The retina is the ocular tissue responsible for the detection of light. Its extensive demand for oxygen, coupled with a concomitant elevated supply, renders this tissue prone to both hypoxia and hyperoxia. In this thesis, we construct mathematical models of the retina, formulated as systems of reaction-diffusion equations, investigating its oxygen-related dynamics in healthy and diseased states. In the healthy state, we model the oxygen distribution across the human retina, examining the efficacy of the protein neuroglobin in the prevention of hypoxia. It has been suggested that neuroglobin could prevent hypoxia, either by transporting oxygen from regions where it is rich to those where it is poor, or by storing oxygen during periods of diminished supply or increased uptake. Numerical solutions demonstrate that neuroglobin may be effective in preventing or alleviating hypoxia via oxygen transport, but that its capacity for oxygen storage is essentially negligible, whilst asymptotic analysis reveals that, contrary to the prevailing assumption, neuroglobin's oxygen affinity is near optimal for oxygen transport. A further asymptotic analysis justifies the common approximation of a piecewise constant oxygen uptake across the retina, placing existing models upon a stronger theoretical foundation. In the diseased state, we explore the effect of hyperoxia upon the progression of the inherited retinal diseases, known collectively as retinitis pigmentosa. Both numerical solutions and asymptotic analyses show that this mechanism may replicate many of the patterns of retinal degeneration seen in vivo, but that others are inaccessible to it, demonstrating both the strengths and weaknesses of the oxygen toxicity hypothesis. It is shown that the wave speed of hyperoxic degeneration is negatively correlated with the local photoreceptor density, high density regions acting as a barrier to the spread of photoreceptor loss. The effects of capillary degeneration and treatment with antioxidants or trophic factors are also investigated, demonstrating that each has the potential to delay, halt or partially reverse photoreceptor loss. In addition to answering questions that are not accessible to experimental investigation, these models generate a number of experimentally testable predictions, forming the first loop in what has the potential to be a fruitful experimental/modelling cycle.

This dissertation clarifies the concept of evolvability, the increased capacity of some organisms or systems to support evolution, especially the evolution of life-like complexity. I survey the literature, which is spread over the fields of population genetics, developmental biology, artificial life, and microbial and molecular evolution. Finding that researchers have often used the term vaguely and incompatibly I identify five distinct kinds or senses of evolvability. I also identify five key constituent ideas, which I discuss in the context of organismic evolvability, a sense of evolvability with deep roots in the traditional fields of animal development and macroevolution. In these fields research into evolvability has historically been hampered by an insufficiently detailed knowledge of development. Research in molecular evolution has produced a thorough knowledge of the folding of RNA into secondary structure, which can be regarded as a model of development. This has motivated new approaches to evolvability based on representing development via a single genotype-phenotype mapping function. I build on these approaches to invent new mathematical methods to formalise the traditional ideas. I create an exact model illustrating a classic example of evolvability, the capacity for repeated segmentation and simple modularity. I analyse this with two new formal approaches. First is the genospace algebra, a propositional calculus based on graph theory. It is a formal language for describing genotype-phenotype maps. It provides a system for making calculations, proofs, and diagrams about mutational structures in genotype space, and it is flexible enough to allow description at arbitrary degrees of resolution. Second is a pair of concepts, the genetic leverage and the genetic fulcrum. The leverage provides a crude numerical measure of evolvability, and the fulcrum provides a heuristic for identifying the genomic and developmental causes of evolvability. Besides its specific relevance to diversification and development, evolvability is also crucial to the fundamental question of how evolution produces ordinary biological life. Simulation systems that implement only a conventional textbook model of evolution -– systems possessing only variation, inheritance, and selection –- fail to evolve anything resembling the complexity of the biological world. Research into evolvability is our best bet to illuminate the "missing ingredient" for life-like evolution.

