Characterization of Deleterious Mutations in Outcrossing Populations

Corresponding author: Hong-Wen Deng, Osteoporosis Research Center, Creighton University, 601 N. 30th St., Suite 6787, Omaha, NE 68131., deng{at}creighton.edu (E-mail).

Communicating editor: M. SLATKIN

	ABSTRACT

TOP ABSTRACT THEORY COMPUTER SIMULATIONS RESULTS DISCUSSION LITERATURE CITED

Deng and Lynch recently proposed estimating the rate and effects of deleterious genomic mutations from changes in the mean and genetic variance of fitness upon selfing/outcrossing in outcrossing/highly selfing populations. The utility of our original estimation approach is limited in outcrossing populations, since selfing may not always be feasible. Here we extend the approach to any form of inbreeding in outcrossing populations. By simulations, the statistical properties of the estimation under a common form of inbreeding (sib mating) are investigated under a range of biologically plausible situations. The efficiencies of different degrees of inbreeding and two different experimental designs of estimation are also investigated. We found that estimation using the total genetic variation in the inbred generation is generally more efficient than employing the genetic variation among the mean of inbred families, and that higher degree of inbreeding employed in experiments yields higher power for estimation. The simulation results of the magnitude and direction of estimation bias under variable or epistatic mutation effects may provide a basis for accurate inferences of deleterious mutations. Simulations accounting for environmental variance of fitness suggest that, under full-sib mating, our extension can achieve reasonably well an estimation with sample sizes of only ~2000–3000.

THE genome of any organism is subject to continuous bombardment of mutations, the majority of which are deleterious. Numerous theories based on the deleterious genomic mutations have been developed to explain some fundamental phenomena in biology. The validity of these theories critically depends on the rate at which deleterious mutations occur per genome per generation (U) and/or the effects of deleterious mutations.

For example, estimates of U are crucial to testing theories for the evolution of sex and recombination (MULLER 1964 ; KONDRASHOV 1985 , KONDRASHOV 1988 ; CHARLESWORTH 1990 ), mate choice (CHARLESWORTH and CHARLESWORTH 1987 ; KONDRASHOV 1988 ; KIRKPATRICK and RYAN 1991 ), outbreeding mechanisms (CHARLESWORTH and CHARLESWORTH 1987 ), diploidy (KONDRASHOV and CROW 1991 ), and the accelerated extinction rate of small populations (LYNCH and GABRIEL 1990 ; LYNCH et al. 1993 , LYNCH et al. 1995A , LYNCH et al. 1995B ). Estimates of the other parameters of spontaneous deleterious mutations are also important. Such parameters include the mean dominance coefficient () the mean selection coefficient () the genomic mutation variance scaled by environmental variance (V_m/V_e), and variation of mutation effects. Estimates of and are important for testing the theories of evolutionary transition from haploidy to diploidy (PERROT et al. 1991 ) and for testing theories of the role of deleterious mutations in extinction of small populations (LANDE 1994 ; LYNCH et al. 1995A ). Joint estimates of U, , and determine the rate of input of genetic variance from mutation per generation (DENG and LYNCH 1996 , DENG and LYNCH 1997 ) and the extent to which the neutral molecular variation is reduced due to the background selection (CHARLESWORTH et al. 1993 , CHARLESWORTH et al. 1995 ; HUDSON and KAPLAN 1995 ). In finite populations, variation of mutation effects plays an important role in the maintenance of polygenic variations (KEIGHTLEY and HILL 1990 ) and in determining the persistence time and extinction rate of small populations (LANDE 1994 ; LYNCH et al. 1993 , LYNCH et al. 1995A , LYNCH et al. 1995B ).

However, few estimates are available (CROW and SIMMONS 1983 ; KONDRASHOV 1988 ; CROW 1993 ), and none of the current estimation approaches can yield unbiased results under realistic situations (DENG and FU 1998A ). A direct estimation approach for bounds of U and , the traditional mutation-accumulation experiment (BATEMAN 1959 ; MUKAI et al. 1972 ), takes extensive time and labor and is feasible only for asexual organisms, special sexual organisms (such as Drosophila where special chromosomal constructs are available), and artificially constructed purely inbred lines. An indirect procedure for estimating U makes use of inbreeding depression data (MORTON et al. 1956 ; CHARLESWORTH et al. 1990 ), but it depends on an unknown of deleterious mutations ( needed is the harmonic/arithmetic mean in outcrossing/selfing populations; DENG and LYNCH 1996 ). Estimation of requires more assumptions (COMSTOCK and ROBINSON 1948 ; HAYMAN 1954 ; MUKAI et al. 1972 ; CABALLERO et al. 1997 ; DENG 1998 ). Even with the additional assumptions, the estimate is biased and weighted by the selection coefficients of individual mutant alleles at different loci. In addition, this indirect estimation of U is very sensitive to an assumed (or estimated) (DENG and FU 1998A ).

