Ab initio molecular dynamics calculations of ion hydration free energies

A non-Bornian analysis of the Gibbs energy of hydration for organic ions

The hydration energy of organic ions can be well evaluated from the distribution of surface field strength, by using a simple semi-empirical equation.

Vibrational spectroscopic studies and DFT calculations on NaCH3CO2(aq) and CH3COOH(aq)

NaCH₃CO₂(aq) and CH₃COOH(aq) were studied using Raman and infrared spectroscopy over a large concentration range, in the terahertz region and up to 4000 cm⁻¹. Band assignments for CH₃CO₂⁻(aq) and CH₃COOH(aq) were carried out under guidance of DFT frequencies.

Calculating the binding free energies of charged species based on explicit-solvent simulations employing lattice-sum methods: an accurate correction scheme for electrostatic finite-size effects

The calculation of a protein-ligand binding free energy based on molecular dynamics (MD) simulations generally relies on a thermodynamic cycle in which the ligand is alchemically inserted into the system, both in the solvated protein and free in solution. The corresponding ligand-insertion free energies are typically calculated in nanoscale computational boxes simulated under periodic boundary conditions and considering electrostatic interactions defined by a periodic lattice-sum. This is distinct from the ideal bulk situation of a system of macroscopic size simulated under non-periodic boundary conditions with Coulombic electrostatic interactions. This discrepancy results in finite-size effects, which affect primarily the charging component of the insertion free energy, are dependent on the box size, and can be large when the ligand bears a net charge, especially if the protein is charged as well. This article investigates finite-size effects on calculated charging free energies using as a test case the binding of the ligand 2-amino-5-methylthiazole (net charge +1 e) to a mutant form of yeast cytochrome c peroxidase in water. Considering different charge isoforms of the protein (net charges -5, 0, +3, or +9 e), either in the absence or the presence of neutralizing counter-ions, and sizes of the cubic computational box (edges ranging from 7.42 to 11.02 nm), the potentially large magnitude of finite-size effects on the raw charging free energies (up to 17.1 kJ mol(-1)) is demonstrated. Two correction schemes are then proposed to eliminate these effects, a numerical and an analytical one. Both schemes are based on a continuum-electrostatics analysis and require performing Poisson-Boltzmann (PB) calculations on the protein-ligand system. While the numerical scheme requires PB calculations under both non-periodic and periodic boundary conditions, the latter at the box size considered in the MD simulations, the analytical scheme only requires three non-periodic PB calculations for a given system, its dependence on the box size being analytical. The latter scheme also provides insight into the physical origin of the finite-size effects. These two schemes also encompass a correction for discrete solvent effects that persists even in the limit of infinite box sizes. Application of either scheme essentially eliminates the size dependence of the corrected charging free energies (maximal deviation of 1.5 kJ mol(-1)). Because it is simple to apply, the analytical correction scheme offers a general solution to the problem of finite-size effects in free-energy calculations involving charged solutes, as encountered in calculations concerning, e.g., protein-ligand binding, biomolecular association, residue mutation, pKa and redox potential estimation, substrate transformation, solvation, and solvent-solvent partitioning.

Relative Free Energies for Hydration of Monovalent Ions from QM and QM/MM Simulations

