Classifying and counting linear phylogenetic invariants for the Jukes-Cantor model

Area of Science:

• Phylogenetics
• Computational Biology
• Graph Theory

Background:

• Linear invariants are crucial for evaluating phylogenetic hypotheses from DNA/RNA sequences, especially with varying evolutionary rates.
• The Jukes-Cantor model is a fundamental one-parameter model for nucleotide substitution.
• Existing algebraic methods for analyzing linear invariants can be complex.

Purpose of the Study:

• To provide a graph-theoretic classification of the vector space of linear invariants for any phylogenetic tree under the Jukes-Cantor model.
• To derive a simple basis for this vector space.
• To establish a formula for the dimension of the invariant space in binary phylogenetic trees.

Main Methods:

• Utilizing graph theory to classify phylogenetic trees and their associated invariant spaces.
• Applying a Hadamard matrix-based technique to analyze linear invariants.
• Relating invariants to edge-disjoint packings of subtrees within the phylogenetic tree.

Main Results:

• A graph-theoretic classification of the vector space of linear invariants (I(T)) for phylogenetic trees (T) under the Jukes-Cantor model is presented.
• An easily described basis for I(T) is provided.
• For binary phylogenetic trees with n leaves, the dimension of I(T) is shown to be 4n - F(2n) - 2, where F(n) is the nth Fibonacci number.

Conclusions:

• The study offers a novel, graph-theoretic approach to understanding linear invariants in phylogenetics.
• The findings complement existing algebraic treatments and provide a new perspective using Hadamard matrices.
• The derived formula for the dimension of the invariant space offers a concrete mathematical relationship for binary trees.