogram Analysis Research Articles

The UWHAM and SWHAM Software Package

We introduce the UWHAM (binless weighted histogram analysis method) and SWHAM (stochastic UWHAM) software package that can be used to estimate the density of states and free energy differences based on the data generated by multi-state simulations. The programs used to solve the UWHAM equations are written in the C++ language and operated via the command line interface. In this paper, first we review the theoretical bases of UWHAM, its stochastic solver RE-SWHAM (replica exchange-like SWHAM)and ST-SWHAM (serial tempering-like SWHAM). Then we provide a tutorial with examples that explains how to apply the UWHAM program package to analyze the data generated by different types of multi-state simulations: umbrella sampling, replica exchange, free energy perturbation simulations, etc. The tutorial examples also show that the UWHAM equations can be solved stochastically by applying the RE-SWHAM and ST-SWHAM programs when the data ensemble is large. If the simulations at some states are far from equilibrium, the Stratified RE-SWHAM program can be applied to obtain the equilibrium distribution of the state of interest. All the source codes and the tutorial examples are available from our group’s web page: https://ronlevygroup.cst.temple.edu/software/UWHAM_and_SWHAM_webpage/index.html.

Ion Permeation through ORAI Proteins

Under depleted calcium levels in the endoplasmic reticulum, calcium release-activated calcium (CRAC) channels generate a sustained calcium influx crucial for several cellular responses such as gene expression, cell migration, and growth. Abnormal CRAC channel activity has been linked to various diseases, including severe combined immunodeficiency and autoimmunity disorders, allergy, and inflammatory bowel disease. CRAC channels are composed of two proteins: Orai proteins, that form the pore subunit, and Stim proteins, that activate the pore. Recent crystal structures of Orai proteins have shed light onto the nature of the permeation pathway of CRAC channels but details of the ion permeation mechanism are yet to be revealed. In this study, we use Umbrella Sampling and the Weighted Histogram Analysis Method to investigate the permeation of calcium, potassium and sodium ions through Orai pores. We extend the calculations to both Orai hexamer and tetramer assemblies, since previous studies have implicated different multimeric states of Orai as functional CRAC channels. In addition, we investigate the effect of rotation of the pore-forming transmembrane helices (suggested to be responsible for gating) on the free energy cost of ion permeation. Our simulations indicate that both hexamer and tetramer assemblies form conducting ion pores in which the free energy barrier is mainly set by the hydrophobic core of the pore. In all cases, helix rotation lowers the energetic cost of translocating an ion.

Fast Solver for Large Scale Multistate Bennett Acceptance Ratio Equations.

The multistate Bennett acceptance ratio method (MBAR) and unbinned weighted histogram analysis method (UWHAM) are widely employed approaches to calculate relative free energies of multiple thermodynamic states that gain statistical precision by employing free energy contributions from configurations sampled at each of the simulated λ states. With the increasing availability of high throughput computing resources, a large number of configurations can be sampled from hundreds or even thousands of states. Combining sampled configurations from all states to calculate relative free energies requires the iterative solution of large scale MBAR/UWHAM equations. In the current work, we describe the development of a fast solver to iteratively solve these large scale MBAR/UWHAM equations utilizing our previous findings that the MBAR/UWHAM equations can be derived as a Rao-Blackwell estimator. The solver is implemented and distributed as a Python module called FastMBAR. Our benchmark results show that FastMBAR is more than 2 times faster than the widely used solver pymbar, when it runs on a central processing unit (CPU) and more than 100 times faster than pymbar when it runs on a graphical processing unit (GPU). The significant speedup achieved by FastMBAR running on a GPU is useful not only for solving large scale MBAR/UWHAM equations but also for estimating uncertainty of calculated free energies using bootstrapping where the MBAR/UWHAM equations need to be solved multiple times.

Prediction of CB[8] host-guest binding free energies in SAMPL6 using the double-decoupling method.

This study reports the results of binding free energy calculations for CB[8] host-guest systems in the SAMPL6 blind challenge (receipt ID 3z83m). Force-field parameters were developed specific for each of host and guest molecules to improve configurational sampling. We used quantum mechanical (QM) implicit solvent calculations and QM force matching to determine non-bonded (partial atomic charges) and bonded terms, respectively. Free energy calculations were carried out using the double-decoupling method (DDM) combined with Hamiltonian replica exchange method (HREM) and Bennett acceptance ratio (BAR) method. The root mean square error (RMSE) of the predicted values using DDM with respect to the experimental results was 4.32kcal/mol. The coefficient of determination (R2) and Kendall rank coefficient (τ) were 0.49 and 0.52, respectively, highest of all submissions. In addition, these were compared to the results obtained by umbrella sampling (US) and weighted histogram analysis method (WHAM). Overall, DDM achieved a higher prediction accuracy than the US method. Results are discussed in terms of parameterization and free energy simulations.

