Combinatorial methods in comparative genomics

To understand evolution, and to discover how different species are related, gene order analysis is a useful tool. Problems in this area can usually be formulated in a combinatorial language. We regard genomes as signed, circular permutations, and evolutionary operations like reversals (reversing the order of a segment of genes) and transpositions (moving a segment of genes) are easy to describe combinatorially. A commonly studied problem is to determine the evolutionary distance between two species. This is estimated by several combinatorial distances between gene order permutations, for instance the breakpoint distance and the reversal distance. We have in this thesis applied combinatorics to several important problems, leading to these results: • An algorithm for computing, within any factor 1+ε > 1, the minimal number of reversals and transpositions, the latter double-counted for certain reasons, needed to transform one gene order permutation into another. • A good approximative formula for the expected number of reversals between two gene order permutations with b breakpoints between them: tappr(b) = log ( 1− b n(1− 1 2n−2 ) )