New Explicit Expressions for Relative Frequencies of Single-Nucleotide Polymorphisms With Application to Statistical Inference on Population Growth

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.

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.

Bayesian tests for random mating in polyploids

Hardy-Weinberg proportions (HWP) are often explored to evaluate the assumption of random mating. However, in autopolyploids, organisms with more than two sets of homologous chromosomes, HWP and random mating are different hypotheses that require different statistical testing approaches. Currently, the only available methods to test for random mating in autopolyploids (i) heavily rely on asymptotic approximations and (ii) assume genotypes are known, ignoring genotype uncertainty. Furthermore, these approaches are all frequentist, and so do not carry the benefits of Bayesian analysis, including ease of interpretability, incorporation of prior information, and consistency under the null. Here, we present Bayesian approaches to test for random mating, bringing the benefits of Bayesian analysis to this problem. Our Bayesian methods also (i) do not rely on asymptotic approximations, being appropriate for small sample sizes, and (ii) optionally account for genotype uncertainty via genotype likelihoods. We validate our methods in simulations and demonstrate on two real datasets how testing for random mating is more useful for detecting genotyping errors than testing for HWP (in a natural population) and testing for Mendelian segregation (in an experimental S1 population). Our methods are implemented in Version 2.0.2 of the hwep R package on the Comprehensive R Archive Network https://cran.r-project.org/package=hwep.

Scaling the discrete-time Wright-Fisher model to biobank-scale datasets

The discrete-time Wright-Fisher (DTWF) model and its diffusion limit are central to population genetics. These models can describe the forward-in-time evolution of allele frequencies in a population resulting from genetic drift, mutation, and selection. Computing likelihoods under the diffusion process is feasible, but the diffusion approximation breaks down for large samples or in the presence of strong selection. Existing methods for computing likelihoods under the DTWF model do not scale to current exome sequencing sample sizes in the hundreds of thousands. Here, we present a scalable algorithm that approximates the DTWF model with provably bounded error. Our approach relies on two key observations about the DTWF model. The first is that transition probabilities under the model are approximately sparse. The second is that transition distributions for similar starting allele frequencies are extremely close as distributions. Together, these observations enable approximate matrix-vector multiplication in linear (as opposed to the usual quadratic) time. We prove similar properties for Hypergeometric distributions, enabling fast computation of likelihoods for subsamples of the population. We show theoretically and in practice that this approximation is highly accurate and can scale to population sizes in the tens of millions, paving the way for rigorous biobank-scale inference. Finally, we use our results to estimate the impact of larger samples on estimating selection coefficients for loss-of-function variants. We find that increasing sample sizes beyond existing large exome sequencing cohorts will provide essentially no additional information except for genes with the most extreme fitness effects.

cloneRate: fast estimation of single-cell clonal dynamics using coalescent theory

MOTIVATION

While evolutionary approaches to medicine show promise, measuring evolution itself is difficult due to experimental constraints and the dynamic nature of body systems. In cancer evolution, continuous observation of clonal architecture is impossible, and longitudinal samples from multiple timepoints are rare. Increasingly available DNA sequencing datasets at single-cell resolution enable the reconstruction of past evolution using mutational history, allowing for a better understanding of dynamics prior to detectable disease. There is an unmet need for an accurate, fast, and easy-to-use method to quantify clone growth dynamics from these datasets.

RESULTS

We derived methods based on coalescent theory for estimating the net growth rate of clones using either reconstructed phylogenies or the number of shared mutations. We applied and validated our analytical methods for estimating the net growth rate of clones, eliminating the need for complex simulations used in previous methods. When applied to hematopoietic data, we show that our estimates may have broad applications to improve mechanistic understanding and prognostic ability. Compared to clones with a single or unknown driver mutation, clones with multiple drivers have significantly increased growth rates (median 0.94 versus 0.25 per year; P = 1.6×10-6). Further, stratifying patients with a myeloproliferative neoplasm (MPN) by the growth rate of their fittest clone shows that higher growth rates are associated with shorter time to MPN diagnosis (median 13.9 versus 26.4 months; P = 0.0026).

AVAILABILITY AND IMPLEMENTATION

We developed a publicly available R package, cloneRate, to implement our methods (Package website: https://bdj34.github.io/cloneRate/). Source code: https://github.com/bdj34/cloneRate/.

Genomic inference of a severe human bottleneck during the Early to Middle Pleistocene transition

Population size history is essential for studying human evolution. However, ancient population size history during the Pleistocene is notoriously difficult to unravel. In this study, we developed a fast infinitesimal time coalescent process (FitCoal) to circumvent this difficulty and calculated the composite likelihood for present-day human genomic sequences of 3154 individuals. Results showed that human ancestors went through a severe population bottleneck with about 1280 breeding individuals between around 930,000 and 813,000 years ago. The bottleneck lasted for about 117,000 years and brought human ancestors close to extinction. This bottleneck is congruent with a substantial chronological gap in the available African and Eurasian fossil record. Our results provide new insights into our ancestry and suggest a coincident speciation event.

Scaling the Discrete-time Wright Fisher model to biobank-scale datasets

The Discrete-Time Wright Fisher (DTWF) model and its large population diffusion limit are central to population genetics. These models describe the forward-in-time evolution of the frequency of an allele in a population and can include the fundamental forces of genetic drift, mutation, and selection. Computing like-lihoods under the diffusion process is feasible, but the diffusion approximation breaks down for large sample sizes or in the presence of strong selection. Unfortunately, existing methods for computing likelihoods under the DTWF model do not scale to current exome sequencing sample sizes in the hundreds of thousands. Here we present an algorithm that approximates the DTWF model with provably bounded error and runs in time linear in the size of the population. Our approach relies on two key observations about Binomial distributions. The first is that Binomial distributions are approximately sparse. The second is that Binomial distributions with similar success probabilities are extremely close as distributions, allowing us to approximate the DTWF Markov transition matrix as a very low rank matrix. Together, these observations enable matrix-vector multiplication in linear (as opposed to the usual quadratic) time. We prove similar properties for Hypergeometric distributions, enabling fast computation of likelihoods for subsamples of the population. We show theoretically and in practice that this approximation is highly accurate and can scale to population sizes in the billions, paving the way for rigorous biobank-scale population genetic inference. Finally, we use our results to estimate how increasing sample sizes will improve the estimation of selection coefficients acting on loss-of-function variants. We find that increasing sample sizes beyond existing large exome sequencing cohorts will provide essentially no additional information except for genes with the most extreme fitness effects.

