Abstract
Multilocus simulation is used to identify genetic models that can account for the observed rates of inbreeding and fitness decline in laboratory populations of Drosophila melanogaster. The experimental populations were maintained under crowded conditions for ~200 generations at a harmonic mean population size of N_{h} ~65–70. With a simulated population size of N = 50, and a mean selective disadvantage of homozygotes at individual loci ~1–2% or less, it is demonstrated that the mean effective population size over a 200generation period may be considerably greater than N, with a ratio matching the experimental estimate of N_{e}/N_{h} ~1.4. The buildup of associative overdominance at electrophoretic marker loci is largely responsible for the stability of gene frequencies and the observed reduction in the rate of inbreeding, with apparent selection coefficients in favor of the heterozygote at neutral marker loci increasing rapidly over the first N generations of inbreeding to values ~5–10%. The observed decline in fitness under competitive conditions in populations of size ~50 in D. melanogaster therefore primarily results from mutant alleles with mean effects on fitness as homozygotes of s_{m} ≤ 0.02. Models with deleterious recessive mutants at the background loci require that the mean selection coefficient against heterozygotes is at most hs_{m} ~0.002, with a minimum mutation rate for a single Drosophila autosome 100 cM in length estimated to be in the range 0.05–0.25, assuming an exponential distribution of s. A typical chromosome would be expected to carry at least 100–200 such mutant alleles contributing to the decline in competitive fitness with slow inbreeding.
THE ability of small isolated populations to survive and reproduce over periods of many generations, and to evolve if necessary in response to changed environmental conditions, depends to a large extent on the nature of the initial genetic variation in fitness and its response to slow inbreeding. This is an aspect of population genetics of major concern to evolutionary and conservation biologists and to those involved in the genetic improvement of domesticated animal species. However, only very limited data are available on the decline in fitness in small populations because of inbreeding at a rate of 1% or less per generation, especially under competitive conditions (Falconer 1989; Frankham 1995).
Latter and Mulley (1995) and Latter et al. (1995) demonstrated that crowded bottle populations of Drosophila melanogaster maintained without reserves at an initial effective size ~50 individuals show appreciable inbreeding depression in fitness. The observed rate of decline in fitness under competitive conditions was ~2% per 1% increase in the inbreeding coefficient, but only ~0.3% measured in a noncompetitive environment. The probability of extinction resulting from genetic causes during 200 generations of inbreeding was at most 25%. The rate of inbreeding observed at electrophoretic marker loci (Adh, Est6) over this period declined from an initial rate ~1% per generation to an overall average of 0.5% per generation, the mean effective population size exceeding the estimated harmonic mean number of parents by a factor ~1.4. This ratio of effective to census size differs appreciably from other available estimates for captive populations of Drosophila, which range from 0.01 to a maximum ~0.7 (Briscoeet al. 1992).
The purpose of the present paper is to show that a marked reduction in the rate of inbreeding in populations of mean population size ~50 is to be expected from simple genetic models involving many linked loci, and from distributions of mutant effects compatible with the observed pattern of inbreeding depression in competitive fitness. The increase in effective population size and reduction in the rate of inbreeding at marker loci is simply explained by linkage to deleterious homozygotes for alleles with small individual effects on fitness, and by the associative overdominance built up by genetic sampling and natural selection during the process of slow inbreeding.
It is possible to use these unique data to estimate the number and mean effect of the deleterious homozygotes responsible for fitness decline with slow inbreeding in D. melanogaster, by using models involving both overdominant loci and deleterious partial recessives. To investigate this problem systematically it is necessary to consider whether overdominance at the Adh or Est6 marker loci has played a major role in reducing the rate of divergence of gene frequency among replicate subpopulations. The first part of the paper therefore is concerned with overdominant loci segregating independently of other loci subject to selection, or linked on a single chromosome, for comparison with neutral loci. Following the demonstration that the effects of selection for heterozygotes at the marker loci must play at most a minor role, the remainder of the paper deals with attempts to simulate both the observed rate of inbreeding at marker loci and the inbreeding depression in fitness measured in crowded populations, with background loci showing either heterozygote superiority or partial dominance for fitness.
MATERIALS AND METHODS
Experimental data: Sixty slowly inbred lines (SILs) of D. melanogaster were kept under crowded conditions in single bottle cultures without reserves for more than 200 discrete generations, 20 lines being initiated from each of the recently sampled Letona and Tyrrells wild populations, and the Canberra laboratory cage population (Latter and Mulley 1995; Latteret al. 1995). The number of parents transferred each generation was a random variable approximating a lognormal distribution with an estimated harmonic mean of 67 ± 2. All SILs survived the first 40 generations of slow inbreeding, 50 survived to generation 120, and 45 to generation 210. The surviving lines from the Tyrrells wild population and the Canberra cage population were then rapidly inbred by fullsib mating with reserve cultures for 50 generations.
The initial rate of inbreeding estimated from the drift variance in gene frequency at the Adh and Est6 marker loci after ~25 generations was 0.96 ± 0.16% per generation, but the rate declined markedly over the ~200generation period to average only 0.52 ± 0.08% per generation. The initial frequencies of the rarer allele in the three base populations ranged from 0.18 to 0.41 (mean 0.30) for Adh, and 0.15 to 0.32 (mean 0.21) for Est6. Weighted regressions of mean gene frequency q_{t} on number of generations of inbreeding, using the data of Table 3 of Latter et al. (1995) with two independent groups of lines for each base population, show a nonsignificant reduction in the mean frequency of the slow allele at the Adh locus (q_{200}/q_{0} = 0.99 ± 0.07), and a significant 27% reduction in the mean frequency of the fast allele at the Est6 locus (q_{200}/q_{0} = 0.73 ± 0.07, 8 d.f., P < 0.01). The mean value of q_{200}/q_{0} for the two marker loci was 0.86 ± 0.05. The standard errors have been calculated using formulas that include the covariance of estimation of q_{200} and q_{0} (Mode and Robinson 1959).
