Using non-Gaussian density functional fits to improve relative free energy calculations

The accurate and reliable computation of relative free energy differences remains an important long-term goal. Major stumbling blocks for achieving this goal reflect the difficulty of sampling in a known fashion along the reaction coordinate and of maximally combining information that has been collected from the simulation along the reaction coordinate. In this paper we examine the utility of a probability density functional type fit to the distribution of work events collected during a nonequilibrium sample along the reaction coordinate. This approach can readily be generalized to equilibrium sampling and has the potential to estimate the quality of a relative free energy estimate as data are being collected. The method may have the greatest utility for nonequilibrium sampling where non-Gaussian work distributions are generally present that are strongly dominated by rare event sampling in the tail region. We believe that the approach can be used to augment the design and the error analysis of relative free energy computations thus improving the ability to reliably and with known accuracy compute a relative free energy.     

FEW: A workflow tool for free energy calculations of ligand binding

In the later stages of drug design projects, accurately predicting relative binding affinities of chemically similar compounds to a biomolecular target is of utmost importance for making decisions based on the ranking of such compounds. So far, the extensive application of binding free energy approaches has been hampered by the complex and time‐consuming setup of such calculations. We introduce the free energy workflow (FEW) tool that facilitates setup and execution of binding free energy calculations with the AMBER suite for multiple ligands. FEW allows performing free energy calculations according to the implicit solvent molecular mechanics (MM‐PB(GB)SA), the linear interaction energy, and the thermodynamic integration approaches. We describe the tool's architecture and functionality and demonstrate in a show case study on Factor Xa inhibitors that the time needed for the preparation and analysis of free energy calculations is considerably reduced with FEW compared to a fully manual procedure. © 2013 Wiley Periodicals, Inc.     

Predicting relative binding affinities of non-peptide HIV protease inhibitors with free energy perturbation calculations

The relative binding free energies in HIV protease of haloperidol thioketal (THK) and three of its derivatives were examined with free energy calculations. THK is a weak inhibitor (IC50 = 15 M) for which two cocrystal structures with HIV type 1 proteases have been solved [Rutenber, E. et al., J. Biol. Chem., 268 (1993) 15343]. A THK derivative with a phenyl group on C2 of the piperidine ring was expected to be a poor inhibitor based on experiments with haloperidol ketal and its 2- phenyl derivative (Caldera, P., personal communication). Our calculations predict that a 5-phenyl THK derivative, suggested based on examination of the crystal structure, will bind significantly better than THK. Although there are large error bars as estimated from hysteresis, the calculations predict that the 5-phenyl substituent is clearly favored over the 2-phenyl derivative as well as the parent compound. The unfavorable free energies of solvation of both phenyl THK derivatives relative to the parent compound contributed to their predicted binding free energies. In a third simulation, the change in binding free energy for 5-benzyl THK relative to THK was calculated. Although this derivative has a lower free energy in the protein, its decreased free energy of solvation increases the predicted G(bind) to the same range as that of the 2-phenyl derivative.     

Multipole electrostatics in hydration free energy calculations

Hydration free energy (HFE) is generally used for evaluating molecular solubility, which is an important property for pharmaceutical and chemical engineering processes. Accurately predicting HFE is also recognized as one fundamental capability of molecular mechanics force field. Here, we present a systematic investigation on HFE calculations with AMOEBA polarizable force field at various parameterization and simulation conditions. The HFEs of seven small organic molecules have been obtained alchemically using the Bennett Acceptance Ratio method. We have compared two approaches to derive the atomic multipoles from quantum mechanical calculations: one directly from the new distributed multipole analysis and the other involving fitting to the electrostatic potential around the molecules. Wave functions solved at the MP2 level with four basis sets (6-311G*, 6-311++G(2d,2p), cc-pVTZ, and aug-cc-pVTZ) are used to derive the atomic multipoles. HFEs from all four basis sets show a reasonable agreement with experimental data (root mean square error 0.63 kcal/mol for aug-cc-pVTZ). We conclude that aug-cc-pVTZ gives the best performance when used with AMOEBA, and 6-311++G(2d,2p) is comparable but more efficient for larger systems. The results suggest that the inclusion of diffuse basis functions is important for capturing intermolecular interactions. The effect of long-range correction to van der Waals interaction on the hydration free energies is about 0.1 kcal/mol when the cutoff is 12?, and increases linearly with the number of atoms in the solute/ligand. In addition, we also discussed the results from a hybrid approach that combines polarizable solute with fixed-charge water in the HFE calculation.     

