The allele-frequency spectrum in a decoupled Moran model with mutation, drift, and directional selection, assuming small mutation rates

The expected sample allele frequencies from populations of changing size via orthogonal polynomials

In this article, discrete and stochastic changes in (effective) population size are incorporated into the spectral representation of a biallelic diffusion process for drift and small mutation rates. A forward algorithm inspired by Hidden-Markov-Model (HMM) literature is used to compute exact sample allele frequency spectra for three demographic scenarios: single changes in (effective) population size, boom-bust dynamics, and stochastic fluctuations in (effective) population size. An approach for fully agnostic demographic inference from these sample allele spectra is explored, and sufficient statistics for stepwise changes in population size are found. Further, convergence behaviours of the polymorphic sample spectra for population size changes on different time scales are examined and discussed within the context of inference of the effective population size. Joint visual assessment of the sample spectra and the temporal coefficients of the spectral decomposition of the forward diffusion process is found to be important in determining departure from equilibrium. Stochastic changes in (effective) population size are shown to shape sample spectra particularly strongly.

Recurrent mutation in the ancestry of a rare variant

Recurrent mutation produces multiple copies of the same allele which may be co-segregating in a population. Yet, most analyses of allele-frequency or site-frequency spectra assume that all observed copies of an allele trace back to a single mutation. We develop a sampling theory for the number of latent mutations in the ancestry of a rare variant, specifically a variant observed in relatively small count in a large sample. Our results follow from the statistical independence of low-count mutations, which we show to hold for the standard neutral coalescent or diffusion model of population genetics as well as for more general coalescent trees. For populations of constant size, these counts are distributed like the number of alleles in the Ewens sampling formula. We develop a Poisson sampling model for populations of varying size and illustrate it using new results for site-frequency spectra in an exponentially growing population. We apply our model to a large data set of human SNPs and use it to explain dramatic differences in site-frequency spectra across the range of mutation rates in the human genome.

Population dynamics of GC-changing mutations in humans and great apes

The nucleotide composition of the genome is a balance between origin and fixation rates of different mutations. For example, it is well-known that transitions occur more frequently than transversions, particularly at CpG sites. Differences in fixation rates of mutation types are less explored. Specifically, recombination-associated GC-biased gene conversion (gBGC) may differentially impact GC-changing mutations, due to differences in their genomic distributions and efficiency of mismatch repair mechanisms. Given that recombination evolves rapidly across species, we explore gBGC of different mutation types across human populations and great ape species. We report a stronger correlation between segregating GC frequency and recombination for transitions than for transversions. Notably, CpG transitions are most strongly affected by gBGC in humans and chimpanzees. We show that the overall strength of gBGC is generally correlated with effective population sizes in humans, with some notable exceptions, such as a stronger effect of gBGC on non-CpG transitions in populations of European descent. Furthermore, species of the Gorilla and Pongo genus have a greatly reduced gBGC effect on CpG sites. We also study the dependence of gBGC dynamics on flanking nucleotides and show that some mutation types evolve in opposition to the gBGC expectation, likely due to hypermutability of specific nucleotide contexts. Our results highlight the importance of different gBGC dynamics experienced by GC-changing mutations and their impact on nucleotide composition evolution.

A nearly-neutral biallelic Moran model with biased mutation and linear and quadratic selection

In this article, a biallelic reversible mutation model with linear and quadratic selection is analysed. The approach reconnects to one proposed by Kimura (1979), who starts from a diffusion model and derives its equilibrium distribution up to a constant. We use a boundary-mutation Moran model, which approximates a general mutation model for small effective mutation rates, and derive its equilibrium distribution for polymorphic and monomorphic variants in small to moderately sized populations. Using this model, we show that biased mutation rates and linear selection alone can cause patterns of polymorphism within and substitution rates between populations that are usually ascribed to balancing or overdominant selection. We illustrate this using a data set of short introns and fourfold degenerate sites from Drosophila simulans and Drosophila melanogaster.

