This dissertation presents mathematical models, numerical methods, and data driven investigations of heritable cellular phenotypes. First, mathematical models and numerical methods for the dynamics of protein misfolding (prions) in yeast cells are presented (Chapters 2 and 3). This prion phenotype, in Saccharomyces cerevisiae yeast cells can be inherited by daughter cells through transmission of protein aggregates during cell division, a non-Mendelian form of inheritance. Prions are not harmful to yeast; this allows for their use as a biological model to gain insight into the mechanisms that govern prion replication and transmission. Second, data driven mathematical approaches to evaluate new biological techniques used to study the evolution of bacterial antibiotic resistance are presented (Chapters 4 and 5). Bacterial antibiotic resistance is a global human health problem. In the U.S. more than 35,000people die from antibiotic-resistant infections and around 3 million get an antibiotic-resistant infection every year [30]. In this work, we used two repositories of infectious isolates collected from patients at Dignity Health Mercy Medical Center in Merced, California, USA, and a nationwide database compiled from clinical isolate genomes reported by the National Center for Biotechnology Information since 2013.