Methods directly evaluating the hydration structure and thermodynamics of physiologically relevant cations (Na(+), K(+), Cl(-), etc.) have wide ranging applications in the fields of inorganic, physical, and biological chemistry. All-atom simulations based on accurate potential energy surfaces appear to offer a viable option for assessing the chemistry of ion solvation. Although MD and free energy simulations of ion solvation with classical force fields have proven their usefulness, a number of challenges still remain. One of them is the difficulty of force field benchmarking and validation against structural and thermodynamic data obtained for a condensed phase. Hybrid quantum mechanical/molecular mechanical (QM/MM) models combined with sampling algorithms have the potential to provide an accurate solvation model and to incorporate the effects from the surrounding, which is often missing in gas-phase ab initio computations. Herein, we report the results from QM/MM free energy simulations of Na(+)/K(+) and Cl(-)/Br(-) hydration where we simultaneously characterized the relative thermodynamics of ion solvation and changes in the solvation structure. The Flexible Inner Region Ensemble Separator (FIRES) method was used to impose a spatial separation between QM region and the outer sphere of solvent molecules treated with the CHARMM27 force field. FEP calculations based on QM/MM simulations utilizing the CHARMM/deMon2k interface were performed with different basis set combinations for K(+)/Na(+) and Cl(-)/Br(-) perturbations to establish the dependence of the computed free energies on the basis set level. The dependence of the computed relative free energies on the size of the QM and MM regions is discussed. The current methodology offers an accurate description of structural and thermodynamic aspects of the hydration of alkali and halide ions in neat solvents and can be used to obtain thermodynamic data on ion solvation in condensed phase along with underlying structural properties of the ion-solvent system.

Hydration structure of salt solutions from ab initio molecular dynamics

The solvation structures of Na(+), K(+), and Cl(-) ions in aqueous solution have been investigated using density functional theory (DFT) based Car-Parrinello (CP) molecular dynamics (MD) simulations. CPMD trajectories were collected for systems containing three NaCl or KCl ion pairs solvated by 122 water molecules using three different but commonly employed density functionals (BLYP, HCTH, and PBE) with electron correlation treated at the level of the generalized gradient approximation (GGA). The effect of including dispersion forces was analyzed through the use of an empirical correction to the DFT-GGA scheme. Special attention was paid to the hydration characteristics, especially the structural properties of the first solvation shell of the ions, which was investigated through ion-water radial distribution functions, coordination numbers, and angular distribution functions. There are significant differences between the present results obtained from CPMD simulations and those provided by classical MD based on either the CHARMM force field or a polarizable model. Overall, the computed structural properties are in fair agreement with the available experimental results. In particular, the observed coordination numbers 5.0-5.5, 6.0-6.4, and 6.0-6.5 for Na(+), K(+), and Cl(-), respectively, are consistent with X-ray and neutron scattering studies but differ somewhat from some of the many other recent computational studies of these important systems. Possible reasons for the differences are discussed.

Interacting ions in biophysics: real is not ideal

Ions in water are important throughout biology, from molecules to organs. Classically, ions in water were treated as ideal noninteracting particles in a perfect gas. Excess free energy of each ion was zero. Mathematics was not available to deal consistently with flows, or interactions with other ions or boundaries. Nonclassical approaches are needed because ions in biological conditions flow and interact. The concentration gradient of one ion can drive the flow of another, even in a bulk solution. A variational multiscale approach is needed to deal with interactions and flow. The recently developed energetic variational approach to dissipative systems allows mathematically consistent treatment of the bio-ions Na(+), K(+), Ca(2+), and Cl(-) as they interact and flow. Interactions produce large excess free energy that dominate the properties of the high concentration of ions in and near protein active sites, ion channels, and nucleic acids: the number density of ions is often >10 M. Ions in such crowded quarters interact strongly with each other as well as with the surrounding protein. Nonideal behavior found in many experiments has classically been ascribed to allosteric interactions mediated by the protein and its conformation changes. The ion-ion interactions present in crowded solutions-independent of conformation changes of the protein-are likely to change the interpretation of many allosteric phenomena. Computation of all atoms is a popular alternative to the multiscale approach. Such computations involve formidable challenges. Biological systems exist on very different scales from atomic motion. Biological systems exist in ionic mixtures (like extracellular and intracellular solutions), and usually involve flow and trace concentrations of messenger ions (e.g., 10(-7) M Ca(2+)). Energetic variational methods can deal with these characteristic properties of biological systems as we await the maturation and calibration of all-atom simulations of ionic mixtures and divalents.

