Multiple Markov transition matrix method: Obtaining the stationary probability distribution from multiple simulations

We herein propose the multiple Markov transition matrix method (MMMM), an algorithm by which to estimate the stationary probability distribution from independent multiple molecular dynamics simulations with different Hamiltonians. Applications to the potential of mean force calculation in combination with the umbrella sampling method are presented. First, the performance of the MMMM is examined in the case of butane. Compared with the weighted histogram analysis method (WHAM), the MMMM has an advantage with respect to the reasonable evaluation of the stationary probability distribution even from nonequilibrium trajectories. This method is then applied to Met‐enkephalin nonequilibrium simulation. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Convergence and error estimation in free energy calculations using the weighted histogram analysis method

The weighted histogram analysis method (WHAM) has become the standard technique for the analysis of umbrella sampling simulations. In this article, we address the challenges (1) of obtaining fast and accurate solutions of the coupled nonlinear WHAM equations, (2) of quantifying the statistical errors of the resulting free energies, (3) of diagnosing possible systematic errors, and (4) of optimally allocating of the computational resources. Traditionally, the WHAM equations are solved by a fixed-point direct iteration method, despite poor convergence and possible numerical inaccuracies in the solutions. Here, we instead solve the mathematically equivalent problem of maximizing a target likelihood function, by using superlinear numerical optimization algorithms with a significantly faster convergence rate. To estimate the statistical errors in one-dimensional free energy profiles obtained from WHAM, we note that for densely spaced umbrella windows with harmonic biasing potentials, the WHAM free energy profile can be approximated by a coarse-grained free energy obtained by integrating the mean restraining forces. The statistical errors of the coarse-grained free energies can be estimated straightforwardly and then used for the WHAM results. A generalization to multidimensional WHAM is described. We also propose two simple statistical criteria to test the consistency between the histograms of adjacent umbrella windows, which help identify inadequate sampling and hysteresis in the degrees of freedom orthogonal to the reaction coordinate. Together, the estimates of the statistical errors and the diagnostics of inconsistencies in the potentials of mean force provide a basis for the efficient allocation of computational resources in free energy simulations.     

Temperature weighted histogram analysis method, replica exchange, and transition paths

We analyzed the data from a replica exchange molecular dynamics simulation using the weighted histogram analysis method to combine data from all of the temperature replicas (T-WHAM) to obtain the room-temperature potential of mean force of the G-peptide (the C-terminal beta-hairpin of the B1 domain of protein G) in regions of conformational space not sampled at room temperature. We were able to determine the potential of mean force in the transition region between a minor alpha-helical population and the major beta-hairpin population and identify a possible transition path between them along which the peptide retains a significant amount of secondary structure. This observation provides new insights into a possible mechanism of formation of beta-sheet secondary structures in proteins. We developed a novel Bayesian statistical uncertainty estimation method for any quantity derived from WHAM and used it to validate the calculated potential of mean force. The feasibility of estimating regions of the potential of mean force with unfavorable free energy at room temperature by T-WHAM analysis of replica exchange simulations was further tested on a system that can be solved analytically and presented some of the same challenges found in more complex chemical systems.     

Binding energies of five molecular pincers calculated by explicit and implicit solvent models

Molecular pincers or tweezers are designed to hold and release the target molecule. Potential applications involve drug distribution in medicine, environment technologies, or microindustrial techniques. Typically, the binding is dominated by van der Waals forces. Modeling of such complexes can significantly enhance their design; yet obtaining accurate complexation energies by theory is difficult. In this study, density functional theory (DFT) computations combined with dielectric continuum solvent model are compared with the potential of mean force approach using umbrella sampling and the weighted histogram analysis method (WHAM) with molecular dynamics (MD) simulations. For DFT, functional and basis set effects are discussed. The computed results are compared to experimental data based on NMR spectroscopic measurements of five synthesized tweezers based on the Tröger's basis. Whereas the DFT computations correctly provided the observed trends in complex stability, they failed to produce realistic magnitudes of complexation energies. Typically, the binding was overestimated by DFT if compared to experiment. The simpler semiempirical PM6‐DH2X scheme proposed lately yielded better magnitudes of the binding energies than DFT but not the right order. The MD‐WHAM simulations provided the most realistic Gibbs binding energies, although the approximate MD force fields were not able to reproduce completely the ordering of relative stabilities of model complexes found by NMR. Yet the modeling provides interesting insight into the complex geometry and flexibility and appears as a useful tool in the tweezers' design. © 2012 Wiley Periodicals, Inc.     

