The Finite Element Tool Kit (FEtk)28 is an evolving collection of parallel adaptive, multilevel finite element software libraries and supports tools for solving coupled systems of partial differential equations (PDE) and integral equations (IE).29 The numerical libraries are written in an object-oriented form of ANSI-C (Figure 4A). Left (Top/Bottom): Potential contours of the electrostatic potential around a biomolecule, computed adaptively using FEtk, and a closeup of the adapted part of the simplex mesh. Right (Top/Bottom): Isosurfaces projected onto a cutting plane through two black holes in an astrophysics problem. FEtk was used to compute the initial bending of space and time around two massive black holes, which involved adaptively solving a coupled nonlinear elliptic PDE.
The primary FEtk ANSI-C software libraries include MALOC (Minimal Abstraction Layer for Object-oriented C) for portability, SG (Socket Graphics) for networked OpenGL-based visualization, and MC (Manifold Code) for adaptively solving coupled systems of nonlinear PDE on two- and three-manifold domains. A 2D MATLAB-based prototyping tool called MCLite is also available for fast development of MC-based software. A related package, PMG (Parallel Algebraic Multigrid) is designed to numerically solve scalar nonlinear PDE problems such as the Poisson-Boltzmann Equation (PBE) on a class of regular domains using algorithms which have optimal or near-optimal space and time complexity.
APBS is a software package for the numerical solution of the PBE, one of the most popular continuum models for describing electrostatic interactions between molecular solutes in salty, aqueous media.30 Continuum electrostatics plays an important role in several areas of biomolecular simulation, including:
- simulation of diffusional processes to determine ligand-protein and protein-protein binding kinetics;
- implicit solvent molecular dynamics of biomolecules;
- solvation and binding energy calculations to determine ligand-protein and protein-protein equilibrium binding constants and aid in rational drug design;
- and biomolecular titration studies.
APBS was designed to efficiently evaluate electrostatic properties for such simulations for a wide range of length scales to enable the investigation of molecules with tens to millions of atoms. APBS uses FEtk and PMG to solve the Poisson-Boltzmann equation numerically.
APBS is available to users as standalone applications or inside PMV for seamless access (Figure 4B), available in CHARMM or AMBER through the NBCR iAPBS interface,31 or as web services using the NBCR Opal toolkit.32 In this example, the electrostatic binding energy for the protein kinase A (PKA) complex with the inhibitor balanol was computed using APBS and displayed in PMV. The PKA complex is shown as a ribbon diagram with side chains or positively (blue) and negatively (red) charged amino acids displayed as Sticks and Balls. The inhibitor molecule is displayed using a fat Sticks and Balls representation and colored by atom types. The electrostatic binding energy is visualized by direct volume rendering with the shown transfer function and using two isosurfaces.
A new software component from NBCR that allows the efficient modeling of diffusion events across the neuromuscular junction in a steady state or time dependent manner called the Smoluchowski equation solver (SMOL) is also based on the FEtk and has been used in conjunction with APBS in a study of the tetrameric complex acetylcholine receptors.33
