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In Paper I of this work [S. Bauer, G. Mathias, and P. Tavan, J. Chem. Phys.140, 104102 (2014)] we have presented a reaction field (RF) method, which accurately solves the Poisson equation for proteins embedded in dielectric solvent continua at a computational effort comparable to that of polarizable molecular mechanics (MM) force fields. Building upon these results, here we suggest a method for linearly scaling Hamiltonian RF/MM molecular dynamics (MD) simulations, which we call "Hamiltonian dielectric solvent" (HADES). First, we derive analytical expressions for the RF forces acting on the solute atoms. These forces properly account for all those conditions, which have to be self-consistently fulfilled by RF quantities introduced in Paper I. Next we provide details on the implementation, i.e., we show how our RF approach is combined with a fast multipole method and how the self-consistency iterations are accelerated by the use of the so-called direct inversion in the iterative subspace. Finally we demonstrate that the method and its implementation enable Hamiltonian, i.e., energy and momentum conserving HADES-MD, and compare in a sample application on Ac-Ala-NHMe the HADES-MD free energy landscape at 300 K with that obtained in Paper I by scanning of configurations and with one obtained from an explicit solvent simulation. [ABSTRACT FROM AUTHOR]

In the preceding paper by Stork and Tavan, [J. Chem. Phys. 126, 165105 (2007)], the authors have reformulated an electrostatic theory which treats proteins surrounded by dielectric solvent continua and approximately solves the associated Poisson equation [B. Egwolf and P. Tavan, J. Chem. Phys. 118, 2039 (2003)]. The resulting solution comprises analytical expressions for the electrostatic reaction field (RF) and potential, which are generated within the protein by the polarization of the surrounding continuum. Here the field and potential are represented in terms of Gaussian RF dipole densities localized at the protein atoms. Quite like in a polarizable force field, also the RF dipole at a given protein atom is induced by the partial charges and RF dipoles at the other atoms. Based on the reformulated theory, the authors have suggested expressions for the RF forces, which obey Newton’s third law. Previous continuum approaches, which were also built on solutions of the Poisson equation, used to violate the reactio principle required by this law, and thus were inapplicable to molecular dynamics (MD) simulations. In this paper, the authors suggest a set of techniques by which one can surmount the few remaining hurdles still hampering the application of the theory to MD simulations of soluble proteins and peptides. These techniques comprise the treatment of the RF dipoles within an extended Lagrangian approach and the optimization of the atomic RF polarizabilities. Using the well-studied conformational dynamics of alanine dipeptide as the simplest example, the authors demonstrate the remarkable accuracy and efficiency of the resulting RF-MD approach. [ABSTRACT FROM AUTHOR]

We present a reaction field (RF) method which accurately solves the Poisson equation for proteins embedded in dielectric solvent continua at a computational effort comparable to that of an electrostatics calculation with polarizable molecular mechanics (MM) force fields. The method combines an approach originally suggested by Egwolf and Tavan [J. Chem. Phys.118, 2039 (2003)] with concepts generalizing the Born solution [Z. Phys.1, 45 (1920)] for a solvated ion. First, we derive an exact representation according to which the sources of the RF potential and energy are inducible atomic anti-polarization densities and atomic shielding charge distributions. Modeling these atomic densities by Gaussians leads to an approximate representation. Here, the strengths of the Gaussian shielding charge distributions are directly given in terms of the static partial charges as defined, e.g., by standard MM force fields for the various atom types, whereas the strengths of the Gaussian anti-polarization densities are calculated by a self-consistency iteration. The atomic volumes are also described by Gaussians. To account for covalently overlapping atoms, their effective volumes are calculated by another self-consistency procedure, which guarantees that the dielectric function ε(r) is close to one everywhere inside the protein. The Gaussian widths σ i of the atoms i are parameters of the RF approximation. The remarkable accuracy of the method is demonstrated by comparison with Kirkwood's analytical solution for a spherical protein [J. Chem. Phys.2, 351 (1934)] and with computationally expensive grid-based numerical solutions for simple model systems in dielectric continua including a di-peptide (Ac-Ala-NHMe) as modeled by a standard MM force field. The latter example shows how weakly the RF conformational free energy landscape depends on the parameters σ i. A summarizing discussion highlights the achievements of the new theory and of its approximate solution particularly by comparison with so-called generalized Born methods. A follow-up paper describes how the method enables Hamiltonian, efficient, and accurate MM molecular dynamics simulations of proteins in dielectric solvent continua. [ABSTRACT FROM AUTHOR]

