This work introduces a continuous smooth permittivity function into Poisson–Boltzmann techniques for continuum approaches to modeling the solvation of small molecules and proteins. The permittivity function is derived using a Gaussian method to describe volume exclusion. The new method allows a rigorous determination of solvent forces within a grid-based technology. The generality of approach is demonstrated by considering a range of applications for small molecules and macromolecules. We also present a very complete statistical analysis of grid errors, and show that the accuracy of our Gaussian-based method is improved over standard techniques. The method has been implemented in a new code called ZAP, which is freely available to academic institutions.1 © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 608–640, 2001

A smooth permittivity function for Poisson–Boltzmann solvation methods

This review attempts a critical stock-taking of the current state of the science aimed at predicting structural features of proteins from their amino acid sequences. At the primary structure level, methods are considered for detection of remotely related sequences and for recognizing amino acid patterns to predict posttranslational modifications and binding sites. The techniques involving secondary structural features include prediction of secondary structure, membrane-spanning regions, and secondary structural class. At the tertiary structural level, methods for threading a sequence into a mainchain fold, homology modeling and assigning sequences to protein families with similar folds are discussed. A literature analysis suggests that, to date, threading techniques are not able to show their superiority over sequence pattern recognition methods. Recent progress in the state of ab initio structure calculation is reviewed in detail. The analysis shows that many structural features can be predicted from the amino acid sequence much better than just a few years ago and with attendant utility in experimental research. Best prediction can be achieved for new protein sequences that can be assigned to well-studied protein families. For single sequences without homologues, the folding problem has not yet been solved.

Protein structure prediction: recognition of primary, secondary, and tertiary structural features from amino acid sequence.

We propose a fast implementation of the boundary element method for solving the Poisson equation, which approximately determines the electrostatic field around solvated molecules of arbitrary shape. The method presented uses computational resources of order O(N) only, where N is the number of elements representing the dielectric boundary at the molecular surface. The method is based on the Fast Multipole Algorithm by Rokhlin and Greengard, which is used to calculate the Coulombic interaction between surface elements in linear time. We calculate the solvation energies of a sphere, a small polar molecule, and a moderately sized protein. The values obtained by the boundary element method agree well with results from finite difference calculations and show a higher degree of consistency due to the absence of grid dependencies. The boundary element method can be taken to a much higher accuracy than is possible with finite difference methods and can therefore be used to verify their validity. © 1995 by John Wiley & Sons, Inc.

The fast multipole boundary element method for molecular electrostatics: An optimal approach for large systems

Efficiency and accuracy are two major concerns in numerical solutions of the Poisson-Boltzmann equation for applications in chemistry and biophysics. Recent developments in boundary element methods, interface methods, adaptive methods, finite element methods, and other approaches for the Poisson-Boltzmann equation as well as related mesh generation techniques are reviewed. We also discussed the chal- lenging problems and possible future work, in particular, for the aim of biophysical applications.

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RecentProgress in NumericalMethods forthePoisson- Boltzmann Equation in Biophysical Applications

A new method for calculating the total conformational free energy of proteins in water solvent is presented. The method consists of a relatively brief simulation by molecular dynamics with explicit solvent (ES) molecules to produce a set of microstates of the macroscopic conformation. Conformational energy and entropy are obtained from the simulation, the latter in the quasi-harmonic approximation by analysis of the covariance matrix. The implicit solvent (IS) dielectric continuum model is used to calculate the average solvation free energy as the sum of the free energies of creating the solute-size hydrophobic cavity, of the van der Waals solute-solvent interactions, and of the polarization of water solvent by the solute's charges. The reliability of the solvation free energy depends on a number of factors: the details of arrangement of the protein's charges, especially those near the surface; the definition of the molecular surface; and the method chosen for solving the Poisson equation. Molecular dynamics simulation in explicit solvent relaxes the protein's conformation and allows polar surface groups to assume conformations compatible with interaction with solvent, while averaging of internal energy and solvation free energy tend to enhance the precision. Two recently developed methods--SIMS, for calculation of a smooth invariant molecular surface, and FAMBE, for solution of the Poisson equation via a fast adaptive multigrid boundary element--have been employed. The SIMS and FAMBE programs scale linearly with the number of atoms. SIMS is superior to Connolly's MS (molecular surface) program: it is faster, more accurate, and more stable, and it smooths singularities of the molecular surface. Solvation free energies calculated with these two programs do not depend on molecular position or orientation and are stable along a molecular dynamics trajectory. We have applied this method to calculate the conformational free energy of native and intentionally misfolded globular conformations of proteins (the EMBL set of deliberately misfolded proteins) and have obtained good discrimination in favor of the native conformations in all instances.

Discrimination between native and intentionally misfolded conformations of proteins: ES/IS, a new method for calculating conformational free energy that uses both dynamics simulations with an explicit solvent and an implicit solvent continuum model.

A general numerical method is presented to solve the nonlinear Poisson-Boltzmann (NLPB) equation for an arbitrarily-shaped solute. The essence of the method is the separation of the calculation of the solvent reaction potential from that of the potential due to the ion distribution. The solvent reaction potential is calculated by using an efficient boundary element method. The ion-induced potential is then calculated by means of an efficient volume integration using an iterative solution of the NLPB equation coupled to the fixed molecular and solvent electrostatic potential

A combined iterative and boundary-element approach for solution of the nonlinear Poisson-Boltzmann equation

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A combined iterative and boundary-element approach for solution of the nonlinear Poisson-Boltzmann equation