T he importance of electrostatic modeling to biophysics is well established; electrostatics have been shown to influence various aspects of nearly all biochemical reactions. Advances in NMR, x-ray, and cryo-electron microscopy techniques for structure elucidation have drastically increased the size and number of biomolecules and molecular complexes for which coordinates are available. However, although the biophysical community continues to examine macromolecular systems of increasing scale, the computational evaluation of electrostatic properties for these systems is limited by methodology that can handle only relatively small systems, typically consisting of fewer than 50,000 atoms. Despite these limitations, such computational methods have been immensely useful in analyses of the stability, dynamics, and association of proteins, nucleic acids, and their ligands (1-3). Here we describe algorithms that open the way to similar analyses of much larger subcellular structures.One of the most widespread models for the evaluation of electrostatic properties is the Poisson-Boltzmann equation (PBE) (4, 5)a second-order nonlinear elliptic partial differential equation that relates the electrostatic potential ( ) to the dielectric properties of the solute and solvent ( ), the ionic strength of the solution and the accessibility of ions to the solute interior ( 2 ), and the distribution of solute atomic partial charges ( f ). To expedite solution of the equation, this nonlinear PBE is often approximated by the linearized PBE (LPBE) by assuming sinh (x) Ϸ (x). Several numerical techniques have been used to solve the nonlinear PBE and LPBE, including boundary element (6-8), finite element (9-11), and finite difference (12-14) algorithms. However, despite the variety of solution methods, none of these techniques has been satisfactorily applied to large molecular structures at the scales currently accessible to modern biophysical methods. To accommodate arbitrarily large biomolecules, algorithms for solving the PBE must be both efficient and amenable to implementation on a parallel platform in a scalable fashion, requirements that current methods have been unable to satisfy. Although boundary element LPBE solvers provide an efficient representation of the problem domain, they are not useful for the nonlinear problem and have not been applied to the PBE on parallel platforms. Similarly, adaptive finite element methods have shown some success in parallel evaluation of both the LPBE and nonlinear PBE (15), but limitations in current solver technology and difficulty with efficient representation of the biomolecular data prohibits their practical application to large biomolecular systems. Finally, unlike the boundary and finite element techniques, finite difference methods have the advantage of very efficient multilevel solvers (12, 16) and applicability to both the linear and nonlinear forms of the PBE; however, existing parallel finite difference algorithms often require costly interprocessor communication that limits both the n...

Continuum solvation models, such as Poisson-Boltzmann and Generalized Born methods, have become increasingly popular tools for investigating the influence of electrostatics on biomolecular structure, energetics and dynamics. However, the use of such methods requires accurate and complete structural data as well as force field parameters such as atomic charges and radii. Unfortunately, the limiting step in continuum electrostatics calculations is often the addition of missing atomic coordinates to molecular structures from the Protein Data Bank and the assignment of parameters to biomolecular structures. To address this problem, we have developed the PDB2PQR web service (http://agave.wustl.edu/pdb2pqr/). This server automates many of the common tasks of preparing structures for continuum electrostatics calculations, including adding a limited number of missing heavy atoms to biomolecular structures, estimating titration states and protonating biomolecules in a manner consistent with favorable hydrogen bonding, assigning charge and radius parameters from a variety of force fields, and finally generating 'PQR' output compatible with several popular computational biology packages. This service is intended to facilitate the setup and execution of electrostatics calculations for both experts and non-experts and thereby broaden the accessibility to the biological community of continuum electrostatics analyses of biomolecular systems.

Real-world observable physical and chemical characteristics are increasingly being calculated from the 3D structures of biomolecules. Methods for calculating pKa values, binding constants of ligands, and changes in protein stability are readily available, but often the limiting step in computational biology is the conversion of PDB structures into formats ready for use with biomolecular simulation software. The continued sophistication and integration of biomolecular simulation methods for systems- and genome-wide studies requires a fast, robust, physically realistic and standardized protocol for preparing macromolecular structures for biophysical algorithms. As described previously, the PDB2PQR web server addresses this need for electrostatic field calculations (Dolinsky et al., Nucleic Acids Research, 32, W665–W667, 2004). Here we report the significantly expanded PDB2PQR that includes the following features: robust standalone command line support, improved pKa estimation via the PROPKA framework, ligand parameterization via PEOE_PB charge methodology, expanded set of force fields and easily incorporated user-defined parameters via XML input files, and improvement of atom addition and optimization code. These features are available through a new web interface (http://pdb2pqr.sourceforge.net/), which offers users a wide range of options for PDB file conversion, modification and parameterization.

