Recent advances in biotechnology have yielded an explosion of data describing biological systems, creating rich opportunities for new insights into cellular inner-workings and therapeutic discoveries. To keep up with this rapid growth and increase in data complexity, we need novel static, integrative, and dynamic methodologies to continue mining these networked systems. In this thesis we introduce new static, integrative, and dynamic computational frameworks for network analysis, and combine existing ones in new ways, to elucidate the biotechnological biases and functional principles governing molecular interactions and their implications in disease. We focus on mining new knowledge from the yeast and human interactomes, since these are currently the most complete data in biology. We perform three lines of experimental work: 1) the macro-scale study, where we model the yeast and human interactomes and show that their interactome data are growing in structurally and functionally principled ways, characterised by a non-random dual topological nature; 2) the micro-scale study, where we zoom into the specifics of wiring patterns around individual genes and uncover a unique core sub-structure within the human interactome, which contains driver genes dubbed to be the main triggers for disease onset; and 3) the data integration study, where we introduce a new computational framework for fusing multiple types of molecular interaction data and use it to construct the first unified model of the cell's functional organisation and cross-communication lines. Similarly, a new field of systems economics has gained recent attention, with more financial and economic network data emerging at an increasing pace. Hence, we introduce a new computational methodology for tracking network dynamics and use it to quantify the micro- and macro-scale topological changes in the world trade network over the past 50 years, and to demonstrate the fundamental relationship between topological perturbations and indicators of countries' political and economic stabilities.

In this thesis, we explore various computational and axiomatisation problems relating to two-dimensional modal logics that exhibit some modest capacity to count. In particular, we consider modal products in which at least one component is the logic of difference (inequality) relations. These formalisms are connected with finite variable fragments of first-order logic and first-order modal logics, extended with some additional counting~quantifiers. The contributions provided herein serve as a case study to better steer investigation into more general principles governing the interactions between modal logics, and into understanding the interactions between first-order quantifiers.

Microtubules, dynamic protein polymers, form networks that are essential for intracellular organisation. Involved in many cellular processes that are vital in development and homeostasis, improper regulation of the microtubule network is implicated in various diseases. This work addresses the relationships between microtubule dynamics and organisation, using image processing and modelling, focussing on two features of microtubule organisation: radiality and alignment. The hypothesis that radiality results from modulation of dynamics at the cell periphery was tested. Firstly, cells in which the small GTPase Rac1 was inhibited were used as a model for perturbed radiality. Measurements of microtubule dynamics in central and peripheral regions showed that Rac1 inhibition alters microtubule dynamics and the orientation of their growth at the cell periphery. Further investigation was carried out with a simple 1-dimensional, two-area dynamics model, which confirmed that a two-area dynamics system is sufficient to target microtubules to a given length. The propensity to grow of any given dynamics parameters is a major determinant of the accuracy of length targeting, while the extent of pausing and the average length have a modulatory effect on accuracy. Simulation of measured dynamics indicated that two-area dynamics may contribute to radiality in reality, but that this mechanism may work in concert with other cortex-specific processes. The alignment of microtubules was quantified with a new application of the Fourier transform. Depletion of +TIP protein EB2 produced highlyaligned microtubules, and inhibition of formins rescued this phenotype. Inhibition of Rac1 produced less-aligned microtubules in otherwise unperturbed cells, while in EB2-depleted cells, microtubules were further aligned. The method was also used to quantify alignment in plant microtubule arrays. This work presents a set of analyses that test ideas as to how the microtubule network is organised, and highlight interesting relationships between dynamics and organisation that will yield exciting future investigation.

