I develop a number of mathematical models for the study of prion phenotype propagation in S. cerevisiae. Prion proteins underlie a host of non-Mendelian phenotypes in yeast and several fatal, neurodegenerative diseases in mammals, most notably bovine spongiform encephalopathy ("mad-cow disease'") and Creutzfeldt-Jakob disease in humans. Two fundamental questions guide this work:

* How does the infectious prion form of a protein initially occur?

* Once present, how do the infectious prion ``aggregates'' persist and amplify across multiple generations of cell division?

A significant body of literature addresses Question 2, and so I begin with a review of this literature and then develop two new models in an attempt to better address in vivo cellular conditions (Chapters 2 and 3). I then return to Question 1, expanding upon a Markov chain formulation of the problem developed in the literature and implement a memory-efficient, numerical solver for the estimation of the time scale of "spontaneous nucleation" (Chapter 4). Taken together, this work is (to my knowledge) the most comprehensive, stochastic treatment of the yeast prion phenotype system and is the first to permit prion strain coexistence, qualitatively match biological experiments on Hsp104 knock-out mutants, and suggest nucleation size differences between prion strains.