Estimating effective population size from samples of sequences: inefficiency of pairwise and segregating sites as compared to phylogenetic estimates.

It is known that under neutral mutation at a known mutation rate a sample of nucleotide sequences, within which there is assumed to be no recombination, allows estimation of the effective size of an isolated population. This paper investigates the case of very long sequences, where each pair of sequences allows a precise estimate of the divergence time of those two gene copies. The average divergence time of all pairs of copies estimates twice the effective population number and an estimate can also be derived from the number of segregating sites. One can alternatively estimate the genealogy of the copies. This paper shows how a maximum likelihood estimate of the effective population number can be derived from such a genealogical tree. The pairwise and the segregating sites estimates are shown to be much less efficient than this maximum likelihood estimate, and this is verified by computer simulation. The result implies that there is much to gain by explicitly taking the tree structure of these genealogies into account.     

Gene genealogies when the sample size exceeds the effective size of the population

We study the properties of gene genealogies for large samples using a continuous approximation introduced by R. A. Fisher. We show that the major effect of large sample size, relative to the effective size of the population, is to increase the proportion of polymorphisms at which the mutant type is found in a single copy in the sample. We derive analytical expressions for the expected number of these singleton polymorphisms and for the total number of polymorphic, or segregating, sites that are valid even when the sample size is much greater than the effective size of the population. We use simulations to assess the accuracy of these predictions and to investigate other aspects of large-sample genealogies. Lastly, we apply our results to some data from Pacific oysters sampled from British Columbia. This illustrates that, when large samples are available, it is possible to estimate the mutation rate and the effective population size separately, in contrast to the case of small samples in which only the product of the mutation rate and the effective population size can be estimated.     

Maximum Likelihood Estimation of Population Parameters

One of the most important parameters in population genetics is θ = 4N(e)μ where N(e) is the effective population size and μ is the rate of mutation per gene per generation. We study two related problems, using the maximum likelihood method and the theory of coalescence. One problem is the potential improvement of accuracy in estimating the parameter θ over existing methods and the other is the estimation of parameter λ which is the ratio of two θ's. The minimum variances of estimates of the parameter θ are derived under two idealized situations. These minimum variances serve as the lower bounds of the variances of all possible estimates of θ in practice. We then show that Watterson's estimate of θ based on the number of segregating sites is asymptotically an optimal estimate of θ. However, for a finite sample of sequences, substantial improvement over Watterson's estimate is possible when θ is large. The maximum likelihood estimate of λ = θ(1)/θ(2) is obtained and the properties of the estimate are discussed.     

Segregating Sites in a Gene Conversion Model with Mutation

Formulae for the expectation and variance of the number of segregating and homogeneous sites in a sample of two chromosomes are found. The model includes gene conversion and infinitely-many-alleles mutation in a coalescent framework. The corresponding infinitely-many-sites model limits are also found. The formulae for the expectation are extended to any sample size. Comparisons are drawn between the pure mutation model and the model where gene conversion has been added.     

Optimal Sequencing Strategies for Surveying Molecular Genetic Diversity

Two commonly used measures of genetic diversity for intraspecies DNA sequence data are based, respectively, on the number of segregating sites, and on the average number of pairwise nucleotide differences. Expressions are derived for their variance in the presence of intragenic recombination for a panmictic population of fixed size that is at neutral equilibrium at the region sequenced. We show that, in contrast to the slow decrease in variance with increasing sample size, if the recombination rate is nonzero, the asymptotic rate of decrease of variance with increasing sequence length, for fixed sample size, is quite rapid. In particular, it is close to that which would be obtained by sequencing independent chromosome regions. The correlation between measures of diversity from linked regions is also examined. For a given total number of bases sequenced in a particular region, optimal sequencing strategies are derived. These typically involve sequencing relatively few (three to 10) long copies of the region. Under optimal strategies, the variances of the two measures are very similar for most parameter values considered. Results concerning optimal sequencing strategies will be sensitive to gross departures from the underlying assumptions, such as population bottlenecks, selective sweeps, and substantial population substructure.     

