From immunology to MRI data anlysis : problems in mathematical biology

Abstract

This thesis represents a collection of four distinct biological projects rising from immunology and metabolomics that required unique and creative mathematical approaches. One project focuses on understanding the role IL-2 plays in immune response regulation and exploring how these effects can be altered. We developed several dynamic models of the receptor signaling network which we analyze analytically and numerically. In a second project focused also on MS, we sought to create a system for grading magnetic resonance images (MRI) with good correlation with disability. The goal is for these MRI scores to provide a better standard for large-scale clinical drug trials, which limits the bias associated with differences in available MRI technology and general grader/participant variability. The third project involves the study of the CRISPR adaptive immune system in bacteria. Bacterial cells recognize and acquire snippets of exogenous genetic material, which they incorporate into their DNA. In this project we explore the optimal design for the CRISPR system given a viral distribution to maximize its probability of survival. The final project involves the study of the benefits for colocalization of coupled enzymes in metabolic pathways. The hypothesized kinetic advantage, known as 'channeling', of putting coupled enzymes closer together has been used as justification for the colocalization of coupled enzymes in biological systems. We developed and analyzed a simple partial differential equation of the diffusion of the intermediate substrate between coupled enzymes to explore the phenomena of channeling. The four projects of my thesis represent very distinct biological problems that required a variety of techniques from diverse areas of mathematics ranging from dynamical modeling to statistics, Fourier series and calculus of variations. In each case, quantitative techniques were used to address biological questions from a mathematical perspective ultimately providing insight back to the biological problems which motivated them.