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An orthorhombic crystal form of subtilisin BPN’ variant s88 exhibits a systematic variation in growth rates of its three unique faces, resulting in pronounced variations in crystal morphology as a function of the ionic strength (Gallagher et al., J. Crystal Growth 193 (1998) 665). We have sought to explain these observations by performing energetics calculations using the full structural details of the protein. The approach of correlating growth rates with the strength of interactions of molecular pairs in the crystal proved unsatisfactory. The attachment of protein molecules to growing crystal faces is a stochastic process: a protein molecule samples numerous configurational (orientational and translational) states before attaching to the crystal in a configuration consistent with the crystal symmetry. This search-and-bind process determines the rate of crystal growth. We present a simplified approach that mimics this dynamical process: we perform docking of a protein molecule with another protein molecule or a cluster of molecules. An analysis of the bound configurations reveals the presence of orientations that are incommensurate with the crystal symmetry but in which a protein molecule can also bind strongly to the crystal face. Such improperly oriented molecules can be thought of as competitive inhibitors, and their presence is shown to affect the growth rate. Thus procedures aimed at decreasing the effect of such competitive inhibition, or conversely increasing the specificity of protein–protein interactions forming the crystal, are expected to offer better control of crystal growth rates.

A molecular basis is presented for characterizing the osmotic second virial coefficient, B 22 , of dilute protein solutions, which provides a measure of the nature of protein–protein interactions and has been shown to be correlated with crystallization behavior. Experimental measurements of the second virial coefficient of lysozyme and bovine α -chymotrypsinogen A were performed by static light scattering, as a function of pH and electrolyte concentration. Although some of the trends can be explained qualitatively by simple colloidal models of protein interactions, a more realistic interpretation based on protein crystallographic structures suggests a different explanation of experimental trends. The interactions accounted for are solute–solute excluded volume (steric), electrostatic and short-range (mainly van der Waals) interactions. The interactions depend strongly on orientation, and this profoundly affects calculated second virial coefficients. We find that molecular configurations in which complementary surfaces are apposed contribute disproportionately to the second virial coefficient, mainly through short-range interactions; electrostatic interactions play a secondary role in many of these configurations. Thus molecular recognition events can play a role in determining the solution thermodynamic properties of proteins, and this provides a plausible basis for explaining the observed relationship with crystallization behavior.

Probabilities of numbers of ligands proximal to an ion lead to simple, general formulae for the free energy of ion selectivity between different media. That free energy does not depend on the definition of an inner shell for ligand-counting, but other quantities of mechanistic interest do. If analysis is restricted to a specific coordination number, then two distinct probabilities are required to obtain the free energy in addition. The normalizations of those distributions produce partition function formulae for the free energy. Quasi-chemical theory introduces concepts of chemical equilibrium, then seeks the probability that is simplest to estimate, that of the most probable coordination number. Quasi-chemical theory establishes the utility of distributions of ligand-number, and sharpens our understanding of quasi-chemical calculations based on electronic structure methods. This development identifies contributions with clear physical interpretations, and shows that evaluation of those contributions can establish a mechanistic understanding of the selectivity in ion channels.

Thermochemistry of gas-phase ion-water clusters together with estimates of the hydration free energy of the clusters and the water ligands are used to calculate the hydration free energy of the ion. Often the hydration calculations use a continuum model of the solvent. The primitive quasichemical approximation to the quasichemical theory provides a transparent framework to anchor such efforts. Here we evaluate the approximations inherent in the primitive quasichemical approach and elucidate the different roles of the bulk medium. We find that the bulk medium can stabilize configurations of the cluster that are usually not observed in the gas phase, while also simultaneously lowering the excess chemical potential of the ion. This effect is more pronounced for soft ions. Since the coordination number that minimizes the excess chemical potential of the ion is identified as the optimal or most probable coordination number, for such soft ions, the optimum cluster size and the hydration thermodynamics obtained without account of the bulk medium on the ion-water clustering reaction can be different from those observed in simulations of the aqueous ion. The ideas presented in this work are expected to be relevant to experimental studies that translate thermochemistry of ion-water clusters to the thermodynamics of the hydrated ion and to evolving theoretical approaches that combine high-level calculations on clusters with coarse-grained models of the medium.