Efficiency of the Neighbor-Joining Method in Reconstructing Deep and Shallow Evolutionary Relationships in Large Phylogenies

The neighbor-joining (NJ) method is widely used in reconstructing large phylogenies because of its computational speed and the high accuracy in phylogenetic inference as revealed in computer simulation studies. However, most computer simulation studies have quantified the overall performance of the NJ method in terms of the percentage of branches inferred correctly or the percentage of replications in which the correct tree is recovered. We have examined other aspects of its performance, such as the relative efficiency in correctly reconstructing shallow (close to the external branches of the tree) and deep branches in large phylogenies; the contribution of zero-length branches to topological errors in the inferred trees; and the influence of increasing the tree size (number of sequences), evolutionary rate, and sequence length on the efficiency of the NJ method. Results show that the correct reconstruction of deep branches is no more difficult than that of shallower branches. The presence of zero-length branches in realized trees contributes significantly to the overall error observed in the NJ tree, especially in large phylogenies or slowly evolving genes. Furthermore, the tree size does not influence the efficiency of NJ in reconstructing shallow and deep branches in our simulation study, in which the evolutionary process is assumed to be homogeneous in all lineages. Received: 7 March 2000 / Accepted: 2 August 2000     

Population genetic inference using a fixed number of segregating sites: a reassessment

Coalescent theory is commonly used to perform population genetic inference at the nucleotide level. Here, we examine the procedure that fixes the number of segregating sites (henceforth the FS procedure). In this approach a fixed number of segregating sites (S) are placed on a coalescent tree (independently of the total and internode lengths of the tree). Thus, although widely used, the FS procedure does not strictly follow the assumptions of coalescent theory and must be considered an approximation of (i) the standard procedure that uses a fixed population mutation parameter theta, and (ii) procedures that condition on the number of segregating sites. We study the differences in the false positive rate for nine statistics by comparing the FS procedure with the procedures (i) and (ii), using several evolutionary models with single-locus and multilocus data. Our results indicate that for single-locus data the FS procedure is accurate for the equilibrium neutral model, but problems arise under the alternative models studied; furthermore, for multilocus data, the FS procedure becomes inaccurate even for the standard neutral model. Therefore, we recommend a procedure that fixes the theta value (or alternatively, procedures that condition on S and take into account the uncertainty of theta) for analysing evolutionary models with multilocus data. With single-locus data, the FS procedure should not be employed for models other than the standard neutral model.     

Sufficiency of the number of segregating sites in the limit under finite-sites mutation

We show that the number of segregating sites is a sufficient statistic for the scaled mutation parameter (θ) in the limit as the number of sites tends to infinity and there is free recombination between sites. We assume that the mutation parameter at each site tends to zero such than the total mutation parameter (θ) is constant in the limit. Our results show that Watterson’s estimator is the maximum likelihood estimator in this case, but that it estimates a composite parameter which is different for different mutation models. Some of our results hold when recombination is limited, because Watterson’s estimator is an unbiased, method-of-moments estimator regardless of the recombination rate. The quantity it estimates depends on the details of how mutations occur at each site.     

Some Statistical Improvements for Estimating Population Size and Mutation Rate from Segregating Sites in DNA Sequences

In population genetics, under a neutral Wright-Fisher model, the scaling parameter straight theta=4Nmu represents twice the average number of new mutants per generation. The effective population size is N and mu is the mutation rate per sequence per generation. Watterson proposed a consistent estimator of this parameter based on the number of segregating sites in a sample of nucleotide sequences. We study the distribution of the Watterson estimator. Enlarging the size of the sample, we asymptotically set a Central Limit Theorem for the Watterson estimator. This exhibits asymptotic normality with a slow rate of convergence. We then prove the asymptotic efficiency of this estimator. In the second part, we illustrate the slow rate of convergence found in the Central Limit Theorem. To this end, by studying the confidence intervals, we show that the asymptotic Gaussian distribution is not a good approximation for the Watterson estimator.     

