MODEL AND METHODS

Gene action and contribution of mutations: A population of N diploid individuals, with random mating and at Hardy-Weinberg equilibrium, is assumed. It is also assumed that the mutation rate per locus is so low that at most two alleles are segregating per locus. Mutations in a diploid individual are assumed to have effects on a metric trait z, with a being the difference in value between homozygotes, and pleiotropic effects on fitness, with s being the difference in fitness between homozygotes. The haploid genome mutation rate is λ and the mutational variance in the quantitative trait is defined as V_m = ½ λE(^a2). There is neither linkage nor epistasis. Linkage disequilibrium between two segregating loci may be common but is unlikely to be an important factor as long as mutations remain at low frequencies in the majority of cases. Overdominance is also ignored. Although associative overdominance on some loci will appear due to a positive correlation in homozygosity between loci, it is significant only within populations with substantial inbreeding (Lynch and Walsh 1998; Charlesworth and Charlesworth 1999; Wang and Hill 1999).

Let the frequencies of the wild-type allele (A) and the mutant allele (a) at a given locus be 1 – x and x, respectively, and so the frequencies of genotypes AA, Aa, and aa assuming Hardy-Weinberg proportions are (1 – x)², 2x(1 – x), and x². If the dominance coefficients of the mutational effect on the trait z and pleiotropic effect on fitness are h′ and h, respectively, values for the trait z of the three genotypes are 0, a h′, and a, and their pleiotropic effects on fitness are 1, 1 – sh, and 1 – s. Under the joint-effect model of pleiotropic and real stabilizing selection (Zhang and Hill 2002) with weak selection (mean fitness W̄ = 1 – V_G/(2V_s,r) ≈ 1), the change in gene frequency resulting from one generation of selection is approximated as Δx = – x(1 – x) s̃L/2 with the overall fitness effect s∼=2s[h+(1−2h)x]+a24Vs,r{4h′2−2x[1−(1−2h′)2(1−2x)2−4(1−2h′)(1+h′)(1−x)]}. (1) The genetic variance in the trait z affected by n independent loci is given as VG=∑i=1nVG,i≡∑i=1n(VA,i+VD,i), (2) with the additive variance, V_A,i, and the dominance variance, V_D,i, contributed by locus i as VA,i=2xi(1−xi)(ai2∕4)[1+(2hi′−1)(1−2xi)]2, (3) VD,i=[ai(2hi′−1)xi(1−xi)]2 (4) (Falconer and Mackay 1996), respectively. The strength of apparent stabilizing selection can be measured as the regression of fitness on squared deviation of the trait value (z) from the optimum (z̄ and evaluated as Vs,t=−VG2∕[2Cov(w,(z−z¯)2)] (5) (Barton 1990; Keightley and Hill 1990; Zhang and Hill 2002).

In Equation 5, the covariance of relative fitness and squared deviation, (Cov(w, (z – z̄)²), can be partitioned into two parts: that due to real stabilizing selection, V_G2/2V_s,r, and that due to the pleiotropic effect on fitness, Cov_p, where Covp=−∑i=1n[2xi(1−xi)hisi(z¯i−hi′ai)2+xi2si(z¯i−ai)2−s¯iVG,i]. (6) The variance of squared deviations can be decomposed as VG2=m4+2VG2 with the fourth moment under selection, m4=∑i=1n{[(1−xi)zi2−4+2xi(1−xi)(z¯i−hi′ai)4+xi2(z¯i−ai)4]−3VG,i2}. (7) In the above expressions z¯i=[(2hi′−1)(1−xi)+1]xiai is the mean effect on the trait and s̄_i = [(2h_i – 1)(1 – x_i + 1]x_is_i) is the mean pleiotropic effect on fitness of locus i.

Distributions of homozygous effects and dominance coefficients of new mutations: Although fine-scale information is still lacking, empirical data (Mackayet al. 1992; Keightley 1994; Lymanet al. 1996; García-Doradoet al. 1999; Garcia-Dorado and Caballero 2000; Chavarriaset al. 2001; Hayes and Goddard 2001; Mackay 2001) indicate that the distributions of effects of new mutations on both fitness and the metric trait are leptokurtic, and mutational effects on the trait are more leptokurtic than their pleiotropic effects are on fitness. As in previous studies (Keightley and Hill 1990; Zhang and Hill 2002), the distribution of mutational effects on the metric trait is assumed to be symmetrical about a = 0, and only deleterious mutations on fitness are assumed to occur, in accord with the classical view (Falconer and Mackay 1996). The effects of mutations, |a| and s, were sampled from a gamma distribution or a function thereof: the “squared gamma distribution” and the “square-root gamma distribution,” where, respectively, √s and s², for example, have a gamma distribution (cf. Zhanget al. 2002). Note that the reflected square-root gamma (½) distribution is the normal (Gaussian) distribution. Here variants of the gamma distribution are employed to show that different distributions that possess the same variance and kurtosis can induce quite different predictions of V_G (see results below).

Dominance coefficients of new mutations are assumed to be either constant or variable across loci. Analyses of available experimental data suggest that dominance coefficients decrease with the size of homozygous mutational effects (Mukaiet al. 1965; Simmons and Crow 1977; Charlesworth 1979; Lopez and Lopez-Fanjul 1993), and the mean dominance coefficient is distributed approximately as h̄ = exp(–Ks)/2 (Caballero and Keightley 1994; Denget al. 2002), which is in rough accord with the few available data (Mackayet al. 1992; Garcia-Dorado and Caballero 2000). As did Caballero and Keightley (1994), we assume dominance coefficients for the pleiotropic effect on fitness are uniformly distributed in the range 0 < h < exp(–Ks), where the constant K is determined so that for a given distribution of s, the average dominance coefficient is h̄. For example, if s follows a gamma (β₂) distribution with mean s̄_P, then K = (β₂/s̄_p)[(2h̄)^–1/β₂ – 1]. Similarly, h′ is assumed to be uniformly distributed in the range 0 < h′ < exp(–K′|a|). Further it is assumed that h ≤ h′ if the degree of dominance varies, as h and h′ may be correlated (Caballero and Keightley 1994). In this study, the following methods are used to compute V_G and V_s,t, as mutual checks.

Single-locus Monte Carlo simulation: Although apparent stabilizing selection experienced by the trait acts on the alleles at all segregating loci, the above analysis shows that its strength can be computed by summing the impact of each locus separately. Using the above basic expressions, we can simulate the process for each segregating mutant until its fixation in or loss from a population. For each new mutant with properties (a, h′, s, h) sampled from a quadrivariate distribution P(a, h′, s, h) as described above, its initial frequency x₀ is set to 1/(2N), where N is the actual size of the population. We then calculate Equations 1, 2, 3, 4, 5, 6, 7. In the next generation, the expected mutant frequency is given by x₁ = x₀ + Δx and the actual frequency is sampled from a binomial distribution with mean x₁ (Presset al. 1989). Equations 1, 2, 3, 4, 5, 6, 7 are recalculated and values of V_A, V_D, m₄, and Cov_p are added to those of the previous generation to compute the lifetime contