Composite likelihood estimation of demographic parameters
- Daniel Garrigan^{1}Email author
https://doi.org/10.1186/1471-2156-10-72
© Garrigan; licensee BioMed Central Ltd. 2009
Received: 07 April 2009
Accepted: 12 November 2009
Published: 12 November 2009
Abstract
Background
Most existing likelihood-based methods for fitting historical demographic models to DNA sequence polymorphism data to do not scale feasibly up to the level of whole-genome data sets. Computational economies can be achieved by incorporating two forms of pseudo-likelihood: composite and approximate likelihood methods. Composite likelihood enables scaling up to large data sets because it takes the product of marginal likelihoods as an estimator of the likelihood of the complete data set. This approach is especially useful when a large number of genomic regions constitutes the data set. Additionally, approximate likelihood methods can reduce the dimensionality of the data by summarizing the information in the original data by either a sufficient statistic, or a set of statistics. Both composite and approximate likelihood methods hold promise for analyzing large data sets or for use in situations where the underlying demographic model is complex and has many parameters. This paper considers a simple demographic model of allopatric divergence between two populations, in which one of the population is hypothesized to have experienced a founder event, or population bottleneck. A large resequencing data set from human populations is summarized by the joint frequency spectrum, which is a matrix of the genomic frequency spectrum of derived base frequencies in two populations. A Bayesian Metropolis-coupled Markov chain Monte Carlo (MCMCMC) method for parameter estimation is developed that uses both composite and likelihood methods and is applied to the three different pairwise combinations of the human population resequence data. The accuracy of the method is also tested on data sets sampled from a simulated population model with known parameters.
Results
The Bayesian MCMCMC method also estimates the ratio of effective population size for the X chromosome versus that of the autosomes. The method is shown to estimate, with reasonable accuracy, demographic parameters from three simulated data sets that vary in the magnitude of a founder event and a skew in the effective population size of the X chromosome relative to the autosomes. The behavior of the Markov chain is also examined and shown to convergence to its stationary distribution, while also showing high levels of parameter mixing. The analysis of three pairwise comparisons of sub-Saharan African human populations with non-African human populations do not provide unequivocal support for a strong non-African founder event from these nuclear data. The estimates do however suggest a skew in the ratio of X chromosome to autosome effective population size that is greater than one. However in all three cases, the 95% highest posterior density interval for this ratio does include three-fourths, the value expected under an equal breeding sex ratio.
Conclusion
The implementation of composite and approximate likelihood methods in a framework that includes MCMCMC demographic parameter estimation shows great promise for being flexible and computationally efficient enough to scale up to the level of whole-genome polymorphism and divergence analysis. Further work must be done to characterize the effects of the assumption of linkage equilibrium among genomic regions that is crucial to the validity of applying the composite likelihood method.
Background
The availability of whole-genome polymorphism data offers both great opportunities and tremendous challenges to the study of population genetics. Complete genotype information from populations allows increased resolution of parameters in complex evolutionary or demographic models. The challenge is to develop computational methods that permit the efficient use of such large-scale datasets. Likelihood-based coalescent methods have proven very flexible for the analysis of DNA sequence polymorphism. However full likelihood methods, such as Markov chain Monte Carlo (MCMC) and Importance Sampling (IS), are not efficient enough to scale up to genome-wide datasets, necessitating the use of approximate methods for estimating likelihoods. One problem with existing MCMC and IS methods is that a proposal function must be employed to efficiently search among candidate coalescent histories. To circumvent this problem, approximate likelihood methods have proven useful. This class of methods reduces the dimensionality of a full DNA polymorphism dataset to a set of summary statistics, thereby also reducing the number of coalescent histories that need to be sampled to obtain an estimate of the likelihood.
One potential drawback of approximate likelihood methods is that a significant amount of information contained in the original data may be lost. A second problem with full MCMC and IS methods is that integrating over the entire space of possible histories for partially linked polymorphisms along a chromosome can quickly become computationally intractable. In this regard, composite likelihood has been shown to be a promising method for the analysis of partially linked polymorphisms [1–3]. Using this method, the likelihood function is computed marginally for each polymorphism (or contiguous sets of linked polymorphisms) and their product is taken to be an approximation of the full likelihood [4]. Because composite likelihood methods are found to yield consistent estimators of population parameters when the number of regions examined becomes very large [5, 6], they may be particularly applicable to whole-genome datasets.
