Solvation Free Energy Calculations Research Articles

Calculation of Solvation Free Energies of Charged Solutes Using Mixed Cluster/Continuum Models

We derive a consistent approach for predicting the solvation free energies of charged solutes in the presence of implicit and explicit solvents. We find that some published methodologies make systematic errors in the computed free energies because of the incorrect accounting of the standard state corrections for water molecules or water clusters present in the thermodynamic cycle. This problem can be avoided by using the same standard state for each species involved in the reaction under consideration. We analyze two different thermodynamic cycles for calculating the solvation free energies of ionic solutes: (1) the cluster cycle with an n water cluster as a reagent and (2) the monomer cycle with n distinct water molecules as reagents. The use of the cluster cycle gives solvation free energies that are in excellent agreement with the experimental values obtained from studies of ion-water clusters. The mean absolute errors are 0.8 kcal/mol for H(+) and 2.0 kcal/mol for Cu(2+). Conversely, calculations using the monomer cycle lead to mean absolute errors that are >10 kcal/mol for H(+) and >30 kcal/mol for Cu(2+). The presence of hydrogen-bonded clusters of similar size on the left- and right-hand sides of the reaction cycle results in the cancellation of the systematic errors in the calculated free energies. Using the cluster cycle with 1 solvation shell leads to errors of 5 kcal/mol for H(+) (6 waters) and 27 kcal/mol for Cu(2+) (6 waters), whereas using 2 solvation shells leads to accuracies of 2 kcal/mol for Cu(2+) (18 waters) and 1 kcal/mol for H(+) (10 waters).

Accelerating QM/MM free energy calculations: representing the surroundings by an updated mean charge distribution.

Reliable studies of enzymatic reactions by combined quantum mechanical/molecular mechanics (QM(ai)/MM) approaches with an ab initio description of the quantum region presents a major challenge to computational chemists. The main problem is the need for very large computer time to evaluate the QM energy, which in turn makes it extremely challenging to perform proper configurational sampling. One of the most obvious options for accelerating QM/MM simulations is the use of an average solvent potential. In fact, the idea of using an average solvent potential is rather obvious and has implicitly been used in Langevin dipole/QM calculations. However, in the case of explicit solvent models the practical implementations are more challenging, and the accuracy of the averaging approach has not been validated. The present study introduces the average effect of the fluctuating solvent charges by using equivalent charge distributions, which are updated every m steps. Several models are evaluated in terms of the resulting accuracy and efficiency. The most effective model divides the system into an inner region with N explicit solvent atoms and an external region with two effective charges. Different models are considered in terms of the division of the solvent system and the update frequency. Another key element of our approach is the use of the free energy perturbation (FEP) and/or linear response approximation treatments that guarantees the evaluation of the rigorous solvation free energy. Special attention is paid to the convergence of the calculated solvation free energies and the corresponding solute polarization. The performance of the method is examined by evaluating the solvation of a water molecule and a formate ion in water and also the dipole moment of water in water solution. Remarkably, it is found that different averaging procedures eventually converge to the same value but some protocols provide optimal ways of obtaining the final QM(ai)/MM converged results. The current method can provide computational time saving of 1000 for properly converging simulations relative to calculations that evaluate the QM(ai)/MM energy every time step. A specialized version of our approach that starts with a classical FEP charging and then evaluates the free energy of moving from the classical potential to the QM/MM potential appears to be particularly effective. This approach should provide a very powerful tool for QM(ai)/MM evaluation of solvation free energies in aqueous solutions and proteins.

Computation of binding free energy with molecular dynamics and grand canonical Monte Carlo simulations

The binding of a ligand to a receptor is often associated with the displacement of a number of bound water molecules. When the binding site is exposed to the bulk region, this process may be sampled adequately by standard unbiased molecular dynamics trajectories. However, when the binding site is deeply buried and the exchange of water molecules with the bulk region may be difficult to sample, the convergence and accuracy in free energy perturbation (FEP) calculations can be severely compromised. These problems are further compounded when a reduced system including only the region surrounding the binding site is simulated. To address these issues, we couple molecular dynamics (MD) with grand canonical Monte Carlo (GCMC) simulations to allow the number of water to fluctuate during an alchemical FEP calculation. The atoms in a spherical inner region around the binding pocket are treated explicitly while the influence of the outer region is approximated using the generalized solvent boundary potential (GSBP). At each step during thermodynamic integration, the number of water in the inner region is equilibrated with GCMC and energy data generated with MD is collected. Free energy calculations on camphor binding to a deeply buried pocket in cytochrome P450cam, which causes about seven water molecules to be expelled, are used to test the method. It concluded that solvation free energy calculations with the GCMC/MD method can greatly improve the accuracy of the computed binding free energy compared to simulations with fixed number of water.

