Sorting by Weighted Reversals, Transpositions, and Inverted Transpositions

Abstract

During evolution, genomes are subject to genome rearrangements that alter the ordering and orientation of genes on the chromosomes. If a genome consists of a single chromosome (like mitochondrial, chloroplast or bacterial genomes), the biologically relevant genome rearrangements are (1) inversions—also called reversals—where a section of the genome is excised, reversed in orientation, and reinserted and (2) transpositions, where a section of the genome is excised and reinserted at a new position in the genome; if this also involves an inversion, one speaks of an inverted transposition. To reconstruct ancient events in the evolutionary history of organisms, one is interested in finding an optimal sequence of genome rearrangements that transforms a given genome into another genome. It is well known that this problem is equivalent to the problem of “sorting” a signed permutation into the identity permutation. The complexity of the problem is still unknown. The best polynomial-time approximation algorithm, recently devised by Hartman and Sharan, has a 1.5 performance ratio. However, it applies only to the case in which reversals and transpositions are weighted equally. Because in most organisms reversals occur more often than transpositions, it is desirable to have the possibility of weighting reversals and transpositions differently. In this paper, we provide a 1.5-approximation algorithm for sorting by weighted reversals, transpositions and inverted transpositions for biologically realistic weights.