Distribution Of Coalescence Times Research Articles

Coalescence and sampling distributions for Feller diffusions

Consider the diffusion process defined by the forward equation ut(t,x)=12{xu(t,x)}xx−α{xu(t,x)}x for t,x≥0 and −∞<α<∞, with an initial condition u(0,x)=δ(x−x0). This equation was introduced and solved by Feller to model the growth of a population of independently reproducing individuals. We explore important coalescent processes related to Feller’s solution. For any α and x0>0 we calculate the distribution of the random variable An(s;t), defined as the finite number of ancestors at a time s in the past of a sample of size n taken from the infinite population of a Feller diffusion at a time t since since its initiation. In a subcritical diffusion we find the distribution of population and sample coalescent trees from time t back, conditional on non-extinction as t→∞. In a supercritical diffusion we construct a coalescent tree which has a single founder and derive the distribution of coalescent times.

Ultrafast genome-wide inference of pairwise coalescence times.

The pairwise sequentially Markovian coalescent (PSMC) algorithm and its extensions infer the coalescence time of two homologous chromosomes at each genomic position. This inference is used in reconstructing demographic histories, detecting selection signatures, studying genome-wide associations, constructing ancestral recombination graphs, and more. Inference of coalescence times between each pair of haplotypes in a large data set is of great interest, as they may provide rich information about the population structure and history of the sample. Here, we introduce a new method, Gamma-SMC, which is more than 10 times faster than current methods. To obtain this speed-up, we represent the posterior coalescence time distributions succinctly as a gamma distribution with just two parameters; in contrast, PSMC and its extensions hold these in a vector over discrete intervals of time. Thus, Gamma-SMC has constant time-complexity per site, without dependence on the number of discrete time states. Additionally, because of this continuous representation, our method is able to infer times spanning many orders of magnitude and, as such, is robust to parameter misspecification. We describe how this approach works, show its performance on simulated and real data, and illustrate its use in studying recent positive selection in the 1000 Genomes Project data set.

On the effects of selection and mutation on species tree inference

The effect of selection acting on regions of the genome on the accuracy of species-level phylogenetic inference using methods that do not explicitly model selection is an open question that is relevant to most, if not all, phylogenomic studies. To address this, we derive a mathematical approximation to the Wright-Fisher model with mutation and selection in the limit as the population size becomes large. In contrast to previous approximations based on diffusion processes, our approximation can be used to study the distribution of coalescent times for an arbitrary number of lineages, allowing calculation of the probability distribution of gene genealogies under the coalescent model. We use these calculations to show that direct selection at strengths typically encountered in practice has only a small effect on the distribution of coalescent times, and hence on the distribution of gene trees. This implies that many coalescent-based methods for estimating the species tree topology will be robust to the presence of selection in a subset of the underlying genes. Selection will, however, bias the estimation of speciation times, causing them to underestimate the true speciation times. Our model captures the effects of selection on the genealogies that generate the observed sequence data, but does not model selective pressures that act only on the subsequent sequences or that negatively impact gene tree estimation.

Graph-based algorithms for Laplace transformed coalescence time distributions.

Extracting information on the selective and demographic past of populations that is contained in samples of genome sequences requires a description of the distribution of the underlying genealogies. Using the Laplace transform, this distribution can be generated with a simple recursive procedure, regardless of model complexity. Assuming an infinite-sites mutation model, the probability of observing specific configurations of linked variants within small haplotype blocks can be recovered from the Laplace transform of the joint distribution of branch lengths. However, the repeated differentiation required to compute these probabilities has proven to be a serious computational bottleneck in earlier implementations. Here, I show that the state space diagram can be turned into a computational graph, allowing efficient evaluation of the Laplace transform by means of a graph traversal algorithm. This general algorithm can, for example, be applied to tabulate the likelihoods of mutational configurations in non-recombining blocks. This work provides a crucial speed up for existing composite likelihood approaches that rely on the joint distribution of branch lengths to fit isolation with migration models and estimate the parameters of selective sweeps. The associated software is available as an open-source Python library, agemo.

