[Paper Review] Distinguishing Convergence on Phylogenetic Networks

JD Mitchell|arXiv (Cornell University)|Jun 23, 2016

Plant and animal studies3 citations

TL;DR

This paper investigates the distinguishability of phylogenetic networks from trees using Abelian group-based models and algebraic geometry. It demonstrates that certain phylogenetic networks and trees—particularly with three or four taxa—can be distinguished via their phylogenetic tensors, especially when applying Hadamard conjugation and Gröbner basis methods to solve polynomial equations in the tensor space.

ABSTRACT

We compare the phylogenetic tensors for various trees and networks for two, three and four taxa. If the probability spaces between one tree or network and another are not identical then there will be phylogenetic tensors that could have arisen on one but not the other. We call these two trees or networks distinguishable from each other. We show that for the binary symmetric model there are no two-taxon trees and networks that are distinguishable from each other, however there are three-taxon trees and networks that are distinguishable from each other. We compare the time parameters for the phylogenetic tensors for various taxon label permutations on a given tree or network. If the time parameters on one taxon label permutation in terms of the other taxon label permutation are all non-negative then we say that the two taxon label permutations are not network identifiable from each other. We show that some taxon label permutations are network identifiable from each other. We show that some four-taxon networks satisfy the four-point condition. Of the two "shapes" of four-taxon rooted trees, one is defined by the cluster, b,c,d, labelling taxa alphabetically from left to right. The network with this shape and convergence between the two taxa with the root as their most recent common ancestor satisfies the four-point condition. The phylogenetic tensors contain polynomial equations that cannot be easily solved for four-taxon or higher trees or networks. We show how methods from algebraic geometry, such as Gröbner bases, can be used to solve the polynomial equations. We show that some four-taxon trees and networks can be distinguished from each other.

Motivation & Objective

• To determine under what conditions phylogenetic networks and trees are distinguishable based on their evolutionary models.
• To investigate the role of convergence in phylogenetic networks and how it affects model distinguishability.
• To apply algebraic geometry techniques, particularly Gröbner bases, to solve polynomial equations arising from phylogenetic tensors.
• To examine the identifiability of taxon label permutations on networks and their implications for model inference.
• To assess whether four-taxon networks satisfy the four-point condition, a key criterion for tree-likeness in phylogenetics.

Proposed method

• Utilizes Abelian group-based models in molecular phylogenetics, particularly focusing on the binary symmetric model.
• Applies Hadamard conjugation to diagonalize rate matrices and transform phylogenetic tensors into a more analyzable basis.
• Compares probability distributions (phylogenetic tensors) across different trees and networks to assess model distinguishability.
• Employs algebraic geometry tools, including Gröbner bases, to solve systems of polynomial equations derived from tensor constraints.
• Analyzes taxon label permutations to test network identifiability, checking whether time parameters remain non-negative under relabeling.
• Evaluates the four-point condition on four-taxon rooted networks to determine tree-like structure and compatibility with tree models.

Experimental results

Research questions

• RQ1 Are there cases where two phylogenetic networks or a network and a tree are distinguishable based on their phylogenetic tensors?
• RQ2 Under what conditions do phylogenetic networks with convergence fail or satisfy the four-point condition?
• RQ3 Can taxon label permutations on a network be identified as distinct under the same model, or are they indistinguishable?
• RQ4 To what extent can algebraic geometry methods like Gröbner bases resolve the distinguishability of four-taxon trees and networks?
• RQ5 How does the binary symmetric model perform in distinguishing networks from trees, especially in small taxon sets?

Key findings

• For the binary symmetric model, no two-taxon trees and networks are distinguishable, but there exist three-taxon configurations where they are distinguishable.
• Some four-taxon networks do not satisfy the four-point condition, indicating they cannot be represented as trees, while others do.
• The network structure defined by the cluster (b,c,d) with convergence between the two taxa sharing the root as their most recent common ancestor satisfies the four-point condition.
• Gröbner basis methods successfully solve the polynomial equations in the phylogenetic tensor space, enabling the distinction of certain four-taxon trees and networks.
• Taxon label permutations on some networks are network identifiable, meaning time parameters remain non-negative under relabeling, while others are not.
• The study confirms that algebraic geometry provides a viable framework for resolving complex phylogenetic distinguishability problems beyond the reach of standard tree-based models.