Inbreeding of Bottlenecked Butterfly Populations: Estimation Using the Likelihood of Changes in Marker Allele Frequencies

Corresponding author: Ilik J. Saccheri, Section of Evolutionary Biology, Institute of Evolutionary and Ecological Sciences, University of Leiden, Kaiserstraat 63, P.O. Box 9516, 2300 RA Leiden, The Netherlands., saccheri{at}rulsfb.leidenuniv.nl (E-mail)

Communicating editor: L. PARTRIDGE

	ABSTRACT

TOP ABSTRACT MATERIALS AND METHODS ANALYSIS RESULTS DISCUSSION LITERATURE CITED

Polymorphic enzyme and minisatellite loci were used to estimate the degree of inbreeding in experimentally bottlenecked populations of the butterfly, Bicyclus anynana (Satyridae), three generations after founding events of 2, 6, 20, or 300 individuals, each bottleneck size being replicated at least four times. Heterozygosity fell more than expected, though not significantly so, but this traditional measure of the degree of inbreeding did not make full use of the information from genetic markers. It proved more informative to estimate directly the probability distribution of a measure of inbreeding, ², the variance in the number of descendants left per gene. In all bottlenecked lines, ² was significantly larger than in control lines (300 founders). We demonstrate that this excess inbreeding was brought about both by an increase in the variance of reproductive success of individuals, but also by another process. We argue that in bottlenecked lines linkage disequilibrium generated by the small number of haplotypes passing through the bottleneck resulted in hitchhiking of particular marker alleles with those haplotypes favored by selection. In control lines, linkage disequilibrium was minimal. Our result, indicating more inbreeding than expected from demographic parameters, contrasts with the findings of previous (Drosophila) experiments in which the decline in observed heterozygosity was slower than expected and attributed to associative overdominance. The different outcomes may both be explained as a consequence of linkage disequilibrium under different regimes of inbreeding. The likelihood-based method to estimate inbreeding should be of wide applicability. It was, for example, able to resolve small differences in ² among replicate lines within bottleneck-size treatments, which could be related to the observed variation in reproductive viability.

FOLLOWING the seminal study of Lewontin and Hubby (HUBBY and LEWONTIN 1966; LEWONTIN and HUBBY 1966 ), molecular genetic markers have been used increasingly to make inferences about the breeding structure and demography of populations (reviewed in FRANKHAM 1996 ). An important application of these techniques, relevant to the genetic management of populations, is the detection and quantification of inbreeding, including historical bottlenecks (reviewed in AMOS and HOELZEL 1992 ). Numerous studies have reported low levels of allelic variability at allozyme loci as evidence of genetic drift (e.g., PRAKASH et al. 1969 ; BONNELL and SELANDER 1974 ; TAYLOR and GORMAN 1975 ; SCHWAEGERLE and SCHAAL 1979 ; BRYANT et al. 1981 ; O'BRIEN et al. 1983 ; HUETTEL et al. 1985 ; PACKER et al. 1991 ; WAYNE et al. 1991 ; BROOKES et al. 1997 ; KRETZMANN et al. 1997 ), usually corroborating other historical demographic information. More recently, hypervariable DNA loci, primarily minisatellites (JEFFREYS et al. 1985A , JEFFREYS et al. 1985B ) and microsatellites (reviewed in BRUFORD and WAYNE 1993 ), have been used to measure inbreeding (e.g., PATENAUDE et al. 1994 ; RASSMANN et al. 1994 ; TAYLOR et al. 1994 ; HOULDEN et al. 1996 ).

Experimental models of such natural episodes include the loss of allozyme variation following experimental bottlenecks in houseflies (MCCOMMAS and BRYANT 1990 ), mosquitofish (LEBERG 1992 ), and fruitflies (MONTGOMERY et al. 1999 ). As an extension to these studies, we have investigated the effect of experimental bottlenecks on genetic markers in the butterfly Bicyclus anynana (Satyridae). Hypervariable single-locus minisatellite markers were developed in addition to the allozyme loci used in most previous studies. The highly polymorphic minisatellite loci were expected to resolve smaller changes in inbreeding. Particularly accurate estimates of inbreeding were needed in this study to interpret the effects of the experimental bottlenecks on quantitative characters. Specifically, the genetic load on egg hatchability was unusually high, and additive genetic variance for wing size and pattern fell more than the purely demographic estimates of inbreeding led us to expect (SACCHERI et al. 1996 ; I. J. SACCHERI, R. A. NICHOLS and P. M. BRAKEFIELD, unpublished results). A potential explanation for these results is that inbreeding additional to that inferred from demographic parameters has occurred. Here we use direct estimates of inbreeding derived from molecular data to evaluate this hypothesis.

