Computer simulations with explicit solvent: recent progress in the thermodynamic decomposition of free energies and in modeling electrostatic effects

Statistical-Mechanics Analyses on Thermodynamics of Protein Folding Constructed by Privalov and Co-Workers

Privalov and co-workers estimated the changes in hydration enthalpy and entropy upon ubiquitin unfolding and their temperature dependences denoted by ΔHhyd(T) and ΔShyd(T), respectively, from experimentally measured enthalpies and entropies of transfer of various model compounds from gaseous phase to water. We calculate ΔHhyd(T) and ΔShyd(T) for ubiquitin by our statistical-mechanics theory where molecular and atomistic models are employed for water and protein structure, respectively. ΔHhyd(T) and ΔShyd(T) calculated are in remarkably good agreement with those estimated by Privalov and co-workers. By examining relative magnitudes and signs of the changes in a variety of constituents of ΔHhyd(T) and ΔShyd(T), we confirm that the hydrophobic effect is an essential force driving a protein to fold. Detailed and comprehensive explanations are given for our claim that the prevailing views of the hydrophobic effect are not capable of elucidating its weakening at low temperatures, whereas our updated view is. We find out problematic points of the changes in enthalpy and entropy upon protein unfolding denoted by ΔH°(T) and ΔS°(T), respectively, which are measured using the differential scanning calorimetry at low pH, suggesting a theoretical method of calculating ΔH°(T) and ΔS°(T) at pH ∼ 7.

A classical density functional theory for solvation across length scales

A central aim of multiscale modeling is to use results from the Schrödinger equation to predict phenomenology on length scales that far exceed those of typical molecular correlations. In this work, we present a new approach rooted in classical density functional theory (cDFT) that allows us to accurately describe the solvation of apolar solutes across length scales. Our approach builds on the Lum-Chandler-Weeks (LCW) theory of hydrophobicity [K. Lum et al., J. Phys. Chem. B 103, 4570 (1999)] by constructing a free energy functional that uses a slowly varying component of the density field as a reference. From a practical viewpoint, the theory we present is numerically simpler and generalizes to solutes with soft-core repulsion more easily than LCW theory. Furthermore, by assessing the local compressibility and its critical scaling behavior, we demonstrate that our LCW-style cDFT approach contains the physics of critical drying, which has been emphasized as an essential aspect of hydrophobicity by recent theories. As our approach is parameterized on the two-body direct correlation function of the uniform fluid and the liquid-vapor surface tension, it straightforwardly captures the temperature dependence of solvation. Moreover, we use our theory to describe solvation at a first-principles level on length scales that vastly exceed what is accessible to molecular simulations.

The Joint Solvation Interaction

The solvent-induced interactions (SIIs) between flexible solutes can be separated into two distinct components: the solvation-induced conformational effect and the joint solvation interaction (JSI). The JSI quantifies the thermodynamic effect of the solvent simultaneously accommodating the solutes, generalizing the typical notion of the hydrophobic interaction. We present a formal definition of the JSI within the framework of the mixture expansion, demonstrate that this definition is equivalent to the SII between rigid solutes, and propose a method, partially connected molecular dynamics, which allows one to compute the interaction with existing free energy algorithms. We also compare the JSI to the more natural generalization of the hydrophobic interaction, the indirect solvent-mediated interaction, and argue that JSI is a more useful quantity for studying solute binding thermodynamics. Direct calculation of the JSI may prove useful in developing our understanding of solvent effects in self-assembly, protein aggregation, and protein folding, for which the isolation of the JSI from the conformational component of the SII becomes important due to the intra-species flexibility.

Doping heteroatoms to form multiple hydrogen bond sites for enhanced interfacial reconstruction and separations

Interfacial challenges in unconventional oil extraction include heavy oil-water-solid multiphase separation and corrosion inhibition. Herein, a novel strategy based on interfacial hydrogen bonding reconstruction is proposed for constructing multifunctional interfacially active materials (MIAMs) to address multi-interfacial separation needs. A simple one-pot method is applied to successfully synthesize four different MIAM varieties, integrating site groups (-NH2, OSO, -COOH, and Si-O-Si) with multiple hydrogen bonds (HBs) into allyl polyether chains. The results indicate that all synthesized MIAMs excel in demulsification, detergency, and corrosion inhibition simultaneously, even at 25 °C. Their dehydration efficiency for different water-in-oil emulsions (even heavy oil emulsion) surpasses 99.9 % even at 16 °C, showing their excellent energy-saving potential for field applications. Furthermore, they demonstrate effective, nondestructive static cleaning (up to 86 %) of adhered oil from solid surfaces at 25 °C and provide corrosion inhibition effects (up to 92.09 %) on mild steel immersed in saturated brine. Mechanistic tests reveal that incorporating multiple HB sites in MIAMs dramatically enhances their effectiveness in interfacial separations. Based on these findings, an HB-dominated noncovalent interaction reconstruction strategy is tentatively proposed to develop advanced materials for low-carbon, efficient interfacial separations.

Calculations of Absolute Free Energies, Enthalpies, and Entropies for Drug Binding

Computer simulation methods can aid in the rational design of drugs aimed at a specific target, typically a protein. The affinity of a drug for its target is given by the free energy of binding. Binding can be further characterized by the enthalpy and entropy changes in the process. Methods exist to determine exact free energies, enthalpies, and entropies that are dependent only on the quality of the potential model and adequate sampling of conformational degrees of freedom. Entropy and enthalpy are roughly an order of magnitude more difficult to calculate than the free energy. This project combines a replica exchange method for enhanced sampling, designed to be efficient for protein-sized systems, with free energy calculations. This approach, replica exchange with dynamical scaling (REDS), uses two conventional simulations at different temperatures so that the entropy can be found from the temperature dependence of the free energy. A third replica is placed between them, with a modified Hamiltonian that allows it to span the temperature range of the conventional replicas. REDS provides temperature-dependent data and aids in sampling. It is applied to the bromodomain-containing protein 4 (BRD4) system. We find that for the force fields used, the free energies are accurate but the entropies and enthalpies are not, with the entropic contribution being too positive. Reproducing the entropy and enthalpy of binding appears to be a more stringent test of the force fields than reproducing the free energy.