The Adaptive Poisson-Boltzmann Solver (APBS) software was developed to solve the equations of continuum electrostatics for large biomolecular assemblages that have provided impact in the study of a broad range of chemical, biological, and biomedical applications. APBS addresses the three key technology challenges for understanding solvation and electrostatics in biomedical applications: accurate and efficient models for biomolecular solvation and electrostatics, robust and scalable software for applying those theories to biomolecular systems, and mechanisms for sharing and analyzing biomolecular electrostatics data in the scientific community. To address new research applications and advancing computational capabilities, we have continually updated APBS and its suite of accompanying software since its release in 2001. In this article, we discuss the models and capabilities that have recently been implemented within the APBS software package including a Poisson-Boltzmann analytical and a semi-analytical solver, an optimized boundary element solver, a geometry-based geometric flow solvation model, a graph theory-based algorithm for determining pK values, and an improved web-based visualization tool for viewing electrostatics.

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the salvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By minimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature. KeywordsGeneralized Poisson-Boltzmann equation; Biomolecular surface formation and evolution; Potential driving geometric flows; Solvation free energy; Multiscale models I IntroductionAmong the various components of molecular interactions, electrostatic interactions are of special importance 6,8,41,44,55,59,71, 142 -144 , 169 ,170 because of their long range and influence on polar or charged molecules -including water, aqueous ions, and amino or nucleic acids. Electrostatic interactions are ubiquitous for any system of charged or polar molecules, such as * Please address correspondence to Guowei Wei. wei@math.msu.edu. (proteins, nucleic acids, lipid bilayers, sugars, etc.) in their aqueous environment. Electrostatic solute-solvent interactions, therefore, are of central importance in analyzing molecular structure and modeling the intramolecular and intermolecular interactions of macromolecules in simulations. There are two types of solvation models: 131, 142 , 143 , 169 explicit ...

Continuum solvation models provide appealing alternatives to explicit solvent methods because of their ability to reproduce solvation effects while alleviating the need for expensive sampling. Our previous work has demonstrated that Poisson-Boltzmann methods are capable of faithfully reproducing polar explicit solvent forces for dilute protein systems; however, the popular solvent-accessible surface area model was shown to be incapable of accurately describing nonpolar solvation forces at atomic-length scales. Therefore, alternate continuum methods are needed to reproduce nonpolar interactions at the atomic scale. In the present work, we address this issue by supplementing the solvent-accessible surface area model with additional volume and dispersion integral terms suggested by scaled particle models and Weeks–Chandler–Andersen theory, respectively. This more complete nonpolar implicit solvent model shows very good agreement with explicit solvent results and suggests that, although often overlooked, the inclusion of appropriate dispersion and volume terms are essential for an accurate implicit solvent description of atomic-scale nonpolar forces.

The pKa-cooperative aims to provide a forum for experimental and theoretical researchers interested in protein pKa values and protein electrostatics in general. The first round of the pKa-cooperative, which challenged computational labs to carry out blind predictions against pKas experimentally determined in the laboratory of Bertrand Garcia-Moreno, was completed and results discussed at the Telluride meeting (July 6–10, 2009). This paper serves as an introduction to the reports submitted by the blind prediction participants that will be published in a special issue of PROTEINS: Structure, Function and Bioinformatics. Here we briefly outline existing approaches for pKa calculations, emphasizing methods that were used by the participants in calculating the blind pKa values in the first round of the cooperative. We then point out some of the difficulties encountered by the participating groups in making their blind predictions, and finally try to provide some insights for future developments aimed at improving the accuracy of pKa calculations.

On page 1332 of the above article the statement "For example, in DelPhi, one may only employ cubical meshes with 63 mesh lines in each direction" is no longer accurate; more recent versions of DelPhi have been extended to allow for the use of larger cubical meshes.