Abstract
The DengLynch method was developed to estimate the rate and effects of deleterious genomic mutations (DGM) in natural populations under the assumption that populations are either completely outcrossing or completely selfing and that populations are at mutationselection (MS) balance. However, in many plant and animal populations, selfing or outcrossing is often incomplete in that a proportion of populations undergo inbreeding while the rest are outcrossing. In addition, the degrees of deviation of populations from MS balance are often not known. Through computer simulations, we investigated the robustness and the applicability of the DengLynch method under different degrees of partial selfing or partial outcrossing and for nonequilibrium populations approaching MS balance at different stages. The investigation was implemented under constant, variable, and epistatic mutation effects. We found that, generally, the estimation by the DengLynch method is fairly robust if the selfing rate (S) is <0.10 in outcrossing populations and if S > 0.8 in selfing populations. The estimation may be unbiased under partial selfing with variable and epistatic mutation effects in predominantly outcrossing populations. The estimation is fairly robust in nonequilibrium populations at different stages approaching MS balance. The dynamics of populations approaching MS balance under various parameters are also studied. Under mutation and selection, populations approach balance at a rapid pace. Generally, it takes 400–2000 generations to reach MS balance even when starting from homogeneous individuals free of DGM. Our investigation here provides a basis for characterizing DGM in partial selfing or outcrossing populations and for nonequilibrium populations.
THE genome of all organisms is subject to continuous bombardment of mutations, the majority of which are deleterious. Numerous theories based on deleterious genomic mutations (DGM) have been developed to explain some fundamental phenomena in biology (Deng 1998b; see references therein). The validity of these theories critically depends on the rate at which DGM occur per genome per generation (U) and/or their effects (mean dominance and selection coefficients h and s). In addition, the knowledge about the mean and the variation of the effects of DGM is vital in evaluating the role of DGM in longterm survival of small populations (Lynch et al. 1993, 1995a,b; Lande 1994).
However, few estimates of the parameters of DGM are available, and even the order of magnitude of these parameters is controversial (Crow and Simmons 1983; Kondrashov 1988; Crow 1993a,b; Keightley 1996). None of the current estimation approaches (Mortonet al. 1956; Bateman 1959; Mukaiet al. 1972; B. Charlesworthet al. 1990; Deng and Lynch 1996) can yield unbiased results under realistic situations (Deng and Fu 1998; Denget al. 1999; H. W. Deng, J. Li, W.M. Chen, Y. Zhou and T. Kibota, unpublished results). Furthermore, the statistical properties (sampling errors and, particularly, the direction and magnitude of bias) of the current estimation approaches are largely unknown under various biologically plausible situations. Nevertheless, our knowledge of these statistical properties is essential, at least for (1) understanding the robustness and applicability of various estimation approaches under various biology conditions; and (2) a sensible decision on the adoption of a particular estimation approach in a particular biological system. Therefore, our knowledge of these statistical properties may not only expand the range of biological systems for which the current estimation approaches can apply but may also provide a quantitative and statistical basis for correctly inferring DGM from these systems. It has been argued that robust characterization of DGM is one of the greatest challenges in the broad field of modern genetics (Crow 1993a,b; Bridgeset al. 1994; Peck and EyreWalker 1997).
