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Corresponding author: Hong-Wen Deng, Osteoporosis Research Ctr., Creighton University, 601 N. 30th St., Ste. 6787, Omaha, NE 68131., deng{at}creighton.edu (E-mail)
Communicating editor: M. SLATKIN
 | ABSTRACT |
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 TOP ABSTRACT THEORYSIMULATIONSRESULTSDISCUSSIONLITERATURE CITED |
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The Deng-Lynch 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 mutation-selection (M-S) 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 M-S balance are often not known. Through computer simulations, we investigated the robustness and the applicability of the Deng-Lynch method under different degrees of partial selfing or partial outcrossing and for nonequilibrium populations approaching M-S balance at different stages. The investigation was implemented under constant, variable, and epistatic mutation effects. We found that, generally, the estimation by the Deng-Lynch 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 M-S balance. The dynamics of populations approaching M-S 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 M-S 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 (
and 
). In addition, the knowledge about the mean and the variation of the effects of DGM is vital in evaluating the role of DGM in long-term survival of small populations (
However, few estimates of the parameters of DGM are available, and even the order of magnitude of these parameters is controversial (
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 Deng-Lynch 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 Deng-Lynch method estimates not only U but also , , and the genetic variance introduced by DGM per generation (
Another important assumption of DENG and LYNCH's (1996) estimation is that the study population is at mutation-selection (M-S) balance. Despite extensive efforts (e.g.,
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
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 M-S 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 M-S balance.
| THEORY |
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| TOPABSTRACTTHEORYSIMULATIONSRESULTSDISCUSSIONLITERATURE CITED |
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The approach to characterize DGM proposed by
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(1) |
In a selfing population,
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(2) |
where x, y, and z are defined, respectively, as
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(3) |
where, in outcrossing populations, 
(O) and 
2(O) are, respectively, the mean and genetic variance of fitness, and (S) and 2(S) are, respectively, the mean and genetic variance of fitness among selfed progeny families. In selfing populations, (S) and 2(S) are, respectively, the mean and genetic variance of fitness, and (O) and 2(O) are, respectively, the mean and genetic variance of fitness among outcrossed progeny.
The experimental procedures in outcrossing and selfing populations are detailed previously ((O) and (S)), and (3) one-way ANOVAs are performed. In the outcrossed parental generation, parental genotypes are treated as main effects and clonal replicates as random effects, so that we obtain estimates of the genetic variance (2(O)). In the selfed offspring generation, selfed families are treated as main effects and selfed progeny genotypes within each family as random effects so that we can obtain the genetic variance among the mean of selfed families (2(S)). In selfing populations, (1) random pairs of genotypes are sampled and outcrossed; (2) the selfed parent and outcrossed progeny genotypes are cloned and assayed in one common environment; and (3) one-way ANOVAs are performed with genotypes as main effects and clonal replicates as random effects, to estimate the genetic variances in the outcrossed progeny (2(O)) and selfed parental (2(S)) generations, together with (O) and (S).
| SIMULATIONS |
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| TOPABSTRACTTHEORYSIMULATIONSRESULTSDISCUSSIONLITERATURE CITED |
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By a deterministic method that was developed by
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 wild-type 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 M-S 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 mutation-selection 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. M-S 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 x 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
Then, mating is simulated within each of the two subpopulations. In the selfing subpopulation, the individual frequencies are changed to
where si,j(k, l) is the probability that an individual having k heterozygous and l homozygous mutations produces an offspring with i heterozygous and j homozygous mutations by selfing,
In the outcrossing subpopulation, assuming that there is no shared mutant allele for any two individuals, the outcrossed progeny will have no homozygous mutational loci. The frequencies of different individuals are
where b(x, y; z) is the frequency of producing an offspring with z heterozygous mutational loci by two parents having x and y heterozygous mutational loci, respectively. It is easy to show that
After mating, selection starts to operate. Relative fitness is used to determine the frequencies after selection, i.e.,
where wi,j is the fitness of an individual with i heterozygous and j homozygous mutations; 
S = 
ijp''i,j(S) wi,j, which is the mean fitness of the selfing subpopulation before selection; and O = ip''i,j(O)wi,j, which is the mean fitness of the outcrossing subpopulation before selection.
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:
where S* = 
and 
T is the mean fitness of the whole population before selection.
Constant mutation effects:
Simulations are performed under different parameter sets. The fitness of an individual having i heterozygous and j homozygous mutations is
In populations that have reached M-S balance, K individuals are sampled as parents. Progenies are produced by outcrossing (in populations with S > 0.5) or selfing (in populations with S < 0.5) the parents. For populations with S > 0.5 (predominantly selfing populations), one outcrossed progeny per parent is obtained and the Deng-Lynch method (
Variable mutation effects:
Mutation effects are unlikely to be constant across loci (
and there is an inverse relationship between h and s
These are consistent with our few data on the distribution of mutation effects (
where sl and sm come from the above exponential distribution with mean .
Epistatic mutation effects:
Although fitness or its component most likely acts multiplicatively (
where n is the effective number of mutations, n = hi + j. 
= s in a selfing population. The parameter ß provides a measure of the synergistic effects of deleterious alleles and the ratio 
provides a measure of the relative contribution of synergistic effects to mean fitness.
Estimation for nonequilibrium populations:
All the above estimations are conducted for populations that have already reached M-S balance. To examine the robustness of the estimation of
Dynamics of populations approaching M-S 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 M-S balance since starting from a homogeneous population free of DGM and the number of generations taken to reach M-S balance.
