The effective population size (N_e) is frequently estimated using temporal changes in allele frequencies at neutral markers. Such temporal changes in allele frequencies are usually estimated from the standardized variance in allele frequencies (F_c). We simulate Wright-Fisher populations to generate expected distributions of F_c and of _c (F_c averaged over several loci). We explore the adjustment of these simulated _c distributions to a chi-square distribution and evaluate the resulting precision on the estimation of N_e for various scenarios. Next, we outline a procedure to test for the homogeneity of the individual F_c across loci and identify markers exhibiting extreme F_c-values compared to the rest of the genome. Such loci are likely to be in genomic areas undergoing selection, driving F_c to values greater (or smaller) than expected under drift alone. Our procedure assigns a P-value to each locus under the null hypothesis (drift is homogeneous throughout the genome) and simultaneously controls the rate of false positive among loci declared as departing significantly from the null. The procedure is illustrated using two published data sets: (i) an experimental wheat population subject to natural selection and (ii) a maize population undergoing recurrent selection.

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THE effective population size (N_e), defined as the size of an ideal Wright-Fisher population undergoing the same rate of genetic change as the population under study, is an essential parameter to predict the evolution of a population due to genetic drift in terms of rates of loss of genetic variation, fixation of deleterious alleles, or inbreeding (WRIGHT 1969). However, obtaining direct estimates of N_e from demographic data has often proved difficult. An alternative is to use indirect methods, for instance, those based on the measurement of temporal changes in allele frequencies at neutral markers (KRIMBAS and TSAKAS 1971; WAPLES 1989a). The foundation of these methods is that the variance of allele frequency due to drift from parents to offspring, V(P₁), depends on N_e as follows: V(P₁) = P₀(1 – P₀)/2N_e, where P₀ is the frequency in the parental population. After t generations of drift, the expected frequency of the allele is E(P_t) = P₀ and the variance of the allele frequency, V(P_t) = E(P_t – P₀)², can be written as a function of N_e: V(P_t) = P₀(1 – P₀)[1 – (1 – 1/2N_e)^t] (CROW and KIMURA 1970). If t is not too large (t << N_e), N_e can be approximated by N_e (P₀(1 – P₀)t)/(2V(P_t)) and therefore an estimator for N_e based on the standardized variance in allele frequency is (V(P_t))/(P₀(1 – P₀)). NEI and TAJIMA (1981) proposed estimating the standardized variance in allele frequency between generation t_x and t_y for each locus l with K_l alleles as

(1)

where p_x(i,l) [respectively p_y(i,l)] represents the frequency of allele i at locus l in the sample of S_x individuals drawn at generation t_x (respectively S_y individuals at t_y). A weighted mean of _c,l-values across several loci,

(2)

is then typically used to estimate N_e via

(3)

(WAPLES 1989a). Note that Equation 1 assumes that alleles frequencies are estimated from samples taken prior to reproduction (so-called plan II sampling scheme). We use that sampling in the remainder of this article; allowing for an alternative sampling scheme is straightforward.

Recently, renewed interest in estimating N_e has led to the development of numerous methods using allele frequencies observed in a series of temporally spaced samples of a population. WILLIAMSON and SLATKIN (1999) and ANDERSON et al. (2000) introduced a maximum-likelihood approach to estimate N_e. BERTHIER et al. (2002) proposed a coalescent-based likelihood approach to estimate N_e and WANG (2001) devised a faster approximate version using a pseudo-maximum-likelihood method. These methods, tested by the authors for some values of population parameters (N_e, l, K_l, and p_x(i,l)), have proved to be slightly more accurate than the F_c method, but are much more computationally intensive. Hence F_c-based estimators of N_e remain frequently used in practice (FUJIIO et al. 1999; TURNER et al. 1999; GOLDRINGER et al. 2001; SHIKANO et al. 2001). Properties of F_c-based estimators of N_e and the quality of the confidence intervals around such estimates depend critically on the distribution of _c-values. Confidence intervals around N_e have been based on the fact that , with n = _l(K_l – 1), is distributed approximately as a chi-square with n d.f. (LEWONTIN and KRAKAUER 1973). Hence, assessing the adjustment of the actual _c distribution to a chi-square distribution is important. The chi-square approximation has been studied for some special cases, but the effects of initial allele frequencies, of the number of alleles, of the number of loci, and of the number of generations as well as the "true" effective population size on the distribution of _c-values and on _e are still poorly known (WAPLES 1989a).

Before averaging estimates of F_c obtained at individual loci to obtain _c and an estimate of N_e, it is desirable to test whether all loci used for that study have experienced the same effective population size. This implicit assumption, which underlies all methods of estimation mentioned above, is rarely tested. Several factors can modify the local effective size at a given locus. The recurrent elimination of deleterious variants linked to a marker locus, known as "background selection," will reduce the effective size locally; this effect depends on the local recombination rates and on genome-wide parameters describing spontaneous mutation and their effect on fitness (CHARLESWORTH et al. 1993). Hitchhiking will also drive higher than expected the temporal variance in allele frequency of markers linked to a positively selected variant (WIEHE and STEPHAN 1993).

