Genome Rearrangements and Sorting by Reversals

Sequence comparison in molecular biology is in the beginning of a major paradigm shift---a shift from gene comparison based on local mutations (i.e., insertions, deletions, and substitutions of nucleotides) to chromosome comparison based on global rearrangements (i.e., inversions and transpositions of fragments). The classical methods of sequence comparison do not work for global rearrangements, and little is known in computer science about the edit distance between sequences if global rearrangements are allowed. In the simplest form, the problem of gene rearrangements corresponds to sorting by reversals, i.e., sorting of an array using reversals of arbitrary fragments. Recently, Kececioglu and Sankoff gave the first approximation algorithm for sorting by reversals with guaranteed error bound 2 and identified open problems related to chromosome rearrangements. One of these problems is Gollan's conjecture on the reversal diameter of the symmetric group. This paper proves the conjecture. Further, the problem of expected reversal distance between two random permutations is investigated. The reversal distance between two random permutations is shown to be very close to the reversal diameter, thereby indicating that reversal distance provides a good separation between related and nonrelated sequences in molecular evolution studies. The gene rearrangement problem forces us to consider reversals of signed permutations, as the genes in DNA could be positively or negatively oriented. An approximation algorithm for signed permutation is presented, which provides a performance guarantee of ${3 \over 2}$. Finally, using the signed permutations approach, an approximation algorithm for sorting by reversals is described which achieves a performance guarantee of ${7 \over 4}$.