Biological tissue differs from other materials in many ways. Perhaps the most crucial difference is its ability to grow. Growth processes may give rise to stresses that exist in a body in the absence of applied loads and these are known as residual stresses. Residual stress is present in many biological systems and can have important consequences on the mechanical response of a body. Mathematical models of biological structures must therefore be able to capture accurately the effects of differential growth and residual stress, since greater understanding of the roles of these phenomena may have applications in many fields. In addition to residual stresses, biological structures often have a complex morphology. The theory of 3-D elasticity is analytically tractable in modelling mechanical properties in simple geometries such as a cylinder. On the other hand, rod theory is well-suited for geometrically-complex deformations, but is unable to account for residual stress. In this thesis, we aim to develop a map between the two frameworks. Firstly, we use 3-D elasticity to determine effective mechanical properties of a growing cylinder and map them into an effective rod. Secondly, we consider a growing filament embedded in an elastic foundation. Here, we estimate the degree of transverse reinforcement the foundation confers on the filament in terms of its material properties. Finally, to gain a greater understanding of the role of residual stress in biological structures, we consider a case study: the chameleon's tongue. In particular we consider the role of residual stress and anisotropy in aiding the rapid projection of the tongue during prey capture. We construct a mechanical model of the tongue and use it to investigate a proposed mechanism of projection by means of an energy balance argument.

Motility of flagellated bacteria has been a topic of increasing scientific interest over the past decades, attracting the attention of mathematicians, physicists, biologists and engineers alike. Bacteria and other micro-organisms cause substantial damage through biofilm growth on submerged interfaces in water cooling systems, ship hulls and medical implants. This gives social and economic motivations for learning about how micro-organisms swim and behave in different environments. Fluid flows on such small scales are dominated by viscosity and therefore behave differently from the inertia-dominated flows that we are more familiar with, making bacterial motility a physically intriguing phenomenon to study as well. We use the boundary element method (BEM) to simulate the motion of singly flagellated bacteria in a viscous, Newtonian fluid. One of our main objectives is to investigate the influence of external surfaces on swimming behaviour. We show that the precise shape of the cell body and flagellum can be important for determining boundary behaviour, in particular, whether bacteria are attracted or repelled from surfaces. Furthermore, we investigate the types of motion that may arise between two parallel plates and in rectangular channels of fluid and show how these relate to the plane boundary interactions. As an extension to original models of flagellar propulsion in bacteria that assume a rotation of the rigid helical flagellum about an axis fixed relative to the cell body, we consider flexibility of the bacterial hook connecting the aforementioned parts of the swimmer. This is motivated by evidence that the hook is much more flexible than the rest of the flagellum, which we therefore treat as a rigid structure. Elastic dynamics of the hook are modelled using the equations for a Kirchhoff rod. In some regimes, the dynamics are well described by a rigid hook model but we find the possibility of additional modes of behaviour.

Sudden cardiac death represents one of the leading causes of death worldwide, with the majority of these deaths caused by arrhythmias derived from ischæmic events. However, the mechanisms leading from ischæmia to re-entry, arrhythmia and eventual death are poorly understood. Furthermore, variability in the action potential of cardiac tissue, while important in determining arrhythmic risk, is only recently being addressed in computational modelling, with little known about the causes and mechanisms underlying it, nor regarding its evolution in response to pathological conditions such as ischæmia. This dissertation investigates the causes of variability in the repolarisation of the action potential of the rabbit ventricular myocyte, and the response of this variability to ischæmia. The effect of variability in ion channel conductances is investigated by means of a complete search of the parameter space revealed by simultaneous variation in multiple parameters describing ion channel conductances in computational models of the rabbit ventricular action potential. Rabbit data and models are used in this thesis due to the similarities to human data, both in terms of electrophysiology generally, and the response to ischæmia specifically. The response of two different model frameworks is assessed to determine similarities and differences between model frameworks that are designed to reproduce the same system. Those models producing action potential durations that fall within an experimentally derived range at multiple pacing rates are used to define model populations that thus reproduce experimental variability in repolarisation. These model populations are used to investigate the effects of ischæmic conditions on population variability. Variability is measured not only for action potential duration, but also for other biomarkers commonly implicated in the development of re-entry. The work presented in this dissertation is significant for: (1) presenting a comprehensive study of the effect of simultaneous variation in ion channel conductances, with details regarding the interactions between conductances and how these interactions change depending on the pacing rate; (2) detailed examination of the differences between two models of the same system; (3) production of the largest extant populations reproducing experimentally observed variability in action potential duration; (4) the first time model populations have been used to investigate the effects of ischæmia on variability.

