Free Energy Perturbation Theory Research Articles

Encoding prior knowledge in ensemble refinement.

The proper balancing of information from experiment and theory is a long-standing problem in the analysis of noisy and incomplete data. Viewed as a Pareto optimization problem, improved agreement with the experimental data comes at the expense of growing inconsistencies with the theoretical reference model. Here, we propose how to set the exchange rate a priori to properly balance this trade-off. We focus on gentle ensemble refinement, where the difference between the potential energy surfaces of the reference and refined models is small on a thermal scale. By relating the variance of this energy difference to the Kullback-Leibler divergence between the respective Boltzmann distributions, one can encode prior knowledge about energy uncertainties, i.e., force-field errors, in the exchange rate. The energy uncertainty is defined in the space of observables and depends on their type and number and on the thermodynamic state. We highlight the relation of gentle refinement to free energy perturbation theory. A balanced encoding of prior knowledge increases the quality and transparency of ensemble refinement. Our findings extend to non-Boltzmann distributions, where the uncertainty in energy becomes an uncertainty in information.

Enthalpies and entropies of hydration from Monte Carlo simulations.

The changes in free energy, enthalpy, and entropy for transfer of a solute from the gas phase into solution are the fundamental thermodynamic quantities that characterize the solvation process. Owing to the development of methods based on free-energy perturbation theory, computation of free energies of solvation has become routine in conjunction with Monte Carlo (MC) statistical mechanics and molecular dynamics (MD) simulations. Computation of the enthalpy change and by inference the entropy change is more challenging. Two methods are considered in this work corresponding to direct averaging for the solvent and solution and to computing the temperature derivative of the free energy in the van't Hoff approach. The application is for neutral organic solutes in TIP4P water using long MC simulations to improve precision. Definitive results are also provided for pure TIP4P water. While the uncertainty in computed free energies of hydration is ca. 0.05 kcal mol-1, it is ca. 0.4 kcal mol-1 for the enthalpy changes from either van't Hoff plots or the direct method with sampling for 5 billion MC configurations. Partial molar volumes of hydration are also computed by the direct method; they agree well with experimental data with an average deviation of 3 cm3 mol-1. In addition, the results permit breakdown of the errors in the free energy changes from the OPLS-AA force field into their enthalpic and entropic components. The excess hydrophobicity of organic solutes is enthalpic in origin.

Assessing Metadynamics and Docking for Absolute Binding Free Energy Calculations Using Severe Acute Respiratory Syndrome Coronavirus 2 Main Protease Inhibitors.

Absolute binding free energy (ABFE) calculations can be an important part of the drug discovery process by identifying molecules that have the potential to be strong binders for a biomolecular target. Recent work has used free energy perturbation (FEP) theory for these calculations, focusing on a set of 16 inhibitors of the severe acute respiratory syndrome coronavirus 2 main protease (Mpro). Herein, the same data set is evaluated by metadynamics (MetaD), four different docking programs, and molecular mechanics with generalized Born and surface area solvation. MetaD yields a Kendall τ distance of 0.28 and Pearson r2 of 0.49, which reflect somewhat less accuracy than that from the ABFE FEP results. Notably, it is demonstrated that an ensemble docking protocol by which each ligand is docked into the 13 crystal structures in this data set provides improved performance, particularly when docking is carried out with Glide XP (Kendall τ distance = 0.20, Pearson r2 = 0.71), Glide SP (Kendall τ distance = 0.19, Pearson r2 = 0.66), or AutoDock 4 (Kendall τ distance = 0.21, Pearson r2 = 0.55). The best results are obtained with "superconsensus" docking by averaging the 52 results for each compound using the 4 docking protocols and all 13 crystal structures (Kendall τ distance = 0.18, Pearson r2 = 0.73).

Computation of Absolute Binding Free Energies for Noncovalent Inhibitors with SARS-CoV-2 Main Protease.

