Many mathematical models in biology and physiology are represented by systems of nonlinear differential equations. In recent years these models have become increasingly large-scale and multiphysics, as increasing amounts of data are available on the properties and behaviour of biological systems. Often an observed behaviour of interest in a model may be written as a linear functional. A key question therefore is to determine which terms in the model have the greatest effect on functionals of interest. An approach for answering this question has recently been developed, called model reduction using a posteriori analysis. The method was initially developed for systems of nonlinear initial value ordinary differential equations, and automatically identifies regions of the computational domain and components of the model solution where an accurate mathematical representation of the model is required to accurately calculate a linear functional of interest. Initial-value ordinary differential equations can be written as a first-order derivative term plus an algebraic 'reaction' term. In previous work on model reduction using a posteriori analysis the algebraic 'reaction' term is removed from the equations in the reduced model. The first contribution of this thesis is to extend the method so that the first-order derivative term is removed from the differential equation instead of the algebraic 'reaction' term, resulting in a quasi-steady state approximation in automatically identified regions of the domain and components of the solution. The second contribution of this thesis is to extend the method to boundary value problems and partial differential equations. We consider differential equations with algebraic terms, first order terms and second order terms, any combination of which may be nonlinear. The method is used to automatically simplify several differential equation models including models of chemotaxis and tissue-level cardiac electrophysiology.

Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. The principal aim of this thesis is to develop a series of mathematical models for somite formation. We begin by reviewing the current models for somitogenesis in the light of new experimental evidence regarding the presence of a segmentation clock and graded expression of FGF8. We conduct a preliminary investigation into the wavefront of FGF8 along the antero-posterior axis and integrate this model into the framework of an existing model for a signalling process. We demonstrate that this new “Clock and Wavefront” model can produce coherent series’ of somites in a manner that is tightly regulated in both space and time, and that it can also mimic the effects seen when FGF8 expression is perturbed locally. We then use the model to make some experimentally testable predictions. The latter part of the thesis concentrates on building more biologically accurate model for the FGF8 gradient. We move to consider a model for the FGF8 gradient which involves a complex network of biochemical interactions with negative feedback between FGF8 and retinoic acid. The resulting system of seven coupled non-linear equations, including both ordinary and partial differential equations, is difficult to analyse. To facilitate our understanding of the non-linear interactions between FGF8 and retinoic acid, we finally consider a reduced model which can display travelling wavefronts of opposing FGF8 and retinoic acid concentrations moving down the antero-posterior axis. The model allowed us to calculate a minimum wave speed for the wavefronts as a function of key model parameters such as the rate of FGF8 and retinoic acid decay; strong dependence on the values of these parameters is a result that is hypothesised to occur in vivo.

Centrosomin (Cnn) is an integral centrosomal protein in Drosophila with orthologues in several species, including humans. The human orthologue of Cnn is required for brain development with Cnn hypothesised to play a similar role in Drosophila. Control of Cnn incorporation into centrosomes is crucial for controlling asymmetric division in certain types of Drosophila stem cells. FRAP experiments on Cnn show that Cnn recovers in a pe- culiar fashion, which suggest that Cnn may be incorporated closest to the centrioles and then spread radially outward, either diffusively or ad- vectively. The aim of this thesis is to understand the mechanism of Cnn incorporation into the Drosophila centrosomes, to determine the mode of transport of the incorporated Cnn, and to obtain parameter estimates for transport and biochemical reactions. A crucial unknown in the modelling process is the distribution of Cnn receptors. We begin by constructing coupled partial differential equation models with either diffusion or advection as the mechanism for incorpo- rated Cnn transport. The simplest receptor distribution we begin with involves a spherical, infinitesimally thick, impermeable shell. We refine the diffusion models using the insights gained from comparing the model out- put with data (gathered during mitosis) and through careful assessment of the behaviour of the data. We show that a Gaussian receptor distribution is necessary to explain the Cnn FRAP data and that the data cannot be explained by other simpler receptor distributions. We predict the exact form of the receptor distribution through data fitting and present pre- liminary experimental results from our collaborators that suggest that a protein called DSpd2 may show a matching distribution. Not only does this provide strong experimental support for a key prediction from our model, but it also suggests that DSpd2 acts as a Cnn receptor. We also show using the mitosis FRAP data that Cnn does not exhibit appreciable radial movement during mitosis, which precludes the use of these data to distinguish between diffusive and advective transport of Cnn. We use long time Cnn FRAP data gathered during S-phase for this purpose. We fit the S-phase FRAP data using the DSpd2 profiles gath- ered for time points corresponding to the Cnn FRAP experiments. We also use data from FRAP experiments where colchicine is injected into the embryos to destroy microtubules (since microtubules are suspected to play a role in advective transport of Cnn). From the analysis of all these data we show that Cnn is transported in part by advection and in part by diffusion. Thus, we are able to provide the first mechanistic description of the Cnn incorporation process. Further, we estimate parameters from the model fitting and predict how some of the parameters may be altered as nuclei progress from S-phase to mitosis. We also generate testable predic- tions regarding the control of the Cnn incorporation process. We believe that this work will be useful to understand the role of Cnn incorporation in centrosome function, particularly in asymmetrically dividing stem cells.

