Bennett Acceptance Ratio Research Articles

Diffusion coefficient and solvation free energy of sucrose in water: a molecular dynamics study

In this work, we have carried out Molecular Dynamics Simulation technique to study the diffusion coefficients, radial distribution functions and solvation free energy of sucrose (C12H22O11) at different temperatures ranging from 298.15 K to 318.15 K. We have taken 2 molecules of sucrose in 1343 molecules of extended simple point charge (SPC/E) water model. The self-diffusion coefficients are obtained by applying linear best fit to the mean squaredisplacement (MSD) plot and binary diffusion coefficients are obtained by using Darken’s relation. Arrhenius equation has been used to show a linear relationship between natural logarithm of binary diffusion coefficient and reciprocal of temperature. The structural analysis of the solution has been carried out with the help of radial distribution functions (RDFs) of its constituents. The solvation free energy of sucrose is studied at 300 K by using free energy perturbation method like Bennett Acceptance Ratio (BAR). The results of binary diffusion coefficients obtained from this simulation work agree very well with the previously reported experimental data.

Small-sample limit of the Bennett acceptance ratio method and the variationally derived intermediates.

Free energy calculations based on atomistic Hamiltonians provide microscopic insight into the thermodynamic driving forces of biophysical or condensed matter systems. Many approaches use intermediate Hamiltonians interpolating between the two states for which the free energy difference is calculated. The Bennett acceptance ratio (BAR) and variationally derived intermediates (VI) methods are optimal estimator and intermediate states in that the mean-squared error of free energy calculations based on independent sampling is minimized. However, BAR and VI have been derived based on several approximations that do not hold for very few sample points. Analyzing one-dimensional test systems, we show that in such cases BAR and VI are suboptimal and that established uncertainty estimates are inaccurate. Whereas for VI to become optimal, less than seven samples per state suffice in all cases; for BAR the required number increases unboundedly with decreasing configuration space densities overlap of the end states. We show that for BAR, the required number of samples is related to the overlap through an inverse power law. Because this relation seems to hold universally and almost independent of other system properties, these findings can guide the proper choice of estimators for free energy calculations.

Thermodynamic and structural description of relative solubility of the flavonoid rutin by DFT calculations and molecular dynamics simulations

Understanding the solubility of flavonoids is of relevance in view of the broad application such as antioxidant and antitumor agents. The solubility of flavonoid rutin has been experimentally investigated in a homologous series of alcohols, acetone, n-heptane and water solvents. In this work theoretical evaluation of Gibbs free energy of solvation was performed at the Density Functional Theory (DFT) and Molecular Dynamics (MD) levels, with the ωB97x-D functional and multistate Bennett acceptance ratio (MBAR) method, respectively. The degree of solubility of a given solute among various solvents is dictated by solute-solvent interactions, which may be represented by the solvation energy. Our calculated relative energy values due to solvent effects, using quantum chemical methods (PCM and SMD solvent models) do not follow strictly the observed solubility trend. However, the PCM model including solute-solvent dispersion interaction, solute-solvent repulsion interaction and solute cavitation energy contribution to the total energy (named DIS,REP,CAV) yielded an overall agreement with experimental solubility profile, with exception of ethanol solvent. A similar satisfactory accordance with experiment was found for MBresults, with a discrepancy also found for 1-butanol, which deserve further investigation. Analysis of spatial and radial distribution functions (SDF and RDF) for pure solvents and the analysis of the contributions of vdW and electrostatic terms for the total free energy reveal that a balance between dispersive and electrostatic interactions between the rutin and the solvent is necessary to solubilize the flavonoid, which corroborates with the values of these energy contributions to the solvation-free energy calculated by the MBAR method. The use of explicit solvent molecules in DFT calculations is precluded due to the size, flexibility, and many sites for solvent interactions of rutin molecule. Our results indicate that in general DFT-PCM (DIS,REP,CAV) and MBAR levels of calculations can satisfactorily predict the solvation free energy and solubility trend for flavonoid rutin along a series of solvents, and also provide a reasonable estimate of relative solubilities of other flavonoids, information that can be relevant for further molecular modeling studies.