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 4.32 kcal/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.     

Theory of binless multi-state free energy estimation with applications to protein-ligand binding

The weighted histogram analysis method (WHAM) is routinely used for computing free energies and expectations from multiple ensembles. Existing derivations of WHAM require observations to be discretized into a finite number of bins. Yet, WHAM formulas seem to hold even if the bin sizes are made arbitrarily small. The purpose of this article is to demonstrate both the validity and value of the multi-state Bennet acceptance ratio (MBAR) method seen as a binless extension of WHAM. We discuss two statistical arguments to derive the MBAR equations, in parallel to the self-consistency and maximum likelihood derivations already known for WHAM. We show that the binless method, like WHAM, can be used not only to estimate free energies and equilibrium expectations, but also to estimate equilibrium distributions. We also provide a number of useful results from the statistical literature, including the determination of MBAR estimators by minimization of a convex function. This leads to an approach to the computation of MBAR free energies by optimization algorithms, which can be more effective than existing algorithms. The advantages of MBAR are illustrated numerically for the calculation of absolute protein-ligand binding free energies by alchemical transformations with and without soft-core potentials. We show that binless statistical analysis can accurately treat sparsely distributed interaction energy samples as obtained from unmodified interaction potentials that cannot be properly analyzed using standard binning methods. This suggests that binless multi-state analysis of binding free energy simulations with unmodified potentials offers a straightforward alternative to the use of soft-core potentials for these alchemical transformations.     

Computing accurate potentials of mean force in electrolyte solutions with the generalized gradient-augmented harmonic Fourier beads method

We establish the accuracy of the novel generalized gradient-augmented harmonic Fourier beads (ggaHFB) method in computing free-energy profiles or potentials of mean force (PMFs) through comparison with two independent conventional techniques. In particular, we employ umbrella sampling with one dimensional weighted histogram analysis method (WHAM) and free molecular dynamics simulation of radial distribution functions to compute the PMF for the Na(+)-Cl(-) ion-pair separation to 16 A in 1.0M NaCl solution in water. The corresponding ggaHFB free-energy profile in six dimensional Cartesian space is in excellent agreement with the conventional benchmarks. We then explore changes in the PMF in response to lowering the NaCl concentration to physiological 0.3 and 0.1M, and dilute 0.0M concentrations. Finally, to expand the scope of the ggaHFB method, we formally develop the free-energy gradient approximation in arbitrary nonlinear coordinates. This formal development underscores the importance of the logarithmic Jacobian correction to reconstruct true PMFs from umbrella sampling simulations with either WHAM or ggaHFB techniques when nonlinear coordinate restraints are used with Cartesian propagators. The ability to employ nonlinear coordinates and high accuracy of the computed free-energy profiles further advocate the use of the ggaHFB method in studies of rare events in complex systems.     

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.     

Estimating statistical distributions using an integral identity

We present an identity for an unbiased estimate of a general statistical distribution. The identity computes the distribution density from dividing a histogram sum over a local window by a correction factor from a mean-force integral, and the mean force can be evaluated as a configuration average. We show that the optimal window size is roughly the inverse of the local mean-force fluctuation. The new identity offers a more robust and precise estimate than a previous one by Adib and Jarzynski [J. Chem. Phys. 122, 014114 (2005)]. It also allows a straightforward generalization to an arbitrary ensemble and a joint distribution of multiple variables. Particularly we derive a mean-force enhanced version of the weighted histogram analysis method. The method can be used to improve distributions computed from molecular simulations. We illustrate the use in computing a potential energy distribution, a volume distribution in a constant-pressure ensemble, a radial distribution function, and a joint distribution of amino acid backbone dihedral angles.     

The potential of mean force on halide ions near the Cu(100) surface

This paper reports an investigation of the phenomenon of specific adsorption of halide ions on a Cu(100) surface using Monte Carlo simulations. The system was modeled by considering each ion in a water lamina placed between two copper walls. The potentials used in simulations were constructed by fitting to results of quantum calculations. The solvent contribution to the potential of mean force (pmf) was calculated for each of the four halide ions using the free energy perturbation method. Given the difficulty of finding a reliable ion–metal potential, several alternatives, representing extremal models, were tested in combination with the solvent mean force on the ions, F⁻, Cl⁻, Br⁻ or I⁻. The results for the pmf on an ion near the metal surface are discussed in the light of the experimental data available. The sensitivity of the results to the type of ion–metal potential used in the simulations is stressed.     