NEI et al. 1975 described the expected effect of a bottleneck of N individuals on average heterozygosity and number of alleles per locus, the most commonly used measures of genetic variation for inferring bottleneck size. They showed that reductions in heterozygosity (1/2N) are expected to be uninformative for all but the most extreme founder numbers; whereas the number of alleles is far more sensitive, even to large (N = 1000) bottlenecks. In practice, estimates of average heterozygosity suffer from very large variance, such that very many loci would have to be surveyed to detect small, but biologically important, differences in average heterozygosity (NEI and ROYCHOUDHURY 1974 ; ARCHIE 1985 ). Average loss of alleles per locus has a complex relationship with bottleneck size (MARUYAMA and FUERST 1985 ) and, being heavily dependent on the prebottleneck numbers of alleles at each locus and the observed sample size, can only provide an approximate qualitative estimate of bottleneck size. In this article we develop a method that makes use of the full allele frequency spectrum and thus makes use of information both from allele loss and from the frequency changes of surviving alleles.

If genetic differentiation is due to a balance between genetic drift and mutation or genetic drift and gene flow, then likelihoods for the population parameters can be calculated from the allele frequency spectrum using the multinomial Dirichlet distribution (see BALDING and NICHOLS 1995). The distribution for genetic drift alone is distinct, and we describe a method for obtaining the likelihood from frequency changes through a bottleneck. We show that it leads to more reliable estimates of inbreeding and has enabled us to detect subtle effects of selection, both among bottleneck-size treatments and even at the finer scale among replicate lines within treatments.

	MATERIALS AND METHODS

TOP ABSTRACT MATERIALS AND METHODS ANALYSIS RESULTS DISCUSSION LITERATURE CITED

Experimental populations:
A laboratory population of B. anynana was established from ~80 gravid females collected from the wild at a single locality (Nkhata Bay) in Malawi in August 1988. Before the experiment the stock was maintained for ~20 generations at a population size of 400–600 adults with some overlap of generations. Daughter populations were derived from this base population according to four treatments, differing in the number of individuals used to start the experimental populations or lines. The founder numbers were 2, 6, 20, and 300 (control treatment), replicated six times for the smallest bottleneck and four times for the rest.

With the exception of the control lines, all lines were established from clutches of known parentage in the following way: A total of 125 clutches were collected from females isolated in copula. From those clutches with >40 fertile eggs (80% of all fertile clutches), the required number of clutches (i.e., 1, 3, or 10) were randomly chosen to found the bottlenecked populations of 1 pair, 3 pairs, and 10 pairs. These populations were then allowed to freely increase in size to a maximum adult population size of 320, controlled by random culling of larvae (see Table 1 for population sizes from P to F₃). It was impractical to establish control lines with clutches obtained from mating pairs collected in copula. However, to control for the small chance that the procedure for choosing clutches to found bottleneck lines was biased with respect to genotype, two types of control were used. Initially, four populations, each consisting of 150 virgin females and 150 virgin males, were established and random mating was allowed for two weeks. For two of these populations we collected eggs from the entire population; these eggs were used to found two control lines (C.1 and C.2). For the remaining two populations we collected clutches from all females individually, but only used those clutches with >40 fertile eggs (85 and 60 clutches from each population, respectively) to found the other two control lines (C.3 and C.4). In subsequent generations, all females were given equal opportunity to lay eggs on potted maize plants placed in the cages for a fixed period of time, all control lines being maintained at an adult population size of 320. Clutches used to found 10-pair and control lines were culled randomly to keep larval densities equivalent in all lines. Husbandry techniques are detailed in SACCHERI 1995 . No differences in inbreeding were detected between the two types of control (see RESULTS).

