GBr(6): a parameterization-free, accurate, analytical generalized born method

The Poisson-Boltzmann (PB) equation is widely used for modeling electrostatic effects and solvation for macromolecules. The generalized Born (GB) model has been developed to mimic PB results at substantial lower computational cost. Here, we report an analytical GB method that reproduces PB results with high accuracy. The analytical approach builds on previous work of Gallicchio and Levy (J. Comput. Chem. 2004, 25, 479), and incorporates an improvement, proposed by Grycuk (J. Chem. Phys. 2003, 119, 4817), of the Coulomb-field approximation used in most GB methods. Tested against PB results, our GB method has an average unsigned relative error of only 0.6% for a representative set of 55 proteins and of 0.4% and 0.3%, respectively, for folded and unfolded conformations of cytochrome b562 sampled in molecular dynamics simulations. The dependencies of the electrostatic solvation free energy on solute and solvent dielectric constants and on salt concentration are fully accounted for in our method.     

The dependence of electrostatic solvation energy on dielectric constants in Poisson-Boltzmann calculations

The Poisson-Boltzmann equation gives the electrostatic free energy of a solute molecule (with dielectric constant epsilon(l)) solvated in a continuum solvent (with dielectric constant epsilon(s)). Here a simple formula is presented that accurately predicts the electrostatic free energy for all combinations of epsilon(l) and epsilon(s) from the calculation on a single set of epsilon(l) and epsilon(s) values.     

Optimization of the GB/SA solvation model for predicting the structure of surface loops in proteins

Implicit solvation models are commonly optimized with respect to experimental data or Poisson-Boltzmann (PB) results obtained for small molecules, where the force field is sometimes not considered. In previous studies, we have developed an optimization procedure for cyclic peptides and surface loops in proteins based on the entire system studied and the specific force field used. Thus, the loop has been modeled by the simplified solvation function E(tot) = E(FF) (epsilon = 2r) + Sigma(i) sigma(i)A(i), where E(FF) (epsilon = nr) is the AMBER force field energy with a distance-dependent dielectric function, epsilon = nr, A(i) is the solvent accessible surface area of atom i, and sigma(i) is its atomic solvation parameter. During the optimization process, the loop is free to move while the protein template is held fixed in its X-ray structure. To improve on the results of this model, in the present work we apply our optimization procedure to the physically more rigorous solvation model, the generalized Born with surface area (GB/SA) (together with the all-atom AMBER force field) as suggested by Still and co-workers (J. Phys. Chem. A 1997, 101, 3005). The six parameters of the GB/SA model, namely, P(1)-P(5) and the surface area parameter, sigma (programmed in the TINKER package) are reoptimized for a "training" group of nine loops, and a best-fit set is defined from the individual sets of optimized parameters. The best-fit set and Still's original set of parameters (where Lys, Arg, His, Glu, and Asp are charged or neutralized) were applied to the training group as well as to a "test" group of seven loops, and the energy gaps and the corresponding RMSD values were calculated. These GB/SA results based on the three sets of parameters have been found to be comparable; surprisingly, however, they are somewhat inferior (e.g, of larger energy gaps) to those obtained previously from the simplified model described above. We discuss recent results for loops obtained by other solvation models and potential directions for future studies.     

Secondary structure bias in generalized Born solvent models: comparison of conformational ensembles and free energy of solvent polarization from explicit and implicit solvation

The effects of the use of three generalized Born (GB) implicit solvent models on the thermodynamics of a simple polyalanine peptide are studied via comparing several hundred nanoseconds of well-converged replica exchange molecular dynamics (REMD) simulations using explicit TIP3P solvent to REMD simulations with the GB solvent models. It is found that when compared to REMD simulations using TIP3P the GB REMD simulations contain significant differences in secondary structure populations, most notably an overabundance of alpha-helical secondary structure. This discrepancy is explored via comparison of the differences in the electrostatic component of the free energy of solvation (DeltaDeltaG(pol)) between TIP3P (via thermodynamic Integration calculations), the GB models, and an implicit solvent model based on the Poisson equation (PE). The electrostatic components of the solvation free energies are calculated using each solvent model for four representative conformations of Ala10. Since the PE model is found to have the best performance with respect to reproducing TIP3P DeltaDeltaG(pol) values, effective Born radii from the GB models are compared to effective Born radii calculated with PE (so-called perfect radii), and significant and numerous deviations in GB radii from perfect radii are found in all GB models. The effect of these deviations on the solvation free energy is discussed, and it is shown that even when perfect radii are used the agreement of GB with TIP3P DeltaDeltaG(pol) values does not improve. This suggests a limit to the optimization of the effective Born radius calculation and that future efforts to improve the accuracy of GB models must extend beyond such optimizations.     

