Combinatorial Hopf algebras in quantum field theory I

This manuscript stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Section 1 is the introduction, and contains as well an elementary invitation to the subject. The rest of part I, comprising Sections 2-6, is devoted to the basics of Hopf algebra theory and examples, in ascending level of complexity. Part II turns around the all-important Faa di Bruno Hopf algebra. Section 7 contains a first, direct approach to it. Section 8 gives applications of the Faa di Bruno algebra to quantum field theory and Lagrange reversion. Section 9 rederives the related Connes-Moscovici algebras. In Part III we turn to the Connes-Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Section10 we describe the first. Then in Section11 we give a simple derivation of (the properly combinatorial part of) Zimmermann's cancellation-free method, in its original diagrammatic form. In Section 12 general incidence algebras are introduced, and the Faa di Bruno bialgebras are described as incidence bialgebras. In Section 13, deeper lore on Rota's incidence algebras allows us to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained; this is the heart of the paper. The structure results for commutative Hopf algebras are found in Sections 14 and 15. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota-Baxter map in renormalization.Comment: 94 pages, LaTeX figures, precisions made, typos corrected, more references adde

Single-nucleotide polymorphisms in the RB1 gene and association with breast cancer in the British population

A substantial proportion of the familial risk of breast cancer may be attributable to genetic variants each contributing a small effect. pRb controls the cell cycle and polymorphisms within it are candidates for such low penetrance susceptibility alleles, since the gene has been implicated in several human tumours, particularly breast cancer. The purpose of this study was to determine whether common variants in the RB1 gene are associated with breast cancer risk. We assessed 15 tagging single-nucleotide polymorphisms (SNPs) using a case–control study design (n⩽4474 cases and n⩽4560 controls). A difference in genotype frequencies was found between cases and controls for rs2854344 in intron 17 (P-trend=0.007) and rs198580 in intron 19 (P-trend=0.018). Carrying the minor allele of these SNPs appears to confer a protective effect on breast cancer risk (odd ratio (OR)=0.86 (0.76–0.96) for rs2854344 and OR=0.80 (0.66–0.96) for rs198580). However, after adjusting for multiple testing these associations were borderline with an adjusted P-trend=0.068 for the most significant SNP (rs2854344). The RB1 gene is not known to contain any coding SNPs with allele frequencies ⩾5% but several intronic variants are in perfect linkage disequilibrium with the associated SNPs. Replication studies are needed to confirm the associations with breast cancer

The associations of anthropometric, behavioural and sociodemographic factors with circulating concentrations of IGF-I, IGF-II, IGFBP-1, IGFBP-2, and IGFBP-3 in a pooled analysis of 16,024 men from 22 studies

Insulin-like growth factors (IGFs) and insulin-like growth factor binding proteins (IGFBPs) have been implicated in the aetiology of several cancers. To better understand whether anthropometric, behavioural, and sociodemographic factors may play a role in cancer risk via IGF signalling, we examined the cross-sectional associations of these exposures with circulating concentrations of IGFs (IGF-I, IGF-II) and IGFBPs (IGFBP-1, IGFBP-2, IGFBP-3). The Endogenous Hormones, Nutritional Biomarkers and Prostate Cancer Collaborative Group dataset includes individual participant data from 16,024 male controls (i.e. without prostate cancer) aged 22-89 years from 22 prospective studies. Geometric means of protein concentrations were estimated using analysis of variance, adjusted for relevant covariates. Older age was associated with higher concentrations of IGFBP-1 and IGFBP-2 and lower concentrations of IGF-I, IGF-II, and IGFBP-3. Higher body mass index was associated with lower concentrations of IGFBP-1 and IGFBP-2. Taller height was associated with higher concentrations of IGF-I and IGFBP-3 and lower concentrations of IGFBP-1. Smokers had higher concentrations of IGFBP-1 and IGFBP-2 and lower concentrations of IGFBP-3 than non-smokers. Higher alcohol consumption was associated with higher concentrations of IGF-II and lower concentrations of IGF-I and IGFBP-2. African Americans had lower concentrations of IGF-II, IGFBP-1, IGFBP-2 and IGFBP-3 and Hispanics had lower IGF-I, IGF-II and IGFBP-3 than non-Hispanic whites. These findings indicate that a range of anthropometric, behavioural, and sociodemographic factors are associated with circulating concentrations of IGFs and IGFBPs in men, which will lead to a greater understanding of the mechanisms through which these factors influence cancer risk. This article is protected by copyright. All rights reserved

