Long-ranged contributions to solvation free energies from theory and short-ranged models

Energy-Scaled Debye-Hückel Theory for the Electrostatic Solvation Free Energy in Size-Asymmetric Electrolyte Solutions

In this report, an energy-scaled Debye-Hückel theory is developed for fast and accurate evaluation of the electrostatic solvation free energy in size-asymmetric electrolyte solutions. A size-asymmetric electrolyte solution is mapped to a dielectric continuum medium with Debye-Hückel-like response. Based on the scaling relation of the electrostatic energy of a spherical ion in the small and large size limits, a Padé polynomial is used to interpolate the electrostatic energy at finite size. The Padé polynomial is further interpreted as the electrostatic energy of an effective Debye-Hückel mean field model, depicted by a modified Debye parameter and a surface charge density due to the size asymmetry of the solvent ions. This theory can distinguish the electrostatic energies and the electrostatic solvation free energies of solutes with the same size but opposite charges. Application to charged hard and charged soft spheres demonstrates the accuracy of our approach.

Quantifying the Relevance of Long-Range Forces for Crystal Nucleation in Water

Understanding nucleation from aqueous solutions is of fundamental importance in a multitude of fields, ranging from materials science to biophysics. The complex solvent-mediated interactions in aqueous solutions hamper the development of a simple physical picture, elucidating the roles of different interactions in nucleation processes. In this work, we make use of three complementary techniques to disentangle the role played by short- and long-range interactions in solvent-mediated nucleation. Specifically, the first approach we utilize is the local molecular field (LMF) theory to renormalize long-range Coulomb electrostatics. Second, we use well-tempered metadynamics to speed up rare events governed by short-range interactions. Third, the deep learning-based State Predictive Information Bottleneck approach is employed in analyzing the reaction coordinate of the nucleation processes obtained from the LMF treatment coupled with well-tempered metadynamics. We find that the two-step nucleation mechanism can largely be captured by the short-range interactions, while the long-range interactions further contribute to the stability of the primary crystal state under ambient conditions. Furthermore, by analyzing the reaction coordinate obtained from the combined LMF-metadynamics treatment, we discern the fluctuations on different time scales, highlighting the need for long-range interactions when accounting for metastability.

A cavity formation energy formula for hard spheres in simple electrolyte solutions

A formula for cavity formation energy of a hard sphere in restricted primitive electrolyte solutions is derived based on the integral equation theory. Specifically, the contact values of radial distribution functions between the hard sphere and the ionic species, determined analytically from the first-order mean spherical approximation theory, are used to evaluate the cavity formation energy. In the large solute-size limit, the scaling relation of the cavity formation energy further leads to an analytical expression for the surface tension of the electrolyte solution near a curved interface. Our theory is applied to hard spheres immersed in restricted primitive electrolyte solutions, where the good agreement of the cavity formation energy with the hyper-netted chain theory demonstrates the accuracy of our theory.

A Statistical Perspective on Microsolvation

The lack of a procedure to determine equilibrium thermodynamic properties of a small system interacting with a bath is frequently seen as a weakness of conventional statistical mechanics. A typical example for such a small system is a solute surrounded by an explicit solvation shell. One way to approach this problem is to enclose the small system of interest in a large bath of explicit solvent molecules, considerably larger than the system itself. The explicit inclusion of the solvent degrees of freedom is obviously limited by the available computational resources. A potential remedy to this problem is a microsolvation approach where only a few explicit solvent molecules are considered and surrounded by an implicit solvent bath. Still, the sampling of the solvent degrees of freedom is challenging with conventional grand canonical Monte Carlo methods, since no single chemical potential for the solvent molecules can be defined in the realm of small-system thermodynamics. In this work, a statistical thermodynamic model based on the grand canonical ensemble is proposed that avoids the conventional system size limitations and accurately characterizes the properties of the system of interest subject to the thermodynamic constraints of the bath. We extend an existing microsolvation approach to a generalized multibath "microstatistical" model and show that the previously derived approaches result as a limit of our model. The framework described here is universal and we validate our method numerically for a Lennard-Jones model fluid.

