A 2-break on edges (u, v) and (x, y) corresponding to (a) reversal, (b) fission, (c) translocation/fusion.

The phylogenetic tree T on five genomes M, R, D, Q, and H. The branches of the tree are denoted as M+, R+, D+, Q+, H+, MR+, and QH+.

The estimated number of intra- and inter-reuses bound(e) and bound(e₁, e₂) between seven branches with varying branch length specified in Figure 4 (data simulated according to FBM). The cells are colored as in Figure 5.

Subtables of Figure 6 for n = 900 (top part) and Figure 7 (bottom part) featuring branches M+, R+, and QH+ as one element of the pair. The cells are colored as in Figure 5.

The scaled inter-reuse for five simulated genomes M, R, D, Q, H on m = 2,000 synteny blocks, n = 900 fragile regions, and the turnover rate x varying from zero to four with the phylogenetic tree and branch lengths shown in Figure 4. The simulations follow FBM (x = 0) and TFBM (x varies from one to four). The plot shows the scaled inter-reuse for only three reference points (corresponding to red, green, and yellow cells) that are somewhat arbitrarily connected by straight segments for better visualization.

Positions of regions broken on the evolutionary path from the rodent-primate-carnivore ancestor (that is, on H+, Q+, and QH+ branches) projected to the human chromosomes.

An example of the breakpoint graph and its transformation into an identity breakpoint graph. (a) Graph representation of a two-chromosomal genome P = (+a + b)(+c + e + -d) as two black-obverse cycles and a unichromosomal genome Q = (+a + b - e + c - d) as a red-obverse cycle. (b) The superposition of the genome graphs P and Q. (c) The breakpoint graph G(P, Q) of the genomes P and Q (with removed obverse edges). The black and red edges in G(P, Q) form c(P, Q) = 2 non-trivial black-red cycles and one trivial black-red cycle. The trivial cycle (a^h, b^t) corresponds to a common adjacency between the genes a and b in the genomes P and Q. The vertices in the non-trivial cycles represent breakpoints corresponding to the endpoints of b(P, Q) = 4 synteny blocks: ab, c, d, and e. By Theorem 1, the distance between the genomes P and Q is d(P, Q) = 4 - 2 = 2. (d) A transformation of the breakpoint graph G(P, Q) into the identity breakpoint graph G(Q, Q), corresponding to a transformation of the genome P into the genome Q with two 2-breaks. The first 2-break transforms P into a genome P' = (+a + b)(+c d - e), while the second 2-break transforms P' into Q. Each 2-break increases the number of black-red cycles in the breakpoint graph by one, implying this transformation is shortest (see Theorem 1).

An example of four genomes with a phylogenetic tree and their multiple breakpoint graph. (a) A phylogenetic tree with four circular genomes P₁, P₂, P₃, P₄ (represented as green, blue, red, and yellow graphs respectively) at the leaves and specified intermediate genomes. The obverse edges are not shown. (b) The multiple breakpoint graph G(P₁, P₂, P₃, P₄) is a superposition of graphs representing genomes P₁, P₂, P₃, P₄.

The number of intra- and inter-reuses between seven branches of the tree in Figure 4, each of length 100, for simulated genomes with n fragile regions (n = 500, 900, 1, 300). The diagonal elements represent intra-reuses while the elements above diagonal represent inter-reuses. In each cell with numbers x : y, x represents the observed reuse while y represents the corresponding lower bound. The cells of the table are colored red (for adjacent branches like M+ and R+), green (for branches that are separated by a single branch like M+ and D+ separated by MR+), and yellow (for branches that are separated by two branches like M+ and H+ separated by MR+ and QH+).

The estimated number of intra- and inter-reuses bound(e) and bound(e₁, e₂) between seven branches of the phylogenetic tree in Figure 4 of five mammalian genomes (real data). The cells are colored as in Figure 5.

The breakpoint intra- and inter-reuse (averaged over 100 simulations) for five simulated genomes M, R, D, Q, H under TFBM model with m = 2, 000 synteny blocks, n fragile regions, the turnover rate x, and the evolutionary tree shown in Figure 4 with the length of each branch equal 100. The cells are colored as in Figure 5.

Empirical and theoretical curves representing the number of reuses R(l) as a function of distance l between pairs of sub-branches of the tree in Figure 4 of the five mammalian genomes (ancestral genomes were computed using MGRA [50]). The empirical curve is averaged over 1, 000 random samplings of shortest rearrangement scenarios, while the theoretical curve represents the best fit with parameters n ≈ 196, x ≈ 1:12, and m ≈ 4, 017 (see Methods).