Computational studies of protein–drug binding affinity changes upon mutations in the drug target

AbstractMutations that lead to drug resistance limit the efficacy of antibiotics, antiviral drugs, targeted cancer therapies, and other treatments. Accurately calculating protein–drug binding affinity changes upon mutations in the drug target is of high interest as this can yield a better understanding into how such mutations drive drug‐resistance, especially when the mutation in question does not directly interfere with binding of the drug. The main aim of this article is to provide an up‐to‐date reference on the computational tools that are available for the calculation of Gibbs energy (free energy) changes upon mutation, their strengths, and limitations. The methods that are discussed include free energy calculations (free energy perturbation, thermodynamic integration, multistate Bennett acceptance ratio), analysis of molecular dynamics simulations (linear interaction energy, molecular mechanics [MM]/Poisson–Boltzmann solvated area, and MM/generalized Born solvated area), and methods that involve quantum mechanical calculations (including QM/MM). The possibility to use machine learning is also introduced. Given that the benefit of accurately calculating binding affinity changes upon mutation depends on comparing calculated values with experimental measurements, a brief survey on experimental methods and observables is provided. Examples of computational studies that go beyond calculating the Gibbs energy changes are given. Factors that need to be addressed by the computational chemist and potential pitfalls are discussed at length.This article is categorized under: Structure and Mechanism > Computational Biochemistry and Biophysics Molecular and Statistical Mechanics > Free Energy Methods Molecular and Statistical Mechanics > Molecular Interactions

Solvation Free Energy of Protonated Lysine: Molecular Dynamics Study

In the present work, we have used an alchemical approach for calculating solvation free energy of protonated lysine in water from molecular dynamics simulations. These approaches use a non-physical pathway between two end states in order to compute free energy difference from the set of simulations. The solute is modeled using bonded and non-bonded interactions described by OPLS-AA potential, while four different water models: TIP3P, SPC, SPC/E and TIP4P are used. The free energy of solvation of protonated lysine in water has been estimated using thermodynamic integration, free energy perturbation, and Bennett acceptance ratio methods at 310 K temperature. The contributions to the free energy due to van der Waals and electrostatics parameters are also separately computed. The estimated values of free energy of solvation using different methods are in well agreement with previously reported experimental value within 14 %.

Molecular Dynamics Simulations Reveal Interactions of an IgG1 Antibody With Selected Fc Receptors.

In a survey of novel interactions between an IgG1 antibody and different Fcγ receptors (FcγR), molecular dynamics simulations were performed of interactions of monoclonal antibody involved complexes with FcγRs. Free energy simulations were also performed of isolated wild-type and substituted Fc regions bound to FcγRs with the aim of assessing their relative binding affinities. Two different free energy calculation methods, Molecular Mechanical/Generalized Born Molecular Volume (MM/GBMV) and Bennett Acceptance Ratio (BAR), were used to evaluate the known effector substitution G236A that is known to selectively increase antibody dependent cellular phagocytosis. The obtained results for the MM/GBMV binding affinity between different FcγRs are in good agreement with previous experiments, and those obtained using the BAR method for the complete antibody and the Fc-FcγR simulations show increased affinity across all FcγRs when binding to the substituted antibody. The FcγRIIa, a key determinant of antibody agonistic efficacy, shows a 10-fold increase in binding affinity, which is also consistent with the published experimental results. Novel interactions between the Fab region of the antibody and the FcγRs were discovered with this in silico approach, and provide insights into the antibody-FcγR binding mechanism and show promise for future improvements of therapeutic antibodies for preclinical studies of biological drugs.

Coarse-Grained Modeling of Multiple Pathways in Conformational Transitions of Multi-Domain Proteins.

