Estimating the Effective Number of Breeders From Heterozygote Excess in Progeny

Corresponding author: Gordon Luikart, Laboratoire de Biologie des Populations d’Altitude, CNRS, UMR 5553, Université Joseph Fourier, F-38041 Grenoble, Cedex 9, France., gluikart{at}ujf-grenoble.fr (E-mail)

Communicating editor: G. B. GOLDING

	ABSTRACT

TOP ABSTRACT PRINCIPLE AND METHODS RESULTS AND DISCUSSION LITERATURE CITED

The heterozygote-excess method is a recently published method for estimating the effective population size (N_e). It is based on the following principle: When the effective number of breeders (N_eb) in a population is small, the allele frequencies will (by chance) be different in males and females, which causes an excess of heterozygotes in the progeny with respect to Hardy-Weinberg equilibrium expectations. We evaluate the accuracy and precision of the heterozygote-excess method using empirical and simulated data sets from polygamous, polygynous, and monogamous mating systems and by using realistic sample sizes of individuals (15–120) and loci (5–30) with varying levels of polymorphism. The method gave nearly unbiased estimates of N_eb under all three mating systems. However, the confidence intervals on the point estimates of N_eb were sufficiently small (and hence the heterozygote-excess method useful) only in polygamous and polygynous populations that were produced by <10 effective breeders, unless samples included > ~60 individuals and 20 multiallelic loci.

THE effective population size (N_e) is an important parameter in evolutionary genetics and conservation biology because it influences the rate of inbreeding and loss of genetic variation. It also influences the efficiency of natural selection in maintaining beneficial alleles and purging deleterious ones. For example, when N_e is very small, genetic drift will often be too strong for natural selection to efficiently maintain or purge alleles. Unfortunately, N_e has proven very difficult to estimate in natural populations (WAPLES 1989 ). Thus, any method with potential for improving our ability to estimate N_e deserves thorough evaluation. N_e can be estimated using demographic or genetic data. However, demographic methods require information such as variance in reproductive success, which is difficult to obtain for many species. Further, demographic estimates of N_e are often higher than the true N_e because demographic methods seldom incorporate all of the factors (e.g., skewed sex ratios, change in population size, etc.) that can make N_e smaller than the population census size (N_c) (FRANKHAM 1995 ).

The (current) effective population size can be estimated using genetic data and the four following statistical methods: the loss of heterozygosity method (e.g., HARRIS and ALLENDORF 1989 ); the temporal method (KRIMBAS and TSASKAS 1971 ; WAPLES 1989 ); the linkage disequilibrium method (HILL 1981 ; WAPLES 1991 ); and, most recently, the heterozygote-excess method (PUDOVKIN et al. 1996 ). PUDOVKIN et al. 1996 demonstrated that the heterozygote-excess method estimates the effective number of breeding parents (N_eb) with no bias and fair precision when the sample size of progeny is infinite and when gametes combine completely at random, i.e., when all male gametes have an equal probability of combining with all female gametes, as in some polygamous, random-mating species (e.g., marine invertebrates such as shellfish).

Here, we extend the evaluation of PUDOVKIN et al. 1996 to include finite samples of individuals (n = 15–120), reasonable numbers of polymorphic loci (5–30) with a wide range of allele frequencies, monogamous mating systems where only pair-mating occurs, and polygynous mating systems where only a few males mate with many females (i.e., skewed sex ratios). Our goal is to delineate the conditions under which the heterozygote-excess method will be useful for estimating N_eb in natural populations (or in captive populations such as fish hatcheries). In this article, we (i) explain the importance of using an unbiased estimator of the expected heterozygosity (H_exp) for calculating _eb from finite samples of progeny, (ii) quantify the bias of _eb, (iii) determine the number of loci and individuals that must be sampled to achieve precise estimates of N_eb, and (iv) test if monogamy generates an interfamily Wahlund effect (i.e., heterozygote deficiency) that counteracts the heterozygote excess generated by small N_eb. To conduct these evaluations, we use data from simulations and natural populations. We focus on populations with a small N_eb (4–100) because a heterozygote excess is generated only when N_eb is small.

	PRINCIPLE AND METHODS

TOP ABSTRACT PRINCIPLE AND METHODS RESULTS AND DISCUSSION LITERATURE CITED

The principle of the heterozygote-excess method is as follows: When the number of breeders is small, the allele frequencies in males and females will be different due to binomial sampling error. This difference generates an excess of heterozygotes in the progeny relative to the proportion of heterozygotes expected under Hardy-Weinberg equilibrium (ROBERTSON 1965 ; RASMUSSEN 1979 ). The proportion of heterozygotes expected in progeny produced by a small and equal number of females and males can be calculated as (FALCONER 1989 , p. 67)

where n is the number of haploid genomes in the mothers or fathers and p and q are the frequencies of alleles at a locus, in the population from which the parents were drawn. Because the excess of heterozygotes expected in progeny increases as the number of parents decreases, PUDOVKIN et al. 1996 suggested using the magnitude of heterozygote excess to estimate the effective number of breeding parents.