The fitness of the surviving SILs under competitive conditions declined by 2.01 ± 0.25% per 1% increase in the estimated inbreeding coefficient, relative to the fitness of panmictic populations formed by intercrossing surviving SILs derived from the same base population. The rate of decline in competitive fitness as a function of F was essentially constant on a logarithmic scale, averaging 2.31 ± 0.23% for one of the wild populations (Tyrrells) and the laboratory cage population at F ~0.5, and 1.94 ± 0.16% at F = 0.93 after 30 generations of fullsib mating in lines initiated from the SILs at generation 210 (Latteret al. 1995; Table 4; Figure 2). The ratio of these two estimates is 1.19 ± 0.15, which does not depart significantly from unity.
Genetic models: The simulation results reported in this paper are based on genetic models involving n loci equally spaced along one or two autosomes each of length 100 cM, with two alleles per locus. The fitness values of genotypes A_{1}A_{1}, A_{1}A_{2}, A_{2}A_{2} at a typical locus are denoted by 1 – s_{1}, 1, 1 – s_{2} for overdominant loci and by 1, 1 – hs, 1 – s for loci showing partial dominance, respectively. Gamete formation in females was simulated either by a Poisson distribution of crossovers randomly placed along each autosome or by a twochiasmata model in which one obligatory chiasma occurred randomly in each of two chromosome arms of equal length for each autosome. Gamete formation in males was assumed to involve no crossing over.
When the effects of the individual loci on reproductive fitness were nonidentical, two alternative distributions were used for the selective disadvantage s in homozygotes (Figure 1). Model I assumes the values of s are drawn from an exponential distribution f(s) = λ exp(–λs), with mean λ^{−1}, variance λ^{−2} and coefficient of variation 1.0. Model II assumes a distribution f(s) = 2λs exp(–λs ^{2}_{s}), with mean √[(π/4)λ^{−1}], variance λ^{−1} (1 – π/4) and coefficient of variation 0.523, and is intermediate between an exponential distribution and an equal effects model.
The values of s_{1} and s_{2} for an overdominant locus were each drawn at random from the specified distribution. The frequency functions for the corresponding equilibrium gene frequencies q_{eq} = s_{1}/(s_{1} + s_{2}) are illustrated in Figure 2. With model I the expected distribution of q_{eq} is uniform with mean 0.5, variance 0.0833, and average heterozygosity 0.3333, provided s_{m} is small. With model II the expected distribution for small s_{m} has frequency function f(q) = 2q(1 – q) (1 – 2q + 2q^{2})^{−2}, with mean 0.5, variance (π – 3)/4, and average heterozygosity 2(1 – π/4) = 0.4292. These expectations are close approximations provided s_{m} ≤ 0.2.
The multiplicative fitness w* of a multilocus genotype was calculated as the product over alln loci of the relevant homozygous or heterozygous fitness values. A pure quadratic model of synergistic interaction was also used, with fitness defined as w = exp[–c (ln w*)^{2}]. For a model involving loci with equal effects on fitness, this corresponds to the pure quadratic interaction version of the fitness functions used by Crow (1970), Sved and Wilton (1989) and Charlesworth et al. (1993).
Simulation techniques: The computer program used in this study was adapted from that described by Latter and Novitski (1969) as computer program B and used by Latter (1969) in the direct simulation of response to artificial selection in D. melanogaster. The haploid genotype of an autosome was encoded as binary digits in a vector of integers (Fraser and Burnell 1970), with recombination simulated by the use of logical operations involving a vector of masks. Offspring were produced by generating a random gamete from male and female parents chosen at random with replacement. Differences in reproductive fitness were simulated as differences in the probability of survival to maturity.
Pseudorandom numbers were produced by one of two simple linear congruential generators of the form X_{n}_{+1} = (aX_{n}) mod m. A value of m = 2^{32} with multipliers a_{1} = 1,812,433,253 or a_{2} = 1,566,083,941 was used, with X_{0} an odd positive integer less than m, giving generators of period 2^{30} (Knuth 1981). The multiplier a_{1} was used routinely and a_{2} for the checking of important results. Random values of the selection coefficients s were generated by the equations s = –s_{m} ln(x) for model I and s = s_{m} [–(4/π) ln(x)]^{1/2} for model II, where s_{m} denotes the mean value of s and x is a pseudorandom number in the interval 0,1.
RESULTS
Independently segregating overdominant marker loci: Selection in finite populations in favor of heterozygotes at loci segregating independently of other selected loci may lead to the retention of heterozygosity if the equilibrium frequency in large populations is close to onehalf, or to an increased rate of fixation of the better homozygote if the equilibrium frequency is close to either extreme (Robertson 1962). In the latter case, there may be an increase in the rate of loss of heterozygosity, and a reduction in the mean frequency of the rarer allele (Hill and Robertson 1968). Table 2 shows the results of direct simulation of 200 generations of slow inbreeding in a population of effective size N for independently segregating autosomal loci, each with two alleles A_{1}, A_{2} that show overdominance. The base population is assumed to have the rarer allele A_{2} at each locus initially at its equilibrium frequency. As pointed out by Hill and Robertson (1968), the expected change in heterozygosity is a function of q_{0}, N(s_{1} + s_{2}) and t/N, where the number of generations of inbreeding is denoted by t. The data in Table 2 were generated by the simultaneous segregation of a number of unlinked loci, with multiplicative determination of the net fitness of each genotype. The number of loci in the table was chosen so that randomly generated completely homozygous genotypes had an expected mean fitness ~25% that of the base population, giving results directly comparable with those in Table 3 for loci linked on a single chromosome. The estimates of ΔH_{m}_{(200)}, the average rate of loss of heterozygosity per generation, can be compared with the value 100/(2N) expected in the absence of selection, i.e., 2% for N = 25 and 1% for N = 50.
Table 2 shows that selection leads to an overall reduction in the rate of loss of heterozygosity for 0.2 ≤ q_{0} ≤ 0.5, and an increase in the rate for q_{0} = 0.1 and N(s_{1} + s_{2}) ≥ 5. The mean frequency of the rarer allele at generation 200 expressed as a fraction of its initial frequency, i.e., q_{200}/q_{0}, is reduced by ~80% or more for q_{0} ≤ 0.2 and N(s_{1} + s_{2}) ≥ 5. Such loci are expected to make little contribution to inbreeding depression in fitness measured relative to the panmictic population formed by intercrossing replicate inbred subpopulations. The inbreeding depression resulting from complete homozygosity at a single locus following slow inbreeding for 200 generations, ID_{(%)}, is expressed as a percentage of that expected because of instantaneous homozygosity in the base population (Table 1). In the extreme case when selection leads to the effective elimination of the less fit homozygotes, as for N = 25, q_{0} = 0.1 and N(s_{1} + s_{2}) = 5, the locus concerned makes virtually no contribution to inbreeding depression.