Accurate, routine calculation of absolute binding free energies (ABFEs) for protein-ligand complexes remains a key goal of computer-aided drug design since it can enable screening and optimization of drug candidates. For development and testing of related methods, it is important to have high-quality datasets. To this end, from our own experimental studies, we have selected a set of 16 inhibitors of the SARS-CoV-2 main protease (Mpro) with structural diversity and well-distributed BFEs covering a 5 kcal/mol range. There is also minimal structural uncertainty since X-ray crystal structures have been deposited for 12 of the compounds. For methods testing, we report ABFE results from 2 μs molecular dynamics (MD) simulations using free energy perturbation (FEP) theory. The correlation of experimental and computed results is encouraging, with a Pearson's r2 of 0.58 and a Kendall τ of 0.24. The results indicate that current FEP-based ABFE calculations can be used for identification of active compounds (hits). While their accuracy for lead optimization is not yet sufficient, this activity remains addressable in separate lead series by relative BFE calculations.

The stepwise dissociation of the Zn(II)-bound Atox1 homodimer and its energetic asymmetry

The copper chaperone Atox1, which is featured by a peculiar CxxC metal-binding motif, is a protein involved in copper trafficking to the Golgi organelle, allowing the delivery of copper ions to specific target proteins in the secretory pathway. Various experimental studies suggest that this cysteine-containing protein may also be implicated in Zn(II) ion binding, functioning as a regulatory factor. However, despite the relevance of this interaction, the mechanism of zinc ion dissociation from the Atox1 homodimer has not yet been elucidated.We herein devised a simulation workflow based on the combination of parallel bias metadynamics and free energy perturbation theory, to predict the dissociation of the zinc ion from the Atox1 homodimer. The freeenergy simulations successfully predicted the dissociation free energy differences of the process, disclosing a mechanism based on a stepwise dissociation pathway of the zinc ion from the Atox1 homodimer.

Free energy perturbation calculations of tetrahydroquinolines complexed to the first bromodomain of BRD4

Alchemical free energy perturbation (FEP) theory is widely used nowadays to calculate protein–ligand binding energies, often in support of drug discovery endeavours. We assess the accuracy and sensitivity of absolute FEP binding energies with respect to the CHARMM/CGenFF and the AMBER/GAFF force field parameterisations for a set of tetrahydroquinoline inhibitors of the first bromodomain of BRD4, a target of keen interest for the development of anti-cancer drugs. We find that AMBER/GAFF is better able than CHARMM/CGenFF to cover the range of and to distinguish between the relative binding energies of the 16 ligands.

The effect of surface functional groups on the wettability of graphene oxide coated alumina substrate: Molecular dynamics simulations

In this article, molecular dynamic simulation was used to simulate the wetting process of graphene oxide coated alumina substrate. By changing the density of hydroxyl and epoxy groups on the surface of graphene oxide, the influence mechanism of two functional groups on the wetting process of water molecules was explored. From the simulation results, the increase of hydroxyl density can enhance the wettability of the substrate and reduce the wettability angle from 64.32° to 54.20°. By calculating the formation of the hydrogen bond in the wetting process, it is found that hydroxyl can significantly raise the hydrogen bond lifetime in the system. This increasing effect is particularly prominent in the observation of the hydrogen bond between water molecules and graphene oxide. The hydrogen bond lifetime increases from 2.20 ps to 12.77 ps. The result also reflects the reason why hydroxyl increases the wettability of the substrate. In addition, the increase of hydroxyl density also brings changes in hydrogen bond behavior and the number of hydrogen bonds. Under the condition of high epoxy density, most of the hydrogen bonds between water molecules and the substrate exist in the form of single hydrogen bonds, and the number is small. When the hydroxyl density increases, double-hydrogen bond, triple-hydrogen bond and single hydrogen bond appear in the model. In this process, the number of hydrogen bonds between the water droplet and the substrate is also increasing. Temperature is an important influence on the formation and lifetime of the hydrogen bond. High temperatures can promote the formation of the hydrogen bond between alumina and water molecules. In addition, it can also decrease the lifetime of all types of hydrogen bonds. Through the free energy perturbation theory, we get the free energy of different models in the wetting process. The results show that high hydroxyl density can improve the free energy of the wetting process, reduce the ability of spontaneous desorption and increase the stability of wetting. High temperatures can reduce the free energy and lead to the occurrence of desorption behavior. This article reveals the wetting mechanism of water molecules on graphene oxide coated alumina substrate, mainly reveals the influence of hydrogen bond and the change of free energy in the wetting process, which lays a theoretical foundation for the application of graphene oxide wetted coating on the aluminum alloy surface.

Discover, Sample, and Refine: Exploring Chemistry with Enhanced Sampling Techniques.

