Abstract
Alternatives to the mutationaccumulation approach have been developed to characterize deleterious genomic mutations. However, they all depend on the assumption that the standing genetic variation in natural populations is solely due to mutationselection (MS) balance and therefore that overdominance does not contribute to heterosis. Despite tremendous efforts, the extent to which this assumption is valid is unknown. With different degrees of violation of the MS balance assumption in large equilibrium populations, we investigated the statistical properties and the robustness of these alternative methods in the presence of overdominance. We found that for dominant mutations, estimates for U (genomic mutation rate) will be biased upward and those for h̄ (mean dominance coefficient) and s̄ (mean selection coefficient), biased downward when additional overdominant mutations are present. However, the degree of bias is generally moderate and depends largely on the magnitude of the contribution of overdominant mutations to heterosis or genetic variation. This renders the estimates of U and s̄ not always biased under variable mutation effects that, when working alone, cause U and s̄ to be underestimated. The contributions to heterosis and genetic variation from overdominant mutations are monotonic but not linearly proportional to each other. Our results not only provide a basis for the correct inference of deleterious mutation parameters from natural populations, but also alleviate the biggest concern in applying the new approaches, thus paving the way for reliably estimating properties of deleterious mutations.
THE genome of any organism is subject to continuous bombardment of mutations, the majority of which are deleterious. Numerous theories based on the assumptions of deleterious genomic mutations have been developed to explain some fundamental phenomena in biology. These phenomena include (but are not limited to) the evolution of sex and recombination (Muller 1964; Kondrashov 1985, 1988; Charlesworth 1990), mate choice (Kirkpatrick and Ryan 1991), diploidy (Kondrashov and Crow 1991), and outbreeding mechanisms (Charlesworth and Charlesworth 1987). Theories also indicate that the parameters of deleterious genomic mutations determine the mutation load in populations at equilibrium (Haldane 1937; Kimuraet al. 1963; Büger and Hofbauer 1994), the role of deleterious mutations in the extinction of small populations (Lande 1994; Lynch et al. 1995, 1996), the rate of input of genetic variance from deleterious mutations per generation (Deng and Lynch 1996, 1997), and the extent to which neutral molecular variation is reduced due to background selection (B. Charlesworthet al. 1993; D. Charlesworthet al. 1995; Hudson and Kaplan 1995). The validity of all these theories critically depends on the parameters of deleterious mutations.
For the rest of the Introduction, the following definitions and distinctions between dominance and overdominance are in order. For a locus with alleles A and a, let the three genotypic values of fitness be
Three essential parameters of deleterious genomic mutations are (1) the genomic mutation rate (U), (2) the mean selection coefficient (s̄), and (3) the mean dominance coefficient (h̄). For the three essential parameters, there are now three approaches for estimation:

The mutationaccumulation (MA) approach (Bateman 1959; Mukai 1964; Mukaiet al. 1972): This technique estimates U and s. Most estimates have come from this approach applied to Drosophila melanogaster (Mukai 1979; Crow and Simmons 1983; Keightley 1994, 1996) and have been very hard to acquire, requiring large and longterm MA and special chromosomal constructs or inbred/asexual lines. The data from MA can also be analyzed by the maximumlikelihood method (Keightley 1994) or the minimumdistance method (GarciaDorado 1997).

The inbreeding depression approach (Mortonet al. 1956; Charlesworthet al. 1990): Requiring a h value that must be assumed or that cannot be estimated without bias (Caballeroet al. 1997; Deng and Fu 1998; Deng 1998a), this technique per se estimates U only. In the highly selfing annual plants Leavenworthia (Charlesworthet al. 1994) and Amsinckia (Johnston and Schoen 1995), U estimates from this approach are in line with earlier ones from MA in Drosophila, suggesting high deleterious genomic mutation rates.

The fitness moments approach (Deng and Lynch 1996, 1997; Deng 1998b): This approach estimates U, h, and s. For two outcrossing species of cyclical parthenogenetic Daphnia (a freshwater microcrustacean), preliminary estimates by this approach generally agree with earlier ones from other species (Deng and Lynch 1997) and those from the direct MA approach in Daphnia (Lynch 1985; Lynchet al. 1998).
