Breakpoint graphs and ancestral genome reconstructions

(A) A phylogenetic tree T with four linear genomes P₁, P₂, P₃, P₄ (represented as green, blue, red, and yellow graphs, respectively) at the leaves. 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 multi-degrees of regular vertices vary from 1 (e.g., vertex b^h) to 3 (e.g., vertex e^h). (C) The same phylogenetic tree T with all intermediate genomes specified and a genome X selected as a root. A T-consistent transformation of X into P₁, P₂, P₃, P₄ can viewed as a transformation of the quadruple (X, X, X, X) into the quadruple (P₁, P₂, P₃, P₄), where a rearrangement at each step is applied to some copies of the same genome in the quadruple. A particular such transformation takes the following steps: (X, X, X, X) (X, X, Q₁, Q₁) (Q₃, Q₃, Q₁, Q₁) (Q₃, Q₃, Q₂, Q₂) (Q₃, Q₃, P₃, Q₂) (Q₃, Q₃, P₃, P₄) (P₁, Q₃, P₃, P₄) (P₁, P₂, P₃, P₄), where r₁ is a reversal in two copies of X; r₂ is a fission in two copies of X; r₃ is a reversal in both copies of Q₁; r₄ is a fission in one copy of Q₂; r₅ is a reversal in the other copy of Q₂; r₆ is a reversal in one copy of Q₃; and r₇ is a translocation in the other copy of Q₃.