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.     

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.     

Calculation of free energy landscapes: A histogram reweighted metadynamics approach

We present an efficient method for the calculation of free energy landscapes. Our approach involves a history‐dependent bias potential, which is evaluated on a grid. The corresponding free energy landscape is constructed via a histogram reweighting procedure a posteriori. Because of the presence of the bias potential, it can be also used to accelerate rare events. In addition, the calculated free energy landscape is not restricted to the actual choice of collective variables and can in principle be extended to auxiliary variables of interest without further numerical effort. The applicability is shown for several examples. We present numerical results for the alanine dipeptide and the Met‐Enkephalin in explicit solution to illustrate our approach. Furthermore, we derive an empirical formula that allows the prediction of the computational cost for the ordinary metadynamics variant in comparison with our approach, which is validated by a dimensionless representation. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

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.     

Systematic and statistical error in histogram-based free energy calculations

A common technique for the numerical calculation of free energies involves estimation of the probability density along a given coordinate from a set of configurations generated via simulation. The process requires discretization of one or more reaction coordinates to generate a histogram from which the continuous probability density is inferred. We show that the finite size of the intervals used to construct the histogram leads to quantifiable systematic error. The width of these intervals also determines the statistical error in the free energy, and the choice of the appropriate interval is therefore driven by the need to balance the two sources of error. We present a method for the construction of the optimal histogram for a given system, and show that the use of this technique requires little additional computational expense. We demonstrate the efficacy of the technique for a model system, and discuss how the principles governing the choice of discretization interval could be used to improve extended sampling techniques.     

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.     

Using the local elevation method to construct optimized umbrella sampling potentials: Calculation of the relative free energies and interconversion barriers of glucopyranose ring conformers in water

A method is proposed to combine the local elevation (LE) conformational searching and the umbrella sampling (US) conformational sampling approaches into a single local elevation umbrella sampling (LEUS) scheme for (explicit‐solvent) molecular dynamics (MD) simulations. In this approach, an initial (relatively short) LE build‐up (searching) phase is used to construct an optimized biasing potential within a subspace of conformationally relevant degrees of freedom, that is then used in a (comparatively longer) US sampling phase. This scheme dramatically enhances (in comparison with plain MD) the sampling power of MD simulations, taking advantage of the fact that the preoptimized biasing potential represents a reasonable approximation to the negative of the free energy surface in the considered conformational subspace. The method is applied to the calculation of the relative free energies of β‐D ‐glucopyranose ring conformers in water (within the GROMOS 45A4 force field). Different schemes to assign sampled conformational regions to distinct states are also compared. This approach, which bears some analogies with adaptive umbrella sampling and metadynamics (but within a very distinct implementation), is shown to be: (i) efficient (nearly all the computational effort is invested in the actual sampling phase rather than in searching and equilibration); (ii) robust (the method is only weakly sensitive to the details of the build‐up protocol, even for relatively short build‐up times); (iii) versatile (a LEUS biasing potential database could easily be preoptimized for small molecules and assembled on a fragment basis for larger ones). © 2009 Wiley Periodicals, Inc. J Comput Chem 2010     

A highly efficient hybrid method for calculating the hydration free energy of a protein

We develop a new method for calculating the hydration free energy (HFE) of a protein with any net charge. The polar part of the energetic component in the HFE is expressed as a linear combination of four geometric measures (GMs) of the protein structure and the generalized Born (GB) energy plus a constant. The other constituents in the HFE are expressed as linear combinations of the four GMs. The coefficients (including the constant) in the linear combinations are determined using the three‐dimensional reference interaction site model (3D‐RISM) theory applied to sufficiently many protein structures. Once the coefficients are determined, the HFE and its constituents of any other protein structure are obtained simply by calculating the four GMs and GB energy. Our method and the 3D‐RISM theory give perfectly correlated results. Nevertheless, the computation time required in our method is over four orders of magnitude shorter.     

