A derivation of all linear invariants for a nonbalanced transversion model

Abstract

The method of linear invariants discovered by Lake is a way of inferring phylogenies by testing statistical hypotheses. The main advantage of the method is that substitution rates for positions along the DNA sequence do not have to be identical. The assumptions and the algebraic background necessary for the applications of the method were clearly laid out in two papers by Cavender, who also described a way to derive a basis for the space of all linear invariants for rooted trees linking four species when the substitution process satisfies the assumption of balanced transversions. Cavender noted a generalization of linear invariants to certain more general substitution models. In this paper we give a simple explicit description of a basis for all linear invariants for a slight variant of Cavender's more general model, which applies to rooted trees linking any number of species. Bases for rooted trees linking five species are enumerated and the method applied to a problem concerning RNA polymerase sequence data.

Slicing Hyperdimensional Oranges: The Geometry of Phylogenetic Estimation

“…This means that the model manifold will be restricted to the linear subspace defined by Prob{[0, 0, 0, 0]} Ϫ Prob{[1, 1, 1, 1]} ϭ 0. Similar considerations of this kind lead to the extraction of linear invariants for phylogenetic trees (Lake, 1987a,b;Nguyen and Speed, 1992;Steel et al, 1993, see below). Conversely, the most general model with general transition matrices for each branch does not have any linear invariants, but it may have higher order invariants.…”

Classifying and Counting Linear Phylogenetic Invariants for the Jukes–Cantor Model

“…The set of phylogenetic invariants for T forms a real vector space, which we denote as I(T). Linear invariants for several models including some more general than Jukes-Cantor have been considered by other authors (see Lake (1987), Felsenstein (1991), Fu and Li (1992), and Nguyen and Speed (1992)); our aim here is to obtain, for the JC model, more information (including an exact enumeration) for linear invariants on any number of sequences, and a more tree-based representation of them; in addition, many of the linear invariants in the JC model do not exist in more general models.…”

Phylogenetic Invariants and Geometry

“…For other models, the complete set of invariants is unknown. However, the set of linear invariants has been classi"ed for the n-taxa, Jukes}Cantor nucleotide model by Steel & Fu (1995) and for the four taxa, six-parameter nucleotide model (Cavender, 1989;Ngyuyen & Speed, 1992). Quintic invariants for the four-taxa, 12-parameter model have also been recently discovered (Hagedorn, 2000b).…”