ANALYSIS OF COEFFICIENTS OF DOMINANCE IN OHNISHI'S EXPERIMENT

In Ohnishi's experiment, mutations were allowed to accumulate for 40 generations in 136 copies of a single original chromosome II of D. melanogaster. The homozygous viability (ν_ii) of each copy i was assayed at different generations as the percentage of wild adults +_i/+_i in the offspring of a cross between five Cy/+_i females and five Cy/+_i males in a half-pint bottle. Viability of heterozygotes (ν_ij) for random pair chromosomes i, j (repulsion heterozygotes) was analogously assayed as the percentage of wild +_i/+_j adults in crosses between five Cy/+_i females and five Cy/+_j males. Furthermore, the viability of heterozygotes carrying a copy of the original chromosome II (+_c), maintained as a control, was also assayed as the percentage of wild adults +_i/+_c from crosses between five Cy/+_i females and five Cy/+_c males. These are assumed to be coupling heterozygotes. However, the control +_c chromosome was maintained by a few single-pair matings between +_c/+_c individuals (O. Ohnishi, personal communication), and the extent to which it might have accumulated new mutations is unknown.

In all cases, the viability of each chromosome +_i was assayed by reference to that of the curly progeny. The interpretation of the results could be greatly complicated if deleterious mutational effects were expressed in the Cy/+_i heterozygotes (Cockerham and Mukai 1978). It was not possible to test for this possibility with Ohnishi's data. However, we have experimental results suggesting that mutations that accumulated in originally isogenic full-sib MA lines, showing deleterious effects in homozygous condition, do not show correlated effects on Cy/+_i heterozygotes (D. Chavarrías, A. García-Dorado and C. López-Fanjul, personal communication).

From now on we refer to all viability measurements relative to the original average ν¯0 , which had a value of 31.71 on Ohnishi's viability scale. Thus, denoting by + the original chromosome and by m a copy of it carrying a new mutation, relative viabilities for genotypes +/+, +/m, and m/m are 1, 1 − hs, and 1 − s, where s and h are the mutation's homozygous deleterious effect and coefficient of dominance, respectively.

Estimation of the average coefficient of dominance from Ohnishi's experiment: Ohnishi (1977c) obtained estimates of the average coefficient of dominance of new mutations by comparing homozygous and heterozygous viability declines, h¯ws=ν¯0−ν¯ij(ν¯0−ν¯ii)+(ν¯0−ν¯jj)=∑sh∑s, (1) where i and j indicate two given MA lines, j = c for coupling heterozygotes, and the summation is over all the accumulated mutations. As stated in the right-hand side of the expression, this is an estimate of the average h of mutations weighted by their homozygous effect, s (Mukai 1969). Estimates obtained by Ohnishi (1974) at different generations for the QN chromosomes (i.e., those with a viability larger than 20/31.71 = 0.63) are given in Table 1. We have computed standard errors (SE) for h¯ws using the approximation for the standard error of the ratio of two variables and assuming that ν¯0 and ν¯c are constant. Averaging over generations and schemes (coupling and repulsion) gives h¯ws=0.45 . The average estimate for non-QN chromosomes (ν_ii < 0.63 excluding those carrying lethals) is 0.085, and that for lethals is 0.046. It is also worthwhile to note that deleterious mutations in chromosomes with homozygous viability >0.95 (of which 29 lines remained by the end of the experiment) tended to appear as partially dominant, with h¯ws=0.74 and 0.67 for coupling and repulsion heterozygotes, respectively.

