Role of attractive methane-water interactions in the potential of mean force between methane molecules in water

How do water-mediated interactions and osmotic second virial coefficients vary with particle size?

We examine quantitatively the solute-size dependences of the effective interactions between nonpolar solutes in water and in a simple liquid. The potential w(r) of mean force and the osmotic second virial coefficients B are calculated with high accuracy from molecular dynamics simulations. As the solute diameter increases from methane's to C60's with the solute-solute and solute-solvent attractive interaction parameters fixed to those for the methane-methane and methane-water interactions, the first minimum of w(r) lowers from -1.1 to -4.7 in units of the thermal energy kT. Correspondingly, the magnitude of B (<0) increases proportional to σα with some power close to 6 or 7, which reinforces the solute-size dependence of B found earlier for a smaller range of σ [H. Naito, R. Okamoto, T. Sumi and K. Koga, J. Chem. Phys., 2022, 156, 221104]. We also demonstrate that the strength of the attractive interactions between solute and solvent molecules can qualitatively change the characteristics of the effective pair interaction between solute particles, both in water and in a simple liquid. If the solute-solvent attractive force is set to be weaker (stronger) than a threshold, the effective interaction becomes increasingly attractive (repulsive) with increasing solute size.

Early Steps in the O2 Scavenger Process in the Aqueous Phase: Hydrazine vs DEHA

We investigate the first direct proton abstraction reactions from reducing agents (RAHs) hydrazine and diethyl hydroxylamine (DEHA), toward dioxygen (O2) in the aqueous phase, spanning ambient to high-temperature conditions. Quantum chemistry methods and molecular dynamics simulations are employed in this study. Quantum chemistry methods are used to analyze the quasi-equilibrium between a reactive conformation and a transition state in the [RAH,O2] cluster. On the other hand, molecular dynamics simulations estimate the probability of observing a reactive conformation of the [RAH,O2] cluster in the solution. In this study, we assume that the energy barrier of the quasi-equilibrium is sufficiently high for the RAH/O2 association process to be at equilibrium. Our findings indicate that the first proton abstraction process from a reactive conformation cluster by DEHA is energetically favored compared to hydrazine. Conversely, the association process of hydrazine and O2 in solution is more favorable than that of DEHA. Consequently, the rate constant for the first proton abstraction process is similar for both hydrazine and DEHA, particularly at high temperatures, with activation energies of approximately 21.5 ± 1.5 kcal mol-1 for both compounds. These results align with recent experiments investigating the complete O2 scavenger process in liquid water with hydrazine and DEHA. Therefore, our findings support the assumption that first proton abstraction reactions are the rate-determining steps in O2 scavenger processes in the aqueous phase.

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.

The behavior of methane-water mixtures under elevated pressures from simulations using many-body potentials

Non-polarizable empirical potentials have been proven to be incapable of capturing the mixing of methane-water mixtures at elevated pressures. Although density functional theory-based ab initio simulations may circumvent this discrepancy, they are limited in terms of the relevant time and length scales associated with mixing phenomena. Here, we show that the many-body MB-nrg potential, designed to reproduce methane-water interactions with coupled cluster accuracy, successfully captures this phenomenon up to 3 GPa and 500 K with varying methane concentrations. Two-phase simulations and long time scales that are required to fully capture the mixing, affordable due to the speed and accuracy of the MBX software, are assessed. Constructing the methane-water equation of state across the phase diagram shows that the stable mixtures are denser than the sum of their parts at a given pressure and temperature. We find that many-body polarization plays a central role, enhancing the induced dipole moments of methane by 0.20 D during mixing under pressure. Overall, the mixed system adopts a denser state, which involves a significant enthalpic driving force as elucidated by a systematic many-body energy decomposition analysis.

Thermodynamics of Hydration from the Perspective of the Molecular Quasichemical Theory of Solutions

The quasichemical organization of the potential distribution theorem, molecular quasichemical theory (QCT), enables practical calculations and also provides a conceptual framework for molecular hydration phenomena. QCT can be viewed from multiple perspectives: (a) as a way to regularize an ill-conditioned statistical thermodynamic problem; (b) as an introduction of and emphasis on the neighborship characteristics of a solute of interest; or (c) as a way to include accurate electronic structure descriptions of near-neighbor interactions in defensible statistical thermodynamics by clearly defining neighborship clusters. The theory has been applied to solutes of a wide range of chemical complexity, ranging from ions that interact with water with both long-ranged and chemically intricate short-ranged interactions, to solutes that interact with water solely through traditional van der Waals interations, and including water itself. The solutes range in variety from monatomic ions to chemically heterogeneous macromolecules. A notable feature of QCT is that, in applying the theory to this range of solutes, the theory itself provides guidance on the necessary approximations and simplifications that can facilitate the calculations. In this Perspective, we develop these ideas and document them with examples that reveal the insights that can be extracted using the QCT formulation.