The Solvation Structure of Na(+) and K(+) in Liquid Water Determined from High Level ab Initio Molecular Dynamics Simulations

Knowledge of the hydration structure of Na(+) and K(+) in the liquid phase has wide ranging implications in the field of biological chemistry. Despite numerous experimental and computational studies, even basic features such as the coordination number of these alkali ions in liquid water, thought to play a critical role in selectivity, continue to be the subject of intensive debates. Simulations based on accurate potential energy surfaces offer one approach to resolve these issues by providing reliable results on ion hydration. In this article, we report the results from molecular dynamics simulations of Na(+) and K(+) hydration based on a novel and rigorous strategy designed to overcome the challenges of QM/MM simulations of solvent molecules in the liquid phase. In this method, which we call Flexible Inner Region Ensemble Separator (FIRES), the ion and a fixed number of nearest water molecules form a dynamical and flexible inner region that is represented with high level ab initio quantum mechanical (QM) methods, while the water molecules from the surrounding bulk form an outer region that is represented by a polarizable molecular mechanical (MM) force field. Simulations yield rigorously correct thermodynamic averages as long as the solvent molecules in the flexible inner and outer regions are not allowed to exchange. Extensive FIRES simulations were carried out based on a QM/MM model in which the Na(+) or K(+) ion and the 12 nearest water molecules were represented by high level ab initio methods (RI-MP2/def2-TZVP and density functional theory with PBE/def2-TZVP), while the surrounding MM water molecules were represented by the polarizable SWM4-NDP potential. On the basis of these results, the ion coordination numbers are estimated to be within the range of 5.7-5.8 for Na(+) and 6.9-7.0 for K(+).

Predicting the acidity constant of a goethite hydroxyl group from first principles

Accurate predictions of the acid-base behavior of hydroxyl groups at mineral surfaces are critical for understanding the trapping of toxic and radioactive ions in soil samples. In this work, we apply ab initio molecular dynamics (AIMD) simulations and potential-of-mean-force techniques to calculate the pK(a) of a doubly protonated oxygen atom bonded to a single Fe atom (Fe(I)OH(2)) on the goethite (101) surface. Using formic acid as a reference system, pK(a) = 7.0 is predicted, suggesting that isolated, positively charged groups of this type are marginally stable at neutral pH. Similarities and differences between AIMD and the more empirical multi-site complexation methodology are highlighted, particularly with respect to the treatment of hydrogen bonding with water and proton sharing among surface hydroxyl groups. We also highlight the importance of an electronic structure method that can accurately predict transition metal ion properties for goethite pK(a) calculations.

Thermodynamics of chemical reactions with COSMO-RS: the extreme case of charge separation or recombination

Many technically relevant chemical processes in the condensed phase involve as elementary reactive steps the formation of ions from neutral species or, as the opposite, recombination of ions. Such reactions that generate or annihilate charge defy the standard gas phase quantum chemical treatment, and also continuum solvation models are only partially able to account for the right amount of stabilization in solution. In this work, for such types of reaction, a solvation treatment involving the COSMO-RS method is assessed, which leads to improved results, i.e., errors of only around 10 kJ/mol for both protic and aprotic solvents. The examples discussed here comprise protolysis reactions and organo halide heterolysis, for both of which a comparison with reliable experimental data is possible. It is observed that for protolysis, the quality of results does not strongly depend on the quantum chemical method used for energy calculation. In contrast, in the case of heterolytic carbon-chlorine bond cleavage, clearly better results are obtained for higher correlated (coupled cluster) methods or the density functional M06-2X, which is well known for its accuracy if applied to organic chemistry. This hints at least that the right answer is obtained for the right reason and not due to a compensation of errors from gas phase thermodynamics with those from the solvation treatment. Problems encountered with certain critical solvents or upon decomposing Gibbs free energies into heats or entropies of reaction are found to relate mostly to the parameterization of the H-bonding term within COSMO-RS.