Local Molecular Field Theory for Coulomb Interactions in Aqueous Solutions

Coulomb interactions play a crucial role in a wide array of processes in aqueous solutions but present conceptual and computational challenges to both theory and simulations. We review recent developments in an approach addressing these challenges─local molecular field (LMF) theory. LMF theory exploits an exact and physically suggestive separation of intermolecular Coulomb interactions into strong short-range and uniformly slowly varying long-range components. This allows us to accurately determine the averaged effects of the long-range components on the short-range structure using effective single particle fields and analytical corrections, greatly reducing the need for complex lattice summation techniques used in most standard approaches. The simplest use of these ideas in aqueous solutions leads to the short solvent (SS) model, where both solvent-solvent and solute-solvent Coulomb interactions have only short-range components. Here we use the SS model to give a simple description of pairing of nucleobases and biologically relevant ions in water.

Self-consistent determination of long-range electrostatics in neural network potentials

Machine learning has the potential to revolutionize the field of molecular simulation through the development of efficient and accurate models of interatomic interactions. Neural networks can model interactions with the accuracy of quantum mechanics-based calculations, but with a fraction of the cost, enabling simulations of large systems over long timescales. However, implicit in the construction of neural network potentials is an assumption of locality, wherein atomic arrangements on the nanometer-scale are used to learn interatomic interactions. Because of this assumption, the resulting neural network models cannot describe long-range interactions that play critical roles in dielectric screening and chemical reactivity. Here, we address this issue by introducing the self-consistent field neural network — a general approach for learning the long-range response of molecular systems in neural network potentials that relies on a physically meaningful separation of the interatomic interactions — and demonstrate its utility by modeling liquid water with and without applied fields.

Machine learning-based neural network potentials often cannot describe long-range interactions. Here the authors present an approach for building neural network potentials that can describe the electronic and nuclear response of molecular systems to long-range electrostatics.

Learning intermolecular forces at liquid-vapor interfaces

By adopting a perspective informed by contemporary liquid-state theory, we consider how to train an artificial neural network potential to describe inhomogeneous, disordered systems. We find that neural network potentials based on local representations of atomic environments are capable of describing some properties of liquid-vapor interfaces but typically fail for properties that depend on unbalanced long-ranged interactions that build up in the presence of broken translation symmetry. These same interactions cancel in the translationally invariant bulk, allowing local neural network potentials to describe bulk properties correctly. By incorporating explicit models of the slowly varying long-ranged interactions and training neural networks only on the short-ranged components, we can arrive at potentials that robustly recover interfacial properties. We find that local neural network models can sometimes approximate a local molecular field potential to correct for the truncated interactions, but this behavior is variable and hard to learn. Generally, we find that models with explicit electrostatics are easier to train and have higher accuracy. We demonstrate this perspective in a simple model of an asymmetric dipolar fluid, where the exact long-ranged interaction is known, and in an ab initio water model, where it is approximated.

A molecular twist on hydrophobicity

A thorough exploration of the molecular basis for hydrophobicity with a comprehensive set of theoretical tools and an extensive set of organic solvent S/water binary systems is discussed in this work. Without a single exception, regardless of the nature or structure of S, all quantum descriptors of bonding yield stabilizing S⋯water interactions, therefore, there is no evidence of repulsion and thus no reason for etymological hydrophobicity at the molecular level. Our results provide molecular insight behind the exclusion of S molecules by water, customarily invoked to explain phase separation and the formation of interfaces, in favor of a complex interplay between entropic, enthalpic, and dynamic factors. S⋯water interfaces are not just thin films separating the two phases; instead, they are non-isotropic regions with density gradients for each component whose macroscopic stability is provided by a large number of very weak dihydrogen contacts. We offer a definition of interface as the region in which the density of the components in the A/B binary system is not constant. At a fundamental level, our results contribute to better current understanding of hydrophobicity.

Notwithstanding the very weak nature of individual contacts, it is the cumulative effect of a large number of interactions (green NCI surfaces) which provides macroscopic stability to the interfaces.