One of the commonly used class of models in population genetics aims to quantify divergence time by measuring the genetic distance between two populations or species. Yet many measures of genetic distance are susceptible to biases introduced by non-equilibrium conditions during the histories of the populations. Specifically, evolutionary forces that reduce within-population variation are known to inflate measures of genetic distance; such forces may include natural selection [7] or temporal fluctuations in the effective population size [8–10]. In contrast to the locus-specific effects of natural selection, fluctuations in effective size, such as population bottlenecks, are expected to influence the frequencies of alleles throughout the entire genome and therefore should be readily detectable using genome-wide polymorphism data. Thus, it is desirable to develop methods that can not only estimate divergence time from genome-wide polymorphism data, but can also simultaneously account for non-equilibrium demographic events, such as population bottlenecks.
One novel implementation of a coalescent-based method that simultaneously estimates divergence time between two populations and accounts for population bottlenecks is described by Li and Stephan [11]. This method achieves the necessary computational economies by summarizing two-population polymorphism data in the form of the joint frequency spectrum. The joint frequency spectrum is a two-dimensional matrix whose elements are the frequencies of the derived nucleotide allele in a joint sample from two populations or species. Using the joint frequency spectrum, mutations can be classified as either fixed, shared, or exclusive to one of the populations [12]. Li and Stephan [11] estimate divergence time and bottleneck parameters from a joint frequency spectrum constructed from 250 X-linked loci, representing samples of African and non-African populations of Drosophila melanogaster.
Results and Discussion
Behavior of the Markov Chain
Markov chain statistics.
Dataset | Statistic | t _{1} | t _{2} | h | α _{1} | α _{2} | α _{3} |
---|---|---|---|---|---|---|---|
AA | ρ _{50} | 0.1520 | 0.1478 | 0.0353 | 0.0600 | 0.0530 | 0.0099 |
PSRF | 1 | 1 | 1 | 1 | 1 | 1 | |
Upper PSRF | 1 | 1 | 1 | 1 | 1 | 1 | |
AE | ρ _{50} | 0.1587 | 0.1382 | 0.0302 | 0.0565 | 0.0349 | 0.0044 |
PSRF | 1 | 1 | 1 | 1 | 1 | 1 | |
Upper PSRF | 1 | 1 | 1 | 1 | 1 | 1 | |
AO | ρ _{50} | 0.1617 | 0.1324 | 0.0133 | 0.0497 | 0.0326 | 0.0081 |
PSRF | 1 | 1 | 1 | 1 | 1 | 1 | |
Upper PSRF | 1 | 1 | 1 | 1 | 1 | 1 | |
A | ρ _{50} | 0.1197 | 0.2637 | 0.0868 | 0.0269 | 0.0538 | 0.0669 |
PSRF | 1 | 1.02 | 1 | 1.02 | 1 | 1 | |
Upper PSRF | 1.01 | 1.03 | 1 | 1.02 | 1 | 1 | |
D | ρ _{50} | 0.1217 | 0.1916 | 0.0728 | 0.0232 | 0.0604 | 0.0570 |
PSRF | 1 | 1.01 | 1 | 1.01 | 1 | 1 | |
Upper PSRF | 1 | 1.02 | 1 | 1.01 | 1 | 1 | |
G | ρ _{50} | 0.1066 | 0.1567 | 0.1097 | 0.0938 | 0.0053 | 0.0805 |
PSRF | 1.13 | 1.19 | 1.11 | 1 | 1.15 | 1.27 | |
Upper PSRF | 1.27 | 1.39 | 1.22 | 1 | 1.30 | 1.53 |
Simulated Datasets
Parameters for simulated data.
Dataset | t _{1} | t _{2} | h | α _{1} | α _{2} | α _{3} |
---|---|---|---|---|---|---|
A | 0.05 | 0.03 | 0.75 | 10 | 0.1 | 1 |
B | 0.05 | 0.03 | 0.75 | 100 | 0.1 | 1 |
C | 0.05 | 0.03 | 0.75 | 100 | 1 | 1 |
D | 0.05 | 0.03 | 1 | 10 | 0.1 | 1 |
E | 0.05 | 0.03 | 1 | 100 | 0.1 | 1 |
F | 0.05 | 0.03 | 1 | 100 | 1 | 1 |
G | 1 | 0.03 | 0.75 | 10 | 0.1 | 1 |
H | 1 | 0.03 | 0.75 | 100 | 0.1 | 1 |
I | 1 | 0.03 | 0.75 | 100 | 1 | 1 |
J | 1 | 0.03 | 1 | 10 | 0.1 | 1 |
K | 1 | 0.03 | 1 | 100 | 0.1 | 1 |
L | 1 | 0.03 | 1 | 100 | 1 | 1 |
Estimates of Human Bottleneck Parameters
Estimates of human demographic parameters.
