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Thesis Defense - Azmi

Use of Smoothing Methods with Stochastic Perturbation for Global Optimization
Computer Science PhD Candidate

Proteins have highly specific three-dimensional structures. It is the structure of a protein that determines how it functions in the cell, i.e. its biological activity. Thus, in order to understand how the cell works it is crucial to know the three-dimensional structure of proteins.

Today, we know of the three-dimensional structure of few hundred proteins, while we know the amino acid sequence of several thousand proteins. Though the latter number is growing rapidly it is a very difficult task to determine the three-dimensional structure of a given protein. Hence, it is desirable if we can predict the three-dimensional structure of a protein with their amino acid sequence as the only clue. This problem is generally referred to as the protein folding problem.

It is believed that this problem corresponds to minimizing the potential energy of the protein. This is generally a very difficult global optimization problem, with a large number of parameters and a huge number of local minimizers including many with function values near that of the global minimizer. To help solve such difficult problems, smoothing the potential energy function have been used with some efficacy. The basic idea of smoothing is to soften the original function by reducing abrupt function value changes while retaining the coarser structure of the original function.

In this work, we first propose two new, simple algebraic ways to smooth the Lennard-Jones and electrostatic energy functions. These two terms are the main contributors to the potential energy function in many molecular models. These simple smoothing schemes are much cheaper than the classic spatial averaging smoothing technique. The proposed smoothing schemes are analyzed. The analysis yields some criteria for picking the smoothing parameters. Afterwards, computational tests on two different proteins show smoothing to be very successful in finding the lowest energy structures.

Finally, we do some mathematical analysis of smooth minimizers of Lennard-Jones clusters. In the limit, the smoothed Lennard-Jones minimizer exhibit an interesting behavior, which we will analyze in some detail.

Committee: Richard Byrd, Professor (Chair)
Robert (Bobby) Schnabel, Professor
Xiao-Chuan Cai, Associate Professor
Gary Hachtel, Department of Electrical Engineering
Kent Goodrich, Department of Mathematics
Department of Computer Science
University of Colorado Boulder
Boulder, CO 80309-0430 USA
May 5, 2012 (14:20)