AFMPB: An adaptive fast multipole Poisson–Boltzmann solver for calculating electrostatics in biomolecular systems☆
Introduction
In the past thirty years, the Poisson–Boltzmann (PB) continuum electrostatic model has been widely accepted as a tool in theoretical studies of interactions of biomolecules such as proteins and DNAs in aqueous solutions. Recent work includes the introduction of more physical boundary conditions so that PB and its linearized version become valid for a wider class of molecular systems [30].
In this paper, we describe an adaptive fast multipole Poisson–Boltzmann (AFMPB) solver for solving the linearized PB equation, and thus for elucidating the electrostatic role in many biological processes, such as enzymatic catalysis, molecular recognition and bio-regulation. Indeed, quite a few PB solvers are available in biochemistry and biophysical communities, including DelPhi, GRASP, MEAD, UHBD and PBEQ based on the finite difference formulation [10], and APBS (adaptive Poisson–Boltzmann solver) based on the finite volume/multigrid framework [1], [11], [19]. In the linearized PB regime, algorithms using the boundary integral equation (BIE) approach have shown great promise for their efficiency on scaling and memory requirements [13], [22], [24]: when Green's theorem and potential theory are applied, the linearized PB equation can be recasted into a set of boundary integral equations where the unknowns are only defined on the surface of the molecule. Therefore, the number of unknowns is reduced when compared with the volumetric discretization in finite difference and finite element methods.
This AFMPB solver reflects our effort on developing more efficient codes using the BIE approach for the linearized PB equation, currently on single processor computing architectures. Several techniques are used in AFMPB to improve its efficiency over existing BIE-based solvers. First, a well-conditioned BIE formulation is used so that the number of iterations in the Krylov subspace methods is bounded, independent of the number of unknowns in the system. Second, a node–patch scheme is applied to discretize the resulting BIE, and the node based scheme reduces the number of unknowns defined on the molecular surface compared with commonly used “constant element” discretizations. Further, the iterative Krylov subspace methods from the SPARSKIT package [7] are applied to the resulting linear systems with simplified calling interface with the use of the reverse communication protocols. Fourth, the adaptive new versions of the fast multipole methods (FMMs) from FMMSuite [4] are applied to the convolution type discretized integrals. Finally, interface computer programs are provided to couple AFMPB with many existing mesh generating (e.g. MSMS [6] or other programs that can generate OFF format type of mesh) and visualization tools (e.g. VMD [5]). Preliminary numerical experiments show that the new AFMPB solver achieves significant speedup and memory savings for calculating the electrostatics of large-scale biomolecular systems on single-processor personal computers. We are currently parallelizing the codes on multi-core/multi-processor computer architectures toward simulating the dynamics of complex biomolecular systems under the influence of electrostatic forces derived from the PB calculation.
This paper is organized as follows. In Section 2, we describe various numerical techniques used in the AFMPB solver, including the boundary integral equation formulation for the linearized PB equation, the node–patch discretization, the Krylov subspace methods, and the adaptive new version of FMM. In Section 3, the overall structure of the codes is discussed, in particular, how it can be coupled with existing tools for pre- and post-processing. In Section 4, test runs are described to illustrate the performance of the solver. Finally, in Appendix A Units, Appendix B Important parameters, Appendix C A sample shell script, we briefly describe the units and several important parameters used in our codes, and provide a sample shell script file for running the package.
Section snippets
Electrostatics in biomolecular systems
The electrostatic force is considered an important factor in understanding the interactions and dynamics of molecular systems in solution. One commonly used continuum model for describing the electrostatic effects of the solvent outside the molecules is the Poisson–Boltzmann (PB) equation and the Poisson equation is used for the inside of the molecules. When electrostatic potentials are small, the linearized PB (LPB) equation is
Code structure and implementation
In this section, we describe the package portability and installation, flow chart, job running, file format and I/O layers, pre- and post-processing, and several related tools.
Spherical cavity
To assess the performance of the AFMPB solver, we first calculate the Born solvation energy of a point charge +50e located at the center of a spherical cavity with a radius of 50 Å. The exact Born solvation energy of the cavity is −4046.0 (energy is in kcal/mol). The surface is discretized at various resolution levels by recursively subdividing an icosahedron. Table 1 summarizes the timing results (on a Dell dual 2.0 GHz P4 desktop with 2 GB memory) along with some critical control
Conclusion and acknowledgements
In this paper, we describe an adaptive fast multipole Poisson–Boltzmann solver for computing the electrostatics in biomolecules. The solver uses the new version of fast multipole methods and Krylov subspace methods to solve a well-conditioned boundary integral equation formulation of the linearized PB equation. Numerical experiments show that the solver is very efficient for relatively large molecules on current desktop computers. We anticipate that the AFMPB solver, when parallelized on
References (30)
Methods in Enzymol.
(2004)- et al.
J. Comput. Phys.
(1999) - et al.
J. Comput. Phys.
(2002) - et al.
J. Comput. Phys.
(1987) - et al.
Computer Physics Communications
(2009) - et al.
J. Comput. Phys.
(1991) - et al.
Biophys. J.
(1997) Wave Motion
(1983)
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This paper and its associated computer program are available via the Computer Physics Communications homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).