Communication: Potentials of mean force study of ionic liquid ion pair aggregation in polar covalent molecule solvents.

Molecular dynamics (MD) simulations were conducted for 1-ethyl-3-methylimidazolium methylsulfate [EMIM][MeSO4] dissolved in six polar covalent molecules [acetic acid, acetone, chloroform, dimethyl sulfoxide (DMSO), isopropyl alcohol, and methanol] to understand the free energies of ionic liquid (IL) ion pairing/aggregation in the limit of infinite dilution. Free energy landscapes or potentials of mean force (PMF) were computed using umbrella sampling and the weighted histogram analysis method. The PMF studies showed the strongest IL ion pairing in chloroform, and the strength of IL ion pairing decreases in the order of chloroform, acetone, propanol, acetic acid, DMSO, and methanol. In the limit of infinite dilution, the free energy curves for IL ion aggregation in co-solvents were characterized by two distinct minima [global (∼3.6 Å) and local (∼5.7 Å)], while free energy values at these minima differed significantly for IL in each co-solvent. The PMF studies were extended for determining the free energy of IL ion aggregation as a function of concentration of methanol. Studies showed that as the concentration of methanol increased, the free energy of ion aggregation decreased, suggesting greater ion pair stability, in agreement with previously reported MD clustering and radial distribution function data.

Scission Free Energies for Wormlike Surfactant Micelles: Development of a Simulation Protocol, Application, and Validation for Personal Care Formulations.

We present a scheme to calculate wormlike micelle scission free energies from a potential of mean force (PMF) derived from a weighted histogram analysis method (WHAM) applied to coarse grained dissipative particle dynamics (DPD) simulations. In contrast to previous related work, we use a specially chosen external potential based on a reaction coordinate that reversibly drives surfactants out of the nascent scission location. For the application to a model body wash formulation, we predict how addition of NaCl and small molecules such as perfume raw materials (PRMs) affect scission energies. The results show qualitative agreement and correct trends compared to recently determined scission energies for the same system; however, a more rigorous parametrization of the underlying DPD potential is required for quantitative agreement.

“Crawling” on the self-assembly system: A molecular simulation of peptide position adjusting over self-assembly block

By combining non-equilibrium molecular dynamics(NEMD), umbrella sampling, and weighted histogram analysis method(WHAM), we calculated the potential of mean force of histidine peptide moving over a self-assembly structure. The reaction coordinate is along the main chain direction of the histidine peptide in the self-assembly structure. It is found that the energy needed for the histidine peptide with 3 and 5 residues while moving along the reaction coordinate is around -2.2 kCal/mol and -7.4 kCal/mol, respectively. And the histidine peptide crawls along the reaction coordinate, performing a snake-like movement. This result could illustrate how histidine peptide adjusts its position during self-assembly process.

Computational modeling and in-vitro/in-silico correlation of phospholipid-based prodrugs for targeted drug delivery in inflammatory bowel disease.

Targeting drugs to the inflamed intestinal tissue(s) represents a major advancement in the treatment of inflammatory bowel disease (IBD). In this work we present a powerful in-silico modeling approach to guide the molecular design of novel prodrugs targeting the enzyme PLA2, which is overexpressed in the inflamed tissues of IBD patients. The prodrug consists of the drug moiety bound to the sn-2 position of phospholipid (PL) through a carbonic linker, aiming to allow PLA2 to release the free drug. The linker length dictates the affinity of the PL-drug conjugate to PLA2, and the optimal linker will enable maximal PLA2-mediated activation. Thermodynamic integration and Weighted Histogram Analysis Method (WHAM)/Umbrella Sampling method were used to compute the changes in PLA2 transition state binding free energy of the prodrug molecule (∆∆Gtr) associated with decreasing/increasing linker length. The simulations revealed that 6-carbons linker is the optimal one, whereas shorter or longer linkers resulted in decreased PLA2-mediated activation. These in-silico results were shown to be in excellent correlation with experimental in-vitro data. Overall, this modern computational approach enables optimization of the molecular design of novel prodrugs, which may allow targeting the free drug specifically to the diseased intestinal tissue of IBD patients.

Stratified UWHAM and Its Stochastic Approximation for Multicanonical Simulations Which Are Far from Equilibrium.

