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.

Three different models, relating to the study of microswimmers immersed in a low Reynolds number fluid, are presented. The underlying, mathematical concepts employed in each are developed using singularity methods of Stokes flow. The first topic concerns the motility of an artificial, three-sphere microswimmer with prescribed, non-reciprocal, internal forces. The swimmer progresses through a low Reynolds number, nonlinear, viscoelastic medium. The model developed illustrates that the presence of the viscoelastic rheology, when compared to a Newtonian environment, increases both the net displacement and swimming efficiency of the microswimmer. The second area concerns biological microswimming, modelling a sperm cell with a hyperactive waveform (vigorous, asymmetric beating), bound to the epithelial walls of the female, reproductive tract. Using resistive-force theory, the model concludes that, for certain regions in parameter space, hyperactivated sperm cells can induce mechanical forces that pull the cell away from the wall binding. This appears to occur via the regulation of the beat amplitude, wavenumber and beat asymmetry. The next topic presents a novel generalisation of slender-body theory that is capable of calculating the approximate flow field around a long, thin, slender body with circular cross sections that vary arbitrarily in radius along a curvilinear centre-line. New, permissible, slender-body shapes include a tapered flagellum and those with ribbed, wave-like structures. Finally, the detailed analytics of the generalised, slender-body theory are exploited to develop a numerical implementation capable of simulating a wider range of slender-body geometries compared to previous studies in the field.

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.

In this thesis we discuss the synchronization of stochastic population oscillators in ecology and neuroscience. Traditionally, the synchronization of oscillators has been studied in deterministic systems with various modes of synchrony induced by coupling between the oscillators. However, recent developments have shown that an ensemble of uncoupled oscillators can be synchronized by a common noise source alone. By considering the effects of noise-induced synchronization on biological oscillators, we are able to explain various biological phenomena in ecological and neurobiological contexts - most importantly, the long-observed Moran effect. Our formulation of the systems as limit cycle oscillators arising from populations of individuals, each with a random element to its behaviour, also allows us to examine the interaction between an external noise source and this intrinsic stochasticity. This provides possible explanations as to why in ecological systems large-amplitude cycles may not be observed in the wild. In neural population oscillators, we were able to observe not just synchronization, but also clustering in some pa- rameter regimes. Finally, we are also able to extend our methods to include coupling in our models. In particular, we examine the competing effects of dispersal and extrinsic noise on the synchronization of a pair of Rosenzweig-Macarthur predator-prey systems. We discover that common environmental noise will ultimately synchronize the oscillators, but that the approach to synchrony depends on whether or not dispersal in the absence of noise supports any stable asynchronous states. We also show how the combination of correlated (shared) and uncorrelated (unshared) noise with dispersal can lead to a multistable steady-state probability density. Similar analysis on a coupled system of neural oscillators would be an interesting project for future work, which, among other future directions of research, is discussed in the concluding section of this thesis.

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.

Cardiac motion is a highly complex and integrated process of vital importance as it sustains the primary function of the heart, that is pumping blood. Cardiac tissue microstructure, in particular the alignment of myocytes (also referred to as fibre direction) and their lateral organisation into laminae (or sheets), has been shown by both experimental and computational research to play an important role in the determination of cardiac motion patterns. However, current models of cardiac electromechanics, although already embedding structural information in the models equations, are not yet able to fully reproduce the connection between structural dynamics and cardiac deformation. The aim of this thesis was to develop an electromechanical modelling framework to investigate the impact of tissue structure on cardiac motion, focussing on left ventricular contraction in rat. The computational studies carried out were complemented with a preliminary validation study based on experimental data of tissue structure rearrangement during contraction from diffusion tensor MRI.

