Estimating Translational and Orientational Entropies Using the k-Nearest Neighbors Algorithm

MPI-parallelization of the grid inhomogeneous solvation theory calculation

The grid inhomogeneous solvation theory (GIST) method requires the often time-consuming calculation of water-water and water-solute energy on a grid. Previous efforts to speed up this calculation include using OpenMP, GPUs, and particle mesh Ewald. This article details how the speed of this calculation can be increased by parallelizing it with MPI, where trajectory frames are divided among multiple processors. This requires very little communication between individual processes during trajectory processing, meaning the calculation scales well to large processor counts. This article also details how the entropy calculation, which must happen after trajectory processing since it requires information from all trajectory frames, is parallelized via MPI. This parallelized GIST method has been implemented in the freely-available CPPTRAJ analysis software.

Significance of the Disulfide Bridge in the Structure and Stability of Metalloprotein Azurin

Metalloproteins make up a class of proteins that incorporate metal ions into their structures, enabling them to perform essential functions in biological systems, such as catalysis and electron transport. Azurin is one such metalloprotein with copper cofactor, having a β-barrel structure with exceptional thermal stability. The copper metal ion is coordinated at one end of the β-barrel structure, and there is a disulfide bond at the opposite end. In this study, we explore the effect of this disulfide bond in the high thermal stability of azurin by analyzing both the native S-S bonded and S-S nonbonded (S-S open) forms using temperature replica exchange molecular dynamics (REMD). Similar to experimental observations, we find a 35 K decrease in denaturation temperature for S-S open azurin compared to that of the native holo form (420 K). As observed in the case of native holo azurin, the unfolding process of the S-S open form also started with disruptions of the α-helix. The free energy surfaces of the unfolding process revealed that the denaturation event of the S-S open form progresses through different sets of conformational ensembles. Subsequently, we compared the stabilities of individual β-sheet strands of both the S-S bonded and the S-S nonbonded forms of azurin. Further, we examined the contacts between individual residues for the central structures from the free energy surfaces of the S-S nonbonded form. The microscopic origin of the lowering in the denaturation temperature is further supplemented by thermodynamic analysis.

Thermodynamic forces from protein and water govern condensate formation of an intrinsically disordered protein domain

Liquid-liquid phase separation (LLPS) can drive a multitude of cellular processes by compartmentalizing biological cells via the formation of dense liquid biomolecular condensates, which can function as membraneless organelles. Despite its importance, the molecular-level understanding of the underlying thermodynamics of this process remains incomplete. In this study, we use atomistic molecular dynamics simulations of the low complexity domain (LCD) of human fused in sarcoma (FUS) protein to investigate the contributions of water and protein molecules to the free energy changes that govern LLPS. Both protein and water components are found to have comparably sizeable thermodynamic contributions to the formation of FUS condensates. Moreover, we quantify the counteracting effects of water molecules that are released into the bulk upon condensate formation and the waters retained within the protein droplets. Among the various factors considered, solvation entropy and protein interaction enthalpy are identified as the most important contributions, while solvation enthalpy and protein entropy changes are smaller. These results provide detailed molecular insights on the intricate thermodynamic interplay between protein- and solvation-related forces underlying the formation of biomolecular condensates.

WaterKit: Thermodynamic Profiling of Protein Hydration Sites

Water desolvation is one of the key components of the free energy of binding of small molecules to their receptors. Thus, understanding the energetic balance of solvation and desolvation resulting from individual water molecules can be crucial when estimating ligand binding, especially when evaluating different molecules and poses as done in High-Throughput Virtual Screening (HTVS). Over the most recent decades, several methods were developed to tackle this problem, ranging from fast approximate methods (usually empirical functions using either discrete atom-atom pairwise interactions or continuum solvent models) to more computationally expensive and accurate ones, mostly based on Molecular Dynamics (MD) simulations, such as Grid Inhomogeneous Solvation Theory (GIST) or Double Decoupling. On one hand, MD-based methods are prohibitive to use in HTVS to estimate the role of waters on the fly for each ligand. On the other hand, fast and approximate methods show an unsatisfactory level of accuracy, with low agreement with results obtained with the more expensive methods. Here we introduce WaterKit, a new grid-based sampling method with explicit water molecules to calculate thermodynamic properties using the GIST method. Our results show that the discrete placement of water molecules is successful in reproducing the position of crystallographic waters with very high accuracy, as well as providing thermodynamic estimates with accuracy comparable to more expensive MD simulations. Unlike these methods, WaterKit can be used to analyze specific regions on the protein surface, (such as the binding site of a receptor), without having to hydrate and simulate the whole receptor structure. The results show the feasibility of a general and fast method to compute thermodynamic properties of water molecules, making it well-suited to be integrated in high-throughput pipelines such as molecular docking.