PUDOVKIN et al. 1996 called H' the proportion of heterozygotes expected to be observed (H_obs) in the progeny (given a limited number of parents), whereas the expected proportion of heterozygotes in the base population under Hardy-Weinberg proportions, 2pq, was designated as H_exp.

Now _eb can be estimated as follows:

The ratio H_exp/(H_obs - H_exp) is the reciprocal of SELANDER 1970 index, D, for excess or deficiency of heterozygotes; thus, an estimate of N_eb is (PUDOVKIN et al. 1996 )

The following more exact equation was derived by PUDOVKIN et al. 1996 :

(1)

The above expression is for two alleles. For a multiallelic locus, one should average D over all k alleles as

(2)

where D_i is the excess of heterozygotes for the ith allele,

H_obs[i] is the total proportion of heterozygotes having allele i, and H_exp[i] is simply 2p_i(1 - p_i) · 2N/(2N - 1), where i is the ith allele.

We note that when the sample size of individuals is finite, H_exp must be estimated using the following unbiased estimator of 2pq (NEI 1987 ): 2N(2pq)/2N - 1, where N is the number of progeny sampled. If 2pq is used instead of the unbiased estimator, _eb will give a biased estimate N_eb (especially when N is small). In nature, N_eb can range from only two to near infinity and _eb can be negative (e.g., in the case of an overall deficit of heterozygotes). In our analyses, we considered N_eb values as infinite if _eb was negative or >10,000 (an arbitrary but reasonably large value). If _eb is infinite, it simply means that the heterozygote-excess signal is obscured by the noise (i.e., sampling error) associated with small samples of loci or individuals.

The simulation model that we used to evaluate the heterozygote-excess method has three main steps. First, it assigns genotypes to the parental generation using random numbers and a predefined allele frequency distribution. We modeled loci with 2, 3, and 5 alleles and the following three allele frequency distributions: equal (e.g., H = 0.8, for 5 alleles), triangular (e.g., 0.33, 0.30, 0.20, 0.13, 0.07, and H = 0.74; see PUDOVKIN et al. 1996 ), and rare (e.g., 0.52, 0.2, 0.10, 0.07, 0.04, with H = 0.67). Second, the simulation model randomly picks a male and female parent and one gamete from each. Under the polygamy model, the gametes from males and females are randomly combined such that one male can potentially mate with several females and vice versa. Under monogamy, the same random male is always paired with the same female. For example, when N_eb = 4, one random male is paired with one random female, and then a second random male is paired with a second random female. Then one of the two pairs is randomly chosen and gametes are drawn to make an offspring. This is repeated until 15–120 offspring have been generated. Here, the variance in reproductive success among the four parents follows a Poisson distribution, thus N_eb = 4. Under polygyny, only one or two males mate with all the females. For example, if one male mates with 99 females, the number of breeders is 100 but the effective number is only ~4 [N_e = , where N_m and N_f are the number of breeding males and females, respectively; CROW and KIMURA 1970 ]. Third, the program computes _eb from a random sample of 15–120 progeny using Equation 1 and Equation 2. These three steps were repeated 500–2000 times per combination of the following parameters: N_eb, sample size of individuals and loci, allele number, allele frequency distribution, and mating system.

	RESULTS AND DISCUSSION

TOP ABSTRACT PRINCIPLE AND METHODS RESULTS AND DISCUSSION LITERATURE CITED

Bias:
Our simulations suggest that the heterozygote-excess method slightly overestimates the N_eb when using finite samples and Nei's unbiased estimator of H_exp. When N_eb was 4, 20, and 100, the harmonic mean estimates of N_eb (from 500 simulations) were 4.1, 22.2, and 103.4, respectively, when sampling 30 individuals and 10 loci (five alleles/locus) with triangular allele frequency distributions and a polygamous breeding system (Table 1). When more individuals were sampled, the bias was slightly lower. Harmonic mean estimates of N_eb were nearly identical for loci with two, three, and five alleles and with widely different allele frequency distributions (Table 1; Figure 1, horizontal bars inside box plots). The largest bias occurred under the polygynous mating system (e.g., = _eb 5.0 when true N_eb 4; Figure 1). This bias was not surprising given the assumption of the heterozygote-excess method that there are equal numbers of male and female parents. Still, the bias was small enough not to substantially diminish the usefulness of the heterozygote-excess method.

