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Keywords:

  • approximate Bayesian computation;
  • effective population size;
  • genetic drift;
  • natural selection;
  • population genetics;
  • time-sampled data

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Data accessibility
  10. Supporting Information

With novel developments in sequencing technologies, time-sampled data are becoming more available and accessible. Naturally, there have been efforts in parallel to infer population genetic parameters from these data sets. Here, we compare and analyse four recent approaches based on the Wright–Fisher model for inferring selection coefficients (s) given effective population size (Ne), with simulated temporal data sets. Furthermore, we demonstrate the advantage of a recently proposed approximate Bayesian computation (ABC)-based method that is able to correctly infer genomewide average Ne from time-serial data, which is then set as a prior for inferring per-site selection coefficients accurately and precisely. We implement this ABC method in a new software and apply it to a classical time-serial data set of the medionigra genotype in the moth Panaxia dominula. We show that a recessive lethal model is the best explanation for the observed variation in allele frequency by implementing an estimator of the dominance ratio (h).


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Data accessibility
  10. Supporting Information

The study of temporal changes in allele frequency originated with two of the early founders of population genetics (Fisher 1922; Wright 1931), in which the fate of an allele was considered under a variety of models – including neutrality, positive and negative selection, and migration. Their celebrated debate on the relative roles of selection and drift in shaping the course of evolution also encompassed time-sampled data (Fisher & Ford 1947; Wright 1948), upon the publication of the time-series analysis of the medionigra phenotype in the moth Panaxia dominula. Following on this, and alternatively taking an experimental evolution approach, Clegg studied the dynamics of gene frequency change in Drosophila melanogaster (Clegg et al. 1976; Cavener & Clegg 1978; Clegg 1978). However, owing to the limited availability of genetic markers, relatively few time-sampled data sets were available for consideration throughout the remainder of the 20th century. Thus, most test statistics for distinguishing the effects of selection from drift were focused on single time point data sets – basing inference on patterns in the site frequency spectrum, linkage disequilibrium and polymorphism/divergence (for a review, see Crisci et al. 2012).

Recently, sequencing data from multiple time points has become increasingly common owing to novel evelopments in sequencing technologies (Schuster 2008) – coming from the fields of both ancient genomics and experimental evolution. This additional temporal component has the promise of providing improved power for inferring population genetic parameters compared with single time point-based analyses, as the trajectory of the allele itself provides valuable information about the underlying selection coefficient.

However, there are a limited number of methods currently available to estimate these parameters from time-sampled data. Moment-based methods (Kimura & Crow 1963; Pamilo & Varvio-Aho, 1980; Nei & Tajima, 1981; Waples 1989; Jorde & Ryman, 2007) have been proposed utilizing the variance of gene frequency changes to infer effective population size (Ne). In addition, likelihood-based methods (Williamson & Slatkin, 1999; Anderson et al. 2000; Berthier et al. 2002; Anderson 2005) have been proposed to calculate the probability of a given data observation given a predefined model. Efforts to incorporate selection into these estimation procedures have only recently begun, and given the rapidly increasing availability of such sequencing data sets, we now have a unique opportunity to readdress the puzzle of distinguishing genetic drift from selection with greater precision and power.

Thus, we present here new software implementing and expanding an approximate Bayesian computation (ABC, Sunnåker et al. 2013) approach to jointly infer per-site effective population sizes (Ne) and selection coefficients (s) from time-sampled data, initially described in Foll et al. (2014) to search for resistance mutations in time-sampled data from the influenza virus. Furthermore, we compare this approach with existing likelihood-based methods (Bollback et al. 2008; Malaspinas et al. 2012; Mathieson & McVean, 2013), in order to inform future users on the most suitable method to be applied for any given data set.

Materials and methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Data accessibility
  10. Supporting Information

Ne-based ABC method

The data X consist of allele frequency trajectories measured at L loci: xi (= 1,…,L). The Ne-based ABC methodology infers both the effective population size Ne shared by all loci and L locus-specific selection coefficients si (= 1,…,L). At a particular locus i, we can approximate the joint posterior distribution as (see Foll et al. 2014 for details):

  • display math

where T(X) = T(X1,…,XL) denotes summary statistics that are a function of all loci together chosen to be informative about Ne, and U(Xi) denotes locus-specific summary statistics chosen to be informative about si. A two-step ABC algorithm as proposed by Bazin et al. (2010) is used to approximate this posterior:

Step 1. Obtain an approximation of the density

  • display math

Step 2. For locus i = 1 to i = L:

  1. Simulate K trajectories Xi,k from a Wright–Fisher model with si randomly sampled from its prior and Ne from the density obtained in step 1.
  2. Compute U(Xi,k) for each simulated trajectory.
  3. Retain the simulations with the smallest Euclidian distance between U(Xi) and U(xi) to obtain a sample from an approximation to P(si|Ne,Xi)P(Ne|X) = P(Ne,si|X).