Data Structures and Algorithms for k-th Nearest Neighbours Conformational Entropy Estimation

Entropy of multivariate distributions may be estimated based on the distances of nearest neighbours from each sample from a statistical ensemble. This technique has been applied on biomolecular systems for estimating both conformational and translational/rotational entropy. The degrees of freedom which mostly define conformational entropy are torsion angles with their periodicity. In this work, tree structures and algorithms to quickly generate lists of nearest neighbours for periodic and non-periodic data are reviewed and applied to biomolecular conformations as described by torsion angles. The effect of dimensionality, number of samples, and number of neighbours on the computational time is assessed. The main conclusion is that using proper data structures and algorithms can greatly reduce the complexity of nearest neighbours lists generation, which is the bottleneck step in nearest neighbours entropy estimation.

Grid inhomogeneous solvation theory for cross-solvation in rigid solvents

Grid Inhomogeneous Solvation Theory (GIST) has proven useful to calculate localized thermodynamic properties of water around a solute. Numerous studies have leveraged this information to enhance structure-based binding predictions. We have recently extended GIST toward chloroform as a solvent to allow the prediction of passive membrane permeability. Here, we further generalize the GIST algorithm toward all solvents that can be modeled as rigid molecules. This restriction is inherent to the method and is already present in the inhomogeneous solvation theory. Here, we show that our approach can be applied to various solvent molecules by comparing the results of GIST simulations with thermodynamic integration (TI) calculations and experimental results. Additionally, we analyze and compare a matrix consisting of 100 entries of ten different solvent molecules solvated within each other. We find that the GIST results are highly correlated with TI calculations as well as experiments. For some solvents, we find Pearson correlations of up to 0.99 to the true entropy, while others are affected by the first-order approximation more strongly. The enthalpy-entropy splitting provided by GIST allows us to extend a recently published approach, which estimates higher order entropies by a linear scaling of the first-order entropy, to solvents other than water. Furthermore, we investigate the convergence of GIST in different solvents. We conclude that our extension to GIST reliably calculates localized thermodynamic properties for different solvents and thereby significantly extends the applicability of this widely used method.

Entropy of Two-Molecule Correlated Translational-Rotational Motions Using the kth Nearest Neighbor Method

The entropy associated with rotations, translations, and their coupled motions provides an important contribution to the free energy of many physicochemical processes such as association and solvation. The kth nearest neighbor method, which offers a convenient way to estimate the entropy in high-dimensional spaces, has been previously applied for translational-rotational entropy estimation. Here, we explore the possibility of extending the kth nearest neighbor method to the computation of the entropy of correlated translation-rotations of two molecules, i.e., in the product space of two translation-rotations, both referred to the same independent reference system, which is relevant for all cases in which the correlated translational-rotational motion of more than two molecules is involved. Numerical tests show that, albeit the relatively high dimensionality (12) of the space, the kth nearest neighbor approach provides an accurate estimate for the entropy of two correlated translational-rotational motions, even when computed from a limited number of samples.

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.

The quaternion-based spatial-coordinate and orientation-frame alignment problems

The general problem of finding a global rotation that transforms a given set of spatial coordinates and/or orientation frames (the `test' data) into the best possible alignment with a corresponding set (the `reference' data) is reviewed. For 3D point data, this `orthogonal Procrustes problem' is often phrased in terms of minimizing a root-mean-square deviation (RMSD) corresponding to a Euclidean distance measure relating the two sets of matched coordinates. This article focuses on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature, where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; focusing on these exact solutions exposes the structure of the entire eigensystem for the traditional 3D spatial-alignment problem. The structure of the less-studied orientation-data context is then explored, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation-frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. The article concludes with a brief discussion of the combined 3D translation-orientation data alignment problem. Appendices are devoted to a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. The supporting information covers novel extensions of quaternion methods to the 4D Euclidean spatial-coordinate alignment and 4D orientation-frame alignment problems, some miscellaneous topics, and additional details of the quartic algebraic eigenvalue problem.