Among the available approaches, Deng and Lynch's (1996) estimation makes use of the data on the changes of both the mean and genetic variance for fitness traits that can be acquired from inbreeding/outbreeding in natural outcrossing/selfing populations. Since the trait under study for the DengLynch method is fitness or its important component(s) under directional selection, it is understood hereafter that the mean and genetic variance referred to are for fitness traits. The DengLynch method estimates not only U but also
Another important assumption of Deng and Lynch's (1996) estimation is that the study population is at mutationselection (MS) balance. Despite extensive efforts (e.g., Houleet al. 1996; Deng 1998a; Denget al. 1998; Charlesworth and Hughes 1999), the validity to which this assumption is reasonable in natural populations is largely unknown and is difficult to measure. It is even more difficult to measure the relative contributions of various potential mechanisms to the deviation from MS balance (if any). The mechanisms and their relative importance in maintaining populations' genetic variability are largely unknown. In addition, we know little about the dynamics of populations approaching MS balance under recurrent mutations and selection. Specifically, we do not know how fast MS balance is established and how long it takes when starting from a homogeneous population (all individuals being free of DGM) subject to deleterious mutations and selection. Li et al. (1999) investigated the robustness of Deng and Lynch's (1996) estimation when the MS assumption is violated by the presence of balancing selection as reflected as overdominance at some genomic loci. Drake et al. (1998) also made some qualitative prediction of the estimation bias of the DengLynch method when substantial genetic variation is actively maintained by selection. However, under directional selection for DGM underlying fitness, the robustness of Deng and Lynch's (1996) estimation in nonequilibrium populations at different stages approaching MS balance is unknown and has never even been qualitatively discussed.
To characterize DGM in natural populations of partial selfing or outcrossing, ideally, we need to develop new methods to estimate mutation parameters by incorporating in estimation the selfing or outcrossing rates that are estimable by the methods of Ritland (1984, 1988), Milligan (1996), and Fu. Our extensive investigation (J. Li and H.W. Deng, unpublished data) has suggested that this line of research effort may not be tenable. This is mainly because there is no accurate and simple analytical distribution to characterize the number of DGM per genome for partial selfing/outcrossing populations at MS balance. The semianalytical distribution developed by Charlesworth et al. (1991) is not simple and requires computation iteration to approximate the number of DGM per genome for partial selfing/outcrossing populations at MS balance. Hence, we cannot develop an analytical method for characterizing DGM based on the semianalytical distribution developed by Charlesworth et al. (1991).
Therefore, we set out to investigate the robustness and statistical properties of Deng and Lynch's (1996) estimation in partial selfing/outcrossing populations and in populations not at MS balance. The investigation is performed by extensive computer simulations under various situations (constant, variable, and epistatic mutation effects) and various parameters. The results on the direction and magnitude of bias and sampling variation of the estimates should provide a basis for robust characterization of DGM in natural populations of partial selfing or outcrossing and in populations under mutation and selection but not at MS balance.
THEORY
The approach to characterize DGM proposed by Deng and Lynch (1996) is a fitnessmoments approach, which utilizes mean and genetic variance of fitness of two generations across inbreeding or outcrossing in outcrossing and selfing populations, respectively. The Deng and Lynch (1996) estimation assumes that mating is either complete outcrossing or complete selfing, and all loci are at MS balance for segregating polymorphism, multiplicative fitness functions, constant mutation effects, infinite population size, and free recombination (or linkage equilibrium) between loci. Under these assumptions, letting a circumflex(^) denote an estimate, in an outcrossing population,
The experimental procedures in outcrossing and selfing populations are detailed previously (Deng and Lynch 1996, 1997; Deng and Fu 1998; Deng 1998b). Briefly, in outcrossing populations, (1) a sample of genotypes are selfed to obtain a number of selfed progenies from each parent to form selfed families; (2) parental genotypes are cloned (genotypes from both generations are assayed together in one environment to estimate
SIMULATIONS
By a deterministic method that was developed by Kondrashov (1985) and later modified by D. Charlesworth et al. (1990), incomplete selfing or outcrossing populations are constructed by computer simulations in which mutation effects are constant, or variable, or epistatic across loci as in Deng and Lynch (1996). Except in the case of epistatic mutation effects, fitness function is assumed to be multiplicative across loci, which is biologically plausible (Deng 1998b and references within). Populations are simulated starting from homogeneous individuals free of DGM. The dynamics of the populations approaching MS balance is recorded. The estimation of Deng and Lynch (1996) is applied to the simulated nonequilibrium populations at different stages approaching the MS balance and the populations that have already reached MS balance. For fitness or its important components that are normally quantitative traits, the phenotypic values are determined not only by genotypic values but also by random environmental sources of variation (Lynch and Walsh 1998). Since the focus here is on the robustness of the assumption under partial selfing/outcrossing and in nonequilibrium populations, the estimation bias is of major interest. Therefore, genotypic values of fitness or its important components are assumed to be measured without error throughout in simulations. In reality, this would require that each genotype be clonally replicated and assayed a very large number of times. Ignoring measurement error for genotypic values will probably reduce the sampling error of estimates but is unlikely to bias the estimation. This is supported by our previous investigation of measurement error for mutation parameter estimation by the DengLynch technique (Deng and Lynch 1996; Deng and Fu 1998; Denget al. 1999).
