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A phase-type distribution is the time to absorption in a continuous- or discrete-time Markov chain. Phase-type distributions can be used as a general framework to calculate key properties of the standard coalescent model and many of its extensions. Here, the ‘phases’ in the phase-type distribution correspond to states in the ancestral process. For example, the time to the most recent common ancestor and the total branch length are phase-type distributed. Furthermore, the site frequency spectrum follows a multivariate discrete phase-type distribution and the joint distribution of total branch lengths in the two-locus coalescent-with-recombination model is multivariate phase-type distributed. In general, phase-type distributions provide a powerful mathematical framework for coalescent theory because they are analytically tractable using matrix manipulations. The purpose of this review is to explain the phase-type theory and demonstrate how the theory can be applied to derive basic properties of coalescent models. These properties can then be used to obtain insight into the ancestral process, or they can be applied for statistical inference. In particular, we show the relation between classical first-step analysis of coalescent models and phase-type calculations. We also show how reward transformations in phase-type theory lead to easy calculation of covariances and correlation coefficients between e.g. tree height, tree length, external branch length, and internal branch length. Furthermore, we discuss how these quantities can be used for statistical inference based on estimating equations. Providing an alternative to previous work based on the Laplace transform, we derive likelihoods for small-size coalescent trees based on phase-type theory. Overall, our main aim is to demonstrate that phase-type distributions provide a convenient general set of tools to understand aspects of coalescent models that are otherwise difficult to derive. Throughout the review, we emph

Genomes are typically mosaics of regions with different evolutionary histories. When speciation events are closely spaced in time, recombination makes the regions sharing the same history small, and the evolutionary history changes rapidly as we move along the genome. When examining rapid radiations such as the early diversification of Neoaves 66 Mya, typically no consistent history is observed across segments exceeding kilobases of the genome. Here, we report an exception. We found that a 21-Mb region in avian genomes, mapped to chicken chromosome 4, shows an extremely strong and discordance-free signal for a history different from that of the inferred species tree. Such a strong discordance-free signal, indicative of suppressed recombination across many millions of base pairs, is not observed elsewhere in the genome for any deep avian relationships. Although long regions with suppressed recombination have been documented in recently diverged species, our results pertain to relationships dating circa 65 Mya. We provide evidence that this strong signal may be due to an ancient rearrangement that blocked recombination and remained polymorphic for several million years prior to fixation. We show that the presence of this region has misled previous phylogenomic efforts with lower taxon sampling, showing the interplay between taxon and locus sampling. We predict that similar ancient rearrangements may confound phylogenetic analyses in other clades, pointing to a need for new analytical models that incorporate the possibility of such events.

Comparative analysis of primate genomes within a phylogenetic context is essential for understanding the evolution of human genetic architecture and primate diversity. We present such a study of 50 primate species spanning 38 genera and 14 families, including 27 genomes first reported here, with many from previously less well represented groups, the New World monkeys and the Strepsirrhini. Our analyses reveal heterogeneous rates of genomic rearrangement and gene evolution across primate lineages. Thousands of genes under positive selection in different lineages play roles in the nervous, skeletal, and digestive systems and may have contributed to primate innovations and adaptations. Our study reveals that many key genomic innovations occurred in the Simiiformes ancestral node and may have had an impact on the adaptive radiation of the Simiiformes and human evolution.

Incomplete lineage sorting (ILS) makes ancestral genetic polymorphisms persist during rapid speciation events, inducing incongruences between gene trees and species trees. ILS has complicated phylogenetic inference in many lineages, including hominids. However, we lack empirical evidence that ILS leads to incongruent phenotypic variation. Here, we performed phylogenomic analyses to show that the South American monito del monte is the sister lineage of all Australian marsupials, although over 31% of its genome is closer to the Diprotodontia than to other Australian groups due to ILS during ancient radiation. Pervasive conflicting phylogenetic signals across the whole genome are consistent with some of the morphological variation among extant marsupials. We detected hundreds of genes that experienced stochastic fixation during ILS, encoding the same amino acids in non-sister species. Using functional experiments, we confirm how ILS may have directly contributed to hemiplasy in morphological traits that were established during rapid marsupial speciation ca. 60 mya.