A theoretical study of aqueous solvation of K comparing ab initio, polarizable, and fixed-charge models

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.

Multibody effects in ion binding and selectivity

Selective binding of ions to biomolecules plays a vital role in numerous biological processes. To understand the specific role of induced effects in selective ion binding, we use quantum chemical and pairwise-additive force-field simulations to study Na(+) and K(+) binding to various small molecules representative of ion binding functional groups in biomolecules. These studies indicate that electronic polarization significantly contributes to both absolute and relative ion-binding affinities. Furthermore, this contribution depends on both the number and the specific chemistries of the coordinating molecules, thus highlighting the complexity of ion-ligand interactions. Specifically, multibody interactions reduce as well as enhance the dipole moments of the ion-coordinating molecules, thereby affecting observables like coordination number distributions of ions. The differential polarization induced in molecules coordinating these two equivalently charged, but different-sized, ions also depends upon the number of coordinating molecules, showing the importance of multibody effects in distinguishing these ions thermodynamically. Because even small differences in ionic radii (0.4 Å for Na(+) and K(+)) produce differential polarization trends critical to distinguishing ions thermodynamically, it is likely that polarization plays an important role in thermodynamically distinguishing other ions and charged chemical and biological functional groups.

High-performance scalable molecular dynamics simulations of a polarizable force field based on classical Drude oscillators in NAMD

Incorporating the influence of induced polarization in large-scale atomistic molecular dynamics (MD) simulations is a critical challenge in the progress toward computations of increased accuracy. One computationally efficient treatment is based on the classical Drude oscillator in which an auxiliary charged particle is attached by a spring to each nucleus. Here, we report the first implementation of this model in the program NAMD. An extended Lagrangian dynamics with a dual-Langevin thermostat scheme applied to the Drude-nucleus pairs is employed to efficiently generate classical dynamic propagation near the self-consistent field limit. Large-scale MD simulations based on the Drude polarizable force field scale very well on massively distributed supercomputing platforms, the computational demand being only about 50-100% higher than for nonpolarizable models. As an illustration, a large-scale 150 mM NaCl aqueous salt solution is simulated, and the calculated ionic conductivity is shown to be in excellent agreement with experiment.

A combined experimental and theoretical study of ion solvation in liquid N-methylacetamide

Most current biomolecular simulations are based on potential energy functions that treat the electrostatic energy as a sum of pairwise Coulombic interactions between effective fixed atomic charges. This approximation, in which many-body induced polarization effects are included in an average way, is expected to be satisfactory for a wide range of systems, but less accurate for processes involving the transfer and partition of ions among heterogeneous environments. The limitations of these potential energy functions are perhaps most obvious in studies of ion permeation through membrane channels. In many cases, the pore is so narrow that the permeating ion must shed most of its surrounding water molecules and the large energetic loss due to dehydration must be compensated by coordination with protein atoms. Interactions of cations with protein backbone carbonyl oxygens, in particular, play a critical role in several important biological channels. As a first step toward meeting the challenge of developing an accurate explicit accounting for induced polarization effects, the present work combines experiments and computation to characterize the interactions of alkali and halide ions with N-methylacetamide chosen to represent the peptide bond. From solubility measurements, we extract the solvation free energies of KCl and NaCl in liquid N-methylacetamide. Polarizable models based on the Drude oscillator are then developed and compared with available experimental and ab initio data. The good agreement for a range of structural and thermodynamic properties in the gas and condensed phases suggests that the polarizable models provide an accurate representation of ion-amide interactions in biological systems.

Assessing the accuracy of approximate treatments of ion hydration based on primitive quasichemical theory

Quasichemical theory (QCT) provides a framework that can be used to partition the influence of the solvent surrounding an ion into near and distant contributions. Within QCT, the solvation properties of the ion are expressed as a sum of configurational integrals comprising only the ion and a small number of solvent molecules. QCT adopts a particularly simple form if it is assumed that the clusters undergo only small thermal fluctuations around a well-defined energy minimum and are affected exclusively in a mean-field sense by the surrounding bulk solvent. The fluctuations can then be integrated out via a simple vibrational analysis, leading to a closed-form expression for the solvation free energy of the ion. This constitutes the primitive form of quasichemical theory (pQCT), which is an approximate mathematical formulation aimed at reproducing the results from the full many-body configurational averages of statistical mechanics. While the results from pQCT from previous applications are reasonable, the accuracy of the approach has not been fully characterized and its range of validity remains unclear. Here, a direct test of pQCT for a set of ion models is carried out by comparing with the results of free energy simulations with explicit solvent. The influence of the distant surrounding bulk on the cluster comprising the ion and the nearest solvent molecule is treated both with a continuum dielectric approximation and with free energy perturbation molecular dynamics simulations with explicit solvent. The analysis shows that pQCT can provide an accurate framework in the case of a small cation such as Li(+). However, the approximation encounters increasing difficulties when applied to larger cations such as Na(+), and particularly for K(+). This suggests that results from pQCT should be interpreted with caution when comparing ions of different sizes.