The propulsion mechanics driving the movement of living cells constitutes one of the most incredible engineering works of nature. Active cell motility via the controlled movement of a flagellum beating is among the phylogentically oldest forms of motility, and has been retained in higher level organisms for spermatozoa transport. Despite this ubiquity and importance, the details of how each structural component within the flagellum is orchestrated to generate bending waves, or even the elastic material response from the sperm flagellum, is far from fully understood. By using microbiomechanical modelling and simulation, we develop bio-inspired mathematical models to allow the exploration of sperm motility and the material response of the sperm flagellum. We successfully construct a simple biomathematical model for the human sperm movement by taking into account the sperm cell and its interaction with surrounding fluid, through resistive-force theory, in addition to the geometrically non-linear response of the flagellum elastic structure. When the surrounding fluid is viscous enough, the model predicts that the sperm flagellum may buckle, leading to profound changes in both the waveforms and the swimming cell trajectories. Furthermore, we show that the tapering of the ultrastructural components found in mammalian spermatozoa is essential for sperm migration in high viscosity medium. By reinforcing the flagellum in regions where high tension is expected this flagellar accessory complex is able to prevent tension-driven elastic instabilities that compromise the spermatozoa progressive motility. We equally construct a mathematical model to describe the structural effect of passive link proteins found in flagellar axonemes, providing, for the first time, an explicit mathematical demonstration of the counterbend phenomenon as a generic property of the axoneme, or any cross-linked filament bundle. Furthermore, we analyse the differences between the elastic cross-link shear and pure material shear resistance. We show that pure material shearing effects from Cosserat rod theory or, equivalently, Timoshenko beam theory or are fundamentally different from elastic cross-link induced shear found in filament bundles, such as the axoneme. Finally, we demonstrate that mechanics and modelling can be utilised to evaluate bulk material properties, such as bending stiffness, shear modulus and interfilament sliding resistance from flagellar axonemes its constituent elements, such as microtubules.

The recent emergence of highly virulent strains of the pathogen causing wheat stem rust has been acknowledged as a threat to global food security. In infected wheat fields, vast amounts of pathogenic fungal spores are produced that can be carried away by wind. For targeted disease surveillance and control it is important to estimate when, where and how many fungal spores are dispersed from infected to susceptible wheat fields. In this study, high-performance computational resources are used to investigate long-distance dispersal revealing atmospheric pathways that connect entire continents. Mechanistic simulations of turbulent atmospheric spore dispersal are conducted. The analyses bring together a variety of data, including international field disease surveys and finely resolved meteorological model data. The UK Met Office's Langrangian stochastic particle dispersion model, NAME, is applied, extended and coupled to other models in a set of case studies. In the first case study, spore dispersal is analysed across Southern/East Africa, the Middle East, and Central/South Asia by simulating billions of stochastic trajectories of fungal spores over dynamically changing host and environmental landscapes. The circumstances under which virulent strains, such as Ug99, pose a risk to globally important wheat producing areas are identified. Simulation results indicate a negligible risk for dispersal from key wheat producing countries on the East African continent (Ethiopia, Kenya) directly to India and Pakistan. However, there is a considerable risk for atmospheric transport from the Arabian Peninsula to South Asia. Spore dispersal trends are quantified between all countries in the domain providing estimates which can be used to improve targeted sampling and control. In the second case study, dispersal from southern Africa to Australia is analysed. Simulation results, as well as data from phenotypic and genotypic analyses, support the hypothesis that extremely long-distance airborne dispersal across the Indian Ocean is possible, albeit rare. This indicates that the pathogen populations on the two continents are connected and underlines the importance of sharing surveillance intelligence between continents. The third case study focusses on Ethiopia, determining likely origins of strain TKTTF that recently caused severe epidemics in East Africa's largest wheat producing country. The analyses suggest inflow into Ethiopia from the Middle East via Yemen, consistent with field survey data. The risk for inflow of pathogens into Ethiopia from key neighbouring countries is ranked for different months of the wheat season. In the last results chapter a pilot study is summarized testing the feasibility of an automated short-term forecasting system for spore dispersal from the latest field disease detection sites. Whilst the functionality and practical relevance of the forecasting system is demonstrated, considerable challenges remain for testing the forecasts. The predictive simulation framework described in this thesis can be applied to any wheat producing area worldwide to assess dispersal risks. The research has broader relevance because long-distance dispersal is a key mechanism for the transmission of several crop and livestock diseases, and also plays an important role in other areas of ecology.