Dataset | t _{1} | t _{2} | h | α _{1} | α _{2} | α _{3} |
---|---|---|---|---|---|---|
AA | 0.0473 | 0.0466 | 1.1582 | 0.6148 | 0.7045 | 2.0666 |
(0.0033-0.1454) | (0.0026-0.1451) | (0.4642-3.4620) | (0.1367-7.5359) | (0.0933-8.9268) | (0.7494-6.6319) | |
AE | 0.0598 | 0.0543 | 1.5091 | 0.5035 | 1.1674 | 2.0612 |
(0.0042-0.1659) | (0.0029-0.1699) | (0.5987-4.1903) | (0.1286-5.8619) | (0.1358-9.6137) | (0.7549-6.5703) | |
AO | 0.0605 | 0.0568 | 1.7542 | 0.4511 | 1.3512 | 2.3385 |
(0.0049-0.1665) | (0.0031-0.1772) | (0.5381-4.6644) | (0.1016-5.5873) | (0.1424-9.9650) | (0.8543-7.8275) |
The divergence time of African and non-African populations (t_{ d }) is consistent across comparisons. The estimated median Africa-Asia divergence time is 0.1010 × 2N_{1} with a 95% HPD interval of 0.0416-0.2077. The estimated median Africa-Europe divergence time is 0.1209 × 2N_{1} with a 95% HPD interval of 0.0524-0.2422. Lastly, the estimated median of the Africa-Oceania divergence time is 0.1254 × 2N_{1} with a 95% HPD interval of 0.0536-0.2458. If the current effective population size of the Mandenka is assumed to be on the order of 10^{4} and that the human generation time is 25 years, these numbers correspond to 50,500 years for the Africa-Asia divergence, 60,450 years for the Africa-Europe divergence, and 62,700 years for the Africa-Oceania divergence. There is no support for the hypothesis that these estimated times significantly differ from one another and therefore, a single Africa/non-Africa divergence event cannot be rejected.
There is some suggestion that the rate of coalescence, after divergence from the African population, may be higher in the Asian population than in the other two non-African populations. This can be seen by examining the intensity of the effective population size reduction phase (F = t_{2}/α_{2}). For the Africa-Asia comparison the estimated median F is 0.1027 with a 95% HPD interval of 0.0010-0.2297, while the estimated median F for the Africa-Europe comparison is 0.0603 (95% HPD interval of 0.0019-0.2035) and 0.05122 for the Africa-Oceania comparison (95% HPD interval of 0.0012-0.1936). While the estimated median F from the Africa-Asia comparison does not lie outside the 95% HPD intervals of the other two comparisons, the difference is much more pronounced than the difference between divergence time estimates, yet cannot be considered conclusive evidence. Lastly, the estimated median ratios of effective population sizes for the X chromosome to that of the autosomes is greater than 3/4 for all comparisons. The model assumes a single value of h for all populations in the model. The highest estimated median h is found in the Africa-Oceania comparison, while the smallest is found in the Africa-Asia comparison (Table 3).
Conclusion
Efficient computational methods for fitting complex demographic or evolutionary models to large genomic datasets present a great challenge to population geneticists. The method presented here uses two approximations to achieve the necessary computational efficiencies. The first is an approximate likelihood method, in which large genomic polymorphism datasets are summarized in terms of the joint frequency spectrum. This approach reduces the number of coalescent genealogies that must be sampled to obtain an estimation of the likelihood, compared with most full likelihood-based approaches. Secondly, this methodology is rendered feasible by a composite likelihood approach, which assumes that all polymorphic sites are in linkage equilibrium and have independent genealogical histories. The method is implemented using a model of allopatric population divergence, with a founder event occurring in the history of one of the two diverging populations.
Simulated datasets are used to investigate the accuracy of the MCMC parameter estimation. The method is found to perform well, although it experiences some difficulty delineating the time of the founding event versus the time of population growth parameters, even though the intensity of the bottleneck is estimated accurately. The problem of bottleneck models being partially identifiable, with respect to the timing and magnitude of the reduction phase, was also observed by several other investigators [16–18]. This suggests that current approaches to estimating bottleneck parameters may be limited to estimating the total amount of drift occurring during the reduction phase of the bottleneck (e.g., the product of the bottleneck duration and the magnitude). Again, it remains to be determined what effect the size of a data set may have on this problem of identifiability.