Throughout, unless otherwise specified in tables or figures, in simulations reported, h (or ) is 0.36 and s (or ) = 0.03, and reported values are the means ± standard deviation (SD). The selected values of h (or ) = 0.36 and s (or ) = 0.03 are consistent with those summarized from the extensive experiments in Drosophila (
| RESULTS |
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| TOPABSTRACTTHEORYSIMULATIONSRESULTSDISCUSSIONLITERATURE CITED |
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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. 
is always upwardly biased for partial selfing/outcrossing (0 < S < 1). This is because the ratio x/z in populations that are critical for estimation (Equation 1 and Equation 2) increases when S deviates from 0 or 1 and approaches 0.5 (Fig 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 
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, = 0.467, and = 0.009 when S = 0.3; and Û = 0.70, = 0.417, and = 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 (Table 2 and Table 3). When S < 0.05 and when S > 0.9, the 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 
is always negative as revealed in our simulations, the sign of Û depends on whether > 0.5 (Equation 1); if < 0.5, Û > 0 and if > 0.5, Û < 0. When is close to 0.5, Û will be very sensitive to the change of , i.e., small changes of may result in quite different Û's. So we obtained large SD of Û when is close to 0.5. When is far from 0.5, the SD decreases. For our simulated parameters, we never found that = 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 is closer to 0.5.
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Estimation under variable mutation effects (Table 2):
The estimates are generally biased, but not always. When 0 < S < 0.5 and when S increases, decreases from ~2 to negative values. Û and change from downwardly biased to upwardly biased. Therefore, in some intermediate selfing rates, , Û, and may be unbiased and this depends on mutation effects, S, and mutation parameters that are generally unknown. For some S values, when S approaches 0.5 from 0, approaches values ~0.5, yielding large bias and sampling error for Û. The biases can be very large in predominantly outcrossing populations (e.g., Û is 67.4 for S = 0.2 and U = 1 and it is negative when S tends toward 0.4; negative values are also obtained for ). However, in a predominantly selfing population, the estimates are not that bad. When 0.5 < S < 1 and when S decreases, increases, from ~2.4 to ~3.2 (for U = 0.5) or 4.8 (for U = 1.0). Û decreases only slightly. Therefore, the estimation under various S in predominantly selfing populations is more robust than the estimation under various S in predominantly outcrossing populations. Again, the estimation is fairly robust when S < 0.10 and when S > 0.8.
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 Deng-Lynch 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 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, 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 Deng-Lynch estimation in nonequilibrium populations (Table 4 and Table 5):
The performance of DENG and LYNCH's (1996) estimation is fairly robust in nonequilibrium populations at different stages approaching M-S balance. Under constant mutation effects (Table 4), and 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.
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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 M-S balance (Fig 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 (Equation 1Equation 2Equation 3) change little and remain essentially the same after the few initial generations. Since is the function of 
/
, it will approximately reach a constant a few generations after the mutation and selection starts to operate. Furthermore, is the function of and / (in outcrossing populations) or / (in selfing populations). Therefore, and quickly become the same as those in populations at M-S balance. Û is a function of and (Equation 1 and Equation 2). When remains relatively constant and the absolute value of increases with the more generations of mutation and selection experienced, Û increases and becomes stable only when approaches equilibrium values.
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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 M-S 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 M-S balance.
Dynamics of nonequilibrium populations approaching M-S balance ( Fig 3 and Fig 4):
S has significant effect on the generations needed to reach M-S balance (Fig 3A). All else being the same, the larger the S, the less the generations are needed to reach M-S balance. This is mainly because as S increases, the mean number of mutations accumulated in the genome necessary to reach M-S balance gets smaller, as is found in our simulations and those of D.
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For populations at M-S 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 (Fig 4). With increasing S and/or decreasing U, mean fitness at M-S 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 M-S balance. Genetic variance at M-S 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,
| DISCUSSION |
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| TOPABSTRACTTHEORYSIMULATIONSRESULTSDISCUSSIONLITERATURE CITED |
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In this article, the robustness and the statistical properties of the
Our results on the robustness of the estimation in partial selfing/outcrossing populations are consistent with that of B.
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.
where µ is the mutation rate from B to b per locus and f is Wright's inbreeding coefficient (
Ignoring the term q2 and letting f 
S/(2 - S + 2Ss) (for relatively large S so that f > h;
From the above equations, the qualitative effects of S, U, h, and s on the number of mutations per individual in partial selfing/outcrossing populations at M-S balance can be easily seen.
In this article, for nonequilibrium populations at various stages approaching M-S 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 (
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 M-S 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 M-S balance when starting from a homogeneous population is of general significance in shedding light on the robustness of the Deng-Lynch method in characterizing DGM. Particularly, even starting from a homogeneous population free of DGM, the number of generations taken to reach M-S balance does not seem to be large for natural populations (Fig 3), and generally 400–2000 generations are needed. Therefore, the
In studying inbreeding depression and mutation load in partial selfing populations, D.
Characterization of DGM is one major challenge in the broad field of genetics (
| ACKNOWLEDGMENTS |
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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.
Manuscript received August 27, 1999; Accepted for publication December 21, 1999.
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| TOPABSTRACTTHEORYSIMULATIONSRESULTSDISCUSSIONLITERATURE CITED |
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