DENG and LYNCH 1996 , DENG and LYNCH 1997 developed an estimation approach, making better use of the data (changes of both the mean and genetic variance for fitness traits) that can be acquired from selfing/outbreeding in outcrossing/highly selfing populations, to estimate not only U, but also , , and V_m, etc. (DENG and LYNCH 1996 , DENG and LYNCH 1997 ). All the estimation approaches applied to natural populations (MORTON et al. 1956 ; CHARLESWORTH et al. 1990 ; DENG and LYNCH 1996 ) assume that all the genetic variation is maintained by mutation-selection balance (see DISCUSSION). Under a range of biologically plausible situations investigated, DENG and LYNCH's (1996) estimation approach almost always yields the best estimation (as reflected by the mean square error, a composite index of bias and sampling variance), when compared with other estimation approaches (DENG and FU 1998A ). However, DENG and LYNCH 1996 , DENG and LYNCH 1997 were mainly concerned with the estimation in outcrossing populations such as those of Daphnia (DENG 1995 ), in which selfing is feasible. Although approximate equations for full-sib mating were developed (Equation 14; DENG and LYNCH 1996 ), they are just a rough approximation and thus inaccurate. Applying these approximate estimations to data from full-sib mating experiments, substantial bias will result even under the ideal assumptions underlying the derivation for estimation (H.-W. DENG, unpublished results). Under the necessary assumptions in DENG and LYNCH 1996 , development of the exact estimation for inbreeding experiments other than selfing is not trivial, and thus was not developed (though the effort was made as evidenced by the approximate estimations given there) in DENG and LYNCH 1996 . Since selfing is not feasible for the majority of outcrossing populations, there is an imperative need to develop estimation procedures with other forms of inbreeding (such as full- and half-sib mating, etc.) that are feasible in almost all outcrossing populations.

In this article we develop exact estimation equations, under the assumptions in DENG and LYNCH 1996 , for any form of inbreeding to characterize deleterious genomic mutations. Additionally, we investigate and compare the statistical properties and the robustness of the estimation under common forms of inbreeding (full- and half-sib mating) and those under selfing. Furthermore, estimation employing the total genetic variation in the inbred generation and that using the genetic variance among the mean of inbred families are compared for their relative efficiencies (multiple inbred progeny from each family is not necessary in the former estimation, while it is mandatory in the latter). These investigations will not only provide a guideline for designing an efficient experiment employing samples from outcrossing populations, but will also provide a basis for accurate inferences of deleterious genomic mutations from the values estimated on the basis of some necessary but unrealistic assumptions.

	THEORY

TOP ABSTRACT THEORY COMPUTER SIMULATIONS RESULTS DISCUSSION LITERATURE CITED

The assumptions of MORTON et al. 1956 , CHARLESWORTH et al. 1990 , and DENG and LYNCH 1996 , DENG and LYNCH 1997 are employed to derive analytical estimations. The population is assumed to be very large for a long enough time such that all loci are at mutation-selection equilibrium for segregating polymorphisms. For any locus, this requires that selective disadvantage of deleterious mutations (selection coefficient s) is on the order of 1/N_e, where N_e is the effective population size (KIMURA et al. 1963 ; LYNCH et al. 1995A , LYNCH et al. 1995B ). This requirement ensures that, besides mutations, selection rather than random genetic drift is the driving force for the standing genetic variation. The frequency of deleterious mutant alleles at any locus is assumed to be small. The fitness function is assumed to be multiplicative, which is biologically plausible (MORTON et al. 1956 ; CROW 1986 ; CRADDOCK et al. 1995 ; FU and RITLAND 1996 ). Mutations at each locus have constant effects s and h (dominance coefficient). The three genotypic values at each locus with mutations are, respectively,

The number of mutations per genome (n) is assumed to be Poisson distributed with a probability density function p(n) = ⁿ ! , where is the mean number of deleterious mutations per genome. Later in this article, we conduct computer simulations to test the effects of violation of some of the above assumptions (such as constant mutation effects and multiplicative fitness function) on estimation.