In studying ion-selectivity in biomaterials, it is common to study ion-protein interactions within a local neighborhood around the ion. This local system analysis for the S 2 site of KcsA, its semisynthetic analog, and valinomycin yields the free energy change in exchanging K + with Na + in quantitative agreement with the value obtained by considering ion-interactions with the entire system. But the energetics of ion binding in the local system and in the entire system differ significantly and lead to different conclusions regarding the physical basis of ion selectivity. For configurations sampled from an all-atom simulation, we show that the selectivity free energy can be decomposed into a contribution arising from interactions of the ion with its local neighborhood, Δ W local , and a term arising from the field imposed on the ion and the binding site by the rest of the medium, Δ W ϕ . The local contribution Δ W local is numerically close to the actual free energy difference because the field contribution is small. The field contribution is small because of cancellation of inversely related ion-medium and site-medium interactions. Our analysis presents a rigorous foundation for the numerical success of the local system analysis and shows that its implications do not always hold for the entire protein.

We study the selective binding of the softer K+ ion over the harder Na+ and Mg++ ions to the 58 nucleotide ribosomal RNA fragment and the S2 site of the KcsA K+ channel using all atom MD simulations. The ribosomal RNA is interesting in that it binds the K+ over the harder Mg++ ion. We calculate the free energy of exchanging K+ with Mg++ in the binding site relative to the change in the aqueous solution. In order to explain the selectivity, we find that it is necessary to consider the change in the internal energy of the ion binding site. Interestingly, similar physics is observed in the KcsA S2 site. There we elucidate the role of coordination number fluctuations, ion-site binding energetics, and site-site interaction energetics in determining the selectivity of the S2 site for K+: once again, the reduced strain in the site in the presence of K+ (relative to Na+) explains selectivity. Thus, despite the uncommon origin and completely different chemical composition of the binding site, in both the cases the binding site is constricted in the presence of the harder ion and causes an increase in the unfavorable electrostatic strain. This strain is what leads to the observed selectivity for the softer K+ ion.

Our understanding of the ``long range'' electrodynamic, electrostatic, and polar interactions that dominate the organization of small objects at separations beyond an interatomic bond length is reviewed. From this basic-forces perspective, a large number of systems are described from which one can learn about these organizing forces and how to modulate them. The many practical systems that harness these nanoscale forces are then surveyed. The survey reveals not only the promise of new devices and materials, but also the possibility of designing them more effectively.

The hydration free energy of an ion is separated into a chemical term, arising due to the interaction of the ion with water molecules within the defined coordination sphere (the inner shell), a packing contribution, accounting for forming an ion-free coordination sphere (the observation volume) in the solvent, and a long range correction, accounting for the interaction of the ion with the solvent outside the coordination sphere. The chemical term is recast as a sum over coordination states, with the nth term depending on the probability of observing n water molecules in the observation volume and the free energy of assembling the n water molecules around the ion in the presence of the outer-shell solvent. Each stepwise increment in the coordination number more fully accounts for the chemical contribution, and this molecular aufbau approach is used to interrogate the thermodynamic importance of various hydration structures X[H(2)O](n) of X(aq) (X = Na(+), K(+), F(-)) within a classical molecular mechanics framework. States with n less than (and at best equal to) the most probable coordination state ñ account for all of the chemical term and evince the role of the ion in drawing water molecules into the coordination sphere. For states with nñ, the influence of the ion is tempered and changes in coordination states due to density fluctuations in water also appear important. Thus the influence of the ion on the solvent matrix is local, and only a subset of water molecules (nor = ñ) contributes dominantly to the hydration thermodynamics. The n = 4 state of Na(+) (ñ = 5) and K(+) (ñ = 7) and the n = 6 state of F(-) (ñ = 6) are thermodynamically dominant; adding a water molecule to the dominant state additionally contributes only about 2-3 k(B)T toward the chemical term, but removing a water molecule is very unfavorable.

