Counting and sampling problems on Eulerian graphs

In this thesis we consider two sets of combinatorial structures defined on an Eulerian graph: the Eulerian orientations and Euler tours. We are interested in the computational problems of counting (computing the number of elements in the set) and sampling (generating a random element of the set). Specifically, we are interested in the question of when there exists an efficient algorithm for counting or sampling the elements of either set. The Eulerian orientations of a number of classes of planar lattices are of practical significance as they correspond to configurations of certain models studied in statistical physics. In 1992 Mihail and Winkler showed that counting Eulerian orientations of a general Eulerian graph is #P-complete and demonstrated that the problem of sampling an Eulerian orientation can be reduced to the tractable problem of sampling a perfect matching of a bipartite graph. We present a proof that this problem remains #Pcomplete when the input is restricted to being a planar graph, and analyse a natural algorithm for generating random Eulerian orientations of one of the afore-mentioned planar lattices. Moreover, we make some progress towards classifying the range of planar graphs on which this algorithm is rapidly mixing by exhibiting an infinite class of planar graphs for which the algorithm will always take an exponential amount of time to converge. The problem of counting the Euler tours of undirected graphs has proven to be less amenable to analysis than that of Eulerian orientations. Although it has been known for many years that the number of Euler tours of any directed graph can be computed in polynomial time, until recently very little was known about the complexity of counting Euler tours of an undirected graph. Brightwell and Winkler showed that this problem is #P-complete in 2005 and, apart from a few very simple examples, e.g., series-parellel graphs, there are no known tractable cases, nor are there any good reasons to believe the problem to be intractable. Moreover, despite several unsuccessful attempts, there has been no progress made on the question of approximability. Indeed, this problem was considered to be one of the more difficult open problems in approximate counting since long before the complexity of exact counting was resolved. By considering a randomised input model, we are able to show that a very simple algorithm can sample or approximately count the Euler tours of almost every d-in/d-out directed graph in expected polynomial time. Then, we present some partial results towards showing that this algorithm can be used to sample or approximately count the Euler tours of almost every 2d-regular graph in expected polynomial time. We also provide some empirical evidence to support the unproven conjecture required to obtain this result. As a sideresult of this work, we obtain an asymptotic characterisation of the distribution of the number of Eulerian orientations of a random 2d-regular graph

Mitochondrial physiology

As the knowledge base and importance of mitochondrial physiology to evolution, health and disease expands, the necessity for harmonizing the terminology concerning mitochondrial respiratory states and rates has become increasingly apparent. The chemiosmotic theory establishes the mechanism of energy transformation and coupling in oxidative phosphorylation. The unifying concept of the protonmotive force provides the framework for developing a consistent theoretical foundation of mitochondrial physiology and bioenergetics. We follow the latest SI guidelines and those of the International Union of Pure and Applied Chemistry (IUPAC) on terminology in physical chemistry, extended by considerations of open systems and thermodynamics of irreversible processes. The concept-driven constructive terminology incorporates the meaning of each quantity and aligns concepts and symbols with the nomenclature of classical bioenergetics. We endeavour to provide a balanced view of mitochondrial respiratory control and a critical discussion on reporting data of mitochondrial respiration in terms of metabolic flows and fluxes. Uniform standards for evaluation of respiratory states and rates will ultimately contribute to reproducibility between laboratories and thus support the development of data repositories of mitochondrial respiratory function in species, tissues, and cells. Clarity of concept and consistency of nomenclature facilitate effective transdisciplinary communication, education, and ultimately further discovery