Under these assumptions, the mean (_O) and the genetic variance [²_w(O)] of fitness in an outcrossing population are found to be, respectively (DENG and LYNCH 1996 ),

(1)

(2)

Equation 1 is a well-known result (HALDANE 1937 ; KIMURA et al. 1963 ; BURGER and HOFBAUER 1994 ). W_max is the expected fitness of a mutation-free genotype in the environmental conditions, where the experimental measurements are taken. W_max serves as a scaling factor so that fitness measurement can be on any scale instead of just from 0.0 to 1.0, and also so that mean environmental effects of experiments do not influence estimation.

Suppose now that the members of an outcrossed population undergo inbreeding with inbreeding coefficient being f in inbred progeny. In the case of selfing, where f = , full-sib mating f = , and half-sib mating f = . For each heterozygous locus in the outcrossed parental generation, an inbred progeny is expected to be heterozygous and homozygous for the deleterious allele with probabilities (1 - f) and f/2, respectively (CROW and KIMURA 1970 ). Thus, under the assumption of free recombination and by the relationship () = (DENG and LYNCH 1996 ), the mean fitness (_I) in the inbred progeny is

(3)

The genetic variance among the mean fitness of the inbred progeny of different inbreeding families [²_w(I)] is

(4)

The total genetic variance of fitness [²_T(I)] in the inbred progeny generation is

(5)

²_T(I) is the sum of ²_w(I) and the genetic variance among the selfed genotypes within selfed families. The derivation of Equation 3Equation 4 is similar to and simpler than that of Equation 5, thus not elaborated here. Equation 5 is derived as follows: Let w(I) represent the fitness of a random genotype from the inbred offspring generation, then ²_T(I) = E(w²(I)) - ²_I , and E(w²(I)) = _n=0 E(w²) P(n) , where E(w²) is the expectation of w²(I) conditional on parents having mutations at n loci. Let x_i denote the fitness of the ith locus (which is heterozygous in the outcrossed parents) in an inbred progeny, under the assumptions of multiplicative fitness and unlinked loci E(w²) = E(ⁿ_i=0x²_i) = ⁿ_i=0E(x²_i) . For each parental locus that is heterozygous for mutations, the inbred progeny is expected to be heterozygous or homozygous for the deleterious alleles with probabilities (1 - f) and f/2, respectively, hence E(x²_i) = + (1 - f)i(1 - hs)² +f , which is independent of the ith locus under the assumption of constant mutation effects across loci. Therefore, we have

From Equation 3 and the relationship = (DENG and LYNCH 1996 ), Equation 5 follows.

To verify our derivation, we substitute f = [the selfing case as considered by DENG and LYNCH 1996 , DENG and LYNCH 1997 ] into Equation 3Equation 4Equation 5. We find that Equation 3 and Equation 4 recover the corresponding Equations 1c–1d of DENG and LYNCH 1996 and Equation 5 reduces to the corresponding Equation A1(1) of DENG and LYNCH 1997 . Additionally, it is noted that if we set f = 0 (outcrossing case), Equation 3 recovers Equation 1 and Equation 4 and Equation 5 are reduced to Equation 2; therefore, the above general equations should be correct. The verification will be further carried out in our computer simulations.

Estimation via the information on ²_w(I) requires the estimation of mean fitness of the inbred progeny from each family and thus the measurement of multiple progeny from each inbreeding family. The mean of the inbred progeny from each family is subject to sampling error as the number of inbred progeny sampled from each family is usually finitely small. Estimation via the information on ²_T(I) does not require multiple inbred progeny from each family. Therefore, estimation of ²_w(I) is subject to two sources of sampling error. One is the number of inbreeding families sampled from populations and the other is the number of inbred progeny sampled from each inbreeding family. However, estimation of ²_T(I) is subject to only one source of sampling error, i.e., the number of inbreeding families sampled from populations. Thus, estimation of the rate and properties of deleterious mutations via estimation of ²_T(I) is likely to be more powerful; it is also likely to be more practical since sometimes multiple inbred progeny are simply not available for each family. This will be investigated and confirmed later, by computer simulations, in the case of selfing. The investigation is necessary since estimation via ²_w(I) is the design that was proposed originally in DENG and LYNCH 1996 . In addition, estimation via ²_w(I) may have an advantage in some practical situations, such as when cloning of genotypes in the progeny generation is not feasible (DENG and LYNCH 1997 ). However, we will focus on studying estimation via ²_T(I) , i.e., estimation will be developed only by Equation 1Equation 2Equation 3 and Equation 5 in this study. Estimation by Equation 1Equation 2Equation 3Equation 4 is straightforward, less powerful, and thus not pursued here.