The average rate of inbreeding at the selected loci in Table 2, ΔF_{m}_{(200)}, measures the reduction in heterozygosity relative to the panmictic population formed at generation 200 by intercrossing a large number of replicate inbred lines (Table 1). This can be directly compared to that given by equations 1–3 of Latter et al. (1995). The rate of loss of heterozygosity, ΔH_{m}_{(200)}, and the rate of inbreeding, ΔF_{m}_{(200)}, are identical for an initial gene frequency of q_{0} = 0.5, when the expected change in mean gene frequency is zero (Table 2). However, the two measures are appreciably different for q_{0} ≤ 0.2, particularly at values of N(s_{1} + s_{2}) ≥ 5, because ΔF_{m}_{(200)} measures the reduction in heterozygosity relative to that in a contemporary panmictic population with gene frequency q_{200}.
The estimates of ΔF_{m}_{(200)} for independently segregating loci in Table 2 suggest that for N = 25–50 and q_{0} = 0.2–0.3, corresponding to the mean initial gene frequencies for the Est6 and Adh marker loci, singlelocus overdominance could account for average rates of inbreeding ~0.5% per generation provided N(s_{1} + s_{2}) ≥ 2.5. However, the associated reduction in the frequency of the rarer allele over 200 generations should be 50% or more for q_{0} = 0.2, and 35% or more for q_{0} = 0.3, significantly greater than the corresponding experimental estimates of 27% and 1% for Est6 and Adh, respectively. There are therefore no grounds for supposing that the observed mean rate of inbreeding of 0.52% reflects natural selection favoring heterozygotes at the two marker loci alone, if it is accepted that gene frequencies at the loci were initially at equilibrium.
The observed rate of inbreeding and change in mean gene frequency at the Adh and Est6 loci might possibly be explained by singlelocus overdominance, if it is postulated that the equilibrium frequency of the rarer allele is higher than its initial frequency. With such a model an initial increase in frequency toward the equilibrium may be partially or completely offset by a subsequent reduction in frequency as inbreeding proceeds (Hill and Robertson 1968). However, this model also predicts that in a large panmictic population formed by intercrossing replicate SILs the mean frequency of the rarer allele should increase appreciably. There is evidence for the Canberra population that this prediction was not realized, the panmictic population CP(80,n) showing a nonsignificant reduction in the frequency of the rarer allele at both marker loci over a period of 60–80 generations of random mating.
Recurrent mutation: The incorporation of mutation into the model makes little difference to the conclusions drawn from the data of Table 2. Recurrent mutation from A_{1} to the rarer allele A_{2} will oppose the effects of selection on mean gene frequency, but the effect is of minor importance even for high mutation rates. For example, in populations with N = 50, q_{0} = 0.2, N(s_{1} + s_{2}) = 5 and a mutation rate u = 10^{−4} to the rarer allele, the mean rate of inbreeding was ΔF_{m}_{(200)} = 0.28 ± 0.03%, and the mean frequency of A_{2} was q_{200}/q_{0} = 0.29 ± 0.02. For N = 50, q_{0} = 0.2, N(s_{1} + s_{2}) = 1 and u = 10^{−4}, the mean rate of inbreeding was ΔF_{m}_{(200)} = 0.70 ± 0.02%, with mean gene frequency q_{200}/q_{0} = 0.84 ± 0.03. Comparison with the results in Table 2 for the same regimes without mutation shows the effects of a high rate of mutation to be detectable, but not sufficient to alter the conclusions which have been reached. The effects of mutation rates of 10^{−5} or less are negligible in this context.
Variable population size: Simulation of variable population size shows the rate of inbreeding and change in mean gene frequency at unlinked loci to be equivalent to those in populations of constant size with the same harmonic mean N_{h}. A lognormal distribution with mean 4.26 and standard deviation 0.34 on the scale of natural logarithms was used to simulate actual breeding population size (Latteret al. 1995), with ratios of N_{e}/N = 0.60 and 0.75 to give harmonic mean effective population sizes of N_{h} = 40 and 50, respectively. For five different combinations of the parameters N, q_{0} and N(s_{1} + s_{2}), the mean estimates of rate of inbreeding and change in gene frequency after 200 generations were not significantly different from the corresponding populations with constant N (results not shown).
Linked overdominant marker loci: Table 3 shows the effects of inbreeding at linked overdominant and neutral loci in an equilibrium base population, for comparison with those for unlinked loci in Table 2. A chromosome of 100 cM in length was assumed, corresponding roughly to a major autosome of D. melanogaster, with loci equally spaced along the chromosome. Every 10^{th} locus was assumed to be neutral in its effects on reproductive fitness. It was assumed that no crossing over occurred in males and that the number of randomly placed crossovers in females was given either by a Poisson distribution or by the twochiasmata model (see materials and methods). No significant mean difference was detected between the two models of recombination, and the results in Table 3 are means over equal numbers of replicates of each.
Comparisons of the corresponding estimates in Tables 2 and 3 show that linkage reduces the average rate of inbreeding ΔF_{m}_{(200)} at selected loci by up to 50%, with the greatest effect at low values of N(s_{1} + s_{2}) where many loci of small effect on fitness are involved. The rate of inbreeding in the presence of linkage is also little affected by the value of q_{0} over the range tested. The change in mean gene frequency at linked loci is also less marked than for independently segregating loci, particularly at the higher value of N = 50. The rate of inbreeding at neutral unlinked loci is expected to be 2% per generation for N = 25 and 1% per generation for N = 50, and for the parameter combinations in Table 3, linkage reduces the rate of inbreeding at neutral loci by as much as 40–50% when N(s_{1} + s_{2}) ≤ 1.
The data of Table 3 clearly establish that the mean rate of inbreeding over a 200generation period at neutral marker loci in populations of effective size N = 50 can be reduced by linkage to ~0.5% per generation. Interpolation in Table 3 also indicates that, with linkage, marker loci subject to overdominant selection with equilibrium frequencies of 0.2–0.3 could show a mean rate of inbreeding ~0.5% per generation, with an overall reduction in gene frequency ~25% or less over 200 generations, given N ~40 and N(s_{1} + s_{2}) ≤ 1. Further simulation has confirmed this conclusion for values of N(s_{1} + s_{2}) = 0.5 and 1.0 (results not shown).
