Escape from Parsimony of a Double-Cut-and-Join Genome Evolution Process

We analyze models of genome evolution based on both restricted and unrestricted double-cut-and-join (DCJ) operations. Not only do our models allow different types of operations generated by DCJs (including reversals, translocations, transpositions, fissions, and fusions) to take different weights during the course of evolution, but they also let these weights fluctuate over time. We compare the number of operations along the evolutionary trajectory with the DCJ distance of the genome from its ancestor at each step, and determine at what point they diverge: the process escapes from parsimony. Adapting the method developed by Berestycki and Durrett, we approximate the number of cycles in the breakpoint graph of a random genome at time t and its ancestral genome by the number of tree components in a random graph (not necessarily an Erdös-Rényi one) constructed from the model of evolution. In both models, the process on a genome of size n is bound to its parsimonious estimate up to $$ steps.