Grid inhomogeneous solvation theory: Hydration structure and thermodynamics of the miniature receptor cucurbit[7]uril

The displacement of perturbed water upon binding is believed to play a critical role in the thermodynamics of biomolecular recognition, but it is nontrivial to unambiguously define and answer questions about this process. We address this issue by introducing grid inhomogeneous solvation theory (GIST), which discretizes the equations of inhomogeneous solvation theory (IST) onto a three-dimensional grid situated in the region of interest around a solute molecule or complex. Snapshots from explicit solvent simulations are used to estimate localized solvation entropies, energies, and free energies associated with the grid boxes, or voxels, and properly summing these thermodynamic quantities over voxels yields information about hydration thermodynamics. GIST thus provides a smoothly varying representation of water properties as a function of position, rather than focusing on hydration sites where solvent is present at high density. It therefore accounts for full or partial displacement of water from sites that are highly occupied by water, as well as for partly occupied and water-depleted regions around the solute. GIST can also provide a well-defined estimate of the solvation free energy and therefore enables a rigorous end-states analysis of binding. For example, one may not only use a first GIST calculation to project the thermodynamic consequences of displacing water from the surface of a receptor by a ligand, but also account, in a second GIST calculation, for the thermodynamics of subsequent solvent reorganization around the bound complex. In the present study, a first GIST analysis of the molecular host cucurbit[7]uril is found to yield a rich picture of hydration structure and thermodynamics in and around this miniature receptor. One of the most striking results is the observation of a toroidal region of high water density at the center of the host's nonpolar cavity. Despite its high density, the water in this toroidal region is disfavored energetically and entropically, and hence may contribute to the known ability of this small receptor to bind guest molecules with unusually high affinities. Interestingly, the toroidal region of high water density persists even when all partial charges of the receptor are set to zero. Thus, localized regions of high solvent density can be generated in a binding site without strong, attractive solute-solvent interactions.     

On the nonpolar hydration free energy of proteins: surface area and continuum solvent models for the solute-solvent interaction energy

Implicit solvent hydration free energy models are an important component of most modern computational methods aimed at protein structure prediction, binding affinity prediction, and modeling of conformational equilibria. The nonpolar component of the hydration free energy, consisting of a repulsive cavity term and an attractive van der Waals solute-solvent interaction term, is often modeled using estimators based on the solvent exposed solute surface area. In this paper, we analyze the accuracy of linear surface area models for predicting the van der Waals solute-solvent interaction energies of native and non-native protein conformations, peptides and small molecules, and the desolvation penalty of protein-protein and protein-ligand binding complexes. The target values are obtained from explicit solvent simulations and from a continuum solvent van der Waals interaction energy model. The results indicate that the standard surface area model, while useful on a coarse-grained scale, may not be accurate or transferable enough for high resolution modeling studies of protein folding and binding. The continuum model constructed in the course of this study provides one path for the development of a computationally efficient implicit solvent nonpolar hydration free energy estimator suitable for high-resolution structural and thermodynamic modeling of biological macromolecules.     

The SGB/NP hydration free energy model based on the surface generalized born solvent reaction field and novel nonpolar hydration free energy estimators

The development and parameterization of a solvent potential of mean force designed to reproduce the hydration thermodynamics of small molecules and macromolecules aimed toward applications in conformation prediction and ligand binding free energy prediction is presented. The model, named SGB/NP, is based on a parameterization of the Surface Generalized Born continuum dielectric electrostatic model using explicit solvent free energy perturbation calculations and a newly developed nonpolar hydration free energy estimator motivated by the results of explicit solvent simulations of the thermodynamics of hydration of hydrocarbons. The nonpolar model contains, in addition to the more commonly used solvent accessible surface area term, a component corresponding to the attractive solute-solvent interactions. This term is found to be important to improve the accuracy of the model, particularly for cyclic and hydrogen bonding compounds. The model is parameterized against the experimental hydration free energies of a set of small organic molecules. The model reproduces the experimental hydration free energies of small organic molecules with an accuracy comparable or superior to similar models employing more computationally demanding estimators and/or a more extensive set of parameters.     

