Problem In Population Genetics Research Articles

Multiple partition structures and harmonic functions on branching graphs

We introduce and study multiple partition structures which are sequences of probability measures on families of Young diagrams subjected to a consistency condition. The multiple partition structures are generalizations of Kingman's partition structures, and are motivated by a problem of population genetics. They are related to harmonic functions and coherent systems of probability measures on a certain branching graph. The vertices of this graph are multiple Young diagrams (or multiple partitions), and the edges depend on the Jack parameter. If the value of the Jack parameter is equal to one the branching graph under considerations reflects the branching rule for the irreducible representations of the wreath product of a finite group with the symmetric group. If the value of the Jack parameter is zero then the coherent systems of probability measures are precisely the multiple partition structures. We establish a bijective correspondence between the set of harmonic functions on the graph and probability measures on the generalized Thoma set. The correspondence is determined by a canonical integral representation of harmonic functions. As a consequence we obtain a representation theorem for multiple partition structures.We give an example of a multiple partition structure which is expected to be relevant for a model of population genetics for the genetic variation of a sample of gametes from a large population. Namely, we construct a probability measure on the wreath product of a finite group with the symmetric group. If the finite group contains the identity element only then it coincides with the well-known Ewens probability measure on the symmetric group. The constructed probability measure defines a multiple partition structure which is a generalization of the Ewens partition structure studied by Kingman. We show that this multiple partition structure can be represented in terms of a multiple analogue of the Poisson-Dirichlet distribution called the multiple Poisson-Dirichlet distribution in the paper.

Timesweeper: accurately identifying selective sweeps using population genomic time series.

Despite decades of research, identifying selective sweeps, the genomic footprints of positive selection, remains a core problem in population genetics. Of the myriad methods that have been developed to tackle this task, few are designed to leverage the potential of genomic time-series data. This is because in most population genetic studies of natural populations, only a single period of time can be sampled. Recent advancements in sequencing technology, including improvements in extracting and sequencing ancient DNA, have made repeated samplings of a population possible, allowing for more direct analysis of recent evolutionary dynamics. Serial sampling of organisms with shorter generation times has also become more feasible due to improvements in the cost and throughput of sequencing. With these advances in mind, here we present Timesweeper, a fast and accurate convolutional neural network-based tool for identifying selective sweeps in data consisting of multiple genomic samplings of a population over time. Timesweeper analyzes population genomic time-series data by first simulating training data under a demographic model appropriate for the data of interest, training a one-dimensional convolutional neural network on said simulations, and inferring which polymorphisms in this serialized data set were the direct target of a completed or ongoing selective sweep. We show that Timesweeper is accurate under multiple simulated demographic and sampling scenarios, identifies selected variants with high resolution, and estimates selection coefficients more accurately than existing methods. In sum, we show that more accurate inferences about natural selection are possible when genomic time-series data are available; such data will continue to proliferate in coming years due to both the sequencing of ancient samples and repeated samplings of extant populations with faster generation times, as well as experimentally evolved populations where time-series data are often generated. Methodological advances such as Timesweeper thus have the potential to help resolve the controversy over the role of positive selection in the genome. We provide Timesweeper as a Python package for use by the community.

Bayesian optimization for demographic inference.

Inference of demographic histories of species and populations is one of the central problems in population genetics. It is usually stated as an optimization problem: find a model's parameters that maximize a certain log-likelihood. This log-likelihood is often expensive to evaluate in terms of time and hardware resources, critically more so for larger population counts. Although genetic algorithm-based solution has proven efficient for demographic inference in the past, it struggles to deal with log-likelihoods in the setting of more than three populations. Different tools are therefore needed to handle such scenarios. We introduce a new optimization pipeline for demographic inference with time consuming log-likelihood evaluations. It is based on Bayesian optimization, a prominent technique for optimizing expensive black box functions. Comparing to the existing widely used genetic algorithm solution, we demonstrate new pipeline's superiority in the limited time budget setting with four and five populations, when using the log-likelihoods provided by the moments tool.

1LocusSim a mobile-friendly simulator for teaching population genetics.