The last two approaches depend on the change in mean (and genetic variance) of fitness traits upon only one generation of mating in large selfing or outcrossing populations. In comparison, the first approach is much more timeconsuming and requires many generations of MA. None of the current experimental designs and statistical methods can estimate mutation parameters without bias. Under a number of biologically plausible conditions, the statistical properties of the above three approaches were compared (Deng and Fu 1998). We found that, generally speaking, the third approach has the best statistical properties as reflected by the minimum mean square error (MSE). MSE is a composite index of both bias and sampling error for biased estimates.
An essential assumption common to the last two approaches is that all the genetic variation in the study population is maintained under MS equilibrium. Accordingly, changes in the mean and genetic variance of fitness (or its components) upon inbreeding or outcrossing are solely due to deleterious dominant mutations maintained by MS balance. Even in large populations, despite tremendous efforts (e.g., Houle 1989, 1994; Houleet al. 1996; Charlesworth and Hughes 1998; Deng 1998a; Denget al. 1998a), the validity of this assumption is unknown. In large populations, alternatives to MS balance, such as functional overdominance or overdominance induced by fluctuating selection, can in principle maintain polymorphisms, although no strong case has emerged for their generality (Deng and Lynch 1996).
The robustness of the approaches applied to natural populations has been investigated under a range of biologically plausible conditions, such as variable and/or epistatic mutation effects, etc. (Charlesworthet al. 1990; Deng and Lynch 1996, 1997; Deng and Fu 1998; Deng 1998b). Generally speaking, U and h̄ are underestimated and s̄ is overestimated. The direction and the magnitude of the bias revealed may provide a numerical basis for the close inference of deleterious genomic mutations. However, estimation under violation of the MS balance assumption has never been investigated. It is intuitive that violation of the MS balance assumption will result in biased estimates (Drakeet al. 1998). However, a critical issue is, What are the statistical properties (the degree of bias and sampling variance, especially the bias) under different degrees of violation of the MS balance assumption?
The MS balance assumption can be violated in several scenarios, such as in small populations subject to random genetic drift or in large populations subject to balancing selection due to functional overdominance and/or fluctuating selection at the allelic level. Each scenario deserves careful consideration and thus separate treatment. The two approaches applicable to natural populations were originally devised for large populations at approximate equilibrium. Hence, we investigate estimation in large natural populations with genetic variance maintained by either MS balance or balancing selection, and with inbreeding depression caused by either dominant or overdominant mutations. The study is conducted by computer simulations using algorithms we devised previously (Deng 1998a) and those we devise here. Other scenarios will be fully investigated in future studies by employing iterative algorithms (i.e., Lynch et al. 1995, 1996) to construct populations (in linkage disequilibrium) of various finite sizes.
The experimental designs to characterize deleterious genomic mutations are different depending on the study population's mating type (Mortonet al. 1956; Charlesworthet al. 1990; Deng and Lynch 1996). In outcrossing populations, the outcrossed parents from natural populations are selfed to obtain selfed progeny.
In selfing populations, selfed parents from natural populations are outcrossed to obtain outcrossed progeny. In this article, we first outline the simulations and develop the associated analytical derivations in outcrossing and selfing populations. Then we present the simulation results in these two types of populations for the fitness moments approach and the inbreeding depression approach for both constant and variable mutation effects. Finally, we discuss the implications of our current results for characterizing deleterious genomic mutations from natural populations.
SIMULATIONS
The direction and the magnitude of the bias under balancing selection with overdominance are of particular interest to geneticists. To focus on this, we assume that genotypic values are measured accurately. In reality, this would require that each genotype be clonally replicated and assayed a large number of times. Ignoring measurement error for genotypic values reduces the sampling error of estimates, but is unlikely to bias either the estimation or the comparison of the techniques, assuming that the same number of genotypes would be handled experimentally. This is supported by our previous investigations (Deng and Lynch 1996; Deng and Fu 1998; Denget al. 1998b). In outcrossing populations, inbreeding (such as sib mating) experiments can be performed for estimation, and selfing is not required (Deng 1998b). To apply the fitness moments approach (Deng and Lynch 1996, 1997), we found (Deng 1998b) that for a given sample size, sampling one selfed progeny is generally more efficient than sampling more selfed progenies from each selfing family. Therefore, for outcrossing populations, selfing experiments in which only one selfed progeny is sampled from each parent are simulated for applying the fitness moments approach.
