| The Poisson-Boltzmann equation (PBE) is one widely-used implicit solvent continuum model for computing biomolecular electrostatics potentials, ionic con-centrations, solvation energies, and molecular binding free energies for a biomolecule in an ionic solvent. Despite its success in many biochemistry and bioengineering applications, the PBE model has been known to have some drawbacks caused by its neglecting ionic size effects. To consider certain ionic size effects in electrostatic calculation, a size-modified PBE (SMPBE) model has been proposed based on the assumption that all water molecules and ions have a uniform volumetric size. However, so far, its numerical solutions were only limited to either a small biolog-ical molecule in a monovalent ion solution or a simple case of one spherical ball containing a central charge immersed in a salt solution. How to solve the SMPBE model effectively and efficiently for a biomolecule in an ionic solvent remains a challenging research issue in the fields of computational biology, computational Mathematics, and high performance scientific computing.To address such a challenge, we first give the SMPBE model a new mathemat-ical analysis. Traditionally, the SMPBE model was known as the Euler equation of an electrostatic free energy variational problem with a Poisson dielectric model be-ing the constraint condition (i.e., a partial differential equation (PDE) constrained optimization problem). However, this PDE constrained optimization problem now has been known to be poorly defined due to the singularities caused by the Dirac-delta atomic point charge distributions from the biomolecule. Some modifications must be done before giving it a mathematical analysis. As a continuation of the early efforts, we modify it as a new well defined minimization problem using so-lution decomposition techniques. We then prove that the SMPBE model is the Euler equation of this new minimization problem. We further propose a solution decomposition scheme for the SMPBE model to split its solution as a sum of three functions G, Ψ, and Φ with G being a given function that collects all the singular-ity points of u,Ψ being a solution of a well defined linear elliptic interface problem and Φ being a solution of a well defined nonlinear elliptic interface problem. This remarkably simplifies the analysis of the SMPBE model. To prove the solution existence and uniqueness of the nonlinear elliptic interface problem, we construct an equivalent variational problem and prove its solution existence and uniqueness. Consequently, a new proof on the solution existence and uniqueness of the SMPBE model is obtained since the solution existence and uniqueness of the linear elliptic interface problem has been given in the literature.Based on this new mathematical analysis of the SMPBE model and its new solution decomposition, we propose an effective minimization protocol for solving the SMPBE model in the finite element method. Different selections of linear and nonlinear iterative methods within this minimization protocol may result in different SMPBE numerical algorithms. As an application of this minimization protocol, we construct a particular SMPBE numerical algorithm for a biomolecule in a symmetric1:1ionic solvent by using a Newton-type minimization method. We then program this SMPBE algorithm as a part of the biomolecule electrostatic computing package developed in Prof. Dexuan Xie’s High Performance Scientific Computing Laboratory at the University of Wisconsin-Milwaukee, USA.In the part of numerical tests, we construct a nonlinear SMPBE ball model with a given analytical solution and use it to validate our SMPBE algorithm. We also make numerical experiments on a central charged ball model to show some physical features captured by the SMPBE model. Furthermore, we make many numerical experiments on six biomolecules (protein, DNA, and protein-DNA com-plex) with different net charges to demonstrate the computing performance of our SMPBE finite element program. Finally, solvation free energies and electrostatic potentials were calculated and compared to the cases of the PBE model.The linearized Poisson-Boltzmann equation (LPBE) is often used to approxi-mate PBE in numerical solution of the PBE model. Based on the solution decom-position of the PBE model, a new scheme for solving the LPBE model is proposed in this paper. And we also program and make numerical study on a new LPBE model. And then, a finite element program is written in Python based on this new scheme of the LPBE model and this new LPBE model in this paper, which is also a part of the biomolecule electrostatic computing package developed in Prof. Dexuan Xie’s High Performance Scientific Computing Laboratory. Numerical re-sults verify the effectiveness and efficiency of this new LPBE scheme, and show that this new LPBE model can significantly improve the solution accuracy of the current LPBE model and broaden its application range.This dissertation is typeset by software LATEX2ε... |
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