Motivated by a quasi-chemical view of protein hydration, we define specific hydration sites on the surface of globular proteins in terms of the local water density at each site relative to bulk water density. The corresponding kinetic definition invokes the average residence time for a water molecule at each site and the average time that site remains unoccupied. Bound waters are identified by high site occupancies using either definition. In agreement with previous molecular dynamics simulation studies, we find only a weak correlation between local water densities and water residence times for hydration sites on the surface of two globular proteins, lysozyme and staphylococcal nuclease. However, a strong correlation is obtained when both the average residence and vacancy times are appropriately taken into account. In addition, two distinct kinetic regimes are observed for hydration sites with high occupancies: long residence times relative to vacancy times for a single water molecule, and short residence times with high turnover involving multiple water molecules. We also correlate water dynamics, characterized by average occupancy and vacancy times, with local heterogeneities in surface charge and surface roughness, and show that both features are necessary to obtain sites corresponding to kinetically bound waters.

A quasi-chemical theory implemented on the basis of molecular simulation is derived and tested for the hydrophobic hydration of CF4(aq). The theory formulated here subsumes a van der Waals treatment of solvation and identifies contributions to the hydration free energy of CF4(aq) that naturally arise from chemical contributions defined by quasi-chemical theory and fluctuation contributions analogous to Debye-Huckel or random phase approximations. The resulting Gaussian statistical thermodynamic model avoids consideration of hypothetical drying-then-rewetting problems and is physically reliable in these applications as judged by the size of the fluctuation contribution. The specific results here confirm that unfavorable tails of binding energy distributions reflect few-body close solute-solvent encounters. The solvent near-neighbors are pushed by the medium into unfavorable interactions with the solute, in contrast to the alternative view that a preformed interface is pulled by the solute-solvent attractive interactions into contact with the solute. The polyatomic model of CF4(aq) studied gives a satisfactory description of the experimental solubilities including the temperature dependence. The proximal distributions evaluated here for polyatomic solutes accurately reconstruct the observed distributions of water near these molecules which are nonspherical. These results suggest that drying is not an essential consideration for the hydrophobic solubilities of CF4, or of C(CH3)4 which is more soluble.

Plateaus in water adsorption isotherms on hydroxylated BeO surfaces suggest significant differences between the hydroxylated (100) and (001) surface structures and reactivities. Density functional theory structures and energies clarify these differences. Using relaxed surface energies, a Wulff construction yields a prism crystal shape exposing long (100) sides and much smaller (001) faces. This is consistent with the BeO prisms observed when beryllium metal is oxidized. A water oxygen atom binds to a single surface beryllium ion in the preferred adsorption geometry on either surface. The water oxygen/beryllium bonding is stronger on the surface with greater beryllium atom exposure, namely the less-stable (001) surface. Water/beryllium coordination facilitates water dissociation. On the (001) surface, the dissociation products are a hydroxide bridging two beryllium ions and a metal coordinated hydride with some surface charge depletion. On the (100) surface, water dissociates into a hydroxide ligating a Be atom and a proton coordinated to a surface oxygen but the lowest energy water state on the (100) surface is the undissociated metal-coordinated water. The (100) fully hydroxylated surface structure has a hydrogen bonding network which facilitates rapid proton shuffling within the network. The corresponding (001) hydroxylated surface is fairly open and lacks internal hydrogen bonding. This supports previous experimental interpretations of the step in water adsorption isotherms. Further, when the (100) surface is heated to 1000 K, hydroxides and protons associate and water desorbs. The more open (001) hydroxylated surface is stable at 1000 K. This is consistent with the experimental disappearance of the isotherm step when heating to 973 K., 21 pages, 2 table, 5 figures

A molecular theory of liquid water is identified and studied on the basis of computer simulation of the TIP3P model of liquid water. This theory would be exact for models of liquid water in which the intermolecular interactions vanish outside a finite spatial range, and therefore provides a precise analysis tool for investigating the effects of longer-ranged intermolecular interactions. We show how local order can be introduced through quasi-chemical theory. Long-ranged interactions are characterized generally by a conditional distribution of binding energies, and this formulation is interpreted as a regularization of the primitive statistical thermodynamic problem. These binding-energy distributions for liquid water are observed to be unimodal. The gaussian approximation proposed is remarkably successful in predicting the Gibbs free energy and the molar entropy of liquid water, as judged by comparison with numerically exact results. The remaining discrepancies are subtle quantitative problems that do have significant consequences for the thermodynamic properties that distinguish water from many other liquids. The basic subtlety of liquid water is found then in the competition of several effects which must be quantitatively balanced for realistic results., Comment: 8 pages, 6 figures