2P-071 溶媒和自由エネルギー計算によるタンパク質のosmolyte効果の分子的及び熱力学的基礎(蛋白質・物性(2),第46回日本生物物理学会年会)

2P-071 溶媒和自由エネルギー計算によるタンパク質のosmolyte効果の分子的及び熱力学的基礎(蛋白質・物性(2),第46回日本生物物理学会年会)

Computation of pKa Values of Substituted Aniline Radical Cations in Dimethylsulfoxide Solution

A newly developed computation strategy was used to calculate the absolute pKa values of 18 substituted aniline radical cations in dimethylsulfoxide (DMSO) solution with the error origin elucidated and deviation minimized. The B3LYP/6-311++G(2df,2p) method was applied and was found to be capable of reproducing the gas-phase proton-transfer free energies of substituted anilines with a precision of 0.83 kcal/mol. The IEF-PCM solvation model with gas-phase optimized structures was adopted in calculating the pKa values of the substituted neutral anilines in DMSO, regenerating the experimental results within a standard deviation of 0.4 pKa unit. When the IEF-PCM solvation model was applied to calculate the standard redox potentials of anilide anions, it showed that the computed values agreed well with experiment, but the redox potentials of substituted anilines were systematically overestimated by 0.304 eV. The cause of this deviation was found to be related to the inaccuracy of the calculated solvation free energies of aniline radical cations. By adjusting the size of the cavity in the IEF-PCM method, we derived a reliable procedure that can reproduce the experimental pKa values of aniline radical cations within 1.2 pKa units to those from experiment.

Protein-Ligand Complexes: Computation of the Relative Free Energy of Different Scaffolds and Binding Modes.

A methodology for the calculation of the free energy difference between a pair of molecules of arbitrary topology is proposed. The protocol relies on a dual-topology paradigm, a softening of the intermolecular interactions, and a constraint that prevents the perturbed molecules from drifting away from each other at the end states. The equivalence and the performance of the methodology against a single-topology approach are demonstrated on a pair of harmonic oscillators, the calculation of the relative solvation free energy of ethane and methanol, and the relative binding free energy of two congeneric inhibitors of cyclooxygenase 2. The stability of two alternative binding modes of an inhibitor of cyclin-dependent kinase 2 is then investigated. Finally, the relative binding free energy of two structurally different inhibitors of cyclin-dependent kinase 2 is calculated. The proposed methodology allows the study of a range of problems that are beyond the reach of traditional relative free energy calculation protocols and should prove useful in drug design studies.

Efficient quantum mechanical calculation of solvation free energies based on density functional theory, numerical atomic orbitals and Poisson–Boltzmann equation

We have successfully coupled the Kohn–Sham with Poisson–Boltzmann equations to predict the solvation free energy, where the Kohn–Sham equations were solved by implementing the flexible pseudo atomic orbitals as in S iesta package. It was found that the calculated solvation free energy is in good agreement with experimental results for small neutral molecules, and its standard error is 1.33 kcal/mol, the correlation coefficient is 0.97. Due to its high efficiency and accuracy, the proposed model can be a promising tool for computing solvation free energies in computer aided drug design in future.

Calculation of the Free Energy of Polarization: Quantifying the Effect of Explicitly Treating Electronic Polarization on the Transferability of Force-Field Parameters

The lack of an explicit description of electronic polarization in nonpolarizable force fields usually results in an incomplete transferability of force-field parameter sets when applied in simulations of the system of interest in either a polar or an apolar environment. For example, the use of nonpolarizable parameter sets optimized to reproduce experimental data on properties of pure liquids of polar compounds commonly yields too low solubilities in water for the corresponding compounds. The reason is that the fixed charge distributions calibrated for the pure liquid might correspond to too low molecular dipole moments in case of hydration. In the current study, we quantitatively show that explicit inclusion of electronic polarization can improve the transferability of biomolecular force-field parameter sets. With this aim, free energies of polarization, DeltaGpola, have been calculated, with DeltaGpola corresponding to the free energy difference between identical systems described by a polarizable and a nonpolarizable model. Using a nonpolarizable model and a polarizable one (based on the charge-on-spring approach) for dimethyl ether (DME), which were both parametrized to reproduce experimental values for pure liquid properties, small values were found for DeltaGpola for the pure liquid or when a DME solute was solvated in the apolar solvent cyclohexane. For the solute hydrated in water, however, DeltaGpola was found to be of the same order of magnitude as the discrepancy between the free energy of hydration from simulation using a nonpolarizable solute model and the experimental value. Thus, introducing polarizabilities clearly improves the transferability of the parameter set. Additionally, in calculations of an anion solvated in DME, DeltaGpola for the solvent adopted relatively large values. From an estimation of the errors in the calculated free energy differences, it was furthermore shown that the calculation of DeltaGpola offers an effective and accurate method to obtain differences in solvation (or excess) free energies between systems described by polarizable and nonpolarizable models when compared to a direct calculation of solvation (or excess) free energies.

