Identifying evolutionary trees and substitution parameters for the general Markov model with invariable sites

Abstract

The general Markov plus invariable sites (GM+I) model of biological sequence evolution is a two-class model in which an unknown proportion of sites are not allowed to change, while the remainder undergo substitutions according to a Markov process on a tree. For statistical use it is important to know if the model is identifiable; can both the tree topology and the numerical parameters be determined from a joint distribution describing sequences only at the leaves of the tree? We establish that for generic parameters both the tree and all numerical parameter values can be recovered, up to clearly understood issues of 'label swapping.' The method of analysis is algebraic, using phylogenetic invariants to study the variety defined by the model. Simple rational formulas, expressed in terms of determinantal ratios, are found for recovering numerical parameters describing the invariable sites.

Identifiability of a Markovian model of molecular evolution with gamma-distributed rates

“…2 ) − (1 − x) 2 10 , and the two summands here have unique local maxima at x = 1 and no local minima, g must have these features as well. Thus, f exhibits the enumerated properties of Rogers' Figure 2.…”

“…Then f (0) = 0 and f (x) = exp exp(−10(x − 1) 2 ) − (1 − x) 2 10 , so f (x) > 0 and f is increasing. Furthermore, we see that f (x) decays quickly enough to 0 as x tends to ∞, so that f (x) has a horizontal asymptote as x tends to ∞.…”

“…That result specializes to give the identifiability of trees for the κ-state GTR models with at most κ − 1 rates-acrosssites classes, including the GTR + I model. Identifiability of numerical model parameters for GTR + I is further explored in [2]. There have also been a number of recent works dealing with nonidentifiability of mixture models which are not of the rates-across-sites type; these include [8], [9], [14], and [15].…”

A short history of algebraic statistics

“…But the amount and type of available data are often very different and present different mathematical and inferential issues. Interesting work in computational biology has been developed by Allman and Rhodes [4] on evolutionary models and Casanellas and co-authors on phylogenetic invariants [19].…”

The Identifiability of Covarion Models in Phylogenetics

“…However, for models with rate variation across sites, where the distribution of rates is not fully known, only recently have the first positive results been obtained [2], [3], [1]. Despite its widespread use in data analysis, identifiability of the GTR+Γ+I model has yet to be addressed rigorously.…”