The hydroxide anion plays an essential role in many chemical and biochemical reactions. But a molecular-scale description of its hydration state, and hence also its transport, in water is currently controversial. The statistical mechanical quasichemical theory of solutions suggests that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{HO}}{\cdot}[{\mathrm{H}}_{2}{\mathrm{O}}]_{3}^{-}\end{equation*}\end{document} is the predominant species in the aqueous phase under standard conditions. This result agrees with recent spectroscopic studies on hydroxide water clusters and with the available thermodynamic hydration free energies. In contrast, a recent ab initio molecular dynamics simulation has suggested that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{HO}}{\cdot}[{\mathrm{H}}_{2}{\mathrm{O}}]_{4}^{-}\end{equation*}\end{document} is the only dominant aqueous solution species. We apply adiabatic ab initio molecular dynamics simulations and find good agreement with both the quasichemical theoretical predictions and experimental results. The present results suggest a picture that is simpler, more traditional, but with additional subtlety. These coordination structures are labile but the tricoordinate species is the prominent case. This conclusion is unaltered with changes in the electronic density functional. No evidence is found for rate-determining activated interconversion of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{HO}}{\cdot}[{\mathrm{H}}_{2}{\mathrm{O}}]_{4}^{-}\end{equation*}\end{document} trap structure to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathrm{HO}}{\cdot}[{\mathrm{H}}_{2}{\mathrm{O}}]_{3}^{-}\end{equation*}\end{document} mediating hydroxide transport. The view of HO - diffusion as the hopping of a proton hole has substantial validity, the rate depending largely on the dynamic disorder of the water hydrogen-bond network.

The HO$^-$(aq) ion participates in myriad aqueous phase chemical processes of biological and chemical interest. A molecularly valid description of its hydration state, currently poorly understood, is a natural prerequisite to modeling chemical transformations involving HO$^-$(aq). Here it is shown that the statistical mechanical quasi-chemical theory of solutions predicts that $\mathrm{HO\cdot[H_2O]_3{}^-}$ is the dominant inner shell coordination structure for HO$^-$(aq) under standard conditions. Experimental observations and other theoretical calculations are adduced to support this conclusion. Hydration free energies of neutral combinations of simple cations with HO$^-$(aq) are evaluated and agree well with experimental values., 10 pages, 1 figure

AIMD results on a liquid krypton-water system are compared to recent XAFS results for the radial hydration structure for a Kr atom in liquid water solution. The comparisons with the liquid solution results are satisfactory and significantly different from the radial distributions extracted from the data on the solid clathrate hydrate phase. The calculations also produce the coordination number distribution that can be examined for metastable coordination structures suggesting possibilities for clathrate-like organization; none are seen in these results. Clathrate pictures of hydrophobic hydration are discussed, as is the quasi-chemical theory that should provide a basis for clathrate pictures. Outer shell contributions are discussed and accurately estimated; they are positive and larger than the positive experimental hydration free energy of Kr(aq), implying that inner shell contributions must be negative. Clathrate-like inner shell coordination structures extracted from the simulation of the liquid, and then subjected to quantum chemical optimization, always decomposed. Interactions with the outer shell material are decisive in stabilizing coordination structures observed in liquid solution and in clathrate phases. The ``what are we to tell students'' question about hydrophobic hydration, often answered with structural models such as clathrate pictures, is then considered; we propose an alternative answer that is consistent with successful molecular theories of hydrophobic effects and based upon distinctive observable properties of liquid water. Considerations of parsimony, for instance Ockham's razor, then suggest that additional structural hypotheses in response to ``what are we to tell students'' aren't required at this stage, 33 pages, 6 figures

Continuum and atomistic descriptions of the partitioning of ions into a self-assembled (D,L)-octapeptide nanotube, cyclo[-(L-Ala-D-Ala)(4)-], are presented. Perturbation free energy calculations, including Ewald electrostatics, are used to estimate the electrostatic component of the excess free energy of charging Li(+), Na(+), Rb(+), and Cl(minus sign) ions inside the nanotube. The radial density and orientational distribution of water around the ion is calculated for the ion at two different positions inside the tube; it is seen that the calculated distributions are sensitive to the location of the ions. Two different continuum electrostatic models are formulated to describe the ion solvation inside the nanotube. When enhanced orientational structuring of water dipoles is evidenced, explicitly including the first solvation shell as part of the low dielectric nanotube environment provides good agreement with molecular dynamics simulations. When water orientational structuring is as in the reference bulk solvent, we find that treating the first shell water explicitly or as a high dielectric continuum leads to similar results. These results are discussed, and their importance for continuum electrostatic modeling of ion channels are highlighted.