Genetic markers:
Eight multi-allelic loci were studied: six allozyme loci [glucose phosphate isomerase (Gpi), isocitrate dehydrogenase-cytoplasmic (Icd-1), glutamate oxaloacetate transaminase-cytoplasmic (Got-1), phosphoglucomutase (Pgm), malate dehydrogenase-cytoplasmic (Mdh-1), alcohol dehydrogenase (Adh)] and two minisatellite loci (detected with Jeffreys probe 33.15), referred to as 33.15-1 and 33.15-2. Molecular methods, including segregation analyses showing Mendelian inheritance of alleles at these marker loci, are described in SACCHERI 1995 and SACCHERI and BRUFORD 1993 . In the founding generation, all 1-pair, 3-pair, and 10-pair line founders were screened at all marker loci; for each control line, enzyme loci were screened in 24 individuals of each sex, and minisatellite loci were screened in 22 females and 21 males. Random F₃ samples of 18 and 9 individuals of each sex from each line were screened at enzyme and minisatellite loci, respectively.

Direct estimate of N_e/N:
We conducted a supplementary experiment to estimate directly the ratio of the effective to the census population size (N_e/N) in control lines. Details of this experiment will be reported elsewhere. Briefly, two replicate cages were established from the same stock of butterflies used for the control lines. For practical reasons the population size was smaller (54 females and 54 males) than the original control lines, but at equivalent density in smaller cages. The butterflies, mixed as 2- to 7-day-old virgins, were each marked with a unique number. The identity of the male and female was recorded for all matings for the following 3 days. Individual egg batches were collected from each female for 3 continuous days and the numbers of hatched and unhatched eggs counted after a 4-day incubation.

The effective population size was calculated using the equations of LANDE and BARROWCLOUGH 1987 ,

where N_em and N_ef are the effective number of males and females as defined by

where x denotes the sex (m or f) and k_x is the number of hatching larvae produced by an individual. Assuming first male sperm precedence, for which there is some evidence in this species (SACCHERI and BRUFORD 1993 ), the estimates of N_e/N thus obtained for the two replicate cages are 0.59 and 0.61. These estimates do not include survivorship from the first larval instar through to reproductive maturity.

	ANALYSIS

TOP ABSTRACT MATERIALS AND METHODS ANALYSIS RESULTS DISCUSSION LITERATURE CITED

Tests for linkage and Hardy-Weinberg equilibrium:
The presence of linkage disequilibrium across all eight loci was tested by a randomization procedure on the basis of the principle that linkage disequilibrium increases the variance of average heterozygosity among individuals (MANLY 1985 , p. 330). For each locus, genotypes were reallocated randomly to individuals, and the variance in heterozygosity among individuals was evaluated for the entire base population and each line in F₃. Samples whose variance in heterozygosity was greater than or equal to that generated by random combinations in >5% of 10,000 iterations were interpreted as showing significant linkage disequilibrium. Similarly, significant deviations from Hardy-Weinberg equilibrium were tested at each locus by generating 10,000 samples of randomly constructed genotypes with the observed allele frequencies. The observed heterozygosity was compared with heterozygosity in randomized samples for the entire base population and each line in F₃.

Estimation of inbreeding:
Conventional methods analyze the data by calculating a summary statistic, typically heterozygosity, and monitoring how it changes through the bottleneck. The change can be compared with expectations given the effective population size (N_e) of the bottleneck generations. There is substantial literature demonstrating that N_e is less than the census population size N because adults do not have equal probabilities of transmitting their genes to the next generation (reviewed in FRANKHAM 1996 ).

In addition to assessing the change of heterozygosity in the bottleneck lines (H_b), relative to the base population heterozygosity (H₀), we have produced a method for making use of the likelihood calculated from the full allele frequency data set. This approach required some development because there is no simple expression for the probability of the data given the bottleneck population sizes. We overcame this problem in the following way.

Our objective is to obtain the likelihood as a function of the marker allele frequencies before and after the bottleneck in terms of a parameter of biological interest ², the variance in number of descendants left per gene. Under constant population sizes, ² is equivalent to the ratio N_e/N.

There is a relatively simple expression for the probability of a vector (S) recording the allele frequency spectrum after the bottleneck given the number of genes (r) that left descendants in our sample. Note that r is usually smaller than the number of genes extant at the start of the bottleneck;

where the summation is over each possible composition () for the r ancestral genes.

The probability of these ancestral genotypes, P(), is obtained from the multinomial distribution, with parameters r and the relative frequencies of each allele. For the control lines these were calculated from allele frequencies in the base population. In the case of the bottleneck lines the genotypes and family size in the founding population were known. The allele frequencies were therefore obtained as an average weighted by the family size. The analysis was therefore carried out over one generation less than for the controls.