B−N Distance Potential of CH3CN−BF3 Revisited: Resolving the Experiment−Theory Structure Discrepancy and Modeling the Effects of Low-Dielectric Environments

We have re-examined the B-N distance potential of CH3CN-BF3 using MP2, DFT, and high-accuracy multicoefficient methods (MCG3 and MC-QCISD). In addition, we have solved a 1-D Schrödinger equation for nuclear motion along the B-N stretching coordinate, thereby obtaining vibrational energy levels, wave functions, and vibrationally averaged B-N distances. For the gas-phase, MCG3//MP2/aug-cc-pVTZ potential, we find an average B-N distance of 1.95 A, which is 0.13 A longer than the corresponding equilibrium value. In turn, this provides solid evidence that the long-standing discrepancy between the experimental (R(B-N) = 2.01 A) and theoretical (R(B-N) = 1.8 A or R(B-N) = 2.2-2.3 A) distances may be genuine, stemming from large amplitude vibrational motion in the B-N stretching coordinate. Furthermore, we have examined the effects of low-dielectric media (epsilon = 1.1-5.0) on the structure of CH3CN-BF3 by calculating solvation free energies (PCM/B97-2/aug-cc-pVTZ) and adding them to the gas-phase, MCG3 potential. These calculations demonstrate that the inner region of the potential is stabilized to a greater extent by these media, and correspondingly, the equilibrium and average B-N distances decrease with increasing dielectric constant. We find that the crystallographic structural result (R(B-N) = 1.63 A) is nearly reproduced with a dielectric constant of only 5.0, and also predict significant structural changes for epsilon values of 1.1-1.5, consistent with results from matrix-isolation-IR experiments.

Prediction of Molecular Solvation Free Energy Based on the Optimization of Atomic Solvation Parameters with Genetic Algorithm

We propose an improved solvent contact model to estimate the solvation free energy of an organic molecule from individual atomic contributions. The modification of the solvation model involves the optimization of three kinds of parameters in the solvation free energy function: atomic fragmental volume, maximum atomic occupancy, and atomic solvation parameters. All of these atomic parameters for 24 atom types are developed by the operation of a standard genetic algorithm in such a way as to minimize the difference between experimental and calculated solvation free energies. The data set for experimental solvation free energies is divided into a training set of 131 compounds and a test set of 24 compounds. Linear regressions with the optimized atomic parameters yield fits with the squared correlation coefficients (r2) of 0.89 and 0.86 for the training set and for the test set, respectively. Overall, the results indicate that the improved solvent contact model with the newly developed atomic parameters would be a useful tool for rapid calculation of molecular solvation free energies in aqueous solution.

Simulation of Ca2+ and Mg2+ Solvation Using Polarizable Atomic Multipole Potential

The alkaline earth metals calcium and magnesium are critically involved in many biomolecular processes. To understand the hydration thermodynamics of these ions, we have performed molecular dynamics simulations using a polarizable potential. Particle-mesh Ewald for point multipoles has been applied to the calculation of electrostatic interactions. The parameters in this model have been determined from an ab initio quantum mechanical calculation of dimer interactions between ions and water. Two methods for ion solvation free energy calculation, free energy perturbation, and the Bennett acceptance ratio have been compared. Both predict results consistent with other theoretical estimations while the Bennett approach leads to a much smaller statistical error. Based on the Born theory and the ion-oxygen radial distribution functions, we estimate the effective size of the ions in solution, concluding that K(+) > Na(+) congruent with Ca(2+) > Mg(2+). There appears to be much stronger perturbation in water structure, dynamics, and dipole moment around the divalent cations than the monovalent K(+) and Na(+). The average water coordination numbers for Ca(2+) and Mg(2+) are 7.3 and 6, respectively. The lifetime of water molecules in the first solvation shell of Mg(2+) is on the order of hundreds of picoseconds, in contrast to only few picoseconds for Ca(2+), K(+), or Na(+).