Biology students often struggle with the fundamental concepts of evolutionary genetics, including genetic drift, mutation and selection. To address this problem, 1LocusSim was developed to simulate the interaction of different factors, such as population size, mutation, selection and dominance, to study their effect on allelic frequency during evolution. With 1LocusSim, students can compare theoretical results with simulation outputs and solve and analyze different problems of population genetics. The 1LocusSim web has a responsive design which means that it has been specifically designed to be used on smartphones. To demonstrate its use, I review the classical overdominance model of population genetics and highlight a characteristic that is often not explicitly stated. Specifically, it is emphasized that the equilibrium of the model does not depend on the homozygous selection coefficients but rather on the ratio of the selection coefficients. This is already clear from the classical formula but maybe not so much for students. Also it implies that the equilibrium can be expressed solely in terms of the dominance coefficient h. To verify this theoretical prediction, I utilize the simulator and calculate the equilibrium for the well-known case of sickle cell anemia. By utilizing this tool, students can learn at their own pace and convenience, anywhere and anytime. 1LocusSim if freely available at https://1LocusSim-biosdev.pythonanywhere.com/. Website implemented under the Bottle micro web-framework for Python, with all major browsers supported. Supplementary data are available at Bioinformatics Advances online.

Robust inference of population size histories from genomic sequencing data.

Unraveling the complex demographic histories of natural populations is a central problem in population genetics. Understanding past demographic events is of general anthropological interest, but is also an important step in establishing accurate null models when identifying adaptive or disease-associated genetic variation. An important class of tools for inferring past population size changes from genomic sequence data are Coalescent Hidden Markov Models (CHMMs). These models make efficient use of the linkage information in population genomic datasets by using the local genealogies relating sampled individuals as latent states that evolve along the chromosome in an HMM framework. Extending these models to large sample sizes is challenging, since the number of possible latent states increases rapidly.Here, we present our method CHIMP (CHMM History-Inference Maximum-Likelihood Procedure), a novel CHMM method for inferring the size history of a population. It can be applied to large samples (hundreds of haplotypes) and only requires unphased genomes as input. The two implementations of CHIMP that we present here use either the height of the genealogical tree (TMRCA) or the total branch length, respectively, as the latent variable at each position in the genome. The requisite transition and emission probabilities are obtained by numerically solving certain systems of differential equations derived from the ancestral process with recombination. The parameters of the population size history are subsequently inferred using an Expectation-Maximization algorithm. In addition, we implement a composite likelihood scheme to allow the method to scale to large sample sizes.We demonstrate the efficiency and accuracy of our method in a variety of benchmark tests using simulated data and present comparisons to other state-of-the-art methods. Specifically, our implementation using TMRCA as the latent variable shows comparable performance and provides accurate estimates of effective population sizes in intermediate and ancient times. Our method is agnostic to the phasing of the data, which makes it a promising alternative in scenarios where high quality data is not available, and has potential applications for pseudo-haploid data.

Sparse Moments of Univariate Step Functions and Allele Frequency Spectra

We study the univariate moment problem of piecewise-constant density functions on the interval [0,1] and its consequences for an inference problem in population genetics. We show that, up to closure, any collection of n moments is achieved by a step function with at most n − 1 breakpoints and that this bound is tight. We use this to show that any point in the n th coalescence manifold in population genetics can be attained by a piecewise constant population history with at most n − 2 changes. Both the moment cones and the coalescence manifold are projected spectrahedra and we describe the problem of finding a nearest point on them as a semidefinite program.

A Deep-Learning Approach for Inference of Selective Sweeps from the Ancestral Recombination Graph.

Detecting signals of selection from genomic data is a central problem in population genetics. Coupling the rich information in the ancestral recombination graph (ARG) with a powerful and scalable deep-learning framework, we developed a novel method to detect and quantify positive selection: Selection Inference using the Ancestral recombination graph (SIA). Built on a Long Short-Term Memory (LSTM) architecture, a particular type of a Recurrent Neural Network (RNN), SIA can be trained to explicitly infer a full range of selection coefficients, as well as the allele frequency trajectory and time of selection onset. We benchmarked SIA extensively on simulations under a European human demographic model, and found that it performs as well or better as some of the best available methods, including state-of-the-art machine-learning and ARG-based methods. In addition, we used SIA to estimate selection coefficients at several loci associated with human phenotypes of interest. SIA detected novel signals of selection particular to the European (CEU) population at the MC1R and ABCC11 loci. In addition, it recapitulated signals of selection at the LCT locus and several pigmentation-related genes. Finally, we reanalyzed polymorphism data of a collection of recently radiated southern capuchino seedeater taxa in the genus Sporophila to quantify the strength of selection and improved the power of our previous methods to detect partial soft sweeps. Overall, SIA uses deep learning to leverage the ARG and thereby provides new insight into how selective sweeps shape genomic diversity.

