Eukaryotes express three or more multisubunit nuclear RNA polymerases (Pols) referred to as Pols I, II, and III, each of which synthesizes a specific subset of RNAs. Consistent with the diversity of their target genes, eukaryotic cells have evolved divergent cohorts of transcription factors and enzymatic properties for each RNA polymerase system. Over the years, many trans-acting factors that orchestrate transcription by the individual Pols have been described; however, little effort has been devoted to characterizing the molecular mechanisms of Pol I activity. To begin to address this gap in our understanding of eukaryotic gene expression, here we establish transient-state kinetic approaches to characterize the nucleotide incorporation mechanism of Pol I. We collected time courses for single turnover nucleotide incorporation reactions over a range of substrate ATP concentrations that provide information on both Pol I's nucleotide addition and nuclease activities. The data were analyzed by model-independent and model-dependent approaches, resulting in, to our knowledge, the first minimal model for the nucleotide addition pathway for Pol I. Using a grid searching approach we provide rigorous bounds on estimated values of the individual elementary rate constants within the proposed model. This work reports the most detailed analysis of Pol I mechanism to date. Furthermore, in addition to their use in transient state kinetic analyses, the computational approaches applied here are broadly applicable to global optimization problems.