In the spontaneous curvature model for two-phase lipid bilayer membranes the shape of vesicles is governed by a combination of an elastic bending energy and an interface energy that penalises the size of phase boundaries. Each lipid phase induces a preferred curvature to the membrane surface, and these curvatures as well as phase boundaries may lead to the development of kinks. In a rotationally symmetric setting we introduce a family of energies for smooth surfaces and phase fields for the lipid components and study convergence to a sharp-interface limit, which depends on the choice of the bending parameters of the phase field model. We prove that, if kinks are excluded, our energies $Gamma$-converge to the commonly used sharp-interface spontaneous curvature energy with the additional assumption of $C^1$-regularity across interfaces. For a choice of parameters such that kinks may appear, we obtain a limit that coincides with the $Gamma$-limit on all reasonable membranes and extends the classical model by assigning a bending energy also to kinks. We illustrate the theoretical result by some numerical examples.

This thesis focuses on the population dynamics of three antigenically diverse RNA viruses: dengue, influenza and HIV-1. It comprises a set of studies highlighting the roles of structural constraints on critical antigenic determinants, interactions between immune responses to different antigenic types, host lifespan, and the degree of mixing between different host populations in determining the epidemiology and within-host dynamics of these pathogen systems. Dengue exists in humans as a collection of four antigenically related serotypes. Although infection by one serotype appears to convey life-long protection to homologous infection, it is believed to be a risk factor for severe disease manifestations upon secondary, heterologous infection due to the phenomenon of Antibody-Dependent Enhancement (ADE). It is not clear if third or fourth infections are possible, and if so, how they contribute to dengue epidemiology. In this thesis, I investigate the effect of third and fourth infections on the transmission dynamics of dengue. By contrast with dengue, human influenza viruses are known to be in rapid antigenic flux, manifesting in the sequential replacement of antigenic types. This pattern of evolution does not appear to be the same in shorter-lived hosts such as swine and birds. In this thesis, I have used a simple multi-locus model to explore the relationship between host lifespan and viral evolution, as well as to elucidate the effects of transmission between hosts of different lifespan in effort to capture the cross-species element of influenza transmission. My final chapter concerns the within-host evolution of HIV-1. I propose a new model for the pathogenesis of HIV-1 where the transition to AIDS is primarily linked to the gradual loss of the ability to make new antibody responses as the CD4+ population declines. Together these studies emphasise that it is the changing profile of immune responses – either at the population level or within the host – that is the principal determinant of the dynamics of the pathogen, rather than the mode and tempo of antigenic innovation.

We report progress made on an analytic investigation of low-frequency cardiorespiratory variability in humans. The work is based on an existing physiological model of chemically-mediated blood-gas control via the central and peripheral chemoreceptors, that of Grodins, Buell & Bart (1967). Scaling and simplification of the Grodins model yields a rich variety of dynamical subsets; the thesis focusses on the dynamics obtained under the normoxic assumption (i.e., when oxygen is decoupled from the system). In general, the method of asymptotic reduction yields submodels that validate or invalidate numerous (and more heuristic) extant efforts in the literature. Some of the physiologically-relevant behaviour obtained here has therefore been reported before, but a large number of features are reported for the first time. A particular novelty is the explicit demonstration of cardiorespiratory coupling via chemosensory control. The physiology and literature reviewed in Chapters 1 and 2 set the stage for the investigation. Chapter 3 scales and simplifies the Grodins model; Chapters 4, 5, 6 consider carbon dioxide dynamics at the central chemoreceptor. Chapter 7 begins analysis of the dynamics mediated by the peripheral receptor. Essentially all of the dynamical behaviour is due to the effect of time delays occurring within the conservation relations (which are ordinary differential equations). The pathophysiology highlighted by the analysis is considerable, and includes central nervous system disorders, heart failure, metabolic diseases, lung disorders, vascular pathologies, physiological changes during sleep, and ascent to high altitude. Chapter 8 concludes the thesis with a summary of achievements and directions for further work.