Maximum likelihood estimators for scaled mutation rates in an equilibrium mutation-drift model

The stationary sampling distribution of a neutral decoupled Moran or Wright-Fisher diffusion with neutral mutations is known to first order for a general rate matrix with small but otherwise unconstrained mutation rates. Using this distribution as a starting point we derive results for maximum likelihood estimates of scaled mutation rates from site frequency data under three model assumptions: a twelve-parameter general rate matrix, a nine-parameter reversible rate matrix, and a six-parameter strand-symmetric rate matrix. The site frequency spectrum is assumed to be sampled from a fixed size population in equilibrium, and to consist of allele frequency data at a large number of unlinked sites evolving with a common mutation rate matrix without selective bias. We correct an error in a previous treatment of the same problem (Burden and Tang, 2017) affecting the estimators for the general and strand-symmetric rate matrices. The method is applied to a biological dataset consisting of a site frequency spectrum extracted from short autosomal introns in a sample of Drosophila melanogaster individuals.

Polymorphism-Aware Species Trees with Advanced Mutation Models, Bootstrap, and Rate Heterogeneity

Molecular phylogenetics has neglected polymorphisms within present and ancestral populations for a long time. Recently, multispecies coalescent based methods have increased in popularity, however, their application is limited to a small number of species and individuals. We introduced a polymorphism-aware phylogenetic model (PoMo), which overcomes this limitation and scales well with the increasing amount of sequence data whereas accounting for present and ancestral polymorphisms. PoMo circumvents handling of gene trees and directly infers species trees from allele frequency data. Here, we extend the PoMo implementation in IQ-TREE and integrate search for the statistically best-fit mutation model, the ability to infer mutation rate variation across sites, and assessment of branch support values. We exemplify an analysis of a hundred species with ten haploid individuals each, showing that PoMo can perform inference on large data sets. While PoMo is more accurate than standard substitution models applied to concatenated alignments, it is almost as fast. We also provide bmm-simulate, a software package that allows simulation of sequences evolving under PoMo. The new options consolidate the value of PoMo for phylogenetic analyses with population data.

Statistical Inference in the Wright-Fisher Model Using Allele Frequency Data

The Wright–Fisher model provides an elegant mathematical framework for understanding allele frequency data. In particular, the model can be used to infer the demographic history of species and identify loci under selection. A crucial quantity for inference under the Wright–Fisher model is the distribution of allele frequencies (DAF). Despite the apparent simplicity of the model, the calculation of the DAF is challenging. We review and discuss strategies for approximating the DAF, and how these are used in methods that perform inference from allele frequency data. Various evolutionary forces can be incorporated in the Wright–Fisher model, and we consider these in turn. We begin our review with the basic bi-allelic Wright–Fisher model where random genetic drift is the only evolutionary force. We then consider mutation, migration, and selection. In particular, we compare diffusion-based and moment-based methods in terms of accuracy, computational efficiency, and analytical tractability. We conclude with a brief overview of the multi-allelic process with a general mutation model.

Variation in the Intensity of Selection on Codon Bias over Time Causes Contrasting Patterns of Base Composition Evolution in Drosophila

Four-fold degenerate coding sites form a major component of the genome, and are often used to make inferences about selection and demography, so that understanding their evolution is important. Despite previous efforts, many questions regarding the causes of base composition changes at these sites in Drosophila remain unanswered. To shed further light on this issue, we obtained a new whole-genome polymorphism data set from D. simulans. We analyzed samples from the putatively ancestral range of D. simulans, as well as an existing polymorphism data set from an African population of D. melanogaster. By using D. yakuba as an outgroup, we found clear evidence for selection on 4-fold sites along both lineages over a substantial period, with the intensity of selection increasing with GC content. Based on an explicit model of base composition evolution, we suggest that the observed AT-biased substitution pattern in both lineages is probably due to an ancestral reduction in selection intensity, and is unlikely to be the result of an increase in mutational bias towards AT alone. By using two polymorphism-based methods for estimating selection coefficients over different timescales, we show that the selection intensity on codon usage has been rather stable in D. simulans in the recent past, but the long-term estimates in D. melanogaster are much higher than the short-term ones, indicating a continuing decline in selection intensity, to such an extent that the short-term estimates suggest that selection is only active in the most GC-rich parts of the genome. Finally, we provide evidence for complex evolutionary patterns in the putatively neutral short introns, which cannot be explained by the standard GC-biased gene conversion model. These results reveal a dynamic picture of base composition evolution.