A hybrid approach combining energy density analysis with the interaction energy decomposition method

We propose a new analysis technique for characterizing molecular interactions that combines an energy decomposition scheme, such as the Kitaura-Morokuma decomposition method, with energy density analysis, which partitions the total energy of the system into atomic contributions. The combined scheme, termed Interaction-EDA, enables us to estimate the local contribution of interaction energy components, such as electrostatic, exchange, polarization, and charge transfer. The evaluation of the local interaction energy is rather important in large systems. For a numerical assessment, the Interaction-EDA method is applied to the process of CO adsorption on Si(100) - (2 x 1) surface.     

Application of the chromatographic step profile method for determination of water film pressure and surface free energy of quartz

Summary Investigation of water adsorption by the step profile method (Glueckauf method) was carried out with the help of a modified gas chromatography equipped with thermal-conductivity detector. On the basis of the adsorption isotherm obtained, the water film pressure and the polar component of the surface free energy of quartz were calculated. The calculated value of the polar component of the surface free energy of quartz agrees with analogous values obtained by other methods.     

Efficiency of alchemical free energy simulations. I. A practical comparison of the exponential formula, thermodynamic integration, and Bennett's acceptance ratio method

We investigate the relative efficiency of thermodynamic integration, three variants of the exponential formula, also referred to as thermodynamic perturbation, and Bennett's acceptance ratio method to compute relative and absolute solvation free energy differences. Our primary goal is the development of efficient protocols that are robust in practice. We focus on minimizing the number of unphysical intermediate states (λ-states) required for the computation of accurate and precise free energy differences. Several indicators are presented which help decide when additional λ-states are necessary. In all tests Bennett's acceptance ratio method required the least number of λ-states, closely followed by the "double-wide" variant of the exponential formula. Use of the exponential formula in only strict "forward" or "backward" mode was not found to be competitive. Similarly, the performance of thermodynamic integration in terms of efficiency was rather poor. We show that this is caused by the use of the trapezoidal rule as method of numerical quadrature. A systematic study focusing on the optimization of thermodynamic integration is presented in a companion paper.     

A conformational analysis method for understanding the energy landscapes of clusters.

A newly developed unbiased structural optimization method, named dynamic lattice searching (DLS), is proposed as an approach for conformational analysis of atomic/molecular clusters and used in understanding the energy landscape of large clusters. The structures of clusters are described in terms of the number of basic tetrahedron (BT) units they contain. We found that the hit numbers of different structural motifs in DLS runs is proportional to the number of BTs. A parameter T(max) is defined to limit the maximal number of atoms moved in a structural transition. Results show that T(max) is a key parameter for modulating the efficiency of the DLS method and has a great influence on the hit number of different motifs in DLS runs. Finally, the effect of potential range on the conformational distribution of the (Morse)(98) cluster is also discussed with different potential-range parameters.     

Comprehensive method based on model free method and IKP method for evaluating kinetic parameters of solid state reactions

This article presents, firstly, a short review of methods for evaluating kinetic parameters of solid state reactions and a critical analysis of the isoconversional principle of model free methods. It shows theoretically that the activation energy for complex reactions is not only a function of the reaction degree but also of heating programs, and points out that any method that attempts to extract the dependences of activation energy on conversion degree without considering the dependences of heating programs is problematic. Then an analysis is given of the invariant kinetic parameters (IKP) method and recommends an incremental version of it. Based on the incremental IKP method and model free method, a comprehensive method is proposed that predicts the degree of the dependences of activation energy on heating programs, selects reliable values of activation energy and extracts the values of variable pre‐exponential factor. This comprehensive method is tested using both simulation data and experimental data, the results of which show it can not only give reliable values of kinetic parameters but also be helpful in explaining inconsistencies of kinetic results in solid state reactions. © 2012 Wiley Periodicals, Inc.     

Electrostatic solvation free energy of amino acid side chain analogs: implications for the validity of electrostatic linear response in water

Electrostatic free energies of solvation for 15 neutral amino acid side chain analogs are computed. We compare three methods of varying computational complexity and accuracy for three force fields: free energy simulations, Poisson-Boltzmann (PB), and linear response approximation (LRA) using AMBER, CHARMM, and OPLS-AA force fields. We find that deviations from simulation start at low charges for solutes. The approximate PB and LRA produce an overestimation of electrostatic solvation free energies for most of molecules studied here. These deviations are remarkably systematic. The variations among force fields are almost as large as the variations found among methods. Our study confirms that success of the approximate methods for electrostatic solvation free energies comes from their ability to evaluate free energy differences accurately.     

