An efficient second-order poisson-boltzmann method

Abstract

Immersed interface method (IIM) is a promising high-accuracy numerical scheme for the Poisson-Boltzmann model that has been widely used to study electrostatic interactions in biomolecules. However, the IIM suffers from instability and slow convergence for typical applications. In this study, we introduced both analytical interface and surface regulation into IIM to address these issues. The analytical interface setup leads to better accuracy and its convergence closely follows a quadratic manner as predicted by theory. The surface regulation further speeds up the convergence for nontrivial biomolecules. In addition, uncertainties of the numerical energies for tested systems are also reduced by about half. More interestingly, the analytical setup significantly improves the linear solver efficiency and stability by generating more precise and better-conditioned linear systems. Finally, we implemented the bottleneck linear system solver on GPUs to further improve the efficiency of the method, so it can be widely used for practical biomolecular applications. © 2019 Wiley Periodicals, Inc.

Figure 1.

Treatment of the reentry surface. The solid line represents the solvent excluded surface. The dash line represents the solvent accessible surface. The dot circle represents the solvent probe. “+” signs represent surface elements with positive curvature and “−” signs represent surface elements with negative curvature.

Figure 2.

Convergence of reaction field energies (kcal/mol) versus grid spacing (Å) for analytical sphere models with both numerical and analytical setups for IIM. The energy results are obtained by averaging 30 grid orientations/offsets for each test case. All the curves are obtained by fitting data to parabolas. Note that the data points at grid spacing 1 Å are not included in the fitting.

Figure 3.

RMSDs between computed normal component of the surface electric field with analytical values (kcal/mol-e-Å) versus grid spacing (Å) for analytical sphere models with both numerical and analytical setups for IIM.

Figure 4.

Convergence trends for reaction field energies of nontrivial biomolecules (kcal/mol) versus grid spacing (Å). The energy results are obtained by averaging of systematic 30 rotations/offsets of the tested molecules. All the curves are obtained by fitting data at discrete grid point to a parabola (y=a+bx²). (Note that for the analytical setup without surface regulation, data at grid spacing equal to 1 Å are not included in the fitting.)

Figure 5.

Timing comparison between two different setups (without regulation).

Figure 6.

Comparison of PB energies (kcal/mol) and solver timing (seconds) between GPU and CPU runs. The energy trend line is 1.001x+1.088, with the medium relative deviation between the two sets 0.066%. The timing trend line is 0.045x+0.921, with the GPU program about 20 times faster than the CPU program.