Boundary value approaches to molecular dynamics simulation

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Vedell, Peter
Major Professor
Zhijun Wu
Robert Jernigan
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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 2nd-order nonlinear ordinary differential equations (ODE's).;In this dissertation, the mathematical framework of AA-MDS, BV-AA-MDS and some numerical methods for BV-AA-MDS---single shooting, multiple shooting, finite differences methods, and stochastic difference equation methods---are described. Important computational limitations of AA-MDS, BV-AA-MDS, and MS for BV-AA-MDS are highlighted and reasons for considering these approaches and methods despite the computational limitations will be provided.;Also, in this dissertation, the application of multiple shooting to BVP's for ODE's corresponding to transitions between two molecular conformations specified by two sets of internal coordinates is proposed. Strategies and issues related to definition of boundary conditions, assignment of initial parameters, and convergence are investigated. Results from the study of transitions between local minima of the potential energy surface of an alanine dipeptide are presented. Implications of the methods and results of this work for application of multiple shooting to the study of conformational transitions in larger systems are discussed.;Defining boundary conditions corresponding to sets of internal coordinates of local minima leads to what is defined to be a full set of 6n boundary conditions, i.e. R = 6n. And, defining parameters of the multiple shooting method as the initial conditions on each subinterval leads to what is defined to be a full set of 6nN parameters. To apply multiple shooting with a full set of parameters to a BVP with a full set of boundary conditions, the number of atoms in the molecule must be limited to avoid excessive computational cost. In this dissertation, for the case of single shooting, an alternate boundary value simulation approach is presented that involves a reduced set of boundary conditions and a reduced set of parameters. We also propose an approach for use a reduced parameter set that is based on an application of principles of normal mode analysis. We provide results from the application of these approaches to the study of transitions between potential energy wells for an alanine dipeptide.;In this dissertation, all-atom distance matrix interpolation (AA-DMI) methods are described. These are methods for generating position trajectories that satisfy certain types of boundary conditions are less computationally demanding than boundary value approaches to AA-MDS, but do provide atomically detailed trajectories. These methods involve an optimization problem with an objective function derived by interpolation of interatomic distances between their values in one conformation and their values in another conformation. Results are presented from the study of conformational transitions of an alanine dipeptide. Future directions of research are discussed. (Abstract shortened by UMI.)

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Mon Jan 01 00:00:00 UTC 2007