Thesis (Ph.D.)--Aberdeen University, 2008. / Includes bibliographical references.

Thesis (doctoral)--Universität Bonn, 1986. / Includes bibliographical references (p. 121-124).

Upon amputation, zebrafish innately regenerate lost or damaged bone by precisely positioning injury-induced, lineage-restricted osteoblast progenitors (pObs). While substantial progress has been made in identifying the cellular and molecular mechanisms underlying this fascinating process, the cell-specific function of these pathways is poorly understood. Understanding how molecular signals initiate osteoblast dedifferentiation, balance progenitor renewal and re-differentiation, and control bone shape during regeneration are of paramount importance for developing human therapies. We show that fin amputation induces a Wnt/β-catenin-dependent epithelial to mesenchymal transformation (EMT) of osteoblasts to generate proliferative Runx2+ pObs. Localized Wnt/β-catenin signaling maintains this progenitor population towards the distal tip of the regenerative blastema. As they become proximally displaced, pObs upregulate sp7 and subsequently mature into re-epithelialized Runx2-/sp7+ osteoblasts that extend pre-existing bone. Autocrine Bone Morphogenetic Protein (BMP) signaling promotes osteoblast differentiation by activating sp7 expression and counters Wnt by inducing Dickkopf-related Wnt antagonists. As such, opposing activities of Wnt and BMP coordinate the simultaneous demand for growth and differentiation during bone regeneration. Previous studies have implicated Hedgehog/Smoothened (Hh/Smo) signaling in controlling the re-establishment of stereotypically branched bony rays during fin regeneration. Using a photoconvertible patched2 reporter, we resolve active Hh/Smo output to a narrow distal regenerate zone comprising pObs and neighboring migratory basal epidermal cells. Hh/Smo activity is driven by epidermal Sonic hedgehog a (Shha) rather than Ob-derived Indian hedgehog a (Ihha), which instead uses non-canonical signaling to support bone maturation. Using high-resolution imaging and BMS-833923, a uniquely effective Smo inhibitor, we show that Shha/Smo promotes branching by escorting pObs into split groups that mirror transiently divided clusters of Shha-expressing epidermis. Epidermal cellular protrusions directly contact pObs only where an otherwise occluding basement membrane remains incompletely assembled. These intimate interactions progressively generate physically separated pOb pools that then regenerate independently to collectively re-form a now branched bone. Our studies elucidate a signaling network model that provides a conceptual framework to understand innate bone repair and regeneration mechanisms and rationally design regenerative therapeutics. This dissertation includes previously published co-authored material. / 10000-01-01

In this thesis we consider the traditional Van der Pol Oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback.

We investigate global pattern-switching effects in 2D cellular solids in which the voids are arranged in a square lattice. Uniaxial compression of these structures triggers an elastic instability which brings about a period-doubling transformation of the void shapes at a critical strain. Specifically, a square array of circular voids forms a pattern of mutually orthogonal ellipses and a similar effect is observed for diamond-shaped voids. The onset of instability is governed by the void fraction and size-effects are found for the experimental samples. We establish empirical laws which characterise the stiffness, strength and stability of cellular structures comprising square arrays of circular voids. A comparison of these with predictions from a discrete model implies underestimation of the resistance of the lattice to buckling, although the size effects are replicated. We find similar pattern-switching effects in the cubic lattice, which is a three-dimensional porous cube. The effect of buckling in this system is to produce a 2D pattern in one plane of voids. In two-phase granular crystals, rearrangement of a square lattice of particles results in a new, period-doubled, structural pattern. This switch can occur via an intermediate phase depending on the size ratio of the particles as shown in experiments and numerical simulations.

The study of bifurcations of vectorfields is concerned with changes in qualitative behaviour that can occur when a non-structurally stable vectorfield is perturbed. In a sense, this is the study of how such a vectorfield fails to be structurally stable. Finding a systematic approach to the study of such questions is a difficult problem. One approach to bifurcations of vectorfields, known as "generic bifurcation theory," is the subject of much of the work of Sotomayor (Sotomayor [1973a], Sotomayor [1973b],Sotomayor [1974]). This approach attempts to construct generic families of k-parameter vectorfields (usually for k=1), for which all the bifurcations can be described. In Sotomayor [1973a] it is stated that the vectorfields associated with the "generic" bifurcations of individual critical elements for k-parameter vectorfields form submanifolds of codimension ≤ k of the Banach space ϰʳ (M) of vectorfields on a compact manifold M. The bifurcations associated with one of these submanifolds of codimension-k are called generic codimension-k bifurcations. In Sotomayor [1974] the construction of these submanifolds and the description of the associated bifurcations of codimension-1 for vectorfields on two dimensional manifolds is presented in detail. The bifurcations that occur are due to the parameterised vectorfield crossing one of these manifolds transversely as the parameter changes. Abraham and Robbin used transversality results for evaluation maps to prove the Kupka-Smale theorem in Abraham and Robbin [1967]. In this thesis, we shall show how an extension of these evaluation transversality techniques will allow us to construct the submanifolds of ϰʳ (M) associated with one type of generic bifurcation of critical elements, and we shall consider how this approach might be extended to include the other well known generic bifurcations. For saddle-node type bifurcations of critical points, we will show that the changes in qualitative behaviour are related to geometric properties of the submanifold Σ₀ of ϰʳ (M) x M, where Σ₀ is the pull-back of the set of zero vectors-or zero section-by the evaluation map for vectorfields. We will look at the relationship between the Taylor series of a vector-field X at a critical point ⍴ and the geometry of Σ₀ at the corresponding point (X,⍴) of ϰʳ (M) x M. This will motivate the non-degeneracy conditions for the saddle-node bifurcations, and will provide a more general geometric picture of this approach to studying bifurcations of critical points. Finally, we shall consider how this approach might be generalised to include other bifurcations of critical elements. / Science, Faculty of / Mathematics, Department of / Graduate

