The Circuit Partition Polynomial with Applications and Relation to the Tutte and Interlace Polynomials

This paper examines several polynomials related to the field of graph theory including the circuit partition polynomial, Tutte polynomial, and the interlace polynomial. We begin by explaining terminology and concepts that will be needed to understand the major results of the paper. Next, we focus on the circuit partition polynomial and its equivalent, the Martin polynomial. We examine the results of these polynomials and their application to the reconstruction of DNA sequences. Then we introduce the Tutte polynomial and its relation to the circuit partition polynomial. Finally, we discuss the interlace polynomial and its relationship to the Tutte and circuit partition polynomials

Elimination and cut-elimination in multiplicative linear logic

We associate to every proof structure in multiplicative linear logic an ideal which represents the logical content of the proof as polynomial equations. We show how cut-elimination in multiplicative proof nets corresponds to instances of the Buchberger algorithm for computing Gr\"obner bases in elimination theory

Revisiting the Complexity of and Algorithms for the Graph Traversal Edit Distance and Its Variants

The graph traversal edit distance (GTED), introduced by Ebrahimpour Boroojeny et al. (2018), is an elegant distance measure defined as the minimum edit distance between strings reconstructed from Eulerian trails in two edge-labeled graphs. GTED can be used to infer evolutionary relationships between species by comparing de Bruijn graphs directly without the computationally costly and error-prone process of genome assembly. Ebrahimpour Boroojeny et al. (2018) propose two ILP formulations for GTED and claim that GTED is polynomially solvable because the linear programming relaxation of one of the ILPs will always yield optimal integer solutions. The claim that GTED is polynomially solvable is contradictory to the complexity of existing string-to-graph matching problems. We resolve this conflict in complexity results by proving that GTED is NP-complete and showing that the ILPs proposed by Ebrahimpour Boroojeny et al. do not solve GTED but instead solve for a lower bound of GTED and are not solvable in polynomial time. In addition, we provide the first two, correct ILP formulations of GTED and evaluate their empirical efficiency. These results provide solid algorithmic foundations for comparing genome graphs and point to the direction of heuristics that estimate GTED efficiently

Guided genome halving: provably optimal solutions provide good insights into the preduplication ancestral genome of Saccharomyces cerevisiae.

International audienceWe present theoretical and practical advances on the Guided Genome Halving problem, a combinatorial optimisation problem which aims at proposing ancestral configurations of extant genomes when one of them has undergone a whole genome duplication. We provide a lower bound on the optimal solution, devise a heuristic algorithm based on subgraph identification, and apply it to yeast gene order data. On some instances, the computation of the bound yields a proof that the obtained solutions are optimal. We analyse a set of optimal solutions, compare them with a manually curated standard ancestor, showing that on yeast data, results coming from different methodologies are largely convergent: the optimal solutions are distant of at most one rearrangement from the reference

Characterization of proton exchange materials for fuel cells by solid state nuclear magnetic resonance

Solid-state nuclear magnetic resonance (NMR) has been used to explore the nanometer-scale structure of Nafion, the widely used fuel cell membrane, and its composites. We have shown that solid-state NMR can characterize chemical structure and composition, domain size and morphology, internuclear distances, molecular dynamics, etc. The newly-developed water channel model of Nafion has been confirmed, and important characteristic length-scales established. Nafion-based organic and inorganic composites with special properties have also been characterized and their structures elucidated. The morphology of Nafion varies with hydration level, and is reflected in the changes in surface-to-volume (S/V) ratio of the polymer obtained by small-angle X-ray scattering (SAXS). The S/V ratios of different Nafion models have been evaluated numerically. It has been found that only the water channel model gives the measured S/V ratios in the normal hydration range of a working fuel cell, while dispersed water molecules and polymer ribbons account for the structures at low and high hydration levels, respectively. Although the cross-section morphology of Nafion has been derived from SAXS data, the structure in the third dimension, which is channel straightness, was not clear. With 2H NMR, D2O can be used as a probe to study channel straightness (persistence length). In drawn Nafion with straight channels, the exchange between bound and free D2O results in a residual quadrupolar splitting of 1-2 kHz; while in normal Nafion with coiled channels, the 2H quadrupolar splitting of D2O is ~ 10-fold smaller. It is explained by the motional averaging of the 2H coupling frequencies in the NMR timescale when D2O diffuses through differently-oriented segments. The simulations of line narrowing and T2 relaxation times of D2O revealed a persistence length within 30 to 80 nm for normal Nafion. The Nafion phosphatranium composite developed by Verkade and Wadhwa, which is a potential candidate for anion exchange membranes, has been characterized by solid-state NMR. The synthesized membrane has two major components, in which phosphatranium cations are bonded to Nafion side-groups via either P or N with a mole ratio of 2:1. Degradation of the phosphatranium cations has not been found in the composite membrane, which implies a good stability of the material. Nafion-silica (NafSil) and Nafion-zirconium phosphate (NafZrP) composites prepared by the in-situ growth of inorganic particles in the channels of Nafion membrane have been characterized. Under typical situations with an inorganic volume fraction of around 15%, elongated nanoparticles are formed inside the water channels. The inorganic particles have cylindrical shapes with a cross-section area of ~ 6 nm2 and surrounded by water layers with a thickness of ~ 0.8 nm. Zirconium phosphates (ZrP) synthesized in and outside Nafion have been characterized in detail by solid-state NMR and X-ray diffraction (XRD). It has been found that typical α-ZrP with water of crystallization transforms to anhydrous α-ZrP and condensed pyrophosphates after drying at 150oC. When grown in Nafion, ZrP favors a structure with two disordered layers and a majority of (HO)P(OZr)3 sites, different from regular α-ZrP, particularly after drying

