Genome Rearrangement Problems Research Articles

Sorting unsigned permutations by weighted reversals, transpositions, and transreversals

Reversals, transpositions and transreversals are common events in genome rearrangement. The genome rearrangement sorting problem is to transform one genome into another using the minimum number of given rearrangement operations. An integer permutation is used to represent a genome in many cases. It can be divided into disjoint strips with each strip denoting a block of consecutive integers. A singleton is a strip of one integer. And the genome rearrangement problem turns into the problem of sorting a permutation into the identity permutation equivalently. Hannenhalli and Pevzner designed a polynomial time algorithm for the unsigned reversal sorting problem on those permutations with O(logn) singletons. In this paper, first we describe one case in which Hannenhalli and Pevzner's algorithm may fail and propose a corrected approach. In addition, we propose a (1 + e)-approximation algorithm for sorting unsigned permutations with O(log n) singletons by reversals of weight 1 and transpositions/transreversals of weight 2.

A Lower Bound on the Transposition Diameter

Sorting permutations by transpositions is an important and difficult problem in genome rearrangements. The transposition diameter $TD(n)$ is the maximum transposition distance among all pairs of permutations in $S_n$. It was previously conjectured [H. Eriksson et al., Discrete Math., 241 (2001), pp. 289–300] that $TD(n)\leq\lceil\frac{n+1}{2}\rceil$. This conjecture was disproved by Elias and Hartman [IEEE/ACM Trans. Comput. Biol. Bioinform., 3 (2006), pp. 369–379] by showing $TD(n)\geq\lfloor\frac{n+1}{2}\rfloor+1$. In this paper we improved the lower bound to $TD(n)\geq\frac{17}{33}n+\frac{1}{33}$ via computation.

Sorting by reversals and block-interchanges with various weight assignments

BackgroundA classical problem in studying genome rearrangements is understanding the series of rearrangement events involved in transforming one genome into another in accordance with the parsimonious principle when two genomes with the same set of genes differ in gene order. The most studied event is the reversal, but an increasing number of reports have considered reversals along with other genome rearrangement events. Some recent studies have investigated the use of reversals and block-interchanges simultaneously with a weight proportion of 1:2. However, there has been less progress towards exploring additional combinations of weights.ResultsIn this paper, we present several approaches to examine genome rearrangement problems by considering reversals and block-interchanges together using various weight assignments. An exact algorithm for the weight proportion of 1:2 is developed, and then, its idea is extended to design approximation algorithms for other weight assignments. The results of our simulations suggest that the performance of our approximation algorithm is superior to its theoretical expectation.ConclusionIf the weight of reversals is no more than that of block-interchanges, our algorithm provides an acceptable solution for the transformation of two permutations. Nevertheless whether there are more tractable results for studying the two events remains open.

A fast algorithm for the multiple genome rearrangement problem with weighted reversals and transpositions

BackgroundDue to recent progress in genome sequencing, more and more data for phylogenetic reconstruction based on rearrangement distances between genomes become available. However, this phylogenetic reconstruction is a very challenging task. For the most simple distance measures (the breakpoint distance and the reversal distance), the problem is NP-hard even if one considers only three genomes.ResultsIn this paper, we present a new heuristic algorithm that directly constructs a phylogenetic tree w.r.t. the weighted reversal and transposition distance. Experimental results on previously published datasets show that constructing phylogenetic trees in this way results in better trees than constructing the trees w.r.t. the reversal distance, and recalculating the weight of the trees with the weighted reversal and transposition distance. An implementation of the algorithm can be obtained from the authors.ConclusionThe possibility of creating phylogenetic trees directly w.r.t. the weighted reversal and transposition distance results in biologically more realistic scenarios. Our algorithm can solve today's most challenging biological datasets in a reasonable amount of time.

Fast Algorithms for Transforming Back and Forth between a Signed Permutation and Its Equivalent Simple Permutation

The problem of sorting signed permutations by reversals is a well-studied problem in computational biology. The first polynomial time algorithm was presented by Hannenhalli and Pevzner in 1995. The algorithm was improved several times, and nowadays the most efficient algorithm has a subquadratic running time. Simple permutations played an important role in the development of these algorithms. Although the latest result of Tannier et al. does not require simple permutations, the preliminary version of their algorithm as well as the first polynomial time algorithm of Hannenhalli and Pevzner use the structure of simple permutations. More precisely, the latter algorithms require a precomputation that transforms a permutation into an equivalent simple permutation. To the best of our knowledge, all published algorithms for this transformation have at least a quadratic running time. For further investigations on genome rearrangement problems, the existence of a fast algorithm for the transformation could be crucial. Another important task is the back transformation, i.e. if we have a sorting on the simple permutation, transform it into a sorting on the original permutation. Again, the naive approach results in an algorithm with quadratic running time. In this paper, we present a linear time algorithm for transforming a permutation into an equivalent simple permutation, and an O(n log n) algorithm for the back transformation of the sorting sequence.

