Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.

A median fin derived from the lateral plate mesoderm and the origin of paired fins

The development of paired appendages was a key innovation during evolution and facilitated the aquatic to terrestrial transition of vertebrates. Largely derived from the lateral plate mesoderm (LPM), one hypothesis for the evolution of paired fins invokes derivation from unpaired median fins via a pair of lateral fin folds located between pectoral and pelvic fin territories1. Whilst unpaired and paired fins exhibit similar structural and molecular characteristics, no definitive evidence exists for paired lateral fin folds in larvae or adults of any extant or extinct species. As unpaired fin core components are regarded as exclusively derived from paraxial mesoderm, any transition presumes both co-option of a fin developmental programme to the LPM and bilateral duplication2. Here, we identify that the larval zebrafish unpaired pre-anal fin fold (PAFF) is derived from the LPM and thus may represent a developmental intermediate between median and paired fins. We trace the contribution of LPM to the PAFF in both cyclostomes and gnathostomes, supporting the notion that this is an ancient trait of vertebrates. Finally, we observe that the PAFF can be bifurcated by increasing bone morphogenetic protein signalling, generating LPM-derived paired fin folds. Our work provides evidence that lateral fin folds may have existed as embryonic anlage for elaboration to paired fins

Pan-cancer analysis of whole genomes

Cancer is driven by genetic change, and the advent of massively parallel sequencing has enabled systematic documentation of this variation at the whole-genome scale. Here we report the integrative analysis of 2,658 whole-cancer genomes and their matching normal tissues across 38 tumour types from the Pan-Cancer Analysis of Whole Genomes (PCAWG) Consortium of the International Cancer Genome Consortium (ICGC) and The Cancer Genome Atlas (TCGA). We describe the generation of the PCAWG resource, facilitated by international data sharing using compute clouds. On average, cancer genomes contained 4-5 driver mutations when combining coding and non-coding genomic elements; however, in around 5% of cases no drivers were identified, suggesting that cancer driver discovery is not yet complete. Chromothripsis, in which many clustered structural variants arise in a single catastrophic event, is frequently an early event in tumour evolution; in acral melanoma, for example, these events precede most somatic point mutations and affect several cancer-associated genes simultaneously. Cancers with abnormal telomere maintenance often originate from tissues with low replicative activity and show several mechanisms of preventing telomere attrition to critical levels. Common and rare germline variants affect patterns of somatic mutation, including point mutations, structural variants and somatic retrotransposition. A collection of papers from the PCAWG Consortium describes non-coding mutations that drive cancer beyond those in the TERT promoter; identifies new signatures of mutational processes that cause base substitutions, small insertions and deletions and structural variation; analyses timings and patterns of tumour evolution; describes the diverse transcriptional consequences of somatic mutation on splicing, expression levels, fusion genes and promoter activity; and evaluates a range of more-specialized features of cancer genomes