View larger version (26K): In this window In a new window Download PPT slide   Figure 1. Harmonic mean eb and the distribution of upper and lower 95% confidence limits on eb computed from 500 independent simulation estimates using 10 loci and samples of 30 individuals. The dotted horizontal line (between boxes) shows the true Neb. The solid horizontal line inside each box shows the harmonic mean eb. The top of each box is the 80th percentile of the distribution of 500 estimates of the upper 95% confidence limit on eb. Each line extending upward from a box is the 95th percentile of the distribution of the upper 95% confidence limit. Bottoms of boxes and lines extending downward from boxes represent the 20th and 5th percentiles, respectively, of the distribution of the lower 95% confidence limit. (*) Infinity is the 95th percentile of the distribution of the upper confidence limit. (**) Both the 95th and 80th percentiles of the distribution of upper confidence limits are infinity. Distributions are compared for a polygamous breeding system (i.e., complete random union of gametes between males and females) and for loci with (i) five alleles at equal frequencies (Even-5), (ii) five alleles at triangular frequencies (Tri-5, as in Table 1), (iii) five alleles including rare alleles (Rare-5, see PRINCIPLE AND METHODS), (iv) three alleles with triangular frequencies (Tri-3), (v) two alleles with triangular frequencies (Tri-2), (vi) Tri-5 allele frequencies for a monogamous breeding system in which males and females are pair-mated, and, finally, (vii) Tri-5 for extreme polygyny where one male breeds all 99 females (and thus Neb 4).

View this table: In this window In a new window   Table 1. Harmonic mean eb and distribution of 95% confidence limits for eb from 500 simulations

The harmonic mean N_eb estimates were similar for the monogamous and polygamous mating systems (_eb was 4.5 and 4.1, respectively, when N_eb was 4.0; Figure 1). This suggests that there is little impact of an interfamily Wahlund effect on the accuracy of N_eb estimates. When only approximately two to three large families exist (e.g., under monogamy with N_eb equaling 4–6), sampling across families does not generate a large heterozygote deficiency (i.e., Wahlund effect). However, a Wahlund heterozygote deficiency is expected when many families exist (A. PUDOVKIN, personal communication). Such a deficiency would at least partially cancel the heterozygote excess caused by small N_eb, and thereby cause biased (high) estimates of N_eb. Although monogamy did not cause a large bias, it did substantially increase the variance among N_eb estimates (see below and Figure 1).

Precision:
To quantify the precision of the N_eb estimates, we used the Student's t-distribution to compute a 95% confidence interval for each _eb (as in PUDOVKIN et al. 1996 ). The confidence interval on _eb contained the true N_eb in 92–96% of the simulation estimates of N_eb when using loci with five alleles (Table 1). As expected, approximately half of the confidence intervals that did not contain the true N_eb were too low (L) and half were too high (H). This suggests that the Student's t-distribution is useful for computing confidence intervals, even though N_eb is not exactly normally distributed. For loci with three alleles or for monogamous mating systems, the confidence intervals also contained the true N_eb ~92–96% of the time. However, when using loci with only two alleles, the confidence intervals were generally too narrow and contained the true N_eb in only 83–89% of the simulation estimates of N_eb (Table 1). Thus, confidence intervals must be interpreted with caution or computed by alternative methods (e.g., bootstrap resampling) when using loci with only two alleles (e.g., many allozyme loci).

Under extreme polygyny (e.g., one male mating with 99 females), the confidence intervals were often too high. For example, when N_eb was four, ~25% of the 500 simulated confidence intervals were slightly higher than the true N_eb, and none were lower than N_eb. Although the confidence intervals were often too high, they were also much narrower under polygyny than under monogamy or polygamy (Figure 1). This narrowness substantially increases the usefulness of the heterozygote-excess method under polygyny. Thus, under extreme polygamy, the heterozygote-excess method will be useful for detecting a small N_eb but will be less useful for quantifying the exact size of N_eb.

To determine the minimum number of loci and individuals that must be sampled to achieve a high probability of obtaining narrow confidence intervals, we plotted the distribution of the (upper and lower) 95% confidence limits obtained from 500 simulation estimates of N_eb. When the true N_eb is only 4, at least 10 loci (with five alleles) and 30 individuals must be sampled to achieve an 80% probability of obtaining an upper 95% confidence limit < ~20 (Figure 2B). In other words, the statistical power is 0.80 when testing the null hypothesis that the true N_eb 20 (and when the true N_eb is actually only 4). The power will be slightly higher when using a one-tailed test and the null hypothesis that true N_eb 20 (the alternative hypothesis is N_eb < 20).