Limiting distribution of X-chromosomal coalescence times under first-cousin consanguineous mating

By providing additional opportunities for coalescence within families, the presence of consanguineous unions in a population reduces coalescence times relative to non-consanguineous populations. First-cousin consanguinity can take one of six forms differing in the configuration of sexes in the pedigree of the male and female cousins who join in a consanguineous union: patrilateral parallel, patrilateral cross, matrilateral parallel, matrilateral cross, bilateral parallel, and bilateral cross. Considering populations with each of the six types of first-cousin consanguinity individually and a population with a mixture of the four unilateral types, we examine coalescent models of consanguinity. We previously computed, for first-cousin consanguinity models, the mean coalescence time for X-chromosomal loci and the limiting distribution of coalescence times for autosomal loci. Here, we use the separation-of-time-scales approach to obtain the limiting distribution of coalescence times for X-chromosomal loci. This limiting distribution has an instantaneous coalescence probability that depends on the probability that a union is consanguineous; lineages that do not coalesce instantaneously coalesce according to an exponential distribution. We study the effects on the coalescence time distribution of the type of first-cousin consanguinity, showing that patrilateral-parallel and patrilateral-cross consanguinity have no effect on X-chromosomal coalescence time distributions and that matrilateral-parallel consanguinity decreases coalescence times to a greater extent than does matrilateral-cross consanguinity.

Studying coalescence at different lengthscales: from films to droplets

The hydrodynamics of thin films is an important factor when it comes to the stability and rheology of multiphasic materials, such as foams, emulsions, and polymer blends. However, there have so far been only limited experimental studies addressing the dynamics of individual free-standing thin films at conditions similar to those encountered on macroscopic scales. In this article, we study a well-characterized system of a water-in-oil emulsion stabilized by a non-ionic surfactant (SPAN80) close to its CMC. We employ a dynamic thin film balance, to study the dynamics of freestanding films under both constant and time-varied pressure drops. We compare with the recently published results of Narayan et al. (2020) on colliding droplets of the same system with a hydrodynamic microfluidic trap, and show for the first time that agreement between the two lengthscales is possible, which indicates that the coalescence is indeed dominated by the dynamics in the film. We then address the scatter in the coalescence times and show that it can be affected by extrinsic factors, as well as by variations in the collision angle. Finally, we discuss the difficulties of extracting insight on the coalescence mechanism from coalescence time distributions when different effects such as impurities, small pressure variations, collision angle variations, and possible Marangoni-related instabilities are at play.

Evaluation of methods for estimating coalescence times using ancestral recombination graphs.

The ancestral recombination graph is a structure that describes the joint genealogies of sampled DNA sequences along the genome. Recent computational methods have made impressive progress toward scalably estimating whole-genome genealogies. In addition to inferring the ancestral recombination graph, some of these methods can also provide ancestral recombination graphs sampled from a defined posterior distribution. Obtaining good samples of ancestral recombination graphs is crucial for quantifying statistical uncertainty and for estimating population genetic parameters such as effective population size, mutation rate, and allele age. Here, we use standard neutral coalescent simulations to benchmark the estimates of pairwise coalescence times from 3 popular ancestral recombination graph inference programs: ARGweaver, Relate, and tsinfer+tsdate. We compare (1) the true coalescence times to the inferred times at each locus; (2) the distribution of coalescence times across all loci to the expected exponential distribution; (3) whether the sampled coalescence times have the properties expected of a valid posterior distribution. We find that inferred coalescence times at each locus are most accurate in ARGweaver, and often more accurate in Relate than in tsinfer+tsdate. However, all 3 methods tend to overestimate small coalescence times and underestimate large ones. Lastly, the posterior distribution of ARGweaver is closer to the expected posterior distribution than Relate's, but this higher accuracy comes at a substantial trade-off in scalability. The best choice of method will depend on the number and length of input sequences and on the goal of downstream analyses, and we provide guidelines for the best practices.

Heterogeneity in effective size across the genome: effects on the inverse instantaneous coalescence rate (IICR) and implications for demographic inference under linked selection.

The relative contribution of selection and neutrality in shaping species genetic diversity is one of the most central and controversial questions in evolutionary theory. Genomic data provide growing evidence that linked selection, i.e. the modification of genetic diversity at neutral sites through linkage with selected sites, might be pervasive over the genome. Several studies proposed that linked selection could be modeled as first approximation by a local reduction (e.g. purifying selection, selective sweeps) or increase (e.g. balancing selection) of effective population size (Ne). At the genome-wide scale, this leads to variations of Ne from one region to another, reflecting the heterogeneity of selective constraints and recombination rates between regions. We investigate here the consequences of such genomic variations of Ne on the genome-wide distribution of coalescence times. The underlying motivation concerns the impact of linked selection on demographic inference, because the distribution of coalescence times is at the heart of several important demographic inference approaches. Using the concept of inverse instantaneous coalescence rate, we demonstrate that in a panmictic population, linked selection always results in a spurious apparent decrease of Ne along time. Balancing selection has a particularly large effect, even when it concerns a very small part of the genome. We also study more general models including genuine population size changes, population structure or transient selection and find that the effect of linked selection can be significantly reduced by that of population structure. The models and conclusions presented here are also relevant to the study of other biological processes generating apparent variations of Ne along the genome.