Additional information can be obtained from the regression of heterozygous on homozygous values (Mukai 1969). For repulsion heterozygotes, if x is the sum of the homozygous viabilities of two parental lines and y is the heterozygous viability, the covariance for a large set of n lines is σ(x,y)≈(Σkhksk2+Σlhisi2)∕n, where summation is over all mutations accumulated, k and l refer to mutations at maternal and paternal chromosomes, respectively, and mutation is assumed to be nonrecurrent. For coupling heterozygotes, the maternal chromosome is the control one and does not contribute to the covariance. Thus, x is the homozygous viability of the MA paternal chromosome, and y is that of the heterozygote. Then, analogously to the repulsion case, σ(x,y)≈∑khksk2∕n , where k stands for the mutations accumulated in the paternal MA chromosome. The between-line component of variance for the homozygous viability is σb2≈∑ksk2∕n=∑lsl2∕n . Thus, an estimate of the average degree of dominance weighted by s² can be obtained in any particular generation as h¯ws2=σ(x,y)zσb2=∑s2h∑s2, (2) where z is the number of homozygotic parental values used (z = 1 for coupling heterozygotes and z = 2 for repulsion heterozygotes). None of these estimates are given in Ohnishi's articles. However, we were able to obtain them from data provided in his Ph.D. thesis (Ohnishi 1974). The correlations r(x, y) between the heterozygous average viability (y) and that (x) of its homozygotic parent (coupling) or midparent (repulsion) are given for the group of QN homozygotic MA lines at generations 10, 20, 30, and 40 (Ohnishi 1974, Table 15). σ_x and σ_y can be estimated from the ANOVA given by Ohnishi (1974, Tables 2a and 12a) as σ^x=MSBhom∕mz , σ^y=MSBhet∕m , where MSB_hom and MSB_het are the between-line means of squares in the ANOVA for homozygotic and heterozygotic assays, respectively, and m is the number of observations per line (three for σ^x and two for σ^y ). Using these values, we can obtain σ^(x,y)=r(x,y)σ^xσ^y for QN chromosomes.

A minor problem is that MSB_het should be the between-line mean of squares for QN heterozygotes. However, the only ANOVA given by Ohnishi for the heterozygous viability refers to nonlethal chromosomes (instead of QN chromosomes). Using this ANOVA, we obtain an overestimate of σ^y and, therefore, of σ^(x,y) . However, the MSB_het for nonlethal chromosomes increased only slightly during the MA process, and the corresponding between-line component of variance was always very small and nonsignificant. Therefore, the bias should be very small.

Using σ^(x,y) and Equation 2, we have computed the estimates for h¯ws2 given in Table 1. Standard errors for the regression of heterozygous on homozygous values were computed using standard equations from linear regression theory. Estimates of h¯ws2 are considerably and significantly smaller than the h¯ws estimates obtained by Ohnishi using Equation 1. As stated above, averaging estimates from different generations gives greater weight to early mutations. Therefore, here we refer to the estimates at generation 40, which are the most informative because more mutations will have accumulated and because they have the smallest standard errors. These are h¯ws=0.348 and h¯ws2=0.065 , which are significantly different (P < 0.001).

All estimates for h¯ws2 in Table 1 were also directly calculated as regression coefficients from the grouped data in Tables 13a and 14a of Ohnishi (1974). The calculations were corrected for the use of phenotypic, instead of genotypic, variances in the regression estimates (Caballeroet al. 1997). The results were very similar to those presented in Table 1 from the correlations and are not given.

At first, it is tempting to explain the substantial difference between both types of estimates in Table 1 by invoking that there is a negative correlation between the homozygous effects and the degree of dominance of new mutations, in which case we should expect h¯ws>h¯ws2 . A negative correlation between s and h is a very likely possibility when severely deleterious mutations are involved (see review by Caballero and Keightley 1994) and, in fact, a sharp decrease in the average h is observed when estimates include non-QN chromosomes (see above). However, the bias of an estimate from Equation 2 is not necessarily large in the analysis of Ohnishi's lines because most severe mutations were excluded from the analysis. After 40 generations, 78 out of the 80 QN chromosomes assayed in heterozygous condition had homozygous viability >0.85.

We suggest that an additional explanation could be, at least in part, responsible for the difference: If some of the change in homozygous and heterozygous average viabilities were due to nonmutational causes, as has been recently suggested (Keightley 1996; García-Dorado 1997; García-Doradoet al. 1999), the estimate from Equation 1 would be biased toward 0.5. Imagine a hypothetical extreme situation in which all the observed decline is due to unknown nonmutational causes affecting both homozygotes and heterozygotes. Then, no between-line variance would be detected (due to the isogenic origin of the lines), and the decline would be ascribed to a large number of mutations with very small deleterious effects (see the Bateman-Mukai method in Mukaiet al. 1972). Using Equation 1, additive gene action (h¯ws=0.5 ) would be inferred for this spurious class of mutations. Since the viability decline is not used in Equation 2, such a bias would not occur with the estimates obtained by the regression method.

