Filament or run-on oligomer formation by metabolic enzymes is now recognized as a widespread phenomenon having potentially unique enzyme regulatory properties and biological roles, and its dysfunction is implicated in human diseases such as cancer, diabetes, and developmental disorders. SgrAI is a bacterial allosteric type II restriction endonuclease that binds to invading phage DNA, may protect the host DNA from off-target cleavage activity, and forms run-on oligomeric filaments with enhanced DNA-cleavage activity and altered DNA sequence specificity. However, the mechanisms of SgrAI filament growth, cooperativity in filament formation, sequestration of enzyme activity, and advantages over other filament mechanisms remain unknown. In this first of a two-part series, we developed methods and models to derive association and dissociation rate constants of DNA-bound SgrAI in run-on oligomers and addressed the specific questions of cooperativity and filament growth mechanisms. We show that the derived rate constants are consistent with the run-on oligomer sizes determined by EM analysis and are most consistent with a noncooperative growth mode of the run-on oligomer. These models and methods are extended in the accompanying article to include the full DNA-cleavage pathway and address specific questions related to the run-on oligomer mechanism including the sequestration of DNA-cleavage activity and trapping of products.