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.

Efficient Computation of the Electrostatic Component of Solvation Free Energy via a Two-Point Padé Approximation

We develop an efficient method to compute the electrostatic component of the solvation free energy via the two-point Padé approximation. The Padé approximant uses four parameters to describe the electrostatic free energy change of the solvation process, which could be readily determined from four thermodynamic properties obtained in two simulations, namely, the first- and second-order free energy gradients of any two states. Therefore, instead of sampling at multiple intermediate states, only two states, e.g., electrostatically fully solvated and desolvated, are needed to determine the Padé approximant and compute the corresponding free energy contribution. Applications to several model systems, including both neutral and charged species, show that the method can accurately produce electrostatic solvation free energy. The method would be very useful to save computational cost in applications in which accurate but expensive energy functions like quantum mechanics are used.

The role of loop dynamics in the prediction of ligand-protein binding enthalpy

The enthalpic and entropic components of ligand-protein binding free energy reflect the interactions and dynamics between ligand and protein. Despite decades of study, our understanding and hence our ability to predict these individual components remains poor. In recent years, there has been substantial effort and success in the prediction of relative and absolute binding free energies, but the prediction of the enthalpic (and entropic) contributions in biomolecular systems remains challenging. Indeed, it is not even clear what kind of performance in terms of accuracy could currently be obtained for such systems. It is, however, relatively straight-forward to compute the enthalpy of binding. We thus evaluated the performance of absolute enthalpy of binding calculations using molecular dynamics simulation for ten inhibitors against a member of the bromodomain family, BRD4-1, against isothermal titration calorimetry data. Initial calculations, with the AMBER force-field showed good agreement with experiment (R² = 0.60) and surprisingly good accuracy with an average of root-mean-square error (RMSE) = 2.49 kcal mol^-1. Of the ten predictions, three were obvious outliers that were all over-predicted compared to experiment. Analysis of various simulation factors, including parameterization, buffer concentration and conformational dynamics, revealed that the behaviour of a loop (the ZA loop on the periphery of the binding site) strongly dictates the enthalpic prediction. Consistent with previous observations, the loop exists in two distinct conformational states and by considering one or the other or both states, the prediction for the three outliers can be improved dramatically to the point where the R² = 0.95 and the accuracy in terms of RMSE improves to 0.90 kcal mol^-1. However, performance across force-fields is not consistent: if OPLS and CHARMM are used, different outliers are observed and the correlation with the ZA loop behaviour is not recapitulated, likely reflecting parameterization as a confounding problem. The results provide a benchmark standard for future study and comparison.

Effect of Conformational Equilibrium on Solvation Properties of 1,2-DCE in Water: A Solvation Thermodynamics and 3D-RISM Study

The contributions of the enthalpy and entropy of solvation for the study of chemical and biological systems are important in the prediction, interpretation, and manipulation of these processes. The relation between solvation Gibbs energies, enthalpies, and entropies of solvation, and their rigorous relation with the conformational equilibrium, are derived for the first time and applied with a computational method, in accordance with the Solvation Thermodynamics previous results, to 1,2-dichloroethane solvation in water. The rigid conformer calculations in solution were performed by using PC+/3D-RISM approach, with the conformational averaged results for enthalpy and solvation Gibbs energy reproducing the experimental results quite successfully. A qualitative agreement in the entropy of solvation predictions was also observed.

State averaged CASSCF in AMOEBA polarizable water model for simulating nonadiabatic molecular dynamics with nonequilibrium solvation effects

This paper presents a state-averaged complete active space self-consistent field (SA-CASSCF) in the atomic multipole optimized energetics for biomolecular application (AMOEBA) polarizable water model, which enables rigorous simulation of non-adiabatic molecular dynamics with nonequilibrium solvation effects. The molecular orbital and configuration interaction coefficients of the solute wavefunction, and the induced dipoles on solvent atoms, are solved by minimizing the state averaged energy variationally. In particular, by formulating AMOEBA water models and the polarizable continuum model (PCM) in a unified way, the algorithms developed for computing SA-CASSCF/PCM energies, analytical gradients, and non-adiabatic couplings in our previous work can be generalized to SA-CASSCF/AMOEBA by properly substituting a specific list of variables. Implementation of this method will be discussed with the emphasis on how the calculations of different terms are partitioned between the quantum chemistry and molecular mechanics codes. We will present and discuss results that demonstrate the accuracy and performance of the implementation. Next, we will discuss results that compare three solvent models that work with SA-CASSCF, i.e., PCM, fixed-charge force fields, and the newly implemented AMOEBA. Finally, the new SA-CASSCF/AMOEBA method has been interfaced with the ab initio multiple spawning method to carry out non-adiabatic molecular dynamics simulations. This method is demonstrated by simulating the photodynamics of the model retinal protonated Schiff base molecule in water.

GPU-Accelerated All-Atom Particle-Mesh Ewald Continuous Constant pH Molecular Dynamics in Amber

Constant pH molecular dynamics (MD) simulations sample protonation states on the fly according to the conformational environment and user specified pH conditions; however, the current accuracy is limited due to the use of implicit-solvent models or a hybrid solvent scheme. Here, we report the first GPU-accelerated implementation, parametrization, and validation of the all-atom continuous constant pH MD (CpHMD) method with particle-mesh Ewald (PME) electrostatics in the Amber22 pmemd.cuda engine. The titration parameters for Asp, Glu, His, Cys, and Lys were derived for the CHARMM c22 and Amber ff14sb and ff19sb force fields. We then evaluated the PME-CpHMD method using the asynchronous pH replica-exchange titration simulations with the c22 force field for six benchmark proteins, including BBL, hen egg white lysozyme (HEWL), staphylococcal nuclease (SNase), thioredoxin, ribonuclease A (RNaseA), and human muscle creatine kinase (HMCK). The root-mean-square deviation from the experimental pK_a's of Asp, Glu, His, and Cys is 0.76 pH units, and the Pearson's correlation coefficient for the pK_a shifts with respect to model values is 0.80. We demonstrated that a finite-size correction or much enlarged simulation box size can remove a systematic error of the calculated pK_a's and improve agreement with experiment. Importantly, the simulations captured the relevant biology in several challenging cases, e.g., the titration order of the catalytic dyad Glu35/Asp52 in HEWL and the coupled residues Asp19/Asp21 in SNase, the large pK_a upshift of the deeply buried catalytic Asp26 in thioredoxin, and the large pK_a downshift of the deeply buried catalytic Cys283 in HMCK. We anticipate that PME-CpHMD will offer proper pH control to improve the accuracies of MD simulations and enable mechanistic studies of proton-coupled dynamical processes that are ubiquitous in biology but remain poorly understood due to the lack of experimental tools and limitation of current MD simulations.