Large-scale conformational transitions in multi-domain proteins are often essential for their functions. To investigate the transitions, it is necessary to explore multiple potential pathways, which involve different intermediate structures. Here, we present a multi-basin (MB) coarse-grained (CG) structure-based Go̅ model for describing transitions in proteins with more than two moving domains. This model is an extension of our dual-basin Go̅ model in which system-dependent parameters are determined systematically using the multistate Bennett acceptance ratio method. In the MB Go̅ model for multi-domain proteins, we assume that intermediate structures may have partial inter-domain native contacts. This approach allows us to search multiple transition pathways that involve distinct intermediate structures using the CG molecular dynamics (MD) simulations. We apply this scheme to an enzyme, adenylate kinase (AdK), which has three major domains and can move along two different pathways. Using the optimized mixing parameters for each pathway, AdK shows frequent transitions between the Open, Closed, and the intermediate basins and samples a wide variety of conformations within each basin. The explored multiple transition pathways could be compared with experimental data and examined in more detail by atomistic MD simulations.

Extension of the Variational Free Energy Profile and Multistate Bennett Acceptance Ratio Methods for High-Dimensional Potential of Mean Force Profile Analysis.

We redevelop the variational free energy profile (vFEP) method using a cardinal B-spline basis to extend the method for analyzing free energy surfaces (FESs) involving three or more reaction coordinates. We also implemented software for evaluating high-dimensional profiles based on the multistate Bennett acceptance ratio (MBAR) method which constructs an unbiased probability density from global reweighting of the observed samples. The MBAR method takes advantage of a fast algorithm for solving the unbinned weighted histogram (UWHAM)/MBAR equations which replaces the solution of simultaneous equations with a nonlinear optimization of a convex function. We make use of cardinal B-splines and multiquadric radial basis functions to obtain smooth, differentiable MBAR profiles in arbitrary high dimensions. The cardinal B-spline vFEP and MBAR methods are compared using three example systems that examine 1D, 2D, and 3D profiles. Both methods are found to be useful and produce nearly indistinguishable results. The vFEP method is found to be 150 times faster than MBAR when applied to periodic 2D profiles, but the MBAR method is 4.5 times faster than vFEP when evaluating unbounded 3D profiles. In agreement with previous comparisons, we find the vFEP method produces superior FESs when the overlap between umbrella window simulations decreases. Finally, the associative reaction mechanism of hammerhead ribozyme is characterized using 3D, 4D, and 6D profiles, and the higher-dimensional profiles are found to have smaller reaction barriers by as much as 1.5 kcal/mol. The methods presented here have been implemented into the FE-ToolKit software package along with new methods for network-wide free energy analysis in drug discovery.

DeepBAR: A Fast and Exact Method for Binding Free Energy Computation.

The fast and accurate calculation of standard binding free energy has many important applications. Existing methodologies struggle at balancing accuracy and efficiency. We introduce a new method to compute binding free energy using deep generative models and the Bennett acceptance ratio method (DeepBAR). Compared to the rigorous potential of mean force (PMF) approach that requires sampling from intermediate states, DeepBAR is an order-of-magnitude more efficient as demonstrated in a series of host-guest systems. Notably, DeepBAR is exact and does not suffer from approximations for entropic contributions used in methods such as the molecular mechanics energy combined with the generalized Born and surface area continuum solvation (MM/GBSA). We anticipate DeepBAR to be a valuable tool for computing standard binding free energy used in drug design.

Evaluating the Multifunctionality of a New Modulator of Zinc-Induced Aβ Aggregation Using a Novel Computational Approach.