Dissolution dynamic nuclear polarisation is an emerging clinical technique which enables the metabolism of hyperpolarised 13C-labelled molecules to be dynamically and non- invasively imaged in tissue. The first molecule to gain clinical approval is [1-13C]pyruvate, the conversion of which to [1-13C]lactate has been shown to detect early treatment re- sponse in cancers and correlate with tumour grade. As the technique has recently been translated into humans, accurate and reliable quantitative methods are required in order to detect, analyse and compare regions of altered metabolism in patients. Furthermore, there is a requirement to understand the biological processes which govern lactate pro- duction in tumours in order to draw reliable conclusions from this data. This work begins with a comprehensive analysis of the quantitative methods which have previously been applied to hyperpolarised 13C data and compares these to some novel approaches. The most appropriate kinetic model to apply to hyperpolarised data is determined and some simple, robust quantitative metrics are identified which are suitable for clinical use. A means of automatically segmenting 5D hyperpolarised imaging data using a fuzzy Markov random field approach is presented in order to reliably identify regions of abnormal metabolic activity. The utility of the algorithm is demonstrated on both in silico and animal data. To gain insight into the processes driving lactate metabolism, a mathematical model is developed which is capable of simulating tumour growth and treatment response under a range of metabolic and tissue conditions, focusing on the interaction between tumour and stroma. Finally, hyperpolarised 13C-pyruvate imaging data from the first human subjects to be imaged in Cambridge is analysed. The ability to detect and quantify lactate production in patients is demonstrated through application of the methods derived in earlier chapters. The mathematical approaches presented in this work have the potential to inform both the analysis and interpretation of clinical hyperpolarised 13C imaging data and to aid in the clinical translation of this technique.

La réponse immunitaire se produit en réaction à une infection, c’est-à-dire à l’introduction d’un pathogène dans l’organisme. Nous nous intéressons à une population de cellules spécifique de la réponse, les lymphocytes T CD8. Nous avons développé un modèle non linéaire structuré en âgedes dynamiques de cette réponse, à l’échelle cellulaire. Nous avons étudié l’existence et la stabilité des états d’équilibre, et obtenu des propriétés mettant en évidence, sous certaines conditions, des dynamiques à long terme correspondant à la situation biologiquement attendue. Nous avons ensuite réalisé une estimation systématique des valeurs de paramètres du modèle, afin de déterminer un ensemble de valeurs pour chaque paramètre qui permet de reproduire convenablement des données expérimentales. Cette démarche permet d’obtenir des informations sur l’influence des paramètres dans le modèle, et sur leurs variations selon la nature du pathogène.Enfin, nous nous sommes intéressés aux dynamiques de la réponse à l’échelle moléculaire, en écrivant un réseau des évènements de signalisation clès depuis l’activation de la cellule en présence d’un antigène, jusqu’à l’entrée en cycle ou l’apoptose de la cellule. Nous avons déterminé un sous modèle centré sur les choix entre survie et apoptose, que nous avons étudié mathématiquement et numériquement. Ce modèle a permis d’étudier les dynamiques de concentrations des protéines impliquées dans la signalisation intra-cellulaire de la réponse T CD8. / Immune response occurs when a pathogen enters the organism and launches an infection. Herewe focused on a specific cell population which takes part in the response : the CD8 T lymphocytes.We first developed an nonlinear age-structured model which accounts for the dynamics of theimmune response at a cell scale. We studied mathematically the existence and stability of steadystates. Hence, we got some properties which highlighted, under certain conditions, some long termdynamics corresponding to biologically expected situation.We then led a systematic estimation of parameter values in the model, in order to find a setof values for each parameter which enabled us to correctly reproduce experimental data. Thanksto this work, we got information of the influence of each parameter in the model and informationof their variations depending on the nature of the pathogen.Eventually, we considered dynamics of the immune response at a molecular scale. We createda network of signal key events, starting from cell activation due to an antigen encounter, to cellcycle entry or apoptosis of the cell. We thus elaborated a sub-model focused on the choice betweensurvival or apoptosis, that we mathematically and numerically studied. Thanks to this model, westudied concentration dynamics of the proteins involved in intracellular signaling in CD8 T cellresponse.