Solvation Thermodynamics of Solutes in Water and Ionic Liquids Using the Multiscale Solvation-Layer Interface Condition Continuum Model

Molecular assembly processes are generally driven by thermodynamic properties in solutions. Atomistic modeling can be very helpful in designing and understanding complex systems, except that bulk solvent is very inefficient to treat explicitly as discrete molecules. In this work, we develop and assess two multiscale solvation models for computing solvation thermodynamic properties. The new SLIC/CDC model combines continuum solvent electrostatics based on the solvent layer interface condition (SLIC) with new statistical thermodynamic models for hydrogen bonding and nonpolar modes: cavity formation, dispersion interactions, combinatorial mixing (CDC). Given the structures of 500 solutes, the SLIC/CDC model predicts Gibbs energies of solvation in water with an average accuracy better than 1 kcal/mol, when compared to experimental measurements, and better than 0.8 kcal/mol, when compared to explicit-solvent molecular dynamics simulations. The individual SLIC/CDC energy mode values agree quantitatively with those computed from explicit-solvent molecular dynamics. The previously published SLIC/SASA multiscale model combines the SLIC continuum electrostatic model with the solvent-accessible surface area (SASA) nonpolar energy mode. With our new, improved parametrization method, the SLIC/SASA model now predicts Gibbs energies of solvation with better than 1.4 kcal/mol average accuracy in aqueous systems, compared to experimental and explicit-solvent molecular dynamics, and better than 1.6 kcal/mol average accuracy in ionic liquids, compared to explicit-solvent molecular dynamics. Both models predict solvation entropies, and are the first implicit-solvation models capable of predicting solvation heat capacities.

Towards a transferable nonelectrostatic model for continuum solvation: The electrostatic and nonelectrostatic energy correction model

In this work, we introduce an electrostatic and non-electrostatic (ENE) correction to the solvation energy based on the Solvent-Accessible Surface Area (SASA) of the solute and the solvent static dielectric constant. The proposed correction was developed for neutral solutes in non-aqueous solvents, considering three different implicit solvation models based on a Self-Consistent Reaction Field treatment of solute-solvent mutual polarization using an Apparent Surface Charge formalism, namely the Integral Equation Formalism of the Polarizable Continuum Model using a continuous surface charge scheme (PCM), the Solvation Model based on solute electron density (SMD), and the generalized Finite-Difference Poisson-Boltzmann (FDPB) model. The proposed correction was parametrized on a diverse training set of 4980 solvation data from the Solv@tum database of experimental solvation energies, and validated on the non-aqueous subset of the MNSOL database comprising 2140 solvation energies. The performances of the proposed ENE models with minimal and extended parameters formulations have been analyzed and the latter variant has been further compared to the widely used Cavity, Dispersion, and Solvent structural effects (CDS) non-electrostatic model originally developed for the SMx family of implicit solvation models. Overall, a very good agreement between the computed solvation energies with the ENE correction and the reference experimental data has been found on both the training and test sets for all continuum solvation models considered. Furthermore, results for the ENE correction are on par with the reference CDS non-electrostatic model for both SMD and FDPB electrostatics, but with the advantage of using a lower number of parameters and thus an improved transferability between different electrostatics treatments.

Automated Molecular Cluster Growing for Explicit Solvation by Efficient Force Field and Tight Binding Methods

An automated and broadly applicable workflow for the description of solvation effects in an explicit manner is introduced. This method, termed quantum cluster growth (QCG), is based on the semiempirical GFN2-xTB/GFN-FF methods, enabling efficient geometry optimizations and MD simulations. Fast structure generation is provided using the intermolecular force field xTB-IFF. Additionally, the approach uses an efficient implicit solvation model for the electrostatic embedding of the growing clusters. The novel QCG procedure presents a robust cluster generation tool for subsequent application of higher-level (e.g., DFT) methods to study solvation effects on molecular geometries explicitly or to average spectroscopic properties over cluster ensembles. Furthermore, the computation of the solvation free energy with a supermolecular approach can be carried out with QCG. The underlying growing process is physically motivated by computing the leading-order solute-solvent interactions first and can account for conformational and chemical changes due to solvation for low-energy barrier processes. The conformational space is explored with the NCI-MTD algorithm as implemented in the CREST program, using a combination of metadynamics and MD simulations. QCG with GFN2-xTB yields realistic solution geometries and reasonable solvation free energies for various systems without introducing many empirical parameters. Computed IR spectra of some solutes with QCG show a better match to the experimental data compared to well-established implicit solvation models.

Widom insertion method in simulations with Ewald summation

We discuss the application of the Widom insertion method for calculation of the chemical potential of individual ions in computer simulations with Ewald summation. Two approaches are considered. In the first approach, an individual ion is inserted into a periodically replicated overall charge neutral system representing an electrolyte solution. In the second approach, an inserted ion is also periodically replicated, leading to the violation of the overall charge neutrality. This requires the introduction of an additional neutralizing background. We find that the second approach leads to a much better agreement with the results of grand canonical Monte Carlo simulation for the total chemical potential of a neutral ionic cluster.