Pixy: Unbiased estimation of nucleotide diversity and divergence in the presence of missing data.

Population genetic analyses often use summary statistics to describe patterns of genetic variation and provide insight into evolutionary processes. Among the most fundamental of these summary statistics are π and dXY , which are used to describe genetic diversity within and between populations, respectively. Here, we address a widespread issue in π and dXY calculation: systematic bias generated by missing data of various types. Many popular methods for calculating π and dXY operate on data encoded in the variant call format (VCF), which condenses genetic data by omitting invariant sites. When calculating π and dXY using a VCF, it is often implicitly assumed that missing genotypes (including those at sites not represented in the VCF) are homozygous for the reference allele. Here, we show how this assumption can result in substantial downward bias in estimates of π and dXY that is directly proportional to the amount of missing data. We discuss the pervasive nature and importance of this problem in population genetics, and introduce a user-friendly UNIX command line utility, pixy, that solves this problem via an algorithm that generates unbiased estimates of π and dXY in the face of missing data. We compare pixy to existing methods using both simulated and empirical data, and show that pixy alone produces unbiased estimates of π and dXY regardless of the form or amount of missing data. In summary, our software solves a long-standing problem in applied population genetics and highlights the importance of properly accounting for missing data in population genetic analyses.

Limiting Genotype Frequencies of Y-Linked Genes with a Mutant Allele in a Two-Sex Population

A two-type two-sex branching process is considered to model the evolution of the number of carriers of an allele and its mutations of a Y-linked gene. The limiting growth rates of the different types of couples and males (depending on the allele, mutated or not, that they carry on) on the set of coexistence of both alleles and on the fixation set of the mutant allele are obtained. In addition, the limiting genotype of the Y-linked gene and the limiting sex frequencies on those sets are established. Finally, the main results have been illustrated with simulated studies contextualized in problems of population genetics.

Taming the diffusion approximation through a controlling-factor WKB method.

The diffusion approximation (DA) is widely used in the analysis of stochastic population dynamics, from population genetics to ecology and evolution. The DA is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied. This failure becomes severe in situations where the direction of selection switches sign. Here we employ the WKB (Wentzel-Kramers-Brillouin) large-deviations method, which requires only the logarithm of a given quantity to be smooth over its state space. Combining the WKB scheme with asymptotic matching techniques, we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations, applicable for much wider regimes of parameters. We also introduce a scalable (independent of population size) WKB-based numerical technique. The method is applied to a central problem in population genetics and evolution, finding the chance of ultimate fixation in a zero-sum, two-types competition.

Inferring the ancestry of parents and grandparents from genetic data.

Inference of admixture proportions is a classical statistical problem in population genetics. Standard methods implicitly assume that both parents of an individual have the same admixture fraction. However, this is rarely the case in real data. In this paper we show that the distribution of admixture tract lengths in a genome contains information about the admixture proportions of the ancestors of an individual. We develop a Hidden Markov Model (HMM) framework for estimating the admixture proportions of the immediate ancestors of an individual, i.e. a type of decomposition of an individual’s admixture proportions into further subsets of ancestral proportions in the ancestors. Based on a genealogical model for admixture tracts, we develop an efficient algorithm for computing the sampling probability of the genome from a single individual, as a function of the admixture proportions of the ancestors of this individual. This allows us to perform probabilistic inference of admixture proportions of ancestors only using the genome of an extant individual. We perform extensive simulations to quantify the error in the estimation of ancestral admixture proportions under various conditions. To illustrate the utility of the method, we apply it to real genetic data.

Deep learning for population size history inference: Design, comparison and combination with approximate Bayesian computation.

