Biased estimates of phylogenetic branch lengths resulting from the discretised Gamma model of site rate heterogeneity

A standard procedure in phylogenetic reconstruction to represent variation in substitution rates between sites in the genome is the discrete Gamma model (DGM). Relative rates are assumed to be distributed according to a discretised Gamma distribution, where the probabilities that a site is included in each discrete class are equal. Here, we identify a serious bias in the branch lengths of reconstructed phylogenies when the DGM is used, whereby branch lengths are usually, and often substantially, overestimated, and the magnitude of this effect increases with the number of sequences in the alignment. We show that the alternative “FreeRate” model, which assumes no parametric distribution and allows the class probabilities to vary, is not subject to the issue. We further establish that reason for the behaviour is the equal class probabilities, not the discretisation itself. We recommend that the DGM be retired from general use. We note that FreeRate is an immediately available replacement, but that it is highly parameterised and known to be difficult to fit, and thus there is scope for innovation in rate heterogeneity models. In a mathematical appendix, we explore the reasons for the phenomenon in more detail. We give an expression for the likelihood of a star tree with uniform branch lengths according to some simplifying assumptions, and use this to estimate the magnitude of the phenomenon for different values of that branch length and alignment size. We show that the effect occurs when the amount of mutation (in substitutions per site) in the sites of the observed alignment differs from that in the unobserved “true” phylogeny, and further that the magnitude of the effect is determined by the mean relative rate of evolution amongst the observed sites. Finally, by investigating the case where the true underlying rate heterogeneity distribution is FreeRate with variable class probabilities but the reconstruction insists on equal probabilities, we show that branches lengths will be overestimated where there is a long-tail of fast-evolving sites in the true rate distribution, the usual situation in real datasets. We finish by giving an example of how to explicitly calculate the magnitude of the effect for a simple case where the true underlying continuous Gamma distribution is an exponential distribution and the reconstruction method uses two equal probability classes.