In this thesis, two algorithms are introduced for use in ultrasound elastography. Ultrasound elastography is a technique developed in the last 20 years by which anomalous regions in soft tissue are located and diagnosed without the need for biopsy. Due to this, the relativity cheap cost of ultrasound imaging and the high level of accuracy in the methods, ultrasound elastography methods have shown great potential for the diagnosis of cancer in soft tissues. The algorithms introduced in this thesis represent an advance in this field. The first algorithm is a two-step iteration procedure consisting of two minimization problems - displacement estimation and elastic parameter calculation that allow for diagnosis of any anomalous regions within soft tissue. The algorithm represents an improvement on existing methods in several ways. A weighting factor is introduced for each different point in the tissue dependent on the confidence in the accuracy of the data at that point, an exponential substitution is made for the elasticity modulus, an adjoint method is used for efficient calculation of the gradient vector and a total variation regularization technique is used. Most importantly, an adaptive mesh refinement strategy is introduced that allows highly efficient calculation of the elasticity distribution of the tissue though using a number of degrees of freedom several orders lower than methods that use a uniform mesh refinement strategy. Results are presented that show the algorithm is robust even in the presence of significant noise and that it can locate a tumour of 4mm in diameter within a 5cm square region of tissue. Also, the algorithm is extended into 3 dimensions and results are presented that show that it can calculate a 3 dimensional elasticity distribution efficiently. This extension into 3-d is a significant advance in the field. The second algorithm is a one-step algorithm that seeks to combine the two problems of elasticity distribution and displacement calculation into one. As in the two-step algorithm, a weighting factor, exponential substitution for the elasticity parameter, adjoint method for calculation of the gradient vector, total variation regularization and adaptive mesh refinement strategy are incorporated. Results are presented that show that this original approach can locate tumours of varying sizes and shapes in the presence of varying levels of added artificial noise and that it can determine the presence of a tumour in images taken from breast tissue in vivo.

Sudden cardiac death is one of the leading causes of mortality in the western world. One of the main factors is myocardial ischaemia, when there is a mismatch between blood demand and supply to the heart, which may lead to disturbed cardiac excitation patterns, known as arrhythmias. Ischaemia is a dynamic and complex process, which is characterised by many electrophysiological changes that vary through space and time. Ischaemia-induced arrhythmic mechanisms, and the safety and efficacy of certain therapies are still not fully understood. Most experimental studies are carried out in animal, due to the ethical and practical limitations of human experiments. Therefore, extrapolation of mechanisms from animal to human is challenging, but can be facilitated by in silico models. Since the first cardiac cell model was built over 50 years ago, computer simulations have provided a wealth of information and insight that is not possible to obtain through experiments alone. Therefore, mathematical models and computational simulations provide a powerful and complementary tool for the study of multi-scale problems. The aim of this thesis is to investigate pro-arrhythmic electrophysiological consequences of acute myocardial ischaemia, using a multi-scale computational modelling and simulation framework. Firstly, we present a novel method, combining computational simulations and optical mapping experiments, to characterise ischaemia-induced spatial differences modulating arrhythmic risk in rabbit hearts. Secondly, we use computer models to extend our investigation of acute ischaemia to human, by carrying out a thorough analysis of recent human action potential models under varied ischaemic conditions, to test their applicability to simulate ischaemia. Finally, we combine state-of-the-art knowledge and techniques to build a human whole ventricles model, in which we investigate how anti-arrhythmic drugs modulate arrhythmic mechanisms in the presence of ischaemia.

The motile behaviour of bacteria underlies many important aspects of their actions, including pathogenicity, foraging efficiency, and ability to form biofilms. In this thesis, we apply mathematical modelling and analysis to various aspects of the planktonic motility of flagellated bacteria, guided by experimental observations. We use data obtained by tracking free-swimming Rhodobacter sphaeroides under a microscope, taking advantage of the availability of a large dataset acquired using a recently developed, high-throughput protocol. A novel analysis method using a hidden Markov model for the identification of reorientation phases in the tracks is described. This is assessed and compared with an established method using a computational simulation study, which shows that the new method has a reduced error rate and less systematic bias. We proceed to apply the novel analysis method to experimental tracks, demonstrating that we are able to successfully identify reorientations and record the angle changes of each reorientation phase. The analysis pipeline developed here is an important proof of concept, demonstrating a rapid and cost-effective protocol for the investigation of myriad aspects of the motility of microorganisms. In addition, we use mathematical modelling and computational simulations to investigate the effect that the microscope sampling rate has on the observed tracking data. This is an important, but often overlooked aspect of experimental design, which affects the observed data in a complex manner. Finally, we examine the role of rotational diffusion in bacterial motility, testing various models against the analysed data. This provides strong evidence that R. sphaeroides undergoes some form of active reorientation, in contrast to the mainstream belief that the process is passive.