Define x, y, and z, respectively, as

(6)

From Equation 1Equation 2Equation 3 and Equation 5, the expected values of x, y, and z are, respectively,

(7a)

(7b)

(7c)

Note, in Equation 7b, if h = 0.5 (pure additive case), there should be no inbreeding depression (E(y) = 0) and estimation of U cannot be obtained. However, a pure additive case almost does not exist as suggested by the universal phenomena of inbreeding depression and heterosis. Rearranging and letting a circumflex (ˆ) denote an estimate throughout, we obtain potential estimators for the mutational parameters:

(8a)

(8b)

(8c)

By substituting f = ¹/₂, the estimation Equation 8aEquation 8bEquation 8c recover those of Equations A1(4a–4c) in DENG and LYNCH 1997 . Different forms of inbreeding have different f's. By substituting f for a specific form of inbreeding employed in the experiment into Equation 8aEquation 8bEquation 8c, estimation of U, h, and s and other derivative parameters can be obtained. These derivative parameters include (not exclusively) the genetic variance introduced into a population by new mutations per generation V_m and the mean number of deleterious mutations per genome (DENG and LYNCH 1996 ).

	COMPUTER SIMULATIONS

TOP ABSTRACT THEORY COMPUTER SIMULATIONS RESULTS DISCUSSION LITERATURE CITED

To verify our analytical derivations under the assumptions made, and also to test the robustness of the estimation with the violation of some essential assumptions for analytical derivation, statistical properties (sampling variance and bias) of the estimation are investigated by computer simulations. These investigations will provide a basis for accurate inference of the genomic mutations with the estimation developed under the necessary but implausible assumptions. Specifically, the following assumptions will be tested: (1) the fitness function is multiplicative and there are no epistatic fitness effects of mutations; (2) the mutation effects s and h are constant across loci; and (3) there are no lethal mutations. Some other practical issues are also investigated by computer simulations: (1) the relative efficiencies of estimation by employing the information of the total genetic variation [²_T(I)] in the inbred generation vs. that of the genetic variation of the mean of inbred families [²_w(I)] (this will be demonstrated by investigating the selfing case); (2) the relative efficiencies of estimation by employing different degrees of inbreeding (this will be investigated in the case of selfing, full-, and half-sib mating); and (3) the power of the estimation when genotypic values cannot be measured without error.

It should be noted that some of the problems were investigated for estimation developed from Equation 1Equation 2Equation 3Equation 4 (DENG and LYNCH 1996 ). However, none of the above problems have been investigated for the estimation developed from Equation 1Equation 2Equation 3 and Equation 5, which employ a different experimental design as explained earlier. Although some of the simulation conclusions will be qualitatively similar to those in DENG and LYNCH 1996 , they will be quantitatively different. These quantitatively different results (especially the different degrees of bias) form the bases for accurate inference of mutations on the basis of different experimental designs and estimations.

Estimation under constant mutation effects:
We assume that a mutation-selection balance has been reached in the parental generation, so that the number of mutations per individual (all in the heterozygous state) is Poisson distributed with an expectation of = . In each situation, simulations are performed for different sets of parameters. For each parameter set, variable K and H individuals are randomly sampled, respectively, from the outcrossed parental and inbred progeny generations. Initially, the genotypic values are assumed to be measured without error and are defined by the multiplicative fitness function used in the derivation. For a genotype with n mutations (randomly determined from the Poisson distribution) from the outcrossed parental generation, the fitness is

For a genotype sampled from the inbred progeny generation, the fitness is

where n₁ and n₂ are, respectively, the numbers of loci with mutations at heterozygous and homozygous states. n₁ and n₂ are determined from two levels of random sampling: (1) A number (n) of loci is randomly determined from the Poisson distribution with mean = ; (2) with inbreeding coefficient f, each of these n loci has a probability of f/2 to be homozygous for the normal A allele, a probability of (1 - f) to be heterozygote Aa, and a probability of f/2 to be homozygote aa. After the genotypic status of each locus is determined as above, n₁/n₂ are just the sum of loci heterozygous/homozygous for mutations. The genetic variances [²_w(O) and/or ²_T(I) ] are just the variances among the genotypes under the assumption that genotypic values are measured without error. This assumption will be relaxed later in simulating the power of the experiments. h_i and s_i are the dominance and selection coefficients of the ith locus with mutations. They will be assumed constant initially and made variable later. We arbitrarily let W_max = 1 throughout, as the values of W_max do not influence the estimation for the mutation parameters. For each set of parameters (U, h, s, K, H), we perform 500 simulations. The average and standard deviations (SD) of the estimates over the 500 independent simulations are reported. Unless otherwise specified, K = H = 200 in simulations.