A class of identifiable phylogenetic birth-death models

In a striking result, Louca and Pennell [S. Louca, M. W. Pennell, Nature 580, 502-505 (2020)] recently proved that a large class of phylogenetic birth-death models is statistically unidentifiable from lineage-through-time (LTT) data: Any pair of sufficiently smooth birth and death rate functions is "congruent" to an infinite collection of other rate functions, all of which have the same likelihood for any LTT vector of any dimension. As Louca and Pennell argue, this fact has distressing implications for the thousands of studies that have utilized birth-death models to study evolution. In this paper, we qualify their finding by proving that an alternative and widely used class of birth-death models is indeed identifiable. Specifically, we show that piecewise constant birth-death models can, in principle, be consistently estimated and distinguished from one another, given a sufficiently large extant timetree and some knowledge of the present-day population. Subject to mild regularity conditions, we further show that any unidentifiable birth-death model class can be arbitrarily closely approximated by a class of identifiable models. The sampling requirements needed for our results to hold are explicit and are expected to be satisfied in many contexts such as the phylodynamic analysis of a global pandemic.

Modes of Selection in Tumors as Reflected by Two Mathematical Models and Site Frequency Spectra

The tug-of-war model was developed in a series of papers of McFarland and co-authors to account for existence of mutually counteracting rare advantageous driver mutations and more frequent slightly deleterious passenger mutations in cancer. In its original version, it was a state-dependent branching process. Because of its formulation, the tug-of-war model is of importance for tackling the problem as to whether evolution of cancerous tumors is "Darwinian" or "non-Darwinian." We define two Time-Continuous Markov Chain versions of the model, including identical mutation processes but adopting different drift and selection components. In Model A, drift and selection process preserves expected fitness whereas in Model B it leads to non-decreasing expected fitness. We investigate these properties using mathematical analysis and extensive simulations, which detect the effect of the so-called drift barrier in Model B but not in Model A. These effects are reflected in different structure of clone genealogies in the two models. Our work is related to the past theoretical work in the field of evolutionary genetics, concerning the interplay among mutation, drift and selection, in absence of recombination (asexual reproduction), where epistasis plays a major role. Finally, we use the statistics of mutation frequencies known as the Site Frequency Spectra (SFS), to compare the variant frequencies in DNA of sequenced HER2+ breast cancers, to those based on Model A and B simulations. The tumor-based SFS are better reproduced by Model A, pointing out a possible selection pattern of HER2+ tumor evolution. To put our models in context, we carried out an exploratory study of how publicly accessible data from breast, prostate, skin and ovarian cancers fit a range of models found in the literature.

Robust detection of natural selection using a probabilistic model of tree imbalance

Neutrality tests such as Tajima's D and Fay and Wu's H are standard implements in the population genetics toolbox. One of their most common uses is to scan the genome for signals of natural selection. However, it is well understood that D and H are confounded by other evolutionary forces-in particular, population expansion-that may be unrelated to selection. Because they are not model-based, it is not clear how to deconfound these tests in a principled way. In this article, we derive new likelihood-based methods for detecting natural selection, which are robust to fluctuations in effective population size. At the core of our method is a novel probabilistic model of tree imbalance, which generalizes Kingman's coalescent to allow certain aberrant tree topologies to arise more frequently than is expected under neutrality. We derive a frequency spectrum-based estimator that can be used in place of D, and also extend to the case where genealogies are first estimated. We benchmark our methods on real and simulated data, and provide an open source software implementation.

Linkage disequilibrium between rare mutations

The statistical associations between mutations, collectively known as linkage disequilibrium, encode important information about the evolutionary forces acting within a population. Yet in contrast to single-site analogues like the site frequency spectrum, our theoretical understanding of linkage disequilibrium remains limited. In particular, little is currently known about how mutations with different ages and fitness costs contribute to expected patterns of linkage disequilibrium, even in simple settings where recombination and genetic drift are the major evolutionary forces. Here, I introduce a forward-time framework for predicting linkage disequilibrium between pairs of neutral and deleterious mutations as a function of their present-day frequencies. I show that the dynamics of linkage disequilibrium become much simpler in the limit that mutations are rare, where they admit a simple heuristic picture based on the trajectories of the underlying lineages. I use this approach to derive analytical expressions for a family of frequency-weighted linkage disequilibrium statistics as a function of the recombination rate, the frequency scale, and the additive and epistatic fitness costs of the mutations. I find that the frequency scale can have a dramatic impact on the shapes of the resulting linkage disequilibrium curves, reflecting the broad range of time scales over which these correlations arise. I also show that the differences between neutral and deleterious linkage disequilibrium are not purely driven by differences in their mutation frequencies and can instead display qualitative features that are reminiscent of epistasis. I conclude by discussing the implications of these results for recent linkage disequilibrium measurements in bacteria. This forward-time approach may provide a useful framework for predicting linkage disequilibrium across a range of evolutionary scenarios.

Nonparametric coalescent inference of mutation spectrum history and demography

As populations boom and bust, the accumulation of genetic diversity is modulated, encoding histories of living populations in present-day variation. Many methods exist to decode these histories, and all must make strong model assumptions. It is typical to assume that mutations accumulate uniformly across the genome at a constant rate that does not vary between closely related populations. However, recent work shows that mutational processes in human and great ape populations vary across genomic regions and evolve over time. This perturbs the mutation spectrum (relative mutation rates in different local nucleotide contexts). Here, we develop theoretical tools in the framework of Kingman's coalescent to accommodate mutation spectrum dynamics. We present mutation spectrum history inference (mushi), a method to perform nonparametric inference of demographic and mutation spectrum histories from allele frequency data. We use mushi to reconstruct trajectories of effective population size and mutation spectrum divergence between human populations, identify mutation signatures and their dynamics in different human populations, and calibrate the timing of a previously reported mutational pulse in the ancestors of Europeans. We show that mutation spectrum histories can be placed in a well-studied theoretical setting and rigorously inferred from genomic variation data, like other features of evolutionary history.