Annotation and comparative analysis of fungal genomes: a hitchhiker's guide to genomics

This thesis describes several genome-sequencing projects such as those from the fungi Laccaria bicolor S238N-H82, Glomus intraradices DAOM 197198, Melampsora laricis-populina 98AG31, Puccinia graminis, Pichia pastoris GS115 and Candida bombicola, as well as the one of the haptophyte Emiliania huxleyi CCMP1516. These species are important organisms in many aspects, for instance: L. bicolor and G. intraradices are symbiotic fungi growing associate with trees and present an important ecological niches for promoting tree growth; M. laricis- populina and P. graminis are two devastating fungi threating plants; the tiny yeast P. pastoris is the major protein production platform in the pharmaceutical industry; the biosurfactant production yeast C. bombicola is likely to provide a low ecotoxicity detergent and E. huxleyi places in a unique phylogeny position of chromalveolate and contributes to the global carbon cycle system. The completion of the genome sequence and the subsequent functional studies broaden our understanding of these complex biological systems and promote the species as possible model organisms. However, it is commonly observed that the genome sequencing projects are launched with lots of enthusiasm but often frustratingly difficult to finish. Part of the reason are the ever-increasing expectations regarding quality delivery (both with respect to data and analyses). The Introductory Chapter aims to provide an overview of how best to conduct a genome sequencing project. It explains the importance of understanding the basic biology and genetics of the target organism. It also discusses the latest developments in new (next) generation high throughput sequencing (HTS) technologies, how to handle the data and their applications. The emergence of the new HTS technologies brings the whole biology research into a new frontier. For instance, with the help of the new sequencing technologies, we were able to sequence the genome of our interest, namely Pichia pastoris. This tiny yeast, the analysis of which forms the bulk of this thesis, is an important heterologous production platform because its methanol assimilation properties makes it ideally suitable for large scale industrial production. The unique protein assembly pathway of P. pastoris also attracts much basic research interests. We used the new HTS method to sequence and assemble the GS115 genome into four chromosomes and made it publicly available to the research community (Chapter 2 and Chapter 3). The public release of the GS115 brought broader interests on the comparison of GS115 and its parental strains. By sequencing the parental strain of GS115 with different new sequencing platforms, we identified several point mutations in the coding genes that likely contribute to the higher protein translocation efficiency in GS115. The sequence divergence and copy number variation of rDNA between strains also explains the difference of protein production efficiency (Chapter 4). Before 2008, the Sanger sequencing method was the only technology to obtain high quality complete genomes of eukaryotes. Because of the high cost of the Sanger method, regarding the other genome projects discussed in this thesis, it was necessary to team up with many other partners and to rely on the U.S. Department of Energy Joint Genome Institute (DOE-JGI) and the Broad Institute to generate the genome sequence. The M. larici-populina srain 98AG31 and the Puccinia graminis f. sp. tritici strain CRL 75-36-700-3 are two devastating basidiomycete ‘rusts’ that infect poplar and wheat. Lineage-specific gene family expansions in these two rusts highlight the possible role in their obligate biotrophic life-style. Two large sets of effector-like small-secreted proteins with different pri- mary sequence structures were identified in each organism. The in planta-induced transcriptomic data showed upregulation of these lineage-specific genes and they are likely involved in the establishing of the rust-host interaction. An additional immunolocalization study on M. larici-populina confirmed the accumulation of some candidate effectors in the haustoria and infection hyphae, which is described in Chapter 5