This thesis investigates how collective decisions in bird flocks arise from simple rules, the factors that give some birds more influence over a flock's direction, and how travelling as a flock affects spatial learning. I used GPS loggers to track pigeons homing alone and in flocks, and applied mathematical modelling to explore the mechanisms underlying group decisions. Across several experiments, the key results were as follows: Flying home with a more experienced individual not only gave a pigeon an immediate advantage in terms of taking a more direct route, but the followers also learned homing routes just as accurately as pigeons flying alone. This shows that using social cues did not interfere with learning about the landscape during a series of paired flights. Pigeons that were faster during solo homing flights also tended to fly at the front of flocks, where they had more influence over the direction taken. Analysis of momentary interactions during paired flights and simulations of pair trajectories support the conclusion that speed increases the likelihood of leading. A pigeon's solo homing efficiency before flock flights did not correlate with leadership in flocks of ten, but leaders did have more efficient solo tracks when tested after a series of flock flights. A possible explanation is that leaders attended more to the landscape and therefore learned faster. Flocks took straighter routes than pigeons flying alone, as would be expected if they effectively pooled information. In addition, pigeons responded more strongly to the direction of several neighbours, during flock flights, than to a single neighbour during paired flights. This behaviour makes sense adaptively because social information will be more reliable when following several conspecifics compared to one. Through a combination of high-resolution tracking and mathematical modelling, this thesis sheds light on the mechanisms of flocking and its navigational consequences.

Angiogenesis, the process by which new vessels form from existing ones, is a key event in the development of a large and malignant vascularised tumour. A rapid expansion in in vivo and in vitro angiogenesis research in recent years has led to increased knowledge about the processes underlying angiogenesis and to promising steps forward in the development of anti-angiogenic therapies for the treatment of various cancers. However, substantial gaps in knowledge persist and the development of effective treatments remains a major challenge. In this thesis we study tumour-induced angiogenesis within the context of a highly controllable experimental environment: the cornea micropocket assay. Using a multidisciplinary approach that combines experiments, image processing and analysis, and mathematical and computational modelling, we aim to provide mechanistic insight into the action of two angiogenic factors which are known to play central roles during tumour-induced angiogenesis: vascular endothelial growth factor A (VEGF-A) and basic fibroblast growth factor (bFGF). Image analysis techniques are used to extract quantitative data, which are both spatially and temporally resolved, from experimental images. These data are then used to develop and parametrise mathematical models describing the evolution of the corneal vasculature in response to both VEGF-A and bFGF. The first models developed in this thesis are one-dimensional continuum models of angiogenesis in which VEGF-A and/or bFGF are released from a pellet implanted into a mouse cornea. We also use an object-oriented framework, designed to facilitate the re-use and extensibility of hybrid multiscale models of angiogenesis and vascular tumour growth, to develop a complementary three-dimensional hybrid model of the same system. The hybrid model incorporates a new non-local cell sensing model which facilitates the formation of well-perfused and functional vascular networks in three dimensions. The continuum models are used to assess the utility of the cornea micropocket assay as a quantitative assay for angiogenesis, to characterise proposed synergies between VEGF-A and bFGF, and to generate experimentally testable predictions regarding the effect of anti-VEGF-A therapies on bFGF-induced angiogenesis. Meanwhile, the hybrid model is used to provide context for the comparison that is drawn between the continuum models and the data, to study the relative distributions of perfused and unperfused vessels in the evolving neovasculature, and to investigate the impact of tip cell sensing dysregulation on the angiogenic response in the cornea. We have found that by exploiting a close link with quantitative data we have been able to extend the predictive and hypothesis-testing capabilities of our models. As such, this thesis demonstrates the potential for integrating mathematical modelling with image analysis techniques to increase insight into the mechanisms underlying angiogenesis.

Over the course of this thesis we will build and develop a model for the dynamics of an environmentally transmitted disease such as cholera. We will also develop methods to analyse and understand that model. The dynamics of a disease in a heterogeneous developing world city have not yet been fully explored, particularly when those dynamics are affected by a natural disaster. Yet, natural disasters such as floods alter infrastructure and population characteristics in a manner that affects disease transmission. Therefore, we shall address this omission from the literature. We will also develop a novel model analysis framework for 'systems epidemiology' where we combine systems biology techniques with epidemiological modelling.

Existem milhares de espécies de serpentes no mundo, muitas com padrões distintos e intricados. Esta diversidade se torna um problema para usuários que precisam criar texturas de pele de serpente para aplicar em modelos 3D, pois a dificuldade em criar estes padrões complexos é considerável. Nós primeiramente propomos uma categorização de padrões de pele de serpentes levando em conta suas características visuais. Então apresentamos um modelo procedural capaz de sintetizar uma vasta gama de textura de padrões de pele de serpentes. O modelo usa processamento de imagem simples (tal como sintetizar bolinhas e listras) bem como autômatos celulares e geradores de ruído para criar texturas realistas para usar em renderizadores modernos. Nossos resultados mostram boa similaridade visual com pele de serpentes reais. As texturas resultantes podem ser usadas não apenas em computação gráfica, mas também em educação sobre serpentes e suas características visuais. Nós também realizamos testes com usuários para avaliar a usabilidade de nossa ferramenta. O escore da Escala de Usabilidade do Sistema foi de 85:8, sugerindo uma ferramenta de texturização altamente efetiva. / There are thousands of snake species in the world, many with intricate and distinct skin patterns. This diversity becomes a problem for users who need to create snake skin textures to apply on 3D models, as the difficulty for creating such complex patterns is considerable. We first propose a categorization of snake skin patterns considering their visual characteristics. We then present a procedural model capable of synthesizing a wide range of texture skin patterns from snakes. The model uses simple image processing (such as synthesizing spots and stripes) as well as cellular automata and noise generators to create realistic textures for use in a modern renderer. Our results show good visual similarity with real skin found in snakes. The resulting textures can be used not only for computer graphics texturing, but also in education about snakes and their visual characteristics. We have also performed a user study to assess the usability of our tool. The score from the System Usability Scale was 85:8, suggesting a highly effective texturing tool.

