The ability of cells to move through their environment and spread on surfaces is fundamental to a host of biological processes; including wound healing, growth and immune surveillance. Controlling cell motion has wide-ranging potential for medical applications; including prevention of cancer metastasis and improved colonisation of clinical implants. The relevance of the topic coupled with the naturally arising interplay of biomechanical and biochemical mechanisms that control cell motility make it an exciting problem for mathematical modellers. Two-phase flow models have been widely used in the literature to model cell motility; however, little is known about the mathematical properties of this framework. The majority of this thesis is dedicated to improving our understanding of the two-phase flow framework. We first present the simplest biologically plausible two-phase model for a cell crawling on a flat surface. Stability analyses and a numerical study reveal a number of features relevant to modelling cell motility. That these features are present in such a stripped-down two-phase flow model is notable. We then proceed to investigate how these features are altered in a series of generalisations to the minimal model. We consider the effect of membrane-regulated polymerization of the cell's actin network, the effect of describing the network as viscoelastic, and the effect of explicitly modelling myosin, which drives contraction of the actin network. Validation of hydrodynamical models for cell crawling and spreading requires data on cell shape. The latter part of the thesis develops an image processing routine for extracting the three-dimensional shape of cells settling on a flat surface from confocal microscopy data. Models for cell and droplet settling available in the literature are reviewed and we demonstrate how these could be compared to our cell data. Finally, we summarise the key results and highlight directions for future work.

Our bodies contain a vast number and diversity of microbes. These microbes interact, and these interactions can define how microbes affect us. Microbial ecology and evolution, therefore, are important for both microbiology and human health. However, our understanding of microbial communities remains limited. There is a need for theory that dissects the complexity and identifies the key factors and processes affecting microbial groups. Here I develop realistic computer simulations and population models of microbial communities. My first project seeks to explain microbial communication (quorum sensing) and argues that quorum sensing is a way to infer when competing genotypes are no longer a threat. The second project proposes an evolutionary explanation for another major microbial trait: adhesion. I argue that adhesion is a weapon allowing cells to compete within microbial groups and push competitors out, particularly when growing on a host epithelium. The third project moves from microbes to the host and asks whether a host can control which microbes grow and persist inside it. I develop a model of the human gut epithelium and show that the gut architecture amplifies the ability of hosts to select helpful microbes over harmful ones using nutrient secretion. In addition to selecting particular microbial strains, a host will also benefit from stable symbiotic communities that behave in a predictable manner. But what determines whether host-associated communities are ecologically stable? My final project uses ecological network theory to show that ecological stability is likely to be a problem for gut communities that are diverse and contain species that cooperate with each other. However, I argue that the host should function as an ecosystem engineer that increases ecological stability by weakening the strong dependence of cooperating species upon one another. While host-associated communities are complex ecological systems, my thesis identifies key factors that affect their form and function.

Determining the dynamics and parameter values that drive tumor growth is of great interest to mathematical modelers, experimentalists and practitioners alike. We provide a basis on which to estimate the growth dynamics of ten different tumors by fitting growth parameters to at least five sets of published experimental data per type of tumor. These timescale tumor growth data are also used to determine which of the most common tumor growth models (exponential, power law, logistic, Gompertz, or von Bertalanffy) provides the best fit for each type of tumor. In order to compute the best-fit parameters, we implemented a hybrid local-global least squares minimization algorithm based on a combination of Nelder-Mead simplex direct search and Monte Carlo Markov Chain methods.

Ecosystems are often made up of interactions between large numbers of species. They are considered complex systems because the behaviour of the system as a whole is not always obvious from the properties of the individual parts. A complex system can be represented by a network: a set of interconnected objects. In the case of ecological networks and food webs, the objects are species and the connections are interactions between species. Many complex systems are dynamic and exhibit intricate time series. Time series analysis has been developed to understand a wide range of natural phenomena. This thesis deals with the structure, dynamics, and robustness of ecological networks, the spatial dynamics of fluctuations in a social system, and the analysis of cardiac time series. Biodiversity on Earth is decreasing largely due to human-induced causes. My work looks at the effect of anthropogenic change on ecological networks. In Chapter Two, I investigate predator adaptation on food-web robustness following species extinctions. I identify a new theoretical category of species that may buffer ecosystems against environmental change. In Chapter Three, I study changes in parasitoid-host (consumer-resource) interaction frequencies between complex and simple environments. I show that the feeding preferences of parasitoid species actively change in response to habitat modification. Ecological networks are embedded in spatially-heterogeneous landscapes. In Chapter Four, I assess the role of geography on population fluctuations in an analogous social system. I demonstrate that fluctuations in the number of venture capital firms registered in cities in the United States of America are consistent with spatial and temporal contagion. Understanding how physiological signals vary through time is of interest to medical practitioners. In Chapter Five, I present a technique for quickly quantifying disorder in high frequency event series. Applying the algorithm to patient cardiac time series provides a rapid way to detect the onset of heart arrhythmia. Increasingly, answers to scientific questions lie at the intersection of traditional disciplines. This thesis applies techniques developed in physics and mathematics to problems in ecology and medicine.