Coordination numbers of K(+) and Na(+) Ions inside the selectivity filter of the KcsA potassium channel: insights from first principles molecular dynamics

Quantum mechanics/molecular mechanics (QM/MM) Car-Parrinello simulations were performed to estimate the coordination numbers of K(+) and Na(+) ions in the selectivity filter of the KcsA channel, and in water. At the DFT/BLYP level, K(+) ions were found to display an average coordination number of 6.6 in the filter, and 6.2 in water. Na(+) ions displayed an average coordination number of 5.2 in the filter, and 5.0 in water. A comparison was made with the average coordination numbers obtained from using classical molecular dynamics (6.7 for K(+) in the filter, 6.6 for K(+) in water, 6.0 for Na(+) in the filter, and 5.2 for Na(+) in water). The observation that different coordination numbers were displayed by the ions in QM/MM simulations and in classical molecular dynamics is relevant to the discussion of selectivity in K-channels.

Structure and dynamics of the hydration shells of the Zn(2+) ion from ab initio molecular dynamics and combined ab initio and classical molecular dynamics simulations

Results of ab initio molecular dynamics (AIMD) simulations (density functional theory+PBE96) of the dynamics of waters in the hydration shells surrounding the Zn(2+) ion (T approximately 300 K, rho approximately 1 gm/cm(3)) are compared to simulations using a combined quantum and classical molecular dynamics [AIMD/molecular mechanical (MM)] approach. Both classes of simulations were performed with 64 solvating water molecules ( approximately 15 ps) and used the same methods in the electronic structure calculation (plane-wave basis set, time steps, effective mass, etc.). In the AIMD/MM calculation, only six waters of hydration were included in the quantum mechanical (QM) region. The remaining 58 waters were treated with a published flexible water-water interaction potential. No reparametrization of the water-water potential was attempted. Additional AIMD/MM simulations were performed with 256 water molecules. The hydration structures predicted from the AIMD and AIMD/MM simulations are found to agree in detail with each other and with the structural results from x-ray data despite the very limited QM region in the AIMD/MM simulation. To further evaluate the agreement of these parameter-free simulations, predicted extended x-ray absorption fine structure (EXAFS) spectra were compared directly to the recently obtained EXAFS data and they agree in remarkable detail with the experimental observations. The first hydration shell contains six water molecules in a highly symmetric octahedral structure is (maximally located at 2.13-2.15 A versus 2.072 A EXAFS experiment). The widths of the peak of the simulated EXAFS spectra agree well with the data (8.4 A(2) versus 8.9 A(2) in experiment). Analysis of the H-bond structure of the hydration region shows that the second hydration shell is trigonally bound to the first shell water with a high degree of agreement between the AIMD and AIMD/MM calculations. Beyond the second shell, the bonding pattern returns to the tetrahedral structure of bulk water. The AIMD/MM results emphasize the importance of a quantum description of the first hydration shell to correctly describe the hydration region. In these calculations the full d(10) electronic structure of the valence shell of the Zn(2+) ion is retained. The simulations show substantial and complex charge relocation on both the Zn(2+) ion and the first hydration shell. The dipole moment of the waters in the first hydration shell is 3.4 D (3.3 D AIMD/MM) versus 2.73 D bulk. Little polarization is found for the waters in the second hydration shell (2.8 D). No exchanges were seen between the first and the second hydrations shells; however, many water transfers between the second hydration shell and the bulk were observed. For 64 waters, the AIMD and AIMD/MM simulations give nearly identical results for exchange dynamics. However, in the larger particle simulations (256 waters) there is a significant reduction in the second shell to bulk exchanges.

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.

On the utilization of energy minimization to the study of ion selectivity

The major pitfalls in studying ion selectivity in binding site models using energy minimization based methods are examined and discussed. It is shown that the properties derived from energy minimization are strongly configuration-dependent and that the results should be interpreted with caution. It is concluded that computational studies of ion selectivity must include thermal fluctuations and entropic effects.