This thesis deals with the mathematical modelling of nutrient uptake by plant roots. It starts with the Nye-Tinker-Barber model for nutrient uptake by a single bare cylindrical root. The model is treated using matched asymptotic expansion and an analytic formula for the rate of nutrient uptake is derived for the first time. The basic model is then extended to include root hairs and mycorrhizae, which have been found experimentally to be very important for the uptake of immobile nutrients. Again, analytic expressions for nutrient uptake are derived. The simplicity and clarity of the analytical formulae for the solution of the single root models allows the extension of these models to more realistic branched roots. These models clearly show that the `volume averaging of branching structure' technique commonly used to extend the Nye-Tinker-Barber with experiments can lead to large errors. The same models also indicate that in the absence of large-scale water movement, due to rainfall, fertiliser fails to penetrate into the soil. This motivates us to build a model for water movement and uptake by branched root structures. This model considers the simultaneous flow of water in the soil, uptake by the roots, and flow within the root branching network to the stems of the plant. The water uptake model shows that the water saturation can develop pseudo-steady-state wet and dry zones in the rooting region of the soil. The dry zone is shown to stop the movement of nutrient from the top of the soil to the groundwater. Finally we present a model for the simultaneous movement and uptake of both nutrients and water. This is discussed as a new tool for interpreting available experimental results and designing future experiments. The parallels between evolution and mathematical optimisation are also discussed.

Acidic pH is a common characteristic of human tumours. It has a significant impact on tumour progression and response to therapies. In this thesis, we utilise mathematical modelling to examine the role of acidosis in the interaction between normal and tumour cell populations. In the first section we investigate the cell–microenvironmental interactions that mediate somatic evolution of cancer cells. The model predicts that selective forces in premalignant lesions act to favour cells whose metabolism is best suited to respond to local changes in oxygen, glucose and pH levels. In particular the emergent cellular phenotype, displaying increased acid production and resistance to acid-induced toxicity, has a significant proliferative advantage because it will consistently acidify the local environment in a way that is toxic to its competitors but harmless to itself. In the second section we analyse the role of acidity in tumour growth. Both vascular and avascular tumour dynamics are investigated, and a number of different behaviours are observed. Whilst an avascular tumour always proceeds to a benign steady state, a vascular tumour may display either benign or invasive dynamics, depending on the value of a critical parameter. Extensions of the model show that cellular quiescence, or non-proliferation, may provide an explanation for experimentally observed cycles of acidity within tumour tissue. Analysis of both models allows assessment of novel therapies directed towards changing the level of acidity within the tumour. Finally we undertake a comparison between experimental tumour pH images and the models of acid dynamics set out in previous chapters. This analysis will allow us to assess and verify the previous modelling work, giving the mathematics a firm biological foundation. Moreover, it provides a methodology of calculating important diagnostic parameters from pH images.

Among the most fundamental features shared by all organisms is the mapping of information encoded in genotypes (genetic material) to generate phenotypes (biological structures, functions, and traits). Hence, elucidating the structure of genotype-phenotype (GP) maps is important for understanding evolution and biology. While it is known that often GP maps are highly degenerate with many different genotypes adopting the same phenotype, the distribution of genotypes over phenotypes is less well studied. In this thesis we investigate the question of the distribution of genotypes over phenotypes, or put differently the distribution of neutral set sizes (NSS), where a neutral set is the collection of all genotypes in a GP map which map to the same phenotype. We focus on examining phenotypic bias in GP maps, where some phenotypes have disproportionally large NSS as compared to others. We find phenotypic bias to be ubiquitous in the broad range of GP maps that we analyse, from the genetic code up to molecular RNA to a model of neuronal connections, and hence we hypothesise bias to be a common property of GP maps. Further, we also consider the implications that this bias has for evolutionary outcomes, and we argue that bias is a significant influencing factor in determining evolutionary outcomes. Finally, we propose a method to predict a phenotype's NNS via estimating the phenotype's structural complexity, without using detailed knowledge about the specifics of the relevant GP map. We achieve this via a novel application of algorithmic information theory and especially Levin's coding theorem.

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