An indication that a nonmutational decline in viability might have occurred in Ohnishi's lines comes from the observed pattern of decline in viability. Figure 1 gives the average viability against generation number of the homozygotes for the control and QN lines, as well as for coupling and repulsion heterozygotes. In all cases the viability decline was much larger in the first half of the experiment than in the second half. In contrast, the proportion of lethals steadily increased during the experiment (Figure 2), being consistent with a steady mutation rate. Also, the between-line variance for QN homozygotes increased at a roughly constant rate (Figure 2), which suggests that the slower final decline in mean viability is not due to diminishing epistasis, to a reduction in the mutation rate, or to a relaxation of the environmental conditions.

• Download figure
• Open in new tab
• Download powerpoint

Figure 1.

Viability decline observed in Ohnishi's (1974) experiment.

Thus, an important fraction of the early viability decline could have been nonmutational. The cause of this hypothetical nonmutational decline is unknown, but it might be due to a simultaneous viability increase in the Cy chromosome by reference to which viability was assayed (Keightley 1996; García-Dorado 1997), to an increase in the ability of the experimenter to detect the Cy phenotype (Fryet al. 1999), or to an environmental effect.

Alternatively, the nature of deleterious mutations could vary between the first and second half of the experiment, with a large initial rate of mutations with very small deleterious effects causing a larger viability decline without appreciable additional increase in variance. It has been suggested (Keightley and Eyre-Walker 1999) that transposition could provide a mechanism for this behavior (see discussion).

Finally, another estimate of the average coefficient of dominance from Ohnishi's experiment, which we denote h¯sd , can be obtained using the ratio of heterozygotic to homozygotic rates of increase of the genetic standard deviations (both given by Ohnishi 1974w, Table 12a). This ratio estimates the square root of the average of h² weighted by s² and, therefore, it is difficult to interpret. If s were constant but h were variable, h¯sd would be an upwardly biased estimate of the true h¯ . In addition to this, if s were variable, a downward bias is expected when there is a negative correlation between h and s. No estimates of this type could be obtained for QN chromosomes. For nonlethal chromosomes we obtained h¯sd=0.058 (variance for coupling and half-variance for repulsion pooled), which suggests small h values. Furthermore, it should be noted that both the rate of increase in variance (1.09 × 10⁻⁶ ± 6.26 × 10⁻⁶) and the genetic variance at generation 40 (1.19 × 10⁻⁴) for heterozygotes of nonlethal chromosomes were nonsignificant, pointing also to reduced values of h.

• Download figure
• Open in new tab
• Download powerpoint

Figure 2.

Percentage of lethal mutations and variance between QN lines in Ohnishi's (1974) experiment. (- - -) Percentage of lethals, (—) Vg × 10,000.

We carried out analytical work and simulations to investigate the possibility that the larger estimates of dominance obtained from the ratio method (Equation 1) were biased due to the hypothesis of a nonmutational decline in viability. These analyses and simulations are explained in what follows.

Parameter estimations according to the two hypotheses: Two different parameter estimations can be made. Bateman-Mukai (BM) estimates (Bateman 1959; Mukaiet al. 1972) can be obtained by considering the observed drop in viability and, therefore, assuming that this drop is not affected by nonmutational factors. Minimum distance (MD) estimates (García-Dorado 1997) can also be obtained that are unconstrained by the observed change in mean and, therefore, would be appropriate whether or not nonmutational factors were involved in the observed decline.

Model based on Bateman-Mukai estimates: For QN chromosomes, relative viability declined at a rate ΔM = 0.00173, and the corresponding rate for the increase of between-line variance was ΔV = 0.594 × 10⁻⁴. From these, a BM lower bound for the rate of mutation per chromosome II and generation is λ (QN) ≥ 0.05, and an upper bound for the average homozygous deleterious effect is E(s) (QN) ≤ 0.034 (Ohnishi 1974). Considering that chromosome II accounts for ~40% of the Drosophila genome, these results suggest an important rate of mildly deleterious mutation, although smaller than those obtained in previous MA experiments by Mukai (1964) and Mukai et al. (1972), which gave λ (QN) ≥ 0.14 and 0.17, respectively. The above estimates imply that, by generation 40, QN MA chromosomes II would carry on the average >2 deleterious mutations and that the proportion of deleterious free QN lines is expected to be <13%, assuming a Poisson distribution of mutations.