We describe a new analysis tool called Stratified unbinned Weighted Histogram Analysis Method (Stratified-UWHAM), which can be used to compute free energies and expectations from a multicanonical ensemble when a subset of the parallel simulations is far from being equilibrated because of barriers between free energy basins which are only rarely (or never) crossed at some states. The Stratified-UWHAM equations can be obtained in the form of UWHAM equations but with an expanded set of states. We also provide a stochastic solver, Stratified RE-SWHAM, for Stratified-UWHAM to remove its computational bottleneck. Stratified-UWHAM and Stratified RE-SWHAM are applied to study three test topics: the free energy landscape of alanine dipeptide, the binding affinity of a host-guest binding complex, and path sampling for a two-dimensional double well potential. The examples show that when some of the parallel simulations are only locally equilibrated, the estimates of free energies and equilibrium distributions provided by the conventional UWHAM (or MBAR) solutions exhibit considerable biases, but the estimates provided by Stratified-UWHAM and Stratified RE-SWHAM agree with the benchmark very well. Lastly, we discuss features of the Stratified-UWHAM approach which is based on coarse-graining in relation to two other maximum likelihood-based methods which were proposed recently, that also coarse-grain the multicanonical data.

Efficient estimation of binding free energies between peptides and an MHC class II molecule using coarse-grained molecular dynamics simulations with a weighted histogram analysis method.

We estimate the binding free energy between peptides and an MHC class II molecule using molecular dynamics (MD) simulations with the weighted histogram analysis method (WHAM). We show that, owing to its more thorough sampling in the available computational time, the binding free energy obtained by pulling the whole peptide using a coarse-grained (CG) force field (MARTINI) is less prone to significant error induced by inadequate-sampling than using an atomistic force field (AMBER). We further demonstrate that using CG MD to pull 3-4 residue peptide segments while leaving the remaining peptide segments in the binding groove and adding up the binding free energies of all peptide segments gives robust binding free energy estimations, which are in good agreement with the experimentally measured binding affinities for the peptide sequences studied. Our approach thus provides a promising and computationally efficient way to rapidly and reliably estimate the binding free energy between an arbitrary peptide and an MHC class II molecule. © 2017 Wiley Periodicals, Inc.

Gibbs Sampler-Based λ-Dynamics and Rao–Blackwell Estimator for Alchemical Free Energy Calculation

λ-dynamics is a generalized ensemble method for alchemical free energy calculations. In traditional λ-dynamics, the alchemical switch variable λ is treated as a continuous variable ranging from 0 to 1 and an empirical estimator is utilized to approximate the free energy. In the present article, we describe an alternative formulation of λ-dynamics that utilizes the Gibbs sampler framework, which we call Gibbs sampler-based λ-dynamics (GSLD). GSLD, like traditional λ-dynamics, can be readily extended to calculate free energy differences between multiple ligands in one simulation. We also introduce a new free energy estimator, the Rao-Blackwell estimator (RBE), for use in conjunction with GSLD. Compared with the current empirical estimator, the advantage of RBE is that RBE is an unbiased estimator and its variance is usually smaller than the current empirical estimator. We also show that the multistate Bennett acceptance ratio equation or the unbinned weighted histogram analysis method equation can be derived using the RBE. We illustrate the use and performance of this new free energy computational framework by application to a simple harmonic system as well as relevant calculations of small molecule relative free energies of solvation and binding to a protein receptor. Our findings demonstrate consistent and improved performance compared with conventional alchemical free energy methods.

BayesWHAM: A Bayesian approach for free energy estimation, reweighting, and uncertainty quantification in the weighted histogram analysis method.

The weighted histogram analysis method (WHAM) is a powerful approach to estimate molecular free energy surfaces (FES) from biased simulation data. Bayesian reformulations of WHAM are valuable in proving statistically optimal use of the data and providing a transparent means to incorporate regularizing priors and estimate statistical uncertainties. In this work, we develop a fully Bayesian treatment of WHAM to generate statistically optimal FES estimates in any number of biasing dimensions under arbitrary choices of the Bayes prior. Rigorous uncertainty estimates are generated by Metropolis-Hastings sampling from the Bayes posterior. We also report a means to project the FES and its uncertainties into arbitrary auxiliary order parameters beyond those in which biased sampling was conducted. We demonstrate the approaches in applications of alanine dipeptide and the unthreading of a synthetic mimic of the astexin-3 lasso peptide. Open-source MATLAB and Python implementations of our codes are available for free public download. © 2017 Wiley Periodicals, Inc.