Molecular docking study and development of an empirical binding free energy model for phosphodiesterase 4 inhibitors

In the present work, several computational methodologies were combined to develop a model for the prediction of PDE4B inhibitors’ activity. The adequacy of applying the ligand docking approach, keeping the enzyme rigid, to the study of a series of PDE4 inhibitors was confirmed by a previous molecular dynamics analysis of the complete enzyme. An exhaustive docking procedure was performed to identify the most probable binding modes of the ligands to the enzyme, including the active site metal ions and the surrounding structural water molecules. The enzyme–inhibitor interaction enthalpies, refined by using the semiempirical molecular orbital approach, were combined with calculated solvation free energies and entropy considerations in an empirical free energy model that enabled the calculation of binding free energies that correlated very well with experimentally derived binding free energies. Our results indicate that both the inclusion of the structural water molecules close to the ions in the binding site and the use of a free energy model with a quadratic dependency on the ligand free energy of solvation are important aspects to be considered for molecular docking investigations involving the PDE4 enzyme family.

Atomic Charge Parameters for the Finite Difference Poisson-Boltzmann Method Using Electronegativity Neutralization.

An optimization of Rappe and Goddard's charge equilibration (QEq) method of assigning atomic partial charges is described. This optimization is designed for fast and accurate calculation of solvation free energies using the finite difference Poisson-Boltzmann (FDPB) method. The optimization is performed against experimental small molecule solvation free energies using the FDPB method and adjusting Rappe and Goddard's atomic electronegativity values. Using a test set of compounds for which experimental solvation energies are available and a rather small number of parameters, very good agreement was obtained with experiment, with a mean unsigned error of about 0.5 kcal/mol. The QEq atomic partial charge assignment method can reflect the effects of the conformational changes and solvent induction on charge distribution in molecules. In the second section of the paper we examined this feature with a study of the alanine dipeptide conformations in water solvent. The different contributions to the energy surface of the dipeptide were examined and compared with the results from fixed CHARMm charge potential, which is widely used for molecular dynamics studies.

Calculation of Solvation Free Energy from Quantum Mechanical Charge Density and Continuum Dielectric Theory

We have combined ultrasoft pseudopotential density functional theory utilizing plane wave basis with a Poisson-Boltzmann/solvent-accessible surface area (PB/SA) model to calculate the solvation free energy of small neutral organic compounds in water. The solute charge density obtained from density functional theory was directly used in solving the Poisson-Boltzmann equation to obtain the reaction field. The polarized electronic wave function of the solute in the solvent was solved by including the reaction field in the density functional Hamiltonian. The quantum mechanical and Poisson-Boltzmann equations were solved self-consistently until the charge density and reaction field converged. Using the solute charge density directly instead of a point-charge representation permitted asymmetric distortion and spreading out of the electron cloud. Because the electron density could leave the van der Waals surface to penetrate into the high-dielectric solvent, the reaction field generated by this density was generally smaller than that obtained by using the point-charge representation. In applying this model to calculate the solvation free energy of 31 small neutral organic molecules spanning a range of 25 kcal/mol, we obtained a root-mean-square error of only 1.3 kcal/mol if we allowed one adjustable parameter to shift the calculated solvation free energy.

Free energy of solvation from molecular dynamics simulation applying Voronoi-Delaunay triangulation to the cavity creation

The free energy of solvation for a large number of representative solutes in various solvents has been calculated from the polarizable continuum model coupled to molecular dynamics computer simulation. A new algorithm based on the Voronoi-Delaunay triangulation of atom-atom contact points between the solute and the solvent molecules is presented for the estimation of the solvent-accessible surface surrounding the solute. The volume of the inscribed cavity is used to rescale the cavitational contribution to the solvation free energy for each atom of the solute atom within scaled particle theory. The computation of the electrostatic free energy of solvation is performed using the Voronoi-Delaunay surface around the solute as the boundary for the polarizable continuum model. Additional short-range contributions to the solvation free energy are included directly from the solute-solvent force field for the van der Waals-type interactions. Calculated solvation free energies for neutral molecules dissolved in benzene, water, CCl4, and octanol are compared with experimental data. We found an excellent correlation between the experimental and computed free energies of solvation for all the solvents. In addition, the employed algorithm for the cavity creation by Voronoi-Delaunay triangulation is compared with the GEPOL algorithm and is shown to predict more accurate free energies of solvation, especially in solvents composed by molecules with nonspherical molecular shapes.