The BM estimates of λ and E(s) are only for QN chromosomes. However, to obtain quantitative predictions from these results (or to simulate the mutational process, see below) we need a more explicit model, specifying the distribution of effects for all deleterious mutations (QN and non-QN, lethals excluded), as well as a model of dominance. Homozygous deleterious effects are assumed to be gamma distributed (shape parameter α, scale parameter β, expected value α/β), which allows us to handle different kurtosis. Noting that only 3 out of the 106 nonlethal chromosomes present at generation 40 were non-QN (which indicates a non-QN mutation rate of ~7 × 10⁻⁴) we assume λ = 0.05 for all deleterious mutations (QN and non-QN, lethals excluded). This value of λ, together with the ΔM and ΔV observed for nonlethal chromosomes (0.00208 and 2.28 × 10⁻⁴, respectively), leads to the values of E(s) and E(s²) presented in Table 2. In turn, these values determine α and β in the gamma distribution given in the same table [E(s) = α/β and E(s²) = α(α + 1)/β²]. Since this set of parameters is based on a BM estimate of λ, we refer to the model they describe as the BM model.

Model based on sharpened Bateman-Mukai estimates: Because the above BM bounds approach unbiased point estimates as the variance of s decreases, we also consider the bounds obtained for the class of QN chromosomes when a few outliers are excluded. The QN set contains 100 lines showing a compact histogram and 3 lines whose mean viability is clearly below this group (ν¯<0.8 , i.e., >3σν¯ below the general average ν¯¯ , where σν¯ is the standard deviation for mean viability in this compact group). Thus, BM bounds can be sharpened further by estimating the mutational rate from the change in mean and variance in the 100 lines belonging to the compact histogram (that we call QN^s lines). These give λ (QN^s) ≥ 0.12 and E(s) (QN^s) ≤ 0.013. This implies that, by generation 40, QN^s chromosomes carry on the average at least 4.8 deleterious mutations, and only 0.8% of the QN^s lines are expected to be deleterious-free. Following the same procedure as before, and using λ = 0.12, we analogously obtain the set of parameters presented in Table 2, which describe the BM^s model.

Model based on minimum distance estimates: An alternative set of parameters unconstrained by the observed decline in mean viability could be obtained from MD estimation to explain Ohnishi's MA results. MD estimates were obtained by García-Dorado (1997) for a viability index (twice the ratio of wild to Cy numbers), using Ohnishi's data of MA lines at generation 40. In this method, mutational effects on viability were assumed to be sampled from a gamma distribution reflected about zero, thus allowing the occurrence of mutations increasing viability. However, the estimate of the probability of these favorable mutations turned out to be zero.

Results for heterozygous effects, available from Ohnishi (1974), referred to the percentage of wild-type flies instead of to twice the ratio of wild to Cy numbers. Thus, MD estimates obtained by García-Dorado (1997) had to be rescaled for the rate of increase in variance of the percentage measurement given by Ohnishi. This was accomplished by using the appropriate scale parameter (β) in the gamma distribution of s to account for the mutational variance in Ohnishi's scale. These parameters describe the MD model and are presented in Table 2. The MD estimates imply that, by generation 40, a QN chromosome would carry 0.16 deleterious mutations on average, and ~85% of the nonlethal chromosomes would be deleterious-free.

The nonmutational viability decline at generation 40 (D) was estimated to be D = 0.056, and this value is considered later as a constant for both homozygotes and heterozygotes. The estimate was obtained from a comparison between the observed drop in viability and the expected one, assuming the MD model. It is strongly supported by the observation that the rate of decline from generations 0–20 exceeded by an amount of 0.00274 per generation that from generations 20–40 (see Figure 1). Assuming this difference is due to nonmutational causes, we obtain an overall D = 0.00274 × 20 = 0.0548, in close agreement with the MD estimate (0.056).

The final viability decay in the control line (D = 0.038) suggests a somewhat smaller nonmutational viability decline but, assuming the standard errors for the control evaluations were similar to those for the lines, this estimated decay was not significantly different from the MD estimate.

Predictions of the average coefficient of dominance for the three models: Ohnishi (1974) estimated h¯ws(QN)=0.348 at generation 40 using Equation 1 (Table 1). From this estimate we need to guess the kind of h values to be expected under each model (BM, BM^s, or MD) for mutants with different deleterious effects. To produce an inverse relationship between s and h, the latter is assumed uniformly distributed between 0 and exp(−ρs) (Caballero and Keightley 1994), where ρ is a constant chosen to give the desired unweighted arithmetic mean dominance (h¯ ) for the whole range of s values. This exponential model for h is used to account for the low dominance usually observed for rare severely deleterious mutations. The values of h¯ predicting the empirical h¯ws(QN)≈0.348 have been found numerically for each model and are given in Table 3, together with the corresponding numerical results for the weighted averages h¯ws and h¯ws2 , and with the empirical estimates.

