Finite element approach to the electrostatics of macromolecules with arbitrary geometries

Abstract

AbstractA new treatment of macromolecular electrostatics has been developed using the 3‐D finite element method to numerically solve the linear Poisson–Boltzmann equation. The procedure is based upon a model where the macromolecule is represented at an atomic level of detail, while the solvent is treated in a continuum approximation. The finite element method has two major advantages over previous methods based upon the finite difference approach. First, charges are located on atomic centers rather than being distributed onto grid points. Second, an isoparameter model allows the use of noncubic grids, providing a more accurate description of molecular shape. The principal disadvantage of the finite element method has been its computational complexity, which arises from the use of large matrices. To overcome this difficulty, a new matrix representation has been formulated and an iterative solution procedure has been adopted. The combination of these two techniques drastically reduces the size of the system matrix and increases the overall computational efficiency of the algorithm, making the new treatment computationally competitive with the finite difference approach. Because of the mathematical rigor and physical sophistication of the finite element algorithm, the new treatment is able to give an accurate description of the electrostatic potential distribution in a macromolecular system. Results on test cases with simple geometries show that the new treatment is able to reach the same level of accuracy achieved by the finite difference method while using a lower grid density. Near changes and surfaces, our method is more accurate than the finite difference method. The overall maximum deviation between computed and analytic potentials is less than 3% except in regions surrounding charges. The applicaions of both the finite element and finite difference methods to the same biomolecular systems produce similar potential distributions that would become identical in the limit of infinitely fine grids. © 1993 John Wiley & Sons, Inc.