BackgroundThe breakpoint median in the set Sn of permutations on n terms is known to have some unusual behavior, especially if the input genomes are maximally different to each other. The mathematical study of the set of medians is complicated by the facts that breakpoint distance is not a metric but a pseudo-metric, and that it does not define a geodesic space.ResultsWe introduce the notion of partial geodesic, or geodesic patch between two permutations, and show that if two permutations are medians, then every permutation on a geodesic patch between them is also a median. We also prove the conjecture that the input permutations themselves are medians.

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We prove that for general models of random gene-order evolution of k ≥ 3 genomes, as the number of genes n goes to ∞, the median value approximates k times the divergence time if the number of rearrangements is less than cn/4 for any c <1. For some c* ≥ 1, if the number of rearrangements is greater than c*n/4, this approximation does not hold.

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