The high concentration of zinc metal ions in Aβ aggregations is one of the most cited hallmarks of Alzheimer's disease (AD), and several substantial pieces of evidence emphasize the key role of zinc metal ions in the pathogenesis of AD. In this study, while designing a multifunctional peptide for simultaneous targeting Aβ aggregation and chelating the zinc metal ion, a novel and comprehensive approach is introduced for evaluating the multifunctionality of a multifunctional drugs based on computational methods. The multifunctional peptide consists of inhibitor and chelator domains, which are included in the C-terminal hydrophobic region of Aβ, and the first four amino acids of human albumin. The ability of the multifunctional peptide in zinc ion chelation has been investigated using molecular dynamics (MD) simulations of the peptide-zinc interaction for 300 ns, and Bennett's acceptance ratio (BAR) method has been used to accurately calculate the chelation free energy. Data analysis demonstrates that the peptide chelating domain can be stably linked to the zinc ion. Besides, the introduced method used for evaluating chelation and calculating the free energy of peptide binding to zinc ions was successfully validated by comparison with previous experimental and theoretical published data. The results indicate that the multifunctional peptide, coordinating with the zinc metal ion, can be effective in Aβ inhibition by preserving the native helical structure of the Aβ42 monomer as well as disrupting the β-sheet structure of Aβ42 aggregates. Detailed assessments of the Aβ42-peptide interactions elucidate that the inhibition of Aβ is achieved by considerable hydrophobic interactions and hydrogen bonding between the multifunctional peptide and the hydrophobic Aβ regions, along with interfering in stable bridges formed inside the Aβ aggregate.

Variational Method for Networkwide Analysis of Relative Ligand Binding Free Energies with Loop Closure and Experimental Constraints.

We describe an efficient method for the simultaneous solution of all free energies within a relative binding free-energy (RBFE) network with cycle closure and experimental/reference constraint conditions using Bennett Acceptance Ratio (BAR) and Multistate BAR (MBAR) analysis. Rather than solving the BAR or MBAR equations for each transformation independently, the simultaneous solution of all transformations are obtained by performing a constrained minimization of a global objective function. The nonlinear optimization of the objective function is subjected to affine linear constraints that couple the free energies between the network edges. The constraints are used to enforce the closure of thermodynamic cycles within the RBFE network, and to enforce an additional set of linear constraint conditions demonstrated here to be subsets of (1 or 2) experimental values. We describe details of the practical implementation of the network BAR/MBAR procedure, including use of generalized coordinates in the minimization of the free-energy objective function, propagation of bootstrap errors from those coordinates, and performance and memory optimization. In some cases it is found that use of restraints in the optimization is more practical than use of generalized coordinates for enforcing constraint conditions. The fast BARnet and MBARnet methods are used to analyze the RBFEs of six prototypical protein-ligand systems, and it is shown that enforcement of cycle closure conditions reduces the error in the predictions only modestly, and further reduction in errors can be achieved when one or two experimental RBFEs are included in the optimization procedure. These methods have been implemented into FE-ToolKit, a new free-energy analysis toolkit. The BARnet/MBARnet framework presented here opens the door to new, more efficient and robust free-energy analysis with enhanced predictive capability for drug discovery applications.

Computing Absolute Free Energy with Deep Generative Models

Fast and accurate evaluation of free energy has broad applications from drug design to material engineering. Computing the absolute free energy is of particular interest since it allows the assessment of the relative stability between states without the use of intermediates. In this letter, we introduce a general framework for calculating the absolute free energy of a state. A key step of the calculation is the definition of a reference state with tractable deep generative models using locally sampled configurations. The absolute free energy of this reference state is zero by design. The free energy for the state of interest can then be determined as the difference from the reference. We applied this approach to both discrete and continuous systems and demonstrated its effectiveness. It was found that the Bennett acceptance ratio method provides more accurate and efficient free energy estimations than approximate expressions based on work. We anticipate the method presented here to be a valuable strategy for computing free energy differences.

