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Let a ∈ Z n >0 , n ≥ 2 , gcd(a) := gcd(a1 , . . . , an ) = 1, b ∈ Z≥0 and denote by k · k∞ the ℓ∞-norm. Consider the knapsack polytope P(a, b) = { x ∈ R n ≥0 : a T x = b and assume that P(a, b) ∩ Z n 6= ; holds. The main result of this thesis states that for any vertex x ∗ of the knapsack polytope P(a, b) there exists a feasible integer point z ∗ ∈ P(a, b) such that, denoting by s the size of the support of z ∗ , i.e. the number of nonzero components in z ∗ and upon assuming s > 0 , the inequality kx ∗ − z ∗ k∞ 2 s−1 s < kak∞ holds. This inequality may be viewed as a transference result which allows strengthening the best known distance (proximity) bounds if integer points are not sparse and, vice versa, strengthening the best known sparsity bounds if feasible integer points are sufficiently far from a vertex of the knapsack polytope. In particular, this bound provides an exponential in s improvement on the previously best known distance bounds in the knapsack scenario. Further, when considering general integer linear programs, we show that a resembling inequality holds for vertices of Gomory's corner polyhedra [49, 96]. In addition, we provide several refinements of the known distance and support bounds under additional assumptions.

The prevalence of Chronic Kidney Disease (CKD) is rising owing to increased diabetes and obesity in our ageing populations. Hence, end-stage renal disease (ESRD) has become a major global health burden. For many ESRD patients, kidney transplantation is not possible. Hence these patients require dialysis to regulate their salt/fluid levels, and to remove toxins from the body. Peritoneal Dialysis (PD) is a well-established and effective form of treatment for ESRD. However, the longevity of PD is limited because of the constant exposure of the peritoneum to bioincompatible PD solutions. This, alongside unavoidable repeated episodes of PD peritonitis, alters the structural and functional integrity of the peritoneal membrane and plays a pivotal role in the development of peritoneal fibrosis leading to failure of the peritoneal tissue to act as a semi-permeable membrane. This results in failure of PD as a therapy for ESRD, limiting a patient's options for renal-replacement therapy. A key process that drives peritoneal fibrosis during PD is mesothelial-to-mesenchymal transition (MMT). MMT is a process whereby mesothelial cells (known to comprise the majority of cells in the peritoneal membrane) undergo myofibroblast differentiation and lay down increased fibrous matrix. Numerous studies have demonstrated that TGF-β1 plays an established role in driving MMT and in the subsequent development of peritoneal fibrosis. Therefore, delineating the regulators of TGF-β1-driven MMT are important in identifying novel mechanisms to prevent peritoneal fibrosis and PD failure. In solid organ fibrosis (lungs, kidneys, skin) TGF-β1-driven myofibroblast differentiation is mediated through increased expression and alterations in the matrix polysaccharide, hyaluronan (HA). Previous studies demonstrate that PD therapy is associated with increased HA generation by mesothelial cells. However, the role of this in mediating TGF-β1-driven in the peritoneum has not been established and the role of increased HA generation during PD is not understood. The aim of this study was to determine the role of the increased HA generated during PD in regulating peritoneal infection, inflammation and fibrosis. This work uses primary human peritoneal mesothelial cells to study the involvement of HA in mediating TGF-β1-driven MMT in an in vitro experimental model. The role of HA in prevention and/or reversal of TGF-β1-driven MMT is also investigated in vitro. The role of HA in modulating peritoneal immunity and inflammation following acute bacterial infection is also examined using an in vivo mouse-model of PD peritonitis. Patients with PD peritonitis had significantly increased HA concentrations in their PD effluent after developing acute bacterial peritonitis compared to non-infected patients. Cell studies established that TGF-β1-driven MMT in primary human mesothelial cells significantly increased HA generation, and this was predominantly driven by Hyaluronan Synthase-1 (HAS1) isoenzyme expression. However, the increased HA was not causally related to TGF-β1- driven MMT; but was simply a consequence of this process. The increased HA also did not mediate prevention and/or reversal of TGF-β1-driven MMT in primary human mesothelial cells. Blocking HA in the peritoneum with PEP1 blocking antibody during bacterial infection lead to a delayed resolution of inflammation, characterized by increased and persistent neutrophil infiltration, dysregulated monocyte recruitment and decreased cytokine and chemokine release. In conclusion, following peritoneal injury, HA generated by mesothelial cells is predominantly HAS1 driven and promotes enhanced peritoneal inflammation and alters leukocyte recruitment following acute bacterial infection.