Estimation under variable mutation effects:
Mutation effects h_i and s_i across loci are unlikely constant. For example, s_i may vary anywhere from 0.0 (neutral mutation) to 1.0 (lethal mutation). The rate of occurrence for 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 by variable mutation effects and variable mutation rates, as in DENG and LYNCH 1996 , DENG and LYNCH 1997 , we adopt an exponentially distributed mutation rate for mutations of variable effect s_i:

(9a)

Also we let

(9b)

As explained in DENG and LYNCH 1996 , these are in rough accordance with the few available data (GREGORY 1965 ; CROW and SIMMONS 1983 ; MACKAY et al. 1992 ; KEIGHTLEY 1994 ) or biochemical arguments (KACSER and BURNS 1981 ). However, true mutational spectra may be such that the dominance of individual mutations is broadly scattered around such a function (CABALLERO and KEIGHTLEY 1994 ).

In simulations, we divide the entire range of s (0.0–1.0) 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_is_i), where p_i is the density of the mutational distribution in the ith class. For an individual from the inbred progeny generation, n_is 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 determined by f, so that probabilities for different genotypes are f/2 for AA, (1 - f) for Aa, and f/2 for aa, respectively.

Estimation with lethal mutations present in the genome:
Due to their low dominance coefficient, lethal mutations are often sheltered from selection by being kept in heterozygous state in outcrossing populations. To investigate the effects of lethal mutations on estimation, we add an additional low genomic mutation rate (0.01U) to lethals (defined as having s = 1.0 and h = 0.02) (Table 3).

Estimation with epistatic mutation effects:
The theory we developed here assumes that deleterious mutations across loci interact multiplicatively. Although there is some good evidence that genes for fitness or its components most likely act multiplicatively (MORTON et al. 1956 ; CROW 1986 ; CRADDOCK et al. 1995 ; FU and RITLAND 1996 ), synergistically epistatic mutation effects can not be ruled out entirely. Thus, we test the robustness of the estimation method under epistatic mutation effects. To evaluate the potential consequences of epistasis on the mutation-parameter estimates derived from our model, we consider the epistatic fitness model described by CHARLESWORTH 1990 and employed by us before (DENG and LYNCH 1996 , DENG and LYNCH 1997 ):

where n = n₁ + (n₂/h) is the effective number of heterozygous mutations per individual. n₁ and n₂ are, respectively, the numbers of loci heterozygous and homozygous for mutations. The parameter ß measures the strength of the synergistic effects of deleterious mutations. With ß = 0, the model reduces to one of multiplicative effects, and with ß > (<) 0, the effects of deleterious alleles are reinforcing (diminishing) epistatic; i.e., as more deleterious alleles are added to the genome, the decline in fitness per additional deleterious allele increases (decreases). Reinforcing epistatic effects were suggested (though not convincingly) by several empirical results (e.g., MUKAI 1969 ) and are most interesting to geneticists, while empirical evidence for diminishing epistatic effects is essentially nonexistent. Therefore, we focus on investigating the effects of reinforcing epistasis on estimation in this study; epistatic effects refer to reinforcing epistasis hereafter. The ratio measures the relative contribution of epistatic effects to mean fitness. The larger the , the larger the relative contribution of epistatic effects to mean fitness.

We implement the epistatic fitness model by assuming mutations of constant effects (s and h). Under mutation-selection equilibrium, n is approximately normally distributed with the mean and variance being functions of U, h, s, and ß, defined by Equation 3 in CHARLESWORTH 1990 . For the outcrossed parental generation, we again assume that all deleterious mutations exist in the heterozygous state before inbreeding, so that the n that is drawn for a parental individual is the number of heterozygous loci in that individual. n₁ and n₂ for an individual from the inbred progeny generation are determined in a similar fashion as before except that n is now randomly determined from a normal distribution instead of a Poisson distribution. The means and variances of fitness for the two generations are then computed, and our estimators, Equation 8aEquation 8bEquation 8c which assume no epistasis, are applied to the data.

