Semester

Spring

Date of Graduation

2025

Document Type

Dissertation

Degree Type

PhD

College

Eberly College of Arts and Sciences

Department

Mathematics

Committee Chair

Harumi Hattori

Committee Co-Chair

Adam Halasz

Committee Member

Adrian Tudorascu

Committee Member

Casian Pantea

Committee Member

Tudor Stanescu

Abstract

This dissertation presents results from two mathematical projects concerned with the biology of cells. Chapter 1 provides biological background and places the two mathematical problems in the context of cell signaling. The larger project, with Prof. H. Hattori on a chemotaxis model is presented in Chapters 3 and 4. Work with Prof. \'{A}. Hal\'{a}sz on a chemical reaction network system with linear multimers and two types of labels is presented in Chapter 2. The chemotaxis system describes the one-dimensional dynamics of a species of cells with two chemical species, a chemo-attractant and chemo-repellent. The goal is to analyze the behavior of the system at steady state (in time), find critical points and investigate their stability properties. The main approach relies on singular perturbation theory with various choices for the fast and slow variables. Chapter 3 provides the general definition of the chemotaxis model, as a time dependent system in one spatial dimension, with diffusion, transport, and chemical interaction terms. A simplifying assumption (damping) is introduced, and the rest of the chapter is devoted to the study of this model at steady state in time. The resulting Ordinary Differential Equation system is investigated using Singular Perturbation Theory (SPT). We focus on the critical points and related orbits. There are four branches of critical points, an all related orbits are homoclinic. In Chapter 4 we return to the full system and perform a steady-state analysis using a different SPT approach. We obtain a structure of critical (point) branches similar to the damping case. However, in addition to homiclinic orbits, there are heteroclinic orbits associated to some of the critical points. In Chapter 2 we develop a mathematical framework to help connect molecular-resolution experimental data to chemical reaction network (CRN) models of dimerization and oligomerizaion. We analyze a CRN system where a base (monomer) species of molecules can form dimers, trimers, and larger oligomers. This type of behavior is often seen with receptors and is directly relevant to the evolution of clinical conditions. One experimental approach relies on the visualisation of individual monomers by adding fluorescent tags. We use the results from the CRN model to analyze the relation between observable patterns (frequency of one- or two-color aggregates) and the distribution of oligomer sizes.

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