We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.
A computational method was developed for delineating connected gene neighborhoods in bacterial and archaeal genomes. These gene neighborhoods are not typically present, in their entirety, in any single genome, but are held together by overlapping, partially conserved gene arrays. The procedure was applied to comparing the orders of orthologous genes, which were extracted from the database of Clusters of Orthologous Groups of proteins (COGs), in 31 prokaryotic genomes and resulted in the identification of 188 clusters of gene arrays, which included 1001 of 2890 COGs. These clusters were projected onto actual genomes to produce extended neighborhoods including additional genes, which are adjacent to the genes from the clusters and are transcribed in the same direction, which resulted in a total of 2387 COGs being included in the neighborhoods. Most of the neighborhoods consist predominantly of genes united by a coherent functional theme, but also include a minority of genes without an obvious functional connection to the main theme. We hypothesize that although some of the latter genes might have unsuspected roles, others are maintained within gene arrays because of the advantage of expression at a level that is typical of the given neighborhood. We designate this phenomenon 'genomic hitchhiking'. The largest neighborhood includes 79 genes (COGs) and consists of overlapping, rearranged ribosomal protein superoperons; apparent genome hitchhiking is particularly typical of this neighborhood and other neighborhoods that consist of genes coding for translation machinery components. Several neighborhoods involve previously undetected connections between genes, allowing new functional predictions. Gene neighborhoods appear to evolve via complex rearrangement, with different combinations of genes from a neighborhood fixed in different lineages.
A phylogenetic tree (also called an "evolutionary tree") is a leaf-labelled tree which represents the evolutionary history for a set of species, and the construction of such trees is a fundamental problem in biology. Here we address the issue of how many sequence sites are required in order to recover the tree with high probability when the sites evolve under standard Markov-style i.i.d. mutation models. We provide analytic upper and lower bounds for the required sequence length, by developing a new (and polynomial time) algorithm. In particular we show that when the mutation probabilities are bounded the required sequence length can grow surprisingly slowly (a power of log n) in the number n of sequences, for almost all trees.
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