Macromolecular association is an integral component of numerous cellular and technologically relevant processes. Most molecular theories of such association neglect the explicit solvent structure and rely on continuum concepts such as surface energies for calculating short-range interactions. We present a new such method for calculating the non-electrostatic component of the interaction-free energy, based on formalisms for calculating dispersion interactions between macromolecules. The interactions are separated into a short-ranged component that is treated atomistically, and a longer range component that is treated within the continuum Lifshitz–Hamaker approach. This description avoids the singularities inherent in the continuum dispersion formulation, and its effectiveness in characterizing the shape complementarity between interacting surfaces is shown to be comparable to that of surface area-based methods of similar parametric complexity. An advantage of the new method is that it allows facile calculation of the interaction free energy as a function of intermolecular separation, including steric effects; this makes it suitable for use in simulations of protein solutions.

Coordination state probabilities for the [Zn(H(2)O)(n)(CH(3)OH)(m)](2+) complex in aqueous methanol solutions are calculated as a function of the bulk solution concentration, and the number of methanol ligands, m = 0, 1, ..., 6 with n+m = 6. Zinc ion solvation free energies, which serve to normalize these probabilities, also reproduce the methanol concentration dependence of the experimentally derived free energy of zinc ion transfer from water to aqueous methanol solutions. Coordination state probabilities, p(n, m), are derived by extending quasi-chemical theory of ion hydration to solvent mixtures and mixed ligands. Free energy contributions to p(n, m) include the free energy of forming the mixed-ligand complex in the ideal gas, obtained by quantum chemical calculations, and the solvation free energy of the complex, approximated by a dielectric continuum model. We find that replacing water ligands with methanol ligands preferentially stabilizes methanol-rich complexes in the ideal gas. Conversely, water-rich complexes are stabilized by the solvation free energy contribution, such that the [Zn(H(2)O)(6)](2+) complex is the dominant species in solution for all methanol concentrations considered. Stabilization of the methanol-rich complexes is a consequence of the local coordination chemistry, dominated by the delocalization of charge on the zinc ion, while the stabilization of water-rich complexes is a consequence of favorable ion-solvent electrostatic interactions and smaller dielectric cavities for the water-rich complexes at fixed total charge in the dielectric continuum model. Our analysis also highlights an entropic contribution associated with the reversible work required to remove n water and m methanol molecules from bulk solution to form the [Zn(H(2)O)(n)(CH(3)OH)(m)](2+) complex, which captures the methanol concentration dependence of the solvation free energy of the zinc ion.

To gain analytical insights into metal binding to proteins, we express the effective Hamiltonian of an ion binding site in a protein as a combination of the restraints from the protein and the instrinsic vacuum Hamiltonian of the ion bound site. The protein restraints are described by the elastic network mode and the vacuum Hamiltonian of the ion bound site is a generalized Hessian. The resultant of the addition of the two quadratic forms is recast as a pure quadratic form. The free energy of the ion bound protein is analytical. We apply our model to the zinc finger peptide. The model quantitatively captures thermodynamics of Zn2+ preference over its competitors Fe2+, Co2+, Ni2+, and Cd2+ in the protein binding site. We predict that a binding site with minimal protein restraints will select for Zn2+ better than a zinc finger peptide over Fe2+ and Ni2+ while the preference for Zn2+ over Cd2+ and Co2+ will be diminished. We show that the inspecting the interplay between the instrinsic modes of vibrations of the the ion-site cluster and the restraints from the proteins provides a platform for completely analytical and rigorously thermodynamic design principles for metalloproteins.

The high-energy tail of the distribution of solute-solvent interaction energies is poorly characterized for condensed systems, but this tail region is of principal interest in determining the excess free energy of the solute. We introduce external fields centered on the solute to modulate the short-range repulsive interaction between the solute and solvent. This regularizes the binding energy distribution and makes it easy to calculate the free energy of the solute with the field. Together with the work done to apply the field in the presence and absence of the solute, we calculate the excess chemical potential of the solute. We present the formal development of this idea and apply it to study liquid water.