In the original algorithm (Bazin et al. 2010), the first step is also achieved using ABC. In our case, we define T(X) as a single statistic given by Jorde and Ryman (2007) Fs′ unbiased estimator of Ne:

  • display math

where x and y are the minor allele frequencies at the two time points separated by t generations, z = (x+y)/2, and inline image is the harmonic mean of the sample sizes nx and ny at the two time points expressed in number of chromosomes (twice the number of individuals for diploids). We average Fs′ values over sites and times to obtain a genomewide estimator of Ne = 1/Fs′ for haploids and Ne = 1/2Fs′ for diploids (Jorde & Ryman 2007). A Bayesian bootstrap approach (Rubin 1981) is used to obtain a distribution for P(Ne|T(X)). Please note that we use the common notation where the effective population size Ne corresponds to number of individuals, and the corresponding number of chromosomes for diploids is 2Ne.

In the second step, simulations are performed using a Wright–Fisher model with an initial allele frequency and sample sizes matching the observed ones (simulation code available in the downloadable software package). At each site, we utilize two summary statistics derived from Fs′: U(Xi) = (Fsd′i, Fsi′i) with Fsd′ and Fsi′ calculated, respectively, between pairs of time points where the allele considered is decreasing and increasing in frequency, such that at a given site Fs′ Fsd′ + Fsi′. For the diploid model, we define the relative fitness as wAA = 1 + s, wAa = 1 + sh and waa = 1 where h denotes the dominance ratio (1 = dominant, 0.5 = codominance, 0 = recessive), and as wA = 1 + s and wa = 1 for the haploid model (Ewens 2004).

We here implement this two-step approach in a new command line C++ program termed Wright–Fisher ABC (WFABC). This estimation procedure is suitable for both haploid and diploid models of selection. Source code and binary executables for Linux, OS X and Windows are freely available from the ‘software’ page of the Jensen Lab website: http://jensenlab.epfl.ch/.

Likelihood-based methods

Currently, there are three likelihood-based methods available for inferring population genetic parameters from time-serial data (Bollback et al. 2008; Malaspinas et al. 2012; Mathieson & McVean 2013), based on a hidden Markov model (HMM) to model the allele frequency trajectory.

Bollback et al. (2008) co-estimate the selection coefficient (s) and the effective population size (Ne) from a diffusion process, by approximating the Wright–Fisher model and computing the maximum likelihood at fixed intervals. Malaspinas et al. (2012) additionally estimate the allele age (t0) and further approximate the Wright–Fisher model through a one-step process. Mathieson and McVean (2013) estimate only the selection coefficient (s) assuming that Ne is known, using an expectation–maximization (EM) algorithm that can be extended to the case of a structured population. The fitness is parameterized as with our Ne-based ABC approach. The likelihood function of the parameters of interest – θ = (γ,Ne) for the Bollback et al. (2008) method and θ = (γ,Ne,t0) for the Malaspinas et al. (2012) method, where γ = 2Nes – is conditioned over all population allelic frequencies inline image at sampling times = (t1,…,tm) and is given as:

  • display math

where ik is the frequency of the minor allele at the sampling time k, and jk is the true minor allele frequency at the sampling time k, where inline image. The first term of the likelihood is the emission probability, which is modelled as a binomial sampling, and the second term represents the transition probabilities in the Markov chain. In both methods, the Markov chain is approximated by a diffusion process, from which the transition probabilities are given as the backward Kolmogorov equations (Ewens 2004).

The major difference between these three likelihood-based methods comes from the implementation of how these probabilities are calculated. The Bollback et al's. (2008) method utilizes numerical approximations to evaluate the likelihood function, first using the Crank–Nicolson approximation (Crank et al. 1947) for the backward Kolmogorov equation and second using numerical integration for the emission probability. Mathieson and McVean (2013) use an EM algorithm to find the maximum-likelihood estimate (MLE) of s based on the MLE for complete observations (i.e. at every generations). Malaspinas et al. (2012) approximate the diffusion process by a one-step process. The state space of the process is the population allele frequencies that are denoted by (z0,…, zH−1), where z0 = 0, zH−1 = 1 and zk−1 < zk. The one-step process only allows transitions between two adjacent states (i.e. from zi to zi−1, or from zi to zi+1); hence, the infinitesimal generator Q can be constructed as a tridiagonal H × H matrix:

  • display math

where βi denotes the transition rate from zi to zi+1, δi the transition rate from to zi to zi−1, and ηi is the rate of no transition such that ηi = 1−(βi + δi). The appropriate choice for the parameters β and δ of the matrix Q is given for both the diploid and the haploid Wright–Fisher models in Malaspinas et al. (2012) and Foll et al. (2014), respectively.