Comparing estimation based on ²_T(I) and on ²_w(I):
We compared estimation by employing the information of the total genetic variation [²_T(I)] in the inbred generation vs. that of the genetic variation of the mean of inbred families [²_w(I)] . This is demonstrated in the case of selfing. We investigated under variable mutation effects in the presence and absence of lethals (Table 5) under two different experimental designs.

View this table: In this window In a new window   Table 5. Comparison of estimation based on 2T(I) vs. that based on 2w(I)

Estimation when genotypic values are measured with error:
In the simulations discussed above, genotypic values are assumed to be known without error. In this case, sampling error of estimates comes only from random sampling of outcrossed and inbred genotypes. In reality, this would require that each genotype be clonally replicated and assayed a very large number of times, since polygenic traits are usually expressed with some environmental variance (FALCONER and MACKAY 1996 ). In Table 6, we consider the estimation by accounting for additional effects of finite clonal replicates for each genotype on the sampling error. The results for full-sib mating are presented. We examine the situation in which the broad-sense heritabilities (H²) of fitness are 0.20, 0.40, and 0.60, respectively, in the parental generation. The environmental variance (including random measurement error and developmental instability) for fitness is defined as ²_e = , where ²_G is the genetic variance of fitness defined by Equation 2. Individual fitnesses are then determined by their genotypic values as described earlier, plus a random environmental deviation drawn from a normal distribution with zero mean and variance ²_e . Estimates of genetic variances for fitness are obtained by conducting one-way ANOVA on the simulated data in the parental and offspring generations, respectively, with genotypes as main, and clonal replicates as random, effects.

	RESULTS

TOP ABSTRACT THEORY COMPUTER SIMULATIONS RESULTS DISCUSSION LITERATURE CITED

Estimation under constant mutation effects:
The parameter estimates for s, h, and U are almost always unbiased with small sampling errors (Table 1). The only exception is when h is very high (h = 0.4), higher than all previously reported h estimates that range from 0.07 to 0.35 (DENG and LYNCH 1997 ). Then has large sampling variance. Estimation of s, h, and U under selfing is better than under full-sib mating, which in turn is better than under half-sib mating (as reflected by SD of the estimates). This is because the magnitude of the change of mean and genetic variance upon inbreeding is larger with higher degree of inbreeding. The estimates obtained from the repeated simulations are consistent with normal distributions (Kolmogorov-Smirnov test, P > 0.50; SOKAL and ROHLF 1995 ), thus SD is employed to reflect the sampling properties of the estimation throughout. The improvement of estimation with an increased degree of inbreeding is small when h is small (h = 0.2). The improvement is very dramatic for U estimation when h (h = 0.4) and U (U = 1.5) are large. Data not shown for simulations where s = 0.050 revealed the same conclusions.

Estimation under variable mutation effects:
All the estimates are biased (Table 2). The bias is relatively small when is small and increases with an increasing . The simulated parameters roughly cover most of the previous experimental estimates. Under the simulated parameters, ranges from ~2 to 3, ranges from ~0.35 to ~0.90 , and ranges from ~0.5 U to 0.8 U. Again, as with constant effects, estimation of , , and U under selfing is better than under full-sib mating, which is better than under half-sib mating. The sampling variance decreases with an increasing degree of inbreeding for estimation, while the bias remains roughly constant. Full-sib mating can generally achieve reasonably good estimates in terms of sampling variance.

Estimation with lethal mutations present in the genome:
The presence of rare lethal mutations (in the simulations shown, an expected number of 0.50 per individual) causes the estimates of to inflate by a factor of ~8, and estimates of U and to decrease by factors of ~2 and 4, respectively. In practical applications of our proposed technique, this type of problem can perhaps be minimized by eliminating individuals that are homozygous for lethals from the final analyses. This is a protocol similar to that employed in mutation-accumulation experiments (MUKAI et al. 1972 ). By dropping inviable inbred progeny (homozygous for lethal mutations) from analyses, the estimation can be greatly improved, even better than when there are no lethals (Table 3). This reflects the common practice of sampling conditional on hatch or birth, etc.