Over the last few decades, enhanced sampling methods have been continuously improved. Here, we exploit this progress and propose a modular workflow for blind reaction discovery and determination of reaction paths. In a three-step strategy, at first we use a collective variable derived from spectral graph theory in conjunction with the explore variant of the on-the-fly probability enhanced sampling method to drive reaction discovery runs. Once different chemical products are determined, we construct an ad-hoc neural network-based collective variable to improve sampling, and finally we refine the results using the free energy perturbation theory and a more accurate Hamiltonian. We apply this strategy to both intramolecular and intermolecular reactions. Our workflow requires minimal user input and extends the power of ab initio molecular dynamics to explore and characterize the reaction space.

Free Energy Computation for an Isomerizing Chromophore in a Molecular Cavity via the Average Solvent Electrostatic Configuration Model: Applications in Rhodopsin and Rhodopsin-Mimicking Systems.

We present a novel technique for computing the free energy differences between two chromophore "isomers" hosted in a molecular environment (a generalized solvent). Such an environment may range from a relatively rigid protein cavity to a flexible solvent environment. The technique is characterized by the application of the previously reported "average electrostatic solvent configuration" method, and it is based on the idea of using the free energy perturbation theory along with a chromophore annihilation procedure in thermodynamic cycle calculations. The method is benchmarked by computing the ground-state room-temperature relative stabilities between (i) the cis and trans isomers of prototypal animal and microbial rhodopsins and (ii) the analogue isomers of a rhodopsin-like light-driven molecular switch in methanol. Furthermore, we show that the same technology can be used to estimate the activation free energy for the thermal isomerization of systems i-ii by replacing one isomer with a transition state. The results show that the computed relative stability and isomerization barrier magnitudes for the selected systems are in line with the available experimental observation in spite of their widely diverse complexity.

Multi-replica biased sampling for photoswitchable π-conjugated polymers.

In recent years, π-conjugated polymers are attracting considerable interest in view of their light-dependent torsional reorganization around the π-conjugated backbone, which determines peculiar light-emitting properties. Motivated by the interest in designing conjugated polymers with tunable photoswitchable pathways, we devised a computational framework to enhance the sampling of the torsional conformational space and, at the same time, estimate ground- to excited-state free-energy differences. This scheme is based on a combination of Hamiltonian Replica Exchange Method (REM), parallel bias metadynamics, and free-energy perturbation theory. In our scheme, each REM samples an intermediate unphysical state between the ground and the first two excited states, which are characterized by time-dependent density functional theory simulations at the B3LYP/6-31G* level of theory. We applied the method to a 5-mer of 9,9-dioctylfluorene and found that upon irradiation, this system can undergo a dihedral inversion from -155° to 155°, crossing a barrier that decreases from 0.1 eV in the ground state (S0) to 0.05 eV and 0.04 eV in the first (S1) and second (S2) excited states. Furthermore, S1 and even more S2 were predicted to stabilize coplanar dihedrals, with a local free-energy minimum located at ±44°. The presence of a free-energy barrier of 0.08 eV for the S1 state and 0.12 eV for the S2 state can trap this conformation in a basin far from the global free-energy minimum located at 155°. The simulation results were compared with the experimental emission spectrum, showing a quantitative agreement with the predictions provided by our framework.

Accounting for the instantaneous disorder in the enzyme-substrate Michaelis complex to calculate the Gibbs free energy barrier of an enzyme reaction.

Many enzyme reactions present instantaneous disorder. These dynamic fluctuations in the enzyme-substrate Michaelis complexes generate a wide range of energy barriers that cannot be experimentally observed, but that determine the measured kinetics of the reaction. These individual energy barriers can be calculated using QM/MM methods, but then the problem is how to deal with this dispersion of energy barriers to provide kinetic information. So far, the most usual procedure has implied the so-called exponential average of the energy barriers. In this paper, we discuss the foundations of this method, and we use the free energy perturbation theory to derive an alternative equation to get the Gibbs free energy barrier of the enzyme reaction. In addition, we propose a practical way to implement it. We have chosen four enzyme reactions as examples. In particular, we have studied the hydrolysis of a glycosidic bond catalyzed by the enzyme Thermus thermophilus β-glycosidase, and the mutant Y284P Ttb-gly, and the hydrogen abstraction reactions from C13 and C7 of arachidonic acid catalyzed by the enzyme rabbit 15-lipoxygenase-1.