The composite likelihood method is applied to three joint frequency spectra datasets constructed from samples of four continental human populations [14]. The results indicate that there is evidence for a reduction in both the African and non-African effective population sizes, but no evidence that this reduction was followed by a recovery in size that is characteristic of population bottlenecks. As noted by Fay and Wu [19], there is expected to be a period of time, following a population bottleneck, during which the X and the autosomes lag in their signal of population growth compared to the Y chromosome and mitochondrial DNA (i.e., a slower accumulation of rare mutations than the two haploid compartments of the genome).
The conclusion gleaned from the present analysis stands in contrast to those of Voight et al. [17], which is the only other resequencing study to test a population bottleneck model explicitly. Voight et al. use a variant on approximate likelihood to infer bottleneck parameters from 50 resequenced autosomal loci; their analysis was performed separately for datasets constructed from an African, European, and Asian sample. Voight et al. conclude that their African sample cannot reject a constant-size population model, while a bottleneck model is supported by the two samples of non-African populations. While it is clear that the analysis presented by Voight et al. supports a reduction in the effective population size of non-African populations, their evidence for a recovery period (growth) from autosomal data appears to depend upon a set of assumptions, including the absolute value of parameters such as the size of the ancestral population and the severity of the bottleneck, for which reliable estimates do not yet exist. Thus, it appears far from conclusive that there is convincing evidence for recent population growth from either autosomal or X-linked non-coding resequence datasets.
The differential recovery time for the X chromosome versus the autosomes has also prompted Pool and Nielsen [20] to suggest that the X chromosome may recover rare variants more quickly than the autosomes following a population bottleneck and that this could elevate the ratio of X-linked to autosomal nucleotide diversity. Indeed, the human data analyzed here show equivalent levels of X-linked and autosomal diversity [21]. The composite likelihood analysis suggests that, for this dataset, the effective population size of the X chromosome is equal to, or greater than, that of the autosomes, even after taking into consideration that effects of a bottleneck. The effect described by Pool and Nielsen [20] would elevate the X to autosomal diversity ratio upwards of 1,000 generations following the recovery period of the bottleneck.
The conclusion presented here, that a bottleneck alone is insufficient to produce the observed elevation in X-linked diversity, indirectly supports the conclusions of Hammer et al. [21], that a systematic skew in the breeding sex-ratio is responsible for the X to autosome diversity ratio. However, if the expected ratio of X chromosome effective population size to that of the autosomes, under a purely neutral demographic model, is h = 9/(8(2 - ϕ)), where ϕ is the proportion of the population that is female, then lim_{ϕ→1}h = 9/8. This expected maximum value for the ratio of X to autosomal effective population size is lower than all three of our median estimates of h. Although the lower bound of the 95% HPD interval for h varies between 0.46 and 0.60 for the three comparisons, the median estimates suggest that there may be additional forces, such as sex-biased migration, that act at a genome-wide scale. Interestingly, the conclusions reached here contrast with those of Keinan et al. [22], who use single nucleotide polymorphism (SNP) genotype data to infer a lower effective population size for the X chromosome. These contrasting conclusions may reflect the different types of data used in the analyses (resequence versus SNP-typing) or the influence of natural selection near regions of the genome with a high density of coding sequence. Undoubtedly, this dichotomy will be resolved by the forthcoming data from the 1,000 Genomes Project.
The combination of approximate and composite likelihood methods is a promising approach for scaling up population genetics analyses to the level of whole-genome polymorphism data, yet much remains to be done to characterize the validity and accuracy of these methods. Projects are underway to examine further the properties of this method and to apply it to a full-genome polymorphism dataset.
Methods
Coalescent Model and Likelihood Estimation
The proposed composite/approximate likelihood method is applied to a model of allopatric population divergence, in which one of the populations experiences a transient founder event. This scenario is modeled by the coalescent process. Looking backwards in time, two populations with effective sizes N_{1} and N_{2} = α_{1}N_{1}. The second population changes its effective population size at time t_{1} to N_{ B }= α_{2}N_{1}. Then at time t_{1} + t_{2}, the two populations are descended from an ancestral population of size N_{ A }= α_{3}N_{1} (Figure 1). Therefore, the divergence time of the two populations is t_{ d }= t_{1} + t_{2} and time is measured in units of 2N_{1} generations before the present. Thus, the model consists of four different rates of coalescence and its behavior is governed by a total of six parameters, which are collectively referred to as the vector λ= {t_{1}, t_{2}, α_{1}, α_{2}, α_{3}}.
