Calculation of the Maxwell stress tensor and the Poisson-Boltzmann force on a solvated molecular surface using hypersingular boundary integrals.

The electrostatic interaction among molecules solvated in ionic solution is governed by the Poisson-Boltzmann equation (PBE). Here the hypersingular integral technique is used in a boundary element method (BEM) for the three-dimensional (3D) linear PBE to calculate the Maxwell stress tensor on the solvated molecular surface, and then the PB forces and torques can be obtained from the stress tensor. Compared with the variational method (also in a BEM frame) that we proposed recently, this method provides an even more efficient way to calculate the full intermolecular electrostatic interaction force, especially for macromolecular systems. Thus, it may be more suitable for the application of Brownian dynamics methods to study the dynamics of protein/protein docking as well as the assembly of large 3D architectures involving many diffusing subunits. The method has been tested on two simple cases to demonstrate its reliability and efficiency, and also compared with our previous variational method used in BEM.

[1]  J. A. McCammon,et al.  Solving the finite difference linearized Poisson‐Boltzmann equation: A comparison of relaxation and conjugate gradient methods , 1989 .

[2]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[3]  Jeng-Tzong Chen,et al.  A Practical Guide to Boundary Element Methods with the Software Library BEMLIB , 2002 .

[4]  O. Huber,et al.  Evaluation of the stress tensor in 3D elastostatics by direct solving of hypersingular integrals , 1993 .

[5]  F. Rizzo,et al.  A General Algorithm for the Numerical Solution of Hypersingular Boundary Integral Equations , 1992 .

[6]  Ray Luo,et al.  Accelerated Poisson–Boltzmann calculations for static and dynamic systems , 2002, J. Comput. Chem..

[7]  B. Honig,et al.  A rapid finite difference algorithm, utilizing successive over‐relaxation to solve the Poisson–Boltzmann equation , 1991 .

[8]  Benzhuo Lu,et al.  Computation of electrostatic forces between solvated molecules determined by the Poisson-Boltzmann equation using a boundary element method. , 2005, The Journal of chemical physics.

[9]  R. Zauhar,et al.  A new method for computing the macromolecular electric potential. , 1985, Journal of molecular biology.

[10]  O. D. Kellogg.  Foundations of potential theory , 1934 .

[11]  H. Zhou,et al.  Boundary element solution of macromolecular electrostatics: interaction energy between two proteins. , 1993, Biophysical journal.

[12]  Nathan A. Baker,et al.  Adaptive multilevel finite element solution of the Poisson–Boltzmann equation I. Algorithms and examples , 2000 .

[13]  L. R. Scott,et al.  Electrostatics and diffusion of molecules in solution: simulations with the University of Houston Brownian dynamics program , 1995 .

[14]  Jan Sladek,et al.  Singular integrals in boundary element methods , 1998 .

[15]  Stephen C. Harvey,et al.  Finite element approach to the electrostatics of macromolecules with arbitrary geometries , 1993, J. Comput. Chem..

[16]  Wei Zu Chen,et al.  A Stochastic Dynamics Simulation Study Associated with Hydration Force and Friction Memory Effect , 2000 .

[17]  J. Andrew McCammon,et al.  Computation of electrostatic forces on solvated molecules using the Poisson-Boltzmann equation , 1993 .

[18]  S. Subramaniam,et al.  Protein electrostatics: rapid multigrid-based Newton algorithm for solution of the full nonlinear Poisson-Boltzmann equation. , 1994, Journal of biomolecular structure & dynamics.

[19]  Andrew J. Bordner,et al.  Boundary element solution of the linear Poisson–Boltzmann equation and a multipole method for the rapid calculation of forces on macromolecules in solution , 2003, J. Comput. Chem..

[20]  C. Tanford.  Macromolecules , 1994, Nature.

[21]  Stuart A. Allison.  Modeling the Electrophoresis of Rigid Polyions. Inclusion of Ion Relaxation , 1996 .

[22]  Wei Zu Chen,et al.  Protein molecular dynamics with electrostatic force entirely determined by a single Poisson‐Boltzmann calculation , 2002, Proteins.

[23]  R. Zauhar,et al.  The incorporation of hydration forces determined by continuum electrostatics into molecular mechanics simulations , 1991 .