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In this dissertation, we explore the effect of variable evolutionary rates on phylogenetic inference. In the first half of the thesis are introduced the biological fundamentals and the statistical framework that will be used throughout the thesis. The basic concepts in phylogenetics and an overview of Bayesian inference are presented in Chapter 1. In Chapter 2, we survey the models that are already used for rate variation. We argue that the CIR process---a diffusion process widely used in finance---is the best suited for applications in phylogenetics, for both mathematical and computational reasons. Chapter 3 shows how evolutionary rate models are incorporated to DNA substitution models. We derive the general formulae for transition probabilities of substitutions when the rate is a continuous-time Markov chain, a diffusion process or a jump process (a diffusion process with discrete jumps). / The second half of the thesis is dedicated to applications of variable evolutionary rate models in two different contexts. In Chapter 4, we use the CIR process to model heterotachy, an evolutionary hypothesis according to which positions of an alignment may evolve at rates that vary with time differently from site to site. A comparison the CIR process with the covarion---a widely-used heterotachous model---on two different data sets allows us to conclude that the CIR provides a significantly better fit. Our approach, based on a Bayesian mixture model, enables us to determine the level of heterotachy at each site. Finally, the impact of variable evolutionary rates on divergence time estimation is explored in Chapter 5. / Several models, including the CIR process are compared on three data sets. We find that autocorrelated models (including the CIR) provide the best fits.

Bibliography: p. 269-279. / viii, 279 p. : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1995

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This study considers the dynamical interaction of two predatory carnivores (Lions (Panthera leo) and Spotted Hyaenas (Crocuta crocuta)) and three of their common prey (Buffalo (Syncerus caffer), Warthog (Phacochoerus africanus) and Kudu (Tragelaphus strepsiceros)). The dependence on spatial structure of species’ interaction stimulated the author to formulate reaction-diffusion models to explain the dynamics of predator-prey relationships in ecology. These models were used to predict and explain the effect of threshold populations, predator additional food and prey refuge on the general species’ dynamics. Vital parameters that model additional food to predators, prey refuge and population thresholds were given due attention in the analyses. The stability of a predator-prey model for an ecosystem faced with a prey out-flux which is analogous to and modelled as an Allee effect was investigated. The results highlight the bounds for the conversion efficiency of prey biomass to predator biomass (fertility gain) for which stability of the three species ecosystem model can be attained. Global stability analysis results showed that the prey (warthog) population density should exceed the sum of its carrying capacity and threshold value minus its equilibrium value i.e., W >(Kw + $) −W . This result shows that the warthog’s equilibrium population density is bounded above by population thresholds, i.e., W < (Kw+$). Besides showing the occurrence under parameter space of the so-called paradox of enrichment, early indicators of chaos can also be deduced. In addition, numerical results revealed stable oscillatory behaviour and stable spirals of the species as predator fertility rate, mortality rate and prey threshold were varied. The stabilising effect of prey refuge due to variations in predator fertility and proportion of prey in the refuge was studied. Formulation and analysis of a robust mathematical model for two predators having an overlapping dietary niche were also done. The Beddington-DeAngelis functional and numerical responses which are relevant in addressing the Principle of Competitive Exclusion as species interact were incorporated in the model. The stabilizing effect of additional food in relation to the relative diffusivity D, and wave number k, was investigated. Stability, dissipativity, permanence, persistence and periodicity of the model were studied using the routine and limit cycle perturbation methods. The periodic solutions (b 1 and b 3), which influence the dispersal rate (') of the interacting species, have been shown to be controlled by the wave number. For stability, and in order to overcome predator natural mortality, the nutritional value of predator additional food has been shown to be of high quality that can enhance predator fertility gain. The threshold relationships between various ecosystem parameters and the carrying capacity of the game park for the prey species were also deduced to ensure ecosystem persistence. Besides revealing irregular periodic travelling wave behaviour due to predator interference, numerical results also show oscillatory temporal dynamics resulting from additional food supplements combined with high predation rates.