Solvation Free Energy as a Measure of Hydrophobicity: Application to Serine Protease Binding Interfaces

Solvation and hydrophobicity play a key role in a variety of biological mechanisms. In substrate binding, but also in structure-based drug design, the thermodynamic properties of water molecules surrounding a given protein are of high interest. One of the main algorithms devised in recent years to quantify thermodynamic properties of water is the grid inhomogeneous solvation theory (GIST), which calculates these features on a grid surrounding the protein. Despite the inherent advantages of GIST, the computational demand is a major drawback, as calculations for larger systems can take days or even weeks. Here, we present a GPU accelerated version of the GIST algorithm, which facilitates efficient estimates of solvation free energy even of large biomolecular interfaces. Furthermore, we show that GIST can be used as a reliable tool to evaluate protein surface hydrophobicity. We apply the approach on a set of nine different proteases calculating localized solvation free energies on the surface of the binding interfaces as a measure of their hydrophobicity. We find a compelling agreement with the hydrophobicity of their substrates, i.e., peptides, binding into the binding cleft, and thus our approach provides a reliable description of hydrophobicity characteristics of these biological interfaces.

Entropy of Simulated Liquids Using Multiscale Cell Correlation

Accurately calculating the entropy of liquids is an important goal, given that many processes take place in the liquid phase. Of almost equal importance is understanding the values obtained. However, there are few methods that can calculate the entropy of such systems, and fewer still to make sense of the values obtained. We present our multiscale cell correlation (MCC) method to calculate the entropy of liquids from molecular dynamics simulations. The method uses forces and torques at the molecule and united-atom levels and probability distributions of molecular coordinations and conformations. The main differences with previous work are the consistent treatment of the mean-field cell approximation to the approriate degrees of freedom, the separation of the force and torque covariance matrices, and the inclusion of conformation correlation for molecules with multiple dihedrals. MCC is applied to a broader set of 56 important industrial liquids modeled using the Generalized AMBER Force Field (GAFF) and Optimized Potentials for Liquid Simulations (OPLS) force fields with 1.14*CM1A charges. Unsigned errors versus experimental entropies are 8.7 J K - 1 mol - 1 for GAFF and 9.8 J K - 1 mol - 1 for OPLS. This is significantly better than the 2-Phase Thermodynamics method for the subset of molecules in common, which is the only other method that has been applied to such systems. MCC makes clear why the entropy has the value it does by providing a decomposition in terms of translational and rotational vibrational entropy and topographical entropy at the molecular and united-atom levels.

A molecular reconstruction approach to site-based 3D-RISM and comparison to GIST hydration thermodynamic maps in an enzyme active site

Computed, high-resolution, spatial distributions of solvation energy and entropy can provide detailed information about the role of water in molecular recognition. While grid inhomogeneous solvation theory (GIST) provides rigorous, detailed thermodynamic information from explicit solvent molecular dynamics simulations, recent developments in the 3D reference interaction site model (3D-RISM) theory allow many of the same quantities to be calculated in a fraction of the time. However, 3D-RISM produces atomic-site, rather than molecular, density distributions, which are difficult to extract physical meaning from. To overcome this difficulty, we introduce a method to reconstruct molecular density distributions from atomic-site density distributions. Furthermore, we assess the quality of the resulting solvation thermodynamics density distributions by analyzing the binding site of coagulation Factor Xa with both GIST and 3D-RISM. We find good qualitative agreement between the methods for oxygen and hydrogen densities as well as direct solute-solvent energetic interactions. However, 3D-RISM predicts lower energetic and entropic penalties for moving water from the bulk to the binding site.

Large-Scale Study of Hydration Environments through Hydration Sites

Hydration sites are locations of interest to water and they can be used to classify the behavior of water around chemical motifs commonly found on the surface of proteins. Inhomogeneous fluid solvation theory (IFST) is a method for calculating hydration free-energy changes from molecular dynamics (MD) trajectories. In this paper, hydration sites are identified from MD simulations of 380 diverse protein structures. The hydration free energies of the hydration sites are calculated using IFST and distributions of these free-energy changes are analyzed. The results show that for some hydration sites near features conventionally regarded as attractive to water, such as hydrogen bond donors, the water molecules are actually relatively weakly bound and are easily displaced. We also construct plots of the spatial density of hydration sites with high, medium, and low hydration free-energy changes which represent weakly and strongly bound hydration sites. It is found that these plots show consistent features around common polar amino acids for all of the proteins studied.