The remainder of this note is organized as follows. In the first part, we study the actual _c distribution and the quality of the chi-square approximation. The actual distribution of _c, its divergence from a chi-square distribution, and the quality of the N_e estimation based on F_c are studied under various scenarios by varying the initial allele frequencies, the number of loci, the number of generations, the sample size at both generations, and the true effective population size. In the second part, we outline a procedure to identify loci with "extreme" individual _c,l-values. We illustrate our approach by reanalyzing two experimental data sets: temporal variation in allele frequencies at 29 markers in an experimental wheat population under natural selection and frequencies at 82 markers in a maize population under recurrent selection.

To investigate the actual distribution of _c, we simulated Wright-Fisher populations using an exact multinomial sampling scheme. We generated expected distributions of temporal variations in allele frequencies conditional on initial allele frequencies. Distributions were based on 3000 independent replicates. In each replication, several loci with the same initial allele frequencies were simulated and _c was computed. All simulations were carried out using Mathematica (WOLFRAM 1996). Our simulations show that substantial departure of the actual distribution of _c from a chi-square distribution, as measured through KULLBACK's (1968) symmetric measure of divergence between both distributions (see Table 1), can be observed under a variety of conditions depending on the parameter values chosen (Table 1). The actual _c distribution is closest to a chi-square and thus N_e is best estimated when biallelic marker loci with equal frequencies are used. Conversely, the discrepancy between the actual _c distribution and the chi-square approximation is large when allele frequencies are strongly unbalanced (P₀ < 0.1 for at least one allele), when the number of alleles per locus is large (K 5) such as for microsatellite markers, or when the number of generations increases (T = t_y – t_x > 15 when N_e = 100 is assumed). Increasing the sample sizes up to 200 or 500 individuals does not diminish the discrepancy, especially for a high number of alleles (data not shown). In most cases, the distribution of simulated values is shrunk compared to the chi-square approximation. In addition, the actual distribution is a bit skewed toward higher values of F_c. As a consequence, the distribution of _e in the simulations is often more narrow than the one based on the chi-square approximation. Confidence intervals at the 95% level based on either the chi-square approximation or the actual _c distribution are given in Table 1. These are exactly the intervals that would be computed in experimental studies. Chi-square confidence intervals are often wider than the confidence intervals based on the simulated _c (Table 1). The width of this interval, which is connected to the precision of the estimation, depends mainly on the number of independent alleles used [L(K – 1)]. Note that the product L(K – 1) is also the number of degrees of freedom of the chi-square used in previous approximations. Confidence intervals derived from simulated data are reduced by 10–25% relative to chi-square-based confidence intervals for scenarios involving unbalanced initial allele frequencies with 5 L(K – 1) 10, or very large number of alleles (K = 10), sample sizes <100, T > 10, or N_e < 75. Otherwise, they are quite close to the chi-square-based intervals (reduction is <10%). Similar results are found with larger sample sizes (200 and 500 individuals).

View this table: In this window In a new window   TABLE 1 Distribution of c: Ne estimation and divergence from chi-square distribution according to different parameters

 
Except for T = 5 generations, ^*_e, computed from ^*_c (an average of _c over 3000 independent replicates), is always higher than the population size N_e used in simulations. This indicates that _c-based estimation tends to return overestimated values of N_e. RICHARDS and LEBERG (1996) and LUIKART et al. (1999) argued that the overestimation of N_e using _c [or POLLAK's (1983) estimator, _k] is due mainly to the loss of alleles in early generations, suggesting that the bias would be greater with increasing drift and when there are rare alleles. LUIKART et al. (1999) focused on the estimation of N_e after very strong bottlenecks (N_e = 4–40), which were not considered here. Whereas our results confirm the existence of a greater bias for rare alleles, for the range of N_e we considered, population size does not appear as critical; however, increasing the number of generations between samples leads to overestimation of N_e. Hence, to improve precision on the F_c-based estimation of N_e, we recommend generating the actual _c distribution using simulations based on the estimated value _e and obtaining a confidence interval directly on _e. With such a gain in accuracy, the performance of the F_c-based estimator of N_e becomes close to those of likelihood-based estimators.

Once N_e has been estimated on the basis of _c by averaging over several marker loci, it is desirable to test whether all loci considered have undergone the same rate of change in allele frequency. Heterogeneity in individual F_c-values might be used as evidence for selection since "while natural selection will operate differently for each locus and each allele at a locus, the effect of breeding structure (migration, genetic drift, inbreeding) is uniform over all loci" (LEWONTIN and KRAKAUER 1973, pp. 176–177). Loci with significantly high _c,l-values should be discarded before (re)computing _c to yield a more reliable estimate of N_e.