Basic ingredients of free energy calculations: A review

Methods to compute free energy differences between different states of a molecular system are reviewed with the aim of identifying their basic ingredients and their utility when applied in practice to biomolecular systems. A free energy calculation is comprised of three basic components: (i) a suitable model or Hamiltonian, (ii) a sampling protocol with which one can generate a representative ensemble of molecular configurations, and (iii) an estimator of the free energy difference itself. Alternative sampling protocols can be distinguished according to whether one or more states are to be sampled. In cases where only a single state is considered, six alternative techniques could be distinguished: (i) changing the dynamics, (ii) deforming the energy surface, (iii) extending the dimensionality, (iv) perturbing the forces, (v) reducing the number of degrees of freedom, and (vi) multi‐copy approaches. In cases where multiple states are to be sampled, the three primary techniques are staging, importance sampling, and adiabatic decoupling. Estimators of the free energy can be classified as global methods that either count the number of times a given state is sampled or use energy differences. Or, they can be classified as local methods that either make use of the force or are based on transition probabilities. Finally, this overview of the available techniques and how they can be best used in a practical context is aimed at helping the reader choose the most appropriate combination of approaches for the biomolecular system, Hamiltonian and free energy difference of interest. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Single-sweep methods for free energy calculations

A simple, efficient, and accurate method is proposed to map multidimensional free energy landscapes. The method combines the temperature-accelerated molecular dynamics (TAMD) proposed in [L. Maragliano and E. Vanden-Eijnden, Chem. Phys. Lett. 426, 168 (2006)] with a variational reconstruction method using radial-basis functions for the representation of the free energy. TAMD is used to rapidly sweep through the important regions of the free energy landscape and to compute the gradient of the free energy locally at points in these regions. The variational method is then used to reconstruct the free energy globally from the mean force at these points. The algorithmic aspects of the single-sweep method are explained in detail, and the method is tested on simple examples and used to compute the free energy of the solvated alanine dipeptide in two and four dihedral angles.     

Calculating zeros: Non-equilibrium free energy calculations

Free energy calculations on three model processes with theoretically known free energy changes have been performed using short simulation times. A comparison between equilibrium (thermodynamic integration) and non-equilibrium (fast growth) methods has been made in order to assess the accuracy and precision of these methods. The three processes have been chosen to represent processes often observed in biomolecular free energy calculations. They involve a redistribution of charges, the creation and annihilation of neutral particles and conformational changes. At very short overall simulation times, the thermodynamic integration approach using discrete steps is most accurate. More importantly, reasonable accuracy can be obtained using this method which seems independent of the overall simulation time. In cases where slow conformational changes play a role, fast growth simulations might have an advantage over discrete thermodynamic integration where sufficient sampling needs to be obtained at every λ-point, but only if the initial conformations do properly represent an equilibrium ensemble. From these three test cases practical lessons can be learned that will be applicable to biomolecular free energy calculations.     

Energy difference space random walk to achieve fast free energy calculations

A method is proposed to efficiently obtain free energy differences. In the present algorithm, free energy calculations proceed by the realization of an energy difference space random walk. Thereby, this algorithm can greatly improve the sampling of the regions in phase space where target states overlap.     

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.     

Alchemical free energy calculations and multiple conformational substates

Thermodynamic integration (TI) was combined with (adaptive) umbrella sampling to improve the convergence of alchemical free energy simulations in which multiple conformational substates are present. The approach, which we refer to as non-Boltzmann TI (NBTI), was tested by computing the free energy differences between three five-atomic model systems, as well as the free energy difference of solvation between leucine and asparagine. In both cases regular TI failed to give converged results, whereas the NBTI results were free from hysteresis and had standard deviations well below +/-0.7 kcal/mole. We also present theoretical considerations that make it possible to compute free energy differences between simple molecules, such as the five-atomic model systems, by numerical integration of the partition functions at the respective end points.     