Studying models of balancing selection using phase-type theory

Balancing selection (BLS) is the evolutionary force that maintains high levels of genetic variability in many important genes. To further our understanding of its evolutionary significance, we analyze models with BLS acting on a biallelic locus: an equilibrium model with long-term BLS, a model with long-term BLS and recent changes in population size, and a model of recent BLS. Using phase-type theory, a mathematical tool for analyzing continuous time Markov chains with an absorbing state, we examine how BLS affects polymorphism patterns in linked neutral regions, as summarized by nucleotide diversity, the expected number of segregating sites, the site frequency spectrum, and the level of linkage disequilibrium (LD). Long-term BLS affects polymorphism patterns in a relatively small genomic neighborhood, and such selection targets are easier to detect when the equilibrium frequencies of the selected variants are close to 50%, or when there has been a population size reduction. For a new mutation subject to BLS, its initial increase in frequency in the population causes linked neutral regions to have reduced diversity, an excess of both high and low frequency derived variants, and elevated LD with the selected locus. These patterns are similar to those produced by selective sweeps, but the effects of recent BLS are weaker. Nonetheless, compared to selective sweeps, nonequilibrium polymorphism and LD patterns persist for a much longer period under recent BLS, which may increase the chance of detecting such selection targets. An R package for analyzing these models, among others (e.g., isolation with migration), is available.

The Impact of Purifying and Background Selection on the Inference of Population History: Problems and Prospects

Current procedures for inferring population history generally assume complete neutrality—that is, they neglect both direct selection and the effects of selection on linked sites. We here examine how the presence of direct purifying selection and background selection may bias demographic inference by evaluating two commonly-used methods (MSMC and fastsimcoal2), specifically studying how the underlying shape of the distribution of fitness effects and the fraction of directly selected sites interact with demographic parameter estimation. The results show that, even after masking functional genomic regions, background selection may cause the mis-inference of population growth under models of both constant population size and decline. This effect is amplified as the strength of purifying selection and the density of directly selected sites increases, as indicated by the distortion of the site frequency spectrum and levels of nucleotide diversity at linked neutral sites. We also show how simulated changes in background selection effects caused by population size changes can be predicted analytically. We propose a potential method for correcting for the mis-inference of population growth caused by selection. By treating the distribution of fitness effect as a nuisance parameter and averaging across all potential realizations, we demonstrate that even directly selected sites can be used to infer demographic histories with reasonable accuracy.

A Computational Approach for Modeling the Allele Frequency Spectrum of Populations with Arbitrarily Varying Size

The allele frequency spectrum (AFS), or site frequency spectrum, is commonly used to summarize the genomic polymorphism pattern of a sample, which is informative for inferring population history and detecting natural selection. In 2013, Chen and Chen developed a method for analytically deriving the AFS for populations with temporally varying size through the coalescence time-scaling function. However, their approach is only applicable to population history scenarios in which the analytical form of the time-scaling function is tractable. In this paper, we propose a computational approach to extend the method to populations with arbitrary complex varying size by numerically approximating the time-scaling function. We demonstrate the performance of the approach by constructing the AFS for two population history scenarios: the logistic growth model and the Gompertz growth model, for which the AFS are unavailable with existing approaches. Software for implementing the algorithm can be downloaded at http://chenlab.big.ac.cn/software/.

Methods for Estimating Demography and Detecting Between-Locus Differences in the Effective Population Size and Mutation Rate

It is known that the effective population size (N_e) and the mutation rate (u) vary across the genome. Here, we show that ignoring this heterogeneity may lead to biased estimates of past demography. To solve the problem, we develop new methods for jointly inferring past changes in population size and detecting variation in N_e and u between loci. These methods rely on either polymorphism data alone or both polymorphism and divergence data. In addition to inferring demography, we can use the methods to study a variety of questions: 1) comparing sex chromosomes with autosomes (for finding evidence for male-driven evolution, an unequal sex ratio, or sex-biased demographic changes) and 2) analyzing multilocus data from within autosomes or sex chromosomes (for studying determinants of variability in N_e and u). Simulations suggest that the methods can provide accurate parameter estimates and have substantial statistical power for detecting difference in N_e and u. As an example, we use the methods to analyze a polymorphism data set from Drosophila simulans. We find clear evidence for rapid population expansion. The results also indicate that the autosomes have a higher mutation rate than the X chromosome and that the sex ratio is probably female-biased. The new methods have been implemented in a user-friendly package.

Mathematical modelling reveals unexpected inheritance and variability patterns of cell cycle parameters in mammalian cells

The cell cycle is the fundamental process of cell populations, it is regulated by environmental cues and by intracellular checkpoints. Cell cycle variability in clonal cell population is caused by stochastic processes such as random partitioning of cellular components to progeny cells at division and random interactions among biomolecules in cells. One of the important biological questions is how the dynamics at the cell cycle scale, which is related to family dependencies between the cell and its descendants, affects cell population behavior in the long-run. We address this question using a “mechanistic” model, built based on observations of single cells over several cell generations, and then extrapolated in time. We used cell pedigree observations of NIH 3T3 cells including FUCCI markers, to determine patterns of inheritance of cell-cycle phase durations and single-cell protein dynamics. Based on that information we developed a hybrid mathematical model, involving bifurcating autoregression to describe stochasticity of partitioning and inheritance of cell-cycle-phase times, and an ordinary differential equation system to capture single-cell protein dynamics. Long-term simulations, concordant with in vitro experiments, demonstrated the model reproduced the main features of our data and had homeostatic properties. Moreover, heterogeneity of cell cycle may have important consequences during population development. We discovered an effect similar to genetic drift, amplified by family relationships among cells. In consequence, the progeny of a single cell with a short cell cycle time had a high probability of eventually dominating the population, due to the heritability of cell-cycle phases. Patterns of epigenetic heritability in proliferating cells are important for understanding long-term trends of cell populations which are either required to provide the influx of maturing cells (such as hematopoietic stem cells) or which started proliferating uncontrollably (such as cancer cells).

All cells in multicellular organisms obey orchestrated sequences of signals to ensure developmental and homeostatic fitness under a variety of external stimuli. However, there also exist self-perpetuating stem-cell populations, the function of which is to provide a steady supply of differentiated progenitors that in turn ensure persistence of organism functions. This “cell production engine” is an important element of biological homeostasis. A similar process, albeit distorted in many respects, plays a major role in cancer development; here the robustness of homeostasis contributes to difficulty in eradication of malignancy. An important role in homeostasis seems to be played by generation of heterogeneity among cell phenotypes, which then can be shaped by selection and other genetic forces. In the present paper, we present a model of a cultured cell population, which factors in relationships among related cells and the dynamics of cell growth and important proteins regulating cell division. We find that the model not only maintains homeostasis, but that it also responds to perturbations in a manner that is similar to that exhibited by the Wright-Fisher model of population genetics. The model-cell population can become dominated by the progeny of the fittest individuals, without invoking advantageous mutations. If confirmed, this may provide an alternative mode of evolution of cell populations.