Characterizing Protein Protonation Microstates Using Monte Carlo Sampling

Proteins are polyelectrolytes with acidic and basic amino acids Asp, Glu, Arg, Lys, and His, making up ≈25% of the residues. The protonation state of residues, cofactors, and ligands defines a "protonation microstate". In an ensemble of proteins some residues will be ionized and others neutral, leading to a mixture of protonation microstates rather than in a single one as is often assumed. The microstate distribution changes with pH. The protein environment also modifies residue proton affinity so microstate distributions change in different reaction intermediates or as ligands are bound. Particular protonation microstates may be required for function, while others exist simply because there are many states with similar energy. Here, the protonation microstates generated in Monte Carlo sampling in MCCE are characterized in HEW lysozyme as a function of pH and bacterial photosynthetic reaction centers (RCs) in different reaction intermediates. The lowest energy and highest probability microstates are compared. The ΔG, ΔH, and ΔS between the four protonation states of Glu35 and Asp52 in lysozyme are shown to be calculated with reasonable precision. At pH 7 the lysozyme charge ranges from 6 to 10, with 24 accepted protonation microstates, while RCs have ≈50,000. A weighted Pearson correlation analysis shows coupling between residue protonation states in RCs and how they change when the quinone in the Q_B site is reduced. Protonation microstates can be used to define input MD parameters and provide insight into the motion of protons coupled to reactions.

Thermodynamic Decomposition of Solvation Free Energies with Particle Mesh Ewald and Long-Range Lennard-Jones Interactions in Grid Inhomogeneous Solvation Theory

Grid Inhomogeneous Solvation Theory (GIST) maps out solvation thermodynamic properties on a fine meshed grid and provides a statistical mechanical formalism for thermodynamic end-state calculations. However, differences in how long-range nonbonded interactions are calculated in molecular dynamics engines and in the current implementation of GIST have prevented precise comparisons between free energies estimated using GIST and those from other free energy methods such as thermodynamic integration (TI). Here, we address this by presenting PME-GIST, a formalism by which particle mesh Ewald (PME)-based electrostatic energies and long-range Lennard-Jones (LJ) energies are decomposed and assigned to individual atoms and the corresponding voxels they occupy in a manner consistent with the GIST approach. PME-GIST yields potential energy calculations that are precisely consistent with modern simulation engines and performs these calculations at a dramatically faster speed than prior implementations. Here, we apply PME-GIST end-state analyses to 32 small molecules whose solvation free energies are close to evenly distributed from 2 kcal/mol to -17 kcal/mol and obtain solvation energies consistent with TI calculations (R² = 0.99, mean unsigned difference 0.8 kcal/mol). We also estimate the entropy contribution from the second and higher order entropy terms that are truncated in GIST by the differences between entropies calculated in TI and GIST. With a simple correction for the high order entropy terms, PME-GIST obtains solvation free energies that are highly consistent with TI calculations (R² = 0.99, mean unsigned difference = 0.4 kcal/mol) and experimental results (R² = 0.88, mean unsigned difference = 1.4 kcal/mol). The precision of PME-GIST also enables us to show that the solvation free energy of small hydrophobic and hydrophilic molecules can be largely understood based on perturbations of the solvent in a region extending a few solvation shells from the solute. We have integrated PME-GIST into the open-source molecular dynamics analysis software CPPTRAJ.

Characterizing the Water Wire in the Gramicidin Channel Found by Monte Carlo Sampling Using Continuum Electrostatics and in Molecular Dynamics Trajectories with Conventional or Polarizable Force Fields

Water molecules play a key role in all biochemical processes. They help define the shape of proteins, and they are reactant or product in many reactions and are released as ligands are bound. They facilitate the transfer of protons through transmembrane proton channel, pump and transporter proteins. Continuum electrostatics (CE) force fields used by program Multiconformation CE (MCCE) capture electrostatic interactions in biomolecules with an implicit solvent, which captures the averaged solvent water equilibrium properties. Hybrid CE methods can use explicit water molecules within the protein surrounded by implicit solvent. These hybrid methods permit the study of explicit hydrogen bond networks within the protein and allow analysis of processes such as proton transfer reactions. Yet hybrid CE methods have not been rigorously tested. Here, we present an explicit treatment of water molecules in the Gramicidin A (gA) channel using MCCE and compare the resulting distributions of water molecules and key hydration features against those obtained with explicit solvent Molecular Dynamics (MD) simulations with the nonpolarizable CHARMM36 and polarizable Drude force fields. CHARMM36 leads to an aligned water wire in the channel characterized by a large absolute net water dipole moment; the MCCE and Drude analysis lead to a small net dipole moment as the water molecules change orientation within the channel. The correct orientation is not as yet known, so these calculations identify an open question.

Exploring the Free-Energy Landscape and Thermodynamics of Protein-Protein Association

We report the free-energy landscape and thermodynamics of the protein-protein association responsible for the drug-induced multimerization of HIV-1 integrase (IN). Allosteric HIV-1 integrase inhibitors promote aberrant IN multimerization by bridging IN-IN intermolecular interactions. However, the thermodynamic driving forces and kinetics of the multimerization remain largely unknown. Here, we explore the early steps in the IN multimerization by using umbrella sampling and unbiased molecular dynamics simulations in explicit solvent. In direct simulations, the two initially separated dimers spontaneously associate to form near-native complexes that resemble the crystal structure of the aberrant tetramer. Most strikingly, the effective interaction of the protein-protein association is very short-ranged: the two dimers associate rapidly within tens of nanoseconds when their binding surfaces are separated by d ≤ 4.3 Å (less than two water diameters). Beyond this distance, the oligomerization kinetics appears to be diffusion controlled with a much longer association time. The free-energy profile also captured the crucial role of allosteric IN inhibitors in promoting multimerization and explained why several C-terminal domain mutations are remarkably resistant to the drug-induced multimerization. The results also show that at small separation, the protein-protein binding process contains two consecutive phases with distinct thermodynamic signatures. First, interprotein water molecules are expelled to the bulk, resulting in a small increase in entropy, as the solvent entropy gain from the water release is nearly cancelled by the loss of side-chain entropies as the two proteins approach each other. At shorter distances, the two dry binding surfaces adapt to each other to optimize their interaction energy at the expense of further protein configurational entropy loss. Although the binding interfaces feature clusters of hydrophobic residues, overall, the protein-protein association in this system is driven by enthalpy and opposed by entropy.