Explicit Representation of Cation-π Interactions in Force Fields with 1/r4 Nonbonded Terms.

The binding energies for cation-π complexation are underestimated by traditional fixed-charge force fields owing to their lack of explicit treatment of ion-induced dipole interactions. To address this deficiency, an explicit treatment of cation-π interactions has been introduced into the OPLS-AA force field. Following prior work with atomic cations, it is found that cation-π interactions can be handled efficiently by augmenting the usual 12-6 Lennard-Jones potentials with 1/r4 terms. Results are provided for prototypical complexes as well as protein-ligand systems of relevance for drug design. Alkali cation, ammonium, guanidinium, and tetramethylammonium were chosen for the representative cations, while benzene and six heteroaromatic molecules were used as the π systems. The required nonbonded parameters were fit to reproduce structure and interaction energies for gas-phase complexes from density functional theory (DFT) calculations at the ωB97X-D/6-311++G(d,p) level. The impact of the solvent was then examined by computing potentials of mean force (pmfs) in both aqueous and tetrahydrofuran (THF) solutions using the free-energy perturbation (FEP) theory. Further testing was carried out for two cases of strong and one case of weak cation-π interactions between druglike molecules and their protein hosts, namely, the JH2 domain of JAK2 kinase and macrophage migration inhibitory factor. FEP results reveal greater binding by 1.5-4.4 kcal/mol from the addition of the explicit cation-π contributions. Thus, in the absence of such treatment of cation-π interactions, errors for computed binding or inhibition constants of 101-103 are expected.

Computing Surface Acidity Constants of Proton Hopping Groups from Density Functional Theory-Based Molecular Dynamics: Application to the SnO2(110)/H2O Interface.

Proton transfer at metal oxide/water interfaces plays an important role in electrochemistry, geochemistry, and environmental science. The key thermodynamic quantity to characterize this process is the surface acidity constant. An ab initio method that combines density functional theory-based molecular dynamics (DFTMD) and free energy perturbation theory has been established for computing surface acidity constants. However, it involves a reversible proton insertion procedure in which frequent proton hopping, e.g., for strong bases and some oxide surfaces (e.g., SnO2), can cause instability issues in electronic structure calculation. In the original implementation, harmonic restraining potentials are imposed on all O-H bonds (denoted by the VrH scheme) to prevent proton hopping and thus may not be applicable for systems involving spontaneous proton hopping. In this work, we introduce an improved restraining scheme with a repulsive potential Vrep to compute the surface acidities of systems in which proton hopping is spontaneous and fast. In this Vrep scheme, a Buckingham-type repulsive potential Vrep is applied between the deprotonation site and all other protons in DFTMD simulations. We first verify the Vrep scheme by calculating the pKa values of H2O and aqueous HS- solution (i.e., strong conjugate bases) and then apply it to the SnO2(110)/H2O interface. It is found that the Vrep scheme leads to a prediction of the point of zero charge (PZC) of 4.6, which agrees well with experiment. The intrinsic individual pKa values of the terminal five-coordinated Sn site (Sn5cOH2) and bridge oxygen site (Sn2ObrH+) are 4.4 and 4.7, respectively, both being almost the same as the PZC. The similarity of the two pKa values indicates that dissociation of terminal water has almost zero free energy at this proton hopping interface (i.e., partial water dissociation), as expected from the acid-base equilibrium on SnO2.

CL-FEP: An End-State Free Energy Perturbation Approach.

The Central Limit Free Energy Perturbation (CL-FEP) approach, based on the Free Energy Perturbation (FEP) theory and the Central Limit (CL) theorem, allows evaluating the FEP identity directly from the energy samples of the end states of a system transformation. In CL-FEP, energies obtained from explicit solvent simulations are used to estimate the absolute free energy change. No fitted parameters are introduced in our implementation, and no stratification is needed to obtain accurate free energy evaluations. We first illustrate the applicability of CL-FEP in four dissimilar benchmark systems from host-guest to proteins-peptide complexes, for which the deviation from the experimental values was below 1 kcal/mol. Next, we extended the validation to three sets of complexes comprising 25 host-guest and protein-ligand systems. There, the mean absolute error of the free energy changes calculated with CL-FEP is between 1.1 and 1.4 kcal/mol among the different sets, which is in the range of accuracy of the most reliable free energy approaches applied to these systems. CL-FEP is an unbiased free energy change estimator that also profits from a bootstrapping algorithm, which makes the evaluation of convergence and confidence an inherent component of the procedure. Finally, we developed CLFEP.pl, an automatic tool to compute free energy changes directly from the energy output of standard molecular dynamics simulations.

