Paths and Cycles in Breakpoint Graphs of Random Multichromosomal Genomes

We study the probability distribution of the distance d = n + chi - kappa - psi between two genomes with n markers distributed on chi chromosomes and with breakpoint graphs containing kappa cycles and psi "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 - 1/2chi - 1/2 log n+chi/2chi. 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, kappa and d depend on the relative proportions of the two kinds of operation.