1990, Waddell and Steel 1997, Yang and Kumar 1996), if at least one of the four eigenvalues of P is negative, the logarithmic matrix function computed by equation (5) is not defined. For any pair of sequences, the GTR distance is defined asĪs noted by ( Rodriguez et al. (1990), Waddell and Steel (1997) proposed an exact estimation procedure to compute GTR distances (also implemented in PAUP* ( Swofford 2003)). On the basis of the seminal work by Lanave et al. In most phylogeny inference packages (eg, PAUP* ( Swofford 2003) and MrBayes ( Ronquist and Huelsenbeck 2003)), homogeneity across the tree is assumed, ie, a single R matrix is optimized for the whole tree. The transition rate matrix R is generally unknown and many inference methods rely on its computation: ( i) distance methods evaluate the GTR distance t ̂ for each sequence pair and require that the corresponding R (see below) is Markovian ( ie, is a real matrix with negative diagonal elements and non-negative elements outside the diagonal), and ( ii) Maximum Likelihood and Bayesian methods require estimating R for computing P( t) that, in turn, is required for computing the Likelihood of a tree ( Felsenstein 2004). Is a solution of equation (1) such that, once R is given, the probability P( t) of substitution between two states can be computed for any t and the evolution of sequences through time is completely described as long as all GTR-assumptions are verified. The dynamic of substitution probabilities for an infinitesimal time dt is described by The GTR model assumes that the equilibrium character state frequencies and the instantaneous transition probabilities remain constant through time.

The GTR model is a stationary Markov process by which substitution probabilities among nucleotides are expressed in the form of a matrix P( t). 1984), (allowing for different nucleotide frequencies and 6 different substitution rates) have been developed. Different models, ranging from the JC model ( Jukes and Cantor 1969) (assuming equal nucleotide frequencies and identical substitution rates) to the General Time Reversible (GTR) model ( Lanave et al. Many methods for reconstructing trees from DNA sequences (eg, distance-matrix methods, Maximum Likelihood and Bayesian approaches) rely on a substitution model that describes how sequences evolve over time. Molecular genetic markers (mostly DNA sequences) have arguably become the most popular and powerful source of data for phylogeny inference. We also discuss the implications of our results on the current implementations of maximum likelihood and Bayesian methods.Īll phylogeny inference methods are based on explicit or implicit assumptions whose validity can possibly be challenged when analysing real data. Our results indicate that, inapplicability of the Waddel and Steel’s procedure can be considered a real practical issue, and illustrate that the probability of this inapplicability is a function of substitution rates and sequence length. We then evaluate both the frequency of the GTR model inapplicability for estimating distances and the accuracy of inferred alignments. Here, we simulate the evolution of DNA sequences along 12 trees characterized by different combinations of tree length, (non-)homogeneity of the substitution rate matrix R, and sequence length.

The procedure described by Waddell and Steel (1997), for estimating distances and instantaneous substitution rate matrices, R, under the GTR model, is known to be inapplicable under some conditions, ie, it leads to the inapplicability of the GTR model. The General Time Reversible (GTR) model of nucleotide substitution is at the core of many distance-based and character-based phylogeny inference methods.

Gtr evolution chart