Prediction of GABARAP interaction with the GABA type A receptor

We have performed docking simulations on GABARAP interacting with the GABA type A receptor using SwarmDock. We have also used a novel method to study hydration sites on the surface of these two proteins; this method identifies regions around proteins where desolvation is relatively easy, and these are possible locations where proteins can bind each other. There is a high degree of consistency between the predictions of these two methods. Moreover, we have also identified binding sites on GABARAP for other proteins, and listed possible binding sites for as yet unknown proteins on both GABARAP and the GABA type A receptor intracellular domain.

Assimilating Radial Distribution Functions To Build Water Models with Improved Structural Properties

The structural properties of three- and four-site water models are improved by extending the ForceBalance parametrization code to include a new methodology allowing for the targeting of any radial distribution function (RDF) during the parametrization of a force field. The mean squared difference (MSD) between the experimental and simulated RDFs contributes to an objective function, allowing for the systematic optimization of force field parameters to reach closer overall agreement with experiment. RDF fitting is applied to develop modified versions of the TIP3P and TIP4P/2005 water models in which the Lennard-Jones potential is replaced by a Buckingham potential. The optimized TIP3P-Buckingham and TIP4P-Buckingham potentials feature 93 and 98% lower MSDs in the OO RDF compared to the TIP3P and TIP4P/2005 models respectively, with marked decreases in the height of the first peak. Additionally, these Buckingham models predict the entropy of water more accurately, reducing the error in the entropy of TIP3P from 11 to 3% and the error in the entropy of TIP4P/2005 from 11 to 2%. These new Buckingham models have improved predictive power for many nonfitted properties particularly in the case of TIP3P. Our work directly demonstrates how the Buckingham potential can improve the description of water's structural properties beyond the Lennard-Jones potential. Moreover, adding a Buckingham potential is a favorable alternative to adding interaction sites in terms of computational speed on modern GPU hardware.

Entropic Stabilization of Cas4 Protein SSO0001 Predicted with Popcoen

Popcoen is a method for configurational entropy estimation of proteins based on machine-learning. Entropy is predicted with an artificial neural network which was trained on simulation trajectories of a large set of representative proteins. Popcoen is extremely fast compared to other approaches based on the sampling of a multitude of microstates. Consequently, Popcoen can be incorporated into a large class of protein software which currently neglects configurational entropy for performance reasons. Here, we apply Popcoen to various conformations of the Cas4 protein SSO0001 of Sulfolobus solfataricus, a protein that assembles to a decamer of known toroidal shape. We provide numerical evidence that the native state (NAT) of a SSO0001 monomer has a similar structure to the protomers of the oligomer, where NAT of the monomer is stabilized mainly entropically. Due to its large amount of configurational entropy, NAT has lower free energy than alternative conformations of very low enthalpy and solvation free-energy. Hence, SSO0001 serves as an example case where neglecting configurational entropy leads to incorrect conclusion. Our results imply that no refolding of the subunits is required during oligomerization which suggests that configurational entropy is employed by nature to largely enhance the rate of assembly.

Quantitative metrics for drug-target ligandability

Ligandability is a prerequisite for druggability and is a much easier concept to understand, model and predict because it does not depend on the complex pharmacodynamic and pharmacokinetic mechanisms in the human body. In this review, we consider a metric for quantifying ligandability from experimental data. We discuss ligandability in terms of the balance between effort and reward. The metric is evaluated for a standard set of well-studied drug targets - some traditionally considered to be ligandable and some regarded as difficult. We suggest that this metric should be used to systematically improve computational predictions of ligandability, which can then be applied to novel drug targets to predict their tractability.