Quadratic interaction: The foregoing conclusions are little affected by the use of a model of pure quadratic interaction in the determination of net fitness. For example, simulation of a single autosome of length 100 cM for N = 50, q_{0} = 0.2, N(s_{1} + s_{2}) = 1, n = 478, with c = 0.24 chosen to give instantaneous inbreeding depression equal to that in Table 3, gave mean rates of inbreeding at neutral and selected loci of 0.58 ± 0.03% and 0.45 ± 0.01% per generation, respectively, which do not differ greatly from the corresponding values in Table 3. The value of q_{200}/q_{0} = 0.85 ± 0.02 is also in close agreement with that given by computer runs with multiplicative interaction. With the same set of parameter values in simulation involving two autosomes each of length 100 cM with 478 loci on each chromosome, and c = 0.15 chosen to give the same single autosome instantaneous inbreeding depression as in Table 3, the estimates of the mean rates of inbreeding for neutral and selected loci were 0.49 ± 0.01% and 0.41 ± 0.02%, respectively, with a value of q_{200}/q_{0} = 0.86 ± 0.02. The values of ΔF_{m}_{(200)} are lower than those for a single autosome with pure quadratic interaction but do not differ greatly from the corresponding values in Table 3 with multiplicative interaction.
Linked overdominant background loci with neutral markers: The foregoing results make it clear that the effects of selection for heterozygotes at the Adh and Est6 marker loci must be at most of minor importance in the SILs of Latter et al. (1995), suggesting that regimes with neutral markers and N ~50, or overdominant markers with N ~40 and N(s_{1} + s_{2}) ≤ 1, are appropriate for further testing of suitable genetic models. Table 4 shows results for neutral marker loci embedded in an autosome with overdominant background loci for which values of s_{1}, s_{2} were drawn at random from the specified distribution f(s). Mutation was ignored, and crossing over in females was simulated by the twochiasmata model. Population size N was assumed to be constant over generations, with equal numbers of male and female parents. The data refer to a base population in which all loci are in equilibrium under multiplicative interaction. Simulation of quadratic interaction involved two autosomes each of length 100 cM with n loci per chromosome.
The strategy of simulation for given N, s_{m}, and f(s) was to choose the value of n that corresponded to the experimental estimate of total inbreeding depression measured under crowded conditions, w_{FS30}/w_{Pan}. The simulation estimates of ΔF_{m}_{(25)}, ΔF_{m}_{(200)}, and Δw_{m}_{(200)} were then compared with the experimental estimates from D. melanogaster given in the last line of Table 4. The w_{FS30}/w_{Pan} estimate of ~0.40 for the homozygous fitness of a major autosome in Drosophila is based on an estimated inbreeding coefficient of F = 0.45, which allows for the lower inbred load per unit length of the X chromosome (Latter and Sved 1994). A value of w_{FS30}/w_{Pan} ~0.17 was used for the simulation of pure quadratic interaction, based on an estimated inbreeding coefficient of F = 0.9 for homozygosity of both major autosomes.
The conclusions from the data of Table 4 are similar for both distributions of homozygous selective disadvantage. With neutral marker loci, a satisfactory fit to the Drosophila data is given by a breeding population size of N = 50, and overdominant background loci with mean homozygote selective disadvantage s_{m} < 0.01 for the multiplicative model, i.e., top four rows of the table, or s_{m} < 0.005 for the synergistic interaction model (top two rows). No significant change in mean gene frequency was detected at the marker loci. The results appear not to be greatly affected by the model of recombination or the initial frequency of alleles at the marker loci. Similar data have been obtained with a Poisson distribution of randomly placed crossovers and with marker alleles at initial frequencies of 0.2, 0.8.
High values of s_{m} in Table 4 are associated with a smaller number of loci contributing to the decline in fitness with inbreeding and, hence, looser linkage between neighboring loci. The mean rate of inbreeding at neutral marker loci, ΔF_{m}_{(200)}, is then significantly greater than that observed when many loci of small effect are involved. Nonlinearity of the decline in fitness as a function of F is extremely marked for high values of s_{m}, estimates of Δw_{m}_{(200)} departing significantly from the Drosophila value of 1.19 for s_{m} ≥ 0.01 with the multiplicative model, and for s_{m} ≥ 0.005 with the pure quadratic model. An estimate R_{2} of Δw_{m}_{(200)} in Table 4 has been compared with the Drosophila estimate of R_{1} = 1.19 ± 0.15 using an approximate test of the difference D = ln(R_{1}) – ln(R_{2}), with SE(D) calculated by the use of Equation 2 of Latter and Mulley (1995). For R_{2} = 0.85 ± 0.02, the value of D is 0.34 ± 0.13.
Figure 3 shows further details of the decline in fitness as a function of F for the regimes of Table 4 involving model I and multiplicative interaction, with s_{m} = 0.10, 0.05, 0.01, and 0.001. The significant values of Δw_{m}_{(200)} for s_{m} ≥ 0.01 in Table 4 correspond to rates of inbreeding depression during slow inbreeding, which are appreciably less than would be predicted from the mean fitness at complete homozygosity and the inbreeding coefficient for neutral loci.
Nonequilibrium base populations: The numerical estimates of rates of inbreeding and fitness decline in Table 4 are little altered by the use of a partially inbred base population, simulating a local population of Drosophila with F ~ 0.05–0.10 (Latter 1981; Latter and Mulley 1995). In addition, comparable results have been generated with a base population assumed to be in equilibrium in its natural environment but transposed to a test environment in which all values of s were resampled from the same distribution f(s). Such populations behave in essentially the same way under slow inbreeding as an equilibrium base population. For regimes with N = 50 and 2Ns_{m} = 0.1 or 0.5, corresponding to the first four lines of Table 4 with multiplicative interaction, the nonequilibrium populations showed initial rates of inbreeding averaging 0.82 ± 0.03 and average rates of inbreeding averaging 0.60 ± 0.02: The values of Δw_{m}_{(200)} were not significantly different from those in Table 4.
