One goal of systems biology is to predict and modify the behavior of biological networks by accurately monitoring and modeling their responses to certain types of perturbations. The construction of mathematical models based on observation of these responses, referred to as reverse engineering, is an important step in elucidating the structure and dynamics of such networks. Continuous models, described by systems of differential equations, have been used to reverse engineer biochemical networks. Of increasing interest is the use of discrete models, which may provide a conceptual description of the network.

In this dissertation we introduce a discrete modeling approach, rooted in computational algebra, to reverse-engineer networks from experimental time series data. The algebraic method uses algorithmic tools, including Groebner-basis techniques, to build the set of all discrete models that fit time series data and to select minimal models from this set. The models used in this work are discrete-time finite dynamical systems, which, when defined over a finite field, are described by systems of polynomial functions. We present novel reverse-engineering algorithms for discrete models, where each algorithm is suitable for different amounts and types of data. We demonstrate the effectiveness of the algorithms on simulated networks and conclude with a description of an ongoing project to reverse-engineer a real gene regulatory network in yeast.