Cavity Particle in Aqueous Solution with a Hydrophobic Solute: Structure, Energetics, and Functionals

Abstract

Endpoints density functional theory (DFT) provides a framework for calculating the excess chemical potential of a solute in solution using solvent distribution functions obtained from both physical endpoints of a hypothetical charging process which transforms the solvent density from that of the pure liquid to the solution state. In this work, the endpoints DFT equations are formulated in terms of the indirect (solvent-mediated) contribution ω­(x) to the solute–solvent potential of mean force, and their connections are established with the conventional DFT expressions which are based on the use of direct correlation functions. ω actually corresponds to the free-energy cost to move a cavity particle (a tagged solvent molecule which interacts with the other solvent molecules but not the solute) from the bulk to the configuration x of a solvent molecule relative to the solute and is a suitable variable to describe the solvent effects on the solute–solvent interactions. HNC and PY type approximations are then used to integrate the DFT charging integral involved in the exact expression for the excess chemical potential. With these approximations, molecular simulations are to be performed at the two endpoints of solute insertion: pure solvent without the solute and the solution system with the fully coupled solute–solvent interaction. An endpoints method thus utilizes the ensembles of intermolecular configurations of physical interest, which are often readily accessible with MD simulations given the present computational power. To illustrate properties of the formulation, we perform simulations of model systems consisting of a cavity particle in an aqueous solution containing a spherical hydrophobic solute of three different sizes from which ω­(x) and the solute chemical potential can be calculated using endpoints DFT expressions. These are compared with corresponding results obtained using the approximations needed in order to evaluate the endpoints DFT charging integral when cavity particle simulation data is not available. We analyze a new approximation (two-points quadratic HNC) to the DFT charging integral which captures the correct behavior of the cavity distributions at both endpoints of the solute insertion. The behavior of the cavity particle in simple and complex liquids plays an important role in various theoretical treatments of the solute chemical potential. For pure Lennard-Jones fluids, the free energy to bring a cavity particle from the bulk to the center of a fluid particle is negative. However, for solutes of varying size, this is not generally true for Lennard-Jones fluids or the systems studied in this work. We carry out energetic and structural analyses of the cavity particle in aqueous solution with hydrophobic solutes of varying size and discuss the results in the context of the hydrophobic effect.

Solvation Thermodynamics from the Perspective of Endpoints DFT

“…In order to assess the accuracy of the endpoints DFT approximations used within the solute excluded volume region, we performed explicit simulations of a cavity particle in solutions containing model hydrophobic solutes of different sizes. 40 We found that, for solutes whose sizes are greater than or equal to ∼1.5× the diameter of a water molecule, ω(x) changes sign from negative to positive as the cavity particle penetrates deeper into the hydrophobic core, whereas the sign remains negative when the HNC or PY approximations are used to estimate ω(x). It was also observed that the water molecules in the first shell adapt well to the hydrogen bonds of the cavity particle located near the solute surface, which may be related to the enhanced fluctuations of water at hydrophobic interfaces.…”

“…We have begun to explore alternative routes to the calculation of ω(x) in the solution system with fully coupled solute−solvent interaction in the interfacial solute−solvent region using reweighting techniques instead of explicitly simulating cavity particles. 40 We now present a more complete description of the endpoints DFT framework for estimating solvation free energies from simulations of the two physical endpoint states, the pure liquid and the solute in solution.…”

“…quantifies the effect on the solute− solvent distribution of the change in the solvent−solvent correlation due to the solute insertion. With eq 2, eq 1 is modified through the use of partial integration into 33,35,40,41…”

“…Given that ϵ ≫ k B T and ρ e (ϵ) ≈ 0 in the deep-core region, the evaluation of ω e (ϵ) from eq 10 at λ = 1 is usually not possible there when molecular simulations are conducted only at λ = 0 and 1 without employing an advanced strategy such as umbrella sampling. 40 It was actually observed, though, that the density of the solvation free energy in the excluded-volume region depends weakly on the structure (conformation) of the solute molecule, 39 and accordingly, the repulsive-interaction effect on the structural change is expected to reflect only the contributions from the "skin" parts that are close to the solute surface. A reweighting technique based on UWHAM (unbinned weighted histogram analysis method) 38,46 can then be used to construct ω e in the "skin" parts efficiently, and the 2P-QHNC functional will be of use to study the solvent effects on the conformational changes of biomolecules.…”

“…This Perspective reviews recent developments of endpoints DFT, highlighting a series of papers we have written together beginning in 2017. − In the following section, we provide a brief introduction to, and overview of, our recent collaborative work. In section , we review the theoretical framework and implementation of endpoints DFT, including the approximations used to evaluate the endpoints DFT equations, especially dimensionality reduction and our construction of a new endpoints density functional based on the two-points quadratic HNC (hypernetted-chain) approximation.…”