Simulated data sets for testing

For real data, it is important to take into account the nonrandom criteria one used to select sites from the genome for analyses. This so-called ascertainment bias is known to be very important for single time point SNP data (Nielsen & Signorovitch 2003), but has not been studied so far for time-sampled data. One of the reasons is that including realistic ascertainment schemes in likelihood-based methods is a difficult task. The one-step process used in Malaspinas et al. (2012) can be adjusted to match the way in which ancient DNA data are generally collected, such that the locus considered is polymorphic at the present time. This condition implies that the process can never reach the absorbing states 0 and 1, and one needs to remove the first and last rows and columns of the Q matrix. In the current implementation of Malaspinas et al. (2012), only this conditional case is available and has been tested for this study. The Bollback et al. (2008) method implements an unconditional model (no ascertainment) as well as the particular case where the allele is known to be beneficial and reaches fixation during the sampling period. There is no ascertainment model implemented in Mathieson and McVean (2013) method. One distinct advantage of this simulation-based approach is the ability to easily incorporate different ascertainment schemes into the estimation procedure, as one simply needs to be able to simulate them. We here present three such nonexclusive schemes: (i) observing a minimum allele frequency over the entire trajectory, (ii) observing a minimum allele frequency at the last time point [including fixation like in the Bollback et al. (2008) method] and (iii) being polymorphic at the last time point [like in the Malaspinas et al. (2012) method]. We note in particular that the first case is something that will be present in any data set but is not available in likelihood-based methods. Because the three available likelihood methods implement different ascertainment processes, and these processes lead to more or less informative data, it is not possible to make a direct comparison of their performance. For this reason, we separately compare them with WFABC.

For this comparative study, we generate simulated data sets using the Wright–Fisher model with a range of parameter values for the effective population size (Ne) and selection coefficient (s). For the diploid Wright–Fisher model, codominant time-serial allele frequency data from 1000 replicates for Ne = (200, 1000, 5000) and s ∈[0,0.4] are generated. To assess the precision and accuracy of these methodologies in estimating two potential empirical cases of small s values and large s values, the selection coefficients are divided into two sets of = (0, 0.005, 0.01, 0.015, 0.02) and = (0, 0.1, 0.2, 0.3, 0.4). For WFABC, we retained the best 1% of 500 000 simulations based on the Euclidian distance between the observed and simulated Fsd′ and Fsi′ statistics and use the mean of the posterior distributions obtained for s using a rejection ABC algorithm (Sunnåker et al. 2013) as a point estimate. First, we use these simulated data sets to demonstrate the performance of WFABC when different sampling time points and different sample sizes are used. Second, we show the influence of the ascertainment procedure using two examples: an unconditional but unrealistic case where all trajectories start with an initial minor allele frequency of 10%, and one ascertained case where a new mutation occurs at the first generation, and only the trajectories reaching a frequency of 5% at least in one sampling time point are kept. We use these simulated data sets to compare the performance of WFABC with the method of Mathieson and McVean (2013).

Finally, we focus on the ascertainment case of the allele segregating at the last sampling time point, as this model is the only realistic one allowing us to compare WFABC with a likelihood method. Depending on the strength of selection and the effective population size, mutations reach fixation more or less rapidly. In order to generate data with the condition of being polymorphic at the last sampling time point, the number of generations is adjusted to have a nonzero probability for this condition, allowing us to efficiently simulate such scenarios. A new mutation occurs at the first generation, and 100 samples are drawn randomly through binomial sampling, with 12 sampling time points. Using these simulated data sets, comparative studies of WFABC with Malaspinas et al. (2012) and Bollback et al. (2008) methods are carried out, with the search range and the prior for the selection coefficient set as s ∈[−0.1,0.1] for small values ranging from 0 to 0.02, and s ∈[−0.2,0.6] for large values ranging from 0 to 0.4. Simulated data sets from the haploid Wright–Fisher model are also generated, but only for one set of Ne = 1000 to validate the performance of the modified haploid version of Malaspinas et al. (2012) model described in Foll et al. (2014). For Malaspinas et al. (2012), we used the quadratic grid option (Gutenkunst et al. 2009) for computing the likelihood and the Nelder–Mead simplex algorithm option to find the maximum likelihood (Nelder & Mead 1965).

Both the Bollback et al. (2008) and Malaspinas et al. (2012) methods estimate parameters based on a single allele frequency trajectory, which thus contains limited information about Ne. WFABC utilizes multiple sites in order to estimate Ne, which is then used as a prior to estimate selection coefficients. In order to implement an equal comparison, we fix Ne to its true value in all scenarios and evaluate the ability of these approaches to estimate selection coefficients.

To demonstrate the advantage of estimating Ne correctly, a final multilocus scenario is generated where both s and Ne values are inferred. For Ne fixed at 1000, 10 000 trajectories are simulated with 500 being under selection with s randomly chosen in [0.05,0.4]. We used a search range of γ ∈[0,2000] and Ne ∈[50,2000] for the Malaspinas et al. (2012) method, and a uniform prior s ∈[−0.2,0.6] for WFABC. For WFABC, all the 10 000 trajectories are used in the first step to obtain a posterior distribution for Ne, which is used in the second step to estimate s at each locus individually as explained above.