The Variationally Derived Intermediates Method to Calculate Free Energy Differences

Accurate and fast free energy calculations, based on atomistic simulations, are key to a first principles understanding of many-particle systems in biophysics, and enable quantitative descriptions of molecular recognition processes, drug binding, transmembrane transport, or functional conformational motions of proteins or complexes. To this aim, alchemical methods, such as the Free Energy Perturbation or the Bennett Acceptance Ratio method, require ‘morphing’ between the Hamiltonians of the involved molecular end states, typically in terms of a linear interpolation. However, linear transformations and soft-core variants thereof are still a very special case amongst all possible transformations. Here, we present the Variationally Derived Intermediates (VI) method that generalizes this approach and yields, under the assumption of uncorrelated sampling points, the intermediates with optimal sampling accuracy, which differ markedly from the established ones. We present applications to the challenging case of solvation free energies, yielding improved accuracies for decoupling electrostatic interactions, and similar ones for Lennard-Jones interactions, compared to state of the art soft-core transformations.

Computing Absolute Free Energy with Deep Generative Models.

Fast and accurate evaluation of free energy has broad applications from drug design to material engineering. Computing the absolute free energy is of particular interest since it allows the assessment of the relative stability between states without intermediates. Here, we introduce a general framework for calculating the absolute free energy of a state. A key step of the calculation is the definition of a reference state with tractable deep generative models using locally sampled configurations. The absolute free energy of this reference state is zero by design. The free energy for the state of interest can then be determined as the difference from the reference. We applied this approach to both discrete and continuous systems and demonstrated its effectiveness. It was found that the Bennett acceptance ratio method provides more accurate and efficient free energy estimations than approximate expressions based on work. We anticipate the method presented here to be a valuable strategy for computing free energy differences.

Variationally derived intermediates for correlated free-energy estimates between intermediate states.

Free-energy difference calculations based on atomistic simulations generally improve in accuracy when sampling from a sequence of intermediate equilibrium thermodynamic states that bridge the configuration space between two states of interest. For reasons of efficiency, usually the same samples are used to calculate the stepwise difference of such an intermediate to both adjacent intermediates. However, this procedure violates the assumption of uncorrelated estimates that is necessary to derive both the optimal sequence of intermediate states and the widely used Bennett acceptance ratio estimator. In this work, via a variational approach, we derive the sequence of intermediate states and the corresponding estimator with minimal mean-squared error that account for these correlations and assess its accuracy.

Repulsive Soft-Core Potentials for Efficient Alchemical Free Energy Calculations.

In alchemical free energy (FE) simulations, annihilation and creation of atoms are generally achieved with the soft-core potential that shifts the interparticle separations. While this soft-core potential eliminates the numerical instability occurring near the two end states of the transformation, it makes the hybrid Hamiltonian vary nonlinearly with respect to the parameter λ, which interpolates between the Hamiltonians representing the two end states. This complicates FE estimation by Bennett acceptance ratio (BAR), free energy perturbation (FEP), and thermodynamic integration (TI) methods, thus reducing their calculation efficiency. In this work, we develop a new type of repulsive soft-core potential, called Gaussian soft-core (GSC) potential, with two parameters controlling its maximum and width. The main advantage of this potential is the linearity of the hybrid Hamiltonian with respect to λ, thus permitting the direct application of BAR, FEP, TI, and other variant FE methods. The accuracy and efficiency of the GSC potential are demonstrated by comparing the free energies of annihilation determined for 13 small molecules and an alchemical mutation of a protein side chain. In addition, in combination with a TI integrand (∂H/∂λ) estimation strategy, we show that GSC can considerably reduce the number of λ simulations compared to the commonly used separation-shifted soft-core potential.

Determining Free-Energy Differences Through Variationally Derived Intermediates.