Guidelines for the analysis of free energy calculations

Journal of Computer-Aided Molecular Design - Free energy calculations based on molecular dynamics simulations show considerable promise for applications ranging from drug discovery to prediction of...     

Free energy derivatives: A new method for probing the convergence problem in free energy calculations

The convergence behavior of free energy calculations has been explored in more detail than in any previously reported work, using a model system of two neon atoms in a periodic box of water. We find that for thermodynamic integration-type free energy calculations as much as a nanosecond or more molecular dynamics sampling is required to obtain a fully converged value for a single λ point of the integrand. The concept of “free energy derivatives” with respect to the individual parameters of the force field is introduced. This formalism allows the total convergence of the simulation to be deconvoluted into components. A determination of the statistical “sampling ratio” from these simulations indicates that for window-type free energy calculations carried out in a periodic waterbox of typical size at least 0.6 ps of sampling should be performed at each window (0.7 ps if constraint contributions to the free energy are being determined). General methods to estimate and reduce the error in thermodynamic integration and free energy perturbation calculations are discussed. We show that the difficulty in applying such methods is determining a reliable estimate of the correlation length from a short series of data. © 1994 by John Wiley & Sons, Inc.     

Adaptively biased molecular dynamics for free energy calculations

We present an adaptively biased molecular dynamics (ABMD) method for the computation of the free energy surface of a reaction coordinate using nonequilibrium dynamics. The ABMD method belongs to the general category of umbrella sampling methods with an evolving biasing potential and is inspired by the metadynamics method. The ABMD method has several useful features, including a small number of control parameters and an O(t) numerical cost with molecular dynamics time t. The ABMD method naturally allows for extensions based on multiple walkers and replica exchange, where different replicas can have different temperatures and/or collective variables. This is beneficial not only in terms of the speed and accuracy of a calculation, but also in terms of the amount of useful information that may be obtained from a given simulation. The workings of the ABMD method are illustrated via a study of the folding of the Ace-GGPGGG-Nme peptide in a gaseous and solvated environment.     

Linear-scaling soft-core scheme for alchemical free energy calculations

Alchemical free energy calculations involving the removal or insertion of atoms into condensed phase systems generally make use of soft-core scaling of nonbonded interactions, designed to circumvent numerical instabilities that arise from weakly interacting "hard" atoms in close proximity. Current methods model soft-core atoms by introducing a nonlinear dependence between the shape of the interaction potential and the strength of the interaction. In this article, we propose a soft-core method that avoids introducing such a nonlinear dependence, through the application of a smooth flattening of the potential energy only in a region that is energetically accessible under normal conditions. We discuss the benefits that this entails and explore a selection of applications, including enhanced methods for the estimation of free energy differences and for the automated optimization of the placement of intermediate states in multistage alchemical calculations.     

A study of pore blockage in silicalite zeolite using free energy perturbation calculations

Binary systems consisting of large coadsorbed molecules (n-hexane, cyclohexane, and benzene) with smaller penetrant molecules (methane) were simulated to investigate the mechanisms of pore blockage in the zeolite silicalite. Benzene and cyclohexane trap the methane molecules in the zeolite channels on the time scales of molecular dynamics simulations. Minimum energy paths for methane diffusion past the blocking molecules were determined, and free energy perturbation calculations were carried out along the paths to get the rate constants of methane hopping past coadsorbed benzene and cyclohexane molecules, which adsorb in the channel intersections. Three principal diffusion pathways were found in both the methane/benzene and methane/cyclohexane systems. Minima which were connected by low-energy pathways were grouped together into macrostates. Using the calculated hopping rates between macrostates, kinetic Monte Carlo was then used to obtain the diffusivity of methane with a coadsorbate benzene loading such that all channel intersections are filled by benzene - conditions where molecular dynamics simulations fail. Passage of methane across cyclohexane molecules involved pushing the cyclohexane molecules into the channels from their preferred channel intersection positions.     