Highly accurate biomolecular electrostatics in continuum dielectric environments

Implicit solvent models based on the Poisson-Boltzmann (PB) equation are frequently used to describe the interactions of a biomolecule with its dielectric continuum environment. A novel, highly accurate Poisson-Boltzmann solver is developed based on the matched interface and boundary (MIB) method, which rigorously enforces the continuity conditions of both the electrostatic potential and its flux at the molecular surface. The MIB based PB solver attains much better convergence rates as a function of mesh size compared to conventional finite difference and finite element based PB solvers. Consequently, highly accurate electrostatic potentials and solvation energies are obtained at coarse mesh sizes. In the context of biomolecular electrostatic calculations it is demonstrated that the MIB method generates substantially more accurate solutions of the PB equation than other established methods, thus providing a new level of reference values for such models. Initial results also indicate that the MIB method can significantly improve the quality of electrostatic surface potentials of biomolecules that are frequently used in the study of biomolecular interactions based on experimental structures.     

Generalized born model with a simple smoothing function

Based on recent developments in generalized Born (GB) theory that employ rapid volume integration schemes (M. S. Lee, F. R. Salabury, Jr., and C. L. Brooks III, J Chem Phys 2002, 116, 10606) we have recast the calculation of the self-electrostatic solvation energy to utilize a simple smoothing function at the dielectric boundary. The present GB model is formulated in this manner to provide consistency with the Poisson-Boltzmann (PB) theory previously developed to yield numerically stable electrostatic solvation forces based on finite-difference methods (W. Im, D. Beglov, and B. Roux, Comp Phys Commun 1998, 111, 59). Our comparisons show that the present GB model is indeed an efficient and accurate approach to reproduce corresponding PB solvation energies and forces. With only two adjustable parameters--a(0) to modulate the Coulomb field term, and a(1) to include a correction term beyond Coulomb field--the PB solvation energies are reproduced within 1% error on average for a variety of proteins. Detailed analysis shows that the PB energy can be reproduced within 2% absolute error with a confidence of about 95%. In addition, the solvent-exposed surface area of a biomolecule, as commonly used in calculations of the nonpolar solvation energy, can be calculated accurately and efficiently using the simple smoothing function and the volume integration method. Our implicit solvent GB calculations are about 4.5 times slower than the corresponding vacuum calculations. Using the simple smoothing function makes the present GB model roughly three times faster than GB models, which attempt to mimic the Lee-Richards molecular volume.     

The (1)H NMR chemical shift for the hydroxy proton of 4-(dimethylamino)-2'-hydroxychalcone in chloroform: a theoretical approach to its inverse dependence on the temperature

[reaction: see text] The inverse dependence of the chemical shift on the temperature experimentally found for the phenolic proton of 4-(dimethylamino)-2'-hydroxychalcone (DMAHC) is theoretically studied. As the temperature decreases, the solvent dielectric constant epsilon increases and the zwitterionic resonance form is more stabilized. Electronic calculations at the DFT level of theory were performed by immersing the solute DMAHC in chloroform cavities of different epsilon values. The values of the calculated chemical shifts for DMAHC as a function of epsilon show that the growing contribution of the zwitterionic structure justifies the experimental results.     

Effective Born radii in the generalized Born approximation: the importance of being perfect

Generalized Born (GB) models provide, for many applications, an accurate and computationally facile estimate of the electrostatic contribution to aqueous solvation. The GB models involve two main types of approximations relative to the Poisson equation (PE) theory on which they are based. First, the self-energy contributions of individual atoms are estimated and expressed as "effective Born radii." Next, the atom-pair contributions are estimated by an analytical function f(GB) that depends upon the effective Born radii and interatomic distance of the atom pairs. Here, the relative impacts of these approximations are investigated by calculating "perfect" effective Born radii from PE theory, and enquiring as to how well the atom-pairwise energy terms from a GB model using these perfect radii in the standard f(GB) function duplicate the equivalent terms from PE theory. In tests on several biological macromolecules, the use of these perfect radii greatly increases the accuracy of the atom-pair terms; that is, the standard form of f(GB) performs quite well. The remaining small error has a systematic and a random component. The latter cannot be removed without significantly increasing the complexity of the GB model, but an alternative choice of f(GB) can reduce the systematic part. A molecular dynamics simulation using a perfect-radii GB model compares favorably with simulations using conventional GB, even though the radii remain fixed in the former. These results quantify, for the GB field, the importance of getting the effective Born radii right; indeed, with perfect radii, the GB model gives a very good approximation to the underlying PE theory for a variety of biomacromolecular types and conformations.     