Large outcrossing populations at equilibrium are constructed with some dominant loci maintained under MS balance and other overdominant loci maintained by balancing selection. In large selfing populations, overdominance does not contribute to the maintenance of genetic variability (because of constant exposure to the homozygous state under selfing), and mutations of overdominant effects are also maintained by MS balance (Charlesworthet al. 1990; Deng 1998a). For both outcrossing and selfing populations, we study first constant, then variable, mutation effects for dominant mutations. For overdominant mutations, we assume that their effects are constant across loci. This treatment may be at least partially justified by the facts that (1) no theoretical and empirical evidence bearing on the genetic effects across overdominant loci exists and (2) what concerns geneticists most is the estimation under different contributions of overdominant loci to heterosis and standing genetic variation in populations, irrespective of their constant or variable effects. Here, heterosis will refer both to inbreeding depression in outcrossing populations and to outbreeding enhancement in inbred populations. The investigation of the methods under their respective assumptions with constant fitness effects can serve as a starting point for comparison with more realistic situations investigated later in this and future studies.
Mutation effects on fitness across all loci are assumed to be multiplicative throughout, an assumption that is biologically plausible (Mortonet al. 1956; Crow 1986; Craddocket al. 1995; Fu and Ritland 1996) and assumed in the original development of the approaches applied to natural populations (Mortonet al. 1956; Charlesworthet al. 1990; Deng and Lynch 1996). Simulations and algorithms are outlined or developed for outcrossing populations and for selfing populations in the following sections.
Outcrossing populations: Loci of constant dominant mutation effects mixed with overdominant loci: At dominant loci at MS balance, the number of mutations per individual (after selection, all in the heterozygous state) is Poisson distributed with an expectation of n̄ = U/(hs) (Deng and Lynch 1996). The population is assumed to be random mating and at linkage equilibrium. Throughout, h and s generally refer to the dominance and selection coefficients of deleterious genomic mutations. In each situation, simulations are performed for different sets of parameters. For each parameter set, K individuals are randomly sampled from both the outcrossed parental and selfed progeny generations (Deng and Lynch 1996; Deng 1998b). Unless otherwise specified, K = 200 for outcrossing populations. The total number of genotypes employed in an experiment for the fitness moments approach in outcrossing populations is then 400. For a genotype with n dominant mutations (randomly determined from the Poisson distribution) from the outcrossed parental generation, the fitness is
For a genotype sampled from the selfed progeny generation, the fitness is
Now consider the overall individual fitness with N polymorphic overdominant loci in the genome in addition to those dominant loci at MS balance. At an overdominant locus with effect h_{o} < 0 and s_{o} in large populations, the equilibrium frequency of the more fit allele B is p = (h_{o} – 1)/(2h_{o} – 1) (Crow 1986) and that of the less fit allele b is q = h_{o}/(2h_{o} – 1). With N such additional overdominant polymorphic loci in the population, the overall fitness of a random parental individual now becomes
Upon selfing, the overall fitness of a selfed progeny whose parent has n_{3} overdominant loci with the Bb genotype and n_{4} overdominant loci with the bb genotype, is
Once the desired samples of K individuals from the parent and selfed progeny generations are simulated, we estimate the parameters of deleterious genomic mutations on the basis of the assumption of pure dominant mutations maintained under MS balance (Deng and Lynch 1997). Let w̄_{o} and
If a value of h is assumed by external knowledge or estimated by other experimental designs and estimation methods, U can then be estimated from the change in the mean fitness upon selfing by Equation 2b (Mortonet al. 1956). The method of Deng (1998a) is employed to estimate h. Unlike Mukai's method (Mukaiet al. 1972), Deng's method does not require construction of homozygous lines. When applied to outcrossing populations, it can achieve about the same quality of estimation as Mukai's method. The data needed are the genotypic value of the parent fitness (w), and the mean genotypic fitness value (z) of the multiple selfed progeny within each selfed family. Let t = 4z – 2w. Then
Two aspects of overdominant mutations concern geneticists most and are directly relevant to characterizing deleterious genomic mutations from natural populations. One is the contribution of dominant mutations to heterosis (the mean fitness of the outcrossed generation to the inbred generation) relative to that of overdominant mutations. The other is the magnitude of genetic variation due to dominant mutations maintained under MS balance relative to that due to overdominant mutations maintained by balancing selection.
The contribution to the total heterosis upon selfing from the dominant mutations can be measured by the index
The index α plays an important role. Compared with a similar index (α, constructed on the original fitness scale) of Dceng (1998a), α here represents the proportion of heterosis on the log fitness scale that is attributable to dominant mutations. Therefore, α ranges from 0to1.If α= 1, the sole cause of heterosis is dominance; if α = 0, it is overdominance. The smaller the α, the larger the contribution to heterosis from overdominant mutations.