Linked overdominant background loci with overdominant markers: The results in Table 3 suggest that loci subject to overdominant selection with N(s_{1} + s_{2}) = 1 could respond to slow inbreeding in a way similar to the experimental allozyme marker loci. Table 5 summarizes the behavior of overdominant marker loci segregating in an initially equilibrium population with N(s_{1} + s_{2}) = 1, q_{0} = 0.2 or 0.3 and background loci with an exponential distribution of gene effects, for comparison with the neutral marker loci of Table 4. The broad features of the results in Table 5 for background loci with 2Ns_{m} ≥ 1 are in good agreement with the Drosophila data, with (1) a reduction in the frequency of the rarer allele at the overdominant marker loci during 200 generations of slow inbreeding; (2) a mean rate of inbreeding which is greatly reduced by comparison with the estimated initial rate; (3) a total fitness decline which closely matches that observed in Drosophila; and (4) an essentially linear decline in fitness as a function of the inbreeding coefficient estimated from the marker loci. Interpolation in the table suggests that values of s_{m} > 0.02 are not capable of explaining the experimental data. Averaging over population sizes of N = 40 and 50 with 2Ns_{m} ≤ 1, the simulated populations show (1) reductions in mean frequency of the rarer allele at marker loci of 22 ± 1% and 13 ± 2% for q_{0} = 0.2 and 0.3, respectively; (2) an initial rate of inbreeding of ΔF_{m}_{(25)} = 0.78 ± 0.01% by comparison with an average value over 200 generations of ΔF_{m}_{(200)} = 0.52 ± 0.01%; and (3) a mean value of Δw_{m}_{(200)} = 0.97 ± 0.01. The model can therefore account for all the main features of the Drosophila data apart from adaptation to the test environment, which requires relaxation of the assumption of an equilibrium base population for at least some loci.
Linked deleterious partially recessive alleles with neutral markers: We turn now to genetic models in which deleterious recessive mutant alleles are solely or largely responsible for the decline in fitness observed in populations of D. melanogaster subject to progressive inbreeding. Unless otherwise indicated, the models involved in this and the next section are based on the following assumptions: (1) mutation occurs at background loci throughout the period of slow inbreeding at a rate u = 5 × 10^{−5} per generation to deleterious recessive alleles having effects hs in heterozygotes and s in homozygotes, with the value of s for each locus randomly chosen from an exponential distribution; (2) mutation at the marker loci and back mutation at the background loci is negligible; (3) population size N is constant with equal numbers of male and female parents; (4) recombination in females is according to the twochiasmata model; (5) fitness is determined multiplicatively; and (6) an infinite equilibrium base population is assumed, and initial gene frequencies at the background loci are given by Equation 5 of Crow (1970), with F = 0 and an upper bound of q_{0} = 1 for small values of s.
Table 6 gives the results of simulation for neutral marker loci, N = 50, and values of h = 0.0, 0.1, and 0.2. The harmonic mean number of parents in the Drosophila SILs was ~65–70, and a simulated population size of 50 is, therefore, considered to be a maximum possible value. With n chosen to give a total inbreeding depression of w_{FS30}/w_{Pan} ~0.40, the simulated populations with 2Ns_{m} ≤ 1.0 show rates of inbreeding at neutral loci and values of Δw_{m}_{(200)} that do not depart significantly from the Drosophila data, provided hs_{m} <0.001. For these regimes, the average rate of inbreeding ΔF_{m}_{(200)} is reduced to ~60% that predicted in the absence of selection. Graphs of the decline in fitness as a function of F are similar to those of Figure 3 for overdominant background loci.
Acceptable parameter values are therefore N = 50 and s_{m} ≤ 0.01, with the mean value of the selection coefficient hs_{m} < 0.001. A mutation rate of u = 5 × 10^{−5} was assumed in all the regimes of Table 6, but similar results are obtained with lower mutation rates and appropriately increased numbers of loci. For example, a regime with N = 50, 2Ns_{m} = 1, h = 0.1, n = 6980 and u = 2.5 × 10^{−5} has given estimates that do not differ significantly from those in Table 6 for n = 3850 and u = 5 × 10^{−5}. The conclusions drawn from Table 6 are also essentially independent of the assumption that the base population is in equilibrium under the effects of selection and mutation pressures. For example, with N = 50, 2Ns_{m} = 1, h = 0.1, n = 2215, and a base population assumed to be in equilibrium in its natural environment but transposed to a test environment in which all values of s are resampled from the same distribution f(s), the estimated rates of inbreeding and decline in fitness do not differ significantly from those in Table 6.
For s_{m} in the range 0.001–0.01, i.e., the first nine rows of Table 6, the unweighted mean level of associative overdominance at segregating marker loci is ~4–5% after 4N generations of slow inbreeding. Approximately 25% of marker loci are still segregating at this stage, of which ~85–90% show average heterozygote superiority. With s_{m} = 0.05 the proportion of marker loci segregating at generation 4N is only 15–20%, of which ~75% show average heterozygote superiority. The mean level of associative overdominance is little altered, and the increased rate of fixation at marker loci is presumably a result of the higher rate of recombination between marker and background loci when mutants of larger effect are involved. In none of the regimes of Table 6 did the mean value of q_{200}/q_{0} at the marker loci depart significantly from unity.
Figures 4 and 5 illustrate the goodnessoffit of the computer model with neutral markers to the experimental data on the rate of inbreeding and decline in competitive fitness, respectively, for a regime with N = 50, h = 0.1, u = 5 × 10^{−5}, 2Ns_{m} = 0.3, n = 6060 and initial gene frequency q_{0} = 0.2 at the neutral marker loci. Figure 6 shows the pattern of change in the mean level of associative overdominance at neutral marker loci in the simulated populations of Figures 4 and 5. Over the first Ngeneration period, the apparent selection coefficients s_{(1)} and s_{(2)} increase rapidly to ~0.05 and ~0.10, respectively, progressing to nearequality at ~0.05 as the mean gene frequency at unfixed loci, q(seg), approaches 0.5. The overall effect of selection at all stages of the inbreeding process is to maintain the current mean gene frequency at segregating neutral loci. The mean gene frequency over fixed and segregating neutral loci, q(m), does not depart significantly from the value q_{0}.
