Matched interface and boundary (MIB) method and its applications to implicit solvent modeling of biomolecules

This dissertation describes the matched interface and boundary (MIB) method [94, 96] for elliptic interface problems, in which the equations have discontinuous coefficients and possible singular source on the interface. This low regularity leads to the slow convergence or divergence of most traditional numerical methods for smooth problems. The complexity of the interface makes it more difficult to develop efficient numerical methods.; The matched interface and boundary method is closely related to the methods with ghost points while it differs from these methods in the implicit enforcing of the interface conditions. A uniform Cartesian grid is used in the formulation of the MIB to take advantages of the conventional high order central finite difference schemes. Attentions are only paid to the irregular grid points near the interface where the difference schemes involve the fictitious values instead of solely solution values. A local coordinate transformation is used to project the interface conditions defined in the gradient direction on the interface to the coordinate directions, which allows the MIB method to handle irregular interfaces. By iterative application of the interface conditions, fourth and sixth order numerical methods for general elliptic interface problems are successfully derived for the first time.; Both immersed interface method (IIM) and MIB methods are used for the accurate solution of the Poisson-Boltzmann equation for electrostatic potentials. Molecular surface generated with MSMS is chosen as the dielectric interface and special techniques are developed to implement this triangulized molecular surface into the finite difference method with Cartesian grid. This is the first application of analytical molecular surface in the finite-difference-based Poisson-Boltzmann solvers, and is also the first method which conserves the continutity of potential flux at the molecular surface. Substantial improvements in the surface potentials and the electrostatic solvation energies are found as a result of this work. It is also found that, by mesh refinements, the solutions of all other traditional Poisson-Boltzmann solvers essentially converge to results of interface methods attained with a coarse mesh. Thus interface methods are relatively more efficient in achieving solutions of the same accuracy. The convergent solvation energies calculated from interface methods at coarse meshes not only permits an accurate energetic analysis of the molecular interactions but also provide an accurate calibration for the Generalized Born method. The accurate potential itself also has significant implications in the analysis of protein/ligand binding and association.