Mathematical methods for implicit solvent models

1. Mathematical methods for implicit solvent models W. H. Geng1, S.N. Yu1, Y.C. Zhou1, Michael Feig2 and G.W. Wei11Department of Mathematics, 2Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI 48824 USA. Introduction Numerical results for 24 proteins Implicit solvent models which treat the solvent as a macroscopic continuum while admitting a microscopic atomic description for biomolecules, are efficient multiscale approaches to complex, large scale biological systems, and involve the Poisson-Boltzmann (PB) equation: Challenges in the Poisson Boltzmann equation Figure 2: Computational results for 24 proteins. The proteins are ordered with increasing radii of gyration. Left: The difference of solvation energy (Kcal/mol) between grids h=0.25Å and h=0.5Å. Right: CPU time in seconds. Figure 1: Numerical challenges for solving the PB equation. Left: Interface discontinuity; Middle: Geometrical singularities on molecular surfaces; Right: Charge singularities inside the molecule. Our approaches We introduce a rigorous mathematical treatment of molecular surface interfaces and geometric singularities by using the matched interface and boundary (MIB) method. Green functions are introduced to remove charge singularities from the numerical solution. The resulting PB solver, called MIBPB, is used for electrostatic force evaluation according to the force density Figure 3: The map of electrostatic potential on the surface of protein 451C. Left: potential computed with MIBPB; Right: Difference of potentials computed with PBEQ and with MIBPB. Conclusion Our 2nd order PBE solver, MIBPB, taking into account for the flux continuity condition of the electrostatic potential, geometric singularities of molecular surfaces and charge singularities, is orders of magnitude more accurate at the same mesh size and about 2~3 times faster at the same accuracy than the earlier PBE solvers. Numerical results for the Kirkwood model References 1. W.H. Geng, S.N. Yu, and G.W. Wei, Treatment of charge singularities in implicit solvent model, J. Chem. Phys., 127, 114106 (2007) 2. S.N. Yu, W.H. Geng, and G.W. Wei, Treatment of geometric singularities in implicit solvent model, J. Chem. Phys., 126, 244108 (2007) 3. 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., 231, 1-30 (2006) Table 1: An analytical test case. Electrostatic solvation energy in kcal/mol and the error of the surface potential in kcal/mol/ec for a sphere with a centered unit charge. The exact solvation energy is -81.98 kcal/mol. The maximal absolute errors in the electrostatic potential are listed. Acknowledgement This work was partially supported by NSF grants DMS-0616704 and IIS-0430987.