Population construction: Assuming that the population size is infinite and the number of loci of each individual is very large, all new mutations may be considered as only occurring on wildtype homozygous loci. Hence, new mutations only change the number of heterozygous loci. The number of new mutations per genome per generation follows a Poisson distribution with mean U. To focus on the effects of partial selfing/outcrossing and MS disequilibrium on the estimation, we assume that all loci are unlinked. The situation of linkage disequilibrium among DGM at different loci is being investigated (H.W. Deng and J. Li, unpublished results).
For each simulation cycle, starting from adult individuals, mutation (in the tth generation), mating [to generate zygotes for the (t + 1)th generation], and selection [in the (t + 1)th generation] are simulated sequentially. In each simulation, a population free of any mutation is employed as the starting population. In each generation, the population includes two subpopulations, in one of which mating is outcrossing and in the other selfing. The mutationselection process is performed with the two subpopulations separately. Then they are combined to generate zygote pools, with proportional contributions being 1 − S and S, respectively, for the outcrossing subpopulation and for the selfing subpopulation. S is the selfing rate in the whole population. Then the cycle is repeated for the next generation. Our data (J. Li and H.W. Deng, unpublished results) show that this simulation is essentially identical to simulating a large population with each individual having a probability of S for selfing and a probability of (1 − S) for outcrossing. MS balance is considered to be reached when the relative changes of both mean and genetic variance of fitness of the whole population in two contiguous generations are <1 × 10^{−9}. Selfing is employed as a special inbreeding form for partial outcrossing populations to demonstrate the qualitative effects of inbreeding on the estimation in outcrossing populations. The effects demonstrated should be an upper bound for inbreeding under the same inbreeding rate in natural populations, as selfing is the most extreme form of inbreeding.
Following Kondrashov (1985) and D. Charlesworth et al. (1990), the simulation procedure is detailed below. Denote p_{i}_{,}_{j}_{(A)},
Then, mating is simulated within each of the two subpopulations. In the selfing subpopulation, the individual frequencies are changed to
After mating, selection starts to operate. Relative fitness is used to determine the frequencies after selection, i.e.,
Before the next cycle begins, the two subpopulations are merged to obtain the frequencies of individuals with different mutations in the whole population for the next generation:
Constant mutation effects: Simulations are performed under different parameter sets. The fitness of an individual having i heterozygous and j homozygous mutations is
Variable mutation effects: Mutation effects are unlikely to be constant across loci (Caballero and Keightley 1994; Deng and Lynch 1996). We utilized the model described by Deng and Lynch (1996), in which the distribution of homozygous effects is roughly exponential with density distribution,
Epistatic mutation effects: Although fitness or its component most likely acts multiplicatively (Deng 1998b and references within), synergistically epistatic mutation effects cannot be ruled out entirely. Hence, the robustness of the DengLynch method is also tested under epistatic mutation effects. The model we consider was described by Charlesworth (1990) and employed earlier by us (Deng and Lynch 1996; Deng 1998b). In this model, the fitness of an individual having i heterozygous and j homozygous mutations (with constant effects) is
Estimation for nonequilibrium populations: All the above estimations are conducted for populations that have already reached MS balance. To examine the robustness of the estimation of Deng and Lynch (1996) for nonequilibrium populations, we applied the estimation to populations at different stages of approaching MS balance when they start from homogeneous populations free of DGM. In simulations, we obtain samples every 25 generations during the course of the populations approaching MS balance. The investigation was conducted for complete outcrossing/selfing populations. These samples are used as the parental generations and corresponding matings (selfing or outcrossing) are performed to yield outcrossed (for selfing populations) or selfed (for outcrossed populations) progenies. Then Deng and Lynch's (1996) estimation is applied to estimate mutation parameters.
