Incorporating variable dielectric environments into the generalized Born model

A generalized Born (GB) model is proposed that approximates the electrostatic part of macromolecular solvation free energy over the entire range of the solvent and solute dielectric constants. The model contains no fitting parameters, and is derived by matching a general form of the GB Green function with the exact Green's function of the Poisson equation for a random charge distribution inside a perfect sphere. The sphere is assumed to be filled uniformly with dielectric medium epsilon(in), and is surrounded by infinite solvent of constant dielectric epsilon(out). This model is as computationally efficient as the conventional GB model based on the widely used functional form due to Still et al. [J. Am. Chem. Soc. 112, 6127 (1990)], but captures the essential physics of the dielectric response for all values of epsilon(in) and epsilon(out). This model is tested against the exact solution on a perfect sphere, and against the numerical Poisson-Boltzmann (PB) treatment on a set of macromolecules representing various structural classes. It shows reasonable agreement with both the exact and the numerical solutions of the PB equation (where available) considered as reference, and is more accurate than the conventional GB model over the entire range of dielectric values.     

Parameterization of the Hamiltonian Dielectric Solvent (HADES) Reaction‐Field Method for the Solvation Free Energies of Amino Acid Side‐Chain Analogs

Optimization of the Hamiltonian dielectric solvent (HADES) method for biomolecular simulations in a dielectric continuum is presented with the goal of calculating accurate absolute solvation free energies while retaining the model’s accuracy in predicting conformational free‐energy differences. The solvation free energies of neutral and polar amino acid side‐chain analogs calculated by using HADES, which may optionally include nonpolar contributions, were optimized against experimental data to reach a chemical accuracy of about 0.5 kcal mol^?1. The new parameters were evaluated for charged side‐chain analogs. The HADES results were compared with explicit‐solvent, generalized Born, Poisson–Boltzmann, and QM‐based methods. The potentials of mean force (PMFs) between pairs of side‐chain analogs obtained by using HADES and explicit‐solvent simulations were used to evaluate the effects of the improved parameters optimized for solvation free energies on intermolecular potentials.     

A classical point charge model study of system size dependence of oxidation and reorganization free energies in aqueous solution

The response of water to a change of charge of a solvated ion is, to a good approximation, linear for the type of iron-like ions frequently used as a model system in classical force field studies of electron transfer. Free energies for such systems can be directly calculated from average vertical energy gaps. Exploiting this feature, we have computed the free energy and the reorganization energy of the M2+/M3+ and M1+/M2+ oxidations in a series of model systems all containing a single Mn+ ion and an increasing number of simple point charge water molecules. Long-range interactions are taken into account by Ewald summation methods. Our calculations confirm the observation made by Hummer, Pratt, and Garcia (J. Phys. Chem. 1996, 100, 1206) that the finite size correction to the estimate of solvation energy (and hence oxidation free energy) in such a setup is effectively proportional to the inverse third power (1/L3) of the length L of the periodic cell. The finite size correction to the reorganization energy is found to scale with 1/L. These simulation results are analyzed using a periodic generalization of the Born cavity model for solvation, yielding three different estimates of the cavity radius, namely, from the infinite system size extrapolation of oxidation free energy and reorganization energy, and from the slope of the linear dependence of oxidation free energy on 1/L3. The cavity radius for the reorganization energy is found to be significantly larger compared to the radius for the oxidation (solvation) free energy. The radius controlling the 1/L3 dependence of oxidation free energy is found to be comparable to the radius for reorganization. The implication of these results for density functional theory-based ab initio molecular dynamics calculation of redox potentials is discussed.     

A self-consistent reaction field approach to liquid photoionization

A cluster-dielectric model with a self-consistent reaction field interaction is proposed and evaluated for liquid photoionization. The model is numerically applied to solvent binding energy shifts for some cations and anions in aqueous solutions. The role of medium interaction with the ion or with the ion dressed with its first solvation shell cluster is investigated by means of a self-consistent Hatree-Fock method employing a reaction field Fock operator. Total binding energy shifts and orbital energy shifts are obtained as functions of the dielectric constant. It is shown that for the range of values of the dielectric constant corresponding to the most common solvents, variation in the shifts results from structural effects rather than from pure dielectric effects. It is found that the Born approximation, which is poor for the prediction of the BE shift for the pure ion, works well for cations dressed with it first solvation shell cluster. For anions the model with only one solvation shell in addition to the dielectric is not as appropriate as for cations, which can be argued from comparison of present data with experimental and simulation data. The relation between the present method and a previously devised statistical method is established.     

