The study of sex allocation is one of the most successful areas in evolutionary biology: its theoretical predictions have been supported by experimental, observational and comparative approaches. Here, I develop sex allocation theory as follows: (1) I use fertility insurance theory to predict the sex ratio strategy of the malaria parasite, in response to human medical interventions that increase mortality and decrease fertility of the parasite’s various sexual stages; (2) Haplodiploidy has been suggested as a driver of the evolution of eusociality, as under this genetic system a female may be more related to her sister than to her own offspring. I examine a model considering queen versus worker control over the sex ratio of the colony and show that haplodiploidy alone does not explain the evolution of helping; (3) I follow up this study of the haplodiploidy hypothesis by examining the idea that split-sex ratios may favour the evolution of eusociality in haplodiploid species. I study the two mechanisms of split sex ratios, that are found in natural populations and may have been important in the transition to eusociality: queen virginity and queen replacement. I focus on the impact of worker reproduction by considering the effect of woker producing a fraction of the colony offspring and by considering variation in the workers’ offspring sex ratio. My analysis shows that worker reproduction does not promote the evolution of helping in haplodiploid species; (4) I examine the evolution and function of a sterile soldier caste in parasitoid wasps from the genus Encyrtidae. Two main functions have been hypothesized for the emergence of soldiers: spiteful mediation of a sex ratio conflict in mixed-sex broods, and altruistic protection and 7 facilitation of the development of relatives. I develop a model considering variation in the oviposition behaviour of females, that may produce single-sex or mixed-sex broods. I show that, in accordance with previous theory, females are expected to produce more soldiers than males, under the sex ratio conflict hypothesis. I also show that one of the consequences of this costly conflict is that females are favoured to produce single-sex broods over mixed-sex broods.

From the developing embryo to the evacuation of football stadiums, the migration and movement of populations of individuals is a vital part of human life. Such movement often occurs in crowded conditions, where the space occupied by each individual impacts on the freedom of others. This thesis aims to analyse and understand the effects of occupied volume (volume exclusion) on the movement of the individual and the population. We consider, as a motivating system, the rearrangement of individuals required to turn a clump of cells into a functioning embryo. Specifically, we consider the migration of cranial neural crest cells in the developing chick embryo. Working closely with experimental collaborators we construct a hybrid model of the system, consisting of a continuum chemoattractant and individual-based cell description and find that multiple cell phenotypes are required for successful migration. In the crowded environment of the migratory system, volume exclusion is highly important and significantly enhances the speed of cell migration in our model, whilst reducing the numbers of individuals that can enter the domain. The developed model is used to make experimental predictions, that are tested in vivo, using cycles of modelling and experimental work to give greater insight into the biological system. Our formulated model is computational, and is thus difficult to analyse whilst considering different parameter regimes. The second part of the thesis is driven by the wish to systematically analyse our model. As such, it concentrates on developing new techniques to derive continuum equations from diffusive and chemotactic individual-based and hybrid models in one and two spatial dimensions with the incorporation of volume exclusion. We demonstrate the accuracy of our techniques under different parameter regimes and using different mechanisms of movement. In particular, we show that our derived continuum equations almost always compare better to data averaged over multiple simulations than the equivalent equations without volume exclusion. Thus we establish that volume exclusion has a substantial effect on the evolution of a migrating population.

We investigate the steady state solutions that can exist for a two dimensional swarm of biological organisms, which have pairwise social interaction forces. The three steady states we investigate using a continuum model are a ribbon migrating swarm, a circular migrating swarm, and a milling swarm. We solve these numerically by reformulating the integral equation that arises from the continuum model as an energy minimization problem. For the ribbon migrating solution, we are able to determine an analytic solution from Carleman's equation which arises after an asymptotic expansion of the social interaction potential. Using this technique we are able to show the existence of a square root singularity that emerges at the boundary of the compactly supported swarm. The analytic solution agrees with the numerical solution for certain parameter values in the social interaction potential. We then demonstrate the existence of solutions for a migrating and milling circular swarm which contain a square root singularity. The milling swarm looks similar to the infinite ribbon, so we are able to use an asymptotic expansion of the potential to obtain an analytic solution in this case as well. The singularities in the density of the swarm suggest that the Morse potential should not be used for modeling biological swarming.

