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Multibreak rearrangements and chromosomal evolution
, 2007
"... Most genome rearrangements (e.g., reversals and translocations) can be represented as 2breaks that break a genome at 2 points and glue the resulting fragments in a new order. Multibreak rearrangements break a genome into multiple fragments and further glue them together in a new order. While multi ..."
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Most genome rearrangements (e.g., reversals and translocations) can be represented as 2breaks that break a genome at 2 points and glue the resulting fragments in a new order. Multibreak rearrangements break a genome into multiple fragments and further glue them together in a new order. While multibreak rearrangements were studied in depth for k = 2 breaks, the kbreak distance problem for arbitrary k remains unsolved. We prove a duality theorem for multibreak distance problem and give a polynomial algorithm for computing this distance. 1
Sorting by Transpositions is Difficult
"... Abstract. In comparative genomics, a transposition is an operation that exchanges two consecutive sequences of genes in a genome. The transposition distance, that is, the minimum number of transpositions needed to transform a genome into another, can be considered as a relevant evolutionary distance ..."
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Abstract. In comparative genomics, a transposition is an operation that exchanges two consecutive sequences of genes in a genome. The transposition distance, that is, the minimum number of transpositions needed to transform a genome into another, can be considered as a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permutations, called the SORTING BY TRANSPOSITIONS problem (SBT), has been introduced by Bafna and Pevzner [3] in 1995. It has naturally been the focus of a number of studies, but the computational complexity of this problem has remained undetermined for 15 years. In this paper, we answer this longstanding open question by proving that the SORTING BY TRANSPOSITIONS problem is NPhard. As a corollary of our result, we also prove that the following problem from [9] is NPhard: given a permutation π, is it possible to sort π using db(π)/3 permutations, where db(π) is the number of breakpoints of π?
Bounding Prefix Transposition Distance for Strings and Permutations
, 2008
"... A transposition is an operation that exchanges two adjacent substrings. When it is restricted so that one of the substrings is a prefix, it is called a prefix transposition. The prefix transposition distance between a pair of strings (permutations) is the shortest sequence of prefix transpositions r ..."
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Cited by 7 (0 self)
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A transposition is an operation that exchanges two adjacent substrings. When it is restricted so that one of the substrings is a prefix, it is called a prefix transposition. The prefix transposition distance between a pair of strings (permutations) is the shortest sequence of prefix transpositions required to transform a given string (permutation) into another given string (permutation). This problem is a variation of the transposition distance problem, related to genome rearrangements. An upper bound of n1 and a lower bound of n/2 are known. We improve the bounds to nlog8 n and 2n/3 respectively. We also give upper and lower bounds for the prefix transposition distance on strings. For example, n/2 prefix transpositions are always sufficient for binary strings. We also prove that the exact prefix transposition distance problem on strings is NP complete.
MultiBreak Rearrangements and Breakpoint Reuses: From . . .
, 2008
"... Multibreak rearrangements break a genome into multiple fragments and further glue them together in a new order. While 2break rearrangements represent standard reversals, fusions, fissions, and translocations, 3break rearrangements represent a natural generalization of transpositions. Alekseyev an ..."
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Cited by 6 (3 self)
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Multibreak rearrangements break a genome into multiple fragments and further glue them together in a new order. While 2break rearrangements represent standard reversals, fusions, fissions, and translocations, 3break rearrangements represent a natural generalization of transpositions. Alekseyev and Pevzner (2007a, 2008a) studied multibreak rearrangements in circular genomes and further applied them to the analysis of chromosomal evolution in mammalian genomes. In this paper, we extend these results to the more difficult case of linear genomes. In particular, we give lower bounds for the rearrangement distance between linear genomes and for the breakpoint reuse rate as functions of the number and proportion of transpositions. We further use these results to analyze comparative genomic architecture of mammalian genomes.
Reversals of Fortune
"... Abstract. The objective function of the genome rearrangement problems allows the integration of other genomelevel problems so that they may be solved simultaneously. Three examples, all of which are hard: 1) Orientation assignment for unsigned genomes. 2) Ortholog identification in the presence of ..."
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Abstract. The objective function of the genome rearrangement problems allows the integration of other genomelevel problems so that they may be solved simultaneously. Three examples, all of which are hard: 1) Orientation assignment for unsigned genomes. 2) Ortholog identification in the presence of multiple copies of genes. 3) Linearisation of‘partially ordered genomes. The comparison of traditional genetic maps by rearrangement algorithms poses all these problems. We combine heuristics for the first two problems with an exact algorithm for the third to solve a moderatesized instance comparing maps of cereal genomes. 1
Multivariate Algorithmics for NPHard String Problems
, 2014
"... String problems arise in various applications ranging from text mining to biological sequence analysis. Many string problems are NPhard. This motivates the search for (fixedparameter) tractable special cases of these problems. We survey parameterized and multivariate algorithmics results for NPha ..."
