This dissertation develops physical and mathematical models for the trypanosome para- site’s mitochondrial DNA and for SARS-CoV-2 mutagenesis. In the case of the trypanosome parasite, we introduce a liquid-crystal based knotting model to improve upon two earlier, purely mathematical models for minicircle topology that were proposed by Chen, Arsuaga, and their colleagues. In chaps. 1 and 3, we provide a detailed review of these models and their extensions, explaining the biological evidence for the model assumptions and discussing their performance in predicting topological quantities like linking probability and mean valence. Then, in chap. 2, we introduce an orientational energy—the Lebwohl-Lasher model—that reproduces the order of its kinetoplast “minicircles” (mitochondrial DNA plasmids).We develop a mean-field theory (MFT) for this model, which predicts a field-parallel pseudo-transition in the quadrupolar hQ0i order parameter. We then expand on this analysis with Markov-chain Monte-Carlo simulations. These results match the MFT predictions for most temperatures and they predict pseudo-transitions at T⇤ = 0.5 and T⇤ = 1.1 (for pc,0 pc,2 hQ2i). The latter crossover is particularly interesting because it breaks the systems symmetry while remaining consistent with the Hohenberg-Mermin-Wagner theorem. Our study of SARS-CoV-2 spike (S) mutagenesis addresses the question of whether or not the S477N substitution in the protein’s important “receptor-binding domain” provides a binding improvement consistent with its empirically observed fitness advantage in worldwide genomic data. We use 3-D modeling, molecular dynamics simulations, and binding free- energy calculations to show that the S477N substitution likely improves fitness, consistent with the population behavior.