Parameterization of the Hamiltonian Dielectric Solvent (HADES) Reaction‐Field Method for the Solvation Free Energies of Amino Acid Side‐Chain Analogs

Optimization of the Hamiltonian dielectric solvent (HADES) method for biomolecular simulations in a dielectric continuum is presented with the goal of calculating accurate absolute solvation free energies while retaining the model’s accuracy in predicting conformational free‐energy differences. The solvation free energies of neutral and polar amino acid side‐chain analogs calculated by using HADES, which may optionally include nonpolar contributions, were optimized against experimental data to reach a chemical accuracy of about 0.5 kcal mol^?1. The new parameters were evaluated for charged side‐chain analogs. The HADES results were compared with explicit‐solvent, generalized Born, Poisson–Boltzmann, and QM‐based methods. The potentials of mean force (PMFs) between pairs of side‐chain analogs obtained by using HADES and explicit‐solvent simulations were used to evaluate the effects of the improved parameters optimized for solvation free energies on intermolecular potentials.     

Continuum solvation models in the linear interaction energy method

The linear interaction energy (LIE) method in combination with two different continuum solvent models has been applied to calculate protein-ligand binding free energies for a set of inhibitors against the malarial aspartic protease plasmepsin II. Ligand-water interaction energies are calculated from both Poisson-Boltzmann (PB) and Generalized Born (GB) continuum models using snapshots from explicit solvent simulations of the ligand and protein-ligand complex. These are compared to explicit solvent calculations, and we find close agreement between the explicit water and PB solvation models. The GB model overestimates the change in solvation energy, and this is caused by consistent underestimation of the effective Born radii in the protein-ligand complex. The explicit solvent LIE calculations and LIE-PB, with our standard parametrization, reproduce absolute experimental binding free energies with an average unsigned error of 0.5 and 0.7 kcal/mol, respectively. The LIE-GB method, however, requires a constant offset to approach the same level of accuracy.     

Surveying implicit solvent models for estimating small molecule absolute hydration free energies

Implicit solvent models are powerful tools in accounting for the aqueous environment at a fraction of the computational expense of explicit solvent representations. Here, we compare the ability of common implicit solvent models (TC, OBC, OBC2, GBMV, GBMV2, GBSW, GBSW/MS, GBSW/MS2 and FACTS) to reproduce experimental absolute hydration free energies for a series of 499 small neutral molecules that are modeled using AMBER/GAFF parameters and AM1-BCC charges. Given optimized surface tension coefficients for scaling the surface area term in the nonpolar contribution, most implicit solvent models demonstrate reasonable agreement with extensive explicit solvent simulations (average difference 1.0-1.7 kcal/mol and R(2)=0.81-0.91) and with experimental hydration free energies (average unsigned errors=1.1-1.4 kcal/mol and R(2)=0.66-0.81). Chemical classes of compounds are identified that need further optimization of their ligand force field parameters and others that require improvement in the physical parameters of the implicit solvent models themselves. More sophisticated nonpolar models are also likely necessary to more effectively represent the underlying physics of solvation and take the quality of hydration free energies estimated from implicit solvent models to the next level.     

How accurate are continuum solvation models for drug-like molecules?

We have estimated the hydration free energy for 20 neutral drug-like molecules, as well as for three series of 6–11 inhibitors to avidin, factor Xa, and galectin-3 with four different continuum solvent approaches (the polarised continuum method the Langevin dipole method, the finite-difference solution of the Poisson equation, and the generalised Born method), and several variants of each, giving in total 24 different methods. All four types of methods have been thoroughly calibrated for a number of experimentally known small organic molecules with a mean absolute deviation (MAD) of 1–6 kJ/mol for neutral molecules and 4–30 kJ/mol for ions. However, for the drug-like molecules, the accuracy seems to be appreciably worse. The reason for this is that drug-like molecules are more polar than small organic molecules and that the uncertainty of the methods is proportional to the size of the solvation energy. Therefore, the accuracy of continuum solvation methods should be discussed in relative, rather than absolute, terms. In fact, the mean unsigned relative deviations of the best solvation methods, 0.09 for neutral and 0.05 for ionic molecules, correspond to 2–20 kJ/mol absolute error for the drug-like molecules in this investigation, or 2–3,000 in terms of binding constants. Fortunately, the accuracy of all methods can be improved if only relative energies within a series of inhibitors are considered, especially if all of them have the same net charge. Then, all except two methods give MADs of 2–5 kJ/mol (corresponding to an uncertainty of a factor of 2–7 in the binding constant). Interestingly, the generalised Born methods typically give better results than the Poison–Boltzmann methods. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Implicit solvent models: Combining an analytical formulation of continuum electrostatics with simple models of the hydrophobic effect