The Effects of Variable Mutation Rates across Sites on the Phylogenetic Estimation of Effective Population Size or Mutation Rate of DNA Sequences

Multiple hits at some sites of human mitochondrial DNA sequences suggest that the commonly assumed infinite-sites model can be violated. Under the neutral Wright-Fisher model without recombination and population subdivision, we investigated, by computer simulations, the effect of multiple hits on the estimation of the essential parameter θ = 4N(e)μ by FU's UPBLUE procedure. We found that with moderate mutation rate heterogeneity, UPBLUE performs very well in terms of unbiasness and efficiency. Under extreme mutation rate heterogeneity, if sample size is reasonably large (e.g., >60), UPBLUE is still very satisfactory; otherwise we developed a new correction equation. Given knowledge of the degree of mutation rate heterogeneity, the performance of UPBLUE with the new correction equation was tested to be fairly satisfactory: there is almost no bias and the sampling variance is only slightly higher than the theoretical minimum variance. Thus, with an appropriate correction, UPBLUE is relatively robust to the multiple hits. In genealogies reconstructed by UPGMA, we found that the total length of branches directly linked to the tips is underestimated, and those far away tend to be overestimated, while the total length of all branches is not biased.     

Ancestral inference on gene trees under selection

The extent to which natural selection shapes diversity within populations is a key question for population genetics. Thus, there is considerable interest in quantifying the strength of selection. A full likelihood approach for inference about selection at a single site within an otherwise neutral fully linked sequence of sites is described here. A coalescent model of evolution is used to model the ancestry of a sample of DNA sequences which have the selected site segregating. The mutation model, for the selected and neutral sites, is the infinitely many-sites model where there is no back or parallel mutation at sites. A unique perfect phylogeny, a gene tree, can be constructed from the configuration of mutations on the sample sequences under this model of mutation. The approach is general and can be used for any bi-allelic selection scheme. Selection is incorporated through modelling the frequency of the selected and neutral allelic classes stochastically back in time, then using a subdivided population model considering the population frequencies through time as variable population sizes. An importance sampling algorithm is then used to explore over coalescent tree space consistent with the data. The method is applied to a simulated data set and the gene tree presented in Verrelli et al. (2002).     

The number of segregating sites in expanding human populations, with implications for estimates of demographic parameters

The frequency distribution of pairwise differences between sequences of mtDNA has recently been used to estimate the size of human populations before and after a hypothetical episode of rapid population growth and the time at which the population grew. To test the internal consistency of this method, we used three different sets of human mtDNA data and the corresponding demographic parameters estimated from the distribution of pairwise differences to determine by simulation the expected number of segregating sites, S, and its empirical distribution. The results indicate that the observed values of S are significantly lower than expected in two of three cases under the assumption of the infinite-sites model. Further simulations in which mutations were allowed to occur more than once at the same site and in which there was variation in mutation rate among sites show that the expected number of segregating sites can be much lower than under the infinite-site assumption. Nevertheless, the observed value of S is still significantly different from the value expected under the expansion hypothesis in two of three cases.      

Multilocus patterns of nucleotide polymorphism and the demographic history of Populus tremula

I have studied nucleotide polymorphism and linkage disequilibrium using multilocus data from 77 fragments, with an average length of fragments of 550 bp, in the deciduous tree Populus tremula (Salicaceae). The frequency spectrum across loci showed a modest excess of mutations segregating at low frequency and a marked excess of high-frequency derived mutations at silent sites, relative to neutral expectations. These excesses were also seen at replacement sites, but were not so pronounced for high-frequency derived mutations. There was a marked excess of low-frequency mutations at replacement sites, likely indicating deleterious amino acid-changing mutations that segregate at low frequencies in P. tremula. I used approximate Bayesian computation (ABC) to evaluate a number of different demographic scenarios and to estimate parameters for the best-fitting model. The data were found to be consistent with a historical reduction in the effective population size of P. tremula through a bottleneck. The timing inferred for this bottleneck is largely consistent with geological data and with data from several other long-lived plant species. The results show that P. tremula harbors substantial levels of nucleotide polymorphism with the posterior mode of the scaled mutation rate, = 0.0177 across loci. The ABC analyses also provided an estimate of the scaled recombination rate that indicates that recombination rates in P. tremula are likely to be 2-10 times higher than the mutation rate. This study reinforces the notion that linkage disequilibrium is low and decays to negligible levels within a few hundred base pairs in P. tremula.     