Using Genomic Data to Infer Historic Population Dynamics of Nonmodel Organisms

Genome sequence data are now being routinely obtained from many nonmodel organisms. These data contain a wealth of information about the demographic history of the populations from which they originate. Many sophisticated statistical inference procedures have been developed to infer the demographic history of populations from this type of genomic data. In this review, we discuss the different statistical methods available for inference of demography, providing an overview of the underlying theory and logic behind each approach. We also discuss the types of data required and the pros and cons of each method. We then discuss how these methods have been applied to a variety of nonmodel organisms. We conclude by presenting some recommendations for researchers looking to use genomic data to infer demographic history.

The Wright-Fisher site frequency spectrum as a perturbation of the coalescent's

The first terms of the Wright-Fisher (WF) site frequency spectrum that follow the coalescent approximation are determined precisely, with a view to understanding the accuracy of the coalescent approximation for large samples. The perturbing terms show that the probability of a single mutant in the sample (singleton probability) is elevated in WF but the rest of the frequency spectrum is lowered. A part of the perturbation can be attributed to a mismatch in rates of merger between WF and the coalescent. The rest of it can be attributed to the difference in the way WF and the coalescent partition children between parents. In particular, the number of children of a parent is approximately Poisson under WF and approximately geometric under the coalescent. Whereas the mismatch in rates raises the probability of singletons under WF, its offspring distribution being approximately Poisson lowers it. The two effects are of opposite sense everywhere except at the tail of the frequency spectrum. The WF frequency spectrum begins to depart from that of the coalescent only for sample sizes that are comparable to the population size. These conclusions are confirmed by a separate analysis that assumes the sample size n to be equal to the population size N. Partly thanks to the canceling effects, the total variation distance of WF minus coalescent is 0.12∕logN for a population sized sample with n=N, which is only 1% for N=2×10⁴. The coalescent remains a good approximation for the site frequency spectrum of-large samples.

Geometry of the Sample Frequency Spectrum and the Perils of Demographic Inference

The sample frequency spectrum (SFS), which describes the distribution of mutant alleles in a sample of DNA sequences, is a widely used summary statistic in population genetics. The expected SFS has a strong dependence on the historical population demography and this property is exploited by popular statistical methods to infer complex demographic histories from DNA sequence data. Most, if not all, of these inference methods exhibit pathological behavior, however. Specifically, they often display runaway behavior in optimization, where the inferred population sizes and epoch durations can degenerate to zero or diverge to infinity, and show undesirable sensitivity to perturbations in the data. The goal of this article is to provide theoretical insights into why such problems arise. To this end, we characterize the geometry of the expected SFS for piecewise-constant demographies and use our results to show that the aforementioned pathological behavior of popular inference methods is intrinsic to the geometry of the expected SFS. We provide explicit descriptions and visualizations for a toy model, and generalize our intuition to arbitrary sample sizes using tools from convex and algebraic geometry. We also develop a universal characterization result which shows that the expected SFS of a sample of size n under an arbitrary population history can be recapitulated by a piecewise-constant demography with only [Formula: see text] epochs, where [Formula: see text] is between [Formula: see text] and [Formula: see text] The set of expected SFS for piecewise-constant demographies with fewer than [Formula: see text] epochs is open and nonconvex, which causes the above phenomena for inference from data.

An efficient algorithm for generating the internal branches of a Kingman coalescent

Coalescent simulations are a widely used approach for simulating sample genealogies, but can become computationally burdensome in large samples. Methods exist to analytically calculate a sample's expected frequency spectrum without simulating full genealogies. However, statistics that rely on the distribution of the length of internal coalescent branches, such as the probability that two mutations of equal size arose on the same genealogical branch, have previously required full coalescent simulations to estimate. Here, we present a sampling method capable of efficiently generating limited portions of sample genealogies using a series of analytic equations that give probabilities for the number, start, and end of internal branches conditional on the number of final samples they subtend. These equations are independent of the coalescent waiting times and need only be calculated a single time, lending themselves to efficient computation. We compare our method with full coalescent simulations to show the resulting distribution of branch lengths and summary statistics are equivalent, but that for many conditions our method is at least 10 times faster.

Non-parametric estimation of population size changes from the site frequency spectrum

Changes in population size is a useful quantity for understanding the evolutionary history of a species. Genetic variation within a species can be summarized by the site frequency spectrum (SFS). For a sample of size n, the SFS is a vector of length n - 1 where entry i is the number of sites where the mutant base appears i times and the ancestral base appears n - i times. We present a new method, CubSFS, for estimating the changes in population size of a panmictic population from an observed SFS. First, we provide a straightforward proof for the expression of the expected site frequency spectrum depending only on the population size. Our derivation is based on an eigenvalue decomposition of the instantaneous coalescent rate matrix. Second, we solve the inverse problem of determining the changes in population size from an observed SFS. Our solution is based on a cubic spline for the population size. The cubic spline is determined by minimizing the weighted average of two terms, namely (i) the goodness of fit to the observed SFS, and (ii) a penalty term based on the smoothness of the changes. The weight is determined by cross-validation. The new method is validated on simulated demographic histories and applied on unfolded and folded SFS from 26 different human populations from the 1000 Genomes Project.

Single and simultaneous binary mergers in Wright-Fisher genealogies

The Kingman coalescent is a commonly used model in genetics, which is often justified with reference to the Wright-Fisher (WF) model. Current proofs of convergence of WF and other models to the Kingman coalescent assume a constant sample size. However, sample sizes have become quite large in human genetics. Therefore, we develop a convergence theory that allows the sample size to increase with population size. If the haploid population size is N and the sample size is N^1∕3-ϵ, ϵ>0, we prove that Wright-Fisher genealogies involve at most a single binary merger in each generation with probability converging to 1 in the limit of large N. Single binary merger or no merger in each generation of the genealogy implies that the Kingman partition distribution is obtained exactly. If the sample size is N^1∕2-ϵ, Wright-Fisher genealogies may involve simultaneous binary mergers in a single generation but do not involve triple mergers in the large N limit. The asymptotic theory is verified using numerical calculations. Variable population sizes are handled algorithmically. It is found that even distant bottlenecks can increase the probability of triple mergers as well as simultaneous binary mergers in WF genealogies.

