^{1}and Thomas L. Beck

^{2,a)}

### Abstract

A recently developed statistical mechanical quasichemical theory (QCT) has led to significant insights into solvation phenomena for both hydrophilic and hydrophobic solutes. The QCT exactly partitions solvation free energies into three components: (1) Inner-shell chemical, (2) outer-shell packing, and (3) outer-shell long-ranged contributions. In this paper, we discuss efficient methods for computing each of the three parts of the free energy. A Bayesian estimation approach is developed to compute the inner-shell chemical and outer-shell packing contributions. We derive upper and lower bounds on the outer-shell long-ranged portion of the free energy by expressing this component in two equivalent ways. Local, high-energy contacts between the solute and solvent are eliminated by spatial conditioning in this free energy piece, leading to near-Gaussian distributions of solute-solvent interactionenergies. Thus, the average of the two mean-field bounds yields an accurate and efficient free energy estimate. Aqueous solvation free energy results are presented for several solutes, including methane, perfluoromethane, water, and sodium and chloride ions. The results demonstrate the accuracy and efficiency of the methods. The approach should prove useful in computing solvation free energies in inhomogeneous, restricted environments.

We would like to gratefully acknowledge the support of the NSF (CHE-0709560), the Army MURI program (DAAD19-02-1-0227), and the DOE Computational Science Graduate Fellowship (DE-FG02-97ER25308) for the support of this work. We acknowledge the Ohio Supercomputer Center for a grant of supercomputer time. We especially thank Lawrence Pratt for many helpful discussions.

I. INTRODUCTION

II. QUASICHEMICAL THEORY

III. BAYESIAN ESTIMATES OF OUTER-SHELL PACKING AND INNER-SHELL CHEMICAL CONTRIBUTIONS

IV. INCLUSION OF A CONTINUOUS MODEL POTENTIAL FOR MOLECULAR DYNAMICS SAMPLING

V. MOLECULAR DYNAMICS SIMULATIONS

VI. RESULTS

VII. CONCLUSIONS AND DISCUSSION

### Key Topics

- Free energy
- 110.0
- Solvents
- 56.0
- Chemical potential
- 27.0
- Chemical solutions
- 23.0
- Mean field theory
- 18.0

## Figures

Distributions of binding energy of methane to SPC water. The probability distribution of the interaction energy between a one-site methane solute and SPC water ( LJ cutoff). Note that the overlapping region is sparsely sampled (occupying less than 0.2% of the uncoupled distribution). The noninteracting data were taken from particle insertions for a solvent-only simulation. The fully coupled system data were collected from 6000 samples over . Two observed single-count outliers at the right of this distribution are included.

Distributions of binding energy of methane to SPC water. The probability distribution of the interaction energy between a one-site methane solute and SPC water ( LJ cutoff). Note that the overlapping region is sparsely sampled (occupying less than 0.2% of the uncoupled distribution). The noninteracting data were taken from particle insertions for a solvent-only simulation. The fully coupled system data were collected from 6000 samples over . Two observed single-count outliers at the right of this distribution are included.

This figure illustrates the construction of the probability of closest solvent approach, . is the probability of observing a cavity of size *and* a solvent particle in the shell . This is equivalent to the conditional probability for observing a solvent molecule in the prescribed shell, given the existence of the cavity, times the cavity probability. This physical picture is used to derive the SPT expression for the HS chemical potential. The inner circle is the unoccupied cavity, while the dashed circle indicates the shell for the observation of solvent closest approach. The solvent molecule centers are indicated by the small gray circles.

This figure illustrates the construction of the probability of closest solvent approach, . is the probability of observing a cavity of size *and* a solvent particle in the shell . This is equivalent to the conditional probability for observing a solvent molecule in the prescribed shell, given the existence of the cavity, times the cavity probability. This physical picture is used to derive the SPT expression for the HS chemical potential. The inner circle is the unoccupied cavity, while the dashed circle indicates the shell for the observation of solvent closest approach. The solvent molecule centers are indicated by the small gray circles.

Division of the minimum solute-solvent distance into successive shells. Derived quantities are shown above the axis (from bottom to top): Probabilities of falling in each shell, , and cavity probabilities . The logical organization of simulation data is shown below the axis (from top to bottom): Bin counts from simulations including HSs of size , total samples from each simulation, , and bin totals .

