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Cosolvent hydrophobic preferential interactions with methane in aqueous methanol solutions are evaluated on the basis of the solute excess chemical potential derived from molecular simulations using the quasi-chemical (QC) theory generalization of the potential distribution theorem (PDT). We find that the methane-methanol preferential interaction parameter derived from QC theory quantitatively captures the favorable solvation of methane in methanol solutions in terms of important local solute-solvent (water and methanol) intermolecular interactions within a defined inner shell around the solute, and nonlocal solute interactions with solvent molecules outside this inner shell. Moreover, a unique inner shell can be defined such that the preferential interaction parameter is derived exclusively from the free energy of cavity formation in the aqueous cosolvent solution without the solute, where this cavity corresponds to the specified inner shell, and the mean interaction or binding energy of the solute with solvent molecules outside this inner shell. This inner-shell definition leads to a description of solute-cosolvent preferential interactions in which the molecular details of those interactions are derived from the effect of cosolvent on cavity statistics in the aqueous cosolvent solution alone. The finding suggests that solution thermodynamic behavior beyond steric exclusion (macromolecular crowding) contribute to the molecular mechanisms by which cosolvent preferential interactions influence protein stability and activity.

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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.

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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.

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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.

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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.

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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.

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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.

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On the basis of a Gaussian quasichemical 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-epsilon 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 the details of their shape are not significant.

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