Colorectal cancer (CRC) is one of the most prevalent and deadly forms of cancer. Its high mortality rate is associated with difficulties in early detection, which is crucial to survival. The onset of CRC is marked by macroscopic changes in intestinal tissue, originating from a deviation in the healthy cell dynamics of glands known as the crypts of Lieberkuhn. It is believed that accumulated genetic alterations confer on mutated cells the ability to persist in the crypts, which can lead to the formation of a benign tumour through localised proliferation. Stress on the crypt walls can lead to buckling, or crypt fission, and the further spread of mutant cells. Elucidating the initial perturbations in crypt dynamics is not possible experimentally, but such investigations could be made using a predictive, computational model. This thesis proposes a new discrete crypt model, which focuses on the interaction between cell- and tissue-level behaviour, while incorporating key subcellular components. The model contains a novel description of the role of the surrounding tissue and musculature, which allows the shape of the crypt to evolve and deform. A two-dimensional (2D) cross-sectional geometry is considered. Simulation results reveal how the shape of the crypt base may contribute mechanically to the asymmetric division events typically associated with the stem cells in this region. The model predicts that epithelial cell migration may arise due to feedback between cell loss at the crypt collar and density-dependent cell division, an hypothesis which can be investigated in a wet lab. Further, in silico experiments illustrate how this framework can be used to investigate the spread of mutations, and conclude that a reduction in cell migration is key to confer persistence on mutant cell populations. A three-dimensional (3D) model is proposed to remove the spatial restrictions imposed on cell migration in 2D, and preliminary simulation results agree with the hypotheses generated in 2D. Computational limitations that currently restrict extension to a realistic 3D geometry are discussed. These models enable investigation of the role that mechanical forces play in regulating tissue homeostasis, and make a significant contribution to the theoretical study of the onset of crypt deformation under pre-cancerous conditions.

Metapopulation ecology has developed to explain the population dynamics that occur in spatially structured landscapes. In this study, I combined an empirical laboratory approach, using metapopulation microcosms of Callosobruchus maculatus and its endospecific parasitoid Anisopteromalus calandrae, with mathematical population models in order to investigate several fundamental metapopulation processes. Population dynamics in these systems can be studied at two scales; the local patch-wise scale and the regional metapopulation scale. Here I demonstrate that in both homogeneous and heterogeneous landscapes knowledge of local scale demographic processes is necessary in order to understand regional metapopulation dynamics. The differences in the rate and net direction of dispersal between patches as a result of the permeability of the matrix in homogeneous systems and density-dependent dispersal in heterogeneous systems were also explored. Metapopulation dynamics rely on a balance between local extinctions and recolonisations. Therefore, increasing local mortality rates is likely to be detrimental to the persistence of the system. Here, the impact of several common harvesting strategies on the persistence of a host-parasitoid metapopulation was examined. Contrary to expectation I discovered that harvesting in these systems increased both local and regional population sizes. The increased population size as a result of increased mortality was explained in terms of a hydra effect, where harvesting relaxed density-dependence acting on local host populations. The results presented in this thesis are relevant for the monitoring, management and conservation of natural metapopulations and the development of sustainable harvesting strategies in structured landscapes.

Successful T cell activation depends on the recognition of antigenic peptides in the context of a Major Histocompatibility Complex molecule (pMHC) by the T cell antigen receptor (TCR), together with additional signals from co-stimulatory receptors such as CD28. Despite their importance, a thorough understanding of how TCR-pMHC binding properties relate to T cell functional responses remains unclear. In addition, there are no consensuses to the exact mechanism leading to CD28 receptor triggering. Activation of CD28 is dependent on the phosphorylation of key tyrosine residues within its cytoplasmic domain by Src family kinases. Just like the TCRs, CD28 receptors are susceptible to perturbations of the local kinase: phosphatase ratio. The K-S model postulates that upon ligand engagement, large RPTPs such as CD45 are segregated from the area of close contact, resulting in increased relative kinase concentration and CD28 receptor triggering. This hypothesis was tested in chapter 3, where elongated forms of CD80 were examined for their ability to costimulate primary T cells. CD28 costimulation was indeed diminished and there was reduced CD45 segregation from the elongated CD80 molecules. Additionally, CD28 habouring key Y170F tyrosine mutations were less susceptible to CD28 signal abrogation by elongated CD80 molecules. Interestingly, elongated CD80 molecules remained much less effective in mediating costimulation even when pMHC molecules were also elongated, suggesting that TCR-pMHC and CD28-CD80 size matching is not critical for costimulation. Despite the well-documented MHC-restriction requirement for TCR recognition, the relative energetic contributions of peptide versus MHC in TCR-pMHC interactions remain elusive. To address this question, the energetic footprints of four TCRs (1G4, JM22, A6 and G10) to HLA-A2 were determined via systematic alanine scanning mutagenesis on the HLA-A2 heavy chain in chapter 4. By targeting exclusive TCR contacting residues on the MHC, we conservatively estimate the contribution of MHCs for the four TCRs to range from 15% to over 70%. Several models have been formulated in an attempt to relate TCR-pMHC binding properties to T cell activation. Validity of the models was tested in chapter 5 using a supra-physiological TCR. By mutating key residues within the cognate pMHC, we generated a panel of TCR-pMHC with affinities that varies up to 105-fold. These reagents were used to stimulate Jurkat and primary T cells transduced with the supra-physiological TCR. Results in the Jurkat T cell system demonstrated the presence of an optimal off-rate (k_off) for TCR-pMHC interaction at low concentrations of pMHC concentration. The results argue against affinity models and the basic kinetic proofreading model for T cell activation.