We assess the contribution of each coordination state to the hydration free energy of a distinguished water molecule, the solute water. We define a coordination sphere, the inner-shell, and separate the hydration free energy into packing, outer-shell, and local, solute-specific (chemical) contributions. The coordination state is defined by the number of solvent water molecules within the coordination sphere. The packing term accounts for the free energy of creating a solute-free coordination sphere in the liquid. The outer-shell contribution accounts for the interaction of the solute with the fluid outside the coordination sphere and it is accurately described by a Gaussian model of hydration for coordination radii greater than the minimum of the oxygen-oxygen pair-correlation function: theory helps identify the length scale to parse chemical contributions from bulk, nonspecific contributions. The chemical contribution is recast as a sum over coordination states. The nth term in this sum is given by the probability p(n) of observing n water molecules inside the coordination sphere in the absence of the solute water times a factor accounting for the free energy, W(n), of forming an n-water cluster around the solute. The p(n) factors thus reflect the intrinsic properties of the solvent while W(n) accounts for the interaction between the solute and inner-shell solvent ligands. We monitor the chemical contribution to the hydration free energy by progressively adding solvent ligands to the inner-shell and find that four-water molecules are needed to fully account for the chemical term. For a chemically meaningful coordination radius, we find that W(4) ≈ W(1) and thus the interaction contribution is principally accounted for by the free energy for forming a one-water cluster, and intrinsic occupancy factors alone account for over half of the chemical contribution. Our study emphasizes the need to acknowledge the intrinsic solvent properties in interpreting the hydration structure of any solute, with particular care in cases where the solute-solvent interaction strength is similar to that between the solvent molecules.

In a zinc-finger protein, we treat a defined metal-residue cluster at atomic detail and the role of the remaining protein and solvent is described by a molecular field. The structure of this field is obtained by treating the protein outside the cluster as an elastic medium. The strength of the field is adjusted self-consistently to reproduce the binding energy distribution of the metal with the cluster in a reference all-atom simulation. Comparison of the binding energy distribution of the metal with the cluster in vacuum and with the molecular field shows that the protein destabilizes the binding site, revealing the role of favorable metal-site interactions in driving peptide folding. We extensively sample configurations of the cluster (with the field) using a semi-empirical Hamiltonian. For configurations from this sample, we study the free energy of replacing Zn2+ with Fe2+, Co2+, and Ni2+ using density functional theory with a large basis set. The calculated selectivities are found to be in good agreement with experimental results. Advantages and limitations of the method are discussed.

We regularize the potential distribution framework to calculate the excess free energy of liquid water simulated with the BLYP-D density functional. Assuming classical statistical mechanical simulations at 350 K model the liquid at 298 K, the calculated free energy is found in fair agreement with experiments, but the excess internal energy and hence also the excess entropy are not. The utility of thermodynamic characterization in understanding the role of high temperatures to mimic nuclear quantum effects and in evaluating ab initio simulations is noted.

In the free energy of hydration of a solute, the chemical contribution is given by the free energy required to expel water molecules from the coordination sphere and the packing contribution is given by the free energy required to create the solute-free coordination sphere (the observation volume) in bulk water. With the simple point charge/extended (SPC/E) water model as a reference, we examine the chemical and packing contributions in the free energy of water simulated using different electron density functionals. The density is fixed at a value corresponding to that for SPC/E water at a pressure of 1 bar. The chemical contribution shows that water simulated at 300 K with BLYP is somewhat more tightly bound than water simulated at 300 K with revised PBE (revPBE) functional or at 350 K with the BLYP and BLYP-D functionals. The packing contribution for various radii of the observation volume is studied. In the size range where the distribution of water molecules in the observation volume is expected to be Gaussian, the packing contribution is expected to scale with the volume of the observation sphere. Water simulated at 300 K with the revPBE and at 350 K with BLYP-D or BLYP conforms to this expectation, but the results suggest an earlier onset of system size effects in the BLYP 350 K and revPBE 300 K systems than that observed for either BLYP-D 350 K or SPC/E. The implication of this observation for constant pressure simulations is indicated. For water simulated at 300 K with BLYP, in the size range where Gaussian distribution of occupation is expected, we instead find non-Gaussian behavior, and the packing contribution scales with surface area of the observation volume, suggesting the presence of heterogeneities in the system.