Solvation Structure and Thermodynamic Mapping (SSTMap): An Open-Source, Flexible Package for the Analysis of Water in Molecular Dynamics Trajectories

We have developed SSTMap, a software package for mapping structural and thermodynamic water properties in molecular dynamics trajectories. The package introduces automated analysis and mapping of local measures of frustration and enhancement of water structure. The thermodynamic calculations are based on Inhomogeneous Fluid Solvation Theory (IST), which is implemented using both site-based and grid-based approaches. The package also extends the applicability of solvation analysis calculations to multiple molecular dynamics (MD) simulation programs by using existing cross-platform tools for parsing MD parameter and trajectory files. SSTMap is implemented in Python and contains both command-line tools and a Python module to facilitate flexibility in setting up calculations and for automated generation of large data sets involving analysis of multiple solutes. Output is generated in formats compatible with popular Python data science packages. This tool will be used by the molecular modeling community for computational analysis of water in problems of biophysical interest such as ligand binding and protein function.

On the accuracy of one- and two-particle solvation entropies

Evaluating solvation entropies directly and combining with direct energy calculations is one way of calculating free energies of solvation and is used by Inhomogeneous Fluid Solvation Theory (IFST). The configurational entropy of a fluid is a function of the interatomic correlations and can thus be expressed in terms of correlation functions. The entropies in this work are directly calculated from a truncated series of integrals over these correlation functions. Many studies truncate all terms higher than the solvent-solute correlations. This study includes an additional solvent-solvent correlation term and assesses the associated free energy when IFST is applied to a fixed Lennard-Jones particle solvated in neon. The strength of the central potential is varied to imitate larger solutes. Average free energy estimates with both levels of IFST are able to reproduce the estimate made using the Free energy Perturbation (FEP) to within 0.16 kcal/mol. We find that the signal from the solvent-solvent correlations is very weak. Our conclusion is that for monatomic fluids simulated by pairwise classical potentials the correction term is relatively small in magnitude. This study shows it is possible to reproduce the free energy from a path based method like FEP, by only considering the endpoints of the path. This method can be directly applied to more complex solutes which break the spherical symmetry of this study.

Elucidating the energetic contributions to the binding free energy

New exact equations are derived for the terms contributing to the binding free energy (ΔG⁰) of a ligand-receptor pair using our recently introduced formalism which we here call perturbation-divergence formalism (PDF). Specifically, ΔG⁰ equals the sum of the average of the perturbation (pertaining to new interactions) and additional dissipative terms. The average of the perturbation includes the sum of the average receptor-ligand interactions and the average of the change of solvation energies upon association. The Kullback-Leibler (KL) divergence quantifies the energetically dissipative terms, which are due to the configurational changes and, using the chain rule of KL divergence, can be decomposed into (i) dissipation due to limiting the external liberation (translation and rotation) of the ligand relative to the receptor and (ii) dissipation due to conformational (internal) changes inside the receptor and the ligand. We also identify all exactly canceling energetic terms which do not contribute to ΔG⁰. Furthermore, the PDF provides a new approach towards dimensionality reduction in the representation of the association process and towards relating the dynamic (high dimensional) with the thermodynamic (one-dimensional) changes.

Solvent effects on ligand binding to a serine protease

ITC experiments and MD simulations reveal the mechanism behind enthalpy/entropy compensation upon trypsin-benzamidine binding at different solvation conditions.

Studying the role of cooperative hydration in stabilizing folded protein states

Understanding and modelling protein folding remains a key scientific and engineering challenge. Two key questions in protein folding are (1) why many proteins adopt a folded state and (2) how these proteins transition from the random coil ensemble to a folded state. In this paper we employ molecular dynamics simulations to address the first of these questions. Computational methods are well-placed to address this issue due to their ability to analyze systems at atomic-level resolution. Traditionally, the stability of folded proteins has been ascribed to the balance of two types of intermolecular interactions: hydrogen-bonding interactions and hydrophobic contacts. In this study, we explore a third type of intermolecular interaction: cooperative hydration of protein surface residues. To achieve this, we consider multiple independent simulations of the villin headpiece domain to quantify the contributions of different interactions to the energy of the native and fully extended states. In addition, we consider whether these findings are robust with respect to the protein forcefield, the water model, and the presence of salt. In all cases, we identify many cooperatively hydrated interactions that are transient but energetically favor the native state. Whilst further work on additional protein structures, forcefields, and water models is necessary, these results suggest a role for cooperative hydration in protein folding that should be explored further. Rational design of cooperative hydration on the protein surface could be a viable strategy for increasing protein stability.