The probability of the r ancestors with allele frequencies having n descendants with allele frequencies S is given by the expression

where n is the number of genes in the sample. The equation is a generalization of the expression given in SLATKIN 1996 to more than two alleles and multiple copies of each allele. This much can be calculated directly, but the total probability of the data consists of a weighted sum of P(S|r) over all possible values of r.

The expression given for the distribution of r in TAVARE 1984 is unwieldy, but it can be simulated using coalescent methodology. If n genes in the sample are descended from r genes at the start of the bottleneck, then the coalescent model explains this by n - r "coalescent events" (e.g., see HUDSON 1990 ). The number of events can be simulated by transforming the standard coalescent process in which events occur at a rate (n + 1 - c)(n - c)/2, where c is the number of events that have already occurred. If the demography could be modeled by a constant population size of N over t generations, then the number of ancestral genes, r, can be obtained by simulating the number of events occurring in a period of length l = . In our case, we know the population size N_i in generation i changes through time and that there is variance in reproductive success. These two complications were taken into account by simulating the number of standard coalescent events in a period of

(1)

Note that ² is the variance in reproductive success per gene, so that in the case of idealized populations of constant size the mean number of descendants is Poisson distributed with mean one, hence ² = 1. When ² = 1 and N_i is constant then expression (1) simplifies to l = as in the conventional coalescent. The distribution of r was checked against discrete simulations for the time period and population sizes involved, and the difference was found to be negligible (data not shown).

The likelihood P(S|²) is obtained from the weighted sum over values of r using the distribution of r, P(r|²):

Estimates of ² obtained from this likelihood function are shown in Figure 3 and Figure 4 as posterior probability density curves. The density is proportional to the product of the likelihood and a noninformative prior (the transformation of a uniform prior between 0 and 1 for the inbreeding coefficient F). Log-likelihood curves were summed over loci to obtain likelihoods for each line and summed across lines to obtain likelihoods for each bottleneck size.

View larger version (27K): In this window In a new window Download PPT slide   Figure 1. Allele frequency distributions in the base population of B. anynana for six allozyme loci and two minisatellite loci, estimated from 300 individuals. Alleles are in order of increasing electrophoretic mobility from a to z. The heterozygosity for each locus is shown in parentheses.

View larger version (87K): In this window In a new window Download PPT slide   Figure 2. Two-locus minisatellite (33.15) genotypes of 18 randomly chosen B. anynana adults from a minimally bottlenecked control line (upper fingerprint) and a one-pair bottleneck line (lower fingerprint), three generations after the founding event (i.e., F3). For loci 1 and 2, respectively, the control line sample contains 16 and 14 heterozygotes, and 12 and 8 electrophoretically distinct alleles, whereas the one-pair line sample contains 10 and 11 heterozygotes, and 3 and 3 electrophoretically distinct alleles.

View larger version (34K): In this window In a new window Download PPT slide   Figure 3. The probability densities for the variance in descendants per gamete 2 for each 1-pair, 3-pair, 10-pair, and control line. For each treatment, line styles for replicates 1–6 are as follows: dotted line with short gaps (1); dashed line with long gaps (2); dashed line with short gaps (3); dotted line with long gaps (4); long and short dashed line with short gaps (5, 1-pair treatment only); long and short dashed line with long gaps (6, 1-pair treatment only). The combined density is drawn as a solid line. Note larger scale for 2 in control line panel.

View larger version (15K): In this window In a new window Download PPT slide   Figure 4. The combined probability densities for the variance in descendants per gamete 2 for control lines (solid line), 1-pair lines (dashed line), 3-pair lines (dotted line), and 10-pair lines (long and short dashed line).

The null hypothesis that the bottlenecked lines had the same value for ² as the control was also evaluated using a likelihood ratio test. The statistic G = 2 ln() is approximately distributed as ², where L_c is the likelihood with ² set to the value that gives the maximum likelihood over all treatments and L_s is the likelihood with the ² value set at the maximum likelihood for each treatment (cf. SOKAL and ROHLF 1995 ).

Point estimates for the coefficient of inbreeding at F₃ relative to the base population (denoted by F₃) for each bottleneck-size treatment were calculated using the familiar equation (FALCONER 1989 )

Substituting,

(2)

where N_f is the number of founders and F_A(3) is the modal value of the transformed probability density for ² for each treatment, taking into account known population sizes.