A grid-based algorithm in conjunction with a gaussian-based model of atoms for describing molecular geometry

A novel grid-based method is presented, which in conjunction with a smooth Gaussian-based model of atoms, is used to compute molecular volume (MV) and surface area (MSA). The MV and MSA are essential for computing nonpolar component of free energies. The objective of our grid-based approach is to identify solute atom pairs that share overlapping volumes in space. Once completed, this information is used to construct a rooted tree using depth-first method to yield the final volume and SA by using the formulations of the Gaussian model described by Grant and Pickup (J. Phys Chem, 1995, 99, 3503). The method is designed to function uninterruptedly with the grid-based finite-difference method implemented in Delphi, a popular and open-source package used for solving the Poisson-Boltzmann equation (PBE). We demonstrate the time efficacy of the method while also validating its performance in terms of the effect of grid-resolution, positioning of the solute within the grid-map and accuracy in identification of overlapping atom pairs. We also explore and discuss different aspects of the Gaussian model with key emphasis on its physical meaningfulness. This development and its future release with the Delphi package are intended to provide a physically meaningful, fast, robust and comprehensive tool for MM/PBSA based free energy calculations. © 2019 Wiley Periodicals, Inc.

Protein-ligand binding enthalpies from near-millisecond simulations: Analysis of a preorganization paradox

Calorimetric studies of protein-ligand binding sometimes yield thermodynamic data that are difficult to understand. Today, molecular simulations can be used to seek insight into such calorimetric puzzles, and, when simulations and experiments diverge, the results can usefully motivate further improvements in computational methods. Here, we apply near-millisecond duration simulations to estimate the relative binding enthalpies of four peptidic ligands with the Grb2 SH2 domain. The ligands fall into matched pairs, where one member of each pair has an added bond that preorganizes the ligand for binding and thus may be expected to favor binding entropically, due to a smaller loss in configurational entropy. Calorimetric studies have shown that the constrained ligands do in fact bind the SH2 domain more tightly than the flexible ones, but, paradoxically, the improvement in affinity for the constrained ligands is enthalpic, rather than entropic. The present enthalpy calculations yield the opposite trend, as they suggest that the flexible ligands bind more exothermically. Additionally, the small relative binding enthalpies are found to be balances of large differences in the energies of structural components such as ligand and the binding site residues. As a consequence, the deviations from experiment in the relative binding enthalpies represent small differences between these large numbers and hence may be particularly susceptible to error, due, for example, to approximations in the force field. We also computed first-order estimates of changes in configurational entropy on binding. These too are, arguably, paradoxical, as they tend to favor binding of the flexible ligands. The paradox is explained in part by the fact that the more rigid constrained ligands reduce the entropy of binding site residues more than their flexible analogs do, at least in the simulations. This result offers a rather general counterargument to the expectation that preorganized ligands should be associated with more favorable binding entropies, other things being equal.

Computational prediction of drug solubility in water-based systems: Qualitative and quantitative approaches used in the current drug discovery and development setting

In this review we will discuss recent advances in computational prediction of solubility in water-based solvents. Our focus is set on recent advances in predictions of biorelevant solubility in media mimicking the human intestinal fluids and on new methods to predict the thermodynamic cycle rather than prediction of solubility in pure water through quantitative structure property relationships (QSPR). While the literature is rich in QSPR models for both solubility and melting point, a physicochemical property strongly linked to the solubility, recent advances in the modelling of these properties make use of theory and computational simulations to better predict these properties or processes involved therein (e.g. solid state crystal lattice packing, dissociation of molecules from the lattice and solvation). This review serves to provide an update on these new approaches and how they can be used to more accurately predict solubility, and also importantly, inform us on molecular interactions and processes occurring during drug dissolution and solubilisation.

Approaches for calculating solvation free energies and enthalpies demonstrated with an update of the FreeSolv database

Solvation free energies can now be calculated precisely from molecular simulations, providing a valuable test of the energy functions underlying these simulations. Here, we briefly review "alchemical" approaches for calculating the solvation free energies of small, neutral organic molecules from molecular simulations, and illustrate by applying them to calculate aqueous solvation free energies (hydration free energies). These approaches use a non-physical pathway to compute free energy differences from a simulation or set of simulations and appear to be a particularly robust and general-purpose approach for this task. We also present an update (version 0.5) to our FreeSolv database of experimental and calculated hydration free energies of neutral compounds and provide input files in formats for several simulation packages. This revision to FreeSolv provides calculated values generated with a single protocol and software version, rather than the heterogeneous protocols used in the prior version of the database. We also further update the database to provide calculated enthalpies and entropies of hydration and some experimental enthalpies and entropies, as well as electrostatic and nonpolar components of solvation free energies.

Bayesian Model Averaging for Ensemble-Based Estimates of Solvation-Free Energies

This paper applies the Bayesian Model Averaging statistical ensemble technique to estimate small molecule solvation free energies. There is a wide range of methods available for predicting solvation free energies, ranging from empirical statistical models to ab initio quantum mechanical approaches. Each of these methods is based on a set of conceptual assumptions that can affect predictive accuracy and transferability. Using an iterative statistical process, we have selected and combined solvation energy estimates using an ensemble of 17 diverse methods from the fourth Statistical Assessment of Modeling of Proteins and Ligands (SAMPL) blind prediction study to form a single, aggregated solvation energy estimate. Methods that possess minimal or redundant information are pruned from the ensemble and the evaluation process repeats until aggregate predictive performance can no longer be improved. We show that this process results in a final aggregate estimate that outperforms all individual methods by reducing estimate errors by as much as 91% to 1.2 kcal mol^-1 accuracy. This work provides a new approach for accurate solvation free energy prediction and lays the foundation for future work on aggregate models that can balance computational cost with prediction accuracy.

Scoring Functions of Protein-Ligand Interactions

Scoring function of protein-ligand interactions is used to recognize the “native” binding pose of a ligand on the protein and to predict the binding affinity, so that the active small molecules can be discriminated from the non-active ones. Scoring function is widely used in computationally molecular docking and structure-based drug discovery. The development and improvement of scoring functions have broad implications in pharmaceutical industry and academic research. During the past three decades, much progress have been made in methodology and accuracy for scoring functions, and many successful cases have be witnessed in virtual database screening. In this chapter, the authors introduced the basic types of scoring functions and their derivations, the commonly-used evaluation methods and benchmarks, as well as the underlying challenges and current solutions. Finally, the authors discussed the promising directions to improve and develop scoring functions for future molecular docking-based drug discovery.