Current studies of the highly complex pathobiology and molecular signatures of human disease require the analysis of large sets of high-throughput data, from clinical to genetic expression experiments, containing a wide range of information types. A number of computational techniques are used to analyse such high-dimensional bioinformatics data. In this thesis we focus on the development of a novel soft clustering technique, DifFUZZY, a fuzzy clustering algorithm applicable to a larger class of problems than other soft clustering approaches. This method is better at handling datasets that contain clusters that are curved, elongated or are of different dispersion. We show how DifFUZZY outperforms a number of frequently used clustering algorithms using a number of examples of synthetic and real datasets. Furthermore, a quality measure based on the diffusion distance developed for DifFUZZY is presented, which is employed to automate the choice of its main parameter. We later apply DifFUZZY and other techniques to data from a clinical study of children from The Gambia with different types of severe malaria. The first step was to identify the most informative features in the dataset which allowed us to separate the different groups of patients. This led to us reproducing the World Health Organisation classification for severe malaria syndromes and obtaining a reduced dataset for further analysis. In order to validate these features as relevant for malaria across the continent and not only in The Gambia, we used a larger dataset for children from different sites in Sub-Saharan Africa. With the use of a novel network visualisation algorithm, we identified pathobiological clusters from which we made and subsequently verified clinical hypotheses. We finish by presenting conclusions and future directions, including image segmentation and clustering time-series data. We also suggest how we could bridge data modelling with bioinformatics by embedding microarray data into cell models. Towards this end we take as a case study a multiscale model of the intestinal crypt using a cell-vertex model.

This thesis is concerned with large-scale systems of Ordinary Differential Equations that model Biomolecular Reaction Networks (BRNs) in Systems and Synthetic Biology. It addresses the strategies of model reduction and decomposition used to overcome the challenges posed by the high dimension and stiffness typical of these models. A number of developments of these strategies are identified, and their implementation on various BRN models is demonstrated. The goal of model reduction is to construct a simplified ODE system to closely approximate a large-scale system. The error estimation problem seeks to quantify the approximation error; this is an example of the trajectory comparison problem. The first part of this thesis applies semi-definite programming (SDP) and dissipativity theory to this problem, producing a single a priori upper bound on the difference between two models in the presence of parameter uncertainty and for a range of initial conditions, for which exhaustive simulation is impractical. The second part of this thesis is concerned with the BRN decomposition problem of expressing a network as an interconnection of subnetworks. A novel framework, called layered decomposition, is introduced and compared with established modular techniques. Fundamental properties of layered decompositions are investigated, providing basic criteria for choosing an appropriate layered decomposition. Further aspects of the layering framework are considered: we illustrate the relationship between decomposition and scale separation by constructing singularly perturbed BRN models using layered decomposition; and we reveal the inter-layer signal propagation structure by decomposing the steady state response to parametric perturbations. Finally, we consider the large-scale SDP problem, where large scale SDP techniques fail to certify a system’s dissipativity. We describe the framework of Structured Storage Functions (SSF), defined where systems admit a cascaded decomposition, and demonstrate a significant resulting speed-up of large-scale dissipativity problems, with applications to the trajectory comparison technique discussed above.

Colorectal cancer (CRC) arises in part from the dysregulation of cellular proliferation, associated with the canonical Wnt pathway, and differentiation, effected by the Notch signalling network. In this thesis, we develop a mathematical model of ordinary differential equations (ODEs) for the coupled interaction of the Notch and Wnt pathways in cells of the human intestinal epithelium. Our central aim is to understand the role of such crosstalk in the genesis and treatment of CRC. An embedding of this model in cells of a simulated colonic tissue enables computational exploration of the cell fate response to spatially inhomogeneous growth cues in the healthy intestinal epithelium. We also examine an alternative, rule-based model from the literature, which employs a simple binary approach to pathway activity, in which the Notch and Wnt pathways are constitutively on or off. Comparison of the two models demonstrates the substantial advantages of the equation-based paradigm, through its delivery of stable and robust cell fate patterning, and its versatility for exploring the multiscale consequences of a variety of subcellular phenomena. Extension of the ODE-based model to include mutant cells facilitates the study of Notch-mediated therapeutic approaches to CRC. We find a marked synergy between the application of γ-secretase inhibitors and Hath1 stabilisers in the treatment of early-stage intestinal polyps. This combined treatment is an efficient means of inducing mitotic arrest in the cell population of the intestinal epithelium through enforced conversion to a secretory phenotype and is highlighted as a viable route for further theoretical, experimental and clinical study.