The method was checked using simulated data with known constant N_e values and census sizes corresponding to those observed in our experiments (results not shown). Marker allele frequencies were set at the founder population values and successive generations were generated by taking multinomial samples from the relative allele frequencies in the previous generation. The posterior distributions of our estimates contained the true value of N_e/N in their 95% credible interval, whereas the estimates of N_e from the reduction in heterozygosity (see below) were biased in the direction of small N_e, particularly by the results from the weakly polymorphic allozyme loci.

The inbreeding that occurred between the founding of the lines and F₃ was also estimated as H_f - H_b/H_f, where H_f is the Hardy-Weinberg heterozygosity from founder allele frequencies, expected if there were no inbreeding. H_f for controls was corrected assuming that H = H₀ in the fraction of the population that was not genotyped.

Simulation:
A computer simulation was constructed to investigate patterns of linkage and selection that might explain the results. The change in heterozygosity and estimates of ² were obtained as for the experimental results. The simulation modeled diploid monogamous organisms. Each diploid genome consisted of five unlinked neutral marker loci. Each marker locus was the middle of five linked loci spaced d cM apart. The other four loci could each have deleterious recessive alleles. One of the nearest pair could have lethals, and the other three could have alleles of less severe effect.

Lethal recessives appeared at frequency _l per gamete and deleterious recessives at frequency _d. Fitness was multiplicative across loci: fitness = w^h, where h is the number of loci homozygous for deleterious alleles. The population size for each treatment mimicked the average in the actual experiments. The number of pairs in the founder, F₁, F₂, and F₃ generations was 1, 36, 150, and 150 (1-pair lines); 3, 110, 150, and 150 (3-pair lines); 10, 150, 150, and 150 (10-pair lines); and 150, 150, 150, and 150 (control). At each marker locus in the founding population, one of eight alleles was allocated at random with frequencies proportional to 3, 4, ... , 10. Deleterious alleles were also allocated at random. In this and each subsequent generation breeding adults were selected with a probability proportional to their fitness and paired.

Possible combinations of the parameters (_i, d, and w) were explored in an attempt to find one that was qualitatively similar to the actual experimental observations in regard to the difference between treatment and controls and, in addition, produced associative overdominance under more prolonged, severe inbreeding.

	RESULTS

TOP ABSTRACT MATERIALS AND METHODS ANALYSIS RESULTS DISCUSSION LITERATURE CITED

Allele distributions and heterozygosity in the base population:
Genotypes of ~300 individuals consisting of all bottleneck line founders and samples of control line founders were used to calculate allele frequency distributions and heterozygosity for each locus in the base population (Figure 1). The distributions illustrate characteristic differences in patterns of variability between allozyme and hypervariable minisatellite loci. The allozyme loci studied have few alleles (2–5), and their allele frequency distributions are dominated by a single allele at high frequency (0.87, on average), resulting in low levels of heterozygosity (0.08–0.39, mean = 0.23). In contrast, the minisatellite loci studied possess large numbers of alleles (26 and 14 for loci 1 and 2, respectively), their allele frequency distributions are multimodal with highest frequencies around 0.20, and heterozygosity at these loci is high (mean = 0.88).

Genetic markers:
The reduction in allelic variability produced by a severe bottleneck is illustrated in Figure 2, which shows the DNA fingerprint of a control line F₃ sample above that of a one-pair line F₃ sample. The one-pair line is more homozygous and contains fewer alleles, three at both loci, consistent with the founder alleles: both parents were heterozygous at loci 1 and 2 and shared one allele at both loci.

There are three F₃ lines whose samples contain alleles not present in the founding parents: one copy of allele a at the Gpi locus in line 3.3; one copy of allele j at 33.15-1 in line 1.1; and four copies of allele b at 33.15-2 in line 10.1. These observations cannot be attributed to incorrect designation of bands. Moreover, the absence of other incompatible alleles within these individuals at other loci, and within the sample as a whole, effectively excludes inadvertent immigration (i.e., contamination) or multiple insemination of founding females as a possible cause. On this basis, approximate estimates of the mutation rate per gamete at the allozyme and minisatellite loci studied are, respectively, 1 x 10^-4 and 7 x 10^-4. The estimate for minisatellites is within the range of accepted values; that for allozymes seems high, but is based only on a single observation.