Binding free energy calculations to rationalize the interactions of huprines with acetylcholinesterase

In the present study, the binding free energy of a family of huprines with acetylcholinesterase (AChE) is calculated by means of the free energy perturbation method, based on hybrid quantum mechanics and molecular mechanics potentials. Binding free energy calculations and the analysis of the geometrical parameters highlight the importance of the stereochemistry of huprines in AChE inhibition. Binding isotope effects are calculated to unravel the interactions between ligands and the gorge of AChE. New chemical insights are provided to explain and rationalize the experimental results. A good correlation with the experimental data is found for a family of inhibitors with moderate differences in the enzyme affinity. The analysis of the geometrical parameters and interaction energy per residue reveals that Asp72, Glu199, and His440 contribute significantly to the network of interactions between active site residues, which stabilize the inhibitors in the gorge. It seems that a cooperative effect of the residues of the gorge determines the affinity of the enzyme for these inhibitors, where Asp72, Glu199, and His440 make a prominent contribution.     

Nonlinear scaling schemes for Lennard-Jones interactions in free energy calculations

Alchemical free energy calculations provide a means for the accurate determination of free energies from atomistic simulations and are increasingly used as a tool for computational studies of protein-ligand interactions. Much attention has been placed on efficient ways to deal with the "endpoint singularity" effect that can cause simulation instabilities when changing the number of atoms. In this study we compare the performance of linear and several nonlinear transformation methods, among them separation shifted "soft core" scaling, for a popular test system, the hydration free energy of an amino acid side chain. All the nonlinear methods yield similar results if extensive sampling is performed, but soft core scaling provides smooth lambda curves that are best suited for commonly used numerical integration schemes. Additionally, results from a more flexible solute, hexane, will also be discussed.     

Using the fast fourier transform in binding free energy calculations

According to implicit ligand theory, the standard binding free energy is an exponential average of the binding potential of mean force (BPMF), an exponential average of the interaction energy between the unbound ligand ensemble and a rigid receptor. Here, we use the fast Fourier transform (FFT) to efficiently evaluate BPMFs by calculating interaction energies when rigid ligand configurations from the unbound ensemble are discretely translated across rigid receptor conformations. Results for standard binding free energies between T4 lysozyme and 141 small organic molecules are in good agreement with previous alchemical calculations based on (1) a flexible complex ( for 24 systems) and (2) flexible ligand with multiple rigid receptor configurations ( for 141 systems). While the FFT is routinely used for molecular docking, to our knowledge this is the first time that the algorithm has been used for rigorous binding free energy calculations. © 2017 Wiley Periodicals, Inc.     

Predicting binding affinities of host-guest systems in the SAMPL3 blind challenge: the performance of relative free energy calculations

Relative free energy calculations based on molecular dynamics simulations are combined with available experimental binding free energies to predict unknown binding affinities of acyclic Cucurbituril complexes in the blind SAMPL3 competition. The predictions yield root mean square errors between 2.6 and 3.2 kcal/mol for seven host-guest systems. Those deviations are comparable to results for solvation free energies of small organic molecules. However, the standard deviations found in our simulations range from 0.4 to 2.4 kcal/mol, which indicates the need for better sampling. Three different approaches are compared. Bennett's Acceptance Ratio Method and thermodynamic integration based on the trapezoidal rule with 12 λ-points exhibit a root mean square error of 2.6 kcal/mol, while thermodynamic integration with Simpson's rule and 11 λ-points leads to a root mean square error of 3.2 kcal/mol. In terms of absolute median errors, Bennett's Acceptance Ratio Method performs better than thermodynamic integration with the trapezoidal rule (1.7 vs. 2.9 kcal/mol). Simulations of the deprotonated forms of the guest molecules exhibit a poorer correspondence to experimental results with a root mean square error of 5.2 kcal/mol. In addition, a decrease of the buffer concentration by approximately 20 mM in the simulations raises the root mean square error to 3.8 kcal/mol.