Ion-dependent protein-surface interactions from intrinsic solvent response

The phyllosilicate mineral muscovite mica is widely used as a surface template for the patterning of macromolecules, yet a molecular understanding of its surface chemistry under varying solution conditions, required to predict and control the self-assembly of adsorbed species, is lacking. We utilize all-atom molecular dynamics simulations in conjunction with an electrostatic analysis based in local molecular field theory that affords a clean separation of long-range and short-range electrostatics. Using water polarization response as a measure of the electric fields that arise from patterned, surface-bound ions that direct the adsorption of charged macromolecules, we apply a Landau theory of forces induced by asymmetrically polarized surfaces to compute protein-surface interactions for two muscovite-binding proteins (DHR10-mica6 and ^C98RhuA). Comparison of the pressure between surface and protein in high-concentration KCl and NaCl aqueous solutions reveals ion-specific differences in far-field protein-surface interactions, neatly capturing the ability of ions to modulate the surface charge of muscovite that in turn selectively attracts one binding face of each protein over all others.

Implicit Solvation Using the Superposition Approximation (IS-SPA): Extension to Peptides in a Polar Solvent

Efficient, accurate, and adaptable implicit solvent models remain a significant challenge in the field of molecular simulation. A recent implicit solvent model, IS-SPA, based on approximating the mean solvent force using the superposition approximation, provides a platform to achieve these goals. IS-SPA was originally developed to handle nonpolar solutes in a polar solvent and did not accurately capture polar solvation. Here, we demonstrate that IS-SPA can accurately capture polar solvation by incorporating solvent orientation and accounting for the contributions from long ranged electrostatics. Solvent orientation is approximated as that of an ideal dipole aligned in a mean electrostatic field and an analytic form of the long ranged electrostatics is derived. Parameters for the model are calculated from explicit solvent simulations of an isolated atom or molecule and include atom-based solvent densities, mean electric field functions, radially symmetric averaged Lennard-Jones forces, and multipoles of the explicit solvent model. Using these parameters, IS-SPA accounts for asymmetry of charge solvation and reproduces the explicit solvent potential of mean force of dimerization of two oppositely charged Lennard-Jones spheres in chloroform with high fidelity. Additionally, the model more accurately captures the effect of explicit solvent on the monomer and dimer configurations of alanine dipeptide in chloroform than a generalized Born or constant density dielectric model. The current version of the algorithm is expected to outperform explicit solvent simulations for aggregation of small peptides at concentrations below 150 mM, well above the typical experimental concentrations for these materials.

Solvation Thermodynamics from the Perspective of Endpoints DFT

Solvation thermodynamics is concerned with the evaluation and physical interpretation of solvation free energies. Endpoints DFT provides a framework for computing solvation free energies by combining molecular simulations with a version of the classical density-functional theory of solutions which focuses on ω, the indirect (solvent-mediated) part of the solute-solvent potential of mean force (indirect PMF). The simulations are performed at the endpoints of a hypothetical charging process which transforms the solvent density from the pure liquid state to that of the solution state. The endpoints DFT expression for solvation free energy can be shown to be equivalent to the standard expression for which the key quantity is the direct correlation function, but it has the advantage that the indirect term ω is more focused on the change in solvent-solvent correlations with respect to the pure liquid as the solute is inserted into the solution. In this Perspective, we review recent developments of endpoints DFT, highlighting a series of papers we have written together beginning in 2017. We emphasize the importance of dimensionality reduction as the key to the evaluation of endpoints DFT expressions and present a recently developed, spatially resolved version of the theory. The role of interfacial water at certain positions which stabilize or destabilize a solute in solution can be analyzed with the spatially resolved version, and it is of considerable interest to investigate how changes in solvation affect protein-ligand binding and conformational landscapes from an endpoints DFT perspective. Endpoints DFT can also be employed in materials science; an example involving the rational design strategy for polymer membrane separation is described. The endpoints DFT method is a scheme to evaluate the solvation free energy by introducing approximations to integrate the classical density functional over a charging parameter. We have further proposed a new functional which captures the correct dependence of the indirect PMF ω at both endpoints of the charging process, and we review how it might be employed in future work.