Unified elucidation of the entropy-driven and -opposed hydrophobic effects

The association of nonpolar solutes is generally believed to be entropy driven, which has been shown to be true for the contact of small molecules, ellipsoids, and plates.

Synergy between experimental and computational studies of aromatic stacking interactions

Aromatic stacking interactions are one of the most common types of non-covalent interactions. However, their fundamental origins and the ability to accurately predict their stability trends are still an active area of research. The study of aromatic stacking interactions has been particularly challenging. The interaction involves a delicate balance of multiple forces, and the aromatic surfaces can readily adopt different interaction geometries. Thus, the collaborative efforts of theoretical and experimental researchers have been essential to understand and build more accurate predictive models of aromatic stacking interactions.

Electrostatic component of binding energy: Interpreting predictions from poisson-boltzmann equation and modeling protocols

Macromolecular interactions are essential for understanding numerous biological processes and are typically characterized by the binding free energy. Important component of the binding free energy is the electrostatics, which is frequently modeled via the solutions of the Poisson-Boltzmann Equations (PBE). However, numerous works have shown that the electrostatic component (ΔΔGelec ) of binding free energy is very sensitive to the parameters used and modeling protocol. This prompted some researchers to question the robustness of PBE in predicting ΔΔGelec . We argue that the sensitivity of the absolute ΔΔGelec calculated with PBE using different input parameters and definitions does not indicate PBE deficiency, rather this is what should be expected. We show how the apparent sensitivity should be interpreted in terms of the underlying changes in several numerous and physical parameters. We demonstrate that PBE approach is robust within each considered force field (CHARMM-27, AMBER-94, and OPLS-AA) once the corresponding structures are energy minimized. This observation holds despite of using two different molecular surface definitions, pointing again that PBE delivers consistent results within particular force field. The fact that PBE delivered ΔΔGelec values may differ if calculated with different modeling protocols is not a deficiency of PBE, but natural results of the differences of the force field parameters and potential functions for energy minimization. In addition, while the absolute ΔΔGelec values calculated with different force field differ, their ordering remains practically the same allowing for consistent ranking despite of the force field used. © 2016 Wiley Periodicals, Inc.

All-Atom Continuous Constant pH Molecular Dynamics With Particle Mesh Ewald and Titratable Water

Development of a pH stat to properly control solution pH in biomolecular simulations has been a long-standing goal in the community. Toward this goal recent years have witnessed the emergence of the so-called constant pH molecular dynamics methods. However, the accuracy and generality of these methods have been hampered by the use of implicit-solvent models or truncation-based electrostatic schemes. Here we report the implementation of the particle mesh Ewald (PME) scheme into the all-atom continuous constant pH molecular dynamics (CpHMD) method, enabling CpHMD to be performed with a standard MD engine at a fractional added computational cost. We demonstrate the performance using pH replica-exchange CpHMD simulations with titratable water for a stringent test set of proteins, HP36, BBL, HEWL, and SNase. With the sampling time of 10 ns per replica, most pK_a's are converged, yielding the average absolute and root-mean-square deviations of 0.61 and 0.77, respectively, from experiment. Linear regression of the calculated vs experimental pK_a shifts gives a correlation coefficient of 0.79, a slope of 1, and an intercept near 0. Analysis reveals inadequate sampling of structure relaxation accompanying a protonation-state switch as a major source of the remaining errors, which are reduced as simulation prolongs. These data suggest PME-based CpHMD can be used as a general tool for pH-controlled simulations of macromolecular systems in various environments, enabling atomic insights into pH-dependent phenomena involving not only soluble proteins but also transmembrane proteins, nucleic acids, surfactants, and polysaccharides.

KECSA-Movable Type Implicit Solvation Model (KMTISM)

Computation of the solvation free energy for chemical and biological processes has long been of significant interest. The key challenges to effective solvation modeling center on the choice of potential function and configurational sampling. Herein, an energy sampling approach termed the “Movable Type” (MT) method, and a statistical energy function for solvation modeling, “Knowledge-based and Empirical Combined Scoring Algorithm” (KECSA) are developed and utilized to create an implicit solvation model: KECSA-Movable Type Implicit Solvation Model (KMTISM) suitable for the study of chemical and biological systems. KMTISM is an implicit solvation model, but the MT method performs energy sampling at the atom pairwise level. For a specific molecular system, the MT method collects energies from prebuilt databases for the requisite atom pairs at all relevant distance ranges, which by its very construction encodes all possible molecular configurations simultaneously. Unlike traditional statistical energy functions, KECSA converts structural statistical information into categorized atom pairwise interaction energies as a function of the radial distance instead of a mean force energy function. Within the implicit solvent model approximation, aqueous solvation free energies are then obtained from the NVT ensemble partition function generated by the MT method. Validation is performed against several subsets selected from the Minnesota Solvation Database v2012. Results are compared with several solvation free energy calculation methods, including a one-to-one comparison against two commonly used classical implicit solvation models: MM-GBSA and MM-PBSA. Comparison against a quantum mechanics based polarizable continuum model is also discussed (Cramer and Truhlar’s Solvation Model 12).

Identification of Protein-Excipient Interaction Hotspots Using Computational Approaches

Protein formulation development relies on the selection of excipients that inhibit protein–protein interactions preventing aggregation. Empirical strategies involve screening many excipient and buffer combinations using force degradation studies. Such methods do not readily provide information on intermolecular interactions responsible for the protective effects of excipients. This study describes a molecular docking approach to screen and rank interactions allowing for the identification of protein–excipient hotspots to aid in the selection of excipients to be experimentally screened. Previously published work with Drosophila Su(dx) was used to develop and validate the computational methodology, which was then used to determine the formulation hotspots for Fab A33. Commonly used excipients were examined and compared to the regions in Fab A33 prone to protein–protein interactions that could lead to aggregation. This approach could provide information on a molecular level about the protective interactions of excipients in protein formulations to aid the more rational development of future formulations.