Genetic modifiers of radon-induced lung cancer risk: a genome-wide interaction study in former uranium miners

PURPOSE: Radon is a risk factor for lung cancer and uranium miners are more exposed than the general population. A genome-wide interaction analysis was carried out to identify genomic loci, genes or gene sets that modify the susceptibility to lung cancer given occupational exposure to the radioactive gas radon. METHODS: Samples from 28 studies provided by the International Lung Cancer Consortium were pooled with samples of former uranium miners collected by the German Federal Office of Radiation Protection. In total, 15,077 cases and 13,522 controls, all of European ancestries, comprising 463 uranium miners were compared. The DNA of all participants was genotyped with the OncoArray. We fitted single-marker and in multi-marker models and performed an exploratory gene-set analysis to detect cumulative enrichment of significance in sets of genes. RESULTS: We discovered a genome-wide significant interaction of the marker rs12440014 within the gene CHRNB4 (OR = 0.26, 95% CI 0.11-0.60, p = 0.0386 corrected for multiple testing). At least suggestive significant interaction of linkage disequilibrium blocks was observed at the chromosomal regions 18q21.23 (p = 1.2 × 10-6), 5q23.2 (p = 2.5 × 10-6), 1q21.3 (p = 3.2 × 10-6), 10p13 (p = 1.3 × 10-5) and 12p12.1 (p = 7.1 × 10-5). Genes belonging to the Gene Ontology term "DNA dealkylation involved in DNA repair" (GO:0006307; p = 0.0139) or the gene family HGNC:476 "microRNAs" (p = 0.0159) were enriched with LD-blockwise significance. CONCLUSION: The well-established association of the genomic region 15q25 to lung cancer might be influenced by exposure to radon among uranium miners. Furthermore, lung cancer susceptibility is related to the functional capability of DNA damage signaling via ubiquitination processes and repair of radiation-induced double-strand breaks by the single-strand annealing mechanism

A saturated map of common genetic variants associated with human height

Common single-nucleotide polymorphisms (SNPs) are predicted to collectively explain 40-50% of phenotypic variation in human height, but identifying the specific variants and associated regions requires huge sample sizes(1). Here, using data from a genome-wide association study of 5.4 million individuals of diverse ancestries, we show that 12,111 independent SNPs that are significantly associated with height account for nearly all of the common SNP-based heritability. These SNPs are clustered within 7,209 non-overlapping genomic segments with a mean size of around 90 kb, covering about 21% of the genome. The density of independent associations varies across the genome and the regions of increased density are enriched for biologically relevant genes. In out-of-sample estimation and prediction, the 12,111 SNPs (or all SNPs in the HapMap 3 panel(2)) account for 40% (45%) of phenotypic variance in populations of European ancestry but only around 10-20% (14-24%) in populations of other ancestries. Effect sizes, associated regions and gene prioritization are similar across ancestries, indicating that reduced prediction accuracy is likely to be explained by linkage disequilibrium and differences in allele frequency within associated regions. Finally, we show that the relevant biological pathways are detectable with smaller sample sizes than are needed to implicate causal genes and variants. Overall, this study provides a comprehensive map of specific genomic regions that contain the vast majority of common height-associated variants. Although this map is saturated for populations of European ancestry, further research is needed to achieve equivalent saturation in other ancestries.A large genome-wide association study of more than 5 million individuals reveals that 12,111 single-nucleotide polymorphisms account for nearly all the heritability of height attributable to common genetic variants