Cancer chromosomal instability (CIN) results from dynamic changes to chromosome number and structure. The resulting diversity in somatic copy number alterations (SCNA) may provide the variation necessary for cancer evolution. Multi-sample phasing and SCNA analysis of 1421 samples from 394 tumours across 24 cancer types revealed ongoing CIN resulting in pervasive SCNA heterogeneity. Parallel evolutionary events, causing disruption to the same genes, such as BCL9, ARNT/HIF1B, TERT and MYC, within separate subclones were present in 35% of tumours. Most recurrent losses occurred prior to whole genome doubling (WGD), a clonal event in 48% of tumours. However, loss of heterozygosity at the human leukocyte antigen locus and loss of 8p to a single haploid copy recurred at significant subclonal frequencies, even in WGD tumours, likely reflecting ongoing karyotype remodeling. Focal amplifications affecting 1q21 (BCL9, ARNT), 5p15.33 (TERT), 11q13.3 (CCND1), 19q12 (CCNE1) and 8q24.1 (MYC) were frequently subclonal and exhibited an illusion of clonality within single samples. Analysis of an independent series of 1024 metastatic samples revealed enrichment for 14 focal SCNAs in metastatic samples, including late gains of 8q24.1 (MYC) in clear cell renal carcinoma and 11q13.3 (CCND1) in HER2-positive breast cancer. CIN may enable ongoing selection of SCNAs, manifested as ordered events, often occurring in parallel, throughout tumour evolution.

In their study of cyclic pattern containment, Domagalski et al. [4] conjecture differential equations for the generating functions of circular permutations avoiding consecutive patterns of length 3. In this note, we prove and significantly generalize these conjectures. We show that, for every consecutive pattern σ beginning with 1, the bivariate generating function counting occurrences of σ in circular permutations can be obtained from the generating function counting occurrences of σ in (linear) permutations. This includes all the patterns for which the latter generating function is known.

Numerical chromosomal instability is a ubiquitous feature of human neoplasms. Due to experimental limitations, fundamental characteristics of karyotypic changes in cancer are poorly understood. Using an experimentally inspired stochastic model, based on the potency and chromosomal distribution of oncogenes and tumor suppressor genes, we show that cancer cells have evolved to exist within a narrow range of chromosome missegregation rates that optimizes phenotypic heterogeneity and clonal survival. Departure from this range reduces clonal fitness and limits subclonal diversity. Mapping of the aneuploid fitness landscape reveals a highly favorable, commonly observed, near-triploid state onto which evolving diploid- and tetraploid-derived populations spontaneously converge, albeit at a much lower fitness cost for the latter. Finally, by analyzing 1,368 chromosomal translocation events in five human cancers, we find that karyotypic evolution also shapes chromosomal translocation patterns by selecting for more oncogenic derivative chromosomes. Thus, chromosomal instability can generate the heterogeneity required for Darwinian tumor evolution.

We present a bijection between 321-and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. This gives a simple combinatorial proof of recent results of Robertson et al. (Ann. Combin. 6 (2003) 427), and Elizalde (Proc. FPSAC 2003). We also show that our bijection preserves additional statistics, which extends the previous results. r

The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits generated by piecewise monotone maps on one-dimensional intervals have forbidden order patterns, i.e., order patterns that do not occur in any orbit. The allowed patterns are then those patterns avoiding the so-called forbidden root patterns and their shifted patterns. The second scope is to study forbidden patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. Due to its simple structure, shift systems are accessible to a more detailed analysis and, at the same time, exhibit all important properties of low-dimensional chaotic dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a dense set of periodic points), allowing to export the results to other dynamical systems via order-isomorphisms.