Fast Poisson-Boltzmann Solver
CDI's Fast Poisson-Boltzmann, (FPB), software package embodies a revolutionary fast boundary element methodology for solving the Poisson-Boltzmann equation characterizing macromolecular electrostatics. Given a set of charges and a molecular surface definition, FPB calculates the electrostatic potential which, once known, can be used to derive a variety of related and measurable properties such as: electrostatic binding and free solvation energies, titration curves, pKa values, electrostatic forces, ionic distributions, electrostatic complementarity between proteins and ligands and protein folding stability. Through appropriate coupling arrangements, the FPB analysis can also be applied to molecular dynamics simulation, protein docking problems, Monte Carlo-based energy minimization and electrophoresis modeling. Simulations of this kind can be used to improve our understanding of the behavior and function of biomolecules and are of direct relevance to drug design. Closely related applications also include the analysis of colloidal systems and material science. At the heart of CDI's FPB analysis is a fast multipole algorithm (FMA) which combines an efficient hierarchical decomposition of the domain with a novel multipole expansion technique specifically tailored for the Poisson-Boltzmann equation. Traditionally, the primary limitation of boundary element methods has been the inability to accommodate truly large problems (more than 10K elements). This constraint arises because every element interacts with every other element; hence for N surface elements the evaluation of the electrostatic potential entails N2 interactions. Using FMA, this count reduces to
The electrostatic properties for a 20 Å (10-10 m) radius sphere containing a centrally located charge, are calculated upon successively finer triangular grids. In large scale BEM calculations, storage of the influence coefficient matrix dominates memory requirements. This figure shows how the number of influence coefficients varies with the number of boundary elements using: (a) a conventional or direct BEM implementation and (b) the FPB algorithm. Results are shown for both zero and finite salt concentrations. Note that because of storage constraints, it was not possible to perform direct calculations using more than 5120 boundary elements (the finite salt calculation then involved nearly 105 million influence coefficients requiring 420MByte storage in real precision). Data beyond this limit was extrapolated from lower element counts. By contrast, when using the FPB code it was possible to accommodate 328,000 (zero salt) or 82,000 (finite salt) elements under the same storage constraints. All results were obtained on a Silicon Graphics single MIPS R10000 processor operating at 180 MHz.
The spherical cavity model with a centrally located charge is again considered. This time the calculation times (Silicon Graphics single MIPS R10000 processor operating at 180 MHz) using a direct BEM implementation and CDI's FPB code, are compared. For more than 5120 boundary elements, the timings for the direct calculation are estimated by extrapolation. At very low BE counts (the number of elements, N < 1000) elements, the additional overhead of the FPB code leads to slightly longer CPU times. These times, however, are very small (1 to 3 seconds). At higher element counts, N > 1000, the FPB code whose associated CPU behaves as
Grid and surface potential for the Carboxypeptidase A molecule which has 4801 atoms and a net charge of -50.6e. The interior and exterior dielectric constants are 2 and 80 respectively; the Debye-Hückel inverse screening length, κ=0.12528 and the molecular surface is discretized into 54,639 elements. Just under 48 million influence coefficients must be stored to evaluate near field interactions. By comparison, a direct implementation would require 14.3 billion coefficients which would tax the storage resources of all readily available modern computer platforms. The computed reaction field energy for this configuration is -10,848 kcal/mol. For κ=0 the computed energy is -10,726 kcal/mol.
Grid and surface potential for the superoxide dismutase molecule containing 2,196 atoms with a net charge of -4.0e. The interior and exterior dielectric constants are 2 and 80 respectively; the Debye-Hückel inverse screening length, κ=0.12528 and the molecular surface is discretized into 44,434 elements. A total of 42.4 million influence coefficients are stored. By comparison, a direct implementation would require 7.9 billion coefficients which would tax the storage resources of all readily available modern computer platforms. The computed reaction field energy for this configuration is -1,684.7 kcal/mol. For κ=0 the computed energy is -1,675.1 kcal/mol.