Tying Up Loose Strands: Defining Equations of the Strand Symmetric Model

TL;DR

This paper characterizes the algebraic structure of the strand symmetric model in phylogenetics, providing a complete set of defining equations that can be used for model analysis and inference.

Contribution

It proves that the known phylogenetic invariants fully define the ideal of the strand symmetric model for any trivalent tree, using computational algebraic geometry techniques.

Findings

The known invariants generate a prime ideal of the correct dimension.

The Zariski closure of the model is the secant variety of a toric variety.

The results enable algebraic analysis of the model for phylogenetic inference.

Abstract

The strand symmetric model is a phylogenetic model designed to reflect the symmetry inherent in the double-stranded structure of DNA. We show that the set of known phylogenetic invariants for the general strand symmetric model of the three leaf claw tree entirely defines the ideal. This knowledge allows one to determine the vanishing ideal of the general strand symmetric model of any trivalent tree. Our proof of the main result is computational. We use the fact that the Zariski closure of the strand symmetric model is the secant variety of a toric variety to compute the dimension of the variety. We then show that the known equations generate a prime ideal of the correct dimension using elimination theory.