Linked deleterious partially recessive alleles with overdominant markers: We finally consider models involving background loci subject to mutation to deleterious recessives, together with a small proportion (1%) of marker loci showing overdominance with N(s_{1} + s_{2}) = 1 and q_{0} = 0.2. The assumptions and simulation procedures are the same as in the preceding section unless otherwise stated. Summary statistics are given in Table 7 for N = 40 and 50, with background loci for which h = 0.0, 0.1, or 0.2 and u = 5 × 10^{−5}. None of the regimes differs significantly from the experimental data in the estimated value of q_{200}/q_{0} or in the initial rate of inbreeding ΔF_{m}_{(25)} estimated from the overdominant marker loci. Interpolation in the table for ΔF_{m}_{(200)} and Δw_{n}_{(200)} leads to the conclusion that values of s_{m} < 0.02 may be compatible with the Drosophila data, provided hs_{m} < 0.002. Comparison of regimes in Tables 6 and 7 with N = 50 and 2Ns_{m} = 1.0 shows the effects of overdominance at the marker loci to be a reduction in the final mean gene frequency q_{200}/q_{0} ~20% and a reduction in the mean rate of inbreeding by approximately 25%.
The foregoing conclusions are little affected by changes in the assumed distribution f(s). A series of regimes comparable to that in Table 7 with N = 50, 2Ns_{m} = 1.0, and h = 0.2, using f(s) = λ exp(–λs), f(s) = 2λs exp(–λs^{2}), and an equal effects model, with N(s_{1} + s_{2}) = 1 and q_{0} = 0.2, 0.3 at the overdominant marker loci, has shown no significant differences in rates of inbreeding or depression in fitness. A satisfactory fit to the Drosophila data has also been demonstrated for a model in which a proportion p of loci are supposed to have values of s in the laboratory resampled at random from the same distribution f(s) = λ exp(–λs) as that assumed for the equilibrium natural population. For N = 50, 2Ns_{m} = 0.5, h = 0.2, n = 5500, p = 0.25 and overdominant markers with N(s_{1} + s_{2}) = 1 and q_{0} = 0.2 or 0.3, the estimates obtained were q_{200}/q_{0} = 0.83 ± 0.05, ΔF_{m}_{(25)} = 0.72 ± 0.04, ΔF_{m}_{(200)} = 0.50 ± 0.02, w_{FS30}/w_{Pan} = 0.39 ± 0.02, and Δw_{m}_{(200)} = 0.96 ± 0.03, none of which departs significantly from the experimental estimates for Drosophila.
DISCUSSION
Associative overdominance and the rate of inbreeding at marker loci: This investigation was prompted by the observation that the rate of inbreeding at electrophoretic marker loci in crowded bottle populations of D. melanogaster (harmonic mean population size N_{h} ~65–70) averaged 0.52 ± 0.08% per generation over a period of 200 generations, corresponding to an estimated mean ratio of N_{e}/N_{h} ~1.4. This is an unusual finding because typical N_{e}/N ratios estimated for vial or bottle populations of Drosophila are ~0.6–0.8, and those for large cage populations are considerably less (Falconer 1989; Briscoeet al. 1992). However, Rumball et al. (1994) also showed that the decline in heterozygosity at electrophoretic markers in vial populations of D. melanogaster under rapid inbreeding is ~80% of that expected, and cited earlier reports that claimed the rate of loss of heterozygosity under inbreeding may be slower than predicted by population genetics theory.
It was suggested by Rumball et al. (1994) and Latter et al. (1995) that the reduction in rate of inbreeding was most likely the result of associative overdominance at the marker loci caused by the segregation of linked deleterious mutations, based on the theoretical and empirical studies of the behavior of linked genes in small populations by Sved (1968, 1972), Ohta (1971), Charlesworth (1991) and Charlesworth et al. (1992, 1993). Sved (1968) showed that the apparent selective values at a neutral locus linked to an equilibrium overdominant locus are highest for the heterozygote if there is linkage disequilibrium but that this apparent overdominant selection does not change gene frequency. The unselected locus is in a state of “pseudoequilibrium,” tending to maintain current allele frequencies but not to return to any previous value if perturbed. A similar phenomenon in expected when a neutral locus is linked to many loci at which partially recessive detrimental alleles occur at equilibrium under recurrent mutation, selective elimination, and random drift resulting from finite population size, where equilibrium gene frequencies predicted from the apparent selection coefficients are close to the prevailing mean gene frequency (Ohta 1971).
It is demonstrated in this paper for a variety of genetic models that the associative overdominance hypothesis accounts satisfactorily for the observed consequences of slow inbreeding, provided the mean selective disadvantage s_{m} of the homozygotes responsible for fitness decline is ~0.01–0.02 or less. This conclusion appears to be independent of assumptions concerning the distribution f(s), at least for those with coefficients of variation in the range 0–1. Effective population sizes for these regimes estimated from the rates of inbreeding over a 200generation period correspond to N_{e}/N_{h} ratios exceeding 1.6 (Tables 4, 5, 6 and 7). The results shown in Figures 4, 5 and 6 are representative of acceptable regimes, with estimates of apparent heterozygote superiority at segregating marker loci ~5% at generation 200. Both neutral marker loci and those showing overdominance were studied in the simulated populations, in view of the significant reduction in the mean frequency of the fast allele at the Est6 locus observed in the Drosophila populations. This change in gene frequency may have been the result of directional selection because Est6F is known to show seasonal fluctuations in frequency in natural populations (Franklin 1981), but it is also compatible with a model involving slight heterozygote superiority at the marker locus in addition to associative overdominance caused by segregation at closely linked loci.
All regimes in Tables 4, 5, 6 and 7 are compatible with the experimental data of Latter et al. (1995) in showing a level of inbreeding depression in nearhomozygous lines of w_{FS30}/w_{Pan} ~0.40 for a single major autosome. Those involving large values of s_{m}, and therefore fewer loci contributing to inbreeding depression, show nonlinearity of log fitness as a function of the estimated inbreeding coefficient F (Figure 3), which is accentuated in models involving synergistic interaction among loci (Table 4). These regimes also show higher average rates of inbreeding at the marker loci than those estimated for Drosophila.
