Moments of genome evolution by Double Cut-and-Join

We study statistical estimators of the number of genomic events separating two genomes under a Double Cut-and Join (DCJ) rearrangement model, by a method of moment estimation. We first propose an exact, closed, analytically invertible formula for the expected number of breakpoints after a given number of DCJs. This improves over the heuristic, recursive and computationally slower previously proposed one. Then we explore the analogies of genome evolution by DCJ with evolution of binary sequences under substitutions, permutations under transpositions, and random graphs. Each of these are presented in the literature with intuitive justifications, and are used to import results from better known fields. We formalize the relations by proving a correspondence between moments in sequence and genome evolution, provided substitutions appear four by four in the corresponding model. Eventually we prove a bounded error on two estimators of the number of cycles in the breakpoint graph after a given number of rearrangements, by an analogy with cycles in permutations and components in random graphs.