There was no evidence of linkage disequilibrium across all eight loci in the base population, and only 2 of 18 F₃ line samples showed significant linkage disequilibrium at the 5% level, neither of which was significant taking into account that these were the most extreme results in 18 comparisons. These tests support the assumption, implicit in the analysis of inbreeding, that all loci are independent. There were no significant deviations from Hardy-Weinberg proportions at allozyme loci. However, there was a significant excess of homozygotes at both minisatellite loci in the base population. The later samples were electrophoresed longer, allowing heterozygotes with similar length restriction fragments to be resolved, and the homozygote excess disappeared. Base population frequency estimates would be slightly affected, but there would be negligible effect on the estimates of inbreeding.

Loss of heterozygosity:
H_b/H₀ (expected Hardy-Weinberg heterozygosity calculated separately for each locus from allele frequencies) for the six allozyme loci, two minisatellite loci, and all eight loci are presented in Table 2, for each line, and in Table 3, for each bottleneck size. Average H_b/H₀ for allozymes shows huge variation among replicates within a bottleneck size. This is due to very large sampling variance at these loci with relatively low levels of polymorphism, as is clearly evident from the extreme variation in H_b/H₀ among allozyme loci within lines (not shown). H_b/H₀ for the hypervariable minisatellite loci is much less variable among replicates (and among loci within lines).

An extreme example of how misleading H_b/H₀ based on a few allozyme loci can be is presented by line 1.5, which is completely monomorphic across the six allozyme loci studied. This should not be interpreted to mean that this line is necessarily more inbred than the other one-pair lines, as is apparent from the intermediate H_b/H₀ values at its minisatellite loci. More generally, in the relationship between average allozyme H_b/H₀ and average minisatellite H_b/H₀, there is no significant correlation at the line level but, not surprisingly, a significant correlation at the treatment (bottleneck size) level.

There are unexpectedly large reductions in mean allozyme H_b/H₀ for all bottleneck sizes and control (see Table 3). The minisatellite estimates coincide more closely with expected values of H_b/H₀. Mean H_b/H₀ over all loci is well below that expected for all bottleneck sizes and control, though not significantly so. The greater differentiation in H_b/H₀ among replicate lines of the two smallest bottlenecks is reflected by larger standard errors.

Variance in reproductive success:
The probability density for the variance in descendants per gamete ² is shown for each control, 1-pair, 3-pair, and 10-pair lines in Figure 3. In control lines ² lies between 2 and 5, confirming that N_ei < N_i in the absence of severe bottlenecks. Remembering that ² can be interpreted as being equivalent to for a constant population size, the control estimates of ² are in reasonably good agreement with the direct estimate of N/N_e ( = 1.7). A number of factors may explain the apparently larger value for the control lines, such as multiple generations, no sex segregation coupled with staggered emergence, longer interval on average between mating and egg laying, and variance in larval to adult survival, as well as some successful fertilizations from second matings.

The four control replicates all show similar distributions for ², indicating that the two methods used to establish control lines were indistinguishable. In contrast, there is a large difference in ² among 1-pair lines. One line in particular (1.2) has a much larger ² estimate, and it is notable that this line had the lowest egg hatching rate in F₃ (see SACCHERI et al. 1996 ). There is also a substantial difference in ² among 3-pair lines, with line 3.4 having the largest variance, which again corresponds to the 3-pair line with the lowest egg hatching rate in F₃. The 10-pair lines show little difference in ².

The combined probability density for ² for each treatment, estimated using all lines from each treatment, is shown in Figure 4. There is unequivocal evidence that ² is larger in all three bottleneck treatments than in the control, as shown by the nonoverlapping curves. The G-test also indicates that this difference is highly significant, both for the comparison between all treatments (²₃ = 29.0, P < 0.001) and for the difference between 10-pair and control treatments (²₁ = 21.7, P < 0.001). The densities for all three bottleneck treatments overlap considerably, such that no strong inference can be made about the relative magnitudes of ² in the treatment lines.

Point estimates of the amount of inbreeding occurring after the founding event (hereafter referred to as additional inbreeding) for each bottleneck size, calculated using the likelihood approach and changes in heterozygosity, are given in Table 4. The demographic estimates included in this table describe the additional inbreeding expected when = 2, a value that is well supported for our control populations. The heterozygosity-based estimates suggest much more additional inbreeding than do likelihood estimates for all bottleneck sizes, but are very imprecise. Values of the total inbreeding coefficient F₃ calculated with Equation 2 were 0.32, 0.12, 0.07, and 0.02 for 1-pair, 3-pair, 10-pair, and control treatments respectively.