Predictions of rates of inbreeding in populations maintained in an optimal environment: The slowly inbred populations studied by Latter et al. (1995) were maintained under competitive conditions in crowded bottle cultures where early egg laying and early emergence of offspring were advantageous. The resulting nearhomozygous genotypes showed inbreeding depression in fitness of 85–90% measured in a similar competitive environment and 25–30% measured under optimal conditions in singlepair vial cultures. We can use the foregoing computer models and parameter values to predict the outcome of experiments designed to measure the realized rate of inbreeding in slowly inbred populations of effective size N_{e} maintained under optimal conditions throughout their entire history. An experiment of this sort has recently been reported by Montgomery et al. (1997). Taking the model with N = 50 and neutral marker loci illustrated in Figures 4, 5 and 6, the observed inbreeding depression of 25–30% in an optimal environment can be simulated by assuming either (1) that the mean homozygote disadvantage is s_{m} = 0.003 in the competitive environment and s_{m} = 0.003 × 0.16 in the optimal environment, all loci being equally reduced in effect; or (2) that 84% of potentially deleterious recessive alleles are without effect on fitness in the optimal environment. The factor 0.16 involved is given by the ratio 0.32/2.01 derived from Figure 2 of Latter and Mulley (1995). These two extreme assumptions have given similar estimates of the rate of inbreeding predicted in an optimal environment, the mean values being 0.90 ± 0.03% per generation in simulated populations of size N_{e} = 50 over 50 generations, and 0.84 ± 0.02% per generation over 200 generations, i.e., 85–90% of the rate predicted by neutral population genetics theory. In the type of experiment conducted by Montgomery et al. (1997), with equal numbers of offspring contributed by each pair of parents throughout the period of slow inbreeding, the intensity of natural selection is effectively halved, and we would therefore expect a rate of inbreeding ~90–95% of the theoretical prediction.
Assumptions involved in the genetic models: Most of the simulation results in this paper involve the following simplifying assumptions. Population size during the period of slow inbreeding is constant with equal numbers of male and female parents and with discrete generations. Loci are assumed to be autosomal and equally spaced along chromosomes 100 cM long, with two alleles per locus. Marker loci are distributed along the entire length of the chromosome. Gamete formation in females is simulated by a twochiasmata model in which one obligatory chiasma occurs randomly in each of two chromosome arms of equal length. Gamete formation in males is assumed to involve no crossing over.
A uniform mutation rate of 5 × 10^{−5} to deleterious recessive alleles is assumed, with constant h for all loci. The rate of mutation at electrophoretic marker loci is assumed negligible. The selective disadvantage of homozygotes is distributed exponentially, with fitness determined multiplicatively and simulated as the probability of survival to reproductive age. SILs are initiated from an infinite base population in equilibrium under natural selection and recurrent mutation. The assumption is also made that the intensity of natural selection in the crowded bottle populations used to maintain the SILs is identical to that in the competition bottles used to measure reproductive fitness.
It has been shown that variable population size, alternative distributions of values of s, and different models of recombination in females have little effect on the outcome of the genetic simulation. The basic conclusions regarding the magnitude of s_{m} also appear to be independent of the type of selection involved at individual loci, the inferences from Tables 4 and 5 for background loci with superior heterozygotes being essentially the same as those from Tables 6 and 7 for loci with deleterious partial recessives. Nonequilibrium base populations, particularly those which are partially inbred to simulate a local population of Drosophila with F ~0.05–0.10, have given similar estimates of rates of inbreeding and fitness decline to those for equilibrium populations. Loci randomly placed along the chromosome have also been shown to lead to the same conclusions as equally spaced loci. For example, a regime comparable to that in Table 6 with 2Ns_{m} = 5.0 and h = 0.0 gave estimates ΔF_{m}_{(25)} = 0.70 ± 0.05, ΔF_{m}_{(200)} = 0.76 ± 0.02 and Δw_{m}_{(200)} = 0.47 ± 0.03, which are essentially the same as those in Table 6 for equally spaced loci.
A quadratic model of synergistic interaction in the determination of fitness has been shown to accentuate the nonlinearity of the decline in fitness on a logarithmic scale associated with high values of s_{m} (Table 4), reinforcing the conclusions drawn from multiplicative models about the importance of genes of small effects on fitness. The data of Latter and Robertson (1962) also suggest little departure from an exponential decline in competitive index with rapid inbreeding, expressed as a function of the theoretical value of F.
Estimates of rates of inbreeding and of levels of associative overdominance in the simulated populations have been averaged over marker loci located throughout the length of the chromosome. Estimates of associative overdominance based on a subdivision into five segments, each 20 cM long, suggest that those shown in Figure 6 are ~90% those expected at the locations of Adh and Est6 on chromosomes 2 and 3, respectively. The rates of inbreeding for small values of 2Ns_{m} in Tables 4, 5, 6 and 7 are therefore expected to be slightly overestimated, but the bias is trivial by comparison with the standard errors associated with the Drosophila data.
Though the level of crowding in the stock bottles during slow inbreeding was comparable to that in the competition bottles used to measure fitness, the two environments differed in some important respects (Latter and Mulley 1995). The stock bottles were maintained by transfer of a variable number of parents each generation following restricted feeding in the parental cultures, and involved both males and females. The fitness measurements are based on the female test of Latter and Robertson (1962) with a constant number of inseminated females in the absence of males, following full feeding in freshculture vials.
If the intensity of natural selection in the competition cultures was less than that in the stock bottles, the rates of inbreeding in Tables 4, 5, 6 and 7, based on a total inbreeding depression of w_{FS30}/w_{Pan} = 0.40, would be expected to be overestimates. To take an extreme position, regimes with n chosen to give w_{FS30}/w_{Pan} = 0.25 have been simulated for comparison. For N = 50, 2Ns_{m} = 1.0, h = 0.1, n = 6168, and neutral markers as in Table 6, estimates were w_{FS30}/w_{Pan} = 0.23 ± 0.01, ΔF_{m}_{(200)} = 0.55 ± 0.03, and Δw_{m}_{(200)} = 0.80 ± 0.03: The rate of inbreeding is significantly reduced by comparison with that of 0.68 ± 0.02 in Table 6, but the estimate of Δw_{m}_{(200)} is only slightly altered. For N = 40, 2Ns_{m} = 2.5, h = 0.1, n = 3946, and overdominant markers as in Table 7, the estimates were w_{FS30}/w_{Pan} = 0.23 ± 0.03, ΔF_{m}_{(200)} = 0.56 ± 0.03, and Δw_{m}_{(200)} = 0.74 ± 0.04. Again, the rate of inbreeding is significantly reduced by comparison with the value of 0.76 ± 0.03 in Table 7, but the value of Δw_{m}_{(200)} is little affected. Because Δw_{m}_{(200)} is the parameter that has been most effective in identifying regimes incompatible with the Drosophila data, the same conclusions flow from estimates based on values of w_{FS30}/w_{Pan} = 0.25 and 0.40.