The evolutionary options of a population are strongly influenced by the avail- ability of adaptive mutants. In this thesis, I use the concept of neutral networks to show that neutral drift can actually increase the accessibility of adaptive mu- tants, and therefore facilitate adaptive evolutionary change. Neutral networks are groups of unique genotypes which all code for the same phenotype, and are connected by simple point mutations. I calculate the size and shape of the networks in a small but exhaustively enumerated space of RNA genotypes by mapping the sequences to RNA secondary structure phenotypes. The qual- itative results are similar to those seen in many other genotype–phenotype map models, despite some significant methodological differences. I show that the boundary of each network has single point–mutation connections to many more phenotypes than the average individual genotype within that network. This means that paths involving a series of neutral point–mutation steps across a network can allow evolution to adaptive phenotypes which would otherwise be extremely unlikely to arise spontaneously. This can be likened to walking along a flat ridge in an adaptive landscape, rather than traversing or jumping across a lower fitness valley. Within this model, when a genotype is made up of just 10 bases, the mean neutral path length is 1.88 point mutations. Furthermore, the map includes some networks that are so convoluted that the path through the network is longer than the direct route between two sequences. A minimum length adaptive walk across the genotype space usually takes as many neutral steps as adaptive ones on its way to the optimum phenotype. Finally I show that the shape of a network can have a very important affect on the number of generations it takes a population to drift across it, and that the more routes between two sequences, the fewer generations required for a population to find an advantageous sequence. My conclusion is that, within the RNA map at least, the size, shape and connectivity of neutral networks all have a profound effect on the way that sequences change and populations evolve, and by not considering them, we risk missing an important evolutionary mechanism.

Wound healing is a complex process, in which a sequence of interrelated events at both the cell and tissue levels interact and contribute to the reduction in wound size. For diabetic patients, many of these processes are compromised, so that wound healing slows down and in some cases halts. In this thesis we develop a series of increasingly detailed mathematical models to describe and investigate healing of full thickness skin wounds. We begin by developing a time-dependent ordinary differential equation model. This phenomenological model focusses on the main processes contributing to closure of a full thickness wound: proliferation in the epidermis and growth and contraction in the dermis. Model simulations suggest that the relative contributions of growth and contraction to healing of the dermis are altered in diabetic wounds. We investigate further the balance between growth and contraction by developing a more detailed, spatially-resolved model using continuum mechanics. Due to the initial large retraction of the wound edge upon injury, we adopt a non-linear elastic framework. Morphoelasticity theory is applied, with the total deformation of the material decomposed into an addition of mass and an elastic response. We use the model to investigate how interactions between growth and stress influence dermal wound healing. The model reveals that contraction alone generates unrealistically high tension in the dermal tissue and, hence, volumetric growth must contribute to healing. We show that, in the simplified case of homogeneous growth, the tissue must grow anisotropically in order to reduce the size of the wound and we postulate mechanosensitive growth laws consistent with this result. After closure the surrounding tissue remodels, returning to its residually stressed state. We identify the steady state growth profile associated with this remodelled state. The model is used to predict the outcome of rewounding experiments as a method of quantifying the amount of stress in the tissue and the application of pressure treatments to control tissue synthesis. The thesis concludes with an extension to the spatially-resolved mechanical model to account for the effects of the biochemical environment. Partial differential equations describing the dynamics of fibroblasts and a regulating growth factor are coupled to equations for the tissue mechanics, described in the morphoelastic framework. By accounting for biomechanical and biochemical stimuli the model allows us to formulate mechanistic laws for growth and contraction. We explore how disruption of mechanical and chemical feedback can lead to abnormal wound healing and use the model to identify specific treatments for normalising healing in these cases.