Fast Boundary Element Method for Biomolecular Electrostatics

    The electrostatics of biomolecular systems in ionic solutions is accurately described by the Poisson-Boltzmann equation (PBE). Given a molecular geometry consisting of a surface definition and the interior charge distribution together with the environmental parameters (the salt concentrations and the dielectric values inside and outside the molecule), then the PBE can be solved for the electrostatic potential which furnishes 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.

    The boundary element method (BEM) provides a highly accurate and compact solution procedure for the PBE. The entire 3D solution is expressed in terms of distributions of the potential and its normal gradient over the molecular surface. The discrete approximation to the governing integral equations is implemented upon a mesh that lies on the molecular surface. This feature, together with the use of analytically exact expressions when evaluating the interactions between elements, contributes to the high accuracy of the method, especially when evaluating properties such as the surface induced charge, ionic pressure and dielectric pressure. The latter two properties are needed to calculate gradients, but are notoriously challenging to evaluate on lattice grid-based PBE solvers.

    For N boundary elements, the conventional BEM is subject to O(N2) computational scalings in storage and CPU because of the mutual interactions between all elements. CDI's Fast Poisson-Boltzmann (FPB) solver utilizes fast multipole methods to reduce this cost to O(NlogN), which, in practice, produces order of magnitude reductions in storage and CPU costs for realistic molecules. What distinguishes FPB from other fast BEM solvers is that the bulk solvent screening effects due to finite salt concentrations are fully accounted for. This is accomplished using the pertinent spherical modified Bessel function multipole expansions associated with the Poisson-Boltzmann equation Green's functions. With the FPB solver, complex configurations with more than 100,000 elements can be routinely studied on PC and workstation class machines. Scalings of computer storage requirements and CPU are shown in Figure 1 for the simple case of a spherical cavity with an interior charge. Applications of FPB to moderate size biomolecules are shown in Figure 2 where the effects of salt concentration upon the surface potential distribution for the parallel right-handed coiled coil tetramer are illustrated and in Figure 3 where the surface potential maps of FinO basic protein and survivin are computed.

  1. Boschitsch, A. & Fenley, M. O. (2004). Hybrid Boundary Element and Finite Difference Method for Solving the Nonlinear Poisson-Boltzmann Equation. Journal of Computational Chemistry 25.
  2. Boschitsch, A. H., Fenley, M. O. & Zhou, H.-X. (2002). Fast Boundary Element Method for the Linear Poisson-Boltzmann Equation. Journal of Physical Chemistry B 106, 2741-2754.
  3. Boschitsch, A. H., Fenley, M. O. & Olson, W. K. (1999). A Fast Adaptive Multipole Algorithm for Calculating Screened Coulomb (Yukawa) Interactions. Journal of Computational Physics 151, 212-241.

Figure 1a. Computation times required to compute the electrostatic potential of the unit spherical cavity containing a centrally located unit charge, immersed in an aqueous solution. Timings were obtained upon a Silicon Graphics single R10000 processor operating at 180 MHz.

Figure 1b. Number of near-field influence coefficients required in the calculation of the electrostatic potential of the unit sphere containing a centrally located unit charge.

(a) Zero Salt.
Color map (-10.2,0,+0.5)

(b) 100 mM NaCL
Color map (-5.1,0,+3.4)

Figure 2. Surface electrostatic potential of the parallel right-handed coiled coil tetramer (PDB entry: 1FE6) at 0 and 100 mM NaCl. The scale is expressed in GRASP notation and units of kBT/e, with negative regions colored red, zero potential colored white and positive regions colored blue. The molecular surface is resolved using 117,541 boundary elements. The screening effect of the salt produces smaller absolute electrostatic potentials and regions of positive electrostatic potential appear.

(a) FinO basic protein (PDB entry: 1DVO).
Color map range: (-6.8,0,+6.8).
101,794 boundary elements.

(b) Survivin (PDB entry: 1F3H).
Color map range: (-13.6,0,+10.2).
145,690 boundary elements.

Figure 3. Mapping of the electrostatic potential (in kBT/e=0.59 kcal/mol/e) on the surface of the FinO basic protein and survivin molecules at 100 mM NaCl, computed with the FPB code. Color maps are in GRASP notation with negative regions colored red, zero potential colored white and positive regions colored blue.

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