An estimation method of binding free energy in terms of ABEEMσπ/MM and continuum electrostatics fused into LIE method

A method is proposed for the estimation of absolute binding free energy of interaction between proteins and ligands. The linear interaction energy method is combined with atom‐bond electronegativity equalization method at σπ level Force field (fused into molecular mechanics) and generalized Born continuum model calculation of electrostatic solvation for the estimation of the absolute free energy of binding. The parameters of this method are calibrated by using a training set of 24 HIV‐1 protease–inhibitor complexes (PDB entry 1AAQ). A correlation coefficient of 0.93 was obtained with a root mean square deviation of 0.70 kcal mol^?1. This approach is further tested on seven inhibitor and protease complexes, and it provides small root mean square deviation between the calculated binding free energy and experimental binding free energy without reparametrization. By comparing the radii of gyration and the hydrogen bond distances between ligand and protein of three training model molecules, the consistent comparison result of binding free energy is obtained. It proves that this method of calculating the binding free energy with appropriate structural analysis can be applied to quickly assess new inhibitors of HIV‐1 proteases. To test whether the parameters of this method can apply to other drug targets, we have validated this method for the drug target cyclooxygenase‐2. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

Calculation of hydration free energy for a solute with many atomic sites using the RISM theory: A robust and efficient algorithm

We have developed an algorithm for solving the reference interaction site model (RISM) equations for water near a solute molecule with many atomic sites (interaction sites). It is a hybrid of the Newton–Raphson and Picard methods and is judiciously constructed. Various considerations are given so that the computer time can be saved as much as possible. The robustness and high efficiency of the algorithm has been demonstrated for calculating hydration free energies of Met-enkephalin (a peptide with 75 sites) with different conformations. The Jacobian matrix is treated as part of the input data, and it has been found that the same matrix can be used for a considerably large set of different conformations of the solute molecule. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1320–1326, 1997     

Discriminant analysis in the presence of interferences: Combined application of target factor analysis and a Bayesian soft-classifier

A method is described for performing discriminant analysis in the presence of interfering background signal. The method is based on performing target factor analysis on a data set comprised of contributions from analyte(s) and interfering components. A library of data from representative analyte classes is tested for possible contributing factors by performing oblique rotations of the principal factors to obtain the best match, in a least-squares sense, between test and predicted vectors. The degree of match between the test and predicted vectors is measured by the Pearson correlation coefficient, r, and the distribution of r for each class is determined. A Bayesian soft classifier is used to calculate the posterior probability based on the distributions of r for each class, which assist the analyst in assessing the presence of one or more analytes. The method is demonstrated by analyses performed on spectra obtained by laser induced breakdown spectroscopy (LIBS). Single and multiple bullet jacketing transfers to steel and porcelain substrates were analyzed to identify the jacketing materials. Additionally, the metal surrounding bullet holes was analyzed to identify the class of bullet jacketing that passed through a stainless steel plate. Of 36 single sample transfers, the copper jacketed (CJ) and non-jacketed (NJ) class on porcelain had an average posterior probability of the metal deposited on the substrate of 1.0. Metal jacketed (MJ) bullet transfers to steel and porcelain were not detected as successfully. Multiple transfers of CJ/NJ and CJ/MJ on the two substrates resulted in posterior probabilities that reflected the presence of both jacketing materials. The MJ/NJ transfers gave posterior probabilities that reflected the presence of the NJ material, but the MJ component was mistaken for CJ on steel, while non-zero probabilities were obtained for both CJ and MJ on porcelain. Jacketing transfer from a bullet to steel as the projectile passed through the steel also proved difficult to analyze. Over 50% of the samples left insufficient transfer to be identified. Transfer from NJ and CJ jacketing was successfully identified by posterior probabilities greater than 0.8.