Many phenomena in the BAEP are difficult to explain on the basis of the accepted hypothesis of its origin (after Jewett, 1970). The alternative mechanism of origin to which these phenomena point is summation of oscillations. Therefore, simulation of the BAEP by a mathematical model consisting of the addition of four sine waves was tested. The model did simulate a normal BAEP as well variations in the waveform produced by reversing click polarity. This simulation gives further clues to the origin of the BAEP. The four sine waves begin simultaneously; corresponding BAEP oscillations must, therefore, originate from a single structure. These oscillations begin in less than half a millisecond after the click. This suggests that the structure from which they arise is outside the brainstem. This alternative mechanism indicates that wave latencies do not reflect nervous conduction between discrete nuclei, and interpretation of BAEP abnormality need to be reconsidered. It also implies that mathematical frequency analysis is more appropriate, but this could be applied only when these methods have been perfected. Meanwhile, through visual analysis and recognition of oscillations, abnormality can be detected and described in terms that may have physiological significance.

In vitro tissue engineering is a method for developing living and functional tissues external to the body, often within a device called a bioreactor to control the chemical and mechanical environment. However, the quality of bone tissue engineered products is currently inadequate for clinical use as the implant cannot bear weight. In an effort to improve the quality of the construct, hydrostatic pressure, the pressure in a fluid at equilibrium that is required to balance the force exerted by the weight of the fluid above, has been investigated as a mechanical stimulus for promoting extracellular matrix deposition and mineralisation within bone tissue. Thus far, little research has been performed into understanding the response of bone tissue cells to mechanical stimulation. In this thesis we investigate an in vitro bone tissue engineering experimental setup, whereby human mesenchymal stem cells are seeded within a collagen gel and cultured in a hydrostatic pressure bioreactor. In collaboration with experimentalists a suite of mathematical models of increasing complexity is developed and appropriate numerical methods are used to simulate these models. Each of the models investigates different aspects of the experimental setup, from focusing on global quantities of interest through to investigating their detailed local spatial distribution. The aim of this work is to increase understanding of the underlying physical processes which drive the growth and development of the construct, and identify which factors contribute to the highly heterogeneous spatial distribution of the mineralised extracellular matrix seen experimentally. The first model considered is a purely temporal model, where the evolution of cells, solid substrate, which accounts for the initial collagen scaffold and deposited extracellular matrix along with attendant mineralisation, and fluid in response to the applied pressure are examined. We demonstrate that including the history of the mechanical loading of cells is important in determining the quantity of deposited substrate. The second and third models extend this non-spatial model, and examine biochemically and biomechanically-induced spatial patterning separately. The first of these spatial models demonstrates that nutrient diffusion along with nutrient-dependent mass transfer terms qualitatively reproduces the heterogeneous spatial effects seen experimentally. The second multiphase model is used to investigate whether the magnitude of the shear stresses generated by fluid flow, can qualitatively explain the heterogeneous mineralisation seen in the experiments. Numerical simulations reveal that the spatial distribution of the fluid shear stress magnitude is highly heterogeneous, which could be related to the spatial heterogeneity in the mineralisation seen experimentally.

Introduction: Antimalarial resistance, particularly artemisinin resistance, is a major threat to P. falciparum malaria elimination efforts worldwide. Urgent intervention is required to tackle artemisinin resistance but field data on which to base planning of strategies are limited. The aims were to collect available field data and develop population level mathematical models of P. falciparum malaria treatment and artemisinin resistance in order to determine the optimal strategies for elimination of artemisinin resistant malaria in Cambodia and treatment of pre-hospital and severe malaria in Cambodia and Bangladesh. Methods: Malaria incidence and parasite clearance data from Cambodia and Bangladesh were collected and analysed and modelling parameters derived. Population dynamic mathematical models of P. falciparum malaria were produced. Results: The modelling demonstrated that elimination of artemisinin resistant P. falciparum malaria would be achievable in Cambodia in the context of artemisinin resistance using high coverages with ACT treatment, ideally combined with LLITNs and adjunctive single dose primaquine. Sustained efforts would be necessary to achieve elimination and effective surveillance is essential, both to identify the baseline malaria burden and to monitor parasite prevalence as interventions are implemented. A modelled policy change to rectal and intravenous artesunate in the context of pre-existing artemisinin resistance would not compromise the efficacy of ACT for malaria elimination. Conclusions: By being developed rapidly in response to specific questions the models presented here are helping to inform planning efforts to combat artemisinin resistance. As further field data become available, their planned on-going development will produce increasingly realistic and informative models which can be expected to play a central role in planning efforts for years to come.

