Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations

This thesis is concerned with uniformly convergent finite element methods for numerically solving singularly perturbed parabolic partial differential equations in one space variable. First, we use Petrov-Galerkin finite element methods to generate three schemes for such problems, each of these schemes uses exponentially fitted elements in space. Two of them are lumped and the other is non-lumped. On meshes which are either arbitrary or slightly restricted, we derive global energy norm and L2 norm error bounds, uniformly in the diffusion parameter. Under some reasonable global assumptions together with realistic local assumptions on the solution and its derivatives, we prove that these exponentially fitted schemes are locally uniformly convergent, with order one, in a discrete L∞norm both outside and inside the boundary layer. We next analyse a streamline diffusion scheme on a Shishkin mesh for a model singularly perturbed parabolic partial differential equation. The method with piecewise linear space-time elements is shown, under reasonable assumptions on the solution, to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside layers and almost order 3/4 inside the boundary layer. Numerical results for the above schemes are presented. Finally, we examine a cell vertex finite volume method which is applied to a model time-dependent convection-diffusion problem. Local errors away from all layers are obtained in the l2 seminorm by using techniques from finite element analysis.

The role of the IL-1 type 1 receptor in the life, death and differentiation of adult hippocampal neural precursor cells

Neurogenesis occurs in two distinct regions of the adult brain; the subgranular zone (SGZ) of the dentate gyrus (DG) in the hippocampus, and the subventricular zone (SVZ) lining the lateral ventricles. It is now well-known that adult hippocampal neurogenesis can be modulated by a number of intrinsic and extrinsic factors e.g. local signalling molecules, exercise, environmental enrichment and learning. Moreover, levels of adult hippocampal neurogenesis decrease with age, at least in rodents, and alterations in hippocampal neurogenesis have been reported in animal models and human studies of neuropsychiatric and neurodegenerative conditions. Neuroinflammation is a common pathological feature of these conditions and is also a potent modulator of adult hippocampal neurogenesis. Recently, the orphan nuclear receptor TLX has been identified as an important regulator of adult hippocampal neurogenesis as its expression is necessary to maintain the neural precursor cell (NPC) pool in the adult DG. Likewise, exposure of animals to voluntary exercise has been consistently demonstrated to promote adult hippocampal neurogenesis. Lentivirus (LV)- mediated gene transfer is a useful tool to elucidate gene function and to explore potential therapeutic candidates across an array of conditions as it facilitates sustained gene expression in both dividing and post-mitotic cell populations. Both intrinsic and extrinsic factors are important regulators of adult hippocampal neurogenesis. Examining how these factors are affected by an inflammatory stimulus, and the subsequent effects on adult hippocampal neurogenesis provides important information for the development of novel treatment strategies for neuropsychiatric and neurodegenerative conditions in which adult hippocampal neurogenesis is impaired. The aims of the series of experiments presented in this thesis were to examine the effect of the pro-inflammatory cytokine interleukin-1β (IL-1β) on adult hippocampal NPCs both in vitro and in vivo. In vitro, we have shown that IL-1β reduces proliferation of adult hippocampal NPCs in a dose and time-dependent manner. In addition, we have demonstrated that TLX expression is reduced by IL-1β. Blockade of IL-1β signalling prevented both the IL-1β-induced reduction in cell proliferation and TLX expression. In vivo, we examined the effect of short term and long term exposure to LV-IL-1β in sedentary mice and in mice exposed to voluntary running. We demonstrated that impaired hippocampal neurogenesis is only evident after long term exposure to IL-1β. In mice exposed to voluntary running, hippocampal neurogenesis is significantly increased following short-term but not long-term exposure to running. Moreover, short-term running effectively prevents any IL-1β-induced effects on hippocampal neurogenesis; however, no such effects are seen following long-term exposure to running.

