A new algebraic approach to genome rearrangement models

TL;DR

This paper introduces a unified algebraic framework for modeling genome rearrangements that incorporates symmetries and allows flexible, efficient maximum likelihood estimation for various genome models.

Contribution

It develops a general genome algebra framework for modeling genomes with symmetries, enabling flexible and efficient likelihood computations for genome rearrangement distances.

Findings

Constructed genome algebra for unsigned circular genomes with dihedral symmetry.

Demonstrated efficient maximum likelihood estimation within this algebraic framework.

Extended the framework to genomes represented by arbitrary groups and symmetries.

Abstract

We present a unified framework for modelling genomes and their rearrangements in a genome algebra, as elements that simultaneously incorporate all physical symmetries. Building on previous work utilising the group algebra of the symmetric group, we explicitly construct the genome algebra for the case of unsigned circular genomes with dihedral symmetry and show that the maximum likelihood estimate (MLE) of genome rearrangement distance can be validly and more efficiently performed in this setting. We then construct the genome algebra for a more general case, that is, for genomes that may be represented by elements of an arbitrary group and symmetry group, and show that the MLE computations can be performed entirely within this framework. There is no prescribed model in this framework; that is, it allows any choice of rearrangements that preserve the set of regions, along with arbitrary weights. Further, since the likelihood function is built from path probabilities -- a generalisation of path counts -- the framework may be utilised for any distance measure that is based on path probabilities.