Model of a Drosophila chromosome: The number of loci on each of the two major autosomes of D. melanogaster is very likely to be ~6000, based on current estimates of the number of mRNA transcripts per polytene chromosome band and the fraction of DNA giving rise to mRNA (John and Miklos 1988; Sved and Wilton 1989). The minimum mutation rate to deleterious alleles has been estimated by Mukai (1964), Mukai et al. (1972) and Ohnishi (1977) to be u = 0.12–0.34 per chromosome based on mutants affecting viability in quasinormal chromosomes, assuming an exponential distribution of s. A mutation rate of ∑u = 0.34 –0.44 per chromosome has been estimated by Houle et al. (1992) for mutants affecting competitive fitness, based on an exponential distribution of s for mutants in quasinormal chromosomes. These figures for gene number and mutation rate imply an average minimum mutation rate per locus of ∑u ~5 × 10^{−5}.
Sved and Wilton (1989) have shown that models of a Drosophila autosome involving deleterious partially recessive mutations and multiplicative interaction are capable of explaining the low levels of competitive fitness (~20–25%) observed for chromosome homozygotes derived from natural populations. Their calculations for a base population at equilibrium under natural selection and mutation pressures show that a wide range of values of s are compatible with this level of inbreeding depression but that values of h > 0.2 are not acceptable unless mutation rates are much higher than the above estimate. The simulation data in Tables 4, 5, 6 and 7 of the present paper strongly suggest that the average selective disadvantage of homozygotes contributing to the decline in fitness with slow inbreeding is s_{m} ~0.02 or less for either the overdominance or partial recessives model. For loci with deleterious recessive mutants, a mean value of hs_{m} ≤ 0.002 is required, the appropriate mean value of h being 0.2 or less in agreement with the predictions of Sved and Wilton (1989). B. Charlesworth and K. A. Hughes (personal communication) have also estimated mean values of h ~0.2 and hs_{m} = 0.008 for variants in quasinormal chromosomes affecting life history traits in natural populations.
If the contribution of overdominant loci to inbreeding depression is small, interpolation in Tables 6 and 7 indicates that a major autosome of D. melanogaster produces deleterious mutations with hs_{m} ≤ 0.002 at a minimum rate ∑u = 0.05–0.25, the data for h = 0.0 leading to the lower estimate and those for h = 0.2 to the higher value. There will, in addition, be mutants of large effect on fitness and those with higher values of h, which make little contribution to inbreeding depression when populations are slowly inbred under conditions of intense natural selection. A typical chromosome is estimated to carry at least 100–200 mutant alleles contributing to inbreeding depression if h = 0.0, the higher figure deriving from acceptable regimes with neutral markers (Table 6) and the lower value from those with overdominant markers (Table 7). For h = 0.1 the frequency of such mutants can be estimated to be at least 200–500.
With an exponential distribution of gene effects and hs_{m} ≤ 0.002, it is to be expected that roughly 40% of the mutants concerned will have values of hs ≤ 0.001. Is it realistic to suggest that selection coefficients of this order are responsible for a major porportion of the decline in fitness associated with slow inbreeding? There is no doubt that selection against mutants with very small effects on fitness can be responsible for conspicuous departures from the predictions of neutral theory. In natural populations of Drosophila, it appears that variants for glucose metabolizing (group I) enzymes are subject to negative selection as heterozygotes, with selection coefficients ~20 times the mutation rate (Latter 1981). The estimated mutation rates for allozymes are ~5–10 × 10^{−6} (Mukai 1970; Tobari and Kojima 1972; Mukai and Cockerham 1977), suggesting selection coefficients against heterozygotes of only 1–2 × 10^{−4}. Their effects on the distributions of gene frequencies and of heterozygosity are nevertheless conspicuous (Latter 1981; Figure 1), potentially accounting for the significant excess of lowfrequency variants at group I enzyme loci in natural populations. The consistently lower mean heterozygosity observed for group I enzymes by comparison with those of group II cannot satisfactorily be explained by differences in mutation rates alone (Latter 1981).
Models with a superior homozygote and mutants which are deleterious as heterozygotes may be appropriate for many loci in Drosophila, but natural selection must frequently tend to preserve an intermediate level of gene activity, with appropriate control of the timing and rate of transcription. A simple multiallelic singlelocus model of this sort has been shown to lead to a high proportion of overdominant polymorphisms in finite populations, with mean heterozygote superiority corresponding to Ns ~1.5 and a mean value of s ~30 times the mutation rate (Latter 1970, 1972). For enzyme polymorphisms detected by electrophoresis, this again suggests s ~1–3 × 10^{−4}.
Voehler et al. (1980) and Langley et al. (1981) have studied the frequency of null alleles occurring at allozyme loci in natural populations of D. melanogaster, concluding that the alleles are in mutationselective balance in the North Carolina and London populations, with an estimated mean selective disadvantage as heterozygotes of 0.0015. It is therefore probable that the selection coefficients for mutant alleles of less drastic effect at these allozyme loci are considerably lower than this figure. It is also relevant to note that an analysis of the data of Mukai et al. (1972) and Ohnishi (1977) using models involving two classes of mutants affecting viability, has suggested a high proportion of mutants with mean homozygous effects ~0.001–0.002 (Keightley 1994). In light of all these findings, it is not surprising that mutants with heterozygous effects ~0.001 or less have been found to be important in contributing to the decline in competitive fitness with slow inbreeding.
Acknowledgments
I thank Dick Frankham, Dick Lewontin, and John Sved for their comments on an earlier version of the manuscript. Deborah Charlesworth and the two reviewers made many suggestions that greatly improved the presentation and discussion of the results.
Footnotes

Communicating editor: D. Charlesworth
 Received December 2, 1996.
 Accepted October 14, 1997.
 Copyright © 1998 by the Genetics Society of America