Counting and sampling problems on Eulerian graphs

In this thesis we consider two sets of combinatorial structures defined on an Eulerian graph: the Eulerian orientations and Euler tours. We are interested in the computational problems of counting (computing the number of elements in the set) and sampling (generating a random element of the set). Specifically, we are interested in the question of when there exists an efficient algorithm for counting or sampling the elements of either set. The Eulerian orientations of a number of classes of planar lattices are of practical significance as they correspond to configurations of certain models studied in statistical physics. In 1992 Mihail and Winkler showed that counting Eulerian orientations of a general Eulerian graph is #P-complete and demonstrated that the problem of sampling an Eulerian orientation can be reduced to the tractable problem of sampling a perfect matching of a bipartite graph. We present a proof that this problem remains #Pcomplete when the input is restricted to being a planar graph, and analyse a natural algorithm for generating random Eulerian orientations of one of the afore-mentioned planar lattices. Moreover, we make some progress towards classifying the range of planar graphs on which this algorithm is rapidly mixing by exhibiting an infinite class of planar graphs for which the algorithm will always take an exponential amount of time to converge. The problem of counting the Euler tours of undirected graphs has proven to be less amenable to analysis than that of Eulerian orientations. Although it has been known for many years that the number of Euler tours of any directed graph can be computed in polynomial time, until recently very little was known about the complexity of counting Euler tours of an undirected graph. Brightwell and Winkler showed that this problem is #P-complete in 2005 and, apart from a few very simple examples, e.g., series-parellel graphs, there are no known tractable cases, nor are there any good reasons to believe the problem to be intractable. Moreover, despite several unsuccessful attempts, there has been no progress made on the question of approximability. Indeed, this problem was considered to be one of the more difficult open problems in approximate counting since long before the complexity of exact counting was resolved. By considering a randomised input model, we are able to show that a very simple algorithm can sample or approximately count the Euler tours of almost every d-in/d-out directed graph in expected polynomial time. Then, we present some partial results towards showing that this algorithm can be used to sample or approximately count the Euler tours of almost every 2d-regular graph in expected polynomial time. We also provide some empirical evidence to support the unproven conjecture required to obtain this result. As a sideresult of this work, we obtain an asymptotic characterisation of the distribution of the number of Eulerian orientations of a random 2d-regular graph.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Diagnosis of genetic diseases in seriously ill children by rapid whole-genome sequencing and automated phenotyping and interpretation.

By informing timely targeted treatments, rapid whole-genome sequencing can improve the outcomes of seriously ill children with genetic diseases, particularly infants in neonatal and pediatric intensive care units (ICUs). The need for highly qualified professionals to decipher results, however, precludes widespread implementation. We describe a platform for population-scale, provisional diagnosis of genetic diseases with automated phenotyping and interpretation. Genome sequencing was expedited by bead-based genome library preparation directly from blood samples and sequencing of paired 100-nt reads in 15.5 hours. Clinical natural language processing (CNLP) automatically extracted childrens deep phenomes from electronic health records with 80% precision and 93% recall. In 101 children with 105 genetic diseases, a mean of 4.3 CNLP-extracted phenotypic features matched the expected phenotypic features of those diseases, compared with a match of 0.9 phenotypic features used in manual interpretation. We automated provisional diagnosis by combining the ranking of the similarity of a patients CNLP phenome with respect to the expected phenotypic features of all genetic diseases, together with the ranking of the pathogenicity of all of the patients genomic variants. Automated, retrospective diagnoses concurred well with expert manual interpretation (97% recall and 99% precision in 95 children with 97 genetic diseases). Prospectively, our platform correctly diagnosed three of seven seriously ill ICU infants (100% precision and recall) with a mean time saving of 22:19 hours. In each case, the diagnosis affected treatment. Genome sequencing with automated phenotyping and interpretation in a median of 20:10 hours may increase adoption in ICUs and, thereby, timely implementation of precise treatments

HLA-DQA1-HLA-DRB1 variants confer susceptibility to pancreatitis induced by thiopurine immunosuppressants

Pancreatitis occurs in approximately 4% of patients treated with the thiopurines azathioprine or mercaptopurine. Its development is unpredictable and almost always leads to drug withdrawal. We identified patients with inflammatory bowel disease (IBD) who had developed pancreatitis within 3 months of starting these drugs from 168 sites around the world. After detailed case adjudication, we performed a genome-wide association study on 172 cases and 2,035 controls with IBD. We identified strong evidence of association within the class II HLA region, with the most significant association identified at rs2647087 (odds ratio 2.59, 95% confidence interval 2.07-3.26, P = 2 x 10(-16)). We replicated these findings in an independent set of 78 cases and 472 controls with IBD matched for drug exposure. Fine mapping of the HLA region identified association with the HLA-DQA1*02:01-HLA-DRB1*07:01 haplotype. Patients heterozygous at rs2647087 have a 9% risk of developing pancreatitis after administration of a thiopurine, whereas homozygotes have a 17% risk