Distribution of genome rearrangement distance under double cut and join

Abstract

Using the double-cut-and-join (DCJ) model for genome rearrangement we use combinatorial techniques to analyze the distribution of genomes under DCJ distance. We present an exponential generating function for the number of genomes that are maximally distant from a given genome and provide a formula for the number of genomes that are any given distance from an arbitrary starting genome. Many mathematical models have been developed to aid biologists and bioinformaticians in their study of the genome rearrangement problem, whose goal is to find the optimal sequence of mutations for the transformation of one genome into another. Using the double-cut-and-join (DCJ) model, Bergeron, Mixtacki, and Stoye [Bergeron et al. 2006] found that the distance between two genomes is completely determined by a bipartite graph created from the genomes. We utilize their data structure to find the distribution of genomes that are distance d from a given genome under DCJ. In Section 2, we introduce genome rearrangement, DCJ, and an important result of the same authors. In Section 3, we present a generating function for the number of maximally distant genomes from a given genome, and in Section 4, we obtain the distribution of all genomes by distance from a given genome.