To understand the basis of ion-specific effects in biology, it is necessary to first understand the hydration structure and thermodynamics of ions. Based on a multi-state organization of the potential distribution theorem, we present new insights on the role of ion-water interactions and water density fluctuations at the size-scale of the ion in determining the ion-hydration structure and thermodynamics. We find that the hydration free energy of the ion depends on three quantities: 1) the hydration free energy of the ion in a specified n-coordinate state, where in the n-coordinate state n water molecules are present within the coordination volume of the ion; 2) the probability, xn, of observing that n-coordinate state around the ion; and 3) the probability, pn, of observing n water molecules in the coordination volume in the absence of the ion. Based on this development we find that only a small subset of water molecules in the first hydration shell of the ion sense the chemical type of the ion. Further, these core-water molecules tend to attenuate the interaction of the ion with the rest of the medium, and thus the higher coordination states of the ion more sensitively reflect density fluctuations of the solvent medium at the size scale of the observation volume. The relevance of this development in understanding ion-pairing and the selective binding of ions to biological molecules is discussed.

The determination of single ion hydration free energies is troubled by the thermodynamic constraint that only the properties of neutral pairs can be uniquely determined. As such, single ion properties depend on extrathermodynamic information, which can differ between experimental and molecular simulation measurements. This comparison is hampered by the quantum mechanical nature of the proton, the reference ion of choice for developing standard tables, and uncertainty in the experimental reference potential to which properties are measured. We revisit the methodology of Latimer et al. [J. Chem. Phys. 7, 108 (1939)], which extracts single ion properties from neutral pair transfer free energies under the assumption that the Born equation provides an accurate description of the charging of monovalent ions. This methodology permits us to make a consistent comparison between experimental and theoretical values for single ion hydration free energies and gives insight into nonpolar contributions to the ion hydration free energy as well as the potential at the center of a hypothetical uncharged ion.

On the basis of a gaussian quasi-chemical model of hydration, a model of non van der Waals character, we explore the role of attractive methane-water interactions in the hydration of methane and in the potential of mean force between two methane molecules in water. We find that the hydration of methane is dominated by packing and a mean-field energetic contribution. Contributions beyond the mean-field term are unimportant in the hydration phenomena for a hydrophobic solute such as methane. Attractive solute-water interactions make a net repulsive contribution to these pair potentials of mean force. With no conditioning, the observed distributions of binding energies are super-gaussian and can be effectively modeled by a Gumbel (extreme value) distribution. This further supports the view that the characteristic form of the unconditioned distribution in the high-e tail is due to energetic interactions with a small number of molecules. Generalized extreme value distributions also effectively model the results with minimal conditioning, but in those cases the distributions are sufficiently narrow that details of their shape aren't significant., 8 pages, 4 figures

The KcsA potassium channel belongs to a class of K+ channels that is selective for K+ over Na+ at rates of K+ transport approaching the diffusion limit. This selectivity is explained thermodynamically in terms of favorable partitioning of K+ relative to Na+ in a narrow selectivity filter in the channel. One mechanism for selectivity based on the atomic structure of the KcsA channel invokes the size difference between K+ and Na+, and the molecular complementarity of the selectivity filter with the larger K+ ion. An alternative view holds that size-based selectivity is precluded because atomic structural fluctuations are greater than the size difference between these two ions. We examine these hypotheses by calculating the distribution of binding energies for Na+ and K+ in a simplified model of the selectivity filter of the KcsA channel. We find that Na+ binds strongly to the selectivity filter with a mean binding energy substantially lower than that for K+. The difference is comparable to the difference in hydration free energies of Na+ and K+ in bulk aqueous solution. Thus, the average filter binding energies do not discriminate Na+ from K+ when measured from the baseline of the difference in bulk hydration free energies. Instead, Na+/K+ discrimination can be attributed to scarcity of good binding configurations for Na+ compared to K+. That relative scarcity is quantified as enhanced binding energy fluctuations, and is consistent with predicted relative constriction of the filter by Na+., 8 pages, 6 figures