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Data accessibility
  10. Supporting Information

Performance of the examined estimation procedures

The performance studies of WFABC are presented in a standard box plot with the box as the first, second and the third quartiles, and the whiskers as the lowest and highest datum within the 1.5 interquartile range of the lower and upper quartiles, respectively. The first box plot shows the estimated selection coefficients for different numbers of sampling time points as 12, 6 or 2 (Fig. S1, Supporting information). As expected, a larger number of sampling time points yields a better estimate of s – however, WFABC is able to estimate s accurately with as small as two sampling time points for moderate values of s < 0.2. The second box plot shows the estimated selection coefficients from WFABC for the sample sizes of 1000, 100 and 20 with Ne = 1000 (Fig. S2, Supporting information). The estimation of s improves as the number of sample sizes increases as expected.

For the comparative studies of WFABC with the Mathieson and McVean (2013) method, the unconditional case with an initial minor allele frequency of 10% is shown in box plots for the small s values (Fig. S3, Supporting information) and the large s values (Fig. S4, Supporting information). For the small s values, the Mathieson and McVean (2013) method performs better than WFABC. However, for larger values of s, the Mathieson and McVean (2013) method shows an increasing trend of underestimation, whereas WFABC remains unbiased (Fig. S4, Supporting information). For the conditional case of ascertaining the simulated data sets with a minimum frequency of 5% at one sampling time point, the performance of WFABC is noticeably better for both the small and large s values (Figs 1 and 2). Figure 1 shows that the Mathieson and McVean (2013) method constantly overestimates small s values. Figure 2 indicates that this bias is compensated by the underestimation shown above for large s values.

image

Figure 1. Box plot for the estimated small selection coefficients from each simulated replicate of the Wright–Fisher diploid model, with Ne = 1000, 90 generations, and the minimum frequency of 5% at one sampling time point. The grey rectangles correspond to Wright–Fisher ABC and the white rectangles to the Mathieson and McVean (2013) method. The red circles are the true values for s, and the blue triangles are the mean of the estimated s values.

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image

Figure 2. Box plot for the estimated large selection coefficients from each simulated replicate of the Wright–Fisher diploid model, with Ne = 1000, 90 generations, and the minimum frequency of 5% at one sampling time point. The grey rectangles correspond to Wright–Fisher ABC and the white rectangles to Mathieson and McVean (2013) method. The red circles are the true values for s, and the blue triangles are the mean of the estimated s values.

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For the conditional case studies of the allele segregating at the last sampling time point, we obtain six sets of results with the varying parameters for the diploid model and two sets of results for the haploid model from the three approaches. Results are given for Ne = 1000 for small s values (Fig. 3) and large s values (Fig. 4). Additionally, tables with the calculations of the root-mean-square error (RMSE) and bias for each set of results are shown for the small s values and the large s values in Tables 1 and 2, respectively. Note that MSE is defined as the sum of the variance and the squared bias of the estimator and therefore incorporates information from both precision (variance) and accuracy (bias).

Table 1. RMSE and bias for the small s and Ne = 1000 scenario for the Wright–Fisher diploid model
 s = 0s = 0.005s = 0.01s = 0.015s = 0.02
  1. RMSE, root-mean-square error; WFABC, Wright–Fisher ABC.

Bias
WFABC−0.0035−0.0046−0.0032−0.0014−0.0015
Malaspinas et al. (2012)−0.0044−0.0050−0.0039−0.0026−0.0022
RMSE
WFABC0.0170.0180.0170.0160.018
Malaspinas et al. (2012)0.0130.0140.0120.0110.011
Table 2. RMSE and bias for the big s and Ne = 1000 scenario for the Wright–Fisher diploid model
 s = 0s = 0.1s = 0.2s = 0.3s = 0.4
  1. RMSE, root-mean-square error; WFABC, Wright–Fisher ABC.

Bias
WFABC0.017−0.0170.0064−0.0044−0.0083
Malaspinas et al. (2012)−0.0024−0.018−0.025−0.056−0.091
RMSE
WFABC0.0590.0600.0500.0460.059
Malaspinas et al. (2012)0.0470.0460.0480.0840.13
image

Figure 3. Box plot for the estimated small selection coefficients from each simulated replicate of the Wright–Fisher diploid model, for Ne = 1000 and 500 generations. The grey rectangles correspond to Wright–Fisher ABC, the white rectangles to the Malaspinas et al. (2012) method and the gold rectangles to the Bollback et al. (2008) method. The red circles are the true values of s, and the blue triangles are the mean of the estimated s values.

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image

Figure 4. Box plot for the estimated large selection coefficients from each simulation replicate of the Wright–Fisher diploid model, for Ne = 1000 and 90 generations. The grey rectangles correspond to Wright–Fisher ABC, the white rectangles to the Malaspinas et al. (2012) method and the gold rectangles to the Bollback et al. (2008) method. The red circles are the true values for s, and the blue triangles are the mean of the estimated s values.