The recent development of approximate analytical formulations of continuum electrostatics opens the possibility of efficient and accurate implicit solvent models for biomolecular simulations. One such formulation (ACE, Schaefer & Karplus, J. Phys. Chem., 1996, 100:1578) is used to compute the electrostatic contribution to solvation and conformational free energies of a set of small solutes and three proteins. Results are compared to finite-difference solutions of the Poisson equation (FDPB) and explicit solvent simulations and experimental data where available. Small molecule solvation free energies agree with FDPB within 1–1.5 kcal/mol, which is comparable to differences in FDPB due to different surface treatments or different force field parameterizations. Side chain conformation free energies of aspartate and asparagine are in qualitative agreement with explicit solvent simulations, while 74 conformations of a surface loop in the protein Ras are accurately ranked compared to FDPB. Preliminary results for solvation free energies of small alkane and polar solutes suggest that a recent Gaussian model could be used in combination with analytical continuum electrostatics to treat nonpolar interactions. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 322–335, 1999     

基于分子动力学模拟和连续介质模型的自由能计算方法*

近些年,基于分子动力学模拟和连续介质模型的自由能计算方法受到了越来越多的关注,其中MM/PBSA就是最具代表性的方法.在MM/PBSA中,体系的焓变采用分子力学(MM)的方法计算得到;溶剂效应中极性部分对自由能的贡献通过解Poisson-Boltzmann(PB)方程的方法计算得到;溶液效应中非极性部分对自由能的贡献则通过分子表面积(SA)计算得到.本文结合我们科研组的工作,就近几年MM/PBSA方法的最新进展做了较为详细的阐述,同时对MM/PBSA的发展前景进行了展望.     

Calculations of solute and solvent entropies from molecular dynamics simulations

The translational, rotational and conformational (vibrational) entropy contributions to ligand-receptor binding free energies are analyzed within the standard formulation of statistical thermodynamics. It is shown that the partitioning of the binding entropy into different components is to some extent arbitrary, but an appropriate method to calculate both translational and rotational entropy contributions to noncovalent association is by estimating the configurational volumes of the ligand in the bound and free states. Different approaches to calculating solute entropies using free energy perturbation calculations, configurational volumes based on root-mean-square fluctuations and covariance matrix based quasiharmonic analysis are illustrated for some simple molecular systems. Numerical examples for the different contributions demonstrate that theoretically derived results are well reproduced by the approximations. Calculation of solvent entropies, either using total potential energy averages or van't Hoff plots, are carried out for the case of ion solvation in water. Although convergence problems will persist for large and complex simulation systems, good agreement with experiment is obtained here for relative and absolute ion hydration entropies. We also outline how solvent and solute entropic contributions are taken into account in empirical binding free energy calculations using the linear interaction energy method. In particular it is shown that empirical scaling of the nonpolar intermolecular ligand interaction energy effectively takes into account size dependent contributions to the binding free energy.     

The importance of excluded solvent volume effects in computing hydration free energies

Continuum dielectric methods such as the Born equation have been widely used to compute the electrostatic component of the solvation free energy, DeltaG(solv)(elec), because they do not need to include solvent molecules explicitly and are thus far less costly compared to molecular simulations. All of these methods can be derived from Gauss Law of Maxwell's equations, which yields an analytical solution for the solvation free energy, DeltaG(Born), when the solute is spherical. However, in Maxwell's equations, the solvent is assumed to be a structureless continuum, whereas in reality, the near-solute solvent molecules are highly structured unlike far-solute bulk solvent. Since we have recently reformulated Gauss Law of Maxwell's equations to incorporate the near-solute solvent structure by considering excluded solvent volume effects, we have used it in this work to derive an analytical solution for the hydration free energy of an ion. In contrast to continuum solvent models, which assume that the normalized induced solvent electric dipole density P(n) is constant, P(n) mimics that observed from simulations. The analytical formula for the ionic hydration free energy shows that the Born radius, which has been used as an adjustable parameter to fit experimental hydration free energies, is no longer ill defined but is related to the radius and polarizability of the water molecule, the hydration number, and the first peak position of the solute-solvent radial distribution function. The resulting DeltaG(solv)(elec) values are shown to be close to the respective experimental numbers.     

