Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Distribution of Segment Lengths in Genome Rearrangements. | Distribution of Segment Lengths in Genome Rearrangements Glenn Tesler Department of Mathematics University of California San Diego USA gptesler@ Submitted Nov 13 2007 Accepted Aug 3 2008 Published Aug 11 2008 Mathematics Subject Classifications 05A15 92D15 92D20 Abstract The study of gene orders for constructing phylogenetic trees was introduced by Dobzhansky and Sturtevant in 1938. Different genomes may have homologous genes arranged in different orders. In the early 1990s Sankoff and colleagues modelled this as ordinary unsigned permutations on a set of numbered genes 1 2 . n with biological events such as inversions modelled as operations on the permutations. Signed permutations may be used when the relative strands of the genes are known and circular permutations may be used for circular genomes. We use combinatorial methods generating functions commutative and noncommutative formal power series asymptotics recursions and enumeration formulas to study the distributions of the number and lengths of conserved segments of genes between two or more unichromosomal genomes including signed and unsigned genomes and linear and circular genomes. This generalizes classical work on permutations from the 1940s-60s by Wolfowitz Kaplansky Riordan Abramson and Moser who studied decompositions of permutations into strips of ascending or descending consecutive numbers. In our setting their work corresponds to comparison of two unsigned genomes known gene orders unknown gene orientations . Maple software implementing our formulas is available at http gptesler strips . 1 Introduction The study of gene orders in phylogenetics was introduced by Dobzhansky and Sturtevant 1938 11 in a study of inversions in Drosophila pseudoobscura. More recently in the late 1980s Jeffrey Palmer and colleagues 21 22 compared the mitochondiral genomes of Funded by a Sloan Research Fellowship in Molecular Biology and NSF Grant DMS-0718810. The author also thanks the .
