Weighted Histogram Analysis Method Research Articles

Dynamic Histogram Analysis To Determine Free Energies and Rates from Biased Simulations.

We present an algorithm to calculate free energies and rates from molecular simulations on biased potential energy surfaces. As input, it uses the accumulated times spent in each state or bin of a histogram and counts of transitions between them. Optimal unbiased equilibrium free energies for each of the states/bins are then obtained by maximizing the likelihood of a master equation (i.e., first-order kinetic rate model). The resulting free energies also determine the optimal rate coefficients for transitions between the states or bins on the biased potentials. Unbiased rates can be estimated, e.g., by imposing a linear free energy condition in the likelihood maximization. The resulting "dynamic histogram analysis method extended to detailed balance" (DHAMed) builds on the DHAM method. It is also closely related to the transition-based reweighting analysis method (TRAM) and the discrete TRAM (dTRAM). However, in the continuous-time formulation of DHAMed, the detailed balance constraints are more easily accounted for, resulting in compact expressions amenable to efficient numerical treatment. DHAMed produces accurate free energies in cases where the common weighted-histogram analysis method (WHAM) for umbrella sampling fails because of slow dynamics within the windows. Even in the limit of completely uncorrelated data, where WHAM is optimal in the maximum-likelihood sense, DHAMed results are nearly indistinguishable. We illustrate DHAMed with applications to ion channel conduction, RNA duplex formation, α-helix folding, and rate calculations from accelerated molecular dynamics. DHAMed can also be used to construct Markov state models from biased or replica-exchange molecular dynamics simulations. By using binless WHAM formulated as a numerical minimization problem, the bias factors for the individual states can be determined efficiently in a preprocessing step and, if needed, optimized globally afterward.

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.

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.

Binding of Vitamin E in Model Membranes Studied by Umbrella Sampling Simulations

Alpha tocopherol (αtoc), the form of vitamin E that is retained in the human body, is a lipid-soluble anti-oxidant. Its presumptive role is to protect polyunsaturated fatty acids (PUFA) from oxidation in cell membranes. The global concentration of αtoc in the plasma membrane is low (generally < 1 mol%), so preferential interaction with PUFA-containing phospholipids has been hypothesized to optimize protection of the lipid species most vulnerable to oxidative assault. To test this hypothesis we are investigating the desorption energy of αtoc in model membranes using umbrella sampling molecular dynamics (USMD) simulations. Insight into the flip-flop of αtoc across the membrane will also be obtained from the free energy barrier at the center of the bilayer. USMD simulations of αtoc molecules in membranes composed of 1-stearoyl-2-docosahexaenoylphosphatidylcholine (SDPC, 18:0-22:6PC) and, as a monounsaturated reference, 1-stearoyl-2-oleoylphosphatidylcholine (SOPC, 18:0-18:1PC) are now ongoing. The position of the chromanol group of αtoc in the membrane-normal direction is chosen to be the reaction coordinate. The simulations consist of 45 windows separated by 1 Å and arranged from the center of bilayer all the way to the bulk water, each sampled for 200 ns. Preliminary calculation of the mean force in SDPC shows that the equilibrium position for the chromanol group is around 12 Å from the bilayer center, consistent with our earlier simulations. More accurate calculation of the potential of mean force (PMF) is being performed using the weighted histogram analysis method (WHAM).

Locally weighted histogram analysis and stochastic solution for large-scale multi-state free energy estimation.

The weighted histogram analysis method (WHAM) including its binless extension has been developed independently in several different contexts, and widely used in chemistry, physics, and statistics, for computing free energies and expectations from multiple ensembles. However, this method, while statistically efficient, is computationally costly or even infeasible when a large number, hundreds or more, of distributions are studied. We develop a locally WHAM (local WHAM) from the perspective of simulations of simulations (SOS), using generalized serial tempering (GST) to resample simulated data from multiple ensembles. The local WHAM equations based on one jump attempt per GST cycle can be solved by optimization algorithms orders of magnitude faster than standard implementations of global WHAM, but yield similarly accurate estimates of free energies to global WHAM estimates. Moreover, we propose an adaptive SOS procedure for solving local WHAM equations stochastically when multiple jump attempts are performed per GST cycle. Such a stochastic procedure can lead to more accurate estimates of equilibrium distributions than local WHAM with one jump attempt per cycle. The proposed methods are broadly applicable when the original data to be "WHAMMED" are obtained properly by any sampling algorithm including serial tempering and parallel tempering (replica exchange). To illustrate the methods, we estimated absolute binding free energies and binding energy distributions using the binding energy distribution analysis method from one and two dimensional replica exchange molecular dynamics simulations for the beta-cyclodextrin-heptanoate host-guest system. In addition to the computational advantage of handling large datasets, our two dimensional WHAM analysis also demonstrates that accurate results similar to those from well-converged data can be obtained from simulations for which sampling is limited and not fully equilibrated.

K+ Block Is the Mechanism of Functional Asymmetry in Bacterial Na(v) Channels.