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.     

Free Energy Level Correction by Monte Carlo Resampling with Weighted Histogram Analysis Method

Free energy calculations may provide vital information for studying various chemical and biological processes. Quantum mechanical methods are required to accurately describe interaction energies, but their computations are often too demanding for conformational sampling. As a remedy, level correction schemes that allow calculating high level free energies based on conformations from lower level simulations have been developed. Here, we present a variation of a Monte Carlo (MC) resampling approach in relation to the weighted histogram analysis method (WHAM). We show that our scheme can generate free energy surfaces that can practically converge to the exact one with sufficient sampling, and that it treats cases with insufficient sampling in a more stable manner than the conventional WHAM-based level correction scheme. It can also provide a guide for checking the uncertainty of the level-corrected surface and a well-defined criterion for deciding the extent of smoothing on the free energy surface for its visual improvement. We demonstrate these aspects by obtaining the free energy maps associated with the alanine dipeptide and proton transfer network of the KillerRed protein in explicit water, and exemplify that the MC resampled WHAM scheme can be a practical tool for producing free energy surfaces of realistic systems.     

Statistical Analysis of Ion Mobility Spectrometry. I. Unbiased and Guided Replica-Exchange Molecular Dynamics

Achieving (bio)macromolecular structural assignment from the interpretation of ion mobility spectrometry (IMS) experiments requires successful comparison with computer modeling. Replica-exchange molecular dynamics simulations with suitable force fields not only offer a convenient framework to locate relevant conformations, especially in the case of multiple-funnel energy landscapes, but they are also well suited to statistical analyses. In the present paper, we discuss two extensions of the method used to improve its efficiency in the context of IMS. Two doubly-protonated polyalanines [RA₄XA₄K + 2H]²⁺ with X = V and D appear as favorable cases for which the calculated collision cross-section distributions naturally agree with the measurements, providing reliable candidate structures. For these compounds, a careful consideration of other order parameters based on the weighted histogram method resolves several otherwise hidden underlying conformational families. In the case of a much larger peptide exhibiting bistability, assignment is more difficult but could be achieved by guiding the sampling with an umbrella potential using the square gyration radius as the biasing coordinate. Applied to triply protonated bradykinine, the two presented methods indicate that different conformations compatible with the measurements are very close in energy.     

Potential of mean force of hydrophobic association: dependence on solute size

The potentials of mean force (PMFs) were determined for systems involving formation of nonpolar dimers composed of methane, ethane, propane, isobutane, and neopentane, respectively, in water, using the TIP3P water model, and in vacuo. A series of umbrella-sampling molecular dynamics simulations with the AMBER force field was carried out for each pair in either water or in vacuo. The PMFs were calculated by using the weighted histogram analysis method (WHAM). The shape of the PMFs for dimers of all five nonpolar molecules is characteristic of hydrophobic interactions with contact and solvent-separated minima and desolvation maxima. The positions of all these minima and maxima change with the size of the nonpolar molecule, that is, for larger molecules they shift toward larger distances. The PMF of the neopentane dimer is similar to those of other small nonpolar molecules studied in this work, and hence the neopentane dimer is too small to be treated as a nanoscale hydrophobic object. The solvent contribution to the PMF was also computed by subtracting the PMF determined in vacuo from the PMF in explicit solvent. The molecular surface area model correctly describes the solvent contribution to the PMF together with the changes of the height and positions of the desolvation barrier for all dimers investigated. The water molecules in the first solvation sphere of the dimer are more ordered compared to bulk water, with their dipole moments pointing away from the surface of the dimer. The average number of hydrogen bonds per water molecule in this first hydration shell is smaller compared to that in bulk water, which can be explained by coordination of water molecules to the hydrocarbon surface. In the second hydration shell, the average number of hydrogen bonds is greater compared to bulk water, which can be explained by increased ordering of water from the first hydration shell; the net effect is more efficient hydrogen bonding between the water molecules in the first and second hydration shells.     