Dielectric response with short-ranged electrostatics

The dielectric nature of polar liquids underpins much of their ability to act as useful solvents, but its description is complicated by the long-ranged nature of dipolar interactions. This is particularly pronounced under the periodic boundary conditions commonly used in molecular simulations. In this article, the dielectric properties of a water model whose intermolecular electrostatic interactions are entirely short-ranged are investigated. This is done within the framework of local molecular-field theory (LMFT), which provides a well-controlled mean-field treatment of long-ranged electrostatics. This short-ranged model gives a remarkably good performance on a number of counts, and its apparent shortcomings are readily accounted for. These results not only lend support to LMFT as an approach for understanding solvation behavior, but also are relevant to those developing interaction potentials based on local descriptions of liquid structure.

Local Molecular Field Theory for Nonequilibrium Systems

We provide a framework for extending equilibrium local molecular field (LMF) theory to a statistical ensemble evolving under a time-dependent applied field. In this context, the self-consistency of the original LMF equation is achieved dynamically, which provides an efficient method for computing the equilibrium LMF potential, in addition to providing the nonequilibrium generalization. As a concrete example, we investigate water confined between hydrophobic or charged walls, systems that are very sensitive to the treatment of long-ranged electrostatics. We then analyze confined water in the presence of a time-dependent applied electric field, generated by a sinusoidal or abrupt variation of the magnitudes of uniform charge densities on each wall. Very accurate results are found from the time-dependent LMF formalism even for strong static fields and for time-dependent systems that are driven far from equilibrium where linear response methods fail.

Charge Density of Cation Determines Inner versus Outer Shell Coordination to Phosphate in RNA

Divalent cations are often required to fold RNA, which is a highly charged polyanion. Condensation of ions, such as Mg²⁺ or Ca²⁺, in the vicinity of RNA renormalizes the effective charges on the phosphate groups, thus minimizing the intra RNA electrostatic repulsion. The prevailing view is that divalent ions bind diffusively in a nonspecific manner. In sharp contrast, we arrive at the exact opposite conclusion using a theory for the interaction of ions with the phosphate groups using RISM theory in conjunction with simulations based on an accurate three-interaction-site RNA model. The divalent ions bind in a nucleotide-specific manner using either the inner (partially dehydrated) or outer (fully hydrated) shell coordination. The high charge density Mg²⁺ ion has a preference to bind to the outer shell, whereas the opposite is the case for Ca²⁺. Surprisingly, we find that bridging interactions, involving ions that are coordinated to two or more phosphate groups, play a crucial role in maintaining the integrity of the folded state. Their importance could become increasingly prominent as the size of the RNA increases. Because the modes of interaction of divalent ions with DNA are likely to be similar, we propose that specific inner and outer shell coordination could play a role in DNA condensation, and perhaps genome organization as well.

Electrostatic contribution to colloidal solvation in terms of the self-energy-modified Boltzmann distribution

Electrostatic interactions make a large contribution to solvation free energy in ionic fluids such as electrolytes and colloidal dispersions. The electrostatic contribution to solvation free energy has been ascribed to the self-energy of a charged particle. Here we apply a variational field theory based on lower bound inequality to the inhomogeneous fluids of one-component charged hard-spheres, thereby verifying that the self-energy is given by the difference between the total correlation function and direct correlation function. Based on the knowledge of the liquid state theory, the self-energy specified in this study not only relates a direct correlation function to the Gaussian smearing of each charged sphere, but also provides the electrostatic contribution to solvation free energy that shows good agreement with simulation results. Furthermore, the Ornstein-Zernike equation leads to a set of generalized Debye-Hückel equations reflecting the Gaussian distributed charges.