Artificial ribonucleases (ARs) are recognised as a potential strategy to selectively target and cleave biologically significant RNA in cells. However, in order to work as true enzymes they must exhibit catalytic turnover. Many of the reported ARs incorporate metal-containing centres (e.g. dysprosium, copper) in order to induce substantial phosphodiester cleavage, which is not amenable to use in vivo. Therefore, new strategic directions employing metal-independent ARs, such as peptidyl-oligonucleotide conjugates (POCs), need to be investigated. Previous work has shown that poor or non-complementary POCs demonstrate catalytic turnover of a HIV-1 substrate; however, sequence specificity is an issue. For POCs to be useful from a therapeutic standpoint they must only cleave specific RNA molecules and do so in a catalytic fashion, therefore removing the requirement for stoichiometric drug delivery and binding. Consequently, novel POC design strategies are required that allow selective RNA targeting but promote dissociation of the POC following phosphodiester cleavage. In this research, three types of different peptidyl-oligonucleotide conjugate designs have been implemented with the attempt to find an appropriate balance between selectivity and catalytic turnover.(i) Selective targeting and quantitative cleavage (97-100%) of a tRNAPhe target was achieved through phosphoramidate attachment of a 17-mer TΨC-targeting oligonucleotide to catalytic amphiphilic peptide sequences containing leucine, arginine and glycine. Although the half-life of tRNAPhe was less than 1 h on exposure to some of these POCs, hybridisation studies reveal that the POCs bind too tightly to their target RNA sequences and thus an excess of POC is required for efficient cleavage activity. The effect of peptide and oligonucleotide sequence variations as well as the role of enhanced conformational freedom via incorporating an abasic deoxyribose linker between the oligonucleotide targeting motif and catalytic peptide is also investigated. (ii) Most of ‘Dual’ peptidyl-oligonucleotide conjugates containing an amphiphilic RNA-cleaving peptide placed between two RNA recognition motifs directed towards the TΨC loop and 3’ acceptor stem of tRNAPhe demonstrate marked RNA binding and cleavage activities. Interestingly, those dual conjugates which showed poor or negligible binding ability in electrophoresis assays, demonstrated sufficient RNA cleavage (70%) within the vicinity of the 65GACAC61 target region. Therefore, weak POC:RNA complexes may exist which could facilitate substrate turnover. (iii) Finally, POCs were designed which induce bulge-loops in their target RNA region upon hybridisation. By introducing regions of non-complementarity into the oligonucleotide sequences, 2- to 5- membered bulges sizes were formed. Via attachment of a catalytic peptide to an internally modified oligonucleotide residue, catalytic peptides were placed directly adjacent to single-stranded RNA regions to promote cleavage by nuclease mimics. Through probing the hybridised complexes with RNase H, the presence of bulges were confirmed for all bulge-loop sizes, which will be followed by cleavage experiments to assess the possibility for reaction catalytic turnover. In conclusion, a variety of POCs have been synthesised, characterised and partially tested for their RNA cleaving and turnover activity. Based on the encouraging results presented POCs could be further developed to target disease specific RNA sequences such as micro- or messenger RNAs.

Crucial to the progression of the field of molecular electronics is the intellectual conception, experimental fabrication and actual physical characterisation of molecular systems that function as electronic components. Future electronic devices could well incorporate such molecular systems in their manufacture, and expectations are high for the exploitation of molecular-scale properties. This thesis will report and discuss the stepwise construction of a variety of functional molecular wires embodied as self-assembled films on gold surfaces, providing evidence for their creation as well as characterisation of electrical properties. The growth of molecular wire systems were observed and monitored using the quartz crystal microbalance (QCM), with further physical and chemical characterisation being obtained from x-ray photoelectron (XPS) and infrared (IR) spectroscopic studies. These investigations demonstrated that amines compete with thiols for self-assembly on gold surfaces. This may have significant consequences for stepwise growth on gold supports when amino-thiols are used as initial building blocks. Electronic characterisation was achieved using the scanning tunnelling microscope (STM). Molecular wires self-assembled and grown on gold-coated surfaces were manipulated to exhibit rectifying behaviour, with current rectification ratios of ca. 55 at ±1 V obtained for a D-π-A type functional molecular wire. The rectifying behaviour was found to conform to the Aviram and Ratner model. Investigations into the nature of 'current jump' events were also undertaken. These characteristic step changes in S1M tunnelling current have been employed in various studies to extract single molecule conductivities. However, the mechanism by which they occur has not been fully determined. Reported herein is evidence that suggests that 'current jumps' most likely represent events whereby STM probes make contact with surface-bound molecules, as opposed to spontaneous molecular attachment.