Dynamics of populations approaching MS balance: During the simulations, under various values of S, U, h, and s, starting from homogeneous populations free of DGM, the dynamics of populations are recorded. The dynamics of populations include the mean and genetic variance of populations at different stages approaching MS balance since starting from a homogeneous population free of DGM and the number of generations taken to reach MS balance.
Throughout, unless otherwise specified in tables or figures, in simulations reported, h (or
RESULTS
Estimation under constant mutation effects (Table 1): Estimated values are unbiased for complete outcrossing (S = 0) or selfing (S = 1.0) populations and become biased when the population deviates from complete outcrossing or selfing. The degree of bias increases with increasing deviation from complete outcrossing or selfing, i.e., when S is getting closer to 0.5.ĥ is always upwardly biased for partial selfing/outcrossing (0 < S < 1). This is because the ratio x/z in populations that are critical for ĥ estimation (Equations 1 and 2) increases when S deviates from 0 or 1 and approaches 0.5 (Figure 1). Recall (Equation 3) that x and z are related to the squared coefficient of variation of genotypic values of fitness in the outcrossed or selfed generations, respectively. Generally, the accuracy of Û and ŝ decreases when populations deviate more from S = 0 or S = 1, as reflected by the increased bias and/or standard deviation. For the same magnitude of deviation from outcrossing or selfing, the biases are larger in predominantly outcrossing populations than in predominantly selfing populations. For example, when U = 0.5, h = 0.36 and s = 0.03, Û = 1.89, ĥ = 0.467, and ŝ = 0.009 when S = 0.3; and Û = 0.70, ĥ = 0.417, and ŝ = 0.027 when S = 0.7. When S = 0.5, application of the estimation method for selfing populations yields less biased estimates than the application of the estimation method for outcrossing populations. This conclusion also holds under variable and epistatic mutation effects (Tables 2 and 3). When S < 0.05 and when S > 0.9, the Deng and Lynch (1996) estimation yields little bias. As is apparent from Table 1, the direction of bias of Û and ŝ may be upward or downward, depending upon the different parameter values of U and S.Û can be negative values with smaller SD when U = 1.0 and S approaches 0.5 starting from intermediate S values (~0.2). This can be explained as follows. Since ŷ is always negative as revealed in our simulations, the sign of Û depends on whether ĥ > 0.5 (Equation 1); if ĥ < 0.5, Û > 0 and if ĥ > 0.5, Û < 0. When ĥ is close to 0.5, Û will be very sensitive to the change of ĥ_{,} i.e., small changes of ĥ may result in quite different Û_{'s.} So we obtained large SD of Û when ĥ is close to 0.5. When ĥ is far from 0.5, the SD decreases. For our simulated parameters, we never found that ĥ = 0.5 (additive mutation effects, for which there is no inbreeding depression and y = 0). Therefore, Û can always be obtained, although the estimation is biased with the bias getting larger when ĥ is closer to 0.5.
Estimation under variable mutation effects (Table 2): The estimates are generally biased, but not always. When 0 < S < 0.5 and when S increases,
Estimation under epistatic effects (Table 3): Under the epistatic mutation model investigated, except under complete selfing, there is no s parameter for a mutation as the mutation effects are epistatic and depend on other mutations. Although the DengLynch method still yields an s estimate, it does not make sense to discuss the bias for s estimation, as the s parameter does not exist for a single DGM under epistatic effects. Therefore, we only summarize estimation for U and h. The estimates are generally biased, but not always. For complete outcrossing populations, Û and ĥ are underestimated. For complete selfing populations, Û is fairly accurate with little bias. When the population deviates from complete outcrossing (S = 0) or complete selfing (S = 1) and S approaches 0.5, ĥ increases and Û has increasingly larger sampling errors, for the same reason outlined before. It can be seen that when populations deviate slightly from S = 0, the effects of epistasis and partial selfing cancel each other to a certain extent, and the estimates of the mutation parameters become less biased, then unbiased, and then the bias gets larger in the other direction. For example, when 0 < S < 0.5 and when S increases, Û is first biased downwardly, then unbiased, and then biased upwardly. Again, the estimation is fairly robust when S < 0.10 and when S > 0.8.