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String problems arise in various applications ranging from text mining to biological sequence analysis. Many string problems are NPhard. This motivates the search for (fixedparameter) tractable special cases of these problems. We survey parameterized and multivariate algorithmics results for NPhard string problems and identify challenges for future research.
Reversal Distances for Strings with Few Blocks or Small Alphabets
"... Abstract. We study the String Reversal Distance problem, an extension of the wellknown Sorting by Reversals problem. String Reversal Distance takes two strings S and T as input, and asks for a minimum number of reversals to obtain T from S. We consider four variants: String Reversal Distance, St ..."
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Abstract. We study the String Reversal Distance problem, an extension of the wellknown Sorting by Reversals problem. String Reversal Distance takes two strings S and T as input, and asks for a minimum number of reversals to obtain T from S. We consider four variants: String Reversal Distance, String Prefix Reversal Distance (in which any reversal must include the first letter of the string), and the signed variants of these problems, namely Signed String Reversal Distance and Signed String Prefix Reversal Distance. We study algorithmic properties of these four problems, in connection with two parameters of the input strings: the number of blocks they contain (a block being maximal substring such that all letters in the substring are equal), and the alphabet size Σ. For instance, we show that Signed String Reversal Distance and Signed String Prefix Reversal Distance are NPhard even if the input strings have only one letter. 1
MultiBreak Rearrangements: from Circular to Linear Genomes
 Lecture Notes in Bioinformatics 4751
, 2007
"... Abstract. Multibreak rearrangements break a genome into multiple fragments and further glue them together in a new order. While 2break rearrangements represent standard reversals, fusions, fissions, and translocations operations; 3break rearrangements are a natural generalization of transpositions ..."
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Abstract. Multibreak rearrangements break a genome into multiple fragments and further glue them together in a new order. While 2break rearrangements represent standard reversals, fusions, fissions, and translocations operations; 3break rearrangements are a natural generalization of transpositions and inverted transpositions. Multibreak rearrangements in circular genomes were studied in depth in [1] and were further applied to the analysis of chromosomal evolution in mammalian genomes [2]. In this paper we extend these results to the more difficult case of linear genomes. In particular, we give lower bounds for the rearrangement distance between linear genomes and use these results to analyze comparative genomic architecture of mammalian genomes. 1
Research Article Sorting Cancer Karyotypes by Elementary Operations
"... Since the discovery of the ‘‘Philadelphia chromosome’ ’ in chronic myelogenous leukemia in 1960, there has been ongoing intensive research of chromosomal aberrations in cancer. These aberrations, which result in abnormally structured genomes, became a hallmark of cancer. Many studies provide evidenc ..."
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Since the discovery of the ‘‘Philadelphia chromosome’ ’ in chronic myelogenous leukemia in 1960, there has been ongoing intensive research of chromosomal aberrations in cancer. These aberrations, which result in abnormally structured genomes, became a hallmark of cancer. Many studies provide evidence for the connection between chromosomal alterations and aberrant genes involved in the carcinogenesis process. An important problem in the analysis of cancer genomes is inferring the history of events leading to the observed aberrations. Cancer genomes are usually described in the form of karyotypes, which present the global changes in the genomes ’ structure. In this study, we propose a mathematical framework for analyzing chromosomal aberrations in cancer karyotypes. We introduce the problem of sorting karyotypes by elementary operations, which seeks a shortest sequence of elementary chromosomal events transforming a normal karyotype into a given (abnormal) cancerous karyotype. Under certain assumptions, we prove a lower bound for the elementary distance, and present a polynomialtime 3approximation algorithm for the problem. We applied our algorithm to karyotypes from the Mitelman database, which records cancer karyotypes reported in the scientific literature. Approximately 94 % of the karyotypes in the database, totaling 58,464 karyotypes, supported our assumptions, and each of them was subjected to our algorithm. Remarkably, even though the algorithm is only guaranteed to generate a 3approximation, it produced a sequence whose length matched the lower bound (and hence optimal) in 99.9 % of the tested karyotypes. Key words: combinatorics, computational molecular biology, gene expression, gene networks, genetic variation, sequence analysis.