Modeling and computation of heterogeneous implicit solvent and its applications for biomolecules
Description of inhomogeneous dielectric properties of a solvent in the vicinity of ions has been attracting research interests in mathematical modeling for many years. From many experimental results, it has been concluded that the dielectric response of a solvent linearly depends on the ionic strength within a certain range. Based on this assumption, a new implicit solvent model is proposed in the form of total free energy functional and a quasi-linear Poisson-Boltzmann equation (QPBE) is derived. Classical Newton’s iteration can be used to solve the QPBE numerically but the corresponding Jacobian matrix is complicated due to the quasi-linear term. In the current work, a systematic formulation of the Jacobian matrix is derived. As an alternative option, an algorithm mixing the Newton’s iteration and the fixed point method is proposed to avoid the complicated Jacobian matrix, and it is a more general algorithm for equation with discontinuous coefficients. Computational efficiency and accuracy for these two methods are investigated based on a set of equation parameters. At last, the QPBE with singular charge source and piece-wisely defined dielectric functions has been applied to analyze electrostatics of macro biomolecules in a complicated solvent. A set of computational algorithms such as interface method, singular charge removal technique and the Newtonfixed- point iteration are employed to solve the QPBE. Biological applications of the proposed model and algorithms are provided, including calculation of electrostatic solvation free energy of proteins, investigation of physical properties of channel pore of an ion channel, and electrostatics analysis for the segment of a DNA strand.
-  B. Honig, A. Nicholls, Classical electrostatics in biology and chemistry, Science 268 (5214) (1995) 1144–9.
-  R. Luo, L. David, M. K. Gilson, Accelerated Poisson-Boltzmann calculations for static and dynamic systems, Journal of Computational Chemistry 23 (13) (2002) 1244–53. [Crossref]
-  J. Warwicker, H. C. Watson, Calculation of the electric potential in the active site cleft due to alpha-helix dipoles, Journal of Molecular Biology 157 (4) (1982) 671–9.
-  W. Im, D. Beglov, B. Roux, Continuum solvation model: electrostatic forces from numerical solutions to the Poisson- Boltzmann equation, Computer Physics Communications 111 (1-3) (1998) 59–75.
-  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 of the United States of America 98 (18) (2001) 10037–10041. [Crossref]
-  A. H. Boschitsch, M. O. Fenley, Hybrid boundary element and finite difference method for solving the nonlinear Poisson- Boltzmann equation, Journal of Computational Chemistry 25 (7) (2004) 935–955. [Crossref]
-  R. J. Zauhar, R. S. Morgan, A new method for computing the macromolecular electric potential, Journal of Molecular Biology 186 (4) (1985) 815–20. [Crossref]
-  I. Klapper, R. Hagstrom, R. Fine, K. Sharp, B. Honig, Focussing of electric fields in the active site of cu-zn superoxide dismutase: Effects of ionic strength and amino acid modification, Protein 1 (1986) 47 – 59.
-  N. V. Prabhu, M. Panda, Q. Y. Yang, K. A. Sharp, Explicit ion, implicit water solvation for molecular dynamics of nucleic acids and highly charged molecules, J. Comput. Chem. 29 (2008) 1113–1130. [Crossref]
-  J. D. Madura, J. M. Briggs, R. C. Wade, M. E. Davis, B. A. Luty, A. Ilin, J. Antosiewicz, M. K. Gilson, B. Bagheri, L. R. Scott, J. A. McCammon, Electrostatics and diffusion of molecules in solution - simulations with the University of Houston Brownian Dynamics program, Computer Physics Communications 91 (1-3) (1995) 57–95.
-  M. Engels, K. Gerwert, D. Bashford, Computational studies on bacteriorhodopsin: Conformation and proton transfer energetics, Biophys. Chem. 56 (1995) 95.