A generalized Born formalism for heterogeneous dielectric environments: application to the implicit modeling of biological membranes

Reliable computer simulations of complex biological environments such as integral membrane proteins with explicit water and lipid molecules remain a challenging task. We propose a modification of the standard generalized Born theory of homogeneous solvent for modeling the heterogeneous dielectric environments such as lipid/water interfaces. Our model allows the representation of biological membranes in the form of multiple layered dielectric regions with dielectric constants that are different from the solute cavity. The proposed new formalism is shown to predict the electrostatic component of solvation free energy with a relative error of 0.17% compared to exact finite-difference solutions of the Poisson equation for a transmembrane helix test system. Molecular dynamics simulations of melittin and bacteriorhodopsin are carried out and performed over 10 ns and 7 ns of simulation time, respectively. The center of melittin along the membrane normal in these stable simulations is in excellent agreement with the relevant experimental data. Simulations of bacteriorhodopsin started from the experimental structure remained stable and in close agreement with experiment. We also examined the free energy profiles of water and amino acid side chain analogs upon membrane insertion. The results with our implicit membrane model agree well with the experimental transfer free energy data from cyclohexane to water as well as explicit solvent simulations of water and selected side chain analogs.     

Calculation of absolute ligand binding free energy to a ribosome-targeting protein as a function of solvent model

A comparative analysis is provided of the effect of different solvent models on the calculation of a potential of mean force (PMF) for determining the absolute binding affinity of the small molecule inhibitor pteroic acid bound to ricin toxin A-chain (RTA). Solvent models include the distance-dependent dielectric constant, several different generalized Born (GB) approximations, and a hybrid explicit/GB-based implicit solvent model. We found that the simpler approximation of dielectric screening and a GB model, with Born radii fitted to a switching-window dielectric-boundary surface Poisson solvent model, severely overpredicted the binding affinity as compared to the experimental value, estimated to range from -4.4 to -6.0 kcal/mol. In contrast, GB models that are parametrized to fit the Lee-Richards molecular surface performed much better, predicting binding free energy within 1-3 kcal/mol of experimental estimates. However, the predicted free-energy profiles of these GB models displayed alternative binding modes not observed in the crystal structure. Finally, the most rigorous and computationally costly approach in this work, which used a hybrid explicit/implicit solvent model, correctly determined a binding funnel in the PMF near the crystallographic bound state and predicted an absolute binding affinity that was 2 kcal/mol more favorable than the estimated experimental binding affinity.     

Use of the dielectric sphere model for the calculation of the effect of a solvent on the kinetics of chemical reactions with charge transer. Communication 1. Energy of reorganization of a polar medium

Conclusions The energy of the reorganization of the solvent in charge transfer reactions was calculated for the dielectric sphere model using the precise solution of the corresponding electrostatic problem. The numerical results are in good agreement with the previously obtained approximate analytic equation.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 3, pp. 613–621, March, 1985.     

The importance of excluded solvent volume effects in computing hydration free energies

Continuum dielectric methods such as the Born equation have been widely used to compute the electrostatic component of the solvation free energy, DeltaG(solv)(elec), because they do not need to include solvent molecules explicitly and are thus far less costly compared to molecular simulations. All of these methods can be derived from Gauss Law of Maxwell's equations, which yields an analytical solution for the solvation free energy, DeltaG(Born), when the solute is spherical. However, in Maxwell's equations, the solvent is assumed to be a structureless continuum, whereas in reality, the near-solute solvent molecules are highly structured unlike far-solute bulk solvent. Since we have recently reformulated Gauss Law of Maxwell's equations to incorporate the near-solute solvent structure by considering excluded solvent volume effects, we have used it in this work to derive an analytical solution for the hydration free energy of an ion. In contrast to continuum solvent models, which assume that the normalized induced solvent electric dipole density P(n) is constant, P(n) mimics that observed from simulations. The analytical formula for the ionic hydration free energy shows that the Born radius, which has been used as an adjustable parameter to fit experimental hydration free energies, is no longer ill defined but is related to the radius and polarizability of the water molecule, the hydration number, and the first peak position of the solute-solvent radial distribution function. The resulting DeltaG(solv)(elec) values are shown to be close to the respective experimental numbers.     

Thermodynamics of individual ions in ethylene glycol-water solvents

A new equation, correlating the cell (or electrode) potential with the dielectric constant of the solvent, has been developed and used to compute the chemical contribution to the transfer thermodynamic quantities of individual ions in various solvents. The results show that the electrostatic contribution to the transfer free energies should in fact account for all the interactions between the charge on the ion and the overall charge on the solvent molecules, of which the Born contribution plays but a minor role. The thermodynamic properties of individual ions have been discussed in the light of ion-solvent interactions as well as the structural effects of the solvents on the transfer process.     

