Studying models of balancing selection using phase-type theory

Phase-type distributions in mathematical population genetics: An emerging framework

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 emphasize the versatility of the phase-type framework, which is also illustrated by our accompanying R-code. All our analyses and figures can be reproduced from code available on GitHub.

How chromosomal inversions reorient the evolutionary process

Inversions are structural mutations that reverse the sequence of a chromosome segment and reduce the effective rate of recombination in the heterozygous state. They play a major role in adaptation, as well as in other evolutionary processes such as speciation. Although inversions have been studied since the 1920s, they remain difficult to investigate because the reduced recombination conferred by them strengthens the effects of drift and hitchhiking, which in turn can obscure signatures of selection. Nonetheless, numerous inversions have been found to be under selection. Given recent advances in population genetic theory and empirical study, here we review how different mechanisms of selection affect the evolution of inversions. A key difference between inversions and other mutations, such as single nucleotide variants, is that the fitness of an inversion may be affected by a larger number of frequently interacting processes. This considerably complicates the analysis of the causes underlying the evolution of inversions. We discuss the extent to which these mechanisms can be disentangled, and by which approach.

The effects of inversion polymorphisms on patterns of neutral genetic diversity

The strong reduction in the frequency of recombination in heterozygotes for an inversion and a standard gene arrangement causes the arrangements to become partially isolated genetically, resulting in sequence divergence between them and changes in the levels of neutral variability at nucleotide sites within each arrangement class. Previous theoretical studies on the effects of inversions on neutral variability have assumed either that the population is panmictic or that it is divided into 2 populations subject to divergent selection. Here, the theory is extended to a model of an arbitrary number of demes connected by migration, using a finite island model with the inversion present at the same frequency in all demes. Recursion relations for mean pairwise coalescent times are used to obtain simple approximate expressions for diversity and divergence statistics for an inversion polymorphism at equilibrium under recombination and drift, and for the approach to equilibrium following the sweep of an inversion to a stable intermediate frequency. The effects of an inversion polymorphism on patterns of linkage disequilibrium are also examined. The reduction in effective recombination rate caused by population subdivision can have significant effects on these statistics. The theoretical results are discussed in relation to population genomic data on inversion polymorphisms, with an emphasis on Drosophila melanogaster. Methods are proposed for testing whether or not inversions are close to recombination-drift equilibrium, and for estimating the rate of recombinational exchange in heterozygotes for inversions; difficulties involved in estimating the ages of inversions are also discussed.

An Ancestral Balanced Inversion Polymorphism Confers Global Adaptation

Since the pioneering work of Dobzhansky in the 1930s and 1940s, many chromosomal inversions have been identified, but how they contribute to adaptation remains poorly understood. In Drosophila melanogaster, the widespread inversion polymorphism In(3R)Payne underpins latitudinal clines in fitness traits on multiple continents. Here, we use single-individual whole-genome sequencing, transcriptomics, and published sequencing data to study the population genomics of this inversion on four continents: in its ancestral African range and in derived populations in Europe, North America, and Australia. Our results confirm that this inversion originated in sub-Saharan Africa and subsequently became cosmopolitan; we observe marked monophyletic divergence of inverted and noninverted karyotypes, with some substructure among inverted chromosomes between continents. Despite divergent evolution of this inversion since its out-of-Africa migration, derived non-African populations exhibit similar patterns of long-range linkage disequilibrium between the inversion breakpoints and major peaks of divergence in its center, consistent with balancing selection and suggesting that the inversion harbors alleles that are maintained by selection on several continents. Using RNA-sequencing, we identify overlap between inversion-linked single-nucleotide polymorphisms and loci that are differentially expressed between inverted and noninverted chromosomes. Expression levels are higher for inverted chromosomes at low temperature, suggesting loss of buffering or compensatory plasticity and consistent with higher inversion frequency in warm climates. Our results suggest that this ancestrally tropical balanced polymorphism spread around the world and became latitudinally assorted along similar but independent climatic gradients, always being frequent in subtropical/tropical areas but rare or absent in temperate climates.