-  M. Holst, N. Baker, F.Wang, Adaptive multilevel finite element solution of the Poisson-Boltzmann equation I. algorithms and examples, Journal of Computational Chemistry 21 (15) (2000) 1319–1342. [Crossref]
-  B. Z. Lu, W. Z. Chen, C. X. Wang, X. J. Xu, Protein molecular dynamics with electrostatic force entirely determined by a single Poisson-Boltzmann calculation, Proteins 48 (3) (2002) 497–504. [PubMed][Crossref]
-  S. Jo, M. Vargyas, J. Vasko-Szedlar, B. Roux, W. Im, Pbeq-solver for online visualization of electrostatic potential of biomolecules, Nucleic Acids Research 36 (2008) W270–W275. [Crossref]
-  B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. States, S. Swaminathan, M. Karplus, Charmm: A program formacromolecular energy, minimization, and dynamics calculations, J. Comput. Chem. 4 (1983) 187–217. [Crossref]
-  Y. C. Zhou, M. Feig, G. W. Wei, Highly accurate biomolecular electrostatics in continuum dielectric environments, Journal of Computational Chemistry 29 (2008) 87–97. [Crossref]
-  W. Geng, S. Yu, G. W. Wei, Treatment of charge singularities in implicit solvent models, Journal of Chemical Physics 127 (2007) 114106. [Crossref]
-  D. Chen, Z. Chen, C. Chen,W. H. Geng, G.W. Wei, MIBPB: A software package for electrostatic analysis, J. Comput. Chem. 32 (2011) 756 – 770. [Crossref]
-  R. J. LeVeque, Z. L. Li, The immersed interface method for elliptic equations with discontinuous coeflcients and singular sources, SIAM J. Numer. Anal. 31 (1994) 1019–1044. [Crossref]
-  Z. L. Li, K. Ito, Maximum principle preserving schemes for interface problems with discontinuous coeflcients, SIAM J. Sci. Comput. 23 (2001) 339–361. [Crossref]
-  W. Geng, R. Krasny, A treecode-accelerated boundary integral Poisson-Boltzmann solver for continuum electrostatics of solvated biomolecules, Journal of Computational Physics 247 (20132-87) 6.
-  A. Hildebrant, R. Blossey, S. Rjasanow, O. Kohlbacher, H. Lenhof, Novel Formulation of nonlocal electrostatics, Physical Review Letter 93 (2004) 108101–1.
-  D. Ben-Yaakov, D. Andelman, R. Podgornik, R. Podgornik, Ion-specific hydration effects: Extending the Poisson-Boltzmann theory, Current Opinion in Colloid and Interface Science 16 (2011) 542–550.
-  I. Borukhov, D. Andelman, H. Orland, Adsorption of large ions from an electrolyte solution: a modified Poisson-Boltzmann equation, Electrochim Acta 46 (2000) 221–9. [Crossref]
-  D. Ben-Yaakov, D. Andelman, R. Podgornik, Dielectric decrement as a source of ion-specific effects, J Chem. Phys. 134 (2011) 074705.
-  M. Z. Bazant, B. D. Storey, A. A. Kornyshev, Double layer in ionic liquids: Overscreening versus crowding, Physical Review Letters 106 (2011) 046102. [Crossref][PubMed]
-  D. Gillespie, W. Nonner, R. S. Eisenberg, Density functional theory of charged, hard-sphere fluids, Phys Rev E 68 (2003) 1–10.
-  B. Li, P. Liu, Z. Xu, S. Zhou, Ionic size effects: generalized Boltzmann distributions, counterion stratification, and modified Debye length , Nonlinearity (2013) 2899–2922.
-  Y. Hyon, B. Eisenberg, C. Liu, A mathematical model for the hard sphere repulsion in ionic solutions, Commun. Math. Sci. 9 (2011) 459–475.
-  L. Hu, G. W. Wei, Nonlinear poisson equation for heterogeneous media, Biophys. J.
-  D. Xie, Y. Jiang, L. Scott, Eflcient algorithms for solving a nonlocal dielectric model for protein in ionic solvent, SIAM Journal on Scientific Computing 38 (2013) B1267–1284. [Crossref]
-  H. Li, B. Lu, An ionic concentration and size dependent dielectric permittivity Poisson-Boltzmann model for biomolecular solvation studies, J. Chem. Phys. 141 (2014) 024115.