Solvent Reorganization Energy and Electronic Coupling for Intramolecular Electron Transfer in Biphenyl-Acceptor Anion Radicals

A novel algorithm was designed and implemented to realize the numerical calculation of the solvent reorganization energy for electron transfer reactions, on the basis of nonequilibrium solvation theory and the dielectric polarizable continuum model. Applying the procedure to the well-investigated intramolecular electron transfer in biphenyl-androstane-naphthyl and biphenyl-androstane-phenanthryl systems, the numerical results of solvent reorganization energy were determined to be around 60 kJ/mol, in good agreement with experimental datKoopman's theorem was adopted for the calculation of the electron transfer coupling element, associated with the linear reaction coordinate approximation. The values for this quantity obtained are acceptable when compared with experimental results.     

Dispersion self-free energies and interaction free energies of finite-sized ions in salt solutions

The role for many-body dipolar (dispersion) potentials in ion-solvent and ion-solvent-interface interactions is explored. Such many-body potentials, accessible in principle from measured dielectric data, are necessary in accounting for Hofmeister specific ion effects. Dispersion self-energy is the quantum electrodynamic analogue of the Born electrostatic self-energy of an ion. We here describe calculations of dispersion self-free energies of four different anions (OH-, Cl-, Br-, and I-) that take finite ion size into account. Three different examples of self-free energy calculations are presented. These are the self-free energy of transfer of an ion to bulk solution, which influences solubility; the dispersion potential acting between one ion and an air-water interface (important for surface tension calculations); and the dispersion potential acting between two ions (relevant to activity coefficient calculations). To illustrate the importance of dispersion self-free energies, we compare the Born and dispersion contributions to the free energy of ion transfer from water to air (oil). We have also calculated the change in interfacial tension with added salt for air (oil)-water interfaces. A new model is used that includes dispersion potentials acting on the ions near the interface, image potentials, and ions of finite size that are allowed to spill over the solution-air interface. It is shown that interfacial free energies require a knowledge of solvent profiles at the interface.     

非平衡溶剂化的连续介质模型和超快过程溶剂效应

在连续介质理论基础上, 根据Jackson的能量积分公式导出非平衡态静电自由能和溶剂化能的正确表达式. 引入“弹簧能”概念, 对平衡态和非平衡态的静电能构成给出了合理解释, 即此能量由溶质自由电荷和溶剂极化电荷的自能、 两者之间的相互作用能和极化电荷的“弹簧能”构成. 对目前几种代表性的非平衡溶剂化理论进行了论证和比较, 指出其中存在的基本理论问题. 根据新的非平衡溶剂化能建立了电子转移反应溶剂重组能的双球模型、 光谱移动的单球孔穴点偶极模型, 多级展开方法和非平衡溶剂效应的数值解方法.在Poisson方程求解中引入类导体屏蔽模型, 建立了任意孔穴极化电荷数值解方法并应用到Closs-Miller电子转移体系, 得到与实验值吻合的溶剂重组能, 解决了传统非平衡溶剂化理论高估溶剂重组能的问题.     

Proton binding to proteins: pK(a) calculations with explicit and implicit solvent models

Ionizable residues play important roles in protein structure and activity, and proton binding is a valuable reporter of electrostatic interactions in these systems. We use molecular dynamics free energy simulations (MDFE) to compute proton pKa shifts, relative to a model compound in solution, for three aspartate side chains in two proteins. Simulations with explicit solvent and with an implicit, dielectric continuum solvent are reported. The implicit solvent simulations use the generalized Born (GB) model, which provides an approximate, analytical solution to Poisson's equation. With explicit solvent, the direction of the pKa shifts is correct in all three cases with one force field (AMBER) and in two out of three cases with another (CHARMM). For two aspartates, the dielectric response to ionization is found to be linear, even though the separate protein and solvent responses can be nonlinear. For thioredoxin Asp26, nonlinearity arises from the presence of two substates that correspond to the two possible orientations of the protonated carboxylate. For this side chain, which is partly buried and has a large pKa upshift, very long simulations are needed to correctly sample several slow degrees of freedom that reorganize in response to the ionization. Thus, nearby Lys57 rotates to form a salt bridge and becomes buried, while three waters intercalate along the opposite edge of Asp26. Such strong and anisotropic reorganization is very difficult to predict with Poisson-Boltzmann methods that only consider electrostatic interactions and employ a single protein structure. In contrast, MDFE with a GB dielectric continuum solvent, used for the first time for pKa calculations, can describe protein reorganization accurately and gives encouraging agreement with experiment and with the explicit solvent simulations.     