Simulation:
The combination _l = 1, _d = 3, d = 3, w = 0.5 produced estimates of ² that were greatest in the one-pair line and smallest in the control. One lethal and three deleterious mutations with an average effect of 0.5 is very high compared to estimates from Drosophila, but apparently consistent with the finding that the total genetic load on egg hatching rate in B. anynana is 6.5 lethal or sterile equivalents per gamete, due predominantly to mutants with large effects (SACCHERI et al. 1996 ). The excess inbreeding is also apparent in the loss of heterozygosity, although conclusive results require many more replicates. Each treatment produced a greater reduction in heterozygosity than expected from demographic parameters (see Table 5). On the other hand, this parameter combination produces clear associative overdominance under intensive inbreeding over more generations. When size is sustained at three pairs, the loss in heterozygosity initially appears to follow expectations but, over longer time periods, the loss is significantly slower. Over generations 3 to 14, the estimate of F from changes in heterozygosity is 0.54 (SE = 0.03) as opposed to an expectation of 0.62.

	DISCUSSION

TOP ABSTRACT MATERIALS AND METHODS ANALYSIS RESULTS DISCUSSION LITERATURE CITED

The major finding of this study has been the detection of greater than expected inbreeding in bottlenecked populations. Our analysis explicitly takes into account founder number, most of the variation in size of founder families, and population sizes in F₁ and F₂, thereby eliminating the most obvious sources of inbreeding. Furthermore, we have quantified ², the variance in reproductive success of a gene, in control lines, and shown that the equivalent variance in bottlenecked lines is significantly greater. Factors supplementary to those operating in the control populations appear to be responsible for this excess inbreeding.

B. anynana carries a heavy genetic load on egg hatching rate (SACCHERI et al. 1996 ) which, in the 1-pair and 3-pair lines, manifested itself both as a reduction in the mean as well as a substantial increase in the variance within lines, largely through an increase in the proportion of completely inviable egg clutches. Variance in egg hatching rate no doubt constitutes a major component of the variance in reproductive success and must therefore have been an important factor contributing to the larger ² in 1-pair and 3-pair lines. However, 10-pair lines, which show an increased ², did not show any detectable change in the distribution of egg hatching rate relative to the control distribution. This is not surprising, as a founder sample of 40 haploid genomes would result in minor changes to the frequencies of deleterious recessives responsible for the marked increase in variance in 1-pair and 3-pair lines. An additional mechanism is therefore required to entirely explain the increase in ².

Though a founding bottleneck of 10 pairs does not change allele frequencies appreciably, it leads to nonrandom association between deleterious alleles and marker alleles. As demonstrated by the simulation, selection at loci carrying deleterious alleles could subsequently produce detectable changes in the frequencies of marker alleles in linkage disequilibrium with selected alleles (i.e., hitchhiking), which could in turn inflate the value of ². In control lines linkage disequilibrium, and therefore the effects of hitchhiking on allele frequencies, would be minimal. While this is a plausible mechanism to explain increases in ² that cannot be attributed directly to inbreeding, as seems particularly necessary in the case of 10-pair lines, it should be noted that our evidence for hitchhiking is indirect.

The potential utility of the method to quantify inbreeding is further illustrated by its demonstrated ability to resolve differences in ² among replicate lines in one-pair and three-pair treatments, which could be related to differences in average egg hatching rates. The association between the largest values of ² and the lowest average egg hatchabilities may reflect the presence of deleterious alleles of particularly strong effect in these lines. In comparison, heterozygosity loss proved much less informative. Our analysis extracts more information from each locus and, by combining likelihood curves from different loci, automatically weights them according to their informativeness.

Our result contrasts with the findings of three previously reported studies (FRANKHAM et al. 1993 ; RUMBALL et al. 1994 ; LATTER et al. 1995 ), demonstrating slower than expected declines of heterozygosity at allozyme loci in continuously inbred laboratory populations of Drosophila melanogaster, which are attributed to associative overdominance. This explanation requires linkage disequilibrium between marker loci and those loci segregating for deleterious recessive alleles. Strong linkage disequilibrium could have been generated in these studies through continued close inbreeding [full-sib or double first-cousin in FRANKHAM et al. 1993 and RUMBALL et al. 1994 ] or small population size [N_e 50 in LATTER et al. 1995 ]. Furthermore, Adh and Est-6, the marker loci used in the study with the lowest rate of inbreeding (LATTER et al. 1995 ), are both located on sections of chromosome that were observed to have very low rates of recombination. This effect was only seen at inbreeding in excess of 0.3 (RUMBALL et al. 1994 ; LATTER et al. 1995 ).

