Interface method and Green's function based Poisson Boltzmann equation solver and interface technique based molecular dynamics

This dissertation describes the development of a higher order interface and Green's function based method for solving the Poisson Boltzmann (PB) equation with both geometric singularities and charge singularities. Meanwhile, in the framework of the implicit solvent model, an interface technique based molecular dynamics method is developed for biomolecules. Some material presented in this thesis is adopted from related reprints and preprints.;The author starts in Chapter 1 from a bird's eye view of the current interface methods, focusing on matched interface and boundary (MIB) method developed in Prof. Wei's group in the past five years.;Chapter 2 reviews the history, status, challenge and application of the PB model. The numerical difficulties of PB equation solvers in handling discontinuous coefficients and solutions, geometric and charge singularities are described there. Meanwhile, as the basis of later chapters, the formulation of PB problem is provided.;Chapter 3 describes the development of the third generation of MIB based PB equation solvers, the MIBPB-III, whose development is based on a Green's function formalism and is a natural continuation of the MIBPB-I and MIBPB-II, the previous two generation of MIBPB with the enforcement of interface flux continuity conditions to the PB equation and the treatment of geometric singularities respectively. In the present Green's function formalism, the MIBPB-III, the charge singularities are transformed into interface flux jump conditions, which are treated on an equal footing as geometric singularities in our MIB framework. The MIBPB-III is able to provide highly accurate electrostatic potentials at a mesh as coarse as 1.2A for proteins.;Chapter 4 describes the application of the MIB method to the development of the first PB based molecular dynamics (MD) method that directly admits sharp molecular surfaces. The classical formulation of electrostatic forces is modified to allow the use of sharp molecular surfaces. Accurate reaction field forces are obtained by directly differentiating the electrostatic potential. Dielectric boundary forces are evaluated at the solvent-solute interface using an accurate Cartesian-grid surface integration method. The electrostatic forces located at reentrant surfaces are appropriately assigned to relevant atoms. Extensive numerical tests are carried out to validate the accuracy and stability of the present electrostatic force calculation. The resulting electrostatic forces are compared with those in the literature. The present MIB and PB based molecular dynamics simulations of biomolecules are demonstrated in conjugation with the AMBER package. It has been shown that there is a pressing need to examine the convergence, stability and reliability of previous PB based MD methods.;Chapter 5 is the summary of author's thesis contributions and a brief description of important future topics in the PB methods. Emphasis is given to potential developments in mathematical modeling and computation of biomolecular systems.