Paths and Cycles in Breakpoint Graphs of Random Multichromosomal Genomes

We study the probability distribution of genomic distance d under the hypothesis of random gene order. We interpret the random order assumption in terms of a stochastic method for constructing the alternating colour cycles in the decomposition of the bicoloured breakpoint graph. For two random genomes of length n and χ chromosomes, we show that the expectation of n + χ– d is $O(\frac{1}{2}\log\frac{n+\chi}{2\chi}+\frac{3}{2}\chi)$. We then discuss how to extend these analyses to the case where intra- and interchromosomal operations have different probabilities.