View larger version (33K): In this window In a new window Download PPT slide   Figure 2. Distributions of 500 (95%) confidence limits on eb computed from 500 independent simulation estimates, as in Figure 1. Dotted horizontal lines represent the true Neb being estimated (4 and 10 for a–c and d–f, respectively). (b and e) "80%" shows the 80th percentile of the distribution of the upper 95% confidence limits. This distribution was generated from 500 independent simulation estimates of Neb. In e, 460 is the 95th percentile of the distribution of the upper 95% confidence limits.

These results show that the heterozygote-excess method is sufficiently powerful for detecting a small N_eb when sampling reasonable numbers of individuals and loci with five alleles. Such results are important for conservation biology and the management of captive and natural populations. The precision of _eb is often increased more by analyzing a larger number of loci than by sampling more individuals. Doubling the number of loci from 10 to 20 (compare the first box plot in Figure 2B and Figure C) generally reduces confidence intervals more than doubling the number of individuals sampled from 15 to 30 (compare the first two box plots in Figure 2B). However, the benefit from doubling the number of loci depends on the number and frequency of alleles (Figure 1).

When true N_eb is 10, we must sample >20 polymorphic loci and 60 individuals to have an 80% probability of obtaining confidence intervals that are <50 (and to have a 95% probability of obtaining confidence intervals <100; Figure 2E). When the true N_eb is 100, >80% of the confidence intervals include infinity, even when sampling 120 individuals and 20 loci with five alleles (data not shown). Clearly, when N_eb is >10, very large samples of loci and individuals are required to achieve a high probability of obtaining reasonably small confidence intervals. Thus, the main limitation of the usefulness of the heterozygote-excess method is its poor precision, i.e., its wide confidence intervals. The confidence intervals are generally too wide for the method to be useful when using diallelic loci, loci with mostly rare alleles, or when studying strictly monogamous species (Figure 1).

When applied to data from natural populations, the heterozygote-excess method often gave estimates of N_eb equal to infinity. For example, _eb was infinity in 5 of 10 cohorts for which the total number of parents was known (or estimated) to be small (i.e., three to a few dozen). Further, only 2 of the 10 estimates gave 95% confidence intervals that did not include infinity as an upper limit (Table 2). This poor precision is not surprising in that only 5–9 polymorphic loci were analyzed, and only 11–25 progeny were sampled. Additional empirical evaluations are needed, but it is extremely difficult to find large data sets containing individuals produced from a known number of parents.

One potential limitation of the method is the requirement for random, representative sampling. For example, if a sample contains only one or few families (due to sampling error) then we could obtain a very low N_eb estimate, even though many families actually exist and N_eb is large. Another obvious limitation is that the method will work only in species with separate sexes. The method will work for haplo-diploid species (e.g., Hymenopterans), but will require the derivation of equations different from those presented here.

Four approaches may help circumvent the problem of poor precision. First, one can compute 80% confidence intervals (in addition to 95% confidence intervals). This will reduce the likelihood that the upper confidence limit will include infinity and be uninformative. Second, one could explore alternative methods for computing confidence intervals (e.g., nonparametric methods such as bootstrap resampling of loci). Third, one could combine estimates of N_eb from several generations or cohorts by computing the harmonic mean of _eb over the multiple generations or cohorts. This can reduce the probability of obtaining infinity for _eb because, when computing a harmonic mean, the low estimates carry far more weight than high ones (e.g., infinity). Finally, one can potentially combine estimates of N_e obtained from several independent N_e estimators by computing the harmonic mean of the N_e estimates. Other promising N_e estimators include those based on gametic disequilibrium (HILL 1981 ) and on temporal variance in allele frequencies (KRIMBAS and TSASKAS 1971 ; WAPLES 1991 ). These two estimators also suffer from low precision (WAPLES 1989 , WAPLES 1991 ; LUIKART 1997 ; LUIKART et al. 1998 ). However, two or more of the estimators may be independent (WAPLES 1991 ; PUDOVKIN et al. 1996 ) and thus could potentially be used simultaneously to achieve a more precise estimate of N_e. More research is needed to evaluate the precision and accuracy that can be achieved by using several N_e estimators simultaneously. Any improvement in our ability to estimate N_e would be significant in light of both the difficulties in assessing N_e and the importance of N_e in population genetics and in conservation biology.

	ACKNOWLEDGMENTS

We thank I. Till-Bottraud and two anonymous reviewers for helpful comments on earlier versions of this manuscript, M. Schwartz for sharing unpublished simulation data, and especially P. Spruell, F. W. Allendorf, A. Estoup, and M. Brown for providing data sets. Support was provided by the French "Bureau Ressources Génétiques," a postdoctoral fellowship (for G.L.) from National Science Foundation/North Atlantic Treaty Organization, and the Laboratoire de Biologie des Populations d'Altitude.

Manuscript received June 17, 1998; Accepted for publication November 20, 1998.

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