Mathematical modeling of biological systems provides a structured framework for exploring and understanding intricate biological phenomena that can be key to unlocking the underlying principles that govern life. Even relatively elementary mathematical models applied to highly complex biological systems have exhibited a capacity to unveil significant patterns and insights, underscoring the substantial utility of computational models in scientific inquiries. In a simplified perspective, mathematical models of biological phenomena can be categorized into three groups: The first group assumes known governing equations and employs mathematical descriptions (e.g. Ordinary or Partial Differential Equations) for modeling dynamics and variations. The second group relies on complete observations to learn some or all system components, often utilizing Machine Learning to uncover useful patterns and relationships in the data. The third group aims to use incomplete observations or assumptions to learn from the system, often seeking to generalize findings to unknown agents in the same or different systems.

The goal of this dissertation is to propose novel mathematical techniques to address some of the intricacies inherent in modeling complex biological systems. This dissertation is structured into three parts, each corresponding to the modeling paradigms mentioned earlier, with individual chapters devoted to various methods and applications. The first part of this dissertation introduces a novel numerical method for modeling Partial Differential Equations on deforming geometries. Subsequently, the dissertation transitions to data-driven modeling of biological tissues in humans and animals, leveraging deep learning to capture meaningful insights that can improve the robustness and accuracy of biological analyses. In the final part of this dissertation, novel deep-learning algorithms are formulated to address data or label scarcity, offering substantial improvements in the analysis of systems where prior knowledge or the amount of observations is limited.

Similar to mathematical biology itself, the methods and systems explored in this dissertation encompass a wide spectrum, spanning various scales (from a single cell to tissues) and species (yeast, mice, humans) within the realm of biological sciences. However, they are unified in their pursuit to enable more robust and accurate analysis of complex biological systems. Despite their distinctions, the frameworks presented in this work underscore my dissertation’s overarching goal: to provide a comprehensive toolkit for modeling complex biological systems across different modeling regimes and limitations. Through these efforts, I hope to contribute to gaining a deeper under- standing of life and its underlying mathematical principles.

I develop a number of mathematical and statistical models for the study of genomic variation within and between species. These variants affect an organism's susceptibility to genetic diseases and are also responsible for speciation events. In particular, my work focuses on the following questions: (1) How does DNA causing genomic variation proliferate through the genome of a species? and (2) for members of the same species, how can we leverage a priori information (i.e, relatedness and sparsity) to improve predictions of genomic variants?

I address these questions in the the context of using noisy and low-quality data. I begin with a review of models of DNA proliferation and detecting these genomic changes in Chapter 1. In Chapter 2, I answer the first question by developing a model which describes non-actively replicating repetitive elements in an organism's genome. Although they comprise a majority of many eukaryotic genomes, these elements are often ignored by models reviewed in Chapter 1. I answer the second question in Chapter 3 by developing a general optimization framework to detect genomic rearrangements in related individuals subject to different sequencing assumptions. In the context of limited and noisy data, this work is one of the only methods (to my knowledge) that simultaneously predicts variants in a group of individuals instead of post-processing this information. Chapter 4 describes some of the convergence properties of the methods introduced in the previous chapter, and Chapter 5 summarizes this work and future projects.

This dissertation investigates the ways that natural languages evolve and what it means in the overall cultural evolution of society. Computational and modeling advances have made possible to explore large-scale text data and test hypothesis of language evolution. Similar to biological systems, natural languages are evolving systems with words as its measurable units. Words have certain functions within a body of text to convey ideas and thought. The frequency distribution of these words can change based on how it is used at a particular point in time and context. This work incorporates two different sources of text data: 109 year’s worth of digitized books and text taken from social media. Given large-scale diachronic corpora, this work focuses on the following topics: 1. Modeling word rank evolution utilizing statistical and data-driven modeling approaches. 2. Exploring the evolution of contextual semantics through the use of distributional semantics and word embedding models. 3. Evaluating the accuracy of reading comprehension tasks by using contemporary machine learning models.

The evolution of language presents a profound problem in natural language processing and cognitive science. The social and cultural aspect of language adds to the problem of how word meanings develop and change. Examining how language evolves, while drawing from both molecular biology and socio-cultural aspects, allows us to explore the ways word meanings form that are influenced by political and social changes.

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.

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.

This dissertation describes the work that I performed with a team to develop the AddTag method for genome editing. First, I introduce genome editing through CRISPR/Cas-induced homology-directed repair, and I introduce the Candida albicans biological system (Chapter 1). Next, I describe how AddTag editing utilizes CRISPR/Cas-induced homology-directed repair to edit the C. albicans genome, and then the process of validating the edits through phenotyping, sequencing, and PCR (Chapter 2). I introduce the ADDTAG software which assists with AddTag editing. First, I describe how ADDTAG identifies genome targets for RNA-guided nucleases (Chapter 3). Then I demonstrate how the software constructs artificial sequences for use as genome repair templates. Lastly, I explain a computational method for producing a set of verification PCR primers for determining if genome edits are successful (Chapter 4).

Blood coagulation is a complex system comprised of numerous biochemical reactions. Due to this complexity, mathematical modeling has been used to increase the overall understanding of the system as a whole, determine previously unknown mechanisms, and to predict system responses. These models, however, may involve uncertainty in both parameter values and kinetic schemes that describe the reactions; this dissertation examines two such ideas. First, we examine the interactions between a specific coagulation factor, FXa, and an experimental tool designed to measure its action, a chromogenic substrate. Second, we examine a more complex mathematical model in regards to its parametric uncertainty. Chapter 1 gives a background on the mathematical tools used in this dissertation and necessary for uncertainty quantification (UQ) and an overview of the two aforementioned systems. In Chapter 2 we demonstrate how an application of UQ identifies a new model for product inhibition between FXa and its chromogenic substrate, which is validated experimentally. In Chapter 3 we conduct an extensive local and global sensitivity analysis for a mathematical model of flow-mediated blood coagulation. We determined that for many cases a local analysis is sufficient to understand the uncertainty in the model’s output, but that for certain cases there are classes of parameters that exhibit strong synergistic behavior, and so a global method that is capable of resolving interaction effects is necessary. These results motivated the work in Chapter 4 where we used global sensitivity analysis on a mathematical model to identify a novel mechanism for recovering a normal clotting response in hemophilia A; the potential mechanism was further supported by experimental validation. Chapter 5 summarizes the conclusions from the preceding chapters and presents ongoing work relating to the two projects.