GENESIS: genome evolution scenarios

We implemented a software tool called GENESIS for three different genome rearrangement problems: Sorting a unichromosomal genome by weighted reversals and transpositions (SwRT), sorting a multichromosomal genome by reversals, translocations, fusions and fissions (SRTl), and sorting a multichromosomal genome by weighted reversals, translocations, fusions, fissions and transpositions (SwRTTl). Source code can be obtained by the authors, or use the web interface http://www.uni-ulm.de/in/theo/research/genesis.html.

Gene-interleaving patterns of synteny in the Saccharomyces cerevisiae genome: are they proof of an ancient genome duplication event?

BackgroundRecent comparative genomic studies claim local syntenic gene-interleaving relationships in Ashbya gossypii and Kluyveromyces waltii are compelling evidence for an ancient whole-genome duplication event in Saccharomyces cerevisiae. We here test, using Hannenhalli-Pevzner rearrangement algorithms that address the multiple genome rearrangement problem, whether syntenic patterns are proof of paleopolyploidization.ResultsWe focus on (1) pairwise comparison of gene arrangement sequences in A. gossypii and S. cerevisiae, (2) reconstruction of gene arrangements ancestral to A. gossypii, S. cerevisiae, and K. waltii, (3) synteny patterns arising within and between lineages, and (4) expected gene orientation of duplicate gene sets. The existence of syntenic patterns between ancestral gene sets and A. gossypii, S. cerevisiae, and K. waltii, and other evidence, suggests that gene-interleaving relationships are the natural consequence of topological rearrangements in chromosomes and that a more gradual scenario of genome evolution involving segmental duplication and recombination constitutes a more parsimonious explanation. Furthermore, phylogenetic trees reconstructed under alternative hypotheses placed the putative whole-genome duplication event after the divergence of the S. cerevisiae and K. waltii lineages, but in the lineage leading to K. waltii. This is clearly incompatible with an ancient genome duplication event in S. cerevisiae.ConclusionBecause the presence of syntenic patterns appears to be a condition that is necessary, but not sufficient, to support the existence of the whole-genome duplication event, our results prompt careful re-evaluation of paleopolyploidization in the yeast lineage and the evolutionary meaning of syntenic patterns.ReviewersThis article was reviewed by Kenneth H. Wolfe (nominated by Nicolas Galtier), Austin L. Hughes (nominated by Eugene Koonin), Mikhail S. Gelfand, and Mark Gerstein.

Reversal Distance for Strings with Duplicates: Linear Time Approximation using Hitting Set

In the last decade there has been an ongoing interest in string comparison problems; to a large extend the interest was stimulated by genome rearrangement problems in computational biology but related problems appear in many other areas of computer science. Particular attention has been given to the problem of sorting by reversals (SBR): given two strings, $A$ and $B$, find the minimum number of reversals that transform the string $A$ into the string $B$ (a reversal $\rho(i,j)$, $i < j$, transforms a string $A=a_1\ldots a_n$ into a string $A'=a_1\ldots a_{i-1} a_{j} a_{j-1} \ldots a_{i} a_{j+1} \ldots a_n$). Closely related is the minimum common string partition problem (MCSP): given two strings, $A$ and $B$, find a minimum size partition of $A$ into substrings $P_1,\ldots,P_l$ (i.e., $A=P_1\ldots P_l$) and a partition of $B$ into substrings $Q_1,\ldots,Q_l$ such that $(Q_1,\ldots,Q_l)$ is a permutation of $(P_1,\ldots,P_l)$. Primarily the SBR problem has been studied for strings in which every symbol appears exactly once (that is, for permutations) and only recently attention has been given to the general case where duplicates of the symbols are allowed. In this paper we consider the problem $k$-SBR, a version of SBR in which each symbol is allowed to appear up to $k$ times in each string, for some $k\geq 1$. The main result of the paper is a $\Theta(k)$-approximation algorithm for $k$-SBR running in time $O(n)$; compared to the previously known algorithm for $k$-SBR, this is an improvement by a factor of $\Theta(k)$ in the approximation ratio, and by a factor of $\Theta(k)$ in the running time. We approach the $k$-SBR by finding an approximation for the $k$-MCSP first and then turning it into a solution for $k$-SBR. Crucial ingredients of our algorithm are the suffix tree data structure and a linear time algorithm for a special case of a disjoint set union problem.