Mapping of the stochastic Lotka-Volterra model to models of population genetics and game theory

The relationship between the M-species stochastic Lotka-Volterra competition (SLVC) model and the M-allele Moran model of population genetics is explored via timescale separation arguments. When selection for species is weak and the population size is large but finite, precise conditions are determined for the stochastic dynamics of the SLVC model to be mappable to the neutral Moran model, the Moran model with frequency-independent selection, and the Moran model with frequency-dependent selection (equivalently a game-theoretic formulation of the Moran model). We demonstrate how these mappings can be used to calculate extinction probabilities and the times until a species' extinction in the SLVC model.

An alternative derivation of the stationary distribution of the multivariate neutral Wright-Fisher model for low mutation rates with a view to mutation rate estimation from site frequency data

Recently, Burden and Tang (2016) provided an analytical expression for the stationary distribution of the multivariate neutral Wright-Fisher model with low mutation rates. In this paper we present a simple, alternative derivation that illustrates the approximation. Our proof is based on the discrete multivariate boundary mutation model which has three key ingredients. First, the decoupled Moran model is used to describe genetic drift. Second, low mutation rates are assumed by limiting mutations to monomorphic states. Third, the mutation rate matrix is separated into a time-reversible part and a flux part, as suggested by Burden and Tang (2016). An application of our result to data from several great apes reveals that the assumption of stationarity may be inadequate or that other evolutionary forces like selection or biased gene conversion are acting. Furthermore we find that the model with a reversible mutation rate matrix provides a reasonably good fit to the data compared to the one with a non-reversible mutation rate matrix.

Reversible polymorphism-aware phylogenetic models and their application to tree inference

We present a reversible Polymorphism-Aware Phylogenetic Model (revPoMo) for species tree estimation from genome-wide data. revPoMo enables the reconstruction of large scale species trees for many within-species samples. It expands the alphabet of DNA substitution models to include polymorphic states, thereby, naturally accounting for incomplete lineage sorting. We implemented revPoMo in the maximum likelihood software IQ-TREE. A simulation study and an application to great apes data show that the runtimes of our approach and standard substitution models are comparable but that revPoMo has much better accuracy in estimating trees, divergence times and mutation rates. The advantage of revPoMo is that an increase of sample size per species improves estimations but does not increase runtime. Therefore, revPoMo is a valuable tool with several applications, from speciation dating to species tree reconstruction.

Inference of directional selection and mutation parameters assuming equilibrium

In a classical study, Wright (1931) proposed a model for the evolution of a biallelic locus under the influence of mutation, directional selection and drift. He derived the equilibrium distribution of the allelic proportion conditional on the scaled mutation rate, the mutation bias and the scaled strength of directional selection. The equilibrium distribution can be used for inference of these parameters with genome-wide datasets of "site frequency spectra" (SFS). Assuming that the scaled mutation rate is low, Wright's model can be approximated by a boundary-mutation model, where mutations are introduced into the population exclusively from sites fixed for the preferred or unpreferred allelic states. With the boundary-mutation model, inference can be partitioned: (i) the shape of the SFS distribution within the polymorphic region is determined by random drift and directional selection, but not by the mutation parameters, such that inference of the selection parameter relies exclusively on the polymorphic sites in the SFS; (ii) the mutation parameters can be inferred from the amount of polymorphic and monomorphic preferred and unpreferred alleles, conditional on the selection parameter. Herein, we derive maximum likelihood estimators for the mutation and selection parameters in equilibrium and apply the method to simulated SFS data as well as empirical data from a Madagascar population of Drosophila simulans.

