A common problem to all applications of linear finite dynamical systems is analyzing the dynamics without enumerating every possible state transition. Of particular interest is the long term dynamical behaviour, and if every element eventually converges on fixed points. In this paper, we study the number of iterations needed for a system to settle on a fixed set of elements. As our main result, we present two upper bounds on iterations needed, and each one may be readily applied to a fixed point system test. The bounds are based on submodule properties of iterated images and reduced systems modulo a prime.

In order to understand biochemical networks we need to know not only how their parts work but also how they interact with each other. The goal of systems biology is to look at biological systems as a whole to understand how interactions of the parts can give rise to complex dynamics. In order to do this efficiently, new techniques have to be developed. This work shows how tools from mathematics are suitable to study problems in systems biology such as modeling, dynamics prediction, reverse engineering and many others. The advantage of using mathematical tools is that there is a large number of theory, algorithms and software available. This work focuses on how algebra can contribute to answer questions arising from systems biology. / Ph. D.

Systems biology is an emergent field focused on developing a system-level understanding of biological systems. In the last decade advances in genomics, transcriptomics and proteomics have gathered a remarkable amount data enabling the possibility of a system-level analysis to be grounded at a molecular level. The reverse-engineering of biochemical networks from experimental data has become a central focus in systems biology. A variety of methods have been proposed for the study and identification of the system's structure and/or dynamics. The objective of this dissertation is to introduce and propose solutions to some of the challenges inherent in reverse-engineering of biological systems. First, previously developed reverse engineering algorithms are studied and compared using data from a simulated network. This study draws attention to the necessity for a uniform benchmark that enables an ob jective comparison and performance evaluation of reverse engineering methods. Since several reverse-engineering algorithms require discrete data as input (e.g. dynamic Bayesian network methods, Boolean networks), discretization methods are being used for this purpose. Through a comparison of the performance of two network inference algorithms that use discrete data (from several different discretization methods) in this work, it has been shown that data discretization is an important step in applying network inference methods to experimental data. Next, a reverse-engineering algorithm is proposed within the framework of polynomial dynamical systems over finite fields. This algorithm is built for the identification of the underlying network structure and dynamics; it uses as input gene expression data and, when available, a priori knowledge of the system. An evolutionary algorithm is used as the heuristic search method for an exploration of the solution space. Computational algebra tools delimit the search space, enabling also a description of model complexity. The performance and robustness of the algorithm are explored via an artificial network of the segment polarity genes in the D. melanogaster. Once a mathematical model has been built, it can be used to run simulations of the biological system under study. Comparison of simulated dynamics with experimental measurements can help refine the model or provide insight into qualitative properties of the systems dynamical behavior. Within this work, we propose an efficient algorithm to describe the phase space, in particular to compute the number and length of all limit cycles of linear systems over a general finite field. This research has been partially supported by NIH Grant Nr. RO1GM068947-01. / Ph. D.