We calculate the hydration free energy of liquid TIP3P water at 298 K and 1 bar using a quasi-chemical theory framework in which interactions between a distinguished water molecule and the surrounding water molecules are partitioned into chemical associations with proximal (inner-shell) waters and classical electrostatic-dispersion interactions with the remaining (outer-shell) waters. The calculated free energy is found to be independent of this partitioning, as expected, and in excellent agreement with values derived from the literature. An analysis of the spatial distribution of inner-shell water molecules as a function of the inner-shell volume reveals that water molecules are preferentially excluded from the interior of large volumes as the occupancy number decreases. The driving force for water exclusion is formulated in terms of a free energy for rearranging inner-shell water molecules under the influence of the field exerted by outer-shell waters in order to accommodate one water molecule at the center. The results indicate a balance between chemical association and molecular packing in liquid water that becomes increasingly important as the inner-shell volume grows in size.

We express the effective Hamiltonian of an ion-binding site in a protein as a combination of the Hamiltonian of the ion-bound site in vacuum and the restraints of the protein on the site. The protein restraints are described by the quadratic elastic network model. The Hamiltonian of the ion-bound site in vacuum is approximated as a generalized Hessian around the minimum energy configuration. The resultant of the two quadratic Hamiltonians is cast into a pure quadratic form. In the canonical ensemble, the quadratic nature of the resultant Hamiltonian allows us to express analytically the excess free energy, enthalpy, and entropy of ion binding to the protein. The analytical expressions allow us to separate the roles of the dynamic restraints imposed by the protein on the binding site and the temperature-independent chemical effects in metal-ligand coordination. For the consensus zinc-finger peptide, relative to the aqueous phase, the calculated free energy of exchanging Zn(2+) with Fe(2+), Co(2+), Ni(2+), and Cd(2+) are in agreement with experiments. The predicted excess enthalpy of ion exchange between Zn(2+) and Co(2+) also agrees with the available experimental estimate. The free energy of applying the protein restraints reveals that relative to Zn(2+), the Co(2+), and Cd(2+)-site clusters are more destabilized by the protein restraints. This leads to an experimentally testable hypothesis that a tetrahedral metal binding site with minimal protein restraints will be less selective for Zn(2+) over Co(2+) and Cd(2+) compared to a zinc finger peptide. No appreciable change is expected for Fe(2+) and Ni(2+). The framework presented here may prove useful in protein engineering to tune metal selectivity.

The thermodynamic properties of protein solutions are determined by the molecular interactions involving both solvent and solute molecules. A quantitative understanding of the relationship would facilitate more systematic procedures for manipulating the properties in a process environment. In this work the molecular basis for the osmotic second virial coefficient, B22, is studied; osmotic effects are critical in membrane transport, and the value of B22 has also been shown to correlate with protein crystallization behavior. The calculations here account for steric, electrostatic, and short-range interactions, with the structural and functional anisotropy of the protein molecules explicitly accounted for. The orientational dependence of the protein interactions is seen to have a pronounced effect on the calculations; in particular, the relatively few protein-protein configurations in which the apposing surfaces display geometric complementarity contribute disproportionately strongly to B22. The importance of electrostatic interactions is also amplified in these high-complementarity configurations. The significance of molecular recognition in determining B22 can explain the correlation with crystallization behavior, and it suggests that alteration of local molecular geometry can help in manipulating protein solution behavior. The results also have implications for the role of protein interactions in biological self-organization.

It is common in studying ion-selectivity in biomaterials to study only the local neighborhood of the ion. This ‘local system analysis’ for the S2 site of KcsA, its semi-synthetic analog, and Valinomycin yields correct selectivity free energy, δμex (S). However considering energetics of binding in the local system and in the entire protein lead to different conclusions regarding the physical basis of ion selectivity.We show that the selectivity free energy can be decomposed into a contribution from the local energetics, Wlocal, and a term arising from the field imposed on the ion and the binding site by the rest of the protein medium, Wmedium unlike the free energy of exchange, both contributions individually depend on the path involved in changing the identity of the ion and are thus not thermodynamic free energies. We present results for different paths of changing the ion. The locally defined free energy change is numerically close to the actual free energy difference because of cancellations of inversely related ion-medium and site-medium interactions. Our analysis presents a rigorousfoundation for the numerical success of the local system analysis and shows that its implications do not always hold for the entire protein.