Analytical electrostatics for biomolecules: beyond the generalized Born approximation

The modeling and simulation of macromolecules in solution often benefits from fast analytical approximations for the electrostatic interactions. In our previous work [G. Sigalov et al., J. Chem. Phys. 122, 094511 (2005)], we proposed a method based on an approximate analytical solution of the linearized Poisson-Boltzmann equation for a sphere. In the current work, we extend the method to biomolecules of arbitrary shape and provide computationally efficient algorithms for estimation of the parameters of the model. This approach, which we tentatively call ALPB here, is tested against the standard numerical Poisson-Boltzmann (NPB) treatment on a set of 579 representative proteins, nucleic acids, and small peptides. The tests are performed across a wide range of solvent/solute dielectrics and at biologically relevant salt concentrations. Over the range of the solvent and solute parameters tested, the systematic deviation (from the NPB reference) of solvation energies computed by ALPB is 0.5-3.5 kcal/mol, which is 5-50 times smaller than that of the conventional generalized Born approximation widely used in this context. At the same time, ALPB is equally computationally efficient. The new model is incorporated into the AMBER molecular modeling package and tested on small proteins.     

Continuum solvation models in the linear interaction energy method

The linear interaction energy (LIE) method in combination with two different continuum solvent models has been applied to calculate protein-ligand binding free energies for a set of inhibitors against the malarial aspartic protease plasmepsin II. Ligand-water interaction energies are calculated from both Poisson-Boltzmann (PB) and Generalized Born (GB) continuum models using snapshots from explicit solvent simulations of the ligand and protein-ligand complex. These are compared to explicit solvent calculations, and we find close agreement between the explicit water and PB solvation models. The GB model overestimates the change in solvation energy, and this is caused by consistent underestimation of the effective Born radii in the protein-ligand complex. The explicit solvent LIE calculations and LIE-PB, with our standard parametrization, reproduce absolute experimental binding free energies with an average unsigned error of 0.5 and 0.7 kcal/mol, respectively. The LIE-GB method, however, requires a constant offset to approach the same level of accuracy.     

A sphere-based model for the electrostatics of globular proteins

We propose a model for the electrostatics of globular proteins in which the low dielectric region is replaced by concentric spheres of the appropriate size. The method uses analytical formulas for the dielectric sphere and allows an efficient and accurate treatment of bulk charges. For surface charges, we propose a numerical determination of the sphere radius based on the solvent exposure of the individual atoms. The present implementation of the sphere model yields a good approximation of finite-difference Poisson solvation and interaction energies for a test set of 12 proteins.     

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.     

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.     

Optical absorption of semiconducting and metallic nanospheres with the confined electron-phonon coupling

We study the optical absorption, especially the (far-) infrared absorption by phonons, of semiconducting and metallic nanospheres. In the nanoscopic sphere, phonons as well as states of electronic excitations are quantized by confinement. It is also known that in the nanoscopic geometry, the confined electron-phonon interaction has a different form from the usual one in the bulk. First, we analyze the phonon and electron contributions to the dielectric response of nanospheres like epsilon(q,omega)=epsilon(ph)(q,omega)+epsilon(el)(q,omega) or 1epsilon(q,omega)=1epsilon(sc-ph)(q,omega)+1epsilon(el)(q,omega) from the confined electron-phonon interaction for three cases: the intrinsic semiconductor, the doped semiconductor, and the metal. From the dielectric response, the optical absorption spectra are calculated within the semiclassical framework concentrating on the (far-) infrared region and compared to the spectra without imposing confinement. Nontrivial differences of the spectra with confined phonons stem from two features: the electron-phonon coupling matrix has a different form and the phase space q of the confined phonon is reduced because of its quantization to q(n). Finally, size distribution effects in an ensemble of isolated nanospheres are briefly discussed. Those effects are found to be important in metallic spheres with rapid sweepings of resonances by a small change of the sphere size.