Investigating the function of PINK1 in cancer development and pathogenesis

PTEN‐induced kinase 1 (PINK1) was identified initially in cancer cells as a gene up‐regulated by overexpression of the central tumour suppressor, PTEN. Loss‐of‐function mutations in PINK1 were discovered subsequently to cause autosomal recessive Parkinsonʹs disease (ARPD). Despite much research focusing on the proposed mechanism(s) through which loss of PINKI function causes neurodegeneration, few studies have focused on a direct role for this serine/threonine kinase in cancer biology. The focus of this thesis was to examine a direct role for PINK1 function in tumourigenesis. Initial studies showed that loss of PINK1 reduces tumour‐associated phenotypes including cell growth, colony formation and invasiveness, in several cell types in vitro, indicating a pro‐tumourigenic role for PINK1 in cancer. Furthermore, results revealed for the first time that PINK1 deletion, examined in mouse embryonic fibroblasts (MEFS) from PINK1 knock‐out animals, causes cell cycle defects, whereby cells arrest at in cytokinesis, giving rise to a highly significant increase in the number of multinucleated cells. This results in several key changes in the expression profile of cell cycle associated protein. In addition, PINK1‐deficient MEFs were found to resist cell cycle exit, with a proportion of cells remaining in proliferative phases upon removal of serum. The ability of cells to progress through mitosis conferred by PINK1 expression was independent of its kinase activity, while the cell cycle exit following serum withdrawal was kinase dependent. Investigations into the mechanism through which loss of PINK1 function gives rise to cell cycle defects revealed that dynamin related protein 1 (Drp1)‐mediated mitochondrial fission is enhanced in PINK1‐ deficient MEFs, and that increased expression of Drp1 on mitochondria and activation of Drp1 is highly significant in PINK1‐deficient multinucleated cells. Deregulated and increased levels and activation of mitochondrial fission via Drp1 was shown to be a major feature of cell cycle defects caused by PINK1 deletion, both during progression through G2/M and cell cycle exit following serum removal. Altered PINK1 localisation was also observed during progression of mitosis, and upon serum deprivation. Thus, PINK1 dissociated from the mitochondria during the mitotic phases and localised to mitochondria upon serum withdrawal. During serum withdrawal deletion of PINK1 disabled the ability of MEFs to increase mitochondrial membrane potential (ΔΨm), and increase autophagy. This was co‐incident with increased mitochondrial fission, and increased localisation of Drp1 to mitochondria following serum deprivation. Together, this indicates an inability of PINK1‐negative cells to respond protectively to this stress‐induced state, primarily via impaired mitochondrial function. In contrast, PINK1 overexpression was found to protect cells from DNA damage following treatment with oxidants. In addition, deletion of PINK1 blocked the ability of cells to re‐enter the cell cycle in response to insulin‐like growth factor‐1 (IGF‐1), a major cancer promoting agonistwhich acts primarily via PI3‐kinase/Akt activation. Furthermore, PINK1 mRNA expression was significantly increased following serum deprivation of MCF‐7 cells, and this was rendered more significant upon additional inhibition of PI3‐kinase. Conversely, IGF‐1 activation of PI3‐kinase/Akt causes a time‐dependent and significant reduction of PINK1 mRNA expression that was PI3‐kinase dependent. Together these results indicate that PINK1 expression is necessary for IGF‐1 signalling and is regulated reciprocally in the absence and presence of IGF‐1, via PI3‐kinase/Akt, a signalling system which has major tumour‐promoting capacity in cancer cell biology. The results of this thesis indicate PINK1 is a candidate tumour-promoting gene which has a significant function in the regulation of the cell cycle, and growth factor responses, at key cell cycle checkpoints, namely, during progression through G2/M and during exit of the cell cycle following removal of serum. Furthermore, the results reveal that the regulation of mitochondrial fission and Drp1 function is mechanistically important in the regulation of cell cycle control by PINK1. As deregulation of the cell cycle is linked to both tumourigenesis and neurodegeneration, the findings of this thesis are of importance not just for understanding cancer biology, but also in the context of PINK1‐associated neurodegeneration.

The role of mitogen activated protein kinase phosphatase-1 in neurotoxic and inflammatory-induced changes in the development of midbrain dopaminergic neurons: relevance to Parkinson’s disease

