Genetic Distance for a General Non-Stationary Markov Substitution Process

Abstract

The genetic distance between biological sequences is a fundamental quantity in molecular evolution. It pertains to questions of rates of evolution, existence of a molecular clock, and phylogenetic inference. Under the class of continuous-time substitution models, the distance is commonly defined as the expected number of substitutions at any site in the sequence. We eschew the almost ubiquitous assumptions of evolution under stationarity and time-reversible conditions and extend the concept of the expected number of substitutions to nonstationary Markov models where the only remaining constraint is of time homogeneity between nodes in the tree. Our measure of genetic distance reduces to the standard formulation if the data in question are consistent with the stationarity assumption. We apply this general model to samples from across the tree of life to compare distances so obtained with those from the general time-reversible model, with and without rate heterogeneity across sites, and the paralinear distance, an empirical pairwise method explicitly designed to address nonstationarity. We discover that estimates from both variants of the general time-reversible model and the paralinear distance systematically overestimate genetic distance and departure from the molecular clock. The magnitude of the distance bias is proportional to departure from stationarity, which we demonstrate to be associated with longer edge lengths. The marked improvement in consistency between the general nonstationary Markov model and sequence alignments leads us to conclude that analyses of evolutionary rates and phylogenies will be substantively improved by application of this model.

Assessing the performance of DNA barcoding using posterior predictive simulations

“…For example, LogDet or paralinear distances have been proposed (Felsenstein 2004) which correct for variation in base composition across a phylogeny. A general nonstationary Markov model has also been proposed that accounts for changes in the substitution process across a tree when calculating pairwise genetic distances among taxa (Kaehler et al 2015). The most flexible option for dealing with process heterogeneity across sites or taxa may be to use patristic distances calculated from branch lengths in a phylogeny for DNA barcoding studies.…”

Embeddability and rate identifiability of Kimura 2-parameter matrices

“…We have obtained that K80 DLC matrices are embeddable if and only if its principal logarithm is a rate matrix (Remark 4.9). In parameter estimation in phylogenetics, it might be relevant to restrict to a subset of matrices where identifiability of the substitution parameters is guaranteed (for a discussion see Zou et al, 2011 andKaehler et al, 2015). The set of all DLC matrices is one of these subsets (see Chang, 1996).…”

Standard codon substitution models overestimate purifying selection for non-stationary data

“…In the simplest sense, our ability 535 to draw inference relies on how well these models represent the process of neutral 536 sequence evolution. As we demonstrated previously (Kaehler et al, 2015), utilising 537 time-reversible nucleotide substitution processes distorts our estimation of the number of 538 events in a manner that is proportional to the extent of non-stationarity. Non-stationarity 539 is common across the tree of life (e.g.…”

“…For non-stationary models, such as GNC, τ does not necessarily equal the expected number of substitutions (Kaehler et al, 2015). Rather, the genetic distance can be calculated as…”

“…Genetic distance across multiple edges is the sum of the dis-182 tances for those edges. Numerical methods for calculation of the integral of the matrix 183 exponential are discussed in Kaehler et al (2015).…”