Some complexities in interpreting apparent effects of hitchhiking: A commentary on Gompert et al. (2022)

We write to address recent claims by regarding the potentially important and underappreciated phenomena of "indirect selection," the observation that neutral regions may be affected by natural selection. We argue both that this phenomenon-generally known as genetic hitchhiking-is neither new nor poorly studied, and that the patterns described by the authors have multiple alternative explanations.

A large and diverse autosomal haplotype is associated with sex-linked colour polymorphism in the guppy

Male colour patterns of the Trinidadian guppy (Poecilia reticulata) are typified by extreme variation governed by both natural and sexual selection. Since guppy colour patterns are often inherited faithfully from fathers to sons, it has been hypothesised that many of the colour trait genes must be physically linked to sex determining loci as a ‘supergene’ on the sex chromosome. Here, we phenotype and genotype four guppy ‘Iso-Y lines’, where colour was inherited along the patriline for 40 generations. Using an unbiased phenotyping method, we confirm the breeding design was successful in creating four distinct colour patterns. We find that genetic differentiation among the Iso-Y lines is repeatedly associated with a diverse haplotype on an autosome (LG1), not the sex chromosome (LG12). Moreover, the LG1 haplotype exhibits elevated linkage disequilibrium and evidence of sex-specific diversity in the natural source population. We hypothesise that colour pattern polymorphism is driven by Y-autosome epistasis.

Extreme colour pattern variation in male Trinidadian guppies are influenced by natural selection and sexual selection. Here, the authors phenotype and genotype four guppy lineages finding that colour pattern is associated with a diverse haplotype on an autosome.

A numerical framework for genetic hitchhiking in populations of variable size

Natural selection on beneficial or deleterious alleles results in an increase or decrease, respectively, of their frequency within the population. Due to chromosomal linkage, the dynamics of the selected site affect the genetic variation at nearby neutral loci in a process commonly referred to as genetic hitchhiking. Changes in population size, however, can yield patterns in genomic data that mimic the effects of selection. Accurately modeling these dynamics is thus crucial to understanding how selection and past population size changes impact observed patterns of genetic variation. Here, we model the evolution of haplotype frequencies with the Wright-Fisher diffusion to study the impact of selection on linked neutral variation. Explicit solutions are not known for the dynamics of this diffusion when selection and recombination act simultaneously. Thus, we present a method for numerically evaluating the Wright-Fisher diffusion dynamics of 2 linked loci separated by a certain recombination distance when selection is acting. We can account for arbitrary population size histories explicitly using this approach. A key step in the method is to express the moments of the associated transition density, or sampling probabilities, as solutions to ordinary differential equations. Numerically solving these differential equations relies on a novel accurate and numerically efficient technique to estimate higher order moments from lower order moments. We demonstrate how this numerical framework can be used to quantify the reduction and recovery of genetic diversity around a selected locus over time and elucidate distortions in the site-frequency-spectra of neutral variation linked to loci under selection in various demographic settings. The method can be readily extended to more general modes of selection and applied in likelihood frameworks to detect loci under selection and infer the strength of the selective pressure.

Revisiting the Notion of Deleterious Sweeps

It has previously been shown that, conditional on its fixation, the time to fixation of a semi-dominant deleterious autosomal mutation in a randomly mating population is the same as that of an advantageous mutation. This result implies that deleterious mutations could generate selective sweep-like effects. Although their fixation probabilities greatly differ, the much larger input of deleterious relative to beneficial mutations suggests that this phenomenon could be important. We here examine how the fixation of mildly deleterious mutations affects levels and patterns of polymorphism at linked sites - both in the presence and absence of interference amongst deleterious mutations - and how this class of sites may contribute to divergence between-populations and species. We find that, while deleterious fixations are unlikely to represent a significant proportion of outliers in polymorphism-based genomic scans within populations, minor shifts in the frequencies of deleterious mutations can influence the proportions of private variants and the value of FST after a recent population split. As sites subject to deleterious mutations are necessarily found in functional genomic regions, interpretations in terms of recurrent positive selection may require reconsideration.