A computational protocol to probe the role of solvation effects on the reduction potential of azurin mutants

Semiquantitative relationships between thermodynamic parameters of Cu2+ reduction experimentally measured for a series of azurin mutants and the solvation free energy of the oxidized state of the proteins were derived. Solvation free energy calculations were carried out within an ONIOM/PCM scheme specifically adapted to this protein series. The method proved to be able to capture the main determinants of the measured reduction parameters, providing satisfactory predictions of the E degrees '.

PKa of Acetate in Water: A Computational Study

Several computational methods including the conductor-like polarizable continuum model, CPCM with both UAKS and UAHF cavities, Cramer and Truhlar's generalized Born solvation model, SM5.4(AM1), SM5.4(PM3), and SM5.43R(mPW1PW91/6-31+G(d)), and mixed QM/MM-Ewald simulations were used to calculate the pK(a) values of acetate and bicarbonate anions in aqueous solution. This work provided a critical and comprehensive assessment of the quality of these theoretical models in the calculation of aqueous solvation free energies for the singly charged acetate and bicarbonate ions, as well as the doubly charged acetate dianion and carbonate dianion. It was shown that QM/MM-Ewald simulations could give an accurate and consistent evaluation of the pK(a) values of acetate and bicarbonate based on both the relative and absolute pK(a) formulas, while other methods could yield satisfactory results only for certain calculations. However, this does not mean that the current QM/MM-Ewald protocol is superior to other methods. The useful information obtained in this investigation is that both the absolute and relative pK(a) formulas should better be tested in accurate calculations of pK(a) values based on any methods.

Absolute solvation free energy of Li+ and Na+ ions in dimethyl sulfoxide solution: A theoretical ab initio and cluster-continuum model study

The solvation of the lithium and sodium ions in dimethyl sulfoxide solution was theoretically investigated using ab initio calculations coupled with the hybrid cluster-continuum model, a quasichemical theory of solvation. We have investigated clusters of ions with up to five dimethyl sulfoxide (DMSO) molecules, and the bulk solvent was described by a dielectric continuum model. Our results show that the lithium and sodium ions have four and five DMSO molecules into the first coordination shell, and the calculated solvation free energies are -135.5 and -108.6 kcal mol(-1), respectively. These data suggest a solvation free energy value of -273.2 kcal mol(-1) for the proton in dimethyl sulfoxide solution, a value that is more negative than the present uncertain experimental value. This and previous studies on the solvation of ions in water solution indicate that the tetraphenylarsonium tetraphenylborate assumption is flawed and the absolute value of the free energy of transfer of ions from water to DMSO solution is higher than the present experimental values.

Neighbor Effect on PPII Conformation in Alanine Peptides

The polyproline II (PPII) conformation is dominant in short alanine oligomers. The noncooperativity of PPII structure in alanine peptides indicates that PPII in water is locally determined and that alanine neighbors are consistent with Flory's isolated pair hypothesis. However, neighbor effects from beta-branched or bulky aromatic residues tend to increase the Phi angle of the nearest neighbor as observed in coil library data. Here we demonstrate directly the neighbor effect using short alanine model peptides GGAAAGG, GGLnALnGG (Ln is norleucine), GGIAAGG, and GGIAIGG. The far-UV CD spectra, NMR 3JalphaN coupling constant, and H-D hydrogen exchange measurements reveal that Ile reduces the PPII content of the probe Ala side chain relative to Ala or norLeu. The free energy differences are consistent with predictions from electrostatic solvation free energy (ESF) calculations. The results indicate that prediction of PPII propensities or scales requires including the neighbor effect.

Ion Solvation in Water from Molecular Dynamics Simulation with the ABEEM/MM Force Field

A systematic study on monovalent ions in water clusters and in aqueous solution is presented for providing insight into their solvation structures, charge distributions, binding energies, as well as dynamic and thermodynamic properties in terms of the atom-bond electronegativity equalization method fused into molecular mechanics (ABEEM/MM) that is to take ABEEM charges into the Coulomb term in MM. For hydrated systems of M+(H2O)n, M+ being Li+, Na+, and K+, as well as X-(H2O)n, X- being F-, Cl-, and Br-, with n = 1-6, parameters for the effective interaction between the ion and the water molecules were determined, so as to reproduce the experimental or ab initio results. The corresponding parameters were tested with molecular dynamics (MD) simulations of these ions in liquid water and with solvation free energy calculations using the perturbation technique. The results of aqueous ionic solution simulations with the ABEEM/MM force field provide a reasonable description of many important properties, which are in good agreement with the experimental measurements. This work demonstrates that the combination of ABEEM/MM-MD provides a powerful tool in analyzing solvation processes of monovalent ions in water.