This thesis develops biophysically-based data-driven mathematical models of intracellular calciumdynamics in ventricularmyocytes for both normal and genetically modified mouse hearts, based on species- and temperature-consistent experimental data. The models were subsequently applied to quantitatively examine the changes in calcium dynamics in mice with cardiomyocyte-specific knockout (KO) of the cardiac sarco/endoplasmic reticulum ATPase (SERCA2) gene, to determine the contributing mechanisms which underlie the ultimate development of heart failure in these animals. In Chapter 1, with emphasis on calcium dynamics and calcium regulation in heart failure, an overview of cardiac electrophysiology, excitation-contraction coupling and mathematical models of cardiac electrophysiology is provided. In Chapter 2, models of calcium dynamics in the ventricular myocytes from the C57BL/6 mouse heart at a physiological temperature is developed and validated based on species- and temperature-consistent measurements. In Chapter 3, the C57BL/6 model framework is re-parameterised to experimental data from the control and SERCA2 KO mice at 4 weeks after gene deletion. The models are then used to quantitatively characterise changes in calcium dynamics in the KO animals and the role of the compensatory mechanisms. In Chapter 4, the model framework is extended to include differential distributions of ion channels in the sarcolemma and the calcium dynamics in the sub-sarcolemmal space, with parameters in these sub-components fitted to experimentally measured calcium dynamics from the control and KO cardiomyocytes at 7-week after gene deletion. Finally in Chapter 5, conclusions are drawn, the limitations of this study are discussed, and the future extensions to this study are described.

This thesis presents a comprehensive work on computerised analysis of fetal heart rate (FHR) features, including feature extraction, feature selection, analysis of influencing factors and setting up/validation of a computerised decision support system. Firstly, a novel feature – pattern readjustment – was extracted and tested. Clinical data were used to train a Support Vector Machine (SVM) to detect pattern readjustment. Then, the association of pattern readjustment and adverse labour outcome was investigated. The validation results with clinical experts show that the pattern readjustment can be accurately detected, while the study on labour outcome shows that the feature is related to fetal acidemia at birth. Secondly, Genetic Algorithms were employed as a feature selection method to select a best subset of FHR features and to use them to predict fetal acidemia with linear and nonlinear SVM. The diagnostic power of the classifier output using selected features was tested on the total set of 7,568 cases. As the classifier output increases, there is a consistent increase of the risk of fetal acidemia. Thirdly, an important influencing factor on FHR features - signal loss – was investigated. A bivariate model was built to estimate error based on signal loss. Validation results show that the bivariate model can accurately predict the error generated by signal loss. The influence of signal loss on labour outcome classification was also investigated. Finally, a computerised decision support system to estimate the risk of fetal acidemia was set up based on the above studies. The system was validated using new retrospective data. Validation results show that the system is capable of predicting adverse labour outcome and providing timely decision support. It is the first time an intrapartum computerised FHR decision support system has been built and validated on this size of dataset. With further improvements, such a system could be implemented clinically in the long term.

This thesis describes the development of a framework for computational modelling of electromechanics in the mouse, with the purpose of being able to integrate cellular and tissue scale observations in the mouse and investigate physiological hypotheses. Specifically, the framework is applied to interpret electromechanical coupling mechanisms and the progression of heart failure in genetically modified mice. Chapter 1 introduces the field of computational biology and provides context for the topics to be investigated. Chapter 2 reviews the biological background and mathematical bases for electromechanical models, as well as their limitations. In Chapter 3, a set of efficient computational methods for coupled cardiac electromechanics was developed. Among these are a modified Newton method combined with a solution predictor which achieves a 98% reduction in computational time for mechanics problems. In Chapter 4, this computational framework is extended to a multiscale electromechanical model of the mouse. This electromechanical model includes our novel cardiac cellular contraction model for mice, which is able to reproduce murine contraction dynamics at body temperature and high pacing frequencies, and provides a novel explanation for the biphasic force-calcium relation seen in cardiac myocytes. Furthermore, our electromechanical model of the left ventricle of the mouse makes novel predictions on the importance of strong velocity-dependent coupling mechanisms in generating a plateau phase of ventricular pressure transients during ejection. In Chapter 5, the framework was applied to investigate the progression of heart failure in genetically modified 'Serca2 knockout' mice, which have a major disruption in mechanisms governing calcium regulation in cardiac myocytes. Our modelling framework was instrumental in showing for the first time the incompatibility between previously measured cellular calcium transients and ventricular ejection. We were then able to integrate new experimental data collected in response to these observations to show the importance of beta-adrenergic stimulation in the progression of heart failure in these knockout mice. Chapter 6 presents the conclusions and discusses possibilities for future work.