Simulating Monovalent and Divalent Ions in Aqueous Solution Using a Drude Polarizable Force Field

An accurate representation of ion solvation in aqueous solution is critical for meaningful computer simulations of a broad range of physical and biological processes. Polarizable models based on classical Drude oscillators are introduced and parametrized for a large set of monoatomic ions including cations of the alkali metals (Li(+), Na(+), K(+), Rb(+) and Cs(+)) and alkaline earth elements (Mg(2+), Ca(2+), Sr(2+) and Ba(2+)) along with Zn(2+) and halide anions (F(-), Cl(-), Br(-) and I(-)). The models are parameterized, in conjunction with the polarizable SWM4-NDP water model [Lamoureux et al., Chem. Phys. Lett. 418, 245 (2006)], to be consistent with a wide assortment of experimentally measured aqueous bulk thermodynamic properties and the energetics of small ion-water clusters. Structural and dynamic properties of the resulting ion models in aqueous solutions at infinite dilution are presented.

Molecular modeling and dynamics studies with explicit inclusion of electronic polarizability. Theory and applications

A current emphasis in empirical force fields is on the development of potential functions that explicitly treat electronic polarizability. In the present article, the commonly used methodologies for modelling electronic polarization are presented along with an overview of selected application studies. Models presented include induced point-dipoles, classical Drude oscillators, and fluctuating charge methods. The theoretical background of each method is followed by an introduction to extended Langrangian integrators required for computationally tractable molecular dynamics simulations using polarizable force fields. The remainder of the review focuses on application studies using these methods. Emphasis is placed on water models, for which numerous examples exist, with a more thorough discussion presented on the recently published models associated with the Drude-based CHARMM and the AMOEBA force fields. The utility of polarizable models for the study of ion solvation is then presented followed by an overview of studies of small molecules (e.g. CCl(4), alkanes, etc) and macromolecule (proteins, nucleic acids and lipid bilayers) application studies. The review is written with the goal of providing a general overview of the current status of the field and to facilitate future application and developments.

Determinants of K+ vs Na+ selectivity in potassium channels

Ion channels, specialized pore-forming proteins, are an indispensable component of the nervous system and play a crucial role in regulating cardiac, skeletal, and smooth muscle contraction. Potassium ion channels, controlling the action potential of a number of excitable cells, are characterized by a remarkable ability to select K(+) over Na(+). Although the molecular basis for this striking ion selectivity has been a subject of extensive investigations using both experimental and theoretical methods, the following outstanding questions remain: (a) To what extent is the number of water molecules bound to the permeating ion (i.e., the hydration number) important for the K(+)/Na(+) competition? (b) Are the chemical type and number of coordinating groups lining the pore critical for the selectivity process? (c) Apart from providing cation-ligating groups, do the channel walls play any other role in the selectivity process? This work reveals that the pore's selectivity for K(+) over Na(+) increases with (i) increasing hydration number of K(+) relative to that of Na(+), (ii) increasing number of K(+)-coordinating dipoles, (iii) increasing number of Na(+)-coordinating dipoles, and (iv) decreasing magnitude of the coordinating dipoles provided by the pore. Thus, a high K(+)/Na(+) selectivity in K(+) channels could be achieved from a combination of several favorable factors involving the native ion, the metal-coordinating ligands, and the protein matrix, viz., (a) an octahydrated permeating K(+), (b) a pore lined with 8 carbonyl ligands, and (c) finely tuned physicomechanical properties of the channel walls providing a low dielectric medium favoring a high hydration number for the permeating K(+) and enough stiffness to force the competing Na(+) to adopt an unfavorable 8-fold coordination. This implies that optimal K(+)/Na(+) selectivity in K(+) channels generally does not arise from solely structural or energetic consideration. The factors affecting ion selectivity revealed herein help to rationalize why valinomycin and the KcsA ion channels are highly K(+)-selective, whereas the NaK channel is nonselective. The calculations predict that other pores containing a different number/chemical type of coordinating groups from those observed in potassium channels could also select K(+) over Na(+).

Trypsin-ligand binding free energies from explicit and implicit solvent simulations with polarizable potential

We have calculated the binding free energies of a series of benzamidine-like inhibitors to trypsin with a polarizable force field using both explicit and implicit solvent approaches. Free energy perturbation has been performed for the ligands in bulk water and in protein complex with molecular dynamics simulations. The binding free energies calculated from explicit solvent simulations are well within the accuracy of experimental measurement and the direction of change is predicted correctly in all cases. We analyzed the molecular dipole moments of the ligands in gas, water and protein environments. Neither binding affinity nor ligand solvation free energy in bulk water shows much dependence on the molecular dipole moments of the ligands. Substitution of the aromatic or the charged group in the ligand results in considerable change in the solvation energy in bulk water and protein whereas the binding affinity varies insignificantly due to cancellation. The effect of chemical modification on ligand charge distribution is mostly local. Replacing benzene with diazine has minimal impact on the atomic multipoles at the amidinium group. We have also utilized an implicit solvent based end-state approach to evaluate the binding free energies of these inhibitors. In this approach, the polarizable multipole model combined with Poisson-Boltzmann/surface area (PMPB/SA) provides the electrostatic interaction energy and the polar solvation free energy. Overall the relative binding free energies obtained from the MM-PMPB/SA model are in good agreement with the experimental data.

