The Poisson-Boltzmann (PB) model is an effective approach for the electrostatics analysis of solvated biomolecules. The nonlinearity associated with the PB equation is critical when the underlying electrostatic potential is strong, but is extremely difficult to solve numerically. In this paper, we construct two operator splitting alternating direction implicit (ADI) schemes to efficiently and stably solve the nonlinear PB equation in a pseudo-transient continuation approach. The operator splitting framework enables an analytical integration of the nonlinear term that suppresses the nonlinear instability. A standard finite difference scheme weighted by piecewise dielectric constants varying across the molecular surface is employed to discretize the nonhomogeneous diffusion term of the nonlinear PB equation, and yields tridiagonal matrices in the Douglas and Douglas-Rachford type ADI schemes. The proposed time splitting ADI schemes are different from all existing pseudo-transient continuation approaches for solving the classical nonlinear PB equation in the sense that they are fully implicit. In a numerical benchmark example, the steady state solutions of the fully-implicit ADI schemes based on different initial values all converge to the time invariant analytical solution, while those of the explicit Euler and semi-implicit ADI schemes blow up when the magnitude of the initial solution is large. For the solvation analysis in applications to real biomolecules with various sizes, the time stability of the proposed ADI schemes can be maintained even using very large time increments, demonstrating the efficiency and stability of the present methods for biomolecular simulation.
@article{bwmeta1.element.doi-10_2478_mlbmb-2013-0006,
author = {Weihua Geng and Shan Zhao},
title = {Fully implicit ADI schemes for solving the nonlinear Poisson-Boltzmann equation},
journal = {Molecular Based Mathematical Biology},
volume = {1},
year = {2013},
pages = {109-123},
zbl = {1276.65063},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_mlbmb-2013-0006}
}
Weihua Geng; Shan Zhao. Fully implicit ADI schemes for solving the nonlinear Poisson-Boltzmann equation. Molecular Based Mathematical Biology, Tome 1 (2013) pp. 109-123. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_mlbmb-2013-0006/
N.A. Baker, D. Sept, S. Joseph, M.J. Holst, J.A. McCammon, Electrostatics of nanosystems: application to microtubules and the ribosome. Proceedings of the National Academy of Sciences 98 (2001) 10037-10041.
N.A. Baker, Improving implicit solvent simulations: a Poisson-centric view. Curr. Opin. Struct. Biol. 15 (2005) 137-143. [PubMed][Crossref]
W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25 (2004) 1674-1697. | Zbl 1061.82025
W. Bao, S. Jin and P.A. Markowich, On time-splitting spectral approximation for the Schrodinger equation in the semiclassical regime. J. Comput. Phys. 175 (2002) 487-524. | Zbl 1006.65112
P.W. Bates, Z. Chen, Y.H. Sun, G.W. Wei, and S. Zhao, Geometric and potential driving formation and evolution of biomolecular surfaces. J. Math. Biol. 59 (2009) 193-231. [WoS] | Zbl 1311.92212
P.W. Bates, G.W. Wei and S. Zhao, Minimal molecular surfaces and their applications. J. Comput. Chem. 29 (2008) 380-391. [WoS]
A.H. Boschitsch, M.O. Fenley, and H.-X. Zhou, Fast boundary element method for the linear Poisson-Boltzmann equation. J. Phys. Chem. B 106 (2002) 2741-2754.
A.H. Boschitsch and M.O. Fenley, Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation. J. Comput. Chem. 25 (2004) 935-955.
Q. Cai, M.J. Hsieh, J. Wang, and R. Luo, Performance of nonlinear finite-difference Poisson-Boltzmann solvers. J. Chem. Theory Comput. 6 (2010) 203-211. [Crossref][WoS]
D. Chen, Z. Chen, C.J. Chen, W.H. Geng, G.W. Wei, MIBPB: A software package for electrostatic analysis. J. Comput. Chem. 32 (2011) 756-770.
J.H. Chen, C.L. Brooks, J. Khandogin, Recent advances in implicit solvent based methods for biomolecular simulations. Curr. Opin. Struct. Biol. 18 (2008) 140-148. [WoS]
Z. Chen, N. Baker, and G.W. Wei, Differential geometry based solvation model I: Eulerian formation, J. Comput. Phys. 229 (2010) 8231-8258. | Zbl 1229.92030
W. Dai and R. Nassar, A generalized Douglas ADI method for solving three-dimensional parabolic differential equations on multilayers. Int. J. Numer. Meth. Heat Fluid Flow 7 (1997) 659-674. [Crossref] | Zbl 0958.65092
W.H. Geng, S.N. Yu, and G.W. Wei, Treatment of charge singularities in implicit solvent models. J. Phys. Chem. 127 (2007) Art. ID 114106, 20pp.
W.H. Geng and R. Krasny, Treecode-accelerated boundary integral Poisson-Boltzmann solver. J. Comput. Phys. (2013) accepted.