Implicit solvation based on generalized Born theory in different dielectric environments

In this paper we are investigating the effect of the dielectric environment on atomic Born radii used in generalized Born (GB) methods. Motivated by the Kirkwood expression for the reaction field of a single off-center charge in a spherical cavity, we are proposing extended formalisms for the calculation of Born radii as a function of external and internal dielectric constants. We demonstrate that reaction field energies calculated from environmentally dependent Born radii lead to much improved agreement with Poisson-Boltzmann solutions for low dielectric external environments, such as biological membranes or organic solvent, compared to previous methods where the calculation of Born radii does not depend on the environment. We also examine how this new approach can be applied for the calculation of transfer free energies from vacuum to a given external dielectric for a system with an internal dielectric larger than one. This has not been possible with standard GB theory but is relevant when scoring minimized or average structures with implicit solvent.     

Mathematical analysis of the boundary-integral based electrostatics estimation approximation for molecular solvation: exact results for spherical inclusions

We analyze the mathematically rigorous BIBEE (boundary-integral based electrostatics estimation) approximation of the mixed-dielectric continuum model of molecular electrostatics, using the analytically solvable case of a spherical solute containing an arbitrary charge distribution. Our analysis, which builds on Kirkwood's solution using spherical harmonics, clarifies important aspects of the approximation and its relationship to generalized Born models. First, our results suggest a new perspective for analyzing fast electrostatic models: the separation of variables between material properties (the dielectric constants) and geometry (the solute dielectric boundary and charge distribution). Second, we find that the eigenfunctions of the reaction-potential operator are exactly preserved in the BIBEE model for the sphere, which supports the use of this approximation for analyzing charge-charge interactions in molecular binding. Third, a comparison of BIBEE to the recent GBε theory suggests a modified BIBEE model capable of predicting electrostatic solvation free energies to within 4% of a full numerical Poisson calculation. This modified model leads to a projection-framework understanding of BIBEE and suggests opportunities for future improvements.     

A modified two-sphere model for solvent reorganization energy in electron transfer

In this work, the solvent reorganization energy is formulated within the framework of classical thermodynamics, by adding some external charges to construct a constrained equilibrium state. The derivation clearly shows that the reorganization energy is exactly the polarization cost for the inertial part of the polarization. We perform our derivation just within the framework of the first law of thermodynamics, and the final form of the reorganization energy is completely the same as that we gave in our recent work by defining a nonequilibrium solvation free energy. With the two-sphere model approximation, our solvent reorganization energy is derived as λ(0) = Δq(2)/2[1/r(D) + 1/r(A) - 2/d][(ε(-1)(op) - ε(-1)(s))/(1 - ε(-1)(s))]. This amends Marcus' model by a factor of (ε(-1)(op) - ε(-1)(s))/(1 - ε(-1)(s)), which is coupled with the solvent polarity. Making use of the modified expression of solvent reorganization energy, two recently reported electron transfer processes are investigated in representative solvents. The results show that our formula can well reproduce the experimental observations.     

Ions in a binary asymmetric dipolar mixture: mole fraction dependent Born energy of solvation and partial solvent polarization structure

Mean spherical approximation (MSA) for electrolyte solution has been extended to investigate the role of partial solvent polarization densities around an ion in a completely asymmetric binary dipolar mixture. The differences in solvent diameters, dipole moments, and ionic size are incorporated systematically within the MSA framework in the present theory for the first time. In addition to the contributions due to difference in dipole moments, the solvent-solvent and ion-solvent size ratios are found to significantly affect the nonideality in binary dipolar mixtures. Subsequently, the theory is used to investigate the role of ion-solvent and solvent-solvent size ratios in determining the nonideality in Born free energy of solvation of a unipositive rigid ion in alcohol-water and dimethyl sulfoxide-acetonitrile mixtures, where the solvent components are represented only by their molecular diameters and dipole moments. Nonideality in Born free energy of solvation in such simplified mixtures is found to be stronger for smaller ions. The slope of the nonideality for smaller alkali metal ions in methanol-water mixture is found to be opposite to that for larger ion, such as quaternary tertiary butyl ammonium ion. For ethanol-water mixtures, the slopes are in the same direction for all the ions studied here. These results are in qualitative agreement with experiments, which is surprising as the present MSA approach does not include the hydrogen bonding and hydrophobic interactions present in the real mixtures. The calculated partial polarization densities around a unipositive ion also show the characteristic deviation from ideality and reveal the microscopic origin of the ion and solvent size dependent preferential solvation. Also, the excess free energy of mixing (in the absence of any ion) for these binary mixtures has been calculated and a good agreement between theory and experiment has been found.