Linear invariants under Jukes’ and Cantor’s one-parameter model

Abstract

Linear invariants are random variables with zero expectations under certain assumptions. In this paper, linear invariants under Jukes’ and Cantor’s one-parameter model, both with and without the assumption that nucleotide frequencies are at equilibrium, are studied using the method developed in a previous paper. Phylogenetic linear invariants (random variables that are linear invariants of some but not all trees of the same number of species) for trees with up to seven species are derived and bases of phylogenetic linear invariant spaces for unrooted trees with four, five and six species are presented. All these bases consist of invariants of simple form. The constraints that specify non-phylogenetic linear invariants (invariants shared by all trees of the same number of species) are determined. Under the assumption that nucleotide frequencies are at equilibrium, it is found that (i) each five-species tree has 17 independent phylogenetic linear invariants, and for two different trees with five species, there are at least three phylogenetic linear invariants of one tree that are not invariants of the other tree; (ii) each six-species tree has 98 independent phylogenetic linear invariants, and for two different trees of six species there are at least nine independent phylogenetic linear invariants that are not invariants of the other tree; and (iii) each seven-species tree has 482 independent phylogenetic linear invariants. It is also found that the number of independent phylogenetic linear invariants is much larger without the assumption of equilibrium than with it, but the reverse is true for the number of non-phylogenetic linear invariants. A class of random variables that are phylogenetic linear invariants with or without the equilibrium assumption is also identified.