BIT Numerical Mathematics

, Volume 47, Issue 4, pp 853–871

A new minimization protocol for solving nonlinear Poisson–Boltzmann mortar finite element equation

Article

Abstract

The nonlinear Poisson–Boltzmann equation (PBE) is a widely-used implicit solvent model in biomolecular simulations. This paper formulates a new PBE nonlinear algebraic system from a mortar finite element approximation, and proposes a new minimization protocol to solve it efficiently. In particular, the PBE mortar nonlinear algebraic system is proved to have a unique solution, and is equivalent to a unconstrained minimization problem. It is then solved as the unconstrained minimization problem by the subspace trust region Newton method. Numerical results show that the new minimization protocol is more efficient than the traditional merit least squares approach in solving the nonlinear system. At least 80 percent of the total CPU time was saved for a PBE model problem.

Key words

Poisson–Boltzmann equation mortar finite element nonlinear system unconstrained minimization biomolecular simulations 

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References

  1. 1.
    Y. Achdou, Y. Maday, and O. Widlund, Iterative substructuring preconditioners for mortar element methods in two dimensions, SIAM J. Numer. Anal., 36 (1999), pp. 551–580.CrossRefMathSciNetGoogle Scholar
  2. 2.
    N. A. Baker, Improving implicit solvent simulations: a Poisson-centric view, Curr. Opin. Struc. Biol., 15 (2005), pp. 137–143.CrossRefGoogle Scholar
  3. 3.
    N. Baker, D. Sept, M. Holst, and J. A. McCammon, The adaptive multilevel finite element solution of the Poisson–Boltzmann equation on massively parallel computers, IBM J. Res. Develop., 45 (2001), pp. 427–438.CrossRefGoogle Scholar
  4. 4.
    N. A. Baker, D. Sept, S. Joseph, M. Holst, and J. A. MoCammon, Electrostatics of nanosystems: Application to microtubules and the ribosome, Proc. Nat. Acad. Sci. USA, 98 (2001), pp. 10037–10041.CrossRefGoogle Scholar
  5. 5.
    R. Bank and M. Holst, A new paradigm for parallel adaptive meshing algorithms, SIAM Rev., 45 (2003), pp. 291–323.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    F. B. Belgacem, The mortar finite element method with Lagrange multipliers, Numer. Math., 84 (1999), pp. 173–197.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    C. Bernardi, Y. Maday, and A. Patera, A new nonconforming approach to domain decomposition: The mortar element method, in H. Brezis et al., eds., Nonlinear Partial Differential Equations and Their Applications, vol. XI, Pitman Res. Notes. Math. Ser., 299, Longman Sci. Tech., Harlow, 1994, pp. 13–51.Google Scholar
  8. 8.
    M. A. Branch, T. F. Coleman, and Y. Li, A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems, SIAM J. Sci. Comput., 21 (1999), pp. 1–23.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 2nd edn., Springer, New York, 2002.MATHGoogle Scholar
  10. 10.
    R. H. Byrd, R. B. Schnabel, and G. A. Shultz, Approximate solution of the trust region problem by minimization over two-dimensional subspaces, Math. Progr., 40 (1988), pp. 247–263.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    W. Chen, Y. Shen and Q. Xia, A mortar finite element approximation for the linear Poisson–Boltzmann equation, Appl. Math. Comput., 164 (2005), pp. 11–23.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    T. F. Coleman and Y. Li, An interior, trust region approach for nonlinear minimization subject to bounds, SIAM J. Optimization, 6 (1996), pp. 418–445.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    T. F. Coleman and Y. Li, On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds, Math. Progr., 67 (1994), pp. 189–224.CrossRefMathSciNetGoogle Scholar
  14. 14.