Coalescent Processes with Skewed Offspring Distributions and Nonequilibrium Demography

Nonequilibrium demography impacts coalescent genealogies leaving detectable, well-studied signatures of variation. However, similar genomic footprints are also expected under models of large reproductive skew, posing a serious problem when trying to make inference. Furthermore, current approaches consider only one of the two processes at a time, neglecting any genomic signal that could arise from their simultaneous effects, preventing the possibility of jointly inferring parameters relating to both offspring distribution and population history. Here, we develop an extended Moran model with exponential population growth, and demonstrate that the underlying ancestral process converges to a time-inhomogeneous psi-coalescent. However, by applying a nonlinear change of time scale-analogous to the Kingman coalescent-we find that the ancestral process can be rescaled to its time-homogeneous analog, allowing the process to be simulated quickly and efficiently. Furthermore, we derive analytical expressions for the expected site-frequency spectrum under the time-inhomogeneous psi-coalescent, and develop an approximate-likelihood framework for the joint estimation of the coalescent and growth parameters. By means of extensive simulation, we demonstrate that both can be estimated accurately from whole-genome data. In addition, not accounting for demography can lead to serious biases in the inferred coalescent model, with broad implications for genomic studies ranging from ecology to conservation biology. Finally, we use our method to analyze sequence data from Japanese sardine populations, and find evidence of high variation in individual reproductive success, but few signs of a recent demographic expansion.

Comparison of Single Genome and Allele Frequency Data Reveals Discordant Demographic Histories

Inference of demographic history from genetic data is a primary goal of population genetics of model and nonmodel organisms. Whole genome-based approaches such as the pairwise/multiple sequentially Markovian coalescent methods use genomic data from one to four individuals to infer the demographic history of an entire population, while site frequency spectrum (SFS)-based methods use the distribution of allele frequencies in a sample to reconstruct the same historical events. Although both methods are extensively used in empirical studies and perform well on data simulated under simple models, there have been only limited comparisons of them in more complex and realistic settings. Here we use published demographic models based on data from three human populations (Yoruba, descendants of northwest-Europeans, and Han Chinese) as an empirical test case to study the behavior of both inference procedures. We find that several of the demographic histories inferred by the whole genome-based methods do not predict the genome-wide distribution of heterozygosity, nor do they predict the empirical SFS. However, using simulated data, we also find that the whole genome methods can reconstruct the complex demographic models inferred by SFS-based methods, suggesting that the discordant patterns of genetic variation are not attributable to a lack of statistical power, but may reflect unmodeled complexities in the underlying demography. More generally, our findings indicate that demographic inference from a small number of genomes, routine in genomic studies of nonmodel organisms, should be interpreted cautiously, as these models cannot recapitulate other summaries of the data.

Forward and backward evolutionary processes and allele frequency spectrum in a cancer cell population

A cancer grows from a single cell, thereby constituting a large cell population. In this work, we are interested in how mutations accumulate in a cancer cell population. We provide a theoretical framework of the stochastic process in a cancer cell population and obtain near exact expressions of allele frequency spectrum or AFS (only continuous approximation is involved) from both forward and backward treatments under a simple setting; all cells undergo cell divisions and die at constant rates, b and d, respectively, such that the entire population grows exponentially. This setting means that once a parental cancer cell is established, in the following growth phase, all mutations are assumed to have no effect on b or d (i.e., neutral or passengers). Our theoretical results show that the difference from organismal population genetics is mainly in the coalescent time scale, and the mutation rate is defined per cell division, not per time unit (e.g., generation). Except for these two factors, the basic logic is very similar between organismal and cancer population genetics, indicating that a number of well established theories of organismal population genetics could be translated to cancer population genetics with simple modifications.

Accuracy of Demographic Inferences from the Site Frequency Spectrum: The Case of the Yoruba Population

Some methods for demographic inference based on the observed genetic diversity of current populations rely on the use of summary statistics such as the Site Frequency Spectrum (SFS). Demographic models can be either model-constrained with numerous parameters, such as growth rates, timing of demographic events, and migration rates, or model-flexible, with an unbounded collection of piecewise constant sizes. It is still debated whether demographic histories can be accurately inferred based on the SFS. Here, we illustrate this theoretical issue on an example of demographic inference for an African population. The SFS of the Yoruba population (data from the 1000 Genomes Project) is fit to a simple model of population growth described with a single parameter (e.g., founding time). We infer a time to the most recent common ancestor of 1.7 million years (MY) for this population. However, we show that the Yoruba SFS is not informative enough to discriminate between several different models of growth. We also show that for such simple demographies, the fit of one-parameter models outperforms the stairway plot, a recently developed model-flexible method. The use of this method on simulated data suggests that it is biased by the noise intrinsically present in the data.

Efficient computation of the joint sample frequency spectra for multiple populations

A wide range of studies in population genetics have employed the sample frequency spectrum (SFS), a summary statistic which describes the distribution of mutant alleles at a polymorphic site in a sample of DNA sequences and provides a highly efficient dimensional reduction of large-scale population genomic variation data. Recently, there has been much interest in analyzing the joint SFS data from multiple populations to infer parameters of complex demographic histories, including variable population sizes, population split times, migration rates, admixture proportions, and so on. SFS-based inference methods require accurate computation of the expected SFS under a given demographic model. Although much methodological progress has been made, existing methods suffer from numerical instability and high computational complexity when multiple populations are involved and the sample size is large. In this paper, we present new analytic formulas and algorithms that enable accurate, efficient computation of the expected joint SFS for thousands of individuals sampled from hundreds of populations related by a complex demographic model with arbitrary population size histories (including piecewise-exponential growth). Our results are implemented in a new software package called momi (MOran Models for Inference). Through an empirical study we demonstrate our improvements to numerical stability and computational complexity.

Coalescence computations for large samples drawn from populations of time-varying sizes

We present new results concerning probability distributions of times in the coalescence tree and expected allele frequencies for coalescent with large sample size. The obtained results are based on computational methodologies, which involve combining coalescence time scale changes with techniques of integral transformations and using analytical formulae for infinite products. We show applications of the proposed methodologies for computing probability distributions of times in the coalescence tree and their limits, for evaluation of accuracy of approximate expressions for times in the coalescence tree and expected allele frequencies, and for analysis of large human mitochondrial DNA dataset.

Robust and scalable inference of population history from hundreds of unphased whole genomes

It has recently been demonstrated that inference methods based on genealogical processes with recombination can uncover past population history in unprecedented detail. However, these methods scale poorly with sample size, limiting resolution in the recent past, and they require phased genomes, which contain switch errors that can catastrophically distort the inferred history. Here we present SMC++, a new statistical tool capable of analyzing orders of magnitude more samples than existing methods while requiring only unphased genomes (its results are independent of phasing). SMC++ can jointly infer population size histories and split times in diverged populations, and it employs a novel spline regularization scheme that greatly reduces estimation error. We apply SMC++ to analyze sequence data from over a thousand human genomes in Africa and Eurasia, hundreds of genomes from a Drosophila melanogaster population in Africa, and tens of genomes from zebra finch and long-tailed finch populations in Australia.