Biological invasions are devastating ecosystems and economies world-wide, while many native species' survival depends on their ability to track climate change. Characterising the spread of biological populations is therefore of utmost importance, and can be studied with spatially explicit, discrete-time integro-difference equations (IDEs), which reflect numerous species' processes of demography and dispersal. While spatial variation has often been ignored when implementing IDE models, real landscapes are rarely spatially uniform and environmental variation is crucial in determining biological spread. To address this, we use novel methods to characterise population spread in heterogeneous landscapes. Asymptotic analysis is used for highly fragmented landscapes, where habitat patches are isolated and smaller than the dispersal scale, and in landscapes with low environmental variation, where the ecological parameters vary by no more than a small factor from their mean values. We find that the choice of dispersal kernel determines the effect of landscape structure on spreading speed, indicating that accurately fitting a kernel to data is important in accurately predicting speed. For the low-variation case, the spreading speeds in the heterogeneous and homogeneous landscapes differ by ϵ², where ϵ governs the degree of variation, suggesting that in many cases, a simpler homogeneous model gives similar spread rates. For irregular landscapes, analytical methods become intractable and numerical simulation is needed to predict spread. Accurate simulation requires high spatial resolution, which, using existing techniques, requires prohibitive amounts of computational resources (RAM, CPU etc). We overcome this by developing and implementing a novel algorithm that uses adaptive mesh refinement. The approximations and simulation algorithm produce accurate results, with the adaptive algorithm providing large improvements in efficiency without significant losses of accuracy compared to non-adaptive simulations. Hence, the adaptive algorithm enables faster simulation at previously unfeasible scales and resolutions, permitting novel areas of scientific research in species spread modelling.

Filamentous fungi are important in an enormous variety of ways to our life, with examples ranging from bioremediation, through the food and drinks industry to human health. These organisms can form huge networks stretching metres and even kilometres. However, their mode of growth is by the extension of individual hyphal tips only a few microns in diameter. Tip growth is mediated by the incorporation of new wall building materials at the soft apex. Just how this process is controlled (in fungi and in cell elongation in other organisms) has been the subject of intense study over many years and has attracted considerable attention from mathematical modellers. In this thesis, we consider mathematical models of fungal tip growth that can be classified as either geometrical or biomechanical. In every model we examine, a 2-D axisymmetric semihemisphere-like curve represents half the medial section of fungal tip geometry. A geometrical model for the role of the Spitzenkorper in the tip growth was proposed by Bartnicki-Garcia et al (1989), where a number of problems with the mathematical derivation were pointed out by Koch (2001). A suggestion is given as an attempt to revise the derivation by introducing a relationship between arc length of a growing tip, deposition of wall-building materials and tip curvature. We also consider two types of geometrical models as proposed by Goriely et al (2005). The first type considers a relationship between the longitudinal curvature and the function used to model deposition of wall-building materials. For these types of models, a generalized formulae for the tip shape is introduced, which allows localization of deposition of wall-building materials to be examined. The second type considers a relationship between longitudinal and latitudinal curvatures and the function used to model deposition of wall-building materials. For these types of models, a new formulation of the function used to model deposition of wall-building materials is introduced. Finally, a biomechanical model as proposed by Goriely et al (2010). Varying arc length of the stretchable region on the tip suggests differences in geometry of tip shape and the effective pressure profile. The hypothesis of orthogonal growth is done by focusing only on the apex of a "germ tube". Following that, it suggests that material points on the tip appear to move in a direction perpendicular to the tip either when surface friction is increased or decreased.

The structure of a protein molecule is highly correlated to its function. Some diseases such as cystic fibrosis are the result of a change in the structure of a protein so that this change interferes or inhibits its function. Often these changes in structure are caused by a misfolding of the protein molecule. To assist computational biologists, there is a database of proteins together with their misfolded versions, called decoys, that can be used to test the accuracy of protein structure prediction algorithms. In our work we use a nested graph model to quantify a selected set of proteins that have two single misfold decoys. The graph theoretic model used is a three tiered nested graph. Measures based on the vertex weights are calculated and we compare the quantification of the proteins with their decoys. Our method is able to separate the misfolded proteins from the correctly folded proteins.