Multidimensional adaptive umbrella sampling: Applications to main chain and side chain peptide conformations

A new adaptive umbrella sampling technique for molecular dynamics simulations is described. The high efficiency of the technique renders multidimensional adaptive umbrella sampling possible and thereby enables uniform sampling of the conformational space spanned by several degrees of freedom. The efficiency is achieved by using the weighted histogram analysis method to combine the results from different simulations, by a suitable extrapolation scheme to define the umbrella potential for regions that have not been sampled, and by a criterion to identify simulations during which the system was not in equilibrium. The technique is applied to two test systems, the alanine dipeptide and the threonine dipeptide, to sample the configurational space spanned by one or two dihedral angles. The umbrella potentials applied at the end of each adaptive umbrella sampling run are equal to the negative of the corresponding potentials of mean force. The trajectories obtained in the simulations can be used to calculate dynamical variables that are of interest. An example is the distribution of the distance between the HN and the Hβ proton that can be important for the interpretation of NMR experiments. Factors influencing the accuracy of the calculated quantities are discussed. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1450–1462, 1997     

Ab initio prediction of protein-ligand binding structures by replica-exchange umbrella sampling simulations

We have developed a prediction method for the binding structures of ligands with proteins. Our method consists of three steps. First, replica-exchange umbrella sampling simulations are performed along the distance between a putative binding site of a protein and a ligand as the reaction coordinate. Second, we obtain the potential of mean force (PMF) of the unbiased system using the weighted histogram analysis method and determine the distance that corresponds to the global minimum of PMF. Third, structures that have this global-minimum distance and energy values around the average potential energy are collected and analyzed using the principal component analysis. We predict the binding structure as the global-minimum free energy state on the free energy landscapes along the two major principal component axes. As test cases, we applied our method to five protein-ligand complex systems. Starting from the configuration in which the protein and the ligand are far away from each other in each system, our method predicted the ligand binding structures in excellent agreement with the experimental data from Protein Data Bank.     

Calibration of force-field dependency in free energy landscapes of peptide conformations by quantum chemical calculations

The free energy landscapes of peptide conformations were calibrated by ab initio quantum chemical calculations, after the enhanced conformational diversity search using the multicanonical molecular dynamics simulations. Three different potentials of mean force for an isolated dipeptide were individually obtained by the multicanonical molecular dynamics simulations using the conventional force fields, AMBER parm94, AMBER parm96, and CHARMm22. Each potential of mean force was then calibrated based upon the umbrella sampling algorithm from the adiabatic energy map that was calculated separately by the ab initio molecular orbital method, and all of the calibrated potentials of mean force coincided well. The calibration method was also applied to the simulations of a peptide dimer in explicit water models, and it was shown that the calibrated free energy landscapes did not depend on the force field used in the classical simulations, as far as the conformational space was sampled well. The current calibration method fuses the classical free energy calculation with the quantum chemical calculation, and it should generally make simulations for biomolecular systems much more reliable when combining with enhanced conformational sampling.     

Implementation of an adaptive umbrella sampling method for the calculation of multidimensional potential of mean force of chemical reactions in solution

We describe the implementation of an adaptive umbrella sampling method, making use of the weighted histogram analysis method, for computing multidimensional potential of mean force for chemical reaction in solution. The approach is illustrated by investigating the effect of aqueous solution on the free energy surface for the proton transfer reaction of [H(3)N-H-NH(3)](+) using a combined quantum mechanical and molecular mechanical AM1/TIP3P potential.     

On the mechanism of surfactant adsorption on solid surfaces: free-energy investigations

The adsorption free-energy of surfactant on solid surfaces has been calculated by molecular dynamics (MD) simulation for a model surfactant/solvent system. The umbrella-sampling with the weight histogram analysis method (WHAM) was applied. The entropic and enthalpic contributions to the full potential of mean force (PMF) were obtained to evaluate the detailed thermodynamics of surfactant adsorption in solid/liquid interfaces. Although we observed that this surfactant adsorption process is driven mainly by a favorable enthalpy change, a highly unfavorable entropic contribution still existed. By decomposing the free energy (including its entropic and enthalpic components) into the solvent-induced contribution and the surfactant-wall term, the effect of surface and solvent on the adsorption free-energy has been distinguished. The contribution to the PMF from the surface effect is thermodynamically favorable, whereas the solvent term displays an obviously unfavorable component with a monotonic increase as the surfactant approaches to the surface. The impact of various interactions from the surfaces (both solvent-philic and solvent-phobic) and the solvent on the adsorption PMF of surfactant has been compared and discussed. Compared to the solvent-philic surface, the solvent-phobic surface generates more stable site for the surfactant adsorption. However, the full PMF profile for the solvent-phobic system shows a clear positive maximum value at the bulk-interface transition region, which leads to a considerable long-range free-energy barrier to the surfactant adsorption. These results have been analyzed in terms of the local interfacial structures. In summary, this comprehensive study is expected to reveal the microscopic interaction mechanisms determining the surfactant adsorption on solid surfaces.