The paper presents an analytical theory and numerical simulations of the dipolar response of hydrated proteins in solution. We calculate the effective dielectric constant representing the average dipole moment induced at the protein by a uniform external field. The dielectric constant shows a remarkable variation among the proteins, changing from 0.5 for ubiquitin to 640 for cytochrome c. The former value implies a negative dipolar susceptibility, that is a dia-electric dipolar response and negative dielectrophoresis. It means that ubiquitin, carrying an average dipole of ≃240 D, is expected to repel from the region of a stronger electric field. This outcome is the result of a negative cross-correlation between the protein and water dipoles, compensating for the positive variance of the intrinsic protein dipole in the overall dipolar susceptibility. In contrast to the neutral ubiquitin, charged proteins studied here show para-electric dipolar response and positive dielectrophoresis. The study suggests that the dipolar response of proteins in solution is strongly affected by the coupling of the protein surface charge to the hydration water. The protein-water dipolar cross-correlations are long-ranged, extending ∼2 nm from the protein surface into the bulk. A similar correlation length of about 1 nm is seen for the electrostatic potential produced by the hydration water inside the protein. The analysis of numerical simulations suggests that the polarization of the protein-water interface is highly heterogeneous and does not follow the standard dielectric results for cavities carved in dielectrics. The polarization of the water shell gains in importance, relative to the intrinsic protein dipole, at high frequencies, above the protein Debye peak. The induced interfacial dipole can be either parallel or antiparallel to the protein dipole, depending on the distribution of the protein surface charge. As a result, the high-frequency absorption of the protein solution can be either higher or lower than the absorption of water. Both scenarios have been experimentally observed in the THz window of radiation. [ABSTRACT FROM AUTHOR]

The authors reformulate and revise an electrostatic theory treating proteins surrounded by dielectric solvent continua [B. Egwolf and P. Tavan, J. Chem. Phys. 118, 2039 (2003)] to make the resulting reaction field (RF) forces compatible with Newton’s third law. Such a compatibility is required for their use in molecular dynamics (MD) simulations, in which the proteins are modeled by all-atom molecular mechanics force fields. According to the original theory the RF forces, which are due to the electric field generated by the solvent polarization and act on the partial charges of a protein, i.e., the so-called qE forces, can be quite accurately computed from Gaussian RF dipoles localized at the protein atoms. Using a slightly different approximation scheme also the RF energies of given protein configurations are obtained. However, because the qE forces do not account for the dielectric boundary pressure exerted by the solvent continuum on the protein, they do not obey the principle that actio equals reactio as required by Newton’s third law. Therefore, their use in MD simulations is severely hampered. An analysis of the original theory has led the authors now to a reformulation removing the main difficulties. By considering the RF energy, which represents the dominant electrostatic contribution to the free energy of solvation for a given protein configuration, they show that its negative configurational gradient yields mean RF forces obeying the reactio principle. Because the evaluation of these mean forces is computationally much more demanding than that of the qE forces, they derive a suggestion how the qE forces can be modified to obey Newton’s third law. Various properties of the thus established theory, particularly issues of accuracy and of computational efficiency, are discussed. A sample application to a MD simulation of a peptide in solution is described in the following paper [M. Stork and P. Tavan, J. Chem. Phys., 126, 165106 (2007). [ABSTRACT FROM AUTHOR]

We report molecular dynamics (MD) simulations of three protein-water systems (ubiquitin, apo-calbindin D9K, and the C-terminal SH2 domain of phospholipase C-γ1), from which we compute the dielectric properties of the solutions. Since two of the proteins studied have a net charge, we develop the necessary theory to account for the presence of charged species in a form suitable for computer simulations. In order to ensure convergence of the time correlation functions needed for the analysis, the minimum length of the MD simulations was 20 ns. The system sizes (box length, number of waters) were chosen so that the resulting protein concentrations are comparable to experimental conditions. A dielectric component analysis was carried out to analyze the contributions from protein and water to the frequency-dependent dielectric susceptibility χ(ω) of the solutions. Additionally, an even finer decomposition into protein, two solvation shells, and the remaining water (bulk water) was carried out. The results of these dielectric decompositions were used to study protein solvation at mesoscopic resolution, i.e., in terms of protein, first and second solvation layers, and bulk water. This study, therefore, complements the structural and dynamical analyses at molecular resolution that are presented in the companion paper. The dielectric component contributions from the second shell and bulk water are very similar in all three systems. We find that the proteins influence the dielectric properties of water even beyond the second solvation shell, in agreement with what was observed for the mean residence times of water molecules in protein solutions. By contrast, the protein contributions, as well as the contributions of the first solvation shell, are system specific. Most importantly, the protein and the first water shell around ubiquitin and apo-calbindin are anticorrelated, whereas the first water shell around the SH2 domain is positively correlated. [ABSTRACT FROM AUTHOR]