Crystal structures of several bacterial Nav channels have been recently published and molecular dynamics simulations of ion permeation through these channels are consistent with many electrophysiological properties of eukaryotic channels. Bacterial Nav channels have been characterized as functionally asymmetric, and the mechanism of this asymmetry has not been clearly understood. To address this question, we combined non-equilibrium simulation data with two-dimensional equilibrium unperturbed landscapes generated by umbrella sampling and Weighted Histogram Analysis Methods for multiple ions traversing the selectivity filter of bacterial NavAb channel. This approach provided new insight into the mechanism of selective ion permeation in bacterial Nav channels. The non-equilibrium simulations indicate that two or three extracellular K+ ions can block the entrance to the selectivity filter of NavAb in the presence of applied forces in the inward direction, but not in the outward direction. The block state occurs in an unstable local minimum of the equilibrium unperturbed free-energy landscape of two K+ ions that can be ‘locked’ in place by modest applied forces. In contrast to K+, three Na+ ions move favorably through the selectivity filter together as a unit in a loose “knock-on” mechanism of permeation in both inward and outward directions, and there is no similar local minimum in the two-dimensional free-energy landscape of two Na+ ions for a block state. The useful work predicted by the non-equilibrium simulations that is required to break the K+ block is equivalent to large applied potentials experimentally measured for two bacterial Nav channels to induce inward currents of K+ ions. These results illustrate how inclusion of non-equilibrium factors in the simulations can provide detailed information about mechanisms of ion selectivity that is missing from mechanisms derived from either crystal structures or equilibrium unperturbed free-energy landscapes.

Temperature and Loading-Dependent Diffusion of Light Hydrocarbons in ZIF-8 as Predicted Through Fully Flexible Molecular Simulations.

Accurate and efficient predictions of hydrocarbon diffusivities in zeolitic imidazolate frameworks (ZIFs) are challenging, due to the small pore size of materials such as ZIF-8 and the wide range of diffusion time scales of hydrocarbon molecules in ZIFs. Here we have computationally measured the hopping rates of 15 different molecules (kinetic diameters of 2.66-5.10 Å) in ZIF-8 via dynamically corrected transition state theory (dcTST). Umbrella sampling combined with the one-dimensional weighted histogram analysis method (WHAM) was used to calculate the diffusion free energy barriers. Both the umbrella sampling and dynamical correction calculations included ZIF-8 flexibility, which is found to be critical in accurately describing molecular diffusion in this material. Comparison of the computed diffusivities to extant experimental results shows remarkable agreement within an order of magnitude for all the molecules. The dcTST method was also applied to study the effect of hydrocarbon loadings. Self and transport diffusion coefficients of methane, ethane, ethylene, propane, propylene, n-butane, and 1-butene in ZIF-8 are reported over a temperature range of 0-150 °C and loadings from infinite dilution to liquid-like loadings.

Molecular Dynamics and Monte Carlo simulations in the microcanonical ensemble: Quantitative comparison and reweighting techniques.

Molecular Dynamics (MD) and Monte Carlo (MC) simulations are the most popular simulation techniques for many-particle systems. Although they are often applied to similar systems, it is unclear to which extent one has to expect quantitative agreement of the two simulation techniques. In this work, we present a quantitative comparison of MD and MC simulations in the microcanonical ensemble. For three test examples, we study first- and second-order phase transitions with a focus on liquid-gas like transitions. We present MD analysis techniques to compensate for conservation law effects due to linear and angular momentum conservation. Additionally, we apply the weighted histogram analysis method to microcanonical histograms reweighted from MD simulations. By this means, we are able to estimate the density of states from many microcanonical simulations at various total energies. This further allows us to compute estimates of canonical expectation values.

Conformational sampling enhancement of replica exchange molecular dynamics simulations using swarm particle intelligence.

In this paper, we present a new method based on swarm particle social intelligence for use in replica exchange molecular dynamics simulations. In this method, the replicas (representing the different system configurations) are allowed communicating with each other through the individual and social knowledge, in additional to considering them as a collection of real particles interacting through the Newtonian forces. The new method is based on the modification of the equations of motion in such way that the replicas are driven towards the global energy minimum. The method was tested for the Lennard-Jones clusters of N = 4, 5, and 6 atoms. Our results showed that the new method is more efficient than the conventional replica exchange method under the same practical conditions. In particular, the new method performed better on optimizing the distribution of the replicas among the thermostats with time and, in addition, ergodic convergence is observed to be faster. We also introduce a weighted histogram analysis method allowing analyzing the data from simulations by combining data from all of the replicas and rigorously removing the inserted bias.

A Stochastic Solution to the Unbinned WHAM Equations.

The weighted histogram analysis method (WHAM) and unbinned versions such as the multistate Bennett acceptance ratio (MBAR) and unbinned WHAM (UWHAM) are widely used to compute free energies and expectations from data generated by independent or coupled parallel simulations. Here we introduce a replica exchange-like algorithm (RE-SWHAM) that can be used to solve the UWHAM equations stochastically. This method is capable of analyzing large data sets generated by hundreds or even thousands of parallel simulations that are too large to be "WHAMMED" using standard methods. We illustrate the method by applying it to obtain free energy weights for each of the 240 states in a simulation of host-guest ligand binding containing ∼3.5 × 10(7) data elements collected from 16 parallel Hamiltonian replica exchange simulations, performed at 15 temperatures. In addition to using much less memory, RE-SWHAM showed a nearly 80-fold improvement in computational time compared with UWHAM.