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.

Molecular and cellular aspects of the SREBP pathway

The SREBP (sterol response element binding proteins) transcription factors are central to regulating de novo biosynthesis of cholesterol and fatty acids. The SREBPs are regulated by retention or escape from the ER to the Golgi where they are proteolytically cleaved into active forms. The SREBP cleavage activating protein (SCAP) and the INSIG proteins are essential in this regulatory process. The aim of this thesis is to further characterise the molecular and cellular aspects surrounding regulation of SREBP processing. SREBP and SCAP are known to interact via their carboxy-terminal regulatory domains (CTDs) but this interaction is poorly characterised. Significant steps were achieved in this thesis towards specific mapping of the interaction site. These included cloning and over expression and partial purification of tagged SREBP1 and SREBP2 CTDs and probing of a SCAP peptide array with the CTDs. Results from the SREBP2 probing were difficult to interpret due to insolubility issues with the protein, however, probing with SREBP1 revealed five potential binding sites which were detected reproducibly. Further research is necessary to overcome SREBP2 insolubility issues and to confirm the identified SREBP1 interaction site(s) on SCAP. INSIG1 has a central role in regulating SREBP processing and in regulating stability of 3-hydroxy-3-methylglutaryl coenzyme A reductase (HMGCR), a rate limiting enzyme in cholesterol biosynthesis. There are two protein isoforms of human INSIG1 produced through the use of two in-frame alternative start sites. Bioinformatic analysis indicated that the presence of two in-frame start sites within the 5-prime region of INSIG1 mRNA is highly conserved and that production of two isoforms of INSIG1is likely a conserved event. Functional differences between these two isoforms were explored. No difference in either the regulation of SREBP processing or HMGCR degradation between the INSIG1 isoforms was observed and the functional significance of the two isoforms is as yet unclear. The final part of this thesis focused on enhancing the cytotoxicity of statins by targeted inhibition of SREBP processing by oxysterols. Statins have significant potential as anti-cancer agents as they inhibit the activity of HMGCR leading to a deficiency in mevalonate which is essential for cell survival. The levels of HMGCR fluctuate widely due to cholesterol feedback of SREBP processing. The relationship between sterol feedback and statin mediated cell death was investigated in depth in HeLa cells. Down regulation of SREBP processing by sterols significantly enhanced the efficacy of statin mediated cell death. Investigation of sterol feedback in additional cancer cell lines showed that sterol feedback was absent in cell lines A- 498, DU-145, MCF-7 and MeWo but was present in cell lines HT-29, HepG2 and KYSE-70. In the latter inhibition of SREBP processing using oxysterols significantly enhanced statin cytotoxicity. The results indicate that this approach is valid to enhance statin cytotoxicity in cancer cells, but may be limited by deregulation of SREBP processing and off target effects of statins, which were observed for some of the cancer cell lines screened.

A qualitative study of trends in patient preferences for the management of partial dentition

Objective: To identify factors influencing attitudes of partially dentate adults towards dental treatment in Ireland. Background: People are retaining more teeth later in life than ever before. Management of partially dentate older adults will be a major requirement for the future and it is important to determine factors which may influence patients’ attitudes to care. Methods: Subjects: A purposive sample of 22 partially dentate patients was recruited; 12 women and 12 men, ranging in age from 45 to 75 years. Data Collection: Semi-structured individual interviews. Results: Dental patients have increasing expectations in relation to (i) a more sophisticated approach to the management of missing teeth and (ii) their right to actively participate in decision making regarding the management of their tooth loss. There is some evidence of a cohort effect with younger patients (45–64 years) having higher expectations. Conclusions: The evidence of a cohort effect within this study in relation to higher patient expectations indicates that both contemporary and future patients are likely to seek a service based on conservation and restoration of missing teeth by fixed prostheses.