The organization of molecules within a cell is central to cellular processes ranging from metabolism and damage repair to migration and replication. Uncovering the emergent properties of this biomolecular organization can improve our understanding of how organisms function and reveal ways to repurpose their components outside of the cell. This dissertation focuses on the role of organization in two widely studied systems: enzyme cascades and active cytoskeletal filaments.Part I of this dissertation studies the emergent properties of the spatial organization of enzyme cascades. Enzyme cascades consist of a series of enzymes that catalyze sequential reactions: the product of one enzyme is the substrate of a subsequent enzyme. Enzyme cascades are a fundamental component of cellular reaction pathways, and spatial organization of the cascading enzymes is often essential to their function. For example, cascading enzymes assembled into multi-enzyme complexes can protect unstable cascade intermediates from the environment by forming tunnels between active sites. We use mathematical modeling to investigate the role of spatial organization in three specific systems. First, we examine enzyme cascade reactions occurring in multi-enzyme complexes where active sites are connected by tunnels. Using stochastic simulations and theoretical results from queueing theory, we demonstrate that the fluctuations arising from the small number of molecules involved can cause non-negligible disruptions to cascade throughput. Second, we develop a set of design principles for a compartmentalized cascade reaction with an unstable intermediate and show that there exists a critical kinetics-dependent threshold at which compartments become useful. Third, we investigate enzyme cascades immobilized on a synthetic DNA origami scaffold and show that the scaffold can create a favorable microenvironment for catalysis. Part II of this dissertation focuses on the organization of active cytoskeletal filaments. Many mechanical processes of a cell, such as cell division, cell migration, and intracellular transport, are driven by the ATP-fueled motion of motor proteins (kinesin, dynein, or myosin) along cytoskeletal filaments (microtubules or actin filaments). Over the past two decades, researchers have been repurposing motor protein-driven propulsion outside of the cell to create systems where cytoskeletal filaments glide on surfaces coated with motor proteins. The study of these systems not only elucidates the mechanisms of force production within the cell, but also opens new avenues for applications ranging from molecular detection to computation. We examine how microtubules gliding on surfaces coated with kinesin motor proteins can generate collective behavior in response to mutualistic interactions between the filaments and motors, thereby maximizing the utilization of system components and production. To this end, we used a microtubule-kinesin system where motors reversibly bind to the surface. In experiments, microtubules gliding on these reversibly bound motors were unable to cross each other and at high enough densities began to align and form long, dense bundles. The kinesin motors accumulated in trails surrounding the microtubule bundles and participated in microtubule transport. In conclusion, our study of the emergent properties of the spatial organization of enzyme cascades and the mutualistic interactions within active systems of motor proteins and cytoskeletal filaments provides insight into both how these systems function within cells and how they can be repurposed outside of them.

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This thesis explores the role of collective cell migration and self-organisation in the development of the embryo and in vitro tissue formation through mathematical and computational approaches. We consider how population heterogeneity, microenvironmental signals and cell-cell interactions facilitate cells to collectively organise and navigate, with the aim to work towards uncovering general rules and principles, rather than delving into the microscopic molecular details. To ensure the biological relevance of our results, we collaborate closely with experimental biologists working on two model systems. First, to understand how neural crest cells obtain directionality, maintain persistence and specialise during their migration, we use computational simulations in parallel with imaging of chick embryos under genetic and surgical perturbations. We show how only a few cells adopting a leader state that enables them to read out chemical signals can lead a population of cells in a follower state over long distances in the embryo. Furthermore, we devise and test an improved mechanism of how cells dynamically switch between leader and follower states in the presence of a chemoattractant gradient. Our computational work guides the choice of new experiments, aids in their interpretation and probes hypotheses in ways the experiments can not. Secondly, to study the self-organisation of mouse skin cells in vitro, we draw on aggregation processes and scaling theory. Dermal and epidermal cells, after being dissociated and mixed, can reconstitute functional (transplantable and hair-growing) skin in culture. Using kinetic aggregation models and scaling analysis we show that the initial clustering of epidermal cells can be described by Smoluchowski coagulation, consistent with the dynamics of the "clustering clusters" universality class. Then, we investigate a potential mechanism for the size-regulation of cell aggregates during the later stages of the skin reconstitution process. Our analysis shows the extent to which this tissue formation follows a single physical process and when the transition to different dynamics occurs, which may be triggered by cellular biochemical changes.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2017. / With an increase in opportunistically-collected data, statistical methods that can accommodate unstructured designs are increasingly useful. Spatial capturerecapture (SCR) has such potential, but its applicability for species that are strongly gregarious is uncertain. It assumes that average animal locations are spatially random and independent, which is violated for gregarious species. I used a data set for African elephants (Loxodonta africana) and data simulation to assess bias and precision of SCR population density estimates given violations in location independence. I found that estimates were negatively biased and likely too precise if non-independence was ignored. Encounter heterogeneity models produced more realistic precision but density estimates were positively biased. Lowest bias was achieved by estimating density of groups, group size, and then multiplying to estimate overall population density. Such findings have important implications for the reliability of population density estimates where data are collected by unstructured means. / LG2017

Singularly perturbed systems with or without delays commonly appear in mathematical modeling of physical and chemical processes, engineering applications, and increasingly, in mathematical biology. There has been intensive work for singularly perturbed systems, yet most of the work so far focused on systems without delays. In this thesis, we provide a new set of tools for the stability analysis for singularly perturbed control systems with time delays. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2015. / FAU Electronic Theses and Dissertations Collection