Date of Award

8-2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Committee Member

Dr. Matthew Macauley, Committee Chair

Committee Member

Dr. Elena Dimitrova

Committee Member

Dr. Svetlana Poznanovikj

Committee Member

Dr. Neil Calkin

Abstract

Discrete models of gene regulatory networks have gained popularity in computational systems biology over the last dozen years. However, not all discrete network models reflect the behaviors of real biological systems. In this work, we focus on two model selection methods and algebraic geometry arising from these model selection methods. The first model selection method involves biologically relevant functions. We begin by introducing k-canalizing functions, a generalization of nested canalizing functions. We extend results on nested canalizing functions and derived a unique extended monomial form of arbitrary Boolean functions. This gives us a stratification of the set of n-variable Boolean functions by canalizing depth. We obtain closed formulas for the number of n-variable Boolean functions with depth k, which simultaneously generalizes enumeration formulas for canalizing, and nested canalizing functions. We characterize the set of k-canalizing functions as an algebraic variety in F2n. 2 . Next, e propose a method for the reverse engineering of networks of k-canalizing functions using techniques from computational algebra, based on our parametrization of k-canalizing functions. We also analyze binary decision diagrams of k-canalizing functions. The second model selection method involves computing minimal polynomial models using Gröbner bases. We built up the connection between staircases and Gröbner bases. We pro-vided a necessary and sufficient condition for the ideal I(V ) to have a unique reduced Gröbner basis, using the concept of a basic staircase. We also provide a sufficient combinatorial characterization of V ⊂ Nnp that yields a unique reduced Grobner basis.

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