To measure the magnitude of genetic variation from dominant mutations maintained under MS balance relative to that from overdominant mutations maintained by balancing selection, we define the index
Dominant loci with variable mutation effects mixed with overdominant loci: Deleterious mutation effects h_{i} and s_{i}, across loci are unlikely to be constant. For example, s_{i} may vary anywhere from 0 (neutral mutation) to 1 (lethal mutation). The rate of mutations with different effects may also vary so that mutations of smaller effects may occur at higher rates. To evaluate the direction and the magnitude of bias introduced jointly by variable mutation effects and overdominant mutations, as in Deng and Lynch (1996), we adopt an exponentially distributed mutation rate for mutations of variable effect s_{i}:
In simulations, we divide the entire range of s (0 – 1) into 100 discrete classes of width 0.01. Within each class, mutations have constant effects (h_{i} and s_{i}). Each individual from the outcrossed parental generation in the simulation is assigned a number (n_{i}) of heterozygous mutations from the ith of these classes by drawing from a Poisson distribution with expectation Up_{i}/(h_{i}s_{i}), where p_{i} is the density of the mutational distribution in the ith class. For an individual from the selfed progeny generation, n_{i}'s are first determined as above. Then for each of the n_{i} loci, the genotype is, as before, determined by randomly sampling from the trinomial probabilities so that probabilities for different genotypes are 1/4 for AA, 1/2 for Aa, and 1/4 for aa, respectively (due to random segregation during selfing of parents). This discrete treatment closely approximates the continuous distribution of mutation effects (H.W. Deng, unpublished data).
Selfing populations: To estimate deleterious genomic mutations, selfed individuals from natural selfing populations are crossed randomly to obtain outcrossed progeny. In selfing populations, new mutations in the genome most likely follow a Poisson distribution, whether they involve dominant or overdominant mutations. In highly selfing populations, mutant alleles will be maintained by MS balance, regardless of their (over)dominance (Deng 1998a). Hence, as in the dominant case, we assume that the number of loci with overdominant mutants (n_{7}), all in the homozygous state, per genome in selfing populations is Poisson distributed with mean n̄_{o} and constant effects h_{o} and s_{o}. If the genomic mutation rate to the overdominant (but less fit) allele a is U_{o}, it can be easily shown that at MS equilibrium, n̄_{o} = U_{o}/(2s_{o}) (Charlesworthet al. 1990; Deng 1998a).
In each situation, a variable number K of individuals is randomly sampled from the selfed parental and outcrossed progeny generations, respectively. For a genotype with n dominant and n_{7} overdominant mutations [randomly determined from the Poisson distribution with mean U/(2s) and U_{o}/(2s_{o}), respectively] from the selfed parental generation, the fitness is
For an outcrossed progeny resulting from crossing two selfed parents (with n_{f}, n_{7} and n_{m}, n_{8} homozygous loci for dominant and overdominant mutations, respectively, where the subscript f indicates female parent and m the male parent), its fitness is
In selfing populations, the indices α and β defined in Equations 4 and 5 can be constructed from the derivations in appendix b for the constant and variable dominant mutation effects, respectively. In simulated populations, the genome contains both dominant and overdominant loci, all at MS equilibrium. In the parental generation, the number of homozygous dominant loci in each individual is determined by random sampling from a Poisson distribution of mean U/(2s), and the number for the overdominant loci is determined from a Poisson distribution with mean n̄_{o} [=U_{o}/(2s_{o})] (Charlesworthet al. 1990; Deng 1998a). In the outcrossed offspring generation, the number of dominant loci in each individual is sampled from a Poisson distribution of mean U/s, and the number of overdominant loci in each individual is determined from a Poisson distribution with mean U_{o}/s_{o} (Charlesworthet al. 1990; Deng 1998a), all in the heterozygous state. The variable dominant mutations are modeled by Equation 6 and are simulated by discrete classes of mutations, in a manner similar to that in outcrossing populations as described earlier.