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Comparing the three approaches for small s values (Fig. 3), WFABC and the Malaspinas et al. (2012) approach produce good estimates of s, whereas the Bollback et al. (2008) method fails to infer different s values. From the box plot, both the median and the mean of WFABC and the Malaspinas et al. (2012) approach are close to the true s value. However, WFABC appears to contain a longer tail of underestimated s values, and the interquartile range boxes are wider. Table 1 provides more quantitative comparisons of their performance. The RMSE values reveal that for all the cases of Ne = 1000 and small s, the Malaspinas et al. (2012) approach is generating more precise estimates. On the other hand, WFABC produces estimates of less bias for all small s values in this set.

Comparing the three approaches for large s values (Fig. 4), WFABC and the Malaspinas et al. (2012) approach produce reasonable estimates of s, whereas the Bollback et al. (2008) method again fails to detect any difference in s values. For s values larger than 0.1, the performance of WFABC is significantly better than the Malaspinas et al. (2012) approach both in accuracy and precision (Table 2) – producing estimates with 10-fold less bias and twofold less error than the Malaspinas et al. (2012) approach. This gap in performance increases from = 0.2 to = 0.4, thus this trend may be extrapolated to higher s values.

Notably, the Bollback et al. (2008) method consistently estimates = 0 for all examined data sets. This poor performance has been evaluated for various conditions including changing grid sizes, conditioning on fixation and varying sampling time points – with no perceptible difference in results. To verify our usage, the method was tested with the exact parameters utilized in the initial paper for the example of bacteriophage MS2 (Bollback et al. 2008), resulting in the successful replication of their results (Figs S5 and S6, Supporting information). Notably, the performance of the statistic depends on the choice of the search range for γ, due to the presence of local peaks in the likelihood function. When the interval is chosen to be narrow and centred around the true s value of 0.4, the estimated s is correctly given owing to the local maximum (replicating their result). However, when the full likelihood surface is examined, the global maximum is present near 0 as observed in all simulated test replicates. For this reason, the Bollback et al. (2008) method is excluded from the further analyses of performance, as well as from the illustrative data application.

As shown, our comparison studies suggest that WFABC and Malaspinas et al. (2012) approaches perform almost equally well for estimating selection coefficients for both small and large values. While WFABC slightly overestimates the selection coefficient in the cases of small s and small Ne (Fig. S7, Table S1, Supporting information), it is notable that the Malaspinas et al. (2012) perform particularly well when s is in the range of 0.01 and 0.02, as observed by Malaspinas et al. (2012). In contrast, WFABC exhibits less bias when s values are large (i.e. > 0.1) for small Ne (Fig. S8, Table S2, Supporting information). Thus, we conclude that the two methods estimate the selection coefficient to a high accuracy and precision for Ne ∈[200,1000] and for the condition of the allele segregating at the last sampling time point. In general, although the difference in performance both in precision and accuracy is minor, the Malaspinas et al. (2012) approach appears to give superior results for small s values, whereas WFABC for large s values.

However, the Malaspinas et al. (2012) approach encounters a limitation in computation efficiency for the set of Ne = 5000 (Figs S9 and S10; Tables S3 and S4, Supporting information). For the large s values, the computation time was too lengthy to complete the 1000 replicates; thus, only the results from WFABC are shown (Fig. S10, Supporting information). Compared with the small Ne values, the estimation of large s is becoming more accurate and precise as Ne gets large (Tables 2, S2 and S4, Supporting information). This trend is the same for small s values (Tables 1, S1 and S3, Supporting information), although the bias appears to switch from overestimation to underestimation as Ne increases. For the Malaspinas et al. (2012) approach, the accuracy of inferring small s values improves as Ne increases, although the precision remains similar (Tables 1, S1 and S3, Supporting information). However, for Ne = 5000 and > 0.01, the 1000 replicates were not complete due to the lengthy computational time; thus, the RMSE and bias values are not available (Table S3, Supporting information).

For the haploid model, WFABC shows superiority in both accuracy and precision for inferring any s values compared with the Malaspinas et al. (2012) approach (Figs S11 and S12, Tables S5 and S6, Supporting information).

Finally, the multilocus scenario demonstrates the great benefit provided by the ability of WFABC to use the information shared by all loci to estimate Ne. The RMSE of the selection coefficients calculated over the 500 trajectories under selection is less than half that obtained using the Malaspinas et al. (2012) approach (0.049 vs. 0.10, see Fig. 5).

image

Figure 5. Correlation between the true and the estimated values of the selection coefficients s for the multilocus scenario using Wright–Fisher ABC (a) and the Malaspinas et al. (2012) methods (b). The red line represents the identity (x) line.

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In summary, WFABC is superior for diploid cases when both s and Ne values are large (i.e. γ = 2Nes is large), for any haploid cases, and when multiple loci are available, whereas the Malaspinas et al. (2012) approach is suitable for cases when γ values are small.

