Comparison Theorems of Phylogenetic Spaces and the Moduli Spaces of Curves

TL;DR

This paper explores the mathematical relationship between phylogenetic models used in biology and the moduli spaces of stable curves, providing a categorical framework that links evolutionary trees and networks to complex algebraic geometry.

Contribution

It introduces a categorical connection between phylogenetic spaces and moduli spaces of stable curves, enhancing the mathematical understanding of evolutionary models.

Findings

Phylogenetic spaces map into boundary components of moduli spaces.

Categorical relationships are established between phylogenetic trees, networks, and stable curves.

The space of network topologies injects into the boundary of moduli spaces.

Abstract

Rapid developments in genetics and biology have led to phylogenetic methods becoming an important direction in the study of cancer and viral evolution. Although our understanding of gene biology and biochemistry has increased and is increasing at a remarkable rate, the theoretical models of genetic evolution still use the phylogenetic tree model that was introduced by Darwin in 1859 and the generalization to phylogenetic networks introduced by Grant in 1971. Darwin's model uses phylogenetic trees to capture the evolutionary relationships of reproducing individuals [6]; Grant's generalization to phylogenetic networks is meant to account for the phenomena of horizontal gene transfer [14]. Therefore, it is important to provide an accurate mathematical description of these models and to understand their connection with other fields of mathematics. In this article, we focus on the graph theoretical aspects of phylogenetic trees and networks and their connection to stable curves. We introduce the building blocks of evolutionary moduli spaces, the dual intersection complex of the moduli spaces of stable curves, and the categorical relationship between the phylogenetic spaces and stable curves in $\overline{\mathfrak{M}}_{0,n}(\mathbb{C})$ and $\overline{\mathfrak{M}}_{0,n}(\mathbb{R})$. We also show that the space of network topologies maps injectively into the boundary of $\overline{\mathfrak{M}}_{g,n}(\mathbb{C})$.