Tumor evolution: Linear, branching, neutral or punctuated?

Intratumor heterogeneity has been widely reported in human cancers, but our knowledge of how this genetic diversity emerges over time remains limited. A central challenge in studying tumor evolution is the difficulty in collecting longitudinal samples from cancer patients. Consequently, most studies have inferred tumor evolution from single time-point samples, providing very indirect information. These data have led to several competing models of tumor evolution: linear, branching, neutral and punctuated. Each model makes different assumptions regarding the timing of mutations and selection of clones, and therefore has different implications for the diagnosis and therapeutic treatment of cancer patients. Furthermore, emerging evidence suggests that models may change during tumor progression or operate concurrently for different classes of mutations. Finally, we discuss data that supports the theory that most human tumors evolve from a single cell in the normal tissue. This article is part of a Special Issue entitled: Evolutionary principles - heterogeneity in cancer?, edited by Dr. Robert A. Gatenby.

Explosive genetic evidence for explosive human population growth

The advent of next-generation sequencing technology has allowed the collection of vast amounts of genetic variation data. A recurring discovery from studying larger and larger samples of individuals had been the extreme, previously unexpected, excess of very rare genetic variants, which has been shown to be mostly due to the recent explosive growth of human populations. Here, we review recent literature that inferred recent changes in population size in different human populations and with different methodologies, with many pointing to recent explosive growth, especially in European populations for which more data has been available. We also review the state-of-the-art methods and software for the inference of historical population size changes that lead to these discoveries. Finally, we discuss the implications of recent population growth on personalized genomics, on purifying selection in the non-equilibrium state it entails and, as a consequence, on the genetic architecture underlying complex disease and the performance of mapping methods in discovering rare variants that contribute to complex disease risk.

The Site Frequency Spectrum for General Coalescents

General genealogical processes such as Λ- and Ξ-coalescents, which respectively model multiple and simultaneous mergers, have important applications in studying marine species, strong positive selection, recurrent selective sweeps, strong bottlenecks, large sample sizes, and so on. Recently, there has been significant progress in developing useful inference tools for such general models. In particular, inference methods based on the site frequency spectrum (SFS) have received noticeable attention. Here, we derive a new formula for the expected SFS for general Λ- and Ξ-coalescents, which leads to an efficient algorithm. For time-homogeneous coalescents, the runtime of our algorithm for computing the expected SFS is O(n2) where n is the sample size. This is a factor of[Formula: see text]faster than the state-of-the-art method. Furthermore, in contrast to existing methods, our method generalizes to time-inhomogeneous Λ- and Ξ-coalescents with measures that factorize as[Formula: see text] and [Formula: see text]respectively, where ζ denotes a strictly positive function of time. The runtime of our algorithm in this setting is[Formula: see text]We also obtain general theoretical results for the identifiability of the Λ measure when ζ is a constant function, as well as for the identifiability of the function ζ under a fixed Ξ measure.

Genetic drift in populations governed by a Galton-Watson branching process

Most population genetics studies have their origins in a Wright-Fisher or some closely related fixed-population model in which each individual randomly chooses its ancestor. Populations which vary in size with time are typically modelled via a coalescent derived from Wright-Fisher, but use a nonlinear time-scaling driven by a deterministically imposed population growth. An alternate, arguably more realistic approach, and one which we take here, is to allow the population size to vary stochastically via a Galton-Watson branching process. We study genetic drift in a population consisting of a number of distinct allele types in which each allele type evolves as an independent Galton-Watson branching process. We find the dynamics of the population is determined by a single parameter κ0=(2m0/σ(2))logλ, where m0 is the initial population size, λ is the mean number of offspring per individual; and σ(2) is the variance of the number of offspring. For 0≲κ0≪1, the dynamics are close to those of Wright-Fisher, with the added property that the population is prone to extinction. For κ0≫1 allele frequencies and ancestral lineages are stable and individual alleles do not fix throughout the population. The existence of a rapid changeover regime at κ0≈1 enables estimates to be made, together with confidence intervals, of the time and population size of the era of mitochondrial Eve.

Inference of Super-exponential Human Population Growth via Efficient Computation of the Site Frequency Spectrum for Generalized Models

The site frequency spectrum (SFS) and other genetic summary statistics are at the heart of many population genetic studies. Previous studies have shown that human populations have undergone a recent epoch of fast growth in effective population size. These studies assumed that growth is exponential, and the ensuing models leave an excess amount of extremely rare variants. This suggests that human populations might have experienced a recent growth with speed faster than exponential. Recent studies have introduced a generalized growth model where the growth speed can be faster or slower than exponential. However, only simulation approaches were available for obtaining summary statistics under such generalized models. In this study, we provide expressions to accurately and efficiently evaluate the SFS and other summary statistics under generalized models, which we further implement in a publicly available software. Investigating the power to infer deviation of growth from being exponential, we observed that adequate sample sizes facilitate accurate inference; e.g., a sample of 3000 individuals with the amount of data expected from exome sequencing allows observing and accurately estimating growth with speed deviating by ≥10% from that of exponential. Applying our inference framework to data from the NHLBI Exome Sequencing Project, we found that a model with a generalized growth epoch fits the observed SFS significantly better than the equivalent model with exponential growth (P-value =3.85×10−6). The estimated growth speed significantly deviates from exponential (P-value ≪10−12), with the best-fit estimate being of growth speed 12% faster than exponential.

Inferring Bottlenecks from Genome-Wide Samples of Short Sequence Blocks

The advent of the genomic era has necessitated the development of methods capable of analyzing large volumes of genomic data efficiently. Being able to reliably identify bottlenecks—extreme population size changes of short duration—not only is interesting in the context of speciation and extinction but also matters (as a null model) when inferring selection. Bottlenecks can be detected in polymorphism data via their distorting effect on the shape of the underlying genealogy. Here, we use the generating function of genealogies to derive the probability of mutational configurations in short sequence blocks under a simple bottleneck model. Given a large number of nonrecombining blocks, we can compute maximum-likelihood estimates of the time and strength of the bottleneck. Our method relies on a simple summary of the joint distribution of polymorphic sites. We extend the site frequency spectrum by counting mutations in frequency classes in short sequence blocks. Using linkage information over short distances in this way gives greater power to detect bottlenecks than the site frequency spectrum and potentially opens up a wide range of demographic histories to blockwise inference. Finally, we apply our method to genomic data from a species of pig (Sus cebifrons) endemic to islands in the center and west of the Philippines to estimate whether a bottleneck occurred upon island colonization and compare our scheme to Li and Durbin’s pairwise sequentially Markovian coalescent (PSMC) both for the pig data and using simulations.

