The application of numerical methods to enable the trivially parallel solution of …

• Abstract
• Introduction
• Multigrid Solution Through Parallel Focusing
• Electrostatic Properties of Microtubules
• Electrostatic Properties of the Ribosome
• Conclusions
• Abbreviations
• References
• Figures

Biology Articles » Biophysics » Molecular Biophysics » Electrostatics of nanosystems: Application to microtubules and the ribosome » Introduction

Introduction

- Electrostatics of nanosystems: Application to microtubules and the ribosome

The 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)

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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)  (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 nature and scale of their execution on parallel platforms (17-21) [see especially Van de Velde (19) for reviews of the various methods].