Variance and limiting distribution of coalescence times in a diploid model of a consanguineous population

Recent modeling studies interested in runs of homozygosity (ROH) and identity by descent (IBD) have sought to connect these properties of genomic sharing to pairwise coalescence times. Here, we examine a variety of features of pairwise coalescence times in models that consider consanguinity. In particular, we extend a recent diploid analysis of mean coalescence times for lineage pairs within and between individuals in a consanguineous population to derive the variance of coalescence times, studying its dependence on the frequency of consanguinity and the kinship coefficient of consanguineous relationships. We also introduce a separation-of-time-scales approach that treats consanguinity models analogously to mathematically similar phenomena such as partial selfing, using this approach to obtain coalescence-time distributions. This approach shows that the consanguinity model behaves similarly to a standard coalescent, scaling population size by a factor 1−3c, where c represents the kinship coefficient of a randomly chosen mating pair. It provides the explanation for an earlier result describing mean coalescence time in the consanguinity model in terms of c. The results extend the potential to make predictions about ROH and IBD in relation to demographic parameters of diploid populations.

Deep learning based methods for estimating distribution of coalescence rates from genome-wide data

Demographic and population structure inference is one of the most important problems in genomics. Population parameters such as effective population sizes, population split times and migration rates are of high interest both themselves and for many applications, e.g. for genome-wide association studies. Hidden Markov Model (HMM) based methods, such as PSMC, MSMC, coalHMM etc., proved to be powerful and useful for estimation of these parameters in many population genetics studies. At the same time, machine and deep learning have began to be used in natural science widely. In particular, deep learning based approaches have already substituted hidden Markov models in many areas, such as speech recognition or user input prediction. We develop a deep learning (DL) approach for local coalescent time estimation from one whole diploid genome. Our DL models are trained on simulated datasets. Importantly, demographic and population parameters can be inferred based on the distribution of coalescent times. We expect that our approach will be useful under complex population scenarios, which cannot be studied with existing HMM based methods. Our work is also a crucial step in developing a deep learning framework which would allow to create population genomics methods for different genomic data representations.

Foam stability in filtered lubricants containing antifoams

Lubricant formulations are filtered to remove deleterious particulate matter. An unintended consequence of this important process is the detrimental effect of fine filtration on the foaming performance of lubricants with antifoam additives. Here we outline a method to study this phenomenon in detail by probing the coalescence stability of single bubbles in filtered antifoam laden lubricants. Initially, we establish the validity of Garrett’s hypothesis for the tested antifoam laden lubricants. Subsequently, we show that the bubble stability in filtered lubricants are positively correlated to the number of filtration cycles - with the most dramatic changes in bubble stability accompanying the initial few cycles of filtration. Further, we show that post filtration, the stability of bubbles in lubricants is inversely correlated to the pore size of the filter and the volume fraction of antifoam in the lubricant prior to filtration. The results also reveal that in the presence of antifoam additives, the bubble coalescence times span multiple Rayleigh distributions. We also provide visual evidence that shows the tested antifoams employ a bridging-stretching mechanism to rupture non-aqueous foams. Finally, a simple probabilistic model is introduced that helps in analyzing the distribution of coalescence times of single bubbles to obtain insights into the volume fraction of antifoams in the lubricant. We believe these results are valuable in guiding the design of lubricants with robust and superior foaming performance.

The Effects of Demography and Genetics on the Neutral Distribution of Quantitative Traits.

Neutral models for quantitative trait evolution are useful for identifying phenotypes under selection. These models often assume normally distributed phenotypes. This assumption may be violated when a trait is affected by relatively few variants or when the effects of those variants arise from skewed or heavy tailed distributions. Molecular phenotypes such as gene expression levels may have these properties. To accommodate deviations from normality, models making fewer assumptions about the underlying genetics and patterns of variation are needed. Here, we develop a general neutral model for quantitative trait variation using a coalescent approach. This model allows interpretation of trait distributions in terms of familiar population genetic parameters because it is based on the coalescent. We show how the normal distribution resulting from the infinitesimal limit, where the number of loci grows large as the effect size per mutation becomes small, depends only on expected pairwise coalescent times. We then demonstrate how deviations from normality depend on demography through the distribution of coalescence times as well as through genetic parameters. In particular, population growth events exacerbate deviations while bottlenecks reduce them. We demonstrate the practical applications of this model by showing how to sample from the neutral distribution of [Formula: see text], the ratio of the variance between subpopulations to that in the overall population. We further show it is likely impossible to distinguish sparsity from skewed or heavy tailed mutational effects using only sampled trait values. The model analyzed here greatly expands the parameter space for neutral trait models.