Comparison of computational efficiency

Apart from accuracy and precision, an important difference in performance between these methods is the computational efficiency. A major advantage of WFABC is the computational speed, which, for example, allows for an evaluation of all observed sites in the genome in order to identify putatively selected outliers (Foll et al. 2014) and to estimate the full distribution of fitness effects of segregating mutations. For the Malaspinas et al. (2012) approach, the computational time becomes heavy when γ is larger than 200 and is no longer feasible when γ is approaching 1000, whereas WFABC has no restriction on the sizes of Ne and s. For the Mathieson and McVean (2013) approach, the CPU time of estimating only the selection coefficient of each site is around 2 s regardless of the size of Ne, but still is slower than WFABC. Therefore, we suggest that the likelihood-based approach is preferable in cases where both the candidate mutation and effective population sizes are known a priori, whereas WFABC is preferable in the absence of this information. The average CPU time spent for each replicate for the diploid model is shown in Table 3. We note that when the Malaspinas et al. (2012) method is also used to estimate Ne, the difference in CPU time between the two methods is even greater.

Table 3. Average CPU time consumed in seconds to run one simulation replicate
 Ne = 200Ne = 1000Ne = 5000
Small sLarge sSmall sLarge sSmall sLarge s
  1. WFABC, Wright–Fisher ABC.

WFABC5315630070
Malaspinas et al. (2012)38025035018 00080 000

Data application

We applied WFABC to a time-serial data set of the medionigra morph in a population of Panaxia dominula (scarlet tiger moth) at Cothill Fen near Oxford. This colony was first studied by Fisher and Ford (1947), and further observations have been collected almost every year until at least 1999 (Cook & Jones 1996; Jones 2000). The moth P. dominula has a 1-year generation time and lives near the Oxford district in isolated colonies. The typical phenotype has a black forewing with white spots and a scarlet hind wing with black patterns (see Fisher & Ford 1947). The medionigra allele produces the medionigra phenotype when heterozygous, and the bimacula phenotype when homozygous, changing the pigment and patterns on the wings to an increasing degree, and is almost never observed (Sheppard & Cook, 1962). Using our notation above, we denote by A the medionigra allele, and the fitness of the three genotypes is given by wAA = + s (bimacula), wAa = + sh (medionigra) and waa = 1.

The respective role of drift and natural selection to explain the rapid decline of the medionigra allele frequency after 1940 (Fig. 6) was the subject of a strong debate between Fisher and Wright (Fisher & Ford 1947; Wright 1948), with Wright arguing that the observed pattern until 1946 could be explained by drift alone with an effective population size of Ne = 150 (Wright 1948). The same data containing further observations have been re-analysed several times (Cook & Jones 1996; O'Hara 2005; Mathieson & McVean 2013) with most studies concluding that the medionigra allele is negatively selected with s = −0.14 (Cook & Jones 1996) or s = −0.11 (Mathieson & McVean 2013) based on a codominant model (h = 1/2). Recently, Mathieson and McVean (2013) found that a fully recessive medionigra allele (= 0) fits with a higher likelihood compared with = 1/2 but with a much larger selection coefficient s ≈ −1. In particular, this recessive lethal model explains better the persistence of the medionigra allele at a low frequency for so many generations (Mathieson & McVean, 2013). However, this large value of s is outside the range for which their approximations are valid, and this hypothesis could not be formally tested.

image

Figure 6. Frequency of the medionigra allele in the Cothill Panaxia dominula moth population (Cook & Jones, 1996; Jones 2000).

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Our ABC approach based on simulations can deal with the full range of s values, and we further extended it here to co-estimate the degree of dominance h. We followed the intuitive idea of Mathieson and McVean (2013) that the number of generations during which the allele persists at low frequency is informative for the degree of dominance h. More formally, we added two summary statistics in our ABC procedure, tl, defined as the number of generations where the allele frequency is below 5% and not lost; th defined as the number of generations where the allele frequency is above 95% and not fixed. For the moth data, we have tl = 54 and th = 0 (see Fig. 6). Using the simulated distributions, Fsd′ and Fsi′ are both normalized by the largest standard deviation max(sd(Fsd′),sd(Fsi′)), as well as tl and th by max(sd(tl),sd(th)). We followed Mathieson and McVean (2013) and ran our ABC method using a fixed population size of 2Ne = 1000, and we plot the corresponding joint posterior distribution for s and h in Fig. 7a. The mode of the joint posterior distribution is at s = −1 and h = 0.043, supporting the idea of a lethal bimacula phenotype (wAA = 0) with a deleterious medionigra phenotype (wAa = 0.96) consistent with previous observations (Sheppard & Cook, 1962). The shape of the joint posterior distribution (Fig. 7a) shows that the medionigra allele is either very strongly selected against and almost completely recessive (h ≈ 0) or codominant with a weaker selection coefficient (s  0.2). Even if the density is larger in the recessive lethal region of the parameter space (bottom left corner in Fig. 7a), the two-dimensional 90% highest posterior region includes the alternative codominant hypothesis. As it has been argued that the effective population size could be of the order of a few hundred (Wright 1948; O'Hara 2005), we also ran the analysis with 2Ne = 100. The joint posterior distribution for s and h in Fig. 7b shows a similar pattern with a mode at s = −1 and = 0. However, the surface is flatter and the two-dimensional 90% highest posterior region now includes = 0, confirming Wright's view that a small enough population size could explain the observed pattern with genetic drift alone (Wright 1948; Mathieson & McVean 2013). We finally ran the analysis using a uniform prior for 2Ne between 100 and 10 000 to take into account its uncertainty in the estimate. In this case, the joint posterior gives a stronger support for a lethal bimacula phenotype with a deleterious medionigra phenotype as compared to 2Ne = 1000 (Fig. S13, Supporting information).