We propose a way to identify, in a series of experimental _c,l measurements, markers exhibiting _c,l-values significantly higher than expected under pure drift based on _e. To do so, each _c,l-value should be compared to an expected distribution based on a genome-wide effective size estimated from the remaining loci and the trajectory of allele frequencies at this locus. We exemplify below our method with two published data sets. First we consider individual _c,l-values estimated from temporal variations of allele frequencies in an experimental composite wheat population undergoing natural selection. A total of 250 and 213 individuals were sampled at generations 1 and 10, respectively, and genotyped at 29 RFLP loci (GOLDRINGER et al. 2001). For each locus l, we pooled the remaining 28 loci to obtain a global _c estimate and an _c-based estimate of N_e (described hereafter as the genome-wide average estimate). The expected distribution of F_c at locus l was then obtained (using typically 3000–5000 independent simulations) conditional on _e (excluding locus l) and the observed initial allele frequencies (at locus l). We then tested whether the observed temporal variance of allele frequencies at locus l, _c,l, was significantly larger than the genome-wide average variations by computing p, the probability for _c,l to be greater than or equal to the observed value at this locus on the basis of the simulated distribution described above. Note that one could also test for the presence of loci exhibiting smaller than expected variations in allele frequency. Some loci exhibited some "excess drift" relative to the rest of the loci and accordingly fairly small P-values: Fba242-C (P = 0.021), Fba280-C (P = 0.042), Fba65-D (P = 0.085), and Fba204-A (P = 0.09). However, the distribution of P-values was fairly uniform (data not shown) and to take into account the fact that multiple loci were examined we computed the expected false discovery rates, also known as q-values, using the distribution of P-values (see STOREY and TIBSHIRANI 2003 for details). The q-values were calculated using the package QVALUE (http://faculty.washington.edu/jstorey/qvalue/index.html). This analysis suggests that declaring only Fba242-C and Fba280-C as "significant" for excess of drift would still yield an estimated rate of false positives of 40% among these two loci.

Next we consider the study published by LABATE et al. (1999), where temporal variations in allele frequencies were surveyed at 82 RFLP loci after 12 generations in maize populations undergoing recurrent selection. A P-value was calculated at each locus (Figure 1), using the method described above. The distribution of P-values was then used to calculate corresponding q-values (Figure 2). In contrast with the previous case, the distribution of P-values is clearly L-shaped (Figure 1) and choosing a q-value cutoff of 0.05 yields 10–11 loci exhibiting significant departures from the genome-wide level of drift. This method proved to be somewhat more conservative than the one used by the authors, who declared 14 loci as outliers (LABATE et al. 1999). Discarding those outlier loci, one can compute a new genome-wide effective size and check that no more loci exhibit F_c-values departing significantly from the null hypothesis of homogenous drift (data not shown).

View larger version (13K): In this window In a new window Download PPT slide   FIGURE 1.— Distribution of P-values for the 82 loci in the maize data set of LABATE et al. (1999).

 

View larger version (11K): In this window In a new window Download PPT slide   FIGURE 2.— Expected proportion of false positives (q-values) among the loci found to be departing significantly from the genome-wide average effective size in the maize data set as a function of the individual threshold used for significance (P-value). Calculations of the q-values were done using the procedure proposed by STOREY and TIBSHIRANI (2003).

 
WAPLES (1989b) proposed a method based on the chi-square test of homogeneity to test the hypothesis that observed changes in allele frequencies can be satisfactorily explained by drift alone. This allows one to examine the variation of a particular allele according to the range of possible N_e-values for the population under study. Yet, this test has not been widely used in experimental studies. Indeed, it is rather complicated to implement, particularly in cases of multiple alleles since it is necessary to consider covariances of frequencies for different alleles sampled at different times. LEWONTIN and KRAKAUER (1973) provided the theoretical grounds for homogeneity tests of variation in allele frequencies, but they emphasized more spatial variation, and the test proposed for temporal variation, which again relies on the assumption of a chi-square distribution of individual F_c,l-values, is much too restrictive [see BEAUMONT and NICHOLS (1996) and VITALIS et al. (2001) for the case of spatial variation in allele frequencies]. The use of simulation-based distributions provides a robust method to test for homogeneity of _c,l-values across loci before pooling estimates. Our procedure yields a more reliable genome-wide estimate of the realized N_e and can be used to detect markers exhibiting F_c-values significantly higher than expected on the basis of mean N_e, thereby providing a (formal) way of assessing if selection is operating on any given genomic segment (see also LUIKART et al. 2003 for a review of available methods for population structure). One potential caveat of our method is that the distribution of F_c under the null hypothesis is generated using information from the data (to estimate the genome-wide N_e). We verified through simulations (see online supplementary material at http://www.genetics.org/supplemental/) that our procedure is actually fairly robust to uncertainty in the estimation of the genome-wide N_e. A program generating the expected individual or mean F_c distributions used in this note is available upon request as a Mathematica notebook from the authors.

We thank F. Hospital, I. Bonnin, and C. Dillmann for helpful discussions and A. Tsitrone, R. Waples, and an anonymous reviewer for their comments on earlier versions of this article. We thank O. Martin for correcting and improving the English of this article.

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