Estimation with epistatic mutation effects:
In general, the biases in estimates of U, h, and s are quite small provided the contribution of epistatic effects to fitness is on the order of <10% (Table 4). With reinforcing epistasis, h and U tend to be underestimated, whereas s tends to be overestimated. When the ratio approaches one, so that synergistic epistasis halves the average fitness of individuals relative to that expected in the absence of epistasis, the bias becomes more substantial. Even with strong epistasis, the estimation of h is altered only slightly and the estimates of U are not downwardly biased by more than ~30%, although the estimates of s can be too high by a factor as large as ~5. Overall, the results suggest that epistasis must be quite strong for our estimation to generate widely unrealistic estimates.

Comparing estimation based on ²_T(I) and on ²_w(I):
When relatively many (10) selfed progeny from each family are sampled (experiment A; Table 5), estimation based on ²_T(I) is about the same as that based on ²_w(I) , but has relatively smaller sampling variance. When only a few (2) selfed progeny from each family are sampled (experiment B), estimation based on ²_T(I) is much better than that based on ²_w(I) , because of both smaller sampling variance and smaller bias. This is consistent with our previous prediction (DENG and LYNCH 1997 ). When based on ²_T(I), the estimation does not change much for experiments A and B, though the sampling effort is much smaller in experiment B. However, when based on ²_w(I) , the estimation is much better for experiment A than for experiment B, due to the better estimation of the mean fitness of larger selfed families. As pointed out in the THEORY section, estimation of ²_w(I) is subject to two sources of sampling error, one being the number of inbreeding families sampled from populations and the other being the number of inbred progeny sampled from each inbreeding family. However, estimation of ²_T(I) is subject to only one source of sampling error, i.e., the number of inbreeding families sampled from populations. Thus, the estimation via ²_T(I) is indeed more powerful most of the time, especially when many inbred progeny are simply not available for each inbred family.

Estimation when genotypic values are measured with error:
The higher the H² (Table 6), the more genotypes (K) sampled, or the more replicates (R) cloned for each genotype at assay, the better the estimation, as reflected by the SDs. The bias remains roughly constant with different experiments of different sample sizes. When H² is reasonably high (>0.40), experiments that employ 100 outcrossed parents and 100 inbred progeny (each from different full-sib matings), with each genotype having at least 10 replicates, can achieve estimation reasonably well. In these experiments, ~2000 individuals need to be assayed. Generally speaking, for a fixed sample size for assay, increasing K can improve estimation more efficiently than increasing R. Even with relatively low H² (0.20), experiments that employ 150 outcrossed parents and 150 inbred progeny (each from different full-sib matings), with each genotype having at least 10 replicates, can achieve estimation reasonably well. In these experiments, ~3000 individuals need to be assayed.

As a specific example, assume that one can measure fitness of 2000 individuals. Then an actual experiment could be roughly as follows: Sample 100 random outcrossed genotypes from the outcrossing population under study. For each of them, sample one full sib to produce 100 full-sib pairs. Mate these 100 full-sib pairs to generate 100 inbred progeny genotypes. Clonally replicate the 100 random outcrossed genotypes and the 100 inbred genotypes to generate 10 clones for each of the genotypes. Then analyses can be performed (DENG and LYNCH 1996 , DENG and LYNCH 1997 ; DENG and FU 1998A ) to estimate the mean and genetic variation in the parental and inbred offspring generations and the mutation parameters can be estimated by Equation 8aEquation 8bEquation 8c. The dependence of the precision on such an experiment is indicated by footnote a in Table 6.

	DISCUSSION

TOP ABSTRACT THEORY COMPUTER SIMULATIONS RESULTS DISCUSSION LITERATURE CITED

In this article, we extended the approach of DENG and LYNCH 1996 , DENG and LYNCH 1997 to any form of inbreeding to characterize deleterious mutations in outcrossing populations. This extension greatly widens the taxa range of outcrossing populations in which characterizing deleterious mutations is feasible. The statistical properties, robustness, and statistical power of our extension are also investigated under several biologically plausible situations. In addition, we compared the estimation under different degrees of inbreeding and two different experimental designs and data analyses. It is revealed that estimation using the total genetic variation ²_T(I) in the inbred generation is generally more efficient than employing the genetic variation among the mean of inbred families ²_w(I) . Additionally, a higher degree of inbreeding employed in experiments yields a higher power for estimation. Our estimation is fairly robust in the presence of synergistic epistasis. For the estimation and experimental design that is different from that in DENG and LYNCH 1996 , the simulation results here on the magnitude and direction of estimation bias may provide a basis for accurate inferences of deleterious mutations. Simulations accounting for environmental variance of fitness suggest that, under full-sib mating, our extension can achieve estimation reasonably well with sample sizes of only ~2000–3000 for assay. These sample sizes should be well within the capacity of many laboratories.