Thermodynamically dominant hydration structures of aqueous ions

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.

Hydration number, topological control, and ion selectivity

The topological control hypothesis presented by Bostick and Brooks [Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 9260] has sought to explain binding selectivity in potassium channels based on the premise that a universal measure of ion solvation in different environments is provided by its average coordination structure in bulk water. This leads to the view that ion selectivity is predominantly controlled by the number of ligands coordinating the ion and that the chemical type of those ligands plays a minor role. The significance of the topological control hypothesis and its ability to predict ion selectivity in protein binding sites are examined. It is shown that the framework encounters increasing difficulties when different protein binding sites with similar coordination numbers are considered.

Ab initio molecular dynamics calculations of ion hydration free energies

We apply ab initio molecular dynamics (AIMD) methods in conjunction with the thermodynamic integration or "lambda-path" technique to compute the intrinsic hydration free energies of Li(+), Cl(-), and Ag(+) ions. Using the Perdew-Burke-Ernzerhof functional, adapting methods developed for classical force field applications, and with consistent assumptions about surface potential (phi) contributions, we obtain absolute AIMD hydration free energies (DeltaG(hyd)) within a few kcal/mol, or better than 4%, of Tissandier et al.'s [J. Phys. Chem. A 102, 7787 (1998)] experimental values augmented with the SPC/E water model phi predictions. The sums of Li(+)/Cl(-) and Ag(+)/Cl(-) AIMD DeltaG(hyd), which are not affected by surface potentials, are within 2.6% and 1.2 % of experimental values, respectively. We also report the free energy changes associated with the transition metal ion redox reaction Ag(+)+Ni(+)-->Ag+Ni(2+) in water. The predictions for this reaction suggest that existing estimates of DeltaG(hyd) for unstable radiolysis intermediates such as Ni(+) may need to be extensively revised.

Assessing the role of polarization in docking

We describe a strategy for including ligand and protein polarization in docking that is based on the conversion of induced dipoles to induced charges. Induced charges have a distinct advantage in that they are readily implemented into a number of different computer programs, including many docking programs and hybrid QM/MM programs; induced charges are also more readily interpreted. In this study, the ligand was treated quantum mechanically to avoid parametrization issues and was polarized by the target protein, which was treated as a set of point charges. The induced dipole at a given target atom, due to polarization by the ligand and neighboring residues, was reformulated as induced charges at the given atom and its bonded neighbors, and these were allowed to repolarize the ligand in an iterative manner. The final set of polarized charges was evaluated in docking using AutoDock 4.0 on 12 protein-ligand systems against the default empirical Gasteiger charges, and against nonpolarized and partially polarized potential-derived charges. One advantage of AutoDock is that the best rmsd structure can be identified not only from the lowest energy pose but also from the largest cluster of poses. Inclusion of polarization does not always lead to the lowest energy pose having a lower rmsd, because docking is designed by necessity to be rapid rather than accurate. However, whenever an improvement in methodology, corresponding to a more thorough treatment of polarization, resulted in an increased cluster size, then there was also a corresponding decrease in the rmsd. The options for implementing polarization within a purely classical docking framework are discussed.

Many-body effects and simulations of potassium channels

The electronic polarizability of an ion or a molecule is a measure of the relative tendency of its electron cloud to be distorted from its normal shape by an electric field. On the molecular scale, in a condensed phase, any species sits in an electric field due to its neighbours, and the resulting polarization is an important contribution to the total interaction energy. Electrostatic interactions are crucial for determining the majority of chemical–physical properties of the system and electronic polarization is a fundamental component of these interactions. Thus, polarization effects should be taken into account if accurate descriptions are desired. In classical computer simulations, the forces required to drive the system are typically based on interatomic interaction potentials derived in part from electronic structure calculations or from experimental data. Owing to the difficulties in including polarization effects in classical force fields, most of them are based just on pairwise additive interaction potentials. At present, major efforts are underway to develop polarizable interaction potentials for biomolecular simulations. In this review, various ways of introducing explicit polarizability into biomolecular models and force fields are reviewed, and the progress that might be achieved in applying such methods to study potassium channels is described.