Once the desired samples of K individuals from the selfed parent and the outcrossed progeny generations are simulated, the estimation developed on the basis of the assumption of pure dominant mutants maintained under MS balance (Deng and Lynch 1996) is applied. Unless otherwise specified, K = 200 for selfing populations. The total sample size is then 400 for the application of the fitness moments approach. Let w̄_{o},
To apply the inbreeding depression approach to estimate U (Charlesworthet al. 1990), the value for h must be assumed or estimated by other experimental designs and methods. Mukai's method (Mukaiet al. 1972) is employed to estimate h. It estimates h approximately by the slope of the regression of the outcrossedprogeny fitness (x) on the fitness sum (y) of the two corresponding parental homozygotes:
In simulations, we arbitrarily let W_{max} = 1, as the values of W_{max} do not influence the estimation for the mutation parameters (Deng and Lynch 1996). For each set of parameters, we perform 500 simulations. Unless otherwise specified, in all the simulations presented (except with pure overdominant mutations in the genome, i.e., when α = β = 0), U = 1.0, h̄ = 0.36, and s̄ = 0.03, which are close to the most often cited values estimated by Mukai et al. (1972; Lynchet al. 1995). The experimental designs have been laid out earlier for different estimations in different populations. Results for other simulation parameters (e.g., U = 0.1–4.0 and s̄ = 0.01–0.05) and experimental designs have also been performed. The results are similar and thus not presented. Because almost all the results are biased, the MSE is presented together with one standard deviation (SD) computed over the repeated simulations.
RESULTS
Outcrossing populations
Constant dominant mutation effects: 1. The fitness moments approach (Table 1): With only deleterious dominant loci in the genome (N = 0 and α=β= 1), the estimates for U, h, and s are unbiased. Recall that N is the number of polymorphic overdominant loci in the population and α and β are, respectively, the proportion of heterosis and genetic variation on the log fitness scale that is attributable to dominance mutations. With overdominant loci coexisting in the genome with deleterious dominant loci (N > 0 and 0 < α, β < 1), Û (^ indicates an estimated value) is an overestimate, while h and s are underestimated. The degree of bias increases with increasing contributions from overdominance to heterosis (decreasing α) and to the standing genetic variation in the population (decreasing β). Generally, the bias is not dramatic so that estimates of the upper bound of U and lower bounds of h and s can be obtained, and these estimates are close to the true parameter values. All the sampling errors are quite small. Even with only overdominant mutations in the genome (α = β = 0), estimates of U, h̄, and s̄ can still be obtained, although the parameter values do not exist for the dominant as mutations. In this case, it is not incorrect to treat Û an upper limit for the true U of zero. This is understandable, because, upon selfing (or outcrossing in selfing populations), overdominant mutations will also cause mean and genetic variance of fitness to change, similar to those changes caused by dominant mutations. This will be similar in every case, and thus will not be repeated. The estimation bias is relatively more sensitive to a change of h_{o} than to a change of s_{o}. With a larger absolute value of h_{o}, the degree of bias increases.
2. The inbreeding depression approach (Table 2): With N = 0 and α = β = 1, the estimates for U and h are nearly unbiased. With N > 0 and 0 < α, β < 1, U is generally overestimated, while h is underestimated. The degree of bias generally increases with decreasing α and β. Compared with the fitness moments approach, the bias is larger for ĥ and smaller for Û. The smaller bias of Û is largely due to the greatly underestimated ĥ. This can be understood from Equation 2b or Figure 1 in Deng and Fu (1998). Although the presence of overdominant mutations will tend to bias Û upward, the bias will be greatly dampened by a greatly underestimated ĥ. However, the estimation of U suffers from large sampling errors, even though the number of genotypes employed (450) is larger than that for the fitness moments approach (400). When both sampling error and bias are considered, the estimation of U by the inbreeding depression approach is generally worse than that by the fitness moments approach, as reflected by the larger MSE. The statistical properties (mean and sampling variance) of Û are relatively unstable with changes of a and β. This instability is largely due to the relatively small sample size employed. When overdominance contributes importantly to the heterosis and standing genetic variation in natural populations (with small α and β), Û is unacceptable even as an estimate for the upper limit because of the large sampling error. ĥ estimated by Deng's (1998b) method can serve well as a lower bound of the true h as evidenced by its small sampling error.
