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...

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.

A multigrid method is presented for the numerical solution of the linearized Poisson-Boltzmann equation arising in molecular biophysics. The equation is discretized with the finite volume method, and the numerical solution of the discrete equations is accomplished with multiple grid techniques originally developed for two-dimensional interface problems occurring in reactor physics. A detailed analysis of the resulting method is presented for several computer architectures, including comparisons to diagonally scaled CG, ICCG, vectorized ICCG and MICCG, and to SOR provided with an optimal relaxation parameter. Our results indicate that the multigrid method is superior to the preconditioned CG methods and SOR, and that the advantage of multigrid grows with the problem size.

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.

In the current study, the three-dimensional (3D) topologies of dyadic clefts and associated membrane organelles were mapped in mouse ventricular myocardium using electron tomography. The morphological details and the distribution of membrane systems, including transverse tubules (T-tubules), junctional sarcoplasmic reticulum (SR) and vicinal mitochondria, were determined and presumed to be crucial for controlling cardiac Ca2+ dynamics. The geometric complexity of T-tubules that varied in diameter with frequent branching was clarified. Dyadic clefts were intricately shaped and remarkably small (average 4.39×105 nm3, median 2.81×105 nm3). Although a dyadic cleft of average size could hold maximum 43 ryanodine receptor (RyR) tetramers, more than one-third of clefts were smaller than the size that is able to package as many as 15 RyR tetramers. The dyadic clefts were also adjacent to one another (average end-to-end distance to the nearest dyadic cleft, 19.9 nm) and were distributed irregularly along T-tubule branches. Electron-dense structures that linked membrane organelles were frequently observed between mitochondrial outer membranes and SR or T-tubules. We, thus, propose that the topology of dyadic clefts and the neighboring cellular micro-architecture are the major determinants of the local control of Ca2+ in the heart, including the establishment of the quantal nature of SR Ca2+ releases (e.g. Ca2+ sparks).

We present a robust and efficient numerical method for solution of the nonlinear Poisson-Boltzmann equation arising in molecular biophysics. The equation is discretized with the box method, and solution of the discrete equations is accomplished with a global inexact-Newton method, combined with linear multilevel techniques we have described in a paper appearing previously in this journal. A detailed analysis of the resulting method is presented, with comparisons to other methods that have been proposed in the literature, including the classical nonlinear multigrid method, the nonlinear conjugate gradient method, and nonlinear relaxation methods such as successive over-relaxation. Both theoretical and numerical evidence suggests that this method will converge in the case of molecules for which many of the existing methods will not. In addition, for problems which the other methods are able to solve, numerical experiments show that the new method is substantially more efficient, and the superiority of this method grows with the problem size. The method is easy to implement once a linear multilevel solver is available, and can also easily be used in conjunction with linear methods other than multigrid.

Abstract:We consider the conformal decomposition of Einstein's constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of non-CMC weak solutions using a combination of a priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions and fixed-point techniques for the Hamiltonian constraint, Riesz-Schauder theory for the momentum constraint, together with a topological fixed-point argument for the coupled system. Although we present general existence results for non-CMC weak solutions when the rescaled background metric is in any of the three Yamabe classes, an important new feature of the results we present for the positive Yamabe class is the absence of the near-CMC assumption, if the freely specifiable part of the data given by the traceless-transverse part of the rescaled extrinsic curvature and the matter fields are sufficiently small, and if the energy density of matter is not identically zero. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, so that it can be arbitrarily far from constant, giving what is apparently the first existence results for non-CMC solutions without the near-CMC assumption. Using a coupled topological fixed-point argument that avoids near-CMC conditions, we establish existence of coupled non-CMC weak solutions with (positive) conformal factor φ ∈ W s, p , where p ∈ (1, ∞) and s( p) ∈ (1 + 3/ p, ∞). In the CMC case, the regularity can be reduced to p ∈ (1, ∞) and s( p) ∈ (3/ p, ∞) ∩ [1, ∞). In the case of s = 2, we reproduce the CMC existence results of , and in the case p = 2, we reproduce the CMC existence results of Maxwell [33], but with a proof that goes through the same analysis framework that we use to obtain the non-CMC results. The non-CMC results on closed manifolds here extend the 1996 non-CMC result of Isenberg and Moncrief in three ways: (1) the near-CMC assumption is removed in the case of the positive Yamabe class; (2) regularity is extended down to the maximum

ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson-Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson-Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson-Boltzmann equation based on certain quasi-uniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson-Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson-Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear PoissonBoltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems. CONTENTS