Short solvent model for ion correlations and hydrophobic association

Coulomb interactions play a major role in determining the thermodynamics, structure, and dynamics of condensed-phase systems, but often present significant challenges. Computer simulations usually use periodic boundary conditions to minimize corrections from finite cell boundaries but the long range of the Coulomb interactions generates significant contributions from distant periodic images of the simulation cell, usually calculated by Ewald sum techniques. This can add significant overhead to computer simulations and hampers the development of intuitive local pictures and simple analytic theory. In this paper, we present a general framework based on local molecular field theory to accurately determine the contributions from long-ranged Coulomb interactions to the potential of mean force between ionic or apolar hydrophobic solutes in dilute aqueous solutions described by standard classical point charge water models. The simplest approximation leads to a short solvent (SS) model, with truncated solvent-solvent and solute-solvent Coulomb interactions and long-ranged but screened Coulomb interactions only between charged solutes. The SS model accurately describes the interplay between strong short-ranged solute core interactions, local hydrogen-bond configurations, and long-ranged dielectric screening of distant charges, competing effects that are difficult to capture in standard implicit solvent models.

Theory and simulations for RNA folding in mixtures of monovalent and divalent cations

RNA molecules cannot fold in the absence of counterions. Experiments are typically performed in the presence of monovalent and divalent cations. How to treat the impact of a solution containing a mixture of both ion types on RNA folding has remained a challenging problem for decades. By exploiting the large concentration difference between divalent and monovalent ions used in experiments, we develop a theory based on the reference interaction site model (RISM), which allows us to treat divalent cations explicitly while keeping the implicit screening effect due to monovalent ions. Our theory captures both the inner shell and outer shell coordination of divalent cations to phosphate groups, which we demonstrate is crucial for an accurate calculation of RNA folding thermodynamics. The RISM theory for ion-phosphate interactions when combined with simulations based on a transferable coarse-grained model allows us to predict accurately the folding of several RNA molecules in a mixture containing monovalent and divalent ions. The calculated folding free energies and ion-preferential coefficients for RNA molecules (pseudoknots, a fragment of the rRNA, and the aptamer domain of the adenine riboswitch) are in excellent agreement with experiments over a wide range of monovalent and divalent ion concentrations. Because the theory is general, it can be readily used to investigate ion and sequence effects on DNA properties.

Analytic theory of finite-size effects in supercell modeling of charged interfaces

The Ewald3D sum with the tinfoil boundary condition (e3dtf) evaluates the electrostatic energy of a finite unit cell inside an infinitely periodic supercell. Although it has been used as a de facto standard treatment of electrostatics for simulations of extended polar or charged interfaces, the finite-size effect on simulated properties has yet to be fully understood. There is, however, an intuitive way to quantify the average effect arising from the difference between the e3dtf and Coulomb potentials on the response of mobile charges to contact surfaces with fixed charges and/or to an applied external electric field. Although any charged interface formed by mobile countercharges that compensate the fixed charges fluctuates upon a change in the acting electric field, the distance between a pair of oppositely charged interfaces is found to be nearly stationary, which allows an analytic finite-size correction to the amount of countercharges. Application of the theory to solvated electric double layers (insulator/electrolyte interfaces) predicts that the state of complete charge compensation is invariant with respect to solvent permittivities, which is confirmed by a proper analysis of simulation data in the literature.

Spatially-Decomposed Free Energy of Solvation Based on the Endpoint Density-Functional Method

A spatially resolved version of the density-functional method for solvation thermodynamics is presented by extending the free-energy functional previously established in the one-dimensional, energy representation and formulating a new expression in a mixed four-dimensional representation (three dimensions for position and one dimension for energy). The space was further divided into a set of discrete regions with respect to the relative position of a solvent molecule from the solute, and the spatially decomposed energetics of solvation were analyzed for small molecules with a methyl, amine, or hydroxyl group and alanine dipeptide in solvent water. It was observed that the density of the solvation free energy is weakly dependent on the solute site in the excluded-volume region and is distinctively favorable in the first shells of the solute atoms that can readily form hydrogen bonds with water. The solvent-reorganization term reduces faster with the separation from the solute than the direct interaction between the solute and solvent, and the latter governs the energetics in the second shell and outer regions. The sum of the contributions to the free energy from the excluded volume and first shell was found to deviate significantly from the total sum over all the regions, implying that the solvation free energy is not spatially localized near the solute in a quantitative sense. Still, a local description was shown to be valid as confirmed by the correlation of the total value of free energy with the corresponding value obtained by integrating the free-energy density to the second shell. The theoretical framework developed in the present work to spatially decompose the solvation free energy can thus be useful to identify stabilizing or destabilizing regions of solvent proximate to a solute and to analyze the role that the displacement of interfacial water plays in the thermodynamics of molecular association.

