#### Date of Award

December 2016

#### Degree Type

Thesis

#### Degree Name

Master of Science

#### Department

Mathematics

#### First Advisor

Dexuan Xie

#### Committee Members

Lei Wang, Bruce Wade, Hans Volkmer

#### Keywords

Biomolecular Electostatics, Boundary Integral Equation, Finite Element Method, Poisson Boltzmann Equation, Treecode

#### Abstract

The Poisson-Boltzmann equation (PBE) is a widely-used model in the calculation of electrostatic potential for solvated biomolecules. PBE is an interface problem defined in the whole space with the interface being a molecular surface of a biomolecule, and has been solved numerically by finite difference, finite element, and boundary integral methods. Unlike the finite difference and finite element methods, the boundary integral method works directly over the whole space without approximating the whole space problem into an artificial boundary value problem. Hence, it is expected to solve PBE in higher accuracy. However, so far, it was only applied to a linear PBE model.

Recently, a solution of PBE was split into three component functions. One of them, G, is a known function that collects all the singularity points of PBE so that the other two components become continuously twice differentiable within the protein and solvent regions. Such an approach has led to efficient PBE finite element solvers. This provided motivation to study the application of this solution decomposition to the development of a new boundary integral algorithm for solving PBE.

Reformulating the interface problem of $\Psi$ into a boundary integral equation is nontrivial because the involved flux interface condition is discontinuous. Development of a fast numerical algorithm for solving the resulted boundary integral equation is an attractive research topic. In this masters thesis, we focus on one key step of our new boundary integral algorithm: how to solve for the second component function $\Psi$ of the PBE solution by a boundary integral method. This work becomes important by itself because the sum of $\Psi$ with $G$ gives the solution of the Poisson dielectric model for the case of a biomolecule in water.

In this project, we obtain the new boundary integral equation and develop an adaptive treecode-accelerated boundary integral algorithm. We then program the new algorithm in Fortran and make various numerical tests to validate our new algorithm and program package. In particular, numerical tests performed against analytic models verify the effectiveness of the solver, and comparisons to experimental data verify its accuracy for real-world applications. In this way, it is demonstrated that this solver and solution decomposition can compute the electrostatics of a biomolecule in water with high numerical accuracy.

#### Recommended Citation

Szatkowski, Andrew Joseph, "An Adapative Treecode-accelerated Boundary Integral Solver for Computing the Electrostatics of a Biomolecule" (2016). *Theses and Dissertations*. 1422.

http://dc.uwm.edu/etd/1422