Date of Award

8-12-2015

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Science

First Advisor

Daniel Berleant

Second Advisor

Nidhal Bouaynaya

Abstract

One of the main challenges facing biologists and mathematicians in the post genomic era is to understand the behavior of molecular networks and harness this understanding into an educated intervention of the cell. The cell maintains its function via an elaborate network of interconnecting positive and negative feedback loops of genes, RNA and proteins that send different signals to a large number of pathways and molecules. These structures are referred to as genetic regulatory networks (GRNs) or molecular networks. GRNs can be viewed as dynamical systems with inherent properties and mechanisms, such as steady-state equilibriums and stability, that determine the behavior of the cell. The biological relevance of the mathematical concepts are important as they may predict the differentiation of a stem cell, the maintenance of a normal cell, the development of cancer and its aberrant behavior, and the design of drugs and response to therapy. Uncovering the underlying GRN structure from gene/protein expression data, e.g., microarrays or perturbation experiments, is called inference or reverse engineering of the molecular network. Because of the high cost and time consuming nature of biological experiments, the number of available measurements or experiments is very small compared to the number of molecules (genes, RNA and proteins). In addition, the observations are noisy, where the noise is due to the measurements imperfections as well as the inherent stochasticity of genetic expression levels. Intra-cellular activities and extra-cellular environmental attributes are also another source of variability. Thus, the inference of GRNs is, in general, an under-determined problem with a highly noisy set of observations. The ultimate goal of GRN inference and analysis is to be able to intervene within the network, in order to force it away from undesirable cellular states and into desirable ones. However, it remains a major challenge to design optimal intervention strategies in order to affect the time evolution of molecular activity in a desirable manner. In this proposal, we address both the inference and control problems of GRNs. In the first part of the thesis, we consider the control problem. We assume that we are given a general topology network structure, whose dynamics follow a discrete-time Markov chain model. We subsequently develop a comprehensive framework for optimal perturbation control of the network. The aim of the perturbation is to drive the network away from undesirable steady-states and to force it to converge to a unique desirable steady-state. The proposed framework does not make any assumptions about the topology of the initial network (e.g., ergodicity, weak and strong connectivity), and is thus applicable to general topology networks. We define the optimal perturbation as the minimum-energy perturbation measured in terms of the Frobenius norm between the initial and perturbed networks. We subsequently demonstrate that there exists at most one optimal perturbation that forces the network into the desirable steady-state. In the event where the optimal perturbation does not exist, we construct a family of sub-optimal perturbations that approximate the optimal solution arbitrarily closely. In the second part of the thesis, we address the inference problem of GRNs from time series data. We model the dynamics of the molecules using a system of ordinary differential equations corrupted by additive white noise. For large-scale networks, we formulate the inference problem as a constrained maximum likelihood estimation problem. We derive the molecular interactions that maximize the likelihood function while constraining the network to be sparse. We further propose a procedure to recover weak interactions based on the Bayesian information criterion. For small-size networks, we investigated the inference of a globally stable 7-gene melanoma genetic regulatory network from genetic perturbation experiments. We considered five melanoma cell lines, who exhibit different motility/invasion behavior under the same perturbation experiment of gene Wnt5a. The results of the simulations validate both the steady state levels and the experimental data of the perturbation experiments of all five cell lines. The goal of this study is to answer important questions that link the response of the network to perturbations, as measured by the experiments, to its structure, i.e., connectivity. Answers to these questions shed novel insights on the structure of networks and how they react to perturbations.

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