Alchemical free energy calculations and umbrella sampling with local molecular field theory

Understanding the thermodynamic driving forces underlying any chemical process requires a description of the underlying free energy surface. However, computation of free energies is difficult, often requiring advanced sampling techniques. Moreover, these computations can be further complicated by the evaluation of any long-ranged interactions in the system of interest, such as Coulomb interactions in charged and polar media. Local molecular field theory is a promising approach to avoid many of the conceptual and computational difficulties associated with long-ranged interactions. We present frameworks for performing alchemical free energy calculations and non-Boltzmann sampling with local molecular field theory. We demonstrate that local molecular field theory can be used to perform these free energy calculations with accuracy comparable to traditional methodologies while eliminating the need for explicit treatment of long-ranged interactions in simulations.

On the connections and differences among three mean-field approximations: a stringent test

This letter attempts to clarify the meaning of three closely related mean-field approximations: random phase approximation (RPA), local molecular field (LMF) approximation, and symmetry-preserving mean-field (SPMF) approximation, and their use of reliability and validity in the field of theory and simulation of liquids when the long-ranged component of the intermolecular interaction plays an important role in determining density fluctuations and correlations. The RPA in the framework of classical density functional theory (DFT) neglects the higher order correlations in the bulk and directly applies the long-ranged part of the potential to correct the pair direct correlation function of the short-ranged system while the LMF approach introduces a nonuniform mimic system under a reconstructed static external potential that accounts for the average effect arising from the long-ranged component of the interaction. Furthermore, the SPMF approximation takes the viewpoint of LMF but instead instantaneously averages the long-ranged component of the potential over the degrees of freedom in the direction with preserved symmetry. The formal connections and the particular differences of the viewpoint among the three approximations are explained and their performances in producing structural properties of liquids are stringently tested using an exactly solvable model. We demonstrate that the RPA treatment often yields uncontrolled poor results for pair distribution functions of the bulk system. On the other hand, the LMF theory produces quite reasonably structural correlations when the pair distribution in the bulk is converted to the singlet particle distribution in the nonuniform system. It turns out that the SPMF approach outperforms the other two at all densities and under extreme conditions where the long-ranged component significantly contributes to the structural correlations.

The effect of electrostatic boundaries in molecular simulations: symmetry matters

Artifacts arise when the long-ranged electrostatic interaction is inappropriately treated in molecular simulations of electrolytes. When the usual Ewald3D sum method with the tinfoil boundary condition (e3dtf) is used for simulations of an interfacial liquid under an external electric field, a straightforward analysis of the liquid structure often suggests unphysical dielectric properties as a consequence of the inaccurate treatment of the electrostatics. In order to understand the underlying mechanism that leads to this apparent violation of thermodynamics, we now derive a new equation in the weak-field limit that, in a mean field view, accounts for the average effect arising from the difference between e3dtf and the sophisticated Ewald2D sum method (e2d). Numerical simulations of a water system in slab geometry confirm the validity of the weak-field limit equation for a series of parameter setup associated with e3dtf. Moreover, a similar procedure applied to a spherically confined water system suggests that corrections to the seemingly inappropriate treatment of the electrostatics in fact vanish. This cancellation of the boundary effect due to symmetry immediately sheds light on the long-lasting problem of the validity of the ad hoc application of e3dtf for bulk systems. In total, we argue that artifacts arising from e3dtf are often predictable and analytical corrections to the straightforward analysis might be applied to reveal consistent thermodynamic properties in liquid simulations.