Solvation thermodynamic mapping of molecular surfaces in AmberTools: GIST

The expulsion of water from surfaces upon molecular recognition and nonspecific association makes a major contribution to the free energy changes of these processes. In order to facilitate the characterization of water structure and thermodynamics on surfaces, we have incorporated Grid Inhomogeneous Solvation Theory (GIST) into the CPPTRAJ toolset of AmberTools. GIST is a grid-based implementation of Inhomogeneous Fluid Solvation Theory, which analyzes the output from molecular dynamics simulations to map out solvation thermodynamic and structural properties on a high-resolution, three-dimensional grid. The CPPTRAJ implementation, called GIST-cpptraj, has a simple, easy-to-use command line interface, and is open source and freely distributed. We have also developed a set of open-source tools, called GISTPP, which facilitate the analysis of GIST output grids. Tutorials for both GIST-cpptraj and GISTPP can be found at ambermd.org. © 2016 Wiley Periodicals, Inc.

New Computational Approach for External Entropy in Protein-Protein Binding

Molecular recognition through the noncovalent association of biomolecules is of central importance in biology and pharmacology. Developing reliable computational methods for estimating binding thermodynamic parameters is therefore of great practical value. However, considerable uncertainty remains regarding the external entropy that is associated with the reduction in the external (positional and orientational) degrees of freedom upon complex formation. Here, we present a novel statistical mechanical method for computing the external entropy by extending the energetic approach we have developed for unimolecular processes to association processes. We find that, in contrary to what is postulated in most of the previous methods, intrinsic couplings between the internal and external degrees of freedom of bound complex cannot in general be neglected in the determination of the external entropy. Nevertheless, there exists the best choice of the external coordinates with which those couplings are minimized. With such a choice of the external coordinates, the lowest upper bound of the external entropy is obtained from a tractable expression, which serves as an estimate of the external entropy. Our method can be implemented in a straightforward manner with molecular dynamics simulations, and its applicability is demonstrated through the application to the barnase-barstar complex.

Accurate Estimation of the Entropy of Rotation-Translation Probability Distributions

The estimation of rotational and translational entropies in the context of ligand binding has been the subject of long-time investigations. The high dimensionality (six) of the problem and the limited amount of sampling often prevent the required resolution to provide accurate estimates by the histogram method. Recently, the nearest-neighbor distance method has been applied to the problem, but the solutions provided either address rotation and translation separately, therefore lacking correlations, or use a heuristic approach. Here we address rotational-translational entropy estimation in the context of nearest-neighbor-based entropy estimation, solve the problem numerically, and provide an exact and an approximate method to estimate the full rotational-translational entropy.

Spatial Decomposition of Translational Water-Water Correlation Entropy in Binding Pockets

A number of computational tools available today compute the thermodynamic properties of water at surfaces and in binding pockets by using inhomogeneous solvation theory (IST) to analyze explicit-solvent simulations. Such methods enable qualitative spatial mappings of both energy and entropy around a solute of interest and can also be applied quantitatively. However, the entropy estimates of existing methods have, to date, been almost entirely limited to the first-order terms in the IST’s entropy expansion. These first-order terms account for localization and orientation of water molecules in the field of the solute but not for the modification of water–water correlations by the solute. Here, we present an extension of the Grid Inhomogeneous Solvation Theory (GIST) approach which accounts for water–water translational correlations. The method involves rewriting the two-point density of water in terms of a conditional density and utilizes the efficient nearest-neighbor entropy estimation approach. Spatial maps of this second order term, for water in and around the synthetic host cucurbit[7]uril and in the binding pocket of the enzyme Factor Xa, reveal mainly negative contributions, indicating solute-induced water–water correlations relative to bulk water; particularly strong signals are obtained for sites at the entrances of cavities or pockets. This second-order term thus enters with the same, negative, sign as the first order translational and orientational terms. Numerical and convergence properties of the methodology are examined.