Parkinson’s disease (PD) is a progressive neurodegenerative disorder characterised by the loss of midbrain dopaminergic neurons from the substantia nigra pars compacta(SNpc), which results in motor, cognitive and psychiatric symptoms. Evidence supports a role for the mitogen-activated protein kinase p38 in the demise of dopaminergic neurons, while mitogen-activated protein kinase phosphatase-1 (MKP-1), which negatively regulates p38 activity, has not yet been investigated in this context. Inflammation may also be associated with the neuropathology of PD due to evidence of increased levels of proinflammatory cytokines such as interleukin-1β (IL-1β) within the SNpc. Because of the specific loss of dopaminergic neurons in a discreet region of the brain, PD is considered a suitable candidate for cell replacement therapy but challenges remain to optimise dopaminergic cell survival and morphological development. The present thesis examined the role of MKP-1 in neurotoxic and inflammatory-induced changes in the development of midbrain dopaminergic neurons. We show that MKP-1 is expressed in dopaminergic neurons cultured from embryonic day (E) 14 rat ventral mesencephalon (VM). Inhibition of dopaminergic neurite growth induced by treatment of rat VM neurons with the dopaminergic neurotoxin 6- hydroxydopamine (6-OHDA) is mediated by p38, and is concomitant with a significant and selective decrease in MKP-1 expression in these neurons. Dopaminergic neurons transfected to overexpress MKP-1 displayed a more complex morphology and contributed to neuroprotection against the effects of 6-OHDA. Therefore, MKP-1 expression can promote the growth and elaboration of dopaminergic neuronal processes and can help protect them from the neurotoxic effects of 6-OHDA. Neural precursor cells (NPCs) have emerged as promising alternative candidates to fetal VM for cell replacement strategies in PD. Here we show that phosphorylated (and thus activated) p38 and MKP-1 are expressed at basal levels in untreated E14 rat VM NPCs (nestin, DCX, GFAP and DAT-positive cells) following proliferation as well as in their differentiated progeny (DCX, DAT, GFAP and βIII-tubulin) in vitro. Challenge with 6-OHDA or IL-1β changed the expression of endogenous phospho-p38 and MKP-1 in these cells in a time-dependent manner, and so the dynamic balance in expression may mediate the detrimental effects of neurotoxicity and inflammation in proliferating and differentiating NPCs. We demonstrate that there was an up-regulation in MKP-1 mRNA expression in adult rat midbrain tissue 4 days post lesion in two rat models of PD; the 6-OHDA medial forebrain bundle (MFB) model and the four-site 6-OHDA striatal lesion model. This was concomitant with a decrease in tyrosine hydroxylase (TH) mRNA expression at 4 and 10 days post-lesion in the MFB model and 10 and 28 days post-lesion in the striatal lesion model. There was no change in mRNA expression of the pro-apoptotic gene, bax and the anti-apoptotic gene, bcl-2 in the midbrain and striatum. These data suggest that the early and transient upregulation of MKP-1 mRNA in the midbrain at 4 days post-6-OHDA administration may be indicative of an attempt by dopaminergic neurons in the midbrain to protect against the neurotoxic effects of 6-OHDA at later time points. Collectively, these findings show that MKP-1 is expressed by developing and adult dopaminergic neurons in the midbrain, and can promote their morphological development. MKP-1 also exerts neuroprotective effects against dopaminergic neurotoxins in vitro, and its expression in dopaminergic neurons can be modulated by inflammatory and neurotoxic insults both in vitro and in vivo. Thus, these data contribute to the information needed to develop therapeutic strategies for protecting midbrain dopaminergic neurons in the context of PD.

Plasmonic gold nanostructures: optical properties and application in mercury detection

This thesis investigates the application of plasmonic gold nanostructures for mercury detection. Various gold and silver single nanostructures and gold nanostructure assemblies were characterised in detail by correlated single nanostructure spectroscopy and electron microscopy. Several routes for mercury detection were explored: plasmon resonance energy transfer (PRET) upon Hg2+ binding to immobilised gold nanoparticle-organic ligand hybrid structures and amalgamation of single immobilised gold nanorods upon chemical and upon electrochemical reduction of Hg2+ ions. The amalgamation approach showed large potential with extraordinary shifts of the nanorods’ scattering spectra upon exposure to reduced mercury; a result of compositional and morphological change induced in the nanorod by amalgamation with mercury. A shift of 5 nm could be recorded for a concentration as low 10 nM Hg2+. Through detailed time-dependent experiments insights into the amalgamation mechanism were gained and a model comprising 5 steps was developed. Finally, spectroelectrochemistry proved to be an excellent way to study in real time in-situ the amalgamation of mercury with gold nanorods paving the way for future work in this field.