Genesis and evolution of the Evx and Mox genes and the extended Hox and ParaHox gene clusters

Background: Hox and ParaHox gene clusters are thought to have resulted from the duplication of a ProtoHox gene cluster early in metazoan evolution. However, the origin and evolution of the other genes belonging to the extended Hox group of homeobox-containing genes, that is, Mox and Evx, remains obscure. We constructed phylogenetic trees with mouse, amphioxus and Drosophila extended Hox and other related Antennapedia-type homeobox gene sequences and analyzed the linkage data available for such genes.Results: We claim that neither Mox nor Evx is a Hox or ParaHox gene. We propose a scenariothat reconciles phylogeny with linkage data, in which an Evx/Mox ancestor gene linked to aProtoHox cluster was involved in a segmental tandem duplication event that generated an arrayof all Hox-like genes, referred to as the `coupled¿ cluster. A chromosomal breakage within thiscluster explains the current composition of the extended Hox cluster (with Evx, Hox and Moxgenes) and the ParaHox cluster.Conclusions: Most studies dealing with the origin and evolution of Hox and ParaHox clustershave not included the Hox-related genes Mox and Evx. Our phylogenetic analyses and theavailable linkage data in mammalian genomes support an evolutionary scenario in which anancestor of Evx and Mox was linked to the ProtoHox cluster, and that a tandem duplication of alarge genomic region early in metazoan evolution generated the Hox and ParaHox clusters, plusthe cluster-neighbors Evx and Mox. The large `coupled¿ Hox-like cluster EvxHox/MoxParaHox wassubsequently broken, thus grouping the Mox and Evx genes to the Hox clusters, and isolating theParaHox cluster.

Restricted walks in regular trees

Let T be the Cayley graph of a finitely generated free group F. Given two vertices in T consider all the walks of a given length between these vertices that at a certain time must follow a number of predetermined steps. We give formulas for the number of such walks by expressing the problem in terms of equations in F and solving the corresponding equations.

Generating trees for permutations avoiding generalized patterns

We construct generating trees with with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation, which allows us to incorporate the adjacency condition about some entries in an occurrence of a generalized pattern. We use these trees to find functional equations for the generating functions enumerating these classes of permutations with respect to different parameters. In several cases we solve them using the kernel method and some ideas of Bousquet-Mélou [2]. We obtain refinements of known enumerative results and find new ones.

The equation xpyq=zr and groups that act freely on Λ-trees

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Polynomial functors and trees

We explore the relationship between polynomial functors and trees. In the first part we characterise trees as certain polynomial functors and obtain a completely formal but at the same time conceptual and explicit construction of two categories of rooted trees, whose main properties we describe in terms of some factorisation systems. The second category is the category Ω of Moerdijk and Weiss. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. Included in Part 1 is also an explicit construction of the free monad on a polynomial endofunctor, given in terms of trees. In the second part we describe polynomial endofunctors and monads as structures built from trees, characterising the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterised by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a projectivity condition, which serves also to characterise polynomial endofunctors and monads among (coloured) collections and operads.

Level-wise node size distribution of randomly generated regular trees

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“Unmixing” Tissue Gene Expression Signatures from Tumor Biopsies

Emergent molecular measurement methods, such as DNA microarray, qRTPCR, andmany others, offer tremendous promise for the personalized treatment of cancer. Thesetechnologies measure the amount of specific proteins, RNA, DNA or other moleculartargets from tumor specimens with the goal of “fingerprinting” individual cancers. Tumorspecimens are heterogeneous; an individual specimen typically contains unknownamounts of multiple tissues types. Thus, the measured molecular concentrations resultfrom an unknown mixture of tissue types, and must be normalized to account for thecomposition of the mixture.For example, a breast tumor biopsy may contain normal, dysplastic and cancerousepithelial cells, as well as stromal components (fatty and connective tissue) and bloodand lymphatic vessels. Our diagnostic interest focuses solely on the dysplastic andcancerous epithelial cells. The remaining tissue components serve to “contaminate”the signal of interest. The proportion of each of the tissue components changes asa function of patient characteristics (e.g., age), and varies spatially across the tumorregion. Because each of the tissue components produces a different molecular signature,and the amount of each tissue type is specimen dependent, we must estimate the tissuecomposition of the specimen, and adjust the molecular signal for this composition.Using the idea of a chemical mass balance, we consider the total measured concentrationsto be a weighted sum of the individual tissue signatures, where weightsare determined by the relative amounts of the different tissue types. We develop acompositional source apportionment model to estimate the relative amounts of tissuecomponents in a tumor specimen. We then use these estimates to infer the tissuespecificconcentrations of key molecular targets for sub-typing individual tumors. Weanticipate these specific measurements will greatly improve our ability to discriminatebetween different classes of tumors, and allow more precise matching of each patient tothe appropriate treatment