Free-energy calculations based on atomistic Hamiltonians and sampling are key to a first-principles understanding of biomolecular processes, material properties, and macromolecular chemistry. Here, we generalize the free-energy perturbation method and derive nonlinear Hamiltonian transformation sequences yielding free-energy estimates with minimal mean squared error with respect to the exact values. Our variational approach applies to finite sampling and holds for any finite number of intermediate states. We show that our sequences are also optimal for the Bennett acceptance ratio (BAR) method, thereby generalizing BAR to small sampling sizes and non-Gaussian error distributions.

Quantifying Protein-Protein Interactions in Molecular Simulations.

Interactions among proteins, nucleic acids, and other macromolecules are essential for their biological functions and shape the physicochemcial properties of the crowded environments inside living cells. Binding interactions are commonly quantified by dissociation constants Kd, and both binding and nonbinding interactions are quantified by second osmotic virial coefficients B2. As a measure of nonspecific binding and stickiness, B2 is receiving renewed attention in the context of so-called liquid–liquid phase separation in protein and nucleic acid solutions. We show that Kd is fully determined by B2 and the fraction of the dimer observed in molecular simulations of two proteins in a box. We derive two methods to calculate B2. From molecular dynamics or Monte Carlo simulations using implicit solvents, we can determine B2 from insertion and removal energies by applying Bennett’s acceptance ratio (BAR) method or the (binless) weighted histogram analysis method (WHAM). From simulations using implicit or explicit solvents, one can estimate B2 from the probability that the two molecules are within a volume large enough to cover their range of interactions. We validate these methods for coarse-grained Monte Carlo simulations of three weakly binding proteins. Our estimates for Kd and B2 allow us to separate out the contributions of nonbinding interactions to B2. Comparison of calculated and measured values of Kd and B2 can be used to (re-)parameterize and improve molecular force fields by calibrating specific affinities, overall stickiness, and nonbinding interactions. The accuracy and efficiency of Kd and B2 calculations make them well suited for high-throughput studies of large interactomes.

Identifying and estimating bias in overlap-sampling free-energy calculations

ABSTRACT We examine several methods for detecting or quantifying bias in free energy calculations performed using Bennett's acceptance ratio (BAR) method for combining data from molecular simulations of two systems. The metrics for bias detection studied here are one based on the emergence, at the BAR estimate, of a zero-slope inflection of the free energy with respect to its optimisation parameter, and another that makes use of calculation of the dissipated work. Bias estimation methods studied here are two approaches based on consideration of the maximum possible contribution from a configuration that may be rarely sampled, and another that appeared recently in the literature that is derived explicitly using the neglected-tail ansatz. All approaches are tested as a function of sampling amount in application to a model multiharmonic system. The dissipated-work method provides the more reliable indicator of the presence of bias, and the maximal sample techniques yield the tightest bounds on the bias, among the methods tested.

An Insight into Solubility of H2S in Choline Based Ionic Liquids from Molecular Dynamics Simulation

Solvation free energy of six choline based ILs were computed under molecular dynamics (MD) simulation using Bennet Acceptance Ratio (BAR) method. H2S removal from the natural gas can be accomplished by absorption process using a solvent. Degree elimination of gases in liquid can be estimated by excess chemical potential (μex), Henry’s law constant (kH), enthalpy (hex), and entropy (sex). The Optimised Potential Liquids Simulation (OPLS) force fields used for ILs were validated against experimental density in order to get valid results of solvation free energy. The percentage error of the predicted density of six choline based ILs were below than 2.0% which is a good agreement with the previous study. The highest value of kH was 109.23 atm gained by choline thiocyanate [Chl][tcy] and the lowest value of kH was 23.25 atm obtained by choline acetate [Chl][act]. There are many features that influence the solubility of H2S for example length of alkyl chain, free volume effect, molar density, type of anions and hydrogen bonding interaction. In order to understand the behaviour of H2S molecules in ILs system, the radial distribution function (RDF) of H2S-ILs and RDF for specific atom were analysed. Basically, the shorter the alkyl chain, increase in molar density of ILs, reduce the free volume thus provide less space for H2S to occupy and decrease insolubility.