Position and Content Paradigms in Genome Rearrangements: The Wild and Crazy World of Permutations in Genomics

Abstract

Modellers of large scale genome rearrangement events, in which segments of DNA are inverted, moved, swapped, or even inserted or deleted, have found a natural syntax in the language of permutations. Despite this, there has been a wide range of modelling choices, assumptions and interpretations that make navigating the literature a significant challenge. Indeed, even authors of papers that use permutations to model genome rearrangement can struggle to interpret each others' work, because of subtle differences in basic assumptions that are often deeply ingrained (and consequently sometimes not even mentioned). In this paper, we describe the different ways in which permutations have been used to model genomes and genome rearrangement events, presenting some features and limitations of each approach, and show how the various models are related. This paper will help researchers navigate the landscape of genome rearrangement models, and make it easier for authors to present clear and consistent models.

A representation-theoretic approach to the calculation of evolutionary distance in bacteria

“…Dividing each of these by 4! and summing over the dihedral group as in (2), we find explicit expressions (constant time in k) for the number of passages G 0 → G ≡ σ using rearrangement moves belonging to M = {(1, 2), (2,3), (3,4), (4, 1)}. In particular we see that we have the limits (as a proportion of total number of passage counts):…”

“…Given a genome as an (unorientated) cyclic order of N genes, the effect of a rearrangement is described as the application of a permutation α ∈ S N understood as moving the gene in location i to location α(i) (this is the "positions" paradigm as described in [3]). Taking the canonical cyclic ordering as reference, each genome is determined by a permutation σ ∈ S N where j = σ(i) is understood as indicating that gene i is in location j = σ(i).…”

A symmetry-inclusive algebraic approach to genome rearrangement

“…Modelling genomes with oriented regions as elements of the hyperoctahedral group is standard; we refer to Bhatia et al [5] and Egri-Nagy et al [12] for detailed treatments. We consider that the genomes of interest share n regions in common, where each region is a contiguous section of DNA (these may also be referred to as synteny blocks or conserved regions).…”

“…As classified by Bhatia et al [5] in their excellent overview, algebraic frameworks for modelling genome rearrangement tend to use either the "content" or the "position" paradigm. In the former paradigm, genomes are represented in terms of the adjacencies between regions; in the latter, positions as well as regions are labelled, and genomes are denoted by maps that link regions to positions.…”

A Path-Deformation Framework for Determining Weighted Genome Rearrangement Distance

“…When the genome is modeled as a map from positions to regions and a rearrangement operator is a bijection on the set of positions, we require that the rearrangement operator act first on a position and then we map the new position to a region using the genome map, and so the function composition is from left to right. For a detailed discussion of right and left actions (see Bhatia et al, 2018). The inversion operator t i,j maps π as follows:…”