Boundary value approaches to molecular dynamics simulation

Determining the relationships between a protein’s structure and its function, identifying a protein’s function in biological systems, and understanding the mechanisms that facilitate a protein’s function are major scientific challenges in modern biological research. The mechanisms that facilitate protein function often involve conformational transitions of proteins or other biomolecules. Successfully meeting the aforementioned challenges will likely require an understanding of the dynamics of conformational transitions of proteins and of other biomolecules. Molecular dynamics is often studied using a computational approach that is called initial value all-atom molecular dynamics simulation (IV-AA-MDS) in this dissertation. In IV-AA-MDS, an empirically determined force field specifies the forces on each atom of a molecular system as a function of the coordinates of the system. The motions of the atoms of the system are governed by Newtonian equations of motion and are tracked dynamically over a period of time. From a mathematical perspective, in IV-AA-MDS, initial conditions for the positions and velocities of the atoms of the system are assigned for an initial point in time and the positions and velocities at a discrete set of points contained in an interval that includes the initial point in time are determined by using a numerical method for solving initial value problems (IVP’s) for systems of 2-order nonlinear ordinary differential equations (ODE’s). The focus of the research of this dissertation is the mathematical modeling of and use of numerical methods for the study of the dynamics of conformational transitions of biomolecules like proteins and small peptides. While an IV-AA-MDS approach could be considered for this purpose, the focus of this dissertation is a related approach that is called boundary value all-atom molecular dynamics simulation (BV-AA-MDS) in this dissertation. This approach includes the application of a numerical method to seek numerical solutions to two-point boundary value problems (BVP’s) for systems of 2-order nonlinear ordinary differential equations (ODE’s). Numerical solutions to two-point BVP’s satisfy boundary conditions at two points in time (a beginning and ending point) and also satisfy Newtonian equations of motion at a discrete set of time points contained in the interval between the two points in time (within limits