The Mathematics of Xenology: Di-cographs, Symbolic Ultrametrics, 2-structures and Tree-representable Systems of Binary Relations

TL;DR

This paper explores the mathematical structures underlying gene relations, extending concepts like ultrametrics and cographs to asymmetric cases to better model horizontal gene transfer in molecular evolution.

Contribution

It introduces a non-symmetric generalization of symbolic ultrametrics and links them with di-cographs and hierarchical structures, providing a new framework for representing gene relations.

Findings

Characterization of non-symmetric ultrametrics and di-cographs

Connection between hierarchical structures and gene relation models

Framework accommodates horizontal gene transfer as an ancestral event

Abstract

The concepts of orthology, paralogy, and xenology play a key role in molecular evolution. Orthology and paralogy distinguish whether a pair of genes originated by speciation or duplication. The corresponding binary relations on a set of genes form complementary cographs. Allowing more than two types of ancestral event types leads to symmetric symbolic ultrametrics. Horizontal gene transfer, which leads to xenologous gene pairs, however, is inherent asymmetric since one offspring copy "jumps" into another genome, while the other continues to be inherited vertically. We therefore explore here the mathematical structure of the non-symmetric generalization of symbolic ultrametrics. Our main results tie non-symmetric ultrametrics together with di-cographs (the directed generalization of cographs), so-called uniformly non-prime 2-structures, and hierarchical structures on the set of strong modules. This yields a characterization of relation structures that can be explained in terms of trees and types of ancestral events. This framework accommodates a horizontal-transfer relation in terms of an ancestral event and thus, is slightly different from the the most commonly used definition of xenology.