On the role of water density fluctuations in the inhibition of a proton channel

Hv1 is a transmembrane four-helix bundle that transports protons in a voltage-controlled manner. Its crucial role in many pathological conditions, including cancer and ischemic brain damage, makes Hv1 a promising drug target. Starting from the recently solved crystal structure of Hv1, we used structural modeling and molecular dynamics simulations to characterize the channel's most relevant conformations along the activation cycle. We then performed computational docking of known Hv1 inhibitors, 2-guanidinobenzimidazole (2GBI) and analogs. Although salt-bridge patterns and electrostatic potential profiles are well-defined and distinctive features of activated versus nonactivated states, the water distribution along the channel lumen is dynamic and reflects a conformational heterogeneity inherent to each state. In fact, pore waters assemble into intermittent hydrogen-bonded clusters that are replaced by the inhibitor moieties upon ligand binding. The entropic gain resulting from releasing these conformationally restrained waters to the bulk solvent is likely a major contributor to the binding free energy. Accordingly, we mapped the water density fluctuations inside the pore of the channel and identified the regions of maximum fluctuation within putative binding sites. Two sites appear as outstanding: One is the already known binding pocket of 2GBI, which is accessible to ligands from the intracellular side; the other is a site located at the exit of the proton permeation pathway. Our analysis of the waters confined in the hydrophobic cavities of Hv1 suggests a general strategy for drug discovery that can be applied to any ion channel.

Hydration thermodynamics beyond the linear response approximation

The solvation energetics associated with the transformation of a solute molecule at infinite dilution in water from an initial state A to a final state B is reconsidered. The two solute states have different potentials energies of interaction, [Formula: see text] and [Formula: see text], with the solvent environment. Throughout the A [Formula: see text] B transformation of the solute, the solvation system is described by a Hamiltonian [Formula: see text] that changes linearly with the coupling parameter ξ. By focusing on the characterization of the probability density [Formula: see text] that the dimensionless perturbational solute-solvent interaction energy [Formula: see text] has numerical value y when the coupling parameter is ξ, we derive a hierarchy of differential equation relations between the ξ-dependent cumulant functions of various orders in the expansion of the appropriate cumulant generating function. On the basis of this theoretical framework we then introduce an inherently nonlinear solvation model for which we are able to find analytical results for both [Formula: see text] and for the solvation thermodynamic functions. The solvation model is based on the premise that there is an upper or a lower bound (depending on the nature of the interactions considered) to the amplitude of the fluctuations of Y in the solution system at equilibrium. The results reveal essential differences in behavior for the model when compared with the linear response approximation to solvation, particularly with regards to the probability density [Formula: see text]. The analytical expressions for the solvation properties show, however, that the linear response behavior is recovered from the new model when the room for the thermal fluctuations in Y is not restricted by the existence of a nearby bound. We compare the predictions of the model with the results from molecular dynamics computer simulations for aqueous solvation, in which either (1) the solute-solvent electrostatic interactions, or (2) the shorter-range attractive interactions are switched-on in the A [Formula: see text] B process.

Statistical Analyses of Hydrophobic Interactions: A Mini-Review

This review focuses on the striking recent progress in solving for hydrophobic interactions between small inert molecules. We discuss several new understandings. First, the inverse temperature phenomenology of hydrophobic interactions, i.e., strengthening of hydrophobic bonds with increasing temperature, is decisively exhibited by hydrophobic interactions between atomic-scale hard sphere solutes in water. Second, inclusion of attractive interactions associated with atomic-size hydrophobic reference cases leads to substantial, nontrivial corrections to reference results for purely repulsive solutes. Hydrophobic bonds are weakened by adding solute dispersion forces to treatment of reference cases. The classic statistical mechanical theory for those corrections is not accurate in this application, but molecular quasi-chemical theory shows promise. Finally, because of the masking roles of excluded volume and attractive interactions, comparisons that do not discriminate the different possibilities face an interpretive danger.