Molten salt eutectics from atomistic simulations

Despite their importance for solar thermal power applications, phase-diagrams of molten salt mixture heat transfer fluids (HTFs) are not readily accessible from first principles. We present a molecular dynamics scheme general enough to identify eutectics of any HTF candidate mixture. The eutectic mixture and temperature are located using the liquid mixture free energy and the pure component solid-liquid free energy differences. The liquid mixture free energy is obtained using thermodynamic integration over particle identity transmutations sampled with molecular dynamics at a single temperature. Drawbacks of conventional phase diagram mapping methodologies are avoided by not considering solid mixtures, thereby evading expensive computations of solid phase free energies. Numerical results for binary and ternary mixtures of alkali nitrates agree well with experimental measurements.

Probing the thermodynamics of competitive ion binding using minimum energy structures

Ion binding is known to affect the properties of biomolecules and is directly involved in many biochemical pathways. Because of the highly polar environments where ions are found, a quantum-mechanical treatment is preferable for understanding the energetics of competitive ion binding. Due to computational cost, a quantum mechanical treatment may involve several approximations, however, whose validity can be difficult to determine. Using thermodynamic cycles, we show how intuitive models for complicated ion binding reactions can be built up from simplified, isolated ion-ligand binding site geometries suitable for quantum mechanical treatment. First, the ion binding free energies of individual, minimum energy structures determine their intrinsic ion selectivities. Next, the relative propensity for each minimum energy structure is determined locally from the balance of ion-ligand and ligand-ligand interaction energies. Finally, the environment external to the binding site exerts its influence both through long-ranged dispersive and electrostatic interactions with the binding site as well as indirectly through shifting the binding site compositional and structural preferences. The resulting picture unifies field-strength, topological control, and phase activation viewpoints into a single theory that explicitly indicates the important role of solute coordination state on overall reaction energetics. As an example, we show that the Na(+) → K(+) selectivities can be recovered by correctly considering the conformational contribution to the selectivity. This can be done even when constraining configuration space to the neighborhood around a single, arbitrarily chosen, minimum energy structure. Structural regions around minima for K(+)- and Na(+)-water clusters are exhibited that display both rigid/mechanical and disordered/entropic selectivity mechanisms for both Na(+) and K(+). Thermodynamic consequences of the theory are discussed with an emphasis on the role of coordination structure in determining experimental properties of ions in complex biological environments.

Path Integral Computation of Quantum Free Energy Differences Due to Alchemical Transformations Involving Mass and Potential

Thermodynamic integration, perturbation theory, and λ-dynamics methods were applied to path integral molecular dynamics calculations to investigate free energy differences due to "alchemical" transformations. Several estimators were formulated to compute free energy differences in solvable model systems undergoing changes in mass and/or potential. Linear and nonlinear alchemical interpolations were used for the thermodynamic integration. We find improved convergence for the virial estimators, as well as for the thermodynamic integration over nonlinear interpolation paths. Numerical results for the perturbative treatment of changes in mass and electric field strength in model systems are presented. We used thermodynamic integration in ab initio path integral molecular dynamics to compute the quantum free energy difference of the isotope transformation in the Zundel cation. The performance of different free energy methods is discussed.

Computing the 7Li NMR chemical shielding of hydrated Li+ using cluster calculations and time-averaged configurations from ab initio molecular dynamics simulations

Ab initio molecular dynamics (AIMD) simulations have been used to predict the time-averaged Li NMR chemical shielding for a Li(+) solution. These results are compared to NMR shielding calculations on smaller Li(+)(H(2)O)(n) clusters optimized in either the gas phase or with a polarizable continuum model (PCM) solvent. The trends introduced by the PCM solvent are described and compared to the time-averaged chemical shielding observed in the AIMD simulations where large explicit water clusters hydrating the Li(+) are employed. Different inner- and outer-coordination sphere contributions to the Li NMR shielding are evaluated and discussed. It is demonstrated an implicit PCM solvent is not sufficient to correctly model the Li shielding, and that explicit inner hydration sphere waters are required during the NMR calculations. It is also shown that for hydrated Li(+), the time averaged chemical shielding cannot be simply described by the population-weighted average of coordination environments containing different number of waters.