Human cancers exhibit the common phenotype of elevated glycolytic metabolism, which causes acidification of the tissue microenvironment and may facilitate tumour invasion. In this thesis, we use mathematical models to address a series of open problems underlying the glycolytic tumour phenotype and its attendant acidity. We first explore tissue-scale consequences of metabolically-derived acid. Incorporating more biological detail into a canonical model of acidity at the tumour-host interface, we extend the range of tumour behaviours captured by the modelling framework. We then carry out an asymptotic travelling wave analysis to express invasive tumour properties in terms of fundamental parameters, and find that interstitial gaps between an advancing tumour and retreating healthy tissue, characteristic of aggressive invasion and comprising a controversial feature of the original model, are less significant under our generalised formulation. Subsequently, we evaluate a potential role of lactate---historically assumed to be a passive byproduct of glycolytic metabolism---in a perfusion-dependent metabolic symbiosis that was recently proposed as a beneficial tumour behaviour. Upon developing a minimal model of dual glucose-lactate consumption in vivo and employing a multidimensional sensitivity analysis, we find that symbiosis may not be straightforwardly beneficial for our model tumour. Moreover, new in vitro experiments, carried out by an experimental collaborator, place U87 glioblastoma tumours in a weakly symbiotic parameter regime despite their clinical malignancy. These results suggest that intratumoural metabolic cooperation is unlikely to be an important role for lactate. Finally, we examine the complex pH regulation system that governs expulsion of metabolically derived acid loads across tumour cell membranes. This system differs from the healthy system by expression of only a few key proteins, yet its dynamics are non-intuitive in the crowded and poorly perfused in vivo environment. We systematically develop a model of tumour pH regulation, beginning with a single-cell scenario and progressing to a spheroid, within a Bayesian framework that incorporates information from in vitro data contributed by a second experimental collaborator. We predict that a net effect of pH regulation is a straightforward transmembrane pH gradient, but also that existing treatments are unable to disrupt the system strongly enough to cause tumour cell death. Taken together, our models help to elucidate previously unresolved features of glycolytic tumour metabolism, and illustrate the utility of a combined mathematical, statistical, and experimental approach for testing biological hypotheses. Opportunities for further investigation are discussed.