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.

Solvation of the Guanidinium Ion in Pure Aqueous Environments: A Theoretical Study from an "Ab Initio"-Based Polarizable Force Field

We report simulation results regarding the hydration process of the guanidinium cation in water droplets and in bulk liquid water, at a low concentration of 0.03 M, performed using a polarizable approach to model both water/water and ion/water interactions. In line with earlier theoretical studies, our simulations show a preferential orientation of guanidinium at water-vacuum interfaces, i.e., a parallel orientation of the guanidinium plane to the aqueous surface. In an apparent contradiction with earlier simulation studies, we show also that guanidinium has a stronger propensity for the cores of aqueous systems than the ammonium cation. However, our bulk simulation conditions correspond to weaker cation concentrations than in earlier studies, by 2 orders of magnitude, and that the same simulations performed using a standard nonpolarizable force field leads to the same conclusion. From droplet data, we extrapolate the guanidinium single hydration enthalpy value to be -82.9 ± 2.2 kcal mol^-1. That is about half as large as the sole experimental estimate reported to date, about -144 kcal mol^-1. Our result yields a guanidinium absolute bulk hydration free energy at ambiant conditions to be -78.4 ± 2.6 kcal mol^-1, a value smaller by 3 kcal mol^-1 compared to ammonium. The relatively large magnitude of our guanidinium hydration free energy estimate suggests the Gdm⁺ protein denaturing properties to result from a competition between the cation hydration effects and the cation/protein interactions, a competition that can be modulated by weak differences in the protein or in the cation chemical environment.

Intramolecular Interactions Overcome Hydration to Drive the Collapse Transition of Gly15

Simulations and experiments show oligo-glycines, polypeptides lacking any side chains, can collapse in water. We assess the hydration thermodynamics of this collapse by calculating the hydration free energy at each of the end points of the reaction coordinate, here taken as the end-to-end distance (r) in the chain. To examine the role of the various conformations for a given r, we study the conditional distribution, P(R_g|r), of the radius of gyration for a given value of r. The free energy change versus R_g, -k_BT ln P(R_g|r), is found to vary more gently compared to the corresponding variation in the excess hydration free energy. Using this observation within a multistate generalization of the potential distribution theorem, we calculate a tight upper bound for the hydration free energy of the peptide for a given r. On this basis, we find that peptide hydration greatly favors the expanded state of the chain, despite primitive hydrophobic effects favoring chain collapse. The net free energy of collapse is seen to be a delicate balance between opposing intrapeptide and hydration effects, with intrapeptide contributions favoring collapse.

On the determination of the crystal-vapor surface free energy, and why a Gaussian expression can be accurate for a system far from Gaussian

The crystal-vapor surface free energy γ is an important physical parameter governing physical processes, such as wetting and adhesion. We explore exact and approximate routes to calculate γ based on cleaving an intact crystal into non-interacting sub-systems with crystal-vapor interfaces. We do this by turning off the interactions, ΔV, between the sub-systems. Using the soft-core scheme for turning off ΔV, we find that the free energy varies smoothly with the coupling parameter λ, and a single thermodynamic integration yields the exact γ. We generate another exact method, and a cumulant expansion for γ by expressing the surface free energy in terms of an average of e(-βΔV) in the intact crystal. The second cumulant, or Gaussian approximation for γ is surprisingly accurate in most situations, even though we find that the underlying probability distribution for ΔV is clearly not Gaussian. We account for this fact by developing a non-Gaussian theory for γ and find that the difference between the non-Gaussian and Gaussian expressions for γ consist of terms that are negligible in many situations. Exact and approximate methods are applied to the (111) surface of a Lennard-Jones crystal and are also tested for more complex molecular solids, the surface of octane and nonadecane. Alkane surfaces were chosen for study because their crystal-vapor surface free energy has been of particular interest for understanding surface freezing in these systems.

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.

