Potential Distribution Methods and Free Energy Models of Molecular Solutions. In: Chipot C, Pohorille A, editors. Free Energy Calculations. Berlin: Springer Berlin Heidelberg; 2007. pp. 323-51. [DOI: 10.1007/978-3-540-38448-9_9]

Hydration Free Energies of Polypeptides from Popular Implicit Solvent Models versus All-Atom Simulation Results Based on Molecular Quasichemical Theory

Calculating the hydration free energy of a macromolecule in all-atom simulations has long remained a challenge, necessitating the use of models wherein the effect of the solvent is captured without explicit account of solvent degrees of freedom. This situation has changed with developments in the molecular quasi-chemical theory (QCT)─an approach that enables calculation of the hydration free energy of macromolecules within all-atom simulations at the same resolution as is possible for small molecular solutes. The theory also provides a rigorous and physically transparent framework to conceptualize and model interactions in molecular solutions and thus provides a convenient framework to investigate the assumptions in implicit solvent models. In this study, we compare the results using molecular QCT versus predictions from EEF1, ABSINTH, and GB/SA implicit solvent models for polyglycine and polyalanine solutes covering a range of chain lengths and conformations. The hydration free energies or the differences in hydration free energies between conformers obtained from the implicit solvent models do not agree with explicit solvent results, with the deviations being largest for the group additive EEF1 and ABSINTH models. GB/SA does better in capturing the qualitative trends seen in explicit solvent results. Analysis founded on QCT reveals the critical importance of the cooperativity of hydration that is inherent in the hydrophilic and hydrophobic contributions to hydration─physics that is not well captured in additive models but somewhat better accounted for by means of a dielectric in the GB/SA approach.

Hydrated Anions: From Clusters to Bulk Solution with Quasi-Chemical Theory

The interactions of hydrated ions with molecular and macromolecular solution and interface partners are strong on a chemical energy scale. Here, we recount the foremost ab initio theory for the evaluation of the hydration free energies of ions, namely, quasi-chemical theory (QCT). We focus on anions, particularly halides but also the hydroxide anion, because they have been outstanding challenges for all theories. For example, this work supports understanding the high selectivity for F^- over Cl^- in fluoride-selective ion channels despite the identical charge and the size similarity of these ions. QCT is built by the identification of inner-shell clusters, separate treatment of those clusters, and then the integration of those results into the broader-scale solution environment. Recent work has focused on a close comparison with mass-spectrometric measurements of ion-hydration equilibria. We delineate how ab initio molecular dynamics (AIMD) calculations on ion-hydration clusters, elementary statistical thermodynamics, and electronic structure calculations on cluster structures sampled from the AIMD calculations obtain just the free energies extracted from the cluster experiments. That theory-experiment comparison has not been attempted before the work discussed here, but the agreement is excellent with moderate computational effort. This agreement reinforces both theory and experiment and provides a numerically accurate inner-shell contribution to QCT. The inner-shell complexes involving heavier halides display strikingly asymmetric hydration clusters. Asymmetric hydration structures can be problematic for the evaluation of the QCT outer-shell contribution with the polarizable continuum model (PCM). Nevertheless, QCT provides a favorable setting for the exploitation of PCM when the inner-shell material shields the ion from the outer solution environment. For the more asymmetrically hydrated, and thus less effectively shielded, heavier halide ions clustered with waters, the PCM is less satisfactory. We therefore investigate an inverse procedure in which the inner-shell structures are sampled from readily available AIMD calculations on the bulk solutions. This inverse procedure is a remarkable improvement; our final results are in close agreement with a standard tabulation of hydration free energies, and the final composite results are independent of the coordination number on the chemical energy scale of relevance, as they should be. Finally, a comparison of anion hydration structure in clusters and bulk solutions from AIMD simulations emphasize some differences: the asymmetries of bulk solution inner-shell structures are moderated compared with clusters but are still present, and inner hydration shells fill to slightly higher average coordination numbers in bulk solution than in clusters.

Hydrophilic Interactions Dominate the Inverse Temperature Dependence of Polypeptide Hydration Free Energies Attributed to Hydrophobicity

We address the association of the hydrophobic driving forces in protein folding with the inverse temperature dependence of protein hydration, wherein stabilizing hydration effects strengthen with increasing temperature in a physiological range. All-atom calculations of the free energy of hydration of aqueous deca-alanine conformers, holistically including backbone and side-chain interactions together, show that attractive peptide-solvent interactions and the thermal expansion of the solvent dominate the inverse temperature signatures that have been interpreted traditionally as the hydrophobic stabilization of proteins in aqueous solution. Equivalent calculations on a methane solute are also presented as a benchmark for comparison. The present study calls for a reassessment of the forces that stabilize folded protein conformations in aqueous solutions and of the additivity of hydrophobic/hydrophilic contributions.