The analysis of non steady state physiological and pathological processes concerns the abstraction, extraction, formalisation and analysis of information from physiological systems that is obscured, hidden or unable to be assessed using traditional methods. Time Series Analysis (TSA) techniques were developed and built into a software program, Easy TSA, with the aim of examining the oscillations of hormonal concentrations in respect to their temporal aspects – periodicity, phase, pulsatility. The Easy TSA program was validated using constructed data sets and used in a clinical study to examine the relationship between insulin and obesity in people without diabetes. In this study fifty-six non-diabetic subjects (28M, 28F) were examined using data from a number of protocols. Fourier Transform and Autocorrelation techniques determined that there was a critical effect of the level of BMI on the frequency, amplitude and regularity of insulin oscillations. Second, information systems formed the background to the development of an algorithm to examine glycaemic variability and a new methodology termed the Glycaemic Risk in Diabetes Equation (GRADE) was developed. The aim was to report an integrated glycaemic risk score from glucose profiles that would complement summary measures of glycaemia, such as the HbA1c. GRADE was applied retrospectively to blood glucose data sets to determine if it was clinically relevant. Subjects with type 1 and type 2 diabetes had higher GRADE scores than the non-diabetic population and the contribution of hypo- and hyperglycaemic episodes to risk was demonstrated. A prospective study was then designed with the aim to apply GRADE in a clinical context and to measure the statistical reproducibility of using GRADE. Fifty-three (Male 26, Female 27) subjects measured their blood glucose 4 times daily for twenty-one days. The results were that lower HbA1c’s correlated with an increased risk of hypoglycaemia and higher HbA1c’s correlated with an increased risk of hyperglycaemia. Some subjects had HbA1c of 7.0 but had median GRADE values ranging from 2.2 to 10.5. The GRADE score summarized diverse glycaemic profiles into a single assessment of risk. Well-controlled glucose profiles yielded GRADE scores <= 5 and higher GRADE scores represented increased clinical risk from hypo or hyperglycaemia. Third, an information system was developed to analyse data-rich multi-variable retinal images using the concept of assessment of change rather than specific lesion recognition. A fully Automated Retinal Image Differencing (ARID) computer system was developed to highlight change between retinal images over time. ARID was validated using a study and then a retrospective study sought to determine if the use of the ARID software was an aid to the retinal screener. One hundred and sixty images (80 image pairs) were obtained from Gloucestershire Diabetic Eye Screening Programme. Images pairs were graded manually and categorised according to how each type of lesion had progressed, regressed, or not changed between image A and image B. After a 30 day washout period image pairs were graded using ARID and the results compared. The comparison of manual grading to grading using ARID (Table 4.3) demonstrated an increased sensitivity and specificity. The mean sensitivity of ARID (87.9%) was increased significantly in comparison to manually grading sensitivity (84.1%) (p<0.05). The specificity of the automated analysis (87.5%) increased significantly from the specificity (56.3%) achieved by manually grading (p<0.05). The conclusion was that automatic display of an ARID differenced image where sequential photographs are available would allow rapid assessment and appropriate triage. Forth, non-linear dynamic systems analysis methods were utilised to build a system to assess the extent of chaos characteristics within the insulin-glucose feedback domain. Biological systems exist that are deterministic yet are neither predictable nor repeatable. Instead they exhibit chaos, where a small change in the initial conditions produces a wholly different outcome. The glucose regulatory system is a dynamic system that maintains glucose homeostasis through the feedback mechanism of glucose, insulin, and contributory hormones and was ideally suited to chaos analysis. To investigate this system a new algorithm was created to assess the Normalised Area of Attraction (NAA). The NAA was calculated by defining an oval using the 95% CI of glucose & Insulin (the limit cycle) on a phasic plot. Thirty non-diabetic subjects and four subjects with type 2 diabetes were analysed. The NAA indicated a smaller range for glucose and insulin excursions with the non-diabetics subjects (p<0.05). The conclusion was that the evaluation of glucose metabolism in terms of homeostatic integrity and not in term of cut-off values may enable a more realistic approach to the effective treatment and prevention of diabetes and its complications.

This thesis is concerned with methodologies for the accurate quantitative modelling of molecular biological systems. The first part is devoted to the chemical Langevin equation (CLE), a stochastic differential equation driven by a multidimensional Wiener process. The CLE is an approximation to the standard discrete Markov jump process model of chemical reaction kinetics. It is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. We observe that the CLE is not a single equation, but a family of equations with shared finite-dimensional distributions. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation, which is just the rank of the stoichiometric matrix. On the practical side, we show that in the case where there are m_1 pairs of reversible reactions and m_2 irreversible reactions, there is another, simple formulation of the CLE with only m_1+m_2 Wiener processes, whereas the standard approach uses 2m_1+m_2. Considerable computational savings are achieved with this latter formulation. A flaw of the CLE model is identified: trajectories may leave the nonnegative orthant with positive probability. The second part addresses the challenge when alternative, structurally different ordinary differential equation models of similar complexity fit the available experimental data equally well. We review optimal experiment design methods for choosing the initial state and structural changes on the biological system to maximally discriminate between the outputs of rival models in terms of L_2-distance. We determine the optimal stimulus (input) profile for externally excitable systems. The numerical implementation relies on sum of squares decompositions and is demonstrated on two rival models of signal processing in starving Dictyostelium amoebae. Such experiments accelerate the perfection of our understanding of biochemical mechanisms.

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.