The apparently contradictory results may be reconciled by considering the difference in experimental design. Associative overdominance can appear after several generations of severe inbreeding when, for example, by chance the only two surviving haplotypes around a marker locus have different marker alleles and deleterious recessives at different loci. The same two haplotypes could be present in our study, but after the first generation the population size increased, and consequently the deleterious alleles would have been purged by selection, leaving behind other more favorable haplotypes. The change in allele frequency at the markers, therefore, could be unexpectedly rapid, as we observed.

While the differences in experimental design may explain the disparity between Drosophila and Bicyclus experiments, other potential contributory factors should not be overlooked: for example, a greater number of chromosomes in butterflies (2n = 26 in B. anynana, D. BOAKYE, personal communication) and very strong selection imposed by much higher genetic load, perhaps at fewer loci (see SACCHERI et al. 1996 ). The observation that the effects of inbreeding depend on the details of population history has practical implications for conservation. Under continuous severe inbreeding, with or without artificial selection [as in FRANKHAM et al. 1993 and RUMBALL et al. 1994 ], associative overdominance can sustain more genetic variation than expected, while at the same time maintaining a high frequency of the deleterious recessives that contribute to the overdominance. The LATTER et al. 1995 experiment involved population sizes at the upper range of what is realistic in most captive breeding programs and demonstrated substantial fitness benefits, which were attributed to the purging of deleterious alleles. Our experiment employed a regime equivalent to a colonization event or acute crisis in population size. There was also evidence of purifying selection (BRAKEFIELD and SACCHERI 1994 ; SACCHERI et al. 1996 ), but at the cost of additional loss of genetic variation in the generations following the bottleneck.

Excess inbreeding following a bottleneck, due to both variance in inbreeding depression among individuals and hitchhiking, has not been previously documented. It seems reasonable, however, to assume that these processes are likely to be integral to postbottleneck population recovery, in natural as well as experimental populations. Regrettably, there is little data with which to assess the generality of this phenomenon, which could play an important role in shifting multi-locus genotype frequencies. Greater excess inbreeding might be expected for natural populations, where selection coefficients associated with deleterious alleles are likely to be higher (JIMENEZ et al. 1994 ).

The excess inbreeding in this study is important in accounting, at least in part, for the greater than expected loss of additive genetic variation in wing size and pattern elements (I. J. SACCHERI, R. A. NICHOLS and P. M. BRAKEFIELD, unpublished results). However, the estimates of unusually high genetic load on egg hatchability cannot be explained away as a consequence of the excess inbreeding. The calculated load is high compared to Drosophila and Musca, irrespective of whether molecular or demographic estimates of inbreeding are used in the regression (SACCHERI et al. 1996 ).

The simulation results confirm that it is possible for selection against deleterious recessives to produce results that are qualitatively similar to those reported here. Each treatment had overlapping curves for ² estimates that were greater than for the control. It is intriguing that a parameter combination could be found that exhibited this pattern and yet also showed associative overdominance on more prolonged, severe inbreeding, an outcome consistent with the experiments of FRANKHAM et al. 1993 , RUMBALL et al. 1994 , and LATTER et al. 1995 on other species. This result suggests that it could be revealing to impose such prolonged, close inbreeding on B. anynana.

	ACKNOWLEDGMENTS

We acknowledge the assistance of Rosie Heywood, Fanja Kesbeke, Els Schlatmann, and Bert de Winter during the breeding operation, and Marta Dibisceglia with DNA fingerprinting. We are also grateful to Bob Wayne, David Balding, Mark Beaumont, Ilkka Hanski, Dick Frankham, and an anonymous reviewer for constructive criticism of a previous version of this manuscript. This work was supported by the Zoological Society of London, University of Leiden, and small National Environment Research Council (NERC) grant GR9/1178 (ML22).

Manuscript received June 10, 1998; Accepted for publication November 20, 1998.

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