Figure 1

Adjacency and breakpoint graphs for 2, 3-cycles and block interchanges In this representation the genome graphs are superimposed over the adjacency graphs. (a) An inversion has 2 breakpoints in each genome, a DCJ distance of 1 and a breakpoint reuse rate of 2d/b=1. (b) A simple transposition with d_DCJ=3-1=2 and b=3 has an inversion-based breakpoint reuse of 4/3. In (c) we note that a block interchange has two overlapping 2-cycles. Performing a DCJ about the first creates a circular intermediate (CI) and about the second, the CI is reabsorbed. Since b=4 and c=2 the total DCJ distance is 4-2=2. Breakpoint reuse for a block interchange in the DCJ paradigm is r=2d_DCJ/b=4/4=1, since in the creation of the CI and its reabsorption, no breakpoints are reused in general. As for simple transpositions, the join creating the CI occurs in the same position as the subsequent cut, so one breakpoint is reused which may not be what occurs in nature! Below the adjacency graph for each example is the corresponding breakpoint graph. Transpositions and block intechanges look deceptively simple in their adjacency graphs. Although a block interchange appears to contain two simple 2-cycles, they are in fact made of unoriented arcs in the breakpoint graph, and similarly for the simple transposition. As a result both simple transpositions and block interchanges contain one hurdle, which increase the HP distance by 1 relative to the DCJ distance. Hence the breakpoint reuse rate for simple transpositions in the HP paradigm is r_GRIMM=2d_HP/b=2 and for block interchanges it is 2d_HP/b=2*3/4=3/2.