Importance of Hydrophilic Hydration and Intramolecular Interactions in the Thermodynamics of Helix-Coil Transition and Helix-Helix Assembly in a Deca-Alanine Peptide

For a model deca-alanine peptide the cavity (ideal hydrophobic) contribution to hydration favors the helix state over extended states and the paired helix bundle in the assembly of two helices. The energetic contributions of attractive protein-solvent interactions are separated into quasi-chemical components consisting of a short-range part arising from interactions with solvent in the first hydration shell and the remaining long-range part that is well described by a Gaussian. In the helix-coil transition, short-range attractive protein-solvent interactions outweigh hydrophobic hydration and favor the extended coil states. Analysis of enthalpic effects shows that it is the favorable hydration of the peptide backbone that favors the unfolded state. Protein intramolecular interactions favor the helix state and are decisive in favoring folding. In the pairing of two helices, the cavity contribution outweighs the short-range attractive protein-water interactions. However, long-range, protein-solvent attractive interactions can either enhance or reverse this trend depending on the mutual orientation of the helices. In helix-helix assembly, change in enthalpy arising from change in attractive protein-solvent interactions favors disassembly. In helix pairing as well, favorable protein intramolecular interactions are found to be as important as hydration effects. Overall, hydrophilic protein-solvent interactions and protein intramolecular interactions are found to play a significant role in the thermodynamics of folding and assembly in the system studied.

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.

A Permeability Study of O2 and the Trace Amine p-Tyramine through Model Phosphatidylcholine Bilayers

We study here the permeability of the hydrophobic O2 molecule through a model DPPC bilayer at 323K and 350K, and of the trace amine p-tyramine through PC bilayers at 310K. The tyramine results are compared to previous experimental work at 298K. Nonequilibrium work methods were used in conjunction to simultaneously obtain both the potential of mean force (PMF) and the position dependent transmembrane diffusion coefficient, D(z), from the simulations. These in turn were used to calculate the permeability coefficient, P, through the inhomogeneous solubility-diffusion model. The results for O2 are consistent with previous simulations, and agree with experimentally measured P values for PC bilayers. A temperature dependence in the permeability of O2 through DPPC was obtained, with P decreasing at higher temperatures. Two relevant species of p-tyramine were simulated, from which the PMF and D(z) were calculated. The charged species had a large energetic barrier to crossing the bilayer of ~ 21 kcal/mol, while the uncharged, deprotonated species had a much lower barrier of ~ 7 kcal/mol. The effective in silico permeability for p-tyramine was calculated by applying three approximations, all of which gave nearly identical results (presented here as a function of the pKa). As the permeability value calculated from simulation was highly dependent on the pKa of the amine group, a further pKa study was performed that also varied the fraction of the uncharged and zwitterionic p-tyramine species. Using the experimental P value together with the simulated results, we were able to label the phenolic group as responsible for the pKa1 and the amine for the pKa2, that together represent all of the experimentally measured pKa values for p-tyramine. This agrees with older experimental results, in contrast to more recent work that has suggested there is a strong ambiguity in the pKa values.

Hydrophobic Ambivalence: Teetering on the Edge of Randomness

Processes ranging from oil-water phase separation to the formation of solid clathrate hydrates send mixed messages regarding whether oil molecules hate or love to be surrounded by water. Recent experimental and theoretical results help decipher these mixed messages by illuminating the conditions under which the stability of a hydrophobic contact is expected to exceed thermal energy fluctuations - thus facilitating hydrophobic self-assembly and the emergence of structure from randomness. Important open questions remain regarding the dependence of hydrophobic interactions on molecular size and temperature, as well as the balance of direct and water-mediated interactions.

Conditional solvation thermodynamics of isoleucine in model peptides and the limitations of the group-transfer model

The hydration thermodynamics of the amino acid X relative to the reference G (glycine) or the hydration thermodynamics of a small-molecule analog of the side chain of X is often used to model the contribution of X to protein stability and solution thermodynamics. We consider the reasons for successes and limitations of this approach by calculating and comparing the conditional excess free energy, enthalpy, and entropy of hydration of the isoleucine side chain in zwitterionic isoleucine, in extended penta-peptides, and in helical deca-peptides. Butane in gauche conformation serves as a small-molecule analog for the isoleucine side chain. Parsing the hydrophobic and hydrophilic contributions to hydration for the side chain shows that both of these aspects of hydration are context-sensitive. Furthermore, analyzing the solute–solvent interaction contribution to the conditional excess enthalpy of the side chain shows that what is nominally considered a property of the side chain includes entirely nonobvious contributions of the background. The context-sensitivity of hydrophobic and hydrophilic hydration and the conflation of background contributions with energetics attributed to the side chain limit the ability of a single scaling factor, such as the fractional solvent exposure of the group in the protein, to map the component energetic contributions of the model-compound data to their value in the protein. But ignoring the origin of cancellations in the underlying components the group-transfer model may appear to provide a reasonable estimate of the free energy for a given error tolerance.

Molecular-scale hydrophobic interactions between hard-sphere reference solutes are attractive and endothermic

The osmotic second virial coefficients, B2, for atomic-sized hard spheres in water are attractive (B2 < 0) and become more attractive with increasing temperature (ΔB2/ΔT < 0) in the temperature range 300 K ≤ T ≤ 360 K. Thus, these hydrophobic interactions are attractive and endothermic at moderate temperatures. Hydrophobic interactions between atomic-sized hard spheres in water are more attractive than predicted by the available statistical mechanical theory. These results constitute an initial step toward detailed molecular theory of additional intermolecular interaction features, specifically, attractive interactions associated with hydrophobic solutes.