Thermodynamic properties of solvated peptides from selective integrated tempering sampling with a new weighting factor estimation algorithm

ABSTRACTConformational sampling under rugged energy landscape is always a challenge in computer simulations. The recently developed integrated tempering sampling, together with its selective variant (SITS), emerges to be a powerful tool in exploring the free energy landscape or functional motions of various systems. The estimation of weighting factors constitutes a critical step in these methods and requires accurate calculation of partition function ratio between different thermodynamic states. In this work, we propose a new adaptive update algorithm to compute the weighting factors based on the weighted histogram analysis method (WHAM). The adaptive-WHAM algorithm with SITS is then applied to study the thermodynamic properties of several representative peptide systems solvated in an explicit water box. The performance of the new algorithm is validated in simulations of these solvated peptide systems. We anticipate more applications of this coupled optimisation and production algorithm to other complicated systems such as the biochemical reactions in solution.

Deconstructing Free Energies in the Law of Matching Water Affinities.

The law of matching water affinities (LMWA) is explored in classical molecular dynamics simulations of several alkali halide ion pairs, spanning the size range from small kosmotropes to large chaotropes. The ion-ion potentials of mean force (PMFs) are computed using three methods: the local molecular field theory (LMFT), the weighted histogram analysis method (WHAM), and integration of the average force. All three methods produce the same total PMF for a given ion pair. In addition, LMFT-based partitioning into van der Waals and local and far-field electrostatic free energies and assessment of the enthalpic, entropic, and ion-water components yield insights into the origins of the observed free energy profiles in water. The results highlight the importance of local electrostatic interactions in determining the shape of the PMFs, while longer-ranged interactions enhance the overall ion-ion attraction, as expected in a dielectric continuum model. The association equilibrium constants are estimated from the smooth WHAM curves and compared to available experimental conductance data. By examining the variations in the average hydration numbers of ions with ion-ion distance, a correlation of the water structure in the hydration shells with the free energy features is found.

Optimally Adjusted Mixture Sampling and Locally Weighted Histogram Analysis

ABSTRACTConsider the two problems of simulating observations and estimating expectations and normalizing constants for multiple distributions. First, we present a self-adjusted mixture sampling method, which accommodates both adaptive serial tempering and a generalized Wang–Landau algorithm. The set of distributions are combined into a labeled mixture, with the mixture weights depending on the initial estimates of log normalizing constants (or free energies). Then, observations are generated by Markov transitions, and free energy estimates are adjusted online by stochastic approximation. We propose two stochastic approximation schemes by Rao–Blackwellization of the scheme commonly used, and derive the optimal choice of a gain matrix, resulting in the minimum asymptotic variance for free energy estimation, in a simple and feasible form. Second, we develop an offline method, locally weighted histogram analysis, for estimating free energies and expectations, using all the simulated data from multiple distributions by either self-adjusted mixture sampling or other sampling algorithms. This method can be computationally much faster, with little sacrifice of statistical efficiency, than a global method currently used, especially when a large number of distributions are involved. We provide both theoretical results and numerical studies to demonstrate the advantages of the proposed methods.

Affinity enhancement of nanobody binding to EGFR: in silico site-directed mutagenesis and molecular dynamics simulation approaches

Epidermal growth factor receptor (EGFR), a transmembrane glycoprotein, is overexpressed in many cancers such as head-neck, breast, prostate, and skin cancers for this reason it is a good target in cancer therapy and diagnosis. In nanobody-based cancer diagnosis and treatment, nanobodies with high affinity toward receptor (e.g. EGFR) results in effective treatment or diagnosis of cancer. In this regard, the main aim of this study is to develop a method based on molecular dynamic (MD) simulations for designing of 7D12 based nanobody with high affinity compared with wild-type nanobody. By surveying electrostatic and desolvation interactions between different residues of 7D12 and EGFR, the critical residues of 7D12 that play the main role in the binding of 7D12 to EGFR were elucidated and based on these residues, five logical variants were designed. Following the 50 ns MD simulations, pull and umbrella sampling simulation were performed for 7D12 and all its variants in complex with EGFR. Binding free energy of 7D12 (and all its variants) with EGFR was obtained by weighted histogram analysis method. According to binding free energy results, GLY101 to GLU mutation showed the highest binding affinity but this variant is unstable after 50 ns MD simulations. ALA100 to GLU mutation shows suitable binding enhancement with acceptable structural stability. Suitable position and orientation of GLU in residue 100 of 7D12 against related amino acids of EGFR formed some extra hydrogen and electrostatic interactions which resulted in binding enhancement.

Molecular simulation of CO chemisorption on Co(0001) in presence of supercritical fluid solvent: A potential of mean force study.