image

Figure 7. Two-dimensional joint posterior distribution for s and h for the moth Panaxia dominula data using 2Ne = 1000 (a) and 2Ne = 100 (b). The grey lines delimit two-dimensional α highest posterior density regions for α = 0.9 (largest region), 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2 and 0.1 (smallest region).

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Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Data accessibility
  10. Supporting Information

Maximum-likelihood estimators have the advantage of being consistent and efficient, but the computational method used to find the maximum likelihood can be critical in complex models. For instance, even though the underlying model of the Bollback et al. (2008) approach is similar to the Malaspinas et al. (2012) approach and the Mathieson and McVean (2013) approach, the difference in performance appears to be coming from the difference in the implementation of computational methods. On the other hand, ABC-based methods reduce data sets into summary statistics, and thus, the performance of these methods is dependent on the chosen statistics. The difference in performance demonstrated through this comparative study between our newly proposed ABC-based method and the likelihood-based methods is very small in most cases. Therefore, we can hypothesize that the two summary statistics – Fsd′ and Fsi′ – are close to being statistically sufficient.

A disadvantage of the likelihood methods arises from the limitations imposed by the assumptions made for diffusion approximation. To approximate the Wright–Fisher model with the diffusion process, one makes the assumption that the Markov process is continuous in state space and time. This assumption requires s to be small and Ne to be large (Durrett, 2008). Thus, likelihood methods based on diffusion approximation are inevitably limited to the cases of large effective population sizes and small selection coefficients and may explain the biases observed at large s values in this study. Additionally, the computational efficiency limits the value of γ to be under 1000 for the Malaspinas et al. (2012) approach, and even if s is small, Ne is again limited to be under 5000. Thus, despite having good performance, the likelihood methods are limited to cases of small s and intermediate Ne values.

Although WFABC gives slightly less precise results in some cases of small s, the two major advantages of this implementation come from its ability to consider complex ascertainment cases and to accurately estimate Ne from multilocus genomewide data. The Mathieson and McVean (2013) method is computationally efficient but is based on the unrealistic assumptions that Ne is known and that the data are completely unascertained. Using a point estimate of Ne obtained from the first step of WFABC in the Mathieson and McVean (2013) method would ignore the uncertainty on this parameter. Malaspinas et al. (2012) only handle the conditional cases of polymorphism at the last sampling time point and estimate Ne separately at each site. WFABC is computational efficient and enables the estimation of a genomewide average Ne from time-sampled data, after which per-site selection coefficients may be estimated efficiently and accurately. There currently exist no such likelihood-based methods to estimate Ne accurately from time-sampled genomic data, as the information is only based on a single allele trajectory. Our simulated multilocus scenario shows that estimating Ne accurately leads to a higher precision in the estimates of selection coefficients s, leading overall to a smaller RMSE as compared to the Malaspinas et al. (2012) method. As the effective population size is an important population genetic parameter that is often unknown in both ancient genomics and experimental evolution, WFABC provides a practical and flexible platform to be utilized in any time-serial data for efficiently inferring a wide range of Ne and s values to high accuracy. Finally, we note that the ABC method also has the advantage of providing a posterior distribution rather than simply a point estimate, allowing for easy-to-build credibility intervals.

The application of WFABC to the Panaxia dominula data confirms that a nearly fully recessive lethal model for the medionigra allele is the best explanation for the observed pattern as hypothesized by Mathieson and McVean (2013). We note that in this case, once the allele frequency is low enough such that heterozygotes almost never occur, it behaves like a neutral model. We used this feature to estimate Ne using Fs′ (Jorde & Ryman, 2007) by considering only time points after 1950, when the medionigra allele first reaches a frequency below inline image and we obtained 2Ne = 927, which is consistent with previous estimates (Fisher & Ford, 1947; Cook & Jones, 1996; O'Hara 2005). This application also demonstrates that WFABC is very flexible, as it can also be used to co-estimate h and accommodate very large selection coefficients (such as s = −1, as in our application here). To confirm the validity of our approach in this case, we simulated 1000 data sets mimicking the P. dominula data (same number of generations, time points, sample sizes and initial allele frequency). We fixed = −1 and h = 0.05 and let 2Ne vary uniformly between 100 and 10 000 for each simulation and estimated s and h using our ABC method (Fig. 8). Both parameter estimates are unbiased and while distinguishing lethality (s = −1) from very strong negative selection (s < −0.5) seems to be difficult, h is estimated with a very small variance.

image

Figure 8. Box plot for the estimated selection coefficient (s, panel a) and dominance ratio (h, panel b) from 1000 simulated data sets mimicking the Panaxia dominula data. The red circles indicate the true simulated values (= −1; = 0.05), and the blue triangles the mean of the estimated values.