Dynamical systems possess an interesting and complex behaviour that have attracted a number of researchers across different fields, such as Biology, Economics and most importantly in Engineering. The complex and unpredictability of nonlinear customary behaviour or the chaotic behaviour, makes it strange to analyse them. This thesis presents the analysis of the system of nonlinear differential equations of the so--called Lu--Chen--Cheng system. The system has similar dynamical behaviour with the famous Lorenz system. The nature of equilibrium points and stability of the system is presented in the thesis. Examples of chaotic dynamical systems are presented in the theory. The thesis shows the dynamical structure of the Lu--Chen--Cheng system depending on the particular values of the system parameters and routes to chaos. This is done by both the qualitative and numerical techniques. The bifurcation diagrams of the Lu--Chen--Cheng system that indicate limit cycles and chaos as one parameter is varied are shown with the help of the largest Lyapunov exponent, which also confirms chaos in the system. It is found out that most of the system's equilibria are unstable especially for positive values of the parameters $a, b$. It is observed that the system is highly sensitive to initial conditions. This study is very important because, it supports the previous findings on chaotic behaviours of different dynamical systems.

This thesis first investigates within-host HIV models for the acute stage. These models incorporate the immune responses and helper T cells produced from the activation of naive CD4 T cells. Because both naive CD4 T cells and helper T cells are susceptible classes, backward bifurcation and bistability may occur. We start with a simple model that ignores the CD8 T cell dynamics, then extend it to include this dynamics. We also extend our model to consider the latent infection of naive CD4 T cells. Backward bifurcation occurs in all these models. We numerically investigate the stability of viral equilibria, and show the bistability caused by backward bifurcation. Increasing the inflow of CTLs prevents the backward bifurcation. With a large homeostatic source of healthy naive CD4 T cells, the disease is easier to establish when the basic reproduction number is less than one. Reducing the reproduction number below one is not sufficient to control the infection of HIV. Secondly, this thesis investigates the development of AIDS caused by viral diversity, as proposed by Wodarz et al. using a model that does not include the details of immune responses. We extend their model to include density dependence, and show that the viral load increases with viral diversity. To study if this result still holds with more realistic HIV dynamics, we incorporate viral diversity into our first model. We conclude theoretically that the total viral load is positively correlated with the number of viral strains, and viral diversity can drive the development of AIDS. We also find that the total CD4 T cell count does not always decrease with viral diversity. Thus further investigation is needed to fully understand the development of AIDS. / Graduate

A finite element model of a fracture test specimen is developed using the STAGS computer code (STructural Analysis of General Shells). The test specimen was an internally pressurized, aluminum cylindrical shell reinforced with two externally bonded aluminum tear straps around its circumference. The shell contained an initial, axial through-crack centered between the straps. The crack propagated slowly in the axial direction as the pressure increased above a certain value until a maximum pressure was attained, and then the crack propagated dynamically. The tear straps sufficiently toughened the shell such that the dynamic crack path bifurcated near the edges of the straps. The bifurcated crack branches ran circumferentially, parallel to the straps causing the shell wall to flap open. The STAGS analysis for the static equilibrium configurations of the fractured shell include geometric nonlinearity and elastic-plastic material behavior. The crack tip opening angle (CTOA) is used in the criterion for ductile crack growth, and the critical value of the CTOA is determined by correlating the STAGS predictions of the stable portion of the crack growth curve (internal pressure versus half crack length) to the test. With the employment of a new STAGS algorithm, the complete axial crack growth curve, including both the stable and unstable portions, through the tear strap is obtained. The complete axial crack growth curve indicates that crack growth through the strap is unlikely. STAGS models with long cracks which bifurcate at various half crack lengths are developed to assess the location of crack bifurcation. Three different stress based crack turning criteria are investigated from the axial crack growth results as a second method for assessing a location of bifurcation. The bifurcation analyses and stress based turning criteria corroborate the experimentally measured bifurcation point. A parametric study is then conducted to determine the influence of tear strap thickness and width on the location of crack bifurcation. / Master of Science

Circadian rhythms are the endogenous, roughly 24-hour rhythms that coordinate an organism's interaction with its cycling environment. The molecular mechanism underlying this physiological process is a cell-autonomous oscillator comprised of a complex regulatory network of interacting DNA, RNA and proteins that is surprisingly conserved across many different species. It is not a trivial task to understand how the positive and negative feedback loops interact to generate an oscillator capable of a) maintaining a 24-hour rhythm in constant conditions; b) entraining to external light and temperature signals; c) responding to pulses of light in a rather particular, predictable manner; and d) compensating itself so that the period is relatively constant over a large range of temperatures, even for mutations that affect the basal period of oscillation. Mathematical modeling is a useful tool for dealing with such complexity, because it gives us an object that can be quickly probed and tested in lieu of the experiment or actual biological system. If we do a good job designing the model, it will help us to understand the biology better by predicting the outcome of future experiments. The difficulty lies in properly designing a model, a task that is made even more difficult by an acute lack of quantitative data. Thankfully, our qualitative understanding of a particular phenomenon, i.e. the observed physiology of the cell, can often be directly related to certain mathematical structures. Bifurcation analysis gives us a glimpse of these structures, and we can use these glimpses to build our models with greater confidence. In this dissertation, I will discuss the particular problem of the circadian clock and describe a number of new methods and tools related to bifurcation analysis. These tools can effectively be applied during the modeling process to build detailed models of biological regulatory with greater ease. / Ph. D.