The phenotypic effect of prions on host cells is influenced by the physical properties of the prion strain and its level of accumulation. In mammalian cell cultures, prion accumulation is determined by the interplay between de novo prion formation, catabolism, cell division, and horizontal cell-to-cell transmission. Understanding this dynamic enables the analytical modeling of protein-based heritability and infectivity. Here, we quantitatively measured these competing effects in a subline of neuroblastoma (N2a) cells and propose a concordant reaction mechanism to explain the kinetics of prion propagation. Our results show that cell division leads to a predictable reduction in steady-state prion levels but not to complete clearance. Scrapie-infected N2a cells were capable of accumulating different steady-state levels of prions, dictated partly by the rate of cell division. We also show that prions in this subline of N2a cells are transmitted primarily from mother to daughter cells, rather than horizontal cell-to-cell transmission. We quantitatively modeled our kinetic results based on a mechanism that assumes a subpopulation of prions is capable of self-catalysis, and the levels of this subpopulation reach saturation in fully infected cells. Our results suggest that the apparent effectiveness of antiprion compounds in culture may be strongly influenced by the growth phase of the target cells.