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Finally, it should be noted that all the methods described and utilized in this study assume that the loci are in linkage equilibrium and take no demographic history into account except the Mathieson and McVean (2013) method which integrates structured populations with a lattice model. Owing to the generally low number of generations in time-series data, one expects only little recombination to occur and long stretched of DNA may be under linkage disequilibrium. Foll et al. (2014) demonstrated that the model is robust to fluctuating population sizes, but this may not hold for all demographic scenarios. It is an important future challenge to expand these methods further for inferring demographic parameters from time-serial data. WFABC has great potential in achieving this challenging task, as ABC-based estimators lend themselves more readily to the incorporation of complex demographic models compared with likelihood-based methods.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Data accessibility
  10. Supporting Information

The computations were performed at the Vital-IT (http://www.vital-it.ch) Center for high-performance computing of the SIB Swiss Institute of Bioinformatics. This work was funded by grants from the Swiss National Science Foundation and a European Research Council (ERC) starting grant to JDJ.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Data accessibility
  10. Supporting Information

M.F. and J.D.J. conceived the idea. M.F. developed the WFABC software, and H.S. extended the Malaspinas et al. (2012) software for haploids. M.F. performed the simulations and the WFABC analyses. H.S. performed the Mathieson and McVean (2013), Bollback et al. (2008) and Malaspinas et al. (2012) analyses. All authors contributed to writing the manuscript.

Data accessibility

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Data accessibility
  10. Supporting Information

This study is primarily based on simulated data created using the WFABC software available from the ‘software’ page at http://jensenlab.epfl.ch/. The Panaxia dominula moth data set has been taken from (Cook & Jones, 1996; Jones 2000) and is also provided in the WFABC package.

Supporting Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Data accessibility
  10. Supporting Information
FilenameFormatSizeDescription
men12280-sup-0001-SuppInfo.docxWord document811K

Fig S1. Box plot for the estimated selection coefficient from WFABC for three different sampling time points.

Fig S2. Box plot for the estimated selection coefficient from WFABC for three different sample sizes.

Fig S3. Box plot for the estimated small s from each simulation replicate of the Wright–Fisher diploid model with Ne = 1000 simulated for 90 generations and the initial minor allele frequency at 10%.

Fig S4. Box plot for the estimated large s from each simulation replicate of the Wright–Fisher diploid model with Ne = 1000 simulated for 90 generations and the initial minor allele frequency at 10%.

Fig S5. Log likelihood of the estimated selection coefficient for the Bollback et al. (2008) method with a small search interval for γ.

Fig S6. Log likelihood of the estimated selection coefficient for the Bollback et al. (2008) method with a large search interval for γ.

Fig S7. Box plot for the estimated small selection coefficient from each simulation replicate of the Wright–Fisher diploid model with Ne = 200 simulated for 300 generations.

Fig S8. Box plot for the estimated large selection coefficient from each simulation replicate of the Wright–Fisher diploid model with Ne = 200 simulated for 80 generations.

Fig S9. Box plot for the estimated small selection coefficient from each simulation replicate of the Wright–Fisher diploid model with Ne = 5000 simulated for 500 generations.

Fig S10. Box plot for the estimated large selection coefficient from each simulation replicate of the Wright–Fisher diploid model with Ne = 5000 simulated for 100 generations.

Fig S11. Box plot for the estimated small selection coefficient from each simulation replicate of the Wright–Fisher haploid model with Ne = 1000 simulated for 300 generations.

Fig S12. Box plot for the estimated large selection coefficient from each simulation replicate of the Wright–Fisher haploid model with Ne = 1000 simulated for 50 generations.

Fig S13. Two-dimensional joint posterior distribution for s and h for the moth P. dominula data using a uniform prior for 2Ne between 100 and 10 000.

Table S1. RMSE and bias for the small s and Ne = 200 scenario for the Wright–Fisher diploid model

Table S2. RMSE and bias for the big s and Ne = 200 scenario for the Wright–Fisher diploid model

Table S3. RMSE and bias for the small s and Ne = 5000 scenario for the Wright–Fisher diploid model

Table S4. RMSE and bias for the big s and Ne = 5000 scenario for the Wright–Fisher diploid model

Table S5. RMSE and bias for the small s and Ne = 1000 scenario for the Wright–Fisher haploid model

Table S6. RMSE and bias for the big s and Ne = 1000 scenario for the Wright–Fisher haploid model

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