Quasi-Chemical Theory with Cluster Sampling from Ab Initio Molecular Dynamics: Fluoride (F-) Anion Hydration

Accurate predictions of the hydration free energy for anions typically has been more challenging than that for cations. Hydrogen bond donation to the anion in hydrated clusters such as F(H₂O) _n ^- can lead to delicate structures. Consequently, the energy landscape contains many local minima, even for small clusters, and these minima present a challenge for computational optimization. Utilization of cluster experimental results for the free energies of gas-phase clusters shows that even though anharmonic effects are interesting they need not be of troublesome magnitudes for careful applications of quasi-chemical theory to ion hydration. Energy-optimized cluster structures for anions can leave the central ion highly exposed, and application of implicit solvation models to these structures can incur more serious errors than those for metal cations. Utilizing cluster structures sampled from ab initio molecular dynamics simulations substantially fixes those issues.

Quasi-chemical theory of F-(aq): The "no split occupancies rule" revisited

We use ab initio molecular dynamics (AIMD) calculations and quasi-chemical theory (QCT) to study the inner-shell structure of F^-(aq) and to evaluate that single-ion free energy under standard conditions. Following the "no split occupancies" rule, QCT calculations yield a free energy value of -101 kcal/mol under these conditions, in encouraging agreement with tabulated values (-111 kcal/mol). The AIMD calculations served only to guide the definition of an effective inner-shell constraint. QCT naturally includes quantum mechanical effects that can be concerning in more primitive calculations, including electronic polarizability and induction, electron density transfer, electron correlation, molecular/atomic cooperative interactions generally, molecular flexibility, and zero-point motion. No direct assessment of the contribution of dispersion contributions to the internal energies has been attempted here, however. We anticipate that other aqueous halide ions might be treated successfully with QCT, provided that the structure of the underlying statistical mechanical theory is absorbed, i.e., that the "no split occupancies" rule is recognized.

Thermodynamics of mixtures of patchy and spherical colloids of different sizes: A multi-body association theory with complete reference fluid information

We present a theory to predict the structure and thermodynamics of mixtures of colloids of different diameters, building on our earlier work [A. Bansal et al., J. Chem. Phys. 145, 074904 (2016)] that considered mixtures with all particles constrained to have the same size. The patchy, solvent particles have short-range directional interactions, while the solute particles have short-range isotropic interactions. The hard-sphere mixture without any association site forms the reference fluid. An important ingredient within the multi-body association theory is the description of clustering of the reference solvent around the reference solute. Here we account for the physical, multi-body clusters of the reference solvent around the reference solute in terms of occupancy statistics in a defined observation volume. These occupancy probabilities are obtained from enhanced sampling simulations, but we also present statistical mechanical models to estimate these probabilities with limited simulation data. Relative to an approach that describes only up to three-body correlations in the reference, incorporating the complete reference information better predicts the bonding state and thermodynamics of the physical solute for a wide range of system conditions. Importantly, analysis of the residual chemical potential of the infinitely dilute solute from molecular simulation and theory shows that whereas the chemical potential is somewhat insensitive to the description of the structure of the reference fluid, the energetic and entropic contributions are not, with the results from the complete reference approach being in better agreement with particle simulations.

McMillan-Mayer theory of solutions revisited: simplifications and extensions

McMillan and Mayer (MM) proved two remarkable theorems in their paper on the equilibrium statistical mechanics of liquid solutions. They first showed that the grand canonical partition function for a solution can be reduced to one with an effectively solute-only form, by integrating out the solvent degrees of freedom. The total effective solute potential in the effective solute grand partition function can be decomposed into components which are potentials of mean force for isolated groups of one, two, three, etc., solute molecules. Second, from the first result, now assuming low solute concentration, MM derived an expansion for the osmotic pressure in powers of the solute concentration, in complete analogy with the virial expansion of gas pressure in powers of the density at low density. The molecular expressions found for the osmotic virial coefficients have exactly the same form as the corresponding gas virial coefficients, with potentials of mean force replacing vacuum potentials. In this paper, we restrict ourselves to binary liquid solutions with solute species A and solvent species B and do three things: (a) By working with a semi-grand canonical ensemble (grand with respect to solvent only) instead of the grand canonical ensemble used by MM, and avoiding graphical methods, we have greatly simplified the derivation of the first MM result, (b) by using a simple nongraphical method developed by van Kampen for gases, we have greatly simplified the derivation of the second MM result, i.e., the osmotic pressure virial expansion; as a by-product, we show the precise relation between MM theory and Widom potential distribution theory, and (c) we have extended MM theory by deriving virial expansions for other solution properties such as the enthalpy of mixing. The latter expansion is proving useful in analyzing ongoing isothermal titration calorimetry experiments with which we are involved. For the enthalpy virial expansion, we have also changed independent variables from semi-grand canonical, i.e., fixed {N(A), μ(B), V, T}, to those relevant to the experiment, i.e., fixed {N(A), N(B), p, T}, where μ denotes chemical potential, N the number of molecules, V the volume, p the pressure, and T the temperature.

Ion-water clusters, bulk medium effects, and ion hydration

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 with and without account of the bulk medium on the ion-water clustering reaction can be different. 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.

Thermodynamically dominant hydration structures of aqueous ions

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 n > ñ, 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 (n < or = 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.