Variable dominant mutation effects: The fitness moments approach (Table 3): With N = 0 and α = β = 1, U and h̄ are underestimated and s̄ is overestimated. With N > 0 and 0 < α, β < 1,
It should be noted that with different h_{o} and s_{o} parameters for overdominant mutations, the same α may correspond to a different β. This can be inferred from the corresponding Equations 4 and 5 and those in appendices a and b. It is also evident in every case as can be seen from the numerical values of Tables 1, 2, 3, 4, 5, 6, 7, 8 for outcrossing and selfing populations and for constant and variable mutation effects. To illustrate the monotonic but nonlinear relationship between α and β, Figure 1 plots the values of α and β for constant and variable mutation effects in both outcrossing and selfing populations.
The inbreeding depression approach (Table 4): With N = 0 and α = β = 1, the estimates for U and h̄ are both biased downward. With N > 0 and 0 < α, β < 1, U is generally underestimated when α and β are relatively large and is only overestimated when α and β are quite small. However, the sampling variance of Û is usually large. On the other hand,
Selfing populations
Constant dominant mutation effects: The fitness moments approach (Table 5): With N = 0 and α = β = 1, the estimates for U, h, and s are unbiased. With N > 0 and 0 < α, β < 1, U is overestimated, while h and s are underestimated. The degree of bias increases with decreasing α and β. However, the bias is not so dramatic that the upper bound of U and lower bounds of h and s can be estimated, and that they are not wildly far away from the true parameter values. The estimation bias is not very sensitive to changes in h_{o} and s_{o}, especially for ĥ and ŝ.
The inbreeding depression approach (Table 6): With N = 0 and α = β = 1, the estimates for U and h are nearly unbiased. With N > 0 and 0 < α, β < 1, U is generally overestimated, while h is underestimated. The degree of bias generally increases with decreasing α and β. Compared with the fitness moments approach, the bias is larger for ĥ and smaller for Û. The smaller bias of Û is largely due to the greatly underestimated ĥ. This can be understood from Equation 2b or Figure 1 in Deng and Fu (1998). Although the presence of overdominant mutations will tend to bias Û upward, the bias will be greatly dampened by a largely underestimated ĥ. Compared with outcrossing populations under constant mutation effects with a comparable sample size of genotypes, the sampling error for Û is relatively small, and hence Û can serve well as an estimate for the upper limit. ĥ estimated by Mukai's method (Mukaiet al. 1972) can also serve well as a lower bound of the true h as evidenced by its small sampling error.
Variable dominant mutation effects: The fitness moments approach (Table 7): With N = 0 and α = β = 1, the estimates for U and h̄ are biased downward and the estimates for s̄ are biased upward. With N > 0 and 0 < α, β < 1,
The inbreeding depression approach (Table 8): With N = 0 and α = β = 1, the estimates for U and h̄ are biased. With N > 0 and 0 < α, β < 1, U is generally underestimated when α and β are relatively large and is only overestimated when α and β are quite small. It should be noted that, as with the case for the outcrossing populations, when overdominant mutations are present but do not contribute substantially to heterosis and genetic
variation, the bias of Û is smaller than under dominant mutations. This is because the directions of estimation bias caused by overdominant mutations and variable effects of dominant mutations are opposite and they cancel each other, resulting in smaller (or no) bias. The
extent of the bias depends on the parameters under estimation and α and β parameter values. The sampling variance of Û is small.
DISCUSSION
Using extensive simulations, we investigated the effect of overdominant mutations on characterizing deleterious dominant mutations by the two existing estimation approaches (Mortonet al. 1956; Charlesworthet al. 1990; Deng and Lynch 1996, 1997; Deng 1998b). We developed two important indices and associated analytical derivations to characterize the relative contributions of overdominant mutations to heterosis and genetic variation. The simulation algorithms and the analytical derivations developed are useful for investigating other issues in genetics concerning the mixture of dominant and overdominant mutations in the genome. Estimates for U are biased upward and those for h̄ and s̄ biased downward by overdominant mutations. However, the degree of bias is generally moderate and depends on the magnitude of the contribution of overdominant mutations to heterosis or genetic variation. This renders the estimates of U and s̄ not invariably biased under variable mutation effects, which when working independently will almost always cause U and s̄ to be underestimated. We also note that the contributions to heterosis and genetic variation from overdominant mutations are monotonic but not linearly proportional to each other. Our results may not only provide a basis for correct inferences about deleterious mutations from natural populations, but may also alleviate the biggest concern and obstacle in applying the inbreeding depression and fitness moments approaches, thus paving the way for efficiently characterizing deleterious genomic mutations from large natural populations.