Modifying the CHARMM36 Lipid Force Field for LJ-PME Simulations

Long-range Lennard-Jones (LJ) interactions were historically excluded from energy calculations in Molecular Dynamics (MD) simulations due to high computational cost. However, these interactions have a significant impact on the properties of nonpolar systems. The LJ particle-mesh Ewald (LJ-PME) method offers a solution to this problem and retains good efficiency. Here we introduce a modification to the additive all-atom CHARMM lipid force field (FF) which makes it compatible with the LJ-PME method. The last generation of the CHARMM36 (C36) lipid FF is suitable for NPT simulations of lipid bilayer systems using the PME method for the nonbonded interactions but systematically yields lower values of surface area per lipid compared to experimental estimates when the long-range LJ interactions are included. Here we develop an automated workflow to optimize the parameters of the nonbonded interactions (partial atomic charges and Lennard-Jones parameters). 1,2-dipalmitoyl-sn-phosphatidylcholine (DPPC) is used as the prototype and various structural properties (surface area per lipid, NMR deuterium order parameters, radial distribution functions of water for selected atoms) are used as targets. In each optimization step, free energy perturbation theory is used to estimate the properties from trajectories generated by the optimal parameters in the previous step. QM calculations of partial atomic charges are also used to guide the optimization. The parametrization improves the above-mentioned properties significantly with small changes to the C36 lipid FF and allows MD simulations of lipid systems to use the LJ-PME method.

Examining Product Specificity in Protein Arginine Methyltransferase 7 (PRMT7) Using Quantum and Molecular Mechanical Simulations.

Protein arginine methyltransferase 7 (PRMT7) catalyzes the formation of monomethylarginine (MMA) but is incapable of performing a dimethylation. Given that PRMT7 performs vital functions in mammalian cells and has been implicated in a variety of diseases, including breast cancer and age-related obesity, elucidating the origin of its strict monomethylation activity is of considerable interest. Three active site residues, Glu172, Phe71, and Gln329, have been reported as particularly important for product specificity and enzymatic activity. To better understand their roles, mixed quantum and molecular mechanical (QM/MM) calculations coupled to molecular dynamics and free energy perturbation theory were carried out for the WT, F71I, and Q329S trypanosomal PRMT7 (TbPRMT7) enzymes bound with S-adenosyl- L-methionine (AdoMet) and an arginine substrate in an unmethylated or methylated form. The Q329S mutation, which experimentally abolished enzymatic activity, was appropriately computed to give an outsized Δ G‡ of 30.1 kcal/mol for MMA formation compared to 16.9 kcal/mol for WT. The F71I mutation, which has been experimentally shown to convert the enzyme from a type III PRMT into a mixed type I/II capable of forming dimethylated arginine products, yielded a reasonable Δ G‡ of 21.9 kcal/mol for the second turnover compared to 28.8 kcal/mol in the WT enzyme. Similar active site orientations for both WT and F71I TbPRMT7 allowed Glu172 and Gln329 to better orient the substrate for SN2 methylation, enhanced the nucleophilicity of the attacking guanidino group by reducing positive charge, and facilitated the binding of the subsequent methylated products.

Reaction mechanism of NO with hydrolysates of NAMI-A: an MD simulation by combining the QM/MM(ABEEM) with the MD-FEP method.