For several decades, heterogeneous catalytic processes have been improved through utilizing supercritical fluids (SCFs) as solvents. While numerous experimental studies have been established across a range of chemistries, such as oxidation, pyrolysis, amination, and Fischer-Tropsch synthesis, still there is little fundamental, molecular-level information regarding the role of the SCF on elementary heterogeneous catalytic steps. In this study, the influence of hexane solvent on the adsorption of carbon monoxide on Co(0001), as the first step in the reaction mechanism of many processes involving syngas conversion, is probed. Simulations are performed at various bulk hexane densities, ranging from ideal gas conditions (no SCF hexane) to various near- and super-critical hexane densities. For this purpose, both density functional theory and molecular dynamics simulations are employed to determine the adsorption energy and free energy change during CO chemisorption. Potential of mean force calculations, utilizing umbrella sampling and the weighted histogram analysis method, provide the first commentary on SCF solvent effects on the energetic aspects of the chemisorption process. Simulation results indicate an enhanced stability of CO adsorption on the catalyst surface in the presence of supercritical hexane within the reduced pressure range of 1.0-1.5 at a constant temperature of 523 K. Furthermore, it is shown that the maximum stability of CO in the adsorbed state as a function of supercritical hexane density at 523 K nearly coincides with the maximum isothermal compressibility of bulk hexane at this temperature.

Accelerating the weighted histogram analysis method by direct inversion in the iterative subspace

The weighted histogram analysis method (WHAM) for free energy calculations is a valuable tool to produce free energy differences with the minimal errors. Given multiple simulations, WHAM obtains from the distribution overlaps the optimal statistical estimator of the density of states, from which the free energy differences can be computed. The WHAM equations are often solved by an iterative procedure. In this work, we use a well-known linear algebra algorithm which allows for more rapid convergence to the solution. We find that the computational complexity of the iterative solution to WHAM, and the closely-related multiple Bennett acceptance ratio method can be improved using the method of direct inversion in the iterative subspace. We give examples from a lattice model, a simple liquid and an aqueous protein solution.

Sampling free energy surfaces as slices by combining umbrella sampling and metadynamics.

Metadynamics (MTD) is a very powerful technique to sample high-dimensional free energy landscapes, and due to its self-guiding property, the method has been successful in studying complex reactions and conformational changes. MTD sampling is based on filling the free energy basins by biasing potentials and thus for cases with flat, broad, and unbound free energy wells, the computational time to sample them becomes very large. To alleviate this problem, we combine the standard Umbrella Sampling (US) technique with MTD to sample orthogonal collective variables (CVs) in a simultaneous way. Within this scheme, we construct the equilibrium distribution of CVs from biased distributions obtained from independent MTD simulations with umbrella potentials. Reweighting is carried out by a procedure that combines US reweighting and Tiwary-Parrinello MTD reweighting within the Weighted Histogram Analysis Method (WHAM). The approach is ideal for a controlled sampling of a CV in a MTD simulation, making it computationally efficient in sampling flat, broad, and unbound free energy surfaces. This technique also allows for a distributed sampling of a high-dimensional free energy surface, further increasing the computational efficiency in sampling. We demonstrate the application of this technique in sampling high-dimensional surface for various chemical reactions using ab initio and QM/MM hybrid molecular dynamics simulations. Further, to carry out MTD bias reweighting for computing forward reaction barriers in ab initio or QM/MM simulations, we propose a computationally affordable approach that does not require recrossing trajectories. © 2016 Wiley Periodicals, Inc.

Freezing Transitions of Nanoconfined Coarse-Grained Water Show Subtle Dependence on Confining Environment.

The solid-to-liquid phase transition in water nanofilms confined between plates, with varying separations and water-plate interactions ranging from strongly hydrophobic to strongly hydrophilic, was simulated using a coarse-grained monatomic water model (mW) and the generalized replica exchange method (gREM). Extensive gREM simulations combined with the statistical temperature weighted histogram analysis method (ST-WHAM) provide a detailed description of the thermodynamic properties intrinsic to the phase transition, including the transition temperature, isobaric heat capacity, phase change enthalpy, entropy, and their dependence on the interplate distance and the plate-water interaction. The ice structure of water nanofilms was characterized at various conditions using the transverse density profile and the distribution of angles formed by hydrogen-bonded neighboring molecules. Flat bilayer ice was observed to be the dominant solid phase at close interplate distance, while puckered bilayer ice, similar to a slab of ice Ih, is the predominant structure at larger interplates. Stable puckered bilayer ice, previously observed to have a low melting point, is observed to have enhanced stability with high melting temperature when confined between hydrophilic plates. These results demonstrate the strong dependence of phase stability and coexistence in nanoconfined systems on the geometry and physical properties of the confining environment.