By using current biological understanding, a conceptually simple, but mathematically complex, model is proposed for the dynamics of the gene circuit responsible for regulating nitrogen catabolite repression (NCR) in yeast. A variety of mathematical ''structure'' theorems are described that allow one to determine the asymptotic dynamics of complicated systems under very weak hypotheses. It is shown that these theorems apply to several subcircuits of the full NCR circuit, most importantly to the URE2-GLN3 subcircuit that is independent of the other constituents but governs the switching behavior of the full NCR circuit under changes in nitrogen source. Under hypotheses that are fully consistent with biological data, it is proven that the dynamics of this subcircuit is simple periodic behavior in synchrony with the cell cycle. Although the current mathematical structure theorems do not apply to the full NCR circuit, extensive simulations suggest that the dynamics is constrained in much the same way as that of the URE2-GLN3 subcircuit. This finding leads to the proposal that mathematicians study genetic circuits to find new geometries for which structure theorems may exist.A s microarray technology has brought systems biology to the theater, the desire to understand gene regulatory networks has brought mathematical modeling to center stage. Recent work has focused on two goals: the determination of recurring network motifs (1) followed by an understanding of their design significance through an analysis of their dynamics (2, 3). Thus, it is now a central objective in mathematics, neuroscience, molecular biology, and medicine to understand how and to what extent the structure of the connections between the components of a system determine its dynamic behavior. We refer to mathematical results that relate these two properties as structure theorems. Our impression is that this body of work has received insufficient attention from biologists, bioinformaticians, engineers, and physicists pursuing gene regulatory networks. The aim here is to demonstrate through an important example, the process of nitrogen catabolite repression (NCR) in yeast, the power of the theory, its current limitations, and some promising new directions. Fig. 1 shows a circuit of five genes whose dynamics ultimately control NCR in yeast. Although much is known about the architecture of the NCR circuit and the interactions among its components, quantitative models do not exist, and neither a molecular-level nor a systems-level understanding is at hand.The power and promise of the structure theorem approach is to be able to infer dynamics from a (possibly annotated) graph of biological interactions, like that shown in Fig. 1. One of the earliest structure theories, called the chemical reaction network theory, was developed by M. Feinberg (4,5). This theory provides a classification of the dynamics of chemical reaction networks based on the associated circuit diagram. This method is completely scalable because the results depend only on the possible re...