The probability that the coalescent event occurs in population 2 is simply Pr (c_{2}) = 1 - Pr(c_{1}). When a separate joint frequency spectrum representing X-linked data is also being considered, then the parameter h scales both t_{1} and t_{2}, such that t_{1}(X) = t_{1}(A)/h and t_{2}(X) = t_{2}(A)/h, where t_{*} (X) is the time of the event for X-linked loci and t_{*} (A) is the time of the event for autosomal loci. Under the assumption of neutral evolution, when the male and females population sizes are equal, h is expected to be 3/4.
where I (b_{ z }∈ B_{ ij }) is a boolean variable indicating membership of branch b_{ z }in the set B_{ ij }.
where S_{ ij }is the number of sites with configuration (i, j) and provided that i + j is not equal to zero or n_{1} + n_{2}, since neither of these two conditions would result in a polymorphism at that site. A total of r = 10^{5} coalescent genealogies are sampled to estimate Pr(i, j|λ).
By taking the product of the marginal likelihoods across all ℒ polymorphic nucleotide sites in equation (7), it is assumed that all sites are in linkage equilibrium and, therefore, have independent coalescent genealogies. This means that equation (7) belongs to a class of approximate methods known as composite likelihood [4]. There is some suggestion that composite likelihood estimators of population parameters are consistent, particularly when the number of regions examined becomes very large. While the data of Wall et al. [14] consist of only 40 independent regions of the genome, the method may be promising for future analyses of whole-genome polymorphism data. One potential consequence of linkage disequilibrium among sites may be that the resulting credible intervals for the MCMC parameter estimation may be too narrow.
Metropolis-coupled Markov chain Monte Carlo (MCMCMC) Parameter Inference
where f(λ) is the prior density of parameter values and f(S) = ∫L(S|λ) f(λ)d λ is a normalizing constant. The prior densities employed in this study were chosen from repeated exploratory runs of the Markov chain. For the analysis of the human resequence data, the prior distributions for the two time parameters are uniform over the interval (0, 1), while the prior for the ratio of X chromosome to autosomal effective population size is uniform over the interval (0, 5). The prior densities for the relative effective population size parameters are exponential with mean 3. Markov chain transition probabilities are governed by the Metropolis-Hastings criterion [24, 25].
[27]. For the purposes of the present study, an incremental heating scheme was used for eight Metropolis-coupled chains; in this scheme, the heating term for chain x is given by β_{ x }= 1/[1 + ΔK(x - 1)]. A temperature increment parameter of ΔK = 1.1 is used in this study, which yields an average swapping rate of 30-40% between chains. Only the state of the non-heated chain is recorded at each step.
The general Metropolis-coupled Markov chain Monte Carlo (MCMCMC) algorithm proceeds as follows:
1) Randomly assign initial parameters value in λ, sampled from f (λ).
2)Sample r genealogies, each with probability Pr ( _{ r }| λ), and calculate f (λ | S) from equation (8).
3) Randomly select a parameter in λ and propose a new value from f (λ) to obtain λ'.
4) With probability U, let λ= λ', otherwise retain λ.
5) Randomly select two chains with states λ_{ x }and λ_{ y }and, with probability V, exchange the values of λ_{ x }and λ_{ y }, otherwise retain the current state of each chain.
6) Go to step 2.
In step 6, the algorithm returns to step 2, rather than step 3. This means that at each step the likelihood of the current state is recalculated, rather than retained from the previous step. Retaining the previous likelihood may result in the chain becoming stuck in a state that, by chance from the Monte Carlo sample, yields an unusually high likelihood; however it is also guaranteed to produce samples from the true posterior distribution, regardless of the Monte Carlo sample size [29]. While the true variance of the target distribution can be obtained with this "sticky" method, it may also impede the convergence behavior of the chain and, hence, require more steps in the chain. In contrast, the practice of recalculating the likelihood ("smooth" MCMC) are expected to result in higher acceptance rates, while the resulting posterior distribution may also have increased variance over that of the true target distribution, but it may also improve convergence behavior [30]. Initial runs using the "sticky" MCMC approach yielded lower overall rates of parameter mixing than did the "smooth" MCMC method. Therefore, only results using the "smooth" MCMC algorithm are reported here.