Quantifying the entropy of binding for water molecules in protein cavities by computing correlations

Protein structural analysis demonstrates that water molecules are commonly found in the internal cavities of proteins. Analysis of experimental data on the entropies of inorganic crystals suggests that the entropic cost of transferring such a water molecule to a protein cavity will not typically be greater than 7.0 cal/mol/K per water molecule, corresponding to a contribution of approximately +2.0 kcal/mol to the free energy. In this study, we employ the statistical mechanical method of inhomogeneous fluid solvation theory to quantify the enthalpic and entropic contributions of individual water molecules in 19 protein cavities across five different proteins. We utilize information theory to develop a rigorous estimate of the total two-particle entropy, yielding a complete framework to calculate hydration free energies. We show that predictions from inhomogeneous fluid solvation theory are in excellent agreement with predictions from free energy perturbation (FEP) and that these predictions are consistent with experimental estimates. However, the results suggest that water molecules in protein cavities containing charged residues may be subject to entropy changes that contribute more than +2.0 kcal/mol to the free energy. In all cases, these unfavorable entropy changes are predicted to be dominated by highly favorable enthalpy changes. These findings are relevant to the study of bridging water molecules at protein-protein interfaces as well as in complexes with cognate ligands and small-molecule inhibitors.

Estimation of Solvation Entropy and Enthalpy via Analysis of Water Oxygen-Hydrogen Correlations

A statistical-mechanical framework for estimation of solvation entropies and enthalpies is proposed, which is based on the analysis of water as a mixture of correlated water oxygens and water hydrogens. Entropic contributions of increasing order are cast in terms of a Mutual Information Expansion that is evaluated to pairwise interactions. In turn, the enthalpy is computed directly from a distance-based hydrogen bonding energy algorithm. The resulting expressions are employed for grid-based analyses of Molecular Dynamics simulations. In this first assessment of the methodology, we obtained global estimates of the excess entropy and enthalpy of water that are in good agreement with experiment and examined the method's ability to enable detailed elucidation of solvation thermodynamic structures, which can provide valuable knowledge toward molecular design.

Interfacial water properties in the presence of surfactants

Water, because of its fundamental role in biology, geology, and many industrial applications and its anomalous behavior compared to that of simple fluids, continues to fascinate and attract extensive scientific interest. Building on previous studies of water in contact with different surfaces, in this study, we report results obtained from molecular dynamics simulations of water near hydrophilic and hydrophobic interfaces in the presence of nonionic and ionic amphiphilic molecules, hexaethylene glycol monododecyl ether (C12E6) and sodium dodecyl sulfate (SDS). We elucidate how these surfactants affect the packing (i.e., density profiles) and orientation of interfacial water. The results highlight the interplay of both surfactant charges and the substrate charge distribution predominantly with respect to the orientation of water molecules, up to distances longer than those expected based on simulation results on flat solid surfaces. We also quantify the dynamics of interfacial water molecules by computing the residence probability for water in contact with various substrates. We compare our results to those previously obtained for interfacial water on silica and graphite and also with experimental sum-frequency vibrational spectroscopy results at the air-water interface in the presence of surfactants. Our analysis could be useful for a better understanding of interfacial water not only near solid substrates but also near self-assembled/aggregated molecules at a variety of interfaces.

Spatial analysis and quantification of the thermodynamic driving forces in protein-ligand binding: binding site variability

The thermodynamic driving forces behind small molecule-protein binding are still not well-understood, including the variability of those forces associated with different types of ligands in different binding pockets. To better understand these phenomena we calculate spatially resolved thermodynamic contributions of the different molecular degrees of freedom for the binding of propane and methanol to multiple pockets on the proteins Factor Xa and p38 MAP kinase. Binding thermodynamics are computed using a statistical thermodynamics based end-point method applied on a canonical ensemble comprising the protein-ligand complexes and the corresponding free states in an explicit solvent environment. Energetic and entropic contributions of water and ligand degrees of freedom computed from the configurational ensemble provide an unprecedented level of detail into the mechanisms of binding. Direct protein-ligand interaction energies play a significant role in both nonpolar and polar binding, which is comparable to water reorganization energy. Loss of interactions with water upon binding strongly compensates these contributions leading to relatively small binding enthalpies. For both solutes, the entropy of water reorganization is found to favor binding in agreement with the classical view of the "hydrophobic effect". Depending on the specifics of the binding pocket, both energy-entropy compensation and reinforcement mechanisms are observed. It is notable to have the ability to visualize the spatial distribution of the thermodynamic contributions to binding at atomic resolution showing significant differences in the thermodynamic contributions of water to the binding of propane versus methanol.