Probing the ecological and evolutionary history of a thermophilic cyanobacterial population via statistical properties of its microdiversity.

Despite extensive DNA sequencing data derived from natural microbial communities, it remains a major challenge to identify the key evolutionary and ecological forces that shape microbial populations. We have focused on the extensive microdiversity of the cyanobacterium Synechococcus sp., which is a dominant member of the dense phototrophic biofilms in the hot springs of Yellowstone National Park. From deep amplicon sequencing of many loci and statistical analyses of these data, we showed previously that the population has undergone an unexpectedly high degree of homologous recombination, unlinking synonymous SNP-pair correlations even on intragenic length scales. Here, we analyze the genic amino acid diversity, which provides new evidence of selection and insights into the evolutionary history of the population. Surprisingly, some features of the data, including the spectrum of distances between genic-alleles, appear consistent with primarily asexual neutral drift. Yet the non-synonymous site frequency spectrum has too large an excess of low-frequency polymorphisms to result from negative selection on deleterious mutations given the distribution of coalescent times that we infer. And our previous analyses showed that the population is not asexual. Taken together, these apparently contradictory data suggest that selection, epistasis, and hitchhiking all play essential roles in generating and stabilizing the diversity. We discuss these as well as potential roles of ecological niches at genomic and genic levels. From quantitative properties of the diversity and comparative genomic data, we infer aspects of the history and inter-spring dispersal of the meta-population since it was established in the Yellowstone Caldera. Our investigations illustrate the need for combining multiple types of sequencing data and quantitative statistical analyses to develop an understanding of microdiversity in natural microbial populations.

Minimal-assumption inference from population-genomic data.

Samples of multiple complete genome sequences contain vast amounts of information about the evolutionary history of populations, much of it in the associations among polymorphisms at different loci. We introduce a method, Minimal-Assumption Genomic Inference of Coalescence (MAGIC), that reconstructs key features of the evolutionary history, including the distribution of coalescence times, by integrating information across genomic length scales without using an explicit model of coalescence or recombination, allowing it to analyze arbitrarily large samples without phasing while making no assumptions about ancestral structure, linked selection, or gene conversion. Using simulated data, we show that the performance of MAGIC is comparable to that of PSMC' even on single diploid samples generated with standard coalescent and recombination models. Applying MAGIC to a sample of human genomes reveals evidence of non-demographic factors driving coalescence.

Modeling Human Population Separation History Using Physically Phased Genomes.

Phased haplotype sequences are a key component in many population genetic analyses since variation in haplotypes reflects the action of recombination, selection, and changes in population size. In humans, haplotypes are typically estimated from unphased sequence or genotyping data using statistical models applied to large reference panels. To assess the importance of correct haplotype phase on population history inference, we performed fosmid pool sequencing and resolved phased haplotypes of five individuals from diverse African populations (including Yoruba, Esan, Gambia, Maasai, and Mende). We physically phased 98% of heterozygous SNPs into haplotype-resolved blocks, obtaining a block N50 of 1 Mbp. We combined these data with additional phased genomes from San, Mbuti, Gujarati, and Centre de'Etude du Polymorphism Humain European populations and analyzed population size and separation history using the pairwise sequentially Markovian coalescent and multiple sequentially Markovian coalescent models. We find that statistically phased haplotypes yield a more recent split-time estimation compared with experimentally phased haplotypes. To better interpret patterns of cross-population coalescence, we implemented an approximate Bayesian computation approach to estimate population split times and migration rates by fitting the distribution of coalescent times inferred between two haplotypes, one from each population, to a standard isolation-with-migration model. We inferred that the separation between hunter-gatherer populations and other populations happened ∼120-140 KYA, with gene flow continuing until 30-40 KYA; separation between west-African and out-of-African populations happened ∼70-80 KYA; while the separation between Maasai and out-of-African populations happened ∼50 KYA.