Under the BM model, the unweighted overall average h for nonlethals should be h¯=0.44 to explain the empirical h¯ws (QN) estimate. However, the expected h¯ws2 (QN) is 0.302, which is considerably larger than the corresponding observed estimate (0.065) obtained in generation 40 (Tables 1 and 3). Similarly, using the BM^s-based model, we need h¯=0.47 to obtain an expected h¯ws (QN) about the empirical value, which again gives too large an expected h¯ws2 (QN) of 0.265. These calculations suggest that different weighting factors (s for h¯ws and s² for h¯ws2 ) are not wholly responsible for the difference between estimates from Equations 1 and 2.

For the MD model, the degree of dominance necessary to explain the observed average h¯ws2 (QN) at generation 40 was found to be h¯=0.20 (Table 3). In the absence of nonmutational viability decline, this gives h¯ws(QN)=0.112 , well below the value estimated from Equation 1. However, when the expected nonmutational decline in viability of 0.056 is considered, the estimate from Equation 1 becomes h¯ws2(QN)=0.381 , in agreement with the observed estimate. Alternatively, using the decline observed in the control line (0.038) as an estimate of the nonmutational viability decline, Equation 1 gives an estimate h¯ws(QN)=0.344 , again in good agreement with the empirical value. Therefore, the MD model with a nonmutational drop is able to explain both h¯ws (QN) and h¯ws2 (QN).

Table 3 also shows the results for all nonlethal (NL) chromosomes. The unweighted average for h is roughly the same as for QN chromosomes. The observed values of h¯ws2 (NL) were obtained as regression coefficients from the grouped data in Tables 13a and 14a of Ohnishi (1974). Observed estimates were considerably lower than the corresponding estimates for QN chromosomes. MD estimates were in close agreement with such low values.

The nonmutational viability decay would also explain the high estimates of h¯ws obtained by Ohnishi (1974) at different generations using the ratio method (Equation 1) for the set of lines with homozygous viability >0.95 (Table 4). Under the nonmutational decline hypothesis, these lines would be a nonrandom subsample of those carrying few or no deleterious mutations. Then, the observed viability decline on their heterozygotic crosses would equal the nonmutational decline (D) and, since ν_ii and ν_jj are chosen close to one, the estimates of h¯ws from Equation 1 would be upwardly biased. The expected value of this estimate was computed under the hypothesis of a nonmutational decline using the observed homozygous viabilities for MA chromosomes with viabilities >0.95 (Tables 13a and 14a in Ohnishi 1974) and substituting ν¯0−ν¯ij=D in Equation 1. We used the MD estimate of D (assuming that all the decline occurred linearly from generations 0–20, i.e., D = 0.028, 0.056, 0.056, and 0.056 at generations 10, 20, 30, and 40, respectively) and, alternatively, the D value estimated from the control (0.021, 0.033, 0.037, and 0.038, respectively). The predictions are shown in Table 4. In general, large values for h¯ws are predicted, in agreement with empirical estimates. The best fitting corresponds to the mean between both alternative approaches, suggesting that the true final nonmutational viability decay could be some value between 0.038 and 0.056.

Simulation results: To get the expected distribution for chromosome viabilities for homozygotes and heterozygotes and to check the above analytical results, we performed Monte Carlo simulations. We simulated a Poisson distribution of the number of mutations per chromosome, with average number equal to 40λ. Mutations accumulated over an original chromosome of viability ν¯0=1 . Viability deleterious effects were gamma distributed and were multiplicative across loci. Chromosome viabilities were analyzed in the (0, 1) scale, but using a log scale made no appreciable difference (data not shown). The model assumed for the dominance of mutations was the same as that used previously in the analytical method. We used the same sampling error obtained by Ohnishi (σe2=7×10−4 and σe2=9×10−4 for homozygotes and heterozygotes, respectively). These were obtained from the environmental variances in Tables 2a and 12a of Ohnishi (1974), divided by the number of observations per line and by the square mean viability at generation 0, i.e., 31.71. Since the heterozygous distribution given by Ohnishi included chromosomes that were lethal when homozygous, additional lethal mutations were simulated in the heterozygous case. These were assumed to occur with the rate λ_L = 0.00466 estimated by Ohnishi (1974) and with degree of dominance h_L = 0.02 (Simmons and Crow 1977). A total of 100 chromosomes were simulated and results were averaged over 100 replicates.