Rapid approximate calculation of water binding free energies in the whole hydration domain of (bio)macromolecules

The evaluation of water binding free energies around solute molecules is important for the thermodynamic characterization of hydration or association processes. Here, a rapid approximate method to estimate water binding free energies around (bio)macromolecules from a single molecular dynamics simulation is presented. The basic idea is that endpoint free‐energy calculation methods are applied and the endpoint quantities are monitored on a three‐dimensional grid around the solute. Thus, a gridded map of water binding free energies around the solute is obtained, that is, from a single short simulation, a map of favorable and unfavorable water binding sites can be constructed. Among the employed free‐energy calculation methods, approaches involving endpoint information pertaining to actual thermodynamic integration calculations or endpoint information as exploited in the linear interaction energy method were examined. The accuracy of the approximate approaches was evaluated on the hydration of a cage‐like molecule representing either a nonpolar, polar, or charged water binding site and on α‐ and β‐cyclodextrin molecules. Among the tested approaches, the linear interaction energy method is considered the most viable approach. Applying the linear interaction energy method on the grid around the solute, a semi‐quantitative thermodynamic characterization of hydration around the whole solute is obtained. Disadvantages are the approximate nature of the method and a limited flexibility of the solute. © 2016 Wiley Periodicals, Inc.     

MST continuum study of the hydration free energies of monovalent ionic species

In this study, we revisit the protocol previously proposed within the framework of the Miertus-Scrocco-Tomasi (MST) continuum model to define the cavity between the solute and solvent for predicting hydration free energies of univalent ions. The protocol relies on the use of a reduced cavity (around 10-15% smaller than the cavity used for neutral compounds) around the atom(s) bearing the formal charge. The suitability of this approach is examined here for a series of 47 univalent ions for which accurate experimental hydration free energies are available. Attention is also paid to the effect of the charge renormalization protocol used to correct uncertainties arising from the electron density located outside the solute cavity. The method presented here provides, with a minimum number of fitted parameters, reasonable estimates within the experimental error of the hydration free energy of ions (average relative error of 4.7%) and is able to reproduce solvation in water of both small and large ions.     

Implicit nonpolar solvent models

We have systematically analyzed a new nonpolar solvent model that separates nonpolar solvation free energy into repulsive and attractive components. Our analysis shows that either molecular surfaces or volumes can be used to correlate with repulsive free energies of tested molecules in explicit solvent with correlation coefficients higher than 0.99. In addition, the attractive free energies in explicit solvent can also be reproduced with the new model with a correlation coefficient higher than 0.999. Given each component optimized, the new nonpolar solvent model is found to reproduce monomer nonpolar solvation free energies in explicit solvent very well. However, the overall accuracy of the nonpolar solvation free energies is lower than that of each component. In the more challenging dimer test cases, the agreement of the new model with explicit solvent is less impressive. Nevertheless, it is found that the new model works reasonably well for reproducing the relative nonpolar free energy landscapes near the global minimum of the dimer complexes.     

Redox entropy of plastocyanin: developing a microscopic view of mesoscopic polar solvation

We report applications of analytical formalisms and molecular dynamics (MD) simulations to the calculation of redox entropy of plastocyanin metalloprotein in aqueous solution. The goal of our analysis is to establish critical components of the theory required to describe polar solvation at the mesoscopic scale. The analytical techniques include a microscopic formalism based on structure factors of the solvent dipolar orientations and density and continuum dielectric theories. The microscopic theory employs the atomistic structure of the protein with force-field atomic charges and solvent structure factors obtained from separate MD simulations of the homogeneous solvent. The MD simulations provide linear response solvation free energies and reorganization energies of electron transfer in the temperature range of 280-310 K. We found that continuum models universally underestimate solvation entropies, and a more favorable agreement is reported between the microscopic calculations and MD simulations. The analysis of simulations also suggests that difficulties of extending standard formalisms to protein solvation are related to the inhomogeneous structure of the solvation shell at the protein-water interface combining islands of highly structured water around ionized residues along with partial dewetting of hydrophobic patches. Quantitative theories of electrostatic protein hydration need to incorporate realistic density profile of water at the protein-water interface.     

