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The Distribution of Distances between Randomly Constructed genomes: Generating Function, Expectation, Variance and Limits

Based on a large repertoire of chromosomal rearrangement operations, the genomic distance d between two genomes with chi(r) and chi(b) linear chromosomes, respectively, both containing the same (or orthologous) n genes or markers, is d=n + max(chi(r),chi(b))-c, where c is the number of cycles in the breakpoint graph of the two genomes. In this paper, we study the exact probability distribution of c. We derive the expectation and variance, and show that, in the limit, the expectation of d is n - (2chirchib)/(2chir+2chib(-1)) - 1/2ln (n + max(chir,chib))/(chir+chib).