Division of the minimum solute-solvent distance into successive shells. Derived quantities are shown above the axis (from bottom to top): Probabilities of falling in each shell, , and cavity probabilities . The logical organization of simulation data is shown below the axis (from top to bottom): Bin counts from simulations including HSs of size , total samples from each simulation, , and bin totals .

Solute/water oxygen rdfs of all systems studied. Solid lines (left scale) show , while the integrated rdfs, , are shown as dashed lines (right scale). The rdfs for sodium and chloride can be visually distinguished because of their large size difference. There the dashed line shows and the dotted line is the corresponding . The subscripts CO, OO, and IO label the carbon/water-oxygen, water-oxygen/water-oxygen, and ion/water-oxygen distances, respectively.

Solute/water oxygen rdfs of all systems studied. Solid lines (left scale) show , while the integrated rdfs, , are shown as dashed lines (right scale). The rdfs for sodium and chloride can be visually distinguished because of their large size difference. There the dashed line shows and the dotted line is the corresponding . The subscripts CO, OO, and IO label the carbon/water-oxygen, water-oxygen/water-oxygen, and ion/water-oxygen distances, respectively.

Solvation free energy contributions for the various systems studied. In each plot, the solid horizontal line labels the comparison result: Test particle insertion for and , the result of White and Meirovitch (Ref. 81) for TIP3P water , and test particle insertion plus charging free energy for the ions ( for and for ). The (×) symbol is for the IS contribution, the (+) symbol labels the OS packing component, and the ( ) symbol is for the OS long-ranged contribution computed from our mean-field average. The bounds on the OS long-ranged contribution are indicated by dotted lines in each figure, and the BAR result for TIP3P water is given by a line as indicated in the figure. Triangles label the sum of all free energy contributions as calculated using the present methodology.

Solvation free energy contributions for the various systems studied. In each plot, the solid horizontal line labels the comparison result: Test particle insertion for and , the result of White and Meirovitch (Ref. 81) for TIP3P water , and test particle insertion plus charging free energy for the ions ( for and for ). The (×) symbol is for the IS contribution, the (+) symbol labels the OS packing component, and the ( ) symbol is for the OS long-ranged contribution computed from our mean-field average. The bounds on the OS long-ranged contribution are indicated by dotted lines in each figure, and the BAR result for TIP3P water is given by a line as indicated in the figure. Triangles label the sum of all free energy contributions as calculated using the present methodology.

Effect of conditioning on the interaction energy distribution. The upper left figure contains no conditioning. The remaining figures are labeled with the hard-particle conditioning radii. TIP3P interaction energy distributions become increasingly Gaussian as increases from zero to —evidenced by plotting the distributions corresponding to the coupled (×) and uncoupled (+) cases.

Effect of conditioning on the interaction energy distribution. The upper left figure contains no conditioning. The remaining figures are labeled with the hard-particle conditioning radii. TIP3P interaction energy distributions become increasingly Gaussian as increases from zero to —evidenced by plotting the distributions corresponding to the coupled (×) and uncoupled (+) cases.

Bayesian estimation of the and profiles for TIP3P water. Each set of lines with the same style shows the mean plus or minus one standard deviation, leading to upper and lower estimates of the free energy. In the top panel, was calculated using all of data from model potential simulations with radii of 0 (first branch), 2.7 (second), and (third). Similarly, used radii of 0, 3.0, and . In the lower panel, only the last of data from simulations with model potential radii of 0, 2.6, 2.9, and were used to calculate and radii of 0, 2.9, 3.0, 3.2, and for .

Bayesian estimation of the and profiles for TIP3P water. Each set of lines with the same style shows the mean plus or minus one standard deviation, leading to upper and lower estimates of the free energy. In the top panel, was calculated using all of data from model potential simulations with radii of 0 (first branch), 2.7 (second), and (third). Similarly, used radii of 0, 3.0, and . In the lower panel, only the last of data from simulations with model potential radii of 0, 2.6, 2.9, and were used to calculate and radii of 0, 2.9, 3.0, 3.2, and for .

## Tables

Shell occupancy data, , for calculating in SPC/E water. The HS conditioning radius indexed by runs along the columns, and the shell index (from zero to ) runs along the rows.

Shell occupancy data, , for calculating in SPC/E water. The HS conditioning radius indexed by runs along the columns, and the shell index (from zero to ) runs along the rows.

LJ force-field parameters.

LJ force-field parameters.

Gaussian approximation error width (kJ/mol) from TIP3P water calculation and bound widening terms.

Gaussian approximation error width (kJ/mol) from TIP3P water calculation and bound widening terms.

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