Although it is intuitive that the two approaches will yield biased estimates (Drakeet al. 1998), it is not clear what the magnitude and the direction of the bias will be for different estimates without the extensive simulations conducted here. Overdominant mutations, when acting together with variable mutation effects and depending on their contributions to heterosis and the standing genetic variation, may actually render estimates of U and s̄ unbiased. It has been stipulated (Deng and Fu 1998; Drakeet al. 1998) that the inbreeding depression and fitness moments approaches may be least affected by overdominant mutations in selfing populations, because overdominant mutations cannot be maintained by balancing selection there. However, as shown in Tables 1, 2, 3, 4, 5, 6, 7, 8, with comparable contributions from overdominant mutations to heterosis and standing genetic variation, the estimation will be affected to a similar degree in outcrossing and selfing populations. We also note that the influence on the estimation from overdominant mutations will depend not only on their contributions to heterosis and the standing genetic variation, but also on the parameters of overdominant mutations such as h_{o} and s_{o}, although such dependence does not seem to be large.
Our simulation results not only reveal the robustness and statistical properties of the current approaches to characterize deleterious dominant mutations in natural populations, but also shed light on the relative efficiencies of the different approaches in different populations. Although the relative efficiencies of all the three available approaches (as outlined in the Introduction) were investigated earlier (Deng and Fu 1998), the investigations were not conducted under conditions of mixed dominant and overdominant mutations in the genome. In the present study, the sample sizes implemented in simulations for the two approaches investigated were deliberately set to be either comparable, or those for the inbreeding depression approach were actually larger. Recall that the number of genotypes employed for the fitness moments approach is 400 in outcrossing and selfing populations, while those for the inbreeding depression approach were 450 and 600, respectively, in outcrossing and selfing populations. However, it can be seen from Tables 1, 2, 3, 4, 5, 6, 7, 8 that the estimation by the fitness moments approach is often better than the inbreeding depression approach. This is especially true for outcrossing populations and for the estimation of h̄. The inbreeding depression approach is sometimes better for the estimation of U; however, the better estimation is achieved because of a greatly biased estimation of h̄. Therefore, it is not the original inbreeding depression approach per se that achieves the better estimation for U. It is actually the greatly underestimated h̄ by the estimation methods chosen that leads to the less biased U in the inbreeding depression approach. Therefore, the estimation of U by the inbreeding depression approach would greatly depend on the methods chosen for the estimation of h̄. With less biased estimates or assumed values for h̄, simulation results not shown here indicate that the U estimation by the inbreeding depression approach is much worse statistically than that of the fitness moments approach.
The issue of dominance and overdominance has been under debate for decades in genetics (Davenport 1908; East 1908; Shull 1908; Crow 1952; Sprague 1983; Wallace 1989; Houle 1989, 1994; Crow 1993; Denget al. 1998a). The debate has farreaching significance for agriculture, human health, evolution, and conservation biology, among other areas. While most of the data are consistent with the dominance hypothesis, overdominance cannot be ruled out in many situations (Simmons and Crow 1977; Charlesworth and Charlesworth 1987; Barrett and Charlesworth 1991; Stuberet al. 1992; Crow 1993; Mitton 1993). Given the current status of the debate, instead of favoring one hypothesis over the other, it may be more sensible to examine the issues concerned under mixed dominant and overdominant mutations in the genome, with mutations of each type having different contributions (e.g., to heterosis and/or genetic variation, etc.). The theoretical machinery for measuring the relative importance of dominance and overdominance has not been available. The development of two important indices, α and β, provides a basis for investigating a number of other genetic issues related to the contribution of dominant and overdominant mutations to inbreeding and the standing genetic variation in natural populations.
It has long been recognized that, when dominant and overdominant mutations coexist, the heterosis and standing genetic variation will be affected by both. However, the disproportional contributions of overdominant mutations to heterosis and to standing genetic variation have not been documented before. This phenomenon may form a basis for discerning the relative importance of dominant and overdominant mutations in the genome. Studies have been initiated along this line of research. It is worthy of note that, for overdominant mutations to contribute relatively importantly to the standing genetic variation, a substantial proportion of heterosis must be caused by overdominant mutations. This is especially true when overdominant mutations contribute to less than half of the heterosis (α > 0.5; Figure 1).