In this article, we focus on estimation that employs experimentally measurable information such as mean and genetic variance in outcrossed and inbred generations. If external knowledge exists on the variation and covariation of h_i and s_i of mutation effects, improved (less biased) estimation accounting for the variation and covariation of h_i and s_i may be obtained. This was done in DENG and LYNCH 1996 for selfing experiments. However, the utility of such estimation is limited because our knowledge on variation and covariation of h_i and s_i essentially does not exist and may be much harder to acquire than to estimate the mean of h_i and s_i. Recently, we developed a method to approximately estimate the variation of h_i in an outcrossed population without having to construct homozygous lines (DENG 1998 ). However, methods to quantify the covariation of h_i and s_i do not exist at present, and the statistical performances of quantifying the variation of s_i from mutation accumulation data (KEIGHTLEY 1994 ) may be poor, as suggested by our preliminary investigations (DENG et al. 1998A , DENG et al. 1998B ). Therefore, estimation incorporating external knowledge of variation and covariation of h_i and s_i is not presented here due to their minimal utility, although it is straightforward to work it out. By employing a more complex design and using more information from the data, estimation that may not be biased by variable mutation effects in both outcrossed and selfed populations can hopefully be developed; furthermore, the covariation between h_i and s_i may be quantified (H.-W. DENG, unpublished results).

One crucial assumption of the estimation developed in this article is that the variation of fitness is maintained by mutation-selection (M-S) balance. This assumption underlies all the previous estimation approaches applied to natural populations (MORTON et al. 1956 ; CHARLESWORTH et al. 1990 ; DENG and LYNCH 1996 , DENG and LYNCH 1997 ). Despite tremendous efforts (e.g., HOULE 1989 , HOULE 1994 ; HOULE et al. 1996 ; CHARLESWORTH and HUGHES 1997 ; DENG et al. 1998), the extent to which this essential assumption is valid is generally unknown. A critical question is how robust the estimations are with different degrees of violation of the M-S balance assumption. Extensive studies have recently been performed (J.-L. LI, J. LI and H.-W. DENG, unpublished results). They have revealed that violation of M-S balance may not be as substantial as envisioned. Under variable dominance mutation effects, the estimation bias is actually reduced, with overdominance mutations maintained by balancing selection present in the genome.

Currently, there are several different approaches to characterize different aspects of deleterious genomic mutations (DENG 1998 ; DENG and FU 1998A , DENG and FU 1998B ). Traditional mutation-accumulation experiments used to be implemented with tremendous labor and time, which may have been far from necessary. If designed properly, mutation-accumulation experiments can be executed much more efficiently with much reduced time and labor (DENG et al. 1998A , DENG et al. 1998B ), so that mutation-accumulation may be adopted by many more empiricists. Under realistic situations, all the current estimations are biased. DENG and LYNCH's (1996) original procedure (estimation via genetic variation among the means of selfed families) generally is statistically better than other currently available estimation approaches (DENG and FU 1998A ). Our extension (estimation via the total genetic variation in the inbred progeny generation; DENG and LYNCH 1997 ) is shown here to be even more powerful than our original procedure (DENG and LYNCH 1996 ). The extension in DENG and LYNCH 1997 is here further extended to any inbreeding experiments in outcrossing populations. Different estimation approaches have different peculiar assumptions that may be difficult to validate in particular experimental settings (DENG and FU 1998A ). In addition, besides the statistical properties, different approaches have different advantages and drawbacks in practice and are best applicable to different organisms and in different situations. The estimates obtained by different approaches in different organisms can be crosschecked and, hopefully, will eventually resolve the issues concerning the genomic mutations.

	ACKNOWLEDGMENTS

I thank Professor M. Lynch for helpful comments on the manuscript and his years of advice, encouragement, and continuous support. I also thank Professor M. Slatkin, Professor A. Kondrashov, and an anonymous reviewer for helpful comments that helped to improve this article. Graduate students J. Li and J.-L. Li helped in running some simulations for this article. This work was partially supported by a grant from National Institutes of Health R01 AR45349 and a Health Future Foundation grant of Creighton University, Nebraska.

Manuscript received April 10, 1998; Accepted for publication July 13, 1998.

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