Linking the structure of a system with its dynamics is an important problem in the theory of finite dynamical systems. For monomial dynamical systems, that is, a system that can be described by monomials, information about the limit cycles can be obtained from the monomials themselves. In particular, this work contains sufficient and necessary conditions for a monomial dynamical system to have only fixed points as limit cycles. / Ph. D.

Systems biology is an interdisciplinary field that focuses on elucidating complex biological processes (systems) by investigating the interactions among its components through an iterative cycle composed of data generation, data analysis and mathematical modeling. Our contributions to systems biology revolve around the following two axes: - Data analysis: Two data analysis projects, which were initiated when I was a co-op at GlaxoSmithKline, are discussed in this thesis. First, next generation sequencing data generated for a phase I clinical trial is analyzed to determine the altered microbial community in human gut before and after antibiotic usage (Chapter 2). To our knowledge, there have not been similar comparative studies in humans on the impacts on the gut microbiome of an antibiotic when administered by different modes. Second, publicly available gene expression data is analyzed to investigate human immune response to tuberculosis (TB) infection (Chapter 3). The novel feature of this study is systematic drug repositioning for the prevention, control and treatment of TB using the Connectivity map. - Mathematical modeling: Polynomial dynamical systems, a state- and time- discrete logical modeling framework, is used to model two biological processes. First, a denitrification pathway in Pseudomonas aeruginosa is modeled to shed light on the reason of greenhouse gas nitrous oxide accumulation (Chapter 4). It is the first mathematical model of denitrification that can predict the effect of phosphate on the denitrification performance of this bacterium. Second, an iron homeostasis pathway linked to iron utilization, oxidative stress response and oncogenic pathways is constructed to investigate how normal breast cells become cancerous (Chapter 5). To date, our intracellular model is the only expanded core iron model that can capture a breast cancer phenotype by overexpression and knockout simulations. / Ph. D.

Agent-based models are computer models made up of agents that can exist in a finite number of states. The state of the system at any given time is determined by rules governing agents' interaction. The rules may be deterministic or stochastic. Optimization is the process of finding a solution that optimizes some value that is determined by simulating the model. Optimal control of an agent-based model is the process of determining a sequence of control inputs to the model that steer the system to a desired state in the most efficient way. In large and complex models, the number of possible control inputs is too large to be enumerated by computers; hence methods must be developed for use with these models in order to find solutions without searching the entire solution space. Heuristic algorithms have been applied to such models with some success. Such algorithms are discussed; case studies of examples from biology are presented. The lack of a standard format for agent-based models is a major issue facing the study of agent-based models; presentation as polynomial dynamical systems is presented as a viable option. Algorithms are adapted and presented for use in this framework. / Master of Science