A method for detecting positive selection at single amino acid sites

A method was developed for detecting the selective force at single amino acid sites given a multiple alignment of protein-coding sequences. The phylogenetic tree was reconstructed using the number of synonymous substitutions. Then, the neutrality was tested for each codon site using the numbers of synonymous and nonsynonymous changes throughout the phylogenetic tree. Computer simulation showed that this method accurately estimated the numbers of synonymous and nonsynonymous substitutions per site, as long as the substitution number on each branch was relatively small. The false-positive rate for detecting the selective force was generally low. On the other hand, the true-positive rate for detecting the selective force depended on the parameter values. Within the range of parameter values used in the simulation, the true-positive rate increased as the strength of the selective force and the total branch length (namely the total number of synonymous substitutions per site) in the phylogenetic tree increased. In particular, with the relative rate of nonsynonymous substitutions to synonymous substitutions being 5.0, most of the positively selected codon sites were correctly detected when the total branch length in the phylogenetic tree was > or = 2.5. When this method was applied to the human leukocyte antigen (HLA) gene, which included antigen recognition sites (ARSs), positive selection was detected mainly on ARSs. This finding confirmed the effectiveness of the present method with actual data. Moreover, two amino acid sites were newly identified as positively selected in non-ARSs. The three-dimensional structure of the HLA molecule indicated that these sites might be involved in antigen recognition. Positively selected amino acid sites were also identified in the envelope protein of human immunodeficiency virus and the influenza virus hemagglutinin protein. This method may be helpful for predicting functions of amino acid sites in proteins, especially in the present situation, in which sequence data are accumulating at an enormous speed.     

Purifying Selection Determines the Short-Term Time Dependency of Evolutionary Rates in SARS-CoV-2 and pH1N1 Influenza

High-throughput sequencing enables rapid genome sequencing during infectious disease outbreaks and provides an opportunity to quantify the evolutionary dynamics of pathogens in near real-time. One difficulty of undertaking evolutionary analyses over short timescales is the dependency of the inferred evolutionary parameters on the timespan of observation. Crucially, there are an increasing number of molecular clock analyses using external evolutionary rate priors to infer evolutionary parameters. However, it is not clear which rate prior is appropriate for a given time window of observation due to the time-dependent nature of evolutionary rate estimates. Here, we characterize the molecular evolutionary dynamics of SARS-CoV-2 and 2009 pandemic H1N1 (pH1N1) influenza during the first 12 months of their respective pandemics. We use Bayesian phylogenetic methods to estimate the dates of emergence, evolutionary rates, and growth rates of SARS-CoV-2 and pH1N1 over time and investigate how varying sampling window and data set sizes affect the accuracy of parameter estimation. We further use a generalized McDonald–Kreitman test to estimate the number of segregating nonneutral sites over time. We find that the inferred evolutionary parameters for both pandemics are time dependent, and that the inferred rates of SARS-CoV-2 and pH1N1 decline by ∼50% and ∼100%, respectively, over the course of 1 year. After at least 4 months since the start of sequence sampling, inferred growth rates and emergence dates remain relatively stable and can be inferred reliably using a logistic growth coalescent model. We show that the time dependency of the mean substitution rate is due to elevated substitution rates at terminal branches which are 2–4 times higher than those of internal branches for both viruses. The elevated rate at terminal branches is strongly correlated with an increasing number of segregating nonneutral sites, demonstrating the role of purifying selection in generating the time dependency of evolutionary parameters during pandemics.     

The coalescent in an island model of population subdivision with variation among demes

A simple genealogical structure is found for a general finite island model of population subdivision. The model allows for variation in the sizes of demes, in contributions to the migrant pool, and in the fraction of each deme that is replaced by migrants every generation. The ancestry of a sample of non-recombining DNA sequences has a simple structure when the sample size is much smaller than the total number of demes in the population. This allows an expression for the probability distribution of the number of segregating sites in the sample to be derived under the infinite-sites mutation model. It also yields easily computed estimators of the migration parameter for each deme in a multi-deme sample. The genealogical process is such that the lineages ancestral to the sample tend to accumulate in demes with low migration rates and/or which contribute disproportionately to the migrant pool. In addition, common ancestor or coalescent events tend to occur in demes of small size. This provides a framework for understanding the determinants of the effective size of the population, and leads to an expression for the probability that the root of a genealogy occurs in a particular geographic region, or among a particular set of demes.     