Proteins, molecules, and macromolecular assemblies in water are surrounded by a nanometer-sized hydration layer with properties very different from bulk water. Here, we use classical molecular dynamics simulations to study the dielectric response of hydration water next to hydrophobic and hydrophilic planar surfaces. We find the interfacial dielectric absorption of water to be strongly anisotropic: compared to bulk water, which shows a broad dielectric absorption maximum around 15 GHz in the imaginary part of the dielectric function, the absorption for electric fields parallel to the surface is of similar strength and shows a slight redshift, while for perpendicular electric fields it is strongly attenuated and blueshifted. This anisotropy is generic for hydrophobic and hydrophilic surfaces. From our spatially resolved dielectric functions and a modified Maxwell-Garnett theory that accounts for anisotropic hydration layers around spherical particles, the dielectric absorption of solutions of organic molecules and micelles is derived to exhibit the experimentally known attenuation in combination with a redshift. These two features are traced back to the subtle interplay of interfacial depolarization effects and the dielectric anisotropy in the hydration layer. By a detailed analysis of the individual water molecule dynamics the perpendicular blueshift is shown not to be linked to accelerated water reorientation, but rather to dielectric boundary effects. Carefully conducted angularly resolved experiments at planar aqueous interfaces will be able to resolve this dielectric anisotropy and thus to confirm the subtle connection between spectral absorption features and the molecular water dynamics in hydration layers. [ABSTRACT FROM AUTHOR]

We study the energetics of burying charges, ion pairs, and ionizable groups in a simple protein model using nonlocal continuum electrostatics. Our primary finding is that the nonlocal response leads to markedly reduced solvent screening, comparable to the use of application-specific protein dielectric constants. Employing the same parameters as used in other nonlocal studies, we find that for a sphere of radius 13.4 Å containing a single +1e charge, the nonlocal solvation free energy varies less than 18 kcal/mol as the charge moves from the surface to the center, whereas the difference in the local Poisson model is ∼35 kcal/mol. Because an ion pair (salt bridge) generates a comparatively more rapidly varying Coulomb potential, energetics for salt bridges are even more significantly reduced in the nonlocal model. By varying the central parameter in nonlocal theory, which is an effective length scale associated with correlations between solvent molecules, nonlocal-model energetics can be varied from the standard local results to essentially zero; however, the existence of the reduction in charge-burial penalties is quite robust to variations in the protein dielectric constant and the correlation length. Finally, as a simple exploratory test of the implications of nonlocal response, we calculate glutamate pKa shifts and find that using standard protein parameters ([variant_greek_epsilon]protein = 2-4), nonlocal results match local-model predictions with much higher dielectric constants. Nonlocality may, therefore, be one factor in resolving discrepancies between measured protein dielectric constants and the model parameters often used to match titration experiments. Nonlocal models may hold significant promise to deepen our understanding of macromolecular electrostatics without substantially increasing computational complexity. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this report we present a general formulation to calculate the van der Waals interaction between two protein molecules in an electrolyte solution using boundary element method of solving linearized Poisson–Boltzmann equation. Our formulation is based upon an inhomogeneous dielectric model of proteins at the residue level. Our results for bovine pancreatic trypsin inhibitor at various relative orientations indicate that the anisotropy of the interaction can be tens of k[sub B]T. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A dielectric continuum model is developed for calculating polarization reorganization energies of electron transfer reactions that occur in proteins. The model is based on an earlier microscopic formulation of the Marcus electron transfer theory. The classical Marcus result, λ=ΔGop1-0-ΔG1-0, for the free energy of polarization reorganization is derived from the microscopic theory. Both ΔGop1-0 and ΔG1-0 denote the electrostatic free energy due to a positive unit charge (+e) distributed in the region representing the electron donor and a negative unit charge (-e) distributed in the region representing the electron acceptor. In calculating ΔGop1-0, the donor and acceptor as well as the environment surrounding them take the optical dielectric constant εop. In calculating ΔG1-0, the donor and acceptor keep the optical dielectric constant but the environment takes the static dielectric constant ε. The environment consists of the protein matrix (where εop=εpop and ε=εp) and the solvent (where εop=εsop and ε=εs). The polarization reorganization free energy can be approximated as the sum of two components λ1 and λ2. In calculating λ1, the protein region is extended outward to infinity. For the case where the donor and acceptor are modeled as spheres (with both radii equal to a and center–center distance at r) and the electron charge is put at either center, a Marcus result, λ1=[(1/εpop)-(1/εp)][(1/a)-(1/r)]e2, is found to be highly accurate (maximum error 4%). In calculating λ2, the protein region is extended inward to fill the donor and acceptor. The magnitude of λ2 is usually much smaller than λ1. A toy electron-transfer protein is studied both by the dielectric continuum model and by implementing the microscopic formulation through computer simulations. Agreement of the results from the two approaches demonstrates the accuracy of... [ABSTRACT FROM AUTHOR]