On some optimization problems in molecular biology

In the last two decades, the study of gene structure and function and molecular genetics have become some of the most prominent sub-fields of molecular biology. Computational molecular biology has emerged as one of the most exciting interdisciplinary fields, riding on the success of the ongoing Human Genome Project, which culminated in the 2001 announcement of the complete sequencing of the human genome. The field has currently benefited from concepts and theoretical results obtained by different scientific research communities, including genetics, biochemistry, and computer science. It is only in the past few years that it has been shown that a large number of molecular biology problems can be formulated as combinatorial optimization problems, including sequence alignment problems, genome rearrangement problems, string selection and comparison problems, and protein structure prediction and recognition. This paper provides a detailed description of some among the most interesting molecular biology problems that can be formulated as combinatorial optimization problems and proposes a new heuristic to find improved solutions for a particular class of them, known as the far from most string problem.

Using median sets for inferring phylogenetic trees

Algorithms for phylogenetic tree reconstruction based on gene order data typically repeatedly solve instances of the reversal median problem (RMP) which is to find for three given gene orders a fourth gene order (called median) with a minimal sum of reversal distances. All existing algorithms of this type consider only one median for each RMP instance even when a large number of medians exist. A careful selection of one of the medians might lead to better phylogenetic trees. We propose a heuristic algorithm amGRP for solving the multiple genome rearrangement problem (MGRP) by repeatedly solving instances of the RMP taking all medians into account. Algorithm amGRP uses a branch-and-bound method that branches over medians from a selected subset of all medians for each RMP instance. Different heuristics for selecting the subsets have been investigated. To show that the medians for RMP vary strongly with respect to different properties that are likely to be relevant for phylogenetic tree reconstruction, the set of all medians has been investigated for artificial datasets and mitochondrial DNA (mtDNA) gene orders. Phylogenetic trees have been computed for a large set of randomly generated gene orders and two sets of mtDNA gene order data for different animal taxa with amGRP and with two standard approaches for solving the MGRP (GRAPPA-DCM and MGR). The results show that amGRP outperforms both other methods with respect to solution quality and computation time on the test data. The source code of amGRP, additional results and the test instances used in this paper are freely available from the authors.

A linear time algorithm for the inversion median problem in circular bacterial genomes

In the median problem, we are given a distance or dissimilarity measure d, three genomes G 1 , G 2 , and G 3 , and we want to find a genome G (a median) such that the sum ∑ i = 1 3 d ( G , G i ) is minimized. The median problem is a special case of the multiple genome rearrangement problem, where one wants to find a phylogenetic tree describing the most “plausible” rearrangement scenario for multiple species. The median problem is NP-hard for both the breakpoint and the reversal distance. To the best of our knowledge, there is no approach yet that takes biological constraints on genome rearrangements into account. In this paper, we make use of the fact that in circular bacterial genomes the predominant mechanism of rearrangement are inversions that are centered around the origin or the terminus of replication and single gene inversions. These constraints simplify the median problem significantly. More precisely, we show that the median problem for the reversal distance can be solved in linear time for circular bacterial genomes.

Advances on sorting by reversals

The problem of sorting signed permutations by reversals is inspired by genome rearrangement problems in computational molecular biology. Given two genomes represented as signed permutations of the same elements (e.g. orthologous genes), the problem consists in finding a most parsimonious scenario of reversals that transforms one genome into the other. Following the first polynomial solution of this problem, several improvements, simplifications, generalizations, tutorials or surveys have been published on the subject. While the reversal distance problem—i.e. the problem of computing the minimum number of reversals in a sorting sequence, without giving the sequence itself—seems to be well explored, the problem of giving a scenario realizing the distance still raises some open questions, one of which by Ozery-Flato and Shamir about whether an algorithm with subquadratic time complexity could ever be achieved for solving the problem. We give a positive answer to this question by describing an algorithm of time complexity O ( n 3 / 2 log n ) .

Genome rearrangements with partially ordered chromosomes

Genomic maps often do not specify the order within some groups of two or more markers. The synthesis of a master map from several sources introduces additional order ambiguity due to markers missing from some sources. We represent each chromosome as a partial order, summarized by a directed acyclic graph (DAG), to account for poor resolution and missing data. The genome rearrangement problem is then to infer a minimum number of translocations and reversals for transforming a set of linearizations, one for each chromosomal DAG in the genome of one species, to linearizations of the DAGs of another species. We augment each DAG to a directed graph (DG) in which all possible linearizations are embedded. The chromosomal DGs representing two genomes are combined to produce a single bicoloured graph. From this we extract a maximal decomposition into alternating coloured cycles, determining an optimal sequence of rearrangements. We test this approach on simulated partially ordered genomes and on marker data from maize and sorghum chromosomal maps.