Robustness of the DengLynch estimation in nonequilibrium populations (Tables 4 and 5): The performance of Deng and Lynch's (1996) estimation is fairly robust in nonequilibrium populations at different stages approaching MS balance. Under constant mutation effects (Table 4), ĥ and ŝ are almost always unbiased for complete outcrossing or selfing populations.Û is downwardly biased when < ~200 (in outcrossing populations) or < ~75 (in selfing populations) generations of mutation and selection are experienced by populations starting from homogeneous individuals free of any DGM. After a moderately large number of generations (>200 and 75 generations, respectively, in outcrossing and selfing populations), the estimation is generally unbiased for constant mutation effects.
The robust performance of the estimation in nonequilibrium populations can be explained by the detailed analyses of the dynamics of x, y, and z (Equation 3) in nonequilibrium populations during the course of approaching MS balance (Figure 2). x, y, and z change monotonously with the number of generations experienced under mutation and selection. The change is relatively large only at the initial generations. However, the ratios x/z and x/y (or z/y) that are important for the estimation of U, h, and s (Equations 1, 2 and 3) change little and remain essentially the same after the few initial generations. Since ĥ is the function of
Similar conclusions hold under variable mutation effects (Table 5). With an increasing number of generations of mutation and selection experienced, the degree of the estimation bias in nonequilibrium populations will quickly approach that in populations at MS balance. If starting from a homogeneous population free of DGM, it generally takes ~200 generations (in outcrossing populations) and ~75 generations (in selfing populations) to reach asymptotic degrees of bias expected for populations at MS balance.
Dynamics of nonequilibrium populations approaching MS balance (Figures 3 and 4): S has significant effect on the generations needed to reach MS balance (Figure 3a). All else being the same, the larger the S, the less the generations are needed to reach MS balance. This is mainly because as S increases, the mean number of mutations accumulated in the genome necessary to reach MS balance gets smaller, as is found in our simulations and those of D. Charlesworth et al. (1990). With an increasing U, the number of generations to reach MS balance increases slightly (Figure 3c). This is because when U increases, it takes more mutations accumulated in the genome and thus more generations to reach MS balance. On the other hand, with an increasing U, mutations occur faster in the population, reducing the number of generations needed to reach MS balance. These two effects cancel each other to a certain extent, causing the number of generations needed to reach MS balance to increase only slightly with an increasing U. h and s have similar effects (Figure 3, b and d) in determining the number of generations needed to reach MS balance. Small values of h and s will result in more generations needed to reach MS balance. This is largely because selection against DGM is not effective with small h and s and more accumulated mutations are needed to reach MS balance.
For populations at MS balance, we found that, as expected, h and s have small effects on mean of fitness and S and U have relatively large effects on mean of fitness (Figure 4). With increasing S and/or decreasing U, mean fitness at MS balance increases. This is due to the increasing efficiency against DGM with an increasing S and mutation pressure decreasing with a decreasing U. For the genetic variance of fitness, increasing S and s and decreasing h result in larger genetic variance at MS balance. Genetic variance at MS balance generally increases with an increasing U when S is large (S = 0.8) and there is an intermediate maximum genetic variance with an intermediate U value when S is small (S = 0.3). This should not be surprising. As expected from the theory (Equations 1b and 10b, Deng and Lynch 1996), under complete selfing, the genetic variance at MS balance is a monotonic function of U; under complete outcrossing, it is not and there is an intermediate maximum level of genetic variance with an intermediate level of U.