Applications in the search for genomic selection signatures in fish

Selection signatures are genomic regions harboring DNA sequences functionally involved in the genetic variation of traits subject to selection. Selection signatures have been intensively studied in recent years because of their relevance to evolutionary biology and their potential association with genes that control phenotypes of interest in wild and domestic populations. Selection signature research in fish has been confined to a smaller scale, due in part to the relatively recent domestication of fish species and limited genomic resources such as molecular markers, genetic mapping, DNA sequences, and reference genomes. However, recent genomic technology advances are paving the way for more studies that may contribute to the knowledge of genomic regions underlying phenotypes of biological and productive interest in fish.

On the retention of gene duplicates prone to dominant deleterious mutations

Recent studies have shown that gene families from different functional categories have been preferentially expanded either by small scale duplication (SSD) or by whole-genome duplication (WGD). In particular, gene families prone to dominant deleterious mutations and implicated in cancers and other genetic diseases in human have been greatly expanded through two rounds of WGD dating back from early vertebrates. Here, we strengthen this intriguing observation, showing that human oncogenes involved in different primary tumors have retained many WGD duplicates compared to other human genes. In order to rationalize this evolutionary outcome, we propose a consistent population genetics model to analyze the retention of SSD and WGD duplicates taking into account their propensity to acquire dominant deleterious mutations. We solve a deterministic haploid model including initial duplicated loci, their retention through sub-functionalization or their neutral loss-of-function or deleterious gain-of-function at one locus. Extensions to diploid genotypes are presented and population size effects are analyzed using stochastic simulations. The only difference between the SSD and WGD scenarios is the initial number of individuals with duplicated loci. While SSD duplicates need to spread through the entire population from a single individual to reach fixation, WGD duplicates are de facto fixed in the small initial post-WGD population arising through the ploidy incompatibility between post-WGD individuals and the rest of the pre-WGD population. WGD duplicates prone to dominant deleterious mutations are then shown to be indirectly selected through purifying selection in post-WGD species, whereas SSD duplicates typically require positive selection. These results highlight the long-term evolution mechanisms behind the surprising accumulation of WGD duplicates prone to dominant deleterious mutations and are shown to be consistent with cancer genome data on the prevalence of human oncogenes with WGD duplicates.

The common ancestor process revisited

We consider the Moran model in continuous time with two types, mutation, and selection. We concentrate on the ancestral line and its stationary type distribution. Building on work by Fearnhead (J. Appl. Probab. 39 (2002), 38-54) and Taylor (Electron. J. Probab. 12 (2007), 808-847), we characterise this distribution via the fixation probability of the offspring of all individuals of favourable type (regardless of the offspring's types). We concentrate on a finite population and stay with the resulting discrete setting all the way through. This way, we extend previous results and gain new insight into the underlying particle picture.

Linking great apes genome evolution across time scales using polymorphism-aware phylogenetic models

The genomes of related species contain valuable information on the history of the considered taxa. Great apes in particular exhibit variation of evolutionary patterns along their genomes. However, the great ape data also bring new challenges, such as the presence of incomplete lineage sorting and ancestral shared polymorphisms. Previous methods for genome-scale analysis are restricted to very few individuals or cannot disentangle the contribution of mutation rates and fixation biases. This represents a limitation both for the understanding of these forces as well as for the detection of regions affected by selection. Here, we present a new model designed to estimate mutation rates and fixation biases from genetic variation within and between species. We relax the assumption of instantaneous substitutions, modeling substitutions as mutational events followed by a gradual fixation. Hence, we straightforwardly account for shared ancestral polymorphisms and incomplete lineage sorting. We analyze genome-wide synonymous site alignments of human, chimpanzee, and two orangutan species. From each taxon, we include data from several individuals. We estimate mutation rates and GC-biased gene conversion intensity. We find that both mutation rates and biased gene conversion vary with GC content. We also find lineage-specific differences, with weaker fixation biases in orangutan species, suggesting a reduced historical effective population size. Finally, our results are consistent with directional selection acting on coding sequences in relation to exonic splicing enhancers.