For any theory to be of great significance, its underlying assumptions must be examined closely and the important parameters must be estimated. There is no doubt that any genome is subject to continuous bombardment of deleterious genomic mutations. However, no amount of theoretical argument can resolve the issues concerning the importance of deleterious genomic mutations without the important parameters being estimated. Indisputably, characterizing deleterious genomic mutations is extremely important. However, even if the importance is realized by more and more scientists and revealed in more and more biological aspects, the estimates are astonishingly few and thus are imperatively needed (Peck and EyreWalker 1997). Among the three approaches currently available, the statistical properties and the robustness of the fitness moments approach are investigated most thoroughly and best known. Investigation of the other two available approaches (the MA approach and the inbreedingdepression approach) is also extremely important and is beginning to appear in studies (Deng and Fu 1998; Denget al. 1998b). Different approaches have different peculiar assumptions whose validity may be difficult to consolidate in a specific experimental setting (Keightley 1994; Peck and EyreWalker 1997; Deng and Fu 1998; Lynchet al. 1998). Examples of these assumptions are MS balance in the fitnessmoments approach and in the inbreedingdepression approach, no line losses because of selection during MA, no gene conversion for the MA chromosome in Drosophila, etc. Applying multiple approaches to the same organism and/or characterizing deleterious mutations in diverse organisms may provide a crosscheck of the results (and of the underlying assumptions to derive these results) and eventually may crystallize the deleterious mutation parameters.
Acknowledgments
H.W. Deng thanks Professor M. Lynch for years of advice, continuous encouragement, and support. We are very grateful to Professor Marjorie A. Asmussen and three anonymous reviewers for their extremely careful comments that helped to improve the article. We thank Drs. Robert R. Recker and Mark Johnson and Ms. Carolyn Meeks for careful editing of the manuscript. The work was partially supported by a grant from National Institutes of Health (R01 AR45349) and a Health Future Foundation grant from Creighton University, Nebraska, and by graduate student tuition waiver to J.L.L. and J.L. from the Department of Biomedical Sciences of Creighton University.
APPENDIX A: DETERMINATION OF α AND β IN OUTCROSSING POPULATIONS
Constant dominant mutation effects: Unless otherwise defined in the appendices, all the notations used are the same as in the text. Multiplicative fitness function and HardyWeinberg and linkage equilibrium are assumed. Let p(n) denote a Poisson distribution withmean n and density function p(n) = n^{n} e^{–}^{n}/n!. Let the subscripts d and o for n, h, and s denote the parameters for dominant and overdominant mutations, respectively. In relation to the parameters for dominant and overdominant mutations, the expectation terms in Equation 4 in the text can be obtained. The expectations of ln(fitness) due to dominant mutations alone in the outcrossed parental and selfed progeny generations are, respectively,
The two terms in Equation 5 in the text can be expressed in terms of the parameters for dominant and overdominant mutations. The derivation is as follows. The expectation of the second moment of the ln(fitness) in the parental generation that is due to dominant mutations only is
With Equations A6 and A8, the index β (defined in Equation 5 in the text) can be expressed in terms of the mutation parameters for outcrossing populations with constant dominant mutation effects.
Variable dominant mutation effects: Although a little more complex, the derivation is a natural extension of the cases of the constant dominant mutation effects. Let IN be number of classes of dominant mutation effects. Dominant mutation effects are constant within each such class and differ among these classes. Although dominant mutation effects are most likely variable, there is no solid knowledge on their distribution (continuous or discrete and the exact form of distribution, etc.). In addition, a discrete distribution can approximate any continuous distribution to any desired degree of accuracy. Therefore, modeling variable dominant mutation effects by a discrete distribution is appropriate,
APPENDIX B: DETERMINATION OF α AND β IN SELFING POPULATIONS
Constant dominant mutation effects: The derivation for selfing populations is relatively straightforward. Noting that the number of dominant loci in a selfed progeny is the sum of those in its selfed parents, we have the four expectation terms for the ln(fitness) in the selfed parental and outcrossed offspring generations that are due to pure dominance (d) or both dominance and overdominance (t):
Variable dominant mutation effects: The derivations and the notations are similar to and simpler than those in outcrossing populations, thus will not be elaborated for selfing populations. The expectations of the first moment of ln(fitness) in the selfed parental and outcrossed offspring generations that are due to dominance alone (d) or both dominance and overdominance (t) are, respectively,
Footnotes

Communicating editor: M. A. Asmussen
 Received April 20, 1998.
 Accepted October 30, 1998.
 Copyright © 1999 by the Genetics Society of America