DISCUSSION
In this article, the robustness and the statistical properties of the Deng and Lynch (1996) method are extensively investigated under incomplete outcrossing/selfing and in nonequilibrium populations at different stages approaching MS balance. It is demonstrated that when S < 0.1 or when S > 0.8, the Deng and Lynch method is generally robust and applicable to yield reasonably reliable estimates. For 0.1 < S < 0.8, relatively larger bias will result and the bias generally increases when S approaches 0.5. Under partial outcrossing and variable or epistatic mutation effects, depending on the unknown mutation effects, selfing rate, and mutation parameters (see results), the estimates of the mutation parameters may not always be biased. The Deng and Lynch estimation in nonequilibrium populations at different stages approaching MS balance is fairly robust and is similar to the performance in populations at MS balance. The underlying mechanisms are investigated and given. The results for only a limited number of mutation parameters are given in this article. Our results not shown for other mutation parameters (U = 0.1–10; s = 0.01–0.10; h = 0.2–0.4) demonstrated similar results and indicated that s and h also influence the estimation bias, although the magnitude is generally small relative to the effects of S and U. Our investigation here provides a basis for applying the Deng and Lynch method to characterize DGM in partial selfing or outcrossing populations and for nonequilibrium populations.
Our results on the robustness of the estimation in partial selfing/outcrossing populations are consistent with that of B. Charlesworth et al. (1990) in testing the robustness of their inbreeding depression method to estimate U. The estimation method of B. Charlesworth et al. (1990) only employs information of change of mean fitness traits upon outcrossing in selfing populations and depends on a prior knowledge of h that is generally not known (Deng and Fu 1998). By computer simulations, B. Charlesworth et al. (1990) found that their inbreeding depression method is fairly robust in populations with selfing rate S > 0.8. These results are consistent in that the estimation methods (B. Charlesworthet al. 1990; Deng and Lynch 1996) developed for complete outcrossing and selfing populations may also be applicable to partial selfing/outcrossing populations, provided that S > 0.8 in predominantly selfing populations or S < 0.1 in predominantly outcrossing populations. There are many populations that are partial outcrossing/selfing (Stebbins 1957; Lande and Schemske 1985; Schemske and Lande 1985; Barrett and Eckert 1990; Charlesworth 1992). In addition, S in many of these populations is either high and close to one or low and close to zero (Lande and Schemske 1985; Schemske and Lande 1985). Our results here thus greatly expand the range of populations for which the DGM can be characterized.
Our results on the mean and genetic variance of fitness with different parameters are partially related to the mean number of mutations per genome in populations. D. Charlesworth et al. (1990) found that the numbers of mutations per individual are affected by S, U, and s. The larger the S or U is or the less the s, the more the number of mutations per genome in populations. Our simulations demonstrate the same pattern as D. Charlesworth et al. (1990). In addition, we find that h also plays an important role in determining the number of mutations per genome at MS balance; the less the h, the more the number of mutations. All these reflect the degree of mutation pressure and/or the effectiveness of selection against DGM. These can be further demonstrated by a theoretical approximation. By the work of Lande and Schemske (1985), for partial selfing population at MS balance, consider a locus with two alleles (wildtype B and mutant allele b), and the frequencies of B and b are, respectively,
In this article, for nonequilibrium populations at various stages approaching MS balance, we examine the dynamics of x, y, and z and their ratios, all of which are functions of the mean and genetic variance of fitness in populations. These detailed examinations provide an explanation for the surprisingly robust performance of the Deng and Lynch estimation in nonequilibrium populations. Although we did not formally investigate the performance of the inbreeding depression approach (Mortonet al. 1956; B. Charlesworthet al. 1990) in our simulated nonequilibrium populations, their performance may not be good. This was suggested by the fact that the inbreeding depression (y) alone changes dramatically before MS balance is reached (Figure 2). The robust performance of the Deng and Lynch (1996) estimation lies in the fact that it employs changes of mean and genetic variance of fitness. The ratios of the mean and genetic variance that are critical for the DengLynch estimation remain fairly stable in nonequilibrium populations at different stages approaching MS balance (Figure 2). Generally, it requires only ~200 or ~75 generations, respectively, in complete outcrossing and selfing populations for these ratios to be stable since the populations experience mutation and selection starting from a homogeneous population free of DGM. In partial selfing/outcrossing populations, the number of generations needed is between ~200 and ~75 generations.