Strong purifying selection at synonymous sites in D. melanogaster

Synonymous sites are generally assumed to be subject to weak selective constraint. For this reason, they are often neglected as a possible source of important functional variation. We use site frequency spectra from deep population sequencing data to show that, contrary to this expectation, 22% of four-fold synonymous (4D) sites in Drosophila melanogaster evolve under very strong selective constraint while few, if any, appear to be under weak constraint. Linking polymorphism with divergence data, we further find that the fraction of synonymous sites exposed to strong purifying selection is higher for those positions that show slower evolution on the Drosophila phylogeny. The function underlying the inferred strong constraint appears to be separate from splicing enhancers, nucleosome positioning, and the translational optimization generating canonical codon bias. The fraction of synonymous sites under strong constraint within a gene correlates well with gene expression, particularly in the mid-late embryo, pupae, and adult developmental stages. Genes enriched in strongly constrained synonymous sites tend to be particularly functionally important and are often involved in key developmental pathways. Given that the observed widespread constraint acting on synonymous sites is likely not limited to Drosophila, the role of synonymous sites in genetic disease and adaptation should be reevaluated.

Evidence for complex selection on four-fold degenerate sites in Drosophila melanogaster

We considered genome-wide four-fold degenerate sites from an African Drosophila melanogaster population and compared them to short introns. To include divergence and to polarize the data, we used its close relatives Drosophila simulans, Drosophila sechellia, Drosophila erecta and Drosophila yakuba as outgroups. In D. melanogaster, the GC content at four-fold degenerate sites is higher than in short introns; compared to its relatives, more AT than GC is fixed. The former has been explained by codon usage bias (CUB) favouring GC; the latter by decreased intensity of directional selection or by increased mutation bias towards AT. With a biallelic equilibrium model, evidence for directional selection comes mostly from the GC-rich ancestral base composition. Together with a slight mutation bias, it leads to an asymmetry of the unpolarized allele frequency spectrum, from which directional selection is inferred. Using a quasi-equilibrium model and polarized spectra, however, only purifying and no directional selection is detected. Furthermore, polarized spectra are proportional to those of the presumably unselected short introns. As we have no evidence for a decrease in effective population size, relaxed CUB must be due to a reduction in the selection coefficient. Going beyond the biallelic model and considering all four bases, signs of directional selection are stronger. In contrast to short introns, complementary bases show strand specificity and allele frequency spectra depend on mutation directions. Hence, the traditional biallelic model to describe the evolution of four-fold degenerate sites should be replaced by more complex models assuming only quasi-equilibrium and accounting for all four bases.

Unconstrained evolution in short introns? - an analysis of genome-wide polymorphism and divergence data from Drosophila

An unconstrained reference sequence facilitates the detection of selection. In Drosophila, sequence variation in short introns seems to be least influenced by selection and dominated by mutation and drift. Here, we test this with genome-wide sequences using an African population (Malawi) of D. melanogaster and data from the related outgroup species D. simulans, D. sechellia, D. erecta and D. yakuba. The distribution of mutations deviates from equilibrium, and the content of A and T (AT) nucleotides shows an excess of variance among introns. We explain this by a complex mutational pattern: a shift in mutational bias towards AT, leading to a slight nonequilibrium in base composition and context-dependent mutation rates, with G or C (GC) sites mutating most frequently in AT-rich introns. By comparing the corresponding allele frequency spectra of AT-rich vs. GC-rich introns, we can rule out the influence of directional selection or biased gene conversion on the mutational pattern. Compared with neutral equilibrium expectations, polymorphism spectra show an excess of low frequency and a paucity of intermediate frequency variants, irrespective of the direction of mutation. Combining the information from different outgroups with the polymorphism data and using a generalized linear model, we find evidence for shared ancestral polymorphism between D. melanogaster and D. simulans, D. sechellia, arguing against a bottleneck in D. melanogaster. Generally, we find that short introns can be used as a neutral reference on a genome-wide level, if the spatially and temporally varying mutational pattern is accounted for.