Population genetic studies in the genomic sequencing era

Recent advances in high-throughput sequencing technologies have revolutionized the field of population genetics. Data now routinely contain genomic level polymorphism information, and the low cost of DNA sequencing enables researchers to investigate tens of thousands of subjects at a time. This provides an unprecedented opportunity to address fundamental evolutionary questions, while posing challenges on traditional population genetic theories and methods. This review provides an overview of the recent methodological developments in the field of population genetics, specifically methods used to infer ancient population history and investigate natural selection using large-sample, large-scale genetic data. Several open questions are also discussed at the end of the review.

Inferring Very Recent Population Growth Rate from Population-Scale Sequencing Data: Using a Large-Sample Coalescent Estimator

Large-sample or population-level sequencing data provide unprecedented opportunities for inferring detailed population histories, especially recent demographic histories. On the other hand, it challenges most existing population genetic methods: Simulation-based approaches require intensive computation, and analytical approaches are often numerically intractable when the sample size is large. We propose a computationally efficient method for simultaneous estimation of population size, the rate, and onset time of population growth in the very recent history, using the pattern of the total number of segregating sites as a function of sample size. Coalescent simulation shows that it can accurately and efficiently estimate the parameters of recent population growth from large-scale data. This approach has the flexibility to model population history with multiple growth stages or other epochs, and it is robust when the sample size is very large or at the population scale, for which the Kingman's coalescent assumption is not valid. This approach is applied to recently published data and estimates the recent population growth rate in the European population to be 1.49% with the onset time 7.26 ka, and the rate in the African population to be 0.735% with the onset time 10.01 ka.

Fundamental limits on the accuracy of demographic inference based on the sample frequency spectrum

The sample frequency spectrum (SFS) of DNA sequences from a collection of individuals is a summary statistic that is commonly used for parametric inference in population genetics. Despite the popularity of SFS-based inference methods, little is currently known about the information theoretic limit on the estimation accuracy as a function of sample size. Here, we show that using the SFS to estimate the size history of a population has a minimax error of at least O(1/log s), where s is the number of independent segregating sites used in the analysis. This rate is exponentially worse than known convergence rates for many classical estimation problems in statistics. Another surprising aspect of our theoretical bound is that it does not depend on the dimension of the SFS, which is related to the number of sampled individuals. This means that, for a fixed number s of segregating sites considered, using more individuals does not help to reduce the minimax error bound. Our result pertains to populations that have experienced a bottleneck, and we argue that it can be expected to apply to many populations in nature.

Leveraging ancestry to improve causal variant identification in exome sequencing for monogenic disorders

Recent breakthroughs in exome-sequencing technology have made possible the identification of many causal variants of monogenic disorders. Although extremely powerful when closely related individuals (eg, child and parents) are simultaneously sequenced, sequencing of a single case is often unsuccessful due to the large number of variants that need to be followed up for functional validation. Many approaches filter out common variants above a given frequency threshold (eg, 1%), and then prioritize the remaining variants according to their functional, structural and conservation properties. Here we present methods that leverage the genetic structure across different populations to improve filtering performance while accounting for the finite sample size of the reference panels. We show that leveraging genetic structure reduces the number of variants that need to be followed up by 16% in simulations and by up to 38% in empirical data of 20 exomes from individuals with monogenic disorders for which the causal variants are known.

Exploring population size changes using SNP frequency spectra

Inferring demographic history is an important task in population genetics. Many existing inference methods are based on pre-defined simplified population models, which are more suitable for hypothesis testing than for exploratory analysis. We developed a novel model-flexible method called stairway plot, which infers population size changes over time using SNP frequency spectra. This method is applicable for whole-genome sequences of hundreds of individuals. Using extensive simulation we demonstrated the usefulness of the method for inferring demographic history, especially recent population size changes. The method was applied to the whole genome sequence data of nine populations from the 1000 Genomes Project, and showed a pattern of human population fluctuations from 10 to 200 thousand years ago.

Can the site-frequency spectrum distinguish exponential population growth from multiple-merger coalescents?

The ability of the site-frequency spectrum (SFS) to reflect the particularities of gene genealogies exhibiting multiple mergers of ancestral lines as opposed to those obtained in the presence of population growth is our focus. An excess of singletons is a well-known characteristic of both population growth and multiple mergers. Other aspects of the SFS, in particular, the weight of the right tail, are, however, affected in specific ways by the two model classes. Using an approximate likelihood method and minimum-distance statistics, our estimates of statistical power indicate that exponential and algebraic growth can indeed be distinguished from multiple-merger coalescents, even for moderate sample sizes, if the number of segregating sites is high enough. A normalized version of the SFS (nSFS) is also used as a summary statistic in an approximate Bayesian computation (ABC) approach. The results give further positive evidence as to the general eligibility of the SFS to distinguish between the different histories.

Efficient inference of population size histories and locus-specific mutation rates from large-sample genomic variation data

With the recent increase in study sample sizes in human genetics, there has been growing interest in inferring historical population demography from genomic variation data. Here, we present an efficient inference method that can scale up to very large samples, with tens or hundreds of thousands of individuals. Specifically, by utilizing analytic results on the expected frequency spectrum under the coalescent and by leveraging the technique of automatic differentiation, which allows us to compute gradients exactly, we develop a very efficient algorithm to infer piecewise-exponential models of the historical effective population size from the distribution of sample allele frequencies. Our method is orders of magnitude faster than previous demographic inference methods based on the frequency spectrum. In addition to inferring demography, our method can also accurately estimate locus-specific mutation rates. We perform extensive validation of our method on simulated data and show that it can accurately infer multiple recent epochs of rapid exponential growth, a signal that is difficult to pick up with small sample sizes. Lastly, we use our method to analyze data from recent sequencing studies, including a large-sample exome-sequencing data set of tens of thousands of individuals assayed at a few hundred genic regions.