-  G. Wei,Multiscalemultiphysics andmultidomain models I: Basic theory, Journal of Theoretical andComputational Chemistry 12 (2013) 1341006.
-  G. W. Wei, Differential geometry based multiscale models, Bulletin of Mathematical Biology 72 (2010) 1562 – 1622. [Crossref]
-  B. S. Eisenberg, Y. K. Hyon, C. Liu, Energy variational analysis of ions inwater and channels: Field theory for primitive models of complex ionic fluids, Journal of Chemical Physics 133 (2010) 104104.
-  D. Chen, G. W. Wei, Quantum dynamics in continuum for proton transport I: Basic formulation, Commun. Comput. Phys. 13 (2013) 285–324.
-  D. Chen, Z. Chen, G.W. Wei, Quantumdynamics in continuumfor proton transport II: Variational solvent-solute intersurface, International Journal for Numerical Methods in Biomedical Engineering 28 (2012) 25–51.
-  D. Chen, G. W. Wei, Quantum dynamics in continuum for proton transport III: Generalized correlation, J Chem. Phys. 136 (2012) 134109.
-  D. Chen, G.W. Wei, Modeling and simulation of electronic structure,material interface and random doping in nano-electronic devices, J. Comput. Phys. 229 (2010) 4431–4460.
-  J. Che, J. Dzubiella, B. Li, J. A. McCammon, Electrostatic free energy and its variations in implicit solvent models, Journal of Physical Chemistry B 112 (10) (2008) 3058–69. [Crossref]
-  B. Li, X. Cheng, Z. Zhang, Dielectric boundary force in molecular solvation with the Poisson-Boltzmann free energy: A shape derivative approach, SIAM J. Applied Math. 71 (2011) 2093–2111. [Crossref]
-  Y. Z. Wei, S. Sridhar, Dielectric spectroscopy up to 20 GHz of LiCl/H2O solutions, J. Chem. Phys. 92 (1990) 923–928.
-  Y. Z. Wei, P. Chiang, S. Sridhar, Ion size effects on the dynamic and static dielectric properties of aqueous alkali solutions, J. Chem. Phys. 96 (1992) 4596.
-  R. Buchner, G. T. Hefter, P. M. May, Dielectric Relaxation of Aqueous NaCl Solutions, J. Chem. Phys. A 103 (1999) 1–9.
-  S. Senapati, A. Chandra, Surface charge induced modifications of the structure and dynamics of mixed dipolar liquids at solid-liquid interfaces: A molecular dynamics simulation study, J. Chem. Phys. 113 (2000) 8817–8826.
-  M. J. Holst, F. Saied, Numerical solution of the nonlinear Poisson-Boltzmann equation: developing more robust and eflcient methods, Journal of Computational Chemistry 16 (3) (1995) 337–64. [Crossref]
-  S. Zhao, Operator splitting ADI schemes for pseudo-time coupled nonlinear solvation simulations, Journal of Computational Physics 257 (2014) 1000–1021.
-  M. K. Gilson, M. E. Davis, B. A. Luty, J. A. McCammon, Computation of electrostatic forces on solvated molecules using the Poisson-Boltzmann equation, Journal of Physical Chemistry 97 (14) (1993) 3591–3600. [Crossref]
-  K. A. Sharp, B. Honig, Calculating total electrostatic energies with the nonlinear Poisson-Boltzmann equatlon, Journal of Physical Chemistry 94 (1990) 7684–7692. [Crossref]
-  Z. Chen, N. A. Baker, G. W. Wei, Differential geometry based solvation models I: Eulerian formulation, J. Comput. Phys. 229 (2010) 8231–8258.
-  Z. Chen, N. A. Baker, G. W. Wei, Differential geometry based solvation models II: Lagrangian formulation, J. Math. Biol. 63 (2011) 1139–1200.[Crossref]