Population structure of the Japanese eel, Anguilla japonica

Mitochondrial DNA (mtDNA) sequences that include (a) a part of the cytochrome b gene, (b) two tRNA genes, and (c) a part of the noncoding D-loop region of 31 Anguilla japonica (Japanese eel) and 1 A. marmorata collected from Taiwan, Japan, and mainland China were determined to evaluate the population structure of Japanese eel. Among 30 genotypes identified from the 31 Japanese eel mtDNAs sequenced, there are 58 variable sites, predominantly clustered at the D-loop region. The phylogenetic tree constructed by the unweighted pair-group method with arithmetic mean shows neither significant genealogical branches nor geographic clusters. Furthermore, the sequence-statistics test reveals little, if any, significant genetic differentiation. These results indicate that the 31 Japanese eels might come from a single population. Analysis of sequence variation in mtDNA by using the relationship between the number of segregating sites and the average number of nucleotide differences under the neutral mutation hypothesis reveals that neutral mutation acts as a major factor influencing the evolutionary divergence of the Japanese eel mitochondrial genome sequenced, especially in the noncoding region.      

Estimating the Age of the Common Ancestor of a DNA Sample Using the Number of Segregating Sites

The number of segregating sites in a sample of DNA sequences and the age of the most recent common ancestor (MRCA) of the sequences in the sample are positively correlated. The value of the former can be used to estimate the value of the latter. Using the coalescent approach, we derive in this paper the joint probability distribution of the number of segregating sites and the age of the MRCA of a sample under the neutral Wright-Fisher model. From this distribution, we are able to compute the likelihood function of the number of segregating sites and the posterior probability of the age of the MRCA of a sample. Three point estimators and one interval estimator of the age of the MRCA are developed; their relationships and properties are investigated. The estimation of the age of the MRCA of human Y chromosomes from a sample of no variation is discussed.     

Statistical Properties of a DNA Sample under the Finite-Sites Model

Statistical properties of a DNA sample from a random-mating population of constant size are studied under the finite-sites model. It is assumed that there is no migration and no recombination occurs within the locus. A Markov process model is used for nucleotide substitution, allowing for multiple substitutions at a single site. The evolutionary rates among sites are treated as either constant or variable. The general likelihood calculation using numerical integration involves intensive computation and is feasible for three or four sequences only; it may be used for validating approximate algorithms. Methods are developed to approximate the probability distribution of the number of segregating sites in a random sample of n sequences, with either constant or variable substitution rates across sites. Calculations using parameter estimates obtained for human D-loop mitochondrial DNAs show that among-site rate variation has a major effect on the distribution of the number of segregating sites; the distribution under the finite-sites model with variable rates among sites is quite different from that under the infinite-sites model.     

Estimating mutation rate: how to count mutations?

Mutation rate is an essential parameter in genetic research. Counting the number of mutant individuals provides information for a direct estimate of mutation rate. However, mutant individuals in the same family can share the same mutations due to premeiotic mutation events, so that the number of mutant individuals can be significantly larger than the number of mutation events observed. Since mutation rate is more closely related to the number of mutation events, whether one should count only independent mutation events or the number of mutants remains controversial. We show in this article that counting mutant individuals is a correct approach for estimating mutation rate, while counting only mutation events will result in underestimation. We also derived the variance of the mutation-rate estimate, which allows us to examine a number of important issues about the design of such experiments. The general strategy of such an experiment should be to sample as many families as possible and not to sample much more offspring per family than the reciprocal of the pairwise correlation coefficient within each family. To obtain a reasonably accurate estimate of mutation rate, the number of sampled families needs to be in the same or higher order of magnitude as the reciprocal of the mutation rate.