DESCARTES' RULE OF SIGNS AND THE IDENTIFIABILITY OF POPULATION DEMOGRAPHIC MODELS FROM GENOMIC VARIATION DATA

The sample frequency spectrum (SFS) is a widely-used summary statistic of genomic variation in a sample of homologous DNA sequences. It provides a highly efficient dimensional reduction of large-scale population genomic data and its mathematical dependence on the underlying population demography is well understood, thus enabling the development of efficient inference algorithms. However, it has been recently shown that very different population demographies can actually generate the same SFS for arbitrarily large sample sizes. Although in principle this nonidentifiability issue poses a thorny challenge to statistical inference, the population size functions involved in the counterexamples are arguably not so biologically realistic. Here, we revisit this problem and examine the identifiability of demographic models under the restriction that the population sizes are piecewise-defined where each piece belongs to some family of biologically-motivated functions. Under this assumption, we prove that the expected SFS of a sample uniquely determines the underlying demographic model, provided that the sample is sufficiently large. We obtain a general bound on the sample size sufficient for identifiability; the bound depends on the number of pieces in the demographic model and also on the type of population size function in each piece. In the cases of piecewise-constant, piecewise-exponential and piecewise-generalized-exponential models, which are often assumed in population genomic inferences, we provide explicit formulas for the bounds as simple functions of the number of pieces. Lastly, we obtain analogous results for the "folded" SFS, which is often used when there is ambiguity as to which allelic type is ancestral. Our results are proved using a generalization of Descartes' rule of signs for polynomials to the Laplace transform of piecewise continuous functions.

Distortion of genealogical properties when the sample is very large

Study sample sizes in human genetics are growing rapidly, and in due course it will become routine to analyze samples with hundreds of thousands, if not millions, of individuals. In addition to posing computational challenges, such large sample sizes call for carefully reexamining the theoretical foundation underlying commonly used analytical tools. Here, we study the accuracy of the coalescent, a central model for studying the ancestry of a sample of individuals. The coalescent arises as a limit of a large class of random mating models, and it is an accurate approximation to the original model provided that the population size is sufficiently larger than the sample size. We develop a method for performing exact computation in the discrete-time Wright-Fisher (DTWF) model and compare several key genealogical quantities of interest with the coalescent predictions. For recently inferred demographic scenarios, we find that there are a significant number of multiple- and simultaneous-merger events under the DTWF model, which are absent in the coalescent by construction. Furthermore, for large sample sizes, there are noticeable differences in the expected number of rare variants between the coalescent and the DTWF model. To balance the trade-off between accuracy and computational efficiency, we propose a hybrid algorithm that uses the DTWF model for the recent past and the coalescent for the more distant past. Our results demonstrate that the hybrid method with only a handful of generations of the DTWF model leads to a frequency spectrum that is quite close to the prediction of the full DTWF model.

General triallelic frequency spectrum under demographic models with variable population size

It is becoming routine to obtain data sets on DNA sequence variation across several thousands of chromosomes, providing unprecedented opportunity to infer the underlying biological and demographic forces. Such data make it vital to study summary statistics that offer enough compression to be tractable, while preserving a great deal of information. One well-studied summary is the site frequency spectrum-the empirical distribution, across segregating sites, of the sample frequency of the derived allele. However, most previous theoretical work has assumed that each site has experienced at most one mutation event in its genealogical history, which becomes less tenable for very large sample sizes. In this work we obtain, in closed form, the predicted frequency spectrum of a site that has experienced at most two mutation events, under very general assumptions about the distribution of branch lengths in the underlying coalescent tree. Among other applications, we obtain the frequency spectrum of a triallelic site in a model of historically varying population size. We demonstrate the utility of our formulas in two settings: First, we show that triallelic sites are more sensitive to the parameters of a population that has experienced historical growth, suggesting that they will have use if they can be incorporated into demographic inference. Second, we investigate a recently proposed alternative mechanism of mutation in which the two derived alleles of a triallelic site are created simultaneously within a single individual, and we develop a test to determine whether it is responsible for the excess of triallelic sites in the human genome.

Factors influencing ascertainment bias of microsatellite allele sizes: impact on estimates of mutation rates

Microsatellite loci play an important role as markers for identification, disease gene mapping, and evolutionary studies. Mutation rate, which is of fundamental importance, can be obtained from interspecies comparisons, which, however, are subject to ascertainment bias. This bias arises, for example, when a locus is selected on the basis of its large allele size in one species (cognate species 1), in which it is first discovered. This bias is reflected in average allele length in any noncognate species 2 being smaller than that in species 1. This phenomenon was observed in various pairs of species, including comparisons of allele sizes in human and chimpanzee. Various mechanisms were proposed to explain observed differences in mean allele lengths between two species. Here, we examine the framework of a single-step asymmetric and unrestricted stepwise mutation model with genetic drift. Analysis is based on coalescent theory. Analytical results are confirmed by simulations using the simuPOP software. The mechanism of ascertainment bias in this model is a tighter correlation of allele sizes within a cognate species 1 than of allele sizes in two different species 1 and 2. We present computations of the expected average allele size difference, given the mutation rate, population sizes of species 1 and 2, time of separation of species 1 and 2, and the age of the allele. We show that when the past demographic histories of the cognate and noncognate taxa are different, the rate and directionality of mutations affect the allele sizes in the two taxa differently from the simple effect of ascertainment bias. This effect may exaggerate or reverse the effect of difference in mutation rates. We reanalyze literature data, which indicate that despite the bias, the microsatellite mutation rate estimate in the ancestral population is consistently greater than that in either human or chimpanzee and the mutation rate estimate in human exceeds or equals that in chimpanzee with the rate of allele length expansion in human being greater than that in chimpanzee. We also demonstrate that population bottlenecks and expansions in the recent human history have little impact on our conclusions.

SNP ascertainment bias in population genetic analyses: why it is important, and how to correct it

Whole genome sequencing and SNP genotyping arrays can paint strikingly different pictures of demographic history and natural selection. This is because genotyping arrays contain biased sets of pre-ascertained SNPs. In this short review, we use comparisons between high-coverage whole genome sequences of African hunter-gatherers and data from genotyping arrays to highlight how SNP ascertainment bias distorts population genetic inferences. Sample sizes and the populations in which SNPs are discovered affect the characteristics of observed variants. We find that SNPs on genotyping arrays tend to be older and present in multiple populations. In addition, genotyping arrays cause allele frequency distributions to be shifted towards intermediate frequency alleles, and estimates of linkage disequilibrium are modified. Since population genetic analyses depend on allele frequencies, it is imperative that researchers are aware of the effects of SNP ascertainment bias. With this in mind, we describe multiple ways to correct for SNP ascertainment bias.