Surface Polarity and Nanoscale Solvation

Linear solvation theories are well established to describe electrostatic hydration of small solutes when the hydration free energy is dominated by the electrostatic free energy of the solute multipole. In contrast, hydration of nanometer solutes is driven by surface hydration. We address the question of whether the linear-response thermodynamics established for small multipolar solutes applies to surface hydration. To this end, molecular dynamics simulations are carried out on a model C180 solute that carries no global multipole, but the surface of which is decorated with radially pointing dipoles. Linear response is dramatically violated in this case. Further, two crossovers in the solvation thermodynamics are discovered as the surface polarity is increased. Both transformations produce strongly nonlinear solvation response. The second, more collective, crossover leads to a dramatic slowing down of the interfacial dynamics, reaching the time-scales of nanoseconds. Our picture offers the possibility of flipping water domains at interfaces of nanoparticles and biomolecules.

A local entropic signature of specific ion hydration

Monovalent ion hydration entropies are analyzed via energetic partitioning of the potential distribution theorem free energy. Extensive molecular dynamics simulations and free energy calculations are performed over a range of temperatures to determine the electrostatic and van der Waals components of the entropy. The far-field electrostatic contribution is negative and small in magnitude, and it does not vary significantly as a function of ion size, consistent with dielectric models. The local electrostatic contribution, however, varies widely as a function of ion size; the sign yields a direct indication of the kosmotropic (strongly hydrated) or chaotropic (weakly hydrated) nature of the ion hydration. The results provide a thermodynamic signature for specific ion effects in hydration and are consistent with experiments that suggest minimal perturbations of water structure outside the first hydration shell. The hydration entropies are also examined in relation to the corresponding entropies for the isoelectronic rare gas pairs; an inverse correlation is observed, as expected from thermodynamic hydration data.

Communication: Direct observation of a hydrophobic bond in loop closure of a capped (-OCH2CH2-)n oligomer in water

The small r variation of the probability density P(r) for end-to-end separations of a -CH(2)CH(3) capped (-OCH(2)CH(2)-)(n) oligomer in water is computed to be closely similar to the CH(4)···CH(4) potential of mean force under the same circumstances. Since the aqueous solution CH(4)···CH(4) potential of mean force is the natural physical definition of a primitive hydrophobic bond, the present result identifies an experimentally accessible circumstance for direct observation of a hydrophobic bond which has not been observed previously because of the low solubility of CH(4) in water. The physical picture is that the soluble chain molecules carry the capping groups into aqueous solution, and permits them to find one another with reasonable frequency. Comparison with the corresponding results without the solvent shows that hydration of the solute oxygen atoms swells the chain molecule globule. This supports the view that the chain molecule globule might have a secondary effect on the hydrophobic interaction that is of first interest here. The volume of the chain molecule globule is important for comparing the probabilities with and without solvent because it characterizes the local concentration of capping groups. Study of other capping groups to enable x-ray and neutron diffraction measurements of P(r) is discussed.

Communication: Thermodynamics of water modeled using ab initio simulations

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.

Ion selectivity in the KcsA potassium channel from the perspective of the ion binding site

To understand the thermodynamic exclusion of Na(+) relative to K(+) from the S(2) site of the selectivity filter, the distribution P(X)(epsilon) (X = K(+) or Na(+)) of the binding energy (epsilon) of the ion with the channel is analyzed using the potential distribution theorem. By expressing the excess chemical potential of the ion as a sum of mean-field epsilon and fluctuation mu(flux,X)(ex) contributions, we find that selectivity arises from a higher value of mu(flux,Na(+))(ex) relative to mu(flux,K(+))(ex). To understand the role of site-site interactions on mu(ex)(flux,X), we decompose P(X)(epsilon) into n-dependent distributions, where n is the number of ion-coordinating ligands within a distance lambda from the ion. For lambda comparable to typical ion-oxygen bond distances, investigations building on this multistate model reveal an inverse correlation between favorable ion-site and site-site interactions: the ion-coordination states that most influence the thermodynamics of the ion are also those for which the binding site is energetically less strained and vice versa. This correlation motivates understanding entropic effects in ion binding to the site and leads to the finding that mu(flux,X)(ex) is directly proportional to the average site-site interaction energy, a quantity that is sensitive to the chemical type of the ligand coordinating the ion. Increasing the coordination number around Na(+) only partially accounts for the observed magnitude of selectivity; acknowledging the chemical type of the ion-coordinating ligand is essential.