Doctor of Philosophy

dissertationCell division is a complex process that involves carefully orchestrated chemical and mechanical events. Tight regulation is vital during division, since a breakdown in control mechanisms can lead to serious disorders such as cancer. A key step in division is the movement of chromosomes to specific locations in the cell with remarkable precision. In higher eukaryotes, the movement of chromosomes has been well observed over the course of hundreds of years. Yet, the mechanisms underlying chromosome motility and the control of precise chromosome localizations in the cell are poorly understood. More recently, a wealth of experimental data has become available for bacterial division. Despite the long supported theory that bacteria and eukaryotes differ widely when undergoing division, it is emerging that similar mechanisms for motility and cell cycle control might be at play in both cell types. Mathematical modeling is useful in the study of these dynamic cellular environments, where it is difficult to experimentally uncover the mechanisms that drive a multitude of mechanical and chemical events. In this dissertation, we develop various mathematical models that address the question of how dynamic polymers can move large objects such as chromosomes in higher eukaryotes and in bacteria. Then, we develop models that address how chemical and mechanical signals can be coordinated to control the precise localization of a chromosome. The mathematical models proposed here employ stochastic differential equations, ordinary differential equations and partial differential equations. The models are numerically simulated to obtain solutions for various parameter values, but we also use tools from bifurcation theory, asymptotic and perturbation methods for our model analysis. Our mathematical models can not only reproduce the experimental data at hand, but also make predictions about the mechanisms underlying chromosome motility in dividing cells

Distribution of lifetimes of kinetochore-microtubule attachments: interplay of energy landscape, molecular motors and microtubule (de-)polymerization

Before a cell divides into two daughter cells, chromosomes are replicated resulting in two sister chromosomes embracing each other. Each sister chromosome is bound to a separate proteinous structure, called kinetochore (kt), that captures the tip of a filamentous protein, called microtubule (MT). Two oppositely oriented MTs pull the two kts attached to two sister chromosomes thereby pulling the two sisters away from each other. Here we theoretically study an even simpler system, namely an isolated kt coupled to a single MT; this system mimics an {\it in-vitro} experiment where a single kt-MT attachment is reconstituted using purified extracts from budding yeast. Our models not only account for the experimentally observed "catch-bond-like" behavior of the kt-MT coupling, but also make new predictions on the probability distribution of the lifetimes of the attachments. In principle, our new predictions can be tested by analyzing the data collected in the {\it in-vitro} experiments provided the experiment is repeated sufficiently large number of times. Our theory provides a deep insight into the effects of (a) size, (b) energetics, and (c) stochastic kinetics of the kt-MT coupling on the distribution of the lifetimes of these attachments.Comment: This is an author-created, un-copyedited version of an article accepted for publication in "Physical Biology" (IOP). IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from i

Stochastic Analysis of a Mammalian Circadian Clock Model: Small Protein Number Effects

The circadian clock, responsible for coordinating organism function with daily and seasonal changes in the day-night cycle, is controlled by a complex protein network that constitutes a robust biochemical oscillator. Deterministic ordinary differential equation models have been used extensively to model the behavior of these central clocks. However, due to the small number of proteins involved in the circadian oscillations, mathematical models that track stochastic variations in the numbers of clock proteins may reveal more complex and biologically relevant behaviors. In this paper, we compare the response of a robust yet detailed deterministic model for the mammalian circadian clock with its corresponding stochastic version that takes into account low protein number noise. We then use signal analysis techniques in order to examine differences in behavior among components of the stochastic system oscillator. This approach reveals differences in the system response between the stochastic and deterministic model and also allows us to extend bifurcation analysis into the stochastic domain. From our analysis of the unfitted stochastic model, we propose novel explanations of some previous experimental results

Sources of inter-individual variability leading to significant changes in anti-PD-1 and anti-PD-L1 efficacy identified in mouse tumor models using a QSP framework

While anti-PD-1 and anti-PD-L1 [anti-PD-(L)1] monotherapies are effective treatments for many types of cancer, high variability in patient responses is observed in clinical trials. Understanding the sources of response variability can help prospectively identify potential responsive patient populations. Preclinical data may offer insights to this point and, in combination with modeling, may be predictive of sources of variability and their impact on efficacy. Herein, a quantitative systems pharmacology (QSP) model of anti-PD-(L)1 was developed to account for the known pharmacokinetic properties of anti-PD-(L)1 antibodies, their impact on CD8+ T cell activation and influx into the tumor microenvironment, and subsequent anti-tumor effects in CT26 tumor syngeneic mouse model. The QSP model was sufficient to describe the variability inherent in the anti-tumor responses post anti-PD-(L)1 treatments. Local sensitivity analysis identified tumor cell proliferation rate, PD-1 expression on CD8+ T cells, PD-L1 expression on tumor cells, and the binding affinity of PD-1:PD-L1 as strong influencers of tumor growth. It also suggested that treatment-mediated tumor growth inhibition is sensitive to T cell properties including the CD8+ T cell proliferation half-life, CD8+ T cell half-life, cytotoxic T-lymphocyte (CTL)-mediated tumor cell killing rate, and maximum rate of CD8+ T cell influx into the tumor microenvironment. Each of these parameters alone could not predict anti-PD-(L)1 treatment response but they could shift an individual mouse’s treatment response when perturbed. The presented preclinical QSP modeling framework provides a path to incorporate potential sources of response variability in human translation modeling of anti-PD-(L)1

Using mathematics to understand biological complexity: from cells to populations

This volume tackles a variety of biological and medical questions using mathematical models to understand complex system dynamics. Working in collaborative teams of six, each with a senior research mentor, researchers developed new mathematical models to address questions in a range of application areas. Topics include retinal degeneration, biopolymer dynamics, the topological structure of DNA, ensemble analysis, multidrug-resistant organisms, tumor growth modeling, and geospatial modeling of malaria. The work is the result of newly formed collaborative groups begun during the Collaborative Workshop for Woman in Mathematical Biology hosted by the Institute of Pure and Applied Mathematics at UCLA in June 2019. Previous workshops in this series have occurred at IMA, NIMBioS, and MBI