Most natural populations have experienced mutation and selection for many generations, and they were generally founded by individuals with different numbers of DGM. It can be shown easily by computer simulations (H.W. Deng and J. Li, unpublished results) that, despite initially different distributions of DGM in founding populations, under mutation and selection, the population dynamics of fitness will quickly (in less than a few dozen generations) converge to that (under the same parameters) of populations approaching MS balance when starting free of DGM. The specific merging point into the dynamics investigated here will largely depend on the detailed initial distributions of DGM in founding populations. Such founding populations may be formed due to migration, population admixture, and population bottlenecks. Therefore, the investigation for nonequilibrium populations approaching MS balance when starting from a homogeneous population is of general significance in shedding light on the robustness of the DengLynch method in characterizing DGM. Particularly, even starting from a homogeneous population free of DGM, the number of generations taken to reach MS balance does not seem to be large for natural populations (Figure 3), and generally 400–2000 generations are needed. Therefore, the Deng and Lynch (1996) method should be robust and applicable for many natural nonequilibrium populations even if they are at different stages of approaching MS balance. In addition, it is not rare for natural populations to be founded by one or a few genotypes (such as in Daphnia or in many plant populations). Therefore, our investigation of nonequilibrium populations approaching MS balance at various stages when starting from homogeneous genotypes is not only of theoretical but also of considerable practical significance in simulating plausible situations in natural populations.
In studying inbreeding depression and mutation load in partial selfing populations, D. Charlesworth et al. (1990) only described some effects of mutation parameters and S on the mean fitness. However, as pointed out by Deng and Lynch (1997), mutation load is manifested not only as the mean but also the genetic variance of populations at MS balance. Studying the dynamics of genetic variance will reveal more aspects and important information about DGM and population evolution (Deng and Lynch 1997). In this study, we demonstrate the dependency of mean and genetic variance at MS balance on mutation parameters and S. As stated earlier, we also demonstrated the detailed dynamics of mean and genetic variances and their ratios in populations approaching MS balance and the number of generations taken to reach MS balance when starting from a homogeneous population free of DGM. These results should be useful not only for the characterization of DGM, but importantly, they should also be useful for a variety of other population genetics topics. For example, the dynamics of mean and genetic variance and the number of generations taken to reach MS balance when starting from a homogeneous population free of DGM are important for conservation genetics.
Characterization of DGM is one major challenge in the broad field of genetics (Crow 1993a,b; Bridgeset al. 1994; Peck and EyreWalker 1997). Several approaches have been developed and data are being accumulated. However, before the results from any approach can be treated as robust, the statistical estimation methods underlying all the approaches must be examined carefully under a variety of biologically plausible conditions. This article represents one of a series of our efforts (Deng and Lynch 1996; Deng 1998b; Liet al. 1999) to examine the robustness and statistical properties of Deng and Lynch's (1996) method for characterization of DGM in natural populations. Our effort in the near future (H.W. Deng and J. Li, unpublished results) will examine the estimation under other conditions such as finite population sizes and linkage disequilibrium between DGM at different loci.
Acknowledgments
We are particularly grateful for the constructive comments from an anonymous reviewer who helped to improve the manuscript. This work was partially supported by a program grant and a faculty development grant from the Health Future Foundation, a grant from HuNan Normal University of People's Republic of China, and by a graduate student tuition waiver to J. Li from the Department of Biomedical Sciences of Creighton University.
Footnotes

Communicating editor: M. Slatkin
 Received August 27, 1999.
 Accepted December 21, 1999.
 Copyright © 2000 by the Genetics Society of America