For the past decades, simulation-based likelihood-free inference methods have enabled researchers to address numerous population genetics problems. As the richness and amount of simulated and real genetic data keep increasing, the field has a strong opportunity to tackle tasks that current methods hardly solve. However, high data dimensionality forces most methods to summarize large genomic data sets into a relatively small number of handcrafted features (summary statistics). Here, we propose an alternative to summary statistics, based on the automatic extraction of relevant information using deep learning techniques. Specifically, we design artificial neural networks (ANNs) that take as input single nucleotide polymorphic sites (SNPs) found in individuals sampled from a single population and infer the past effective population size history. First, we provide guidelines to construct artificial neural networks that comply with the intrinsic properties of SNP data such as invariance to permutation of haplotypes, long scale interactions between SNPs and variable genomic length. Thanks to a Bayesian hyperparameter optimization procedure, we evaluate the performance of multiple networks and compare them to well-established methods like Approximate Bayesian Computation (ABC). Even without the expert knowledge of summary statistics, our approach compares fairly well to an ABC approach based on handcrafted features. Furthermore, we show that combining deep learning and ABC can improve performance while taking advantage of both frameworks. Finally, we apply our approach to reconstruct the effective population size history of cattle breed populations.

How Many Subpopulations Is Too Many? Exponential Lower Bounds for Inferring Population Histories.

Reconstruction of population histories is a central problem in population genetics. Existing coalescent-based methods, such as the seminal work of Li and Durbin, attempt to solve this problem using sequence data but have no rigorous guarantees. Determining the amount of data needed to correctly reconstruct population histories is a major challenge. Using a variety of tools from information theory, the theory of extremal polynomials, and approximation theory, we prove new sharp information-theoretic lower bounds on the problem of reconstructing population structure-the history of multiple subpopulations that merge, split, and change sizes over time. Our lower bounds are exponential in the number of subpopulations, even when reconstructing recent histories. We demonstrate the sharpness of our lower bounds by providing algorithms for distinguishing and learning population histories with matching dependence on the number of subpopulations. Along the way and of independent interest, we essentially determine the optimal number of samples needed to learn an exponential mixture distribution information-theoretically, proving the upper bound by analyzing natural (and efficient) algorithms for this problem.

Exact limits of inference in coalescent models

Recovery of population size history from molecular sequence data is an important problem in population genetics. Inference commonly relies on a coalescent model linking the population size history to genealogies. The high computational cost of estimating parameters from these models usually compels researchers to select a subset of the available data or to rely on insufficient summary statistics for statistical inference. We consider the problem of recovering the true population size history from two possible alternatives on the basis of coalescent time data previously considered by Kim et al. (2015). We improve upon previous results by giving exact expressions for the probability of correctly distinguishing between the two hypotheses as a function of the separation between the alternative size histories, the number of individuals, loci, and the sampling times. In more complicated settings we estimate the exact probability of correct recovery by Monte Carlo simulation. Our results give considerably more pessimistic inferential limits than those previously reported. We also extended our analyses to pairwise SMC and SMC’ models of recombination. This work is relevant for optimal design when the inference goal is to test scientific hypotheses about population size trajectories in coalescent models with and without recombination.

Inferring Continuous and Discrete Population Genetic Structure Across Space.

A classic problem in population genetics is the characterization of discrete population structure in the presence of continuous patterns of genetic differentiation. Especially when sampling is discontinuous, the use of clustering or assignment methods may incorrectly ascribe differentiation due to continuous processes (e.g., geographic isolation by distance) to discrete processes, such as geographic, ecological, or reproductive barriers between populations. This reflects a shortcoming of current methods for inferring and visualizing population structure when applied to genetic data deriving from geographically distributed populations. Here, we present a statistical framework for the simultaneous inference of continuous and discrete patterns of population structure. The method estimates ancestry proportions for each sample from a set of two-dimensional population layers, and, within each layer, estimates a rate at which relatedness decays with distance. This thereby explicitly addresses the "clines versus clusters" problem in modeling population genetic variation, and remedies some of the overfitting to which nonspatial models are prone. The method produces useful descriptions of structure in genetic relatedness in situations where separated, geographically distributed populations interact, as after a range expansion or secondary contact. We demonstrate the utility of this approach using simulations and by applying it to empirical datasets of poplars and black bears in North America.

DiploS/HIC: An Updated Approach to Classifying Selective Sweeps.