A method for computing the solubility limit of solids: application to sodium chloride in water and alcohols

We present an adaptable method to compute the solubility limit of solids by molecular simulation, which avoids the difficulty of reference state calculations. In this way, the method is highly adaptable to molecules of complex topology. Results are shown for solubility calculations of sodium chloride in water and light alcohols at atmospheric conditions. The pseudosupercritical path integration method is used to calculate the free energy of the solid and gives results that are in good agreement with previous studies that reference the Einstein crystal. For the solution phase calculations, the self-adaptive Wang-Landau transition-matrix Monte Carlo method is used within the context of an expanded isothermal-isobaric ensemble. The method shows rapid convergence properties and the uncertainty in the calculated chemical potential was 1% or less for all cases. The present study underpredicts the solubility limit of sodium chloride in water, suggesting a shortcoming of the molecular models. Importantly, the proper trend for the chemical potential in various solvents was captured, suggesting that relative solubilities can be computed by the method.

Cobalt-porphyrin catalyzed electrochemical reduction of carbon dioxide in water. 2. Mechanism from first principles

We apply first principles computational techniques to analyze the two-electron, multistep, electrochemical reduction of CO(2) to CO in water using cobalt porphyrin as a catalyst. Density functional theory calculations with hybrid functionals and dielectric continuum solvation are used to determine the steps at which electrons are added. This information is corroborated with ab initio molecular dynamics simulations in an explicit aqueous environment which reveal the critical role of water in stabilizing a key intermediate formed by CO(2) bound to cobalt. By use of potential of mean force calculations, the intermediate is found to spontaneously accept a proton to form a carboxylate acid group at pH < 9.0, and the subsequent cleavage of a C-OH bond to form CO is exothermic and associated with a small free energy barrier. These predictions suggest that the proposed reaction mechanism is viable if electron transfer to the catalyst is sufficiently fast. The variation in cobalt ion charge and spin states during bond breaking, DFT+U treatment of cobalt 3d orbitals, and the need for computing electrochemical potentials are emphasized.

Quasichemical and structural analysis of polarizable anion hydration

Quasichemical theory is utilized to analyze the relative roles of solute polarization and size in determining the structure and thermodynamics of bulk anion hydration for the Hofmeister series Cl(-), Br(-), and I(-). Excellent agreement with experiment is obtained for whole salt hydration free energies using the polarizable AMOEBA force field. The total hydration free energies display a stronger dependence on ion size than on polarizability. The quasichemical approach exactly partitions the solvation free energy into inner-shell, outer-shell packing, and outer-shell long-ranged contributions by means of a hard-sphere condition. The inner-shell contribution becomes slightly more favorable with increasing ion polarizability, indicating electrostriction of the nearby waters. Small conditioning radii, even well inside the first maximum of the ion-water(oxygen) radial distribution function, result in Gaussian behavior for the long-ranged contribution that dominates the ion hydration free energy. This in turn allows for a mean-field treatment of the long-ranged contribution, leading to a natural division into first-order electrostatic, induction, and van der Waals terms. The induction piece exhibits the strongest ion polarizability dependence, while the larger-magnitude first-order electrostatic piece yields an opposing but weaker polarizability dependence. The van der Waals piece is small and positive, and it displays a small ion specificity. The sum of the inner-shell, packing, and long-ranged van der Waals contributions exhibits little variation along the anion series for the chosen conditioning radii, targeting electrostatic effects (influenced by ion size) as the largest determinant of specificity. In addition, a structural analysis is performed to examine the solvation anisotropy around the anions. As opposed to the hydration free energies, the solvation anisotropy depends more on ion polarizability than on ion size: increased polarizability leads to increased anisotropy. The water dipole moments near the ion are similar in magnitude to bulk water, while the ion dipole moments are found to be significantly larger than those observed in quantum mechanical studies. Possible impacts of the observed over-polarization of the ions on simulated anion surface segregation are discussed.