Multiscale method for modeling binding phenomena involving large objects: application to kinesin motor domains motion along microtubules

Many biological phenomena involve the binding of proteins to a large object. Because the electrostatic forces that guide binding act over large distances, truncating the size of the system to facilitate computational modeling frequently yields inaccurate results. Our multiscale approach implements a computational focusing method that permits computation of large systems without truncating the electrostatic potential and achieves the high resolution required for modeling macromolecular interactions, all while keeping the computational time reasonable. We tested our approach on the motility of various kinesin motor domains. We found that electrostatics help guide kinesins as they walk: N-kinesins towards the plus-end, and C-kinesins towards the minus-end of microtubules. Our methodology enables computation in similar, large systems including protein binding to DNA, viruses, and membranes.

Solvation thermodynamics of amino acid side chains on a short peptide backbone

The hydration process of side chain analogue molecules differs from that of the actual amino acid side chains in peptides and proteins owing to the effects of the peptide backbone on the aqueous solvent environment. A recent molecular simulation study has provided evidence that all nonpolar side chains, attached to a short peptide backbone, are considerably less hydrophobic than the free side chain analogue molecules. In contrast to this, the hydrophilicity of the polar side chains is hardly affected by the backbone. To analyze the origin of these observations, we here present a molecular simulation study on temperature dependent solvation free energies of nonpolar and polar side chains attached to a short peptide backbone. The estimated solvation entropies and enthalpies of the various amino acid side chains are compared with existing side chain analogue data. The solvation entropies and enthalpies of the polar side chains are negative, but in absolute magnitude smaller compared with the corresponding analogue data. The observed differences are large; however, owing to a nearly perfect enthalpy-entropy compensation, the solvation free energies of polar side chains remain largely unaffected by the peptide backbone. We find that a similar compensation does not apply to the nonpolar side chains; while the backbone greatly reduces the unfavorable solvation entropies, the solvation enthalpies are either more favorable or only marginally affected. This results in a very small unfavorable free energy cost, or even free energy gain, of solvating the nonpolar side chains in strong contrast to solvation of small hydrophobic or nonpolar molecules in bulk water. The solvation free energies of nonpolar side chains have been furthermore decomposed into a repulsive cavity formation contribution and an attractive dispersion free energy contribution. We find that cavity formation next to the peptide backbone is entropically favored over formation of similar sized nonpolar side chain cavities in bulk water, in agreement with earlier work in the literature on analysis of cavity fluctuations at nonpolar molecular surfaces. The cavity and dispersion interaction contributions correlate quite well with the solvent accessible surface area of the nonpolar side chains attached to the backbone. This correlation however is weak for the overall solvation free energies owing to the fact that the cavity and dispersion free energy contributions are almost exactly cancelling each other.

A review of methods for the calculation of solution free energies and the modelling of systems in solution

Over the past decade, pharmaceutical companies have seen a decline in the number of drug candidates successfully passing through clinical trials, though billions are still spent on drug development. Poor aqueous solubility leads to low bio-availability, reducing pharmaceutical effectiveness. The human cost of inefficient drug candidate testing is of great medical concern, with fewer drugs making it to the production line, slowing the development of new treatments. In biochemistry and biophysics, water mediated reactions and interactions within active sites and protein pockets are an active area of research, in which methods for modelling solvated systems are continually pushed to their limits. Here, we discuss a multitude of methods aimed towards solvent modelling and solubility prediction, aiming to inform the reader of the options available, and outlining the various advantages and disadvantages of each approach.

From Hydrophobic to Hydrophilic Solvation: An Application to Hydration of Benzene

We report a computer simulation study on the hydration of benzene, which, despite being hydrophobic, is a weak hydrogen bond acceptor. The effect of benzene-water hydrogen bonding on the hydration free energy has been analyzed in terms of solute-solvent energies and entropies. Our calculations show that benzene-water hydrogen bonding restricts the number of arrangements possible for the water molecules resulting in a more unfavorable (negative) solute-solvent entropy change than observed for a 'nonpolar benzene' not capable of accepting water hydrogen bonds. More favorable hydration free energies of aromatic hydrocarbons in comparison with aliphatic hydrocarbons observed experimentally as well as in our calculations must therefore be a result of more favorable solute-solvent interaction energies. This result supports the view that lower aqueous solubilities of nonpolar molecules compared to polar molecules are due to a lack of favorable electrostatic interactions with water molecules. The calculated hydration free energy, enthalpy, entropy, and hydration heat capacity of benzene are in good agreement with experimentally reported values.

Multiscale Coarse-Graining of Mixed Phospholipid/Cholesterol Bilayers

Coarse-grained (CG) models for mixed dimyristoylphosphatidylcholine (DMPC)/cholesterol lipid bilayers are constructed using the recently developed multiscale coarse-graining (MS-CG) method. The MS-CG method permits a systematic fit of the bonded and nonbonded interactions and system pressure to trajectory and force data derived from an underlying reference all-atom molecular dynamics (MD) simulation. The CG sites for lipid and cholesterol molecules are associated with the centers-of-mass of atomic groups because of the simplicity in the evaluation of the forces acting on them from the atomistic MD data. Corresponding models with four-site and seven-site representations of the cholesterol molecule were also developed. The latter CG models differed by the bonding scheme of CG sites to represent intramolecular interactions. A one-site MS-CG model based on the TIP3P potential was used for water, with the interaction site placed at the molecular geometrical center, and the analytical fit of the model is presented. The MS-CG models were then used to conduct simulations in the constant NPT ensemble which reproduce accurately the structural properties as seen in the full all-atom MD simulation.