Table 5 gives the number of mutations per chromosome II and generation within different s intervals obtained from the simulated data, as well as the corresponding average coefficient of dominance. Both BM and BM^s models predict more frequent deleterious mutations than the MD model, the difference being much more important for small deleterious effects. The degree of dominance of mildly deleterious mutations (s < 0.05) is similar (h ≈ 0.45) under the three models. However, for the MD model, it decreases quickly for increasing deleterious effects.

The observed and simulated distributions for homozygous and heterozygous chromosome viabilities under the different models are presented in Figures 3 and 4, respectively.

Both BM and BM^s models failed to predict the shape of the average viability distribution of the homozygous and heterozygous chromosomes. Although sharpening BM estimates (i.e., reducing the variance of s by removing extreme lines) produces larger estimates of the rate of occurrence of mutations with small effect, BM and BM^s models gave very similar distributions for the average viability, both in the homozygous and in the heterozygous condition. This suggests that MA experiments often lack the power to estimate the mutation rate for very small deleterious effects (s < 0.001, see Table 5).

MD parameters (including the rate of nonmutational viability decline) were originally estimated from the observed homozygous average viability distribution. Thus, it is not surprising that the observed and simulated distributions for homozygotic viability agreed with each other to some extent. However, the model also adequately predicted (although with a slight shift to the left) the shape of the observed distribution of average viabilities in the heterozygous state, which was not used in MD estimation.

The simulations were also used to obtain estimates for the average coefficient of dominance analogous to the numerical values computed from analytical expressions and given in Table 3. Simulated and analytical estimates were in close agreement (data not shown).

We also computed h¯sd (the ratio of heterozygotic to homozygotic rates of increase in genetic standard deviations) from our simulated nonlethal chromosome lines. Using the BM or BM^s models, we obtained too large h¯sd values (0.533 or 0.430, respectively), while using the MD model, we obtained a small h¯sd=0.072 , in agreement with the empirical value (0.058) estimated from Ohnishi's data.

• Download figure
• Open in new tab
• Download powerpoint

Figure 3.

Observed (Ohnishi 1974) and simulated distribution of chromosome homozygous viabilities using parameter estimates in Table 2 (see text for explanations). BM, Bateman-Mukai model; BM^s, sharpened Bateman-Mukai model; MD, minimum distance model.

• Download figure
• Open in new tab
• Download powerpoint

Figure 4.

Observed (Ohnishi 1974) and simulated distribution of chromosome heterozygous viabilities using parameter estimates in Table 2 (see text for explanations). See Figure 3 legend for definitions of abbreviations.

A complication mentioned above and not considered in the previous analyses is the possible effect of mutations in Cy/+_i genotypes. The assumption so far has been that h = 0 for all mutations in Cy/+_i. Although this is supported by new experimental evidence (D. Chavarrías, A. García-Dorado and C. López-Fanjul, personal communication), we evaluated this possible effect by simulation. We assumed that mutations expressed the same effect on Cy/+_i as on +_i/+_j genotypes. Thus, the viability of a homozygote (ν_ii) was scaled by the heterozygous effects of chromosome i (ν_ci, where c is a chromosome with no mutations). The heterozygous viability (ν_ij) was scaled by the average heterozygous effects of chromosomes i and j ([ν_ci + ν_cj]/2).

The simulation estimates corresponding to those of Table 3 for h¯ws (QN) and h¯ws2 (QN) were 0.31 and 0.20 for BM, 0.31 and 0.18 for BM^s, and 0.43 and 0.04 for MD, respectively. Therefore, the effect of an expression of mutations on Cy/+_i genotypes was a general reduction in the estimates of h (except when a nonmutational decline is accounted for; cf. Table 3), and the BM and BM^s models were still unable to explain all the observations. Furthermore, an expression of mutations in the Cy genotype had a very small effect on the distributions of homozygous and heterozygous chromosome viabilities presented in Figures 3 and 4 (data not shown), so the qualitative conclusions reached above are not altered. Finally, it is worth noting that an expression of mutations in the Cy genotype implies that Ohnishi's estimates of h¯ws from the ratio method (which are already high) are underestimations, so the true mean h would be exceedingly high.