    M. E. Davis, J. D. Madura, B. A. Luty, and J. A. McCammon, Electrostatics and diffusion of molecules in solution: Simulations with the University of Houston Browian dynamics program, Comp. Phys. Comm., 62 (1991), pp. 187–197.CrossRefGoogle Scholar
  15. 15.
    L. Debnath and P. Mikusinnski, Introduction to Hilbert Spaces with Applications, Academic Press, New York, 1990.MATHGoogle Scholar
  16. 16.
    S. C. Eisenstat and H. F. Walker, Globally convergent inexact Newton methods, SIAM J. Optim., 4 (1994), pp. 393–422.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    I. Ekeland and R. Témam, Convex Analysis and Variational Problems, SIAM, Philadelphia, 1999.MATHGoogle Scholar
  18. 18.
    K. E. Forsten, R. E. Kozack, D. A. Lauffenburger, and S. Subramaniam, Numerical solution of the nonlinear Poisson–Boltzmann equation for a membrane-electrolyte system, J. Phys. Chem., 98 (1994), pp. 5580–5586.CrossRefGoogle Scholar
  19. 19.
    G. H. Golub and C. F. van Loan, Matrix Computations, 3rd edn., John Hopkins University Press, Baltimore, MD, 1996.MATHGoogle Scholar
  20. 20.
    J. Hoffman and A. Logg, PUFFIN Web page: http://www.fenics.org/puffin/, 2004.Google Scholar
  21. 21.
    J. Hoffman, J. Jansson, A. Logg, and G. N. Wells, DOLFIN Web page: http://www.fenics.org/dolfin, 2003.Google Scholar
  22. 22.
    M. Holst and F. Saied, Numerical solution of the nonlinear Poisson–Boltzmann equation: Developing more robust and efficient methods, J. Comput. Chem., 16 (1995), pp. 337–364.CrossRefGoogle Scholar
  23. 23.
    M. Holst, N. Baker, and F. Wang, Adaptive multilevel finite element solution of the Poisson–Boltzmann equation; I: Algorithms and examples, J. Comput. Chem., 21 (2000), pp. 1319–1342.CrossRefGoogle Scholar
  24. 24.
    B. Honig and A. Nicholls, Classical electrostatics in biology and chemistry, Science, 268 (1995), pp. 1144–1149.CrossRefGoogle Scholar
  25. 25.
    B. Al-Lazikani, J. Jung, Z. Xiang, and B. Honig, Protein structure prediction, Current Option in Chemical Biology, 5 (2001), pp. 51–56.CrossRefGoogle Scholar
  26. 26.
    B. P. Lamichhane and B. I. Wohlmuth, Mortar finite elements for interface problems, Computing, 72 (2004), pp. 333–348.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    The MathWorks, Inc., MATLAB Partial Differential Equation Toolbox 1.0.8 Web page: http://www.mathworks.com/products/pde/, 2005.Google Scholar
  28. 28.
    J. J. Moré and D. C. Sorensen, Computing a Trust Region Step, SIAM J. Sci. Stat. Comput., 3 (1983), pp. 553–572.CrossRefGoogle Scholar
  29. 29.
    J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, 2000.Google Scholar
  30. 30.
    W. Rocchia, E. Alexov, and B. Hoing, Extending the applicability of the nonlinear Poisson–Boltzmann equation: Multiple dielectric constants and multivalent ions, J. Phys. Chem. B, 105 (2001), pp. 6507–6514.CrossRefGoogle Scholar
  31. 31.
    B. Roux and T. Simonson, Implicit solvent models, Biophys. Chem., 78 (1999), pp. 1–20.CrossRefGoogle Scholar
  32. 32.
    C. Tanford, Physical Chemistry of Macromolecules, John Wiley & Sons, New York, NY, 1961.Google Scholar
  33. 33.
    Y. N. Vorobjev and H. A. Scheraga, A fast adaptive multigrid boundary element method for macromolecular electrostatic computations in a solvent, J. Comput. Chem., 18 (1997), pp. 569–583.CrossRefGoogle Scholar
  34. 34.
    J. Wagoner and N. A. Baker, Solvatin forces on biomolecular structures: a comparison of explicit solvent and Poisson–Boltzmann models, J. Comput. Chem., 25 (2004), pp. 1623–1629.CrossRefGoogle Scholar
  35. 35.
    B. I. Wohlmuth, A V-cycle multigrid approach for mortar finite elements, SIAM J. Numer. Anal., 42 (2005), pp. 2476–2405.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of WisconsinMilwaukeeUSA
  2. 2.Department of Applied MathematicsHunan UniversityChangshaP.R. China

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