Identifying selective sweeps in populations that have complex demographic histories remains a difficult problem in population genetics. We previously introduced a supervised machine learning approach, S/HIC, for finding both hard and soft selective sweeps in genomes on the basis of patterns of genetic variation surrounding a window of the genome. While S/HIC was shown to be both powerful and precise, the utility of S/HIC was limited by the use of phased genomic data as input. In this report we describe a deep learning variant of our method, diploS/HIC, that uses unphased genotypes to accurately classify genomic windows. diploS/HIC is shown to be quite powerful even at moderate to small sample sizes.

Stability of two-species communities: Drift, environmental stochasticity, storage effect and selection

The dynamics of two competing species in a finite size community is one of the most studied problems in population genetics and community ecology. Stochastic fluctuations lead, inevitably, to the extinction of one of the species, but the relevant timescale depends on the underlying dynamics. The persistence time of the community has been calculated both for neutral models, where the only driving force of the system is drift (demographic stochasticity), and for models with strong selection. Following recent analyses that stress the importance of environmental stochasticity in empirical systems, we present here a general theory of the persistence time of a two-species community where drift, environmental variations and time independent selective advantage are all taken into account.

Approximate maximum likelihood estimation for population genetic inference.

In many population genetic problems, parameter estimation is obstructed by an intractable likelihood function. Therefore, approximate estimation methods have been developed, and with growing computational power, sampling-based methods became popular. However, these methods such as Approximate Bayesian Computation (ABC) can be inefficient in high-dimensional problems. This led to the development of more sophisticated iterative estimation methods like particle filters. Here, we propose an alternative approach that is based on stochastic approximation. By moving along a simulated gradient or ascent direction, the algorithm produces a sequence of estimates that eventually converges to the maximum likelihood estimate, given a set of observed summary statistics. This strategy does not sample much from low-likelihood regions of the parameter space, and is fast, even when many summary statistics are involved. We put considerable efforts into providing tuning guidelines that improve the robustness and lead to good performance on problems with high-dimensional summary statistics and a low signal-to-noise ratio. We then investigate the performance of our resulting approach and study its properties in simulations. Finally, we re-estimate parameters describing the demographic history of Bornean and Sumatran orang-utans.

Allele Age Under Non-Classical Assumptions is Clarified by an Exact Computational Markov Chain Approach

Determination of the age of an allele based on its population frequency is a well-studied problem in population genetics, for which a variety of approximations have been proposed. We present a new result that, surprisingly, allows the expectation and variance of allele age to be computed exactly (within machine precision) for any finite absorbing Markov chain model in a matter of seconds. This approach makes none of the classical assumptions (e.g., weak selection, reversibility, infinite sites), exploits modern sparse linear algebra techniques, integrates over all sample paths, and is rapidly computable for Wright-Fisher populations up to Ne = 100,000. With this approach, we study the joint effect of recurrent mutation, dominance, and selection, and demonstrate new examples of “selective strolls” where the classical symmetry of allele age with respect to selection is violated by weakly selected alleles that are older than neutral alleles at the same frequency. We also show evidence for a strong age imbalance, where rare deleterious alleles are expected to be substantially older than advantageous alleles observed at the same frequency when population-scaled mutation rates are large. These results highlight the under-appreciated utility of computational methods for the direct analysis of Markov chain models in population genetics.

William Friedman, Geneticist Turned Cryptographer.

William Friedman (1891-1969), trained as a plant geneticist at Cornell University, was employed at Riverbank Laboratories by the eccentric millionaire George Fabyan to work on wheat breeding. Friedman, however, soon became intrigued by and started working on a pet project of Fabyan's involving the conjecture that Francis Bacon, a polymath known for the study of ciphers, was the real author of Shakespeare's plays. Thus, beginning in ∼1916, Friedman turned his attention to the so called "Baconian cipher," and developed decryption techniques that bore similarity to approaches for solving problems in population genetics. His most significant, indeed pathbreaking, work used ideas from genetics and statistics, focusing on analysis of the frequencies of letters in language use. Although he had transitioned from being a geneticist to a cryptographer, his earlier work had resonance in his later pursuits. He soon began working directly for the United States government and produced solutions used to solve complex military ciphers, in particular to break the Japanese Purple code during World War II. Another important legacy of his work was the establishment of the Signal Intelligence Service and eventually the National Security Agency.