Demographic inference using genetic data from a single individual: Separating population size variation from population structure

The rapid development of sequencing technologies represents new opportunities for population genetics research. It is expected that genomic data will increase our ability to reconstruct the history of populations. While this increase in genetic information will likely help biologists and anthropologists to reconstruct the demographic history of populations, it also represents new challenges. Recent work has shown that structured populations generate signals of population size change. As a consequence it is often difficult to determine whether demographic events such as expansions or contractions (bottlenecks) inferred from genetic data are real or due to the fact that populations are structured in nature. Given that few inferential methods allow us to account for that structure, and that genomic data will necessarily increase the precision of parameter estimates, it is important to develop new approaches. In the present study we analyze two demographic models. The first is a model of instantaneous population size change whereas the second is the classical symmetric island model. We (i) re-derive the distribution of coalescence times under the two models for a sample of size two, (ii) use a maximum likelihood approach to estimate the parameters of these models (iii) validate this estimation procedure under a wide array of parameter combinations, (iv) implement and validate a model rejection procedure by using a Kolmogorov–Smirnov test, and a model choice procedure based on the AIC, and (v) derive the explicit distribution for the number of differences between two non-recombining sequences. Altogether we show that it is possible to estimate parameters under several models and perform efficient model choice using genetic data from a single diploid individual.

Detecting past changes of effective population size.

Understanding and predicting population abundance is a major challenge confronting scientists. Several genetic models have been developed using microsatellite markers to estimate the present and ancestral effective population sizes. However, to get an overview on the evolution of population requires that past fluctuation of population size be traceable. To address the question, we developed a new model estimating the past changes of effective population size from microsatellite by resolving coalescence theory and using approximate likelihoods in a Monte Carlo Markov Chain approach. The efficiency of the model and its sensitivity to gene flow and to assumptions on the mutational process were checked using simulated data and analysis. The model was found especially useful to provide evidence of transient changes of population size in the past. The times at which some past demographic events cannot be detected because they are too ancient and the risk that gene flow may suggest the false detection of a bottleneck are discussed considering the distribution of coalescence times. The method was applied on real data sets from several Atlantic salmon populations. The method called VarEff (Variation of Effective size) was implemented in the R package VarEff and is made available at https://qgsp.jouy.inra.fr and at http://cran.r-project.org/web/packages/VarEff.

Coalescence Times for the Bienaymé-Galton-Watson Process

We investigate the distribution of the coalescence time (most recent common ancestor) for two individuals picked at random (uniformly) in the current generation of a continuous-time Bienaymé-Galton-Watson process founded t units of time ago. We also obtain limiting distributions as t → ∞ in the subcritical case. We extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.

Coalescence Times for the Bienaymé-Galton-Watson Process

We investigate the distribution of the coalescence time (most recent common ancestor) for two individuals picked at random (uniformly) in the current generation of a continuous-time Bienaymé-Galton-Watson process foundedtunits of time ago. We also obtain limiting distributions ast→ ∞ in the subcritical case. We extend our results for two individuals to the joint distribution of coalescence times for any finite number of individuals sampled in the current generation.

The Influence of Gene Flow on Species Tree Estimation: A Simulation Study

Gene flow among populations or species and incomplete lineage sorting (ILS) are two evolutionary processes responsible for generating gene tree discordance and therefore hindering species tree estimation. Numerous studies have evaluated the impacts of ILS on species tree inference, yet the ramifications of gene flow on species trees remain less studied. Here, we simulate and analyse multilocus sequence data generated with ILS and gene flow to quantify their impacts on species tree inference. We characterize species tree estimation errors under various models of gene flow, such as the isolation-migration model, the n-island model, and gene flow between non-sister species or involving ancestral species, and species boundaries crossed by a single gene copy (allelic introgression) or by a single migrant individual. These patterns of gene flow are explored on species trees of different sizes (4 vs. 10 species), at different time scales (shallow vs. deep), and with different migration rates. Species trees are estimated with the multispecies coalescent model using Bayesian methods (BEST and *BEAST) and with a summary statistic approach (MPEST) that facilitates phylogenomic-scale analysis. Even in cases where the topology of the species tree is estimated with high accuracy, we find that gene flow can result in overestimates of population sizes (species tree dilation) and underestimates of species divergence times (species tree compression). Signatures of migration events remain present in the distribution of coalescent times for gene trees, and with sufficient data it is possible to identify those loci that have crossed species boundaries. These results highlight the need for careful sampling design in phylogeographic and species delimitation studies as gene flow, introgression, or incorrect sample assignments can bias the estimation of the species tree topology and of parameter estimates such as population sizes and divergence times.