The arithmetic topology of genetic alignments

TL;DR

This paper introduces a new mathematical framework using weighted simplicial complexes to analyze genetic variation in sequence alignments, revealing novel insights into viral dynamics.

Contribution

It extends pairwise genetic relations to k-ary dissimilarities and develops a topological approach to capture complex biological interactions.

Findings

Captures new layers of viral genetic dynamics.

Applies to SARS-CoV-2 and H1N1 genomic data.

Provides a mathematical foundation for biological interpretation.

Abstract

We propose a novel mathematical paradigm for the study of genetic variation in sequence alignments. This framework originates from extending the notion of pairwise relations, upon which current analysis is based on, to k-ary dissimilarity. This dissimilarity naturally leads to a generalization of simplicial complexes by endowing simplices with weights, compatible with the boundary operator. We introduce the notion of k-stances and dissimilarity complex, the former encapsulating arithmetic as well as topological structure expressing these k-ary relations. We study basic mathematical properties of dissimilarity complexes and show how this approach captures an entirely new layer of biologically relevant viral dynamics in the context of SARS-CoV-2 and H1N1 flu genomic data.