Sorting Genomes by Translocations and Deletions, Dealing with Duplicated Genes

The problem of sorting by reciprocal translocations (abbreviated as SBT) arises from the field of comparative genomics, which is to find a shortest sequence of reciprocal translocations that transforms one genome Pi into another genome Gamma, with the restriction that Pi and Gamma contain the same genes. SBT has been proved to be polynomial-time solvable, and several polynomial algorithms have been developed, including Bergeron´s SBT algorithm. Recently, an algorithm allowing to compare genomes with different genes but without duplications is given by extending Bergeron´s SBT algorithm. Here, we will improve the approach to handle genomes with duplicated genes. Obviously, the existence of duplications will make the problem more difficult. We should decide which one of the duplications in the subject genome Pi should be assigned to a particular homolog in the target genome Gamma; resolving such a problem appears to require exponential time. In this paper, by a similar method developed by Mark Marron et. al., which was used to sort two chromosomes containing different genes, we present an approximation algorithm for computing near minimal sorting sequences involving translocations and deletions between two (multi-chromosomes) genomes containing different genes allowing non-tails duplications. We derive the error bound for the polynomial-time distance computation. This is the first constant approximation algorithm for SBT with duplications.