We study the probability distribution of the distance d = n + χ − κ − ψ between two genomes with n markers distributed on χ chromosomes and with breakpoint graphs containing κ cycles and ψ “good” paths, under the hypothesis of random gene order. We interpret the random order assumption in terms of a stochastic method for constructing the bicolored breakpoint graph. We show that the limiting expectation of E[d] = n − 1/2χ − 1/2 log n+χ/2χ. We also calculate the variance, the effect of different numbers of chromosomes in the two genomes, and the number of plasmids, or circular chromosomes, generated by the random breakpoint graph construction. A more realistic model allows intra- and interchromosomal operations to have different probabilities, and simulations show that for a fixed number of rearrangements, κ and d depend on the relative proportions of the two kinds of operation.