A 1.375-Approximation Algorithm for Sorting by Transpositions

Sorting permutations by transpositions is an important problem in genome rearrangements. A transposition is a rearrangement operation in which a segment is cut out of the permutation and pasted in a different location. The complexity of this problem is still open and it has been a 10-year-old open problem to improve the best known 1.5-approximation algorithm. In this paper, we provide a 1.375-approximation algorithm for sorting by transpositions. The algorithm is based on a new upper bound on the diameter of 3-permutations. In addition, we present some new results regarding the transposition diameter: we improve the lower bound for the transposition diameter of the symmetric group and determine the exact transposition diameter of simple permutations.

A simpler and faster 1.5-approximation algorithm for sorting by transpositions

An important problem in genome rearrangements is sorting permutations by transpositions. The complexity of the problem is still open, and two rather complicated 1.5-approximation algorithms for sorting linear permutations are known (Bafna and Pevzner, 98 and Christie, 99). The fastest known algorithm is the quadratic algorithm of Bafna and Pevzner. In this paper, we observe that the problem of sorting circular permutations by transpositions is equivalent to the problem of sorting linear permutations by transpositions. Hence, all algorithms for sorting linear permutations by transpositions can be used to sort circular permutations. Our main result is a new O ( n 3 / 2 log n ) 1.5-approximation algorithm, which is considerably simpler than the previous ones, and whose analysis is significantly less involved.

Efficient sorting of genomic permutations by translocation, inversion and block interchange

Finding genomic distance based on gene order is a classic problem in genome rearrangements. Efficient exact algorithms for genomic distances based on inversions and/or translocations have been found but are complicated by special cases, rare in simulations and empirical data. We seek a universal operation underlying a more inclusive set of evolutionary operations and yielding a tractable genomic distance with simple mathematical form. We study a universal double-cut-and-join operation that accounts for inversions, translocations, fissions and fusions, but also produces circular intermediates which can be reabsorbed. The genomic distance, computable in linear time, is given by the number of breakpoints minus the number of cycles (b-c) in the comparison graph of the two genomes; the number of hurdles does not enter into it. Without changing the formula, we can replace generation and re-absorption of a circular intermediate by a generalized transposition, equivalent to a block interchange, with weight two. Our simple algorithm converts one multi-linear chromosome genome to another in the minimum distance.

Divide-and-conquer approach for the exemplar breakpoint distance

A one-to-one correspondence between the sets of genes in the two genomes being compared is necessary for the notions of breakpoint and reversal distances. To compare genomes where there are paralogous genes, Sankoff formulated the exemplar distance problem as a general version of the genome rearrangement problem. Unfortunately, the problem is NP-hard even for the breakpoint distance. This paper proposes a divide-and-conquer approach for calculating the exemplar breakpoint distance between two genomes with multiple gene families. The combination of our approach and Sankoff's branch-and-bound technique leads to a practical program to answer this question. Tests with both simulated and real datasets show that our program is much more efficient than the existing program that is based only on the branch-and-bound technique. Code for the program is available from the authors.

A 1.5-approximation algorithm for sorting by transpositions and transreversals

One of the most promising ways to determine evolutionary distance between two organisms is to compare the order of appearance of orthologous genes in their genomes. The resulting genome rearrangement problem calls for finding a shortest sequence of rearrangement operations that sorts one genome into the other. In this paper we provide a 1.5-approximation algorithm for the problem of sorting by transpositions and transreversals, improving on a five-year-old 1.75 ratio for this problem. Our algorithm is also faster than current approaches and requires O ( n 3 / 2 log n ) time for n genes.

Packing cycles in undirected graphs

Given an undirected graph G with n nodes and m edges, we address the problem of finding a largest collection of edge-disjoint cycles in G. The problem, dubbed Cycle Packing, is very closely related to a few genome rearrangement problems in computational biology. In this paper, we study the complexity and approximability of Cycle Packing, about which very little is known although the problem is natural and has practical applications. We show that the problem is APX -hard but can be approximated within a factor of O(log n) by a simple greedy approach. We do not know whether the O(log n) factor is tight, but we give a nontrivial example for which the ratio achieved by greedy is not constant, namely Ω( logn/( loglogn) ) . We also show that, for “not too sparse” graphs, i.e., graphs for which m=Ω(n 1+1/t+δ) for some positive integer t and for any fixed δ>0, we can achieve an approximation arbitrarily close to 2 t/3 in polynomial time. In particular, for any ε>0, this yields a 4/3+ ε approximation when m=Ω(n 3/2+δ) , therefore also for dense graphs. Finally, we briefly discuss a natural linear programming relaxation for the problem.