Solvation thermodynamics of neutral and charged solutes using the solvation-layer interface condition continuum dielectric model

We demonstrate that the solvation-layer interface condition (SLIC) continuum dielectric model for molecular electrostatics, combined with a simple solvent-accessible-surface-area (SASA)-proportional model for nonpolar solvent effects, accurately predicts solvation entropies of neutral and charged small molecules. The SLIC/SASA model has only seven fitting parameters in total and achieves this accuracy using a training set with only 20 compounds. Despite this simplicity, solvation free energies and entropies are nearly as accurate as those predicted by the more sophisticated Langevin dipoles solvation model. Surprisingly, the model automatically reproduces the negligible contribution of electrostatics to the solvation of hydrophobic compounds. Opportunities for improvement include nonpolar solvation, anion solvation entropies, and heat capacities. More molecular realism may be needed for these quantities. To enable a future, explicit-solvent-based assessment of the SLIC/SASA implicit-solvent model, we predict solvation entropies for the Mobley test set, which are available as Supporting Information.     

The Binding Energy Distribution Analysis Method (BEDAM) for the Estimation of Protein-Ligand Binding Affinities

The Binding Energy Distribution Analysis Method (BEDAM) for the computation of receptor-ligand standard binding free energies with implicit solvation is presented. The method is based on a well established statistical mechanics theory of molecular association. It is shown that, in the context of implicit solvation, the theory is homologous to the test particle method of solvation thermodynamics with the solute-solvent potential represented by the effective binding energy of the protein-ligand complex. Accordingly, in BEDAM the binding constant is computed by means of a weighted integral of the probability distribution of the binding energy obtained in the canonical ensemble in which the ligand is positioned in the binding site but the receptor and the ligand interact only with the solvent continuum. It is shown that the binding energy distribution encodes all of the physical effects of binding. The balance between binding enthalpy and entropy is seen in our formalism as a balance between favorable and unfavorable binding modes which are coupled through the normalization of the binding energy distribution function. An efficient computational protocol for the binding energy distribution based on the AGBNP2 implicit solvent model, parallel Hamiltonian replica exchange sampling and histogram reweighting is developed. Applications of the method to a set of known binders and non-binders of the L99A and L99A/M102Q mutants of T4 lysozyme receptor are illustrated. The method is able to discriminate without error binders from non-binders, and the computed standard binding free energies of the binders are found to be in good agreement with experimental measurements. Analysis of the results reveals that the binding affinities of these systems reflect the contributions from multiple conformations spanning a wide range of binding energies.     

The performance of ANI-ML potentials for ligand-n(H2O) interaction energies and estimation of hydration free energies from end-point MD simulations

Here, we investigate the performance of “Accurate NeurAl networK engINe for Molecular Energies” (ANI), trained on small organic compounds, on bulk systems including non-covalent interactions and applicability to estimate solvation (hydration) free energies using the interaction between the ligand and explicit solvent (water) from single-step MD simulations. The method is adopted from ANI using the Atomic Simulation Environment (ASE) and predicts the non-covalent interaction energies at the accuracy of wb97x/6-31G(d) level by a simple linear scaling for the conformations sampled by molecular dynamics (MD) simulations of ligand-n(H₂O) systems. For the first time, we test ANI potentials' abilities to reproduce solvation free energies using linear interaction energy (LIE) formulism by modifying the original LIE equation. Our results on ~250 different complexes show that the method can be accurate and have a correlation of R² = 0.88–0.89 (MAE <1.0 kcal/mol) to the experimental solvation free energies, outperforming current end-state methods. Moreover, it is competitive to other conventional free energy methods such as FEP and BAR with 15-20 × fold reduced computational cost.