Computational Calorimetry: High-Precision Calculation of Host-Guest Binding Thermodynamics

We present a strategy for carrying out high-precision calculations of binding free energy and binding enthalpy values from molecular dynamics simulations with explicit solvent. The approach is used to calculate the thermodynamic profiles for binding of nine small molecule guests to either the cucurbit[7]uril (CB7) or β-cyclodextrin (βCD) host. For these systems, calculations using commodity hardware can yield binding free energy and binding enthalpy values with a precision of ∼0.5 kcal/mol (95% CI) in a matter of days. Crucially, the self-consistency of the approach is established by calculating the binding enthalpy directly, via end point potential energy calculations, and indirectly, via the temperature dependence of the binding free energy, i.e., by the van't Hoff equation. Excellent agreement between the direct and van't Hoff methods is demonstrated for both host-guest systems and an ion-pair model system for which particularly well-converged results are attainable. Additionally, we find that hydrogen mass repartitioning allows marked acceleration of the calculations with no discernible cost in precision or accuracy. Finally, we provide guidance for accurately assessing numerical uncertainty of the results in settings where complex correlations in the time series can pose challenges to statistical analysis. The routine nature and high precision of these binding calculations opens the possibility of including measured binding thermodynamics as target data in force field optimization so that simulations may be used to reliably interpret experimental data and guide molecular design.

Electrostatics, structure prediction, and the energy landscapes for protein folding and binding

While being long in range and therefore weakly specific, electrostatic interactions are able to modulate the stability and folding landscapes of some proteins. The relevance of electrostatic forces for steering the docking of proteins to each other is widely acknowledged, however, the role of electrostatics in establishing specifically funneled landscapes and their relevance for protein structure prediction are still not clear. By introducing Debye-Hückel potentials that mimic long-range electrostatic forces into the Associative memory, Water mediated, Structure, and Energy Model (AWSEM), a transferable protein model capable of predicting tertiary structures, we assess the effects of electrostatics on the landscapes of thirteen monomeric proteins and four dimers. For the monomers, we find that adding electrostatic interactions does not improve structure prediction. Simulations of ribosomal protein S6 show, however, that folding stability depends monotonically on electrostatic strength. The trend in predicted melting temperatures of the S6 variants agrees with experimental observations. Electrostatic effects can play a range of roles in binding. The binding of the protein complex KIX-pKID is largely assisted by electrostatic interactions, which provide direct charge-charge stabilization of the native state and contribute to the funneling of the binding landscape. In contrast, for several other proteins, including the DNA-binding protein FIS, electrostatics causes frustration in the DNA-binding region, which favors its binding with DNA but not with its protein partner. This study highlights the importance of long-range electrostatics in functional responses to problems where proteins interact with their charged partners, such as DNA, RNA, as well as membranes.

Optimizing the affinity and specificity of ligand binding with the inclusion of solvation effect

Solvation effect is an important factor for protein-ligand binding in aqueous water. Previous scoring function of protein-ligand interactions rarely incorporates the solvation model into the quantification of protein-ligand interactions, mainly due to the immense computational cost, especially in the structure-based virtual screening, and nontransferable application of independently optimized atomic solvation parameters. In order to overcome these barriers, we effectively combine knowledge-based atom-pair potentials and the atomic solvation energy of charge-independent implicit solvent model in the optimization of binding affinity and specificity. The resulting scoring functions with optimized atomic solvation parameters is named as specificity and affinity with solvation effect (SPA-SE). The performance of SPA-SE is evaluated and compared to 20 other scoring functions, as well as SPA. The comparative results show that SPA-SE outperforms all other scoring functions in binding affinity prediction and "native" pose identification. Our optimization validates that solvation effect is an important regulator to the stability and specificity of protein-ligand binding. The development strategy of SPA-SE sets an example for other scoring function to account for the solvation effect in biomolecular recognitions.

The entropic contributions in vitamin B12 enzymes still reflect the electrostatic paradigm

The catalytic power of enzymes containing coenzyme B12 has been, in some respects, the "last bastion" for the strain hypothesis. Our previous study of this system established by a careful sampling that the major part of the catalytic effect is due to the electrostatic interaction between the ribose of the ado group and the protein and that the strain contribution is very small. This finding has not been sufficiently appreciated due to misunderstandings of the power of the empirical valence bond (EVB) calculations and the need of sufficient sampling. Furthermore, some interesting new experiments point toward entropic effects as the source of the catalytic power, casting doubt on the validity of the electrostatic idea, at least, in the case of B12 enzymes. Here, we focus on the observation of the entropic effects and on analyzing their origin. We clarify that our EVB approach evaluates free energies rather than enthalpies and demonstrate by using the restraint release (RR) approach that the observed entropic contribution to the activation barrier is of electrostatic origin. Our study illustrates the power of the RR approach by evaluating the entropic contributions to catalysis and provides further support to our paradigm for the origin of the catalytic power of B12 enzymes. Overall, our study provides major support to our electrostatic preorganization idea and also highlights the basic requirements from ab initio quantum mechanics/molecular mechanics calculations of activation free energies of enzymatic reactions.

A cryospectroscopic infrared and Raman study of the CX⋯π halogen bonding motif: complexes of the CF3Cl, CF3Br, and CF3I with ethyne, propyne and 2-butyne

Experimental information on the C-X⋯π halogen bonding motif was obtained by studying the formation of molecular complexes of CF3Cl, CF3Br and CF3I with ethyne, propyne and 2-butyne in liquid krypton, using FTIR and Raman spectroscopy. For CF3Br, experimental evidence was found for the formation of 1:1 complexes with propyne and 2-butyne only, while for CF3I spectroscopic features confirming the existence of the halogen bonded complexes were observed for ethyne, propyne and 2-butyne. In addition, at higher concentrations of CF3I and 2-butyne, weak absorptions due to a 2:1 complex were also observed. The experimental complexation enthalpies, obtained by using spectra recorded at temperatures between 120 K and 140 K, are -5.9(3) kJ mol(-1) for CF3I.ethyne, -5.6(3) kJ mol(-1) for CF3Br.propyne, -8.1(2) kJ mol(-1) for CF3I.propyne, -7.3(2) kJ mol(-1) for CF3Br.2-butyne, -10.9(2) kJ mol(-1) for CF3I.2-butyne and -20.9(7) kJ mol(-1) for (CF3I)2.2-butyne. The experimental study is supported by theoretical data obtained from ab initio calculations at the MP2/aug-cc-pVDZ(-PP) and MP2/aug-cc-pVTZ(-PP) levels, and Monte Carlo Free Energy Perturbation (MC-FEP) simulations. The experimental and theoretical values on the C-X⋯π halogen-bonding motifs studied are compared with previously reported data for the complexes with ethene and propene and with preliminary results obtained for benzene and toluene.