Nitrosylation reaction mechanisms of the hydrolysates of NAMI-A and hydrolysis reactions of ruthenium nitrosyl complexes were investigated in the triplet state and the singlet state. Activation free energies were calculated by combining the QM/MM(ABEEM) method with free energy perturbation theory, and the explicit solvent environment was simulated by an ABEEMσπ polarizable force field. Our results demonstrate that nitrosylation reactions of the hydrolysates of NAMI-A occur in both the triplet and the singlet states. The Ru-N-O angle of the triplet ruthenium nitrosyl complexes is in the range of 132.0°-138.2°. However, all the ruthenium nitrosyl complexes at the singlet state show an almost linear Ru-N-O angle. The nitrosylation reaction happens prior to the hydrolysis reaction for the first-step hydrolysates. The activation free energies of the nitrosylation reactions show that the H2 O-NO exchange reaction of [RuCl4 (Im)(H2 O)] in the singlet spin sate is the most likely one. Comparing with the activation free energies of the hydrolysis reactions of the ruthenium nitrosyl complexes, the results indicate that the rate of the DMSO-H2 O exchange reaction of [RuCl3 (NO)(Im)(DMSO)] is faster than that of [RuCl3 (H2 O)(Im)(DMSO)] in both the triplet spin state and the singlet spin state. © 2018 Wiley Periodicals, Inc.

A Versatile Computational Strategy To Characterize the Free-Energy Landscape of Excited States in Oligofluorenes.

The π-rich rings of conjugated polymers and molecular rotors shape their typical properties, allowing a variety of chemical and photoresponsive phenomena. Herein, we present a computational method in the framework of classical simulations to estimate the free-energy gap between ground and excited states of oligofluorenes, which is a class of molecular rotors widely used in optoelectronic devices, because of the inner torsional rotation triggered by light irradiation. We devised multiple sets of free-energy simulations in combination with free-energy perturbation theory to predict the free-energy gap between the ground state and the first excited state. The computed excitation energies show good agreement with experiments. The approach presented herein allows one to achieve comprehensive sampling of the conformational landscape and accurate estimates of the excited state free-energy landscapes at the same time.

Dual QM and MM Approach for Computing Equilibrium Isotope Fractionation Factor of Organic Species in Solution.

A dual QM and MM approach for computing equilibrium isotope effects has been described. In the first partition, the potential energy surface is represented by a combined quantum mechanical and molecular mechanical (QM/MM) method, in which a solute molecule is treated quantum mechanically, and the remaining solvent molecules are approximated classically by molecular mechanics. In the second QM/MM partition, differential nuclear quantum effects responsible for the isotope effect are determined by a statistical mechanical double-averaging formalism, in which the nuclear centroid distribution is sampled classically by Newtonian molecular dynamics and the quantum mechanical spread of quantized particles about the centroid positions is treated using the path integral (PI) method. These partitions allow the potential energy surface to be properly represented such that the solute part is free of nuclear quantum effects for nuclear quantum mechanical simulations, and the double-averaging approach has the advantage of sampling efficiency for solvent configuration and for path integral convergence. Importantly, computational precision is achieved through free energy perturbation (FEP) theory to alchemically mutate one isotope into another. The PI-FEP approach is applied to model systems for the 18O enrichment found in cellulose of trees to determine the isotope enrichment factor of carbonyl compounds in water. The present method may be useful as a general tool for studying isotope fractionation in biological and geochemical systems.

Cost-Effective Method for Free-Energy Minimization in Complex Systems with Elaborated Ab Initio Potentials.

We describe a method to locate stationary points in the free-energy hypersurface of complex molecular systems using high-level correlated ab initio potentials. In this work, we assume a combined QM/MM description of the system although generalization to full ab initio potentials or other theoretical schemes is straightforward. The free-energy gradient (FEG) is obtained as the mean force acting on relevant nuclei using a dual level strategy. First, a statistical simulation is carried out using an appropriate, low-level quantum mechanical force-field. Free-energy perturbation (FEP) theory is then used to obtain the free-energy derivatives for the target, high-level quantum mechanical force-field. We show that this composite FEG-FEP approach is able to reproduce the results of a standard free-energy minimization procedure with high accuracy, while simultaneously allowing for a drastic reduction of both computational and wall-clock time. The method has been applied to study the structure of the water molecule in liquid water at the QCISD/aug-cc-pVTZ level of theory, using the sampling from QM/MM molecular dynamics simulations at the B3LYP/6-311+G(d,p) level. The obtained values for the geometrical parameters and for the dipole moment of the water molecule are within the experimental error, and they also display an excellent agreement when compared to other theoretical estimations. The developed methodology represents therefore an important step toward the accurate determination of the mechanism, kinetics, and thermodynamic properties of processes in solution, in enzymes, and in other disordered chemical systems using state-of-the-art ab initio potentials.