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Computing phylogenetic invariants for time-reversible models: from TN93 to its submodels

Marta Casanellas, Jennifer Garbett, Roser Homs, Annachiara Korchmaros, Niharika Chakrabarty Paul

TL;DR

This work extends algebraic phylogenetics to algebraic time-reversible models by deriving invariants for TN93 submodels F81 and F84 through a π-orthogonal basis, and by restricting TN93 invariants to the model’s linear space. It provides a unified treatment of tripods and quartets, showing that for tripods the submodel variety equals the TN93 variety intersected with the submodel's symmetry space, while for quartets it forms an irreducible component of this intersection. The paper derives symmetry equations, rank-based flattening invariants, and local complete intersections around the no-evolution point, delivering explicit defining equations for the spaces of phylogenetic mixtures under F81 and F84. The results enable a more tractable algebraic description of model spaces and mixture spaces, with implications for model selection and quartet-based phylogenetic inference, and lay groundwork for extending ATR methods to larger trees and amino-acid models.

Abstract

Phylogenetic invariants are equations that vanish on algebraic varieties associated with Markov processes that model molecular substitutions on phylogenetic trees. For practical applications, it is essential to understand these equations across a wide range of substitution models. Recent work has shown that, for equivariant models, phylogenetic invariants can be derived from those of the general Markov model by restricting to the linear space defined by the model (namely, the space of mixtures of distributions on the model). Following this philosophy, we describe the space of mixtures and phylogenetic invariants for time-reversible models that are not equivariant. Specifically, we study two submodels of the Tamura-Nei nucleotide substitution model (Felsenstein 81 and 84) using an orthogonal change of basis recently introduced for algebraic time-reversible models. For tripods, we prove that the algebraic variety of each submodel coincides with the variety of Tamura-Nei intersected with the linear space of the submodel. In the case of quartets, we show that it is an irreducible component of this intersection. Moreover, we demonstrate that it suffices to consider only the binomial equations defining the linear space, which correspond to the natural symmetries of the model in the new coordinates. For each submodel, we explicitly provide equations defining a local complete intersection that characterizes the phylogenetic variety on a dense open subset containing the biologically relevant points.

Computing phylogenetic invariants for time-reversible models: from TN93 to its submodels

TL;DR

This work extends algebraic phylogenetics to algebraic time-reversible models by deriving invariants for TN93 submodels F81 and F84 through a π-orthogonal basis, and by restricting TN93 invariants to the model’s linear space. It provides a unified treatment of tripods and quartets, showing that for tripods the submodel variety equals the TN93 variety intersected with the submodel's symmetry space, while for quartets it forms an irreducible component of this intersection. The paper derives symmetry equations, rank-based flattening invariants, and local complete intersections around the no-evolution point, delivering explicit defining equations for the spaces of phylogenetic mixtures under F81 and F84. The results enable a more tractable algebraic description of model spaces and mixture spaces, with implications for model selection and quartet-based phylogenetic inference, and lay groundwork for extending ATR methods to larger trees and amino-acid models.

Abstract

Phylogenetic invariants are equations that vanish on algebraic varieties associated with Markov processes that model molecular substitutions on phylogenetic trees. For practical applications, it is essential to understand these equations across a wide range of substitution models. Recent work has shown that, for equivariant models, phylogenetic invariants can be derived from those of the general Markov model by restricting to the linear space defined by the model (namely, the space of mixtures of distributions on the model). Following this philosophy, we describe the space of mixtures and phylogenetic invariants for time-reversible models that are not equivariant. Specifically, we study two submodels of the Tamura-Nei nucleotide substitution model (Felsenstein 81 and 84) using an orthogonal change of basis recently introduced for algebraic time-reversible models. For tripods, we prove that the algebraic variety of each submodel coincides with the variety of Tamura-Nei intersected with the linear space of the submodel. In the case of quartets, we show that it is an irreducible component of this intersection. Moreover, we demonstrate that it suffices to consider only the binomial equations defining the linear space, which correspond to the natural symmetries of the model in the new coordinates. For each submodel, we explicitly provide equations defining a local complete intersection that characterizes the phylogenetic variety on a dense open subset containing the biologically relevant points.

Paper Structure

This paper contains 22 sections, 25 theorems, 43 equations, 1 figure, 11 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Tamura-Nei nucleotide substitution model and its submodels
  4. Phylogenetic invariants in a $\pi$-orthogonal basis
  5. Flattening matrices
  6. The space of mixtures
  7. The tripod and symmetry equations
  8. Symmetry equations for quartets
  9. Equations derived from rank conditions on flattenings
  10. Rank conditions for F84
  11. Rank conditions for F81
  12. The space of phylogenetic mixtures
  13. Mixtures for F84
  14. Mixtures for F81
  15. Local complete intersection
  16. ...and 7 more sections

Key Result

Proposition 2.6

Let $T$ be a trivalent phylogenetic tree evolving under a submodel $\mathcal{M}$ of TN93 with fixed root distribution $\pi$. Then the dimension of the phylogenetic variety $CV_T^\mathcal{M}$ is $\vert E\vert+1$ if $\mathcal{M}=$F81 and $2\vert E\vert+1$ if $\mathcal{M}=$F84.

Figures (1)

  • Figure 1: Unrooted trivalent trees depicting $T_{12}$, $T_{13}$, and $T_{14}$, from left to right.

Theorems & Definitions (68)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.9: Allman2008
  • ...and 58 more