W.H. Geng, Parallel higher-order boundary integral electrostatics computation on molecular surfaces with curved triangulation. J. Comput. Phys. 241 (2013) 253-265. [WoS]
W.H. Geng and J. Ferosh, A GPU-accelerated direct-sum boundary integral Poisson-Boltzmann solver. Comput. Phys. Commun. 184 (2007) 1490-1496. | Zbl 1310.78017
M.J. Holst, Multilevel methods for the Poisson-Boltzmann equation, Ph.D. thesis, University of Illinois at Urbana- Champaign, Illinois, USA, 1993.
M.J. Holst and F. Saied, Numerical solution of the nonlinear Poisson-Boltzmann equation: Developing more robust and efficient methods. J. Comput. Chem. 16 (1995) 337-364.
W. Im, D. Beglov and B. Roux, Continuum solvation model: Computation of electrostatic forces from numerical solutions to the Poisson-Boltzmann equation. Comput. Phys. Commun. 111 (1998) 59-75. | Zbl 0935.78019
A. Juffer, E. Botta, B. van Keulen, A. van der Ploeg, H. Berendsen, The electric potential of a macromolecule in a solvent: a fundamental approach. J. Comput. Phys. 97 (1991) 144-171. | Zbl 0743.65094
J.G. Kirkwood, Theory of solution of molecules containing widely separated charges with special application to Zwitterions. J. Chem. Phys. 7 (1934) 351-361. [Crossref] | Zbl 0009.27504
B. Lee and F.M. Richards, Interpretation of protein structures: estimation of static accessibility. J. Mol. Biol. 55 (1973) 379-400.
B.Z. Lu, X.L. Cheng, J.F. Huang, J.A. McCammon, Order N algorithm for computation of electrostatic interactions in biomolecular systems. Proceedings of the National Academy of Sciences 103 (2006) 19314-19319.
R. Luo, L. David, M.K. Gilson, Accelerated Poisson-Boltzmann calculations for static and dynamic systems. J. Comput. Chem. 23 (2002) 1244-1253.
B.A. Luty, M.E. Davis, and J.A. McCammon, Solving the finite-difference nonlinear Poisson-Boltzmann equation. J. Comput. Chem. 13 (1992) 1114-1118.
Z.H. Qiao, Z.L. Li, T. Tang, A finite difference scheme for solving the nonlinear Poisson-Boltzmann equation modeling charged spheres. J. Comput. Math. 24 (2006) 252–264. | Zbl 1105.78015
F.M. Richards, Areas, volumes, packing and protein structure. Annu. Rev. Biophys. Bioeng. 6 (1977), 151-176. [PubMed][Crossref]
W. Rocchia, E. Alexov, and B. Honig, Extending the applicability of the nonlinear Poisson-Boltzmann equation: Multiple dielectric constants and multivalent ions. J. Phys. Chem. B 105 (2001) 6507-6514.
S. A. Sauter and C. Schwab, Boundary Element Methods, Springer, 2010.
A. Sayyed-Ahmad, K. Tuncay, and P.J. Ortoleva, Efficient solution technique for solving the Poisson-Boltzmann equation. J. Comput. Chem. 25 (2004) 1068-1074.
A.I. Shestakov, J.L. Milovich, and A. Noy, Solution of the nonlinear Poisson-Boltzmann equation using pseudotransient continuation and the finite element method. J. Colloid Interface Sci. 247 (2002) 62-79.
J.C. Strikwerda, Finite difference schemes and partial differential equations, Second Edition, SIAM, 2004. | Zbl 1071.65118
G.W. Wei, Differential geometry based multiscale models. Bullet. Math. Bio. 72 (2010) 1562-1622. | Zbl 1198.92001
R. Yokota, J.P. Bardhan, M.G. Knepley, L.A. Barba, T. Hamada, Biomolecular electrostatics using a fast multipole BEM on up to 512 GPUS and a billion unknowns. Comput. Phys. Commun. 182 (2011) 1272-1283. [WoS] | Zbl 1259.78044
S. Yu, S. Zhao, and G.W. Wei, Local spectral time-splitting method for first and second order partial differential equations. J. Comput. Phys. 206 (2005) 727-780. | Zbl 1075.65130
S. Zhao, Pseudo-time coupled nonlinear models for biomolecular surface representation and solvation analysis. Int. J. Numer. Method Bio. Engrg. 27 (2011) 1964-1981. | Zbl 1241.92007
S. Zhao, Operator splitting ADI schemes for pseudo-time coupled nonlinear solvation simulations. J. Comput. Phys. (2013) submitted.
S. Zhao, J. Ovadia, X. Liu, Y.-T. Zhang, and Q. Nie, Operator splitting implicit integration factor methods for stiff reaction-diffusion-advection systems. J. Comput. Phys. 230 (2011) 5996-6009. [WoS] | Zbl 1220.65120
Y.C. Zhou, S. Zhao, M. Feig and G. W. Wei, High order matched interface and boundary (MIB) schemes for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys. 213 (2006) 1-30. | Zbl 1089.65117