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Phytoplankton are photosynthetic microbes that form the basis of the marine food web and are estimated to produce over half of all oxygen in the Earth's atmosphere. Recent advances in high-throughput DNA sequencing technologies have allowed scientists to sample the set of genes actively transcribed from communities of microbes in-situ. This set of transcripts (the metatranscriptome) provides a snapshot of actively transcribed genes at the time of sampling, and can provide insights into microbial metabolism and their relationship with their environment. In this thesis we present the computational analysis of eukaryotic phytoplankton metatranscriptome data sampled from representative marine environments; the simulation of metatranscriptome data for benchmarking computational tools; and analysis carried out on a newly sequenced eukaryotic phytoplankton genome. Transcripts a�liated with ribosomal proteins and associated with translation dominated in all but the Equatorial Paci�c metatranscriptome sample. Hierarchical clustering of the metatranscriptome samples by taxa produced two groups: the diatom dominated and the alveolate dominated. However, clustering by Gene Ontology terms clustered the samples by environment type (tropical, temperate and polar), producing a gradient of translation-associated transcripts which increased as the in-situ temperature of the samples decreased. A strong i correlation (R = 0:9) was detected between the relative proportion of transcripts associated with temperature and the in-situ temperature. Laboratory experiments on model diatom species under control conditions con�rmed that as the in-situ temperature decreases, these model diatoms produce more transcripts and consequently more ribosomal proteins. A translational e�ciency experiment demonstrated that the rate of translation decreased under low temperatures for a model diatom species. This suggested that the increased production of ribosomes acts as a compensatory mechanism under low temperatures. As more ribosomes require more phosphate-rich rRNAs we hypothesised that this could have an impact on biogeochemical cycles (E.g. the Red�eld ratio of Nitrate (N) to Phosphate (P)). This was modelled by our collaborators from the University of Exeter, who produced a global phytoplankton cell model of resource allocation. They showed how the N:P ratio di�ers across latitudinal temperature zones and predicted the impact of increasing temperature on global N:P.

Cancer is often associated with a high number of large-scale, structural rearrangements. In a highly selective environment, some `driver' mutations conferring clonal growth advantage will be positively selected, accounting for further cancer development. Clarifying their nature, as well as their contribution to the pathology is a major current focus of biomedical research. Next generation sequencing technologies can be used nowadays to generate high-resolution data-sets of these alterations in cancer genomes. This project has been developed along two main lines: 1) the reconstruction of cancer aberrant karyotypes, together with their underlying evolutionary history; 2) the elucidation of some combinatorial properties associated with gene duplications. We applied graph theory to the problem of reconstructing the final cancer genome sequence; additionally, we developed an algorithmic approach for the reconstruction of a multi-step evolution consistent with read coverage and paired end data, giving insights on the possible molecular mechanisms underlying rearrangements. Looking at the combinatorics of both tandem and inverted duplication, we developed an algebraic formalism for the representation of these processes. This allowed us to both explore the geometric properties of sequences arising by Tandem Duplication (TD), and obtain a recursion for the number of tandem duplications evolutions after n events. Such results are missing for inverted duplications, whose combinatorial properties have been nevertheless deeply elucidated. Our results have allowed: 1) the identification, through an original approach, of potential rearrangement mechanisms associated with cancer development, and 2) the definition and mathematical description of the complete evolutionary space of specific rearrangement classes.

As computer systems become ever more powerful and parallel, processing larger and larger sets of data, there is increased need for ensuring that scientific software applications are tolerant to faults in both hardware and software. New algorithms which take advantage of knowledge about the structure and calculation of important mathematical problems would enable increasingly more efficient and fault tolerant computation to be performed with minimal overhead. This thesis demonstrates how improvements to two important application areas in High Performance Computing (HP C) - that of Monte Carlo methods and Sparse Linear Algebra - can result in software with greater fault tolerance alongside low overheads. It proposes models that employ variations on existing techniques dealing with layout topologies in grids and a form of Error-Correcting Code (ECC) to provide an increased degree of fault tolerance in calculations. The models make efficient use of the variations to produce schemes that are both robust and based on straightforward approaches which can be implemented in a simple manner. The theory behind the models is developed and evaluated and basic implementations are created to gauge the performance and viability of the schemes. Both models perform well in the majority of cases with low overheads in the range of 0-10%, and both are eminently scalable. Furthermore, the methods with highest overhead in the Sparse Linear Algebra schemes are found to increase in performance for larger data sets that are more sparse - those that would require the extra protection afforded by software fault tolerance the most.

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