For each dataset, ten independently seeded replicates of the Markov chain are run for 10^{5} steps, not including an initial "burn-in" period of 10^{3} steps. A C++ program, written to perform the MCMCMC method called mc3 is freely available over the internet at http://www.rochester.edu/College/BIO/labs/Garrigan/software.htm. The OpenMP application program interface http://www.openmp.org/ is used to distribute parallelized Markov chains across the shared memory of eight cores in dual Intel Quad-core Xeon processors running at 2.66 GHz. The duration of the Metropolis-coupled runs are typically 40-80 hours, depending upon the dataset or the MCMC algorithm. The mc3 program allows users to input the desired number of Metropolis-coupled chains to run on their particular computer.
The potential scale reduction factor (PSRF) is used to quantify the convergence of the ten replicate runs to the stationary distribution [31], as implemented in the CODA package for the free statistical programming environment Rhttp://www.r-project.org. The CODA package is also used to calculate posterior probability densities, parameter autocorrelations and cross-correlations, as well as the 95% highest posterior density (HPD) intervals.
Assessing the Accuracy of the Method
Twelve simulated joint frequency spectra are generated under a two-population divergence model with known parameters, given in Table 1. Marginal posterior probability distributions for the model parameters are then estimated, using the method outlined above. For each simulated data set, a joint frequency spectrum of 1000 unlinked single nucleotide polymorphism loci is generated for both the X chromosome and the autosomes, resulting in a total of 2000 unlinked polymorphic sites. The number of sampled chromosomes in each simulation is n_{1} = 20 and n_{2} = 20, for both the X-linked and autosomal joint frequency spectra. The prior distributions used for the analysis of simulated data sets A-F are the same as those given above for the analysis of the human resequence data, except the prior for α_{1} was exponentially distributed with mean 50. For simulated data sets G-L, only the prior for t_{1} was altered to a uniform distribution over the interval (0, 5).
Application to Human Resequence Data
The composite likelihood method is then applied to the X-linked and autosomal resequence data of Wall et al. [14]. The data consist of 14 X chromosomes and 28 autosomes sampled from the Mandenka population, a sub-Saharan Africa food-producing population, 16 X chromosomes and 32 autosomes sampled from the Han Chinese population, 16 X chromosomes and 32 autosomes sampled from the Basque population from France, and 14 X chromosomes and 18 autosomes sampled from the Melanesian population. For each population, 20 autosomal and 20 X-linked loci are resequenced for a total of ~210 kilobases from each individual. Using these four datasets, three pairwise analyses are performed: (1) Mandenka-Han Chinese for the Africa-Asia (AA) comparison with 318 X-linked polymorphisms and 655 autosomal polymorphisms, (2) Mandenka-Basque for the Africa-Europe (AE) comparison with a total of 328 X-linked polymorphisms and 648 autosomal polymorphisms, and (3) Mandenka-Melanesia for the Africa-Oceania (AO) comparison with 148 X-linked polymorphisms and 614 autosomal polymorphisms.
One interesting question is whether the parameters of a putative non-African bottleneck or divergence times are consistent across all three comparisons, suggesting a common historical event shared by all non-African populations, or if there is clear evidence for distinct, independent historical divergence/bottleneck events for different non-African populations. These resequence data were chosen because they likely represent neutrally evolving regions of the genome and do not suffer from the ascertainment bias that may artificially skew the frequency spectra of other single nucleotide polymorphism datasets [14].
Another motivation to utilize the resequence data of Wall et al. [14] is that Hammer et al. [21] find that, after correcting for mutation rate, nucleotide diversity for the X-linked loci is nearly equal to the nucleotide diversity for the autosomal loci. Hammer et al. [21] conclude that high variance in male reproductive success may account for the nearly equal effective population sizes of the X and autosomes; however, Pool and Nielsen [20] show that the X versus autosomal effective size may be equal during the growth phase following a population bottleneck. The intention here is to use the method to ascertain whether rapid population growth following a bottleneck, or a decrease in male effective population size, may result in increased levels of X chromosome genetic diversity.
Declarations
Acknowledgements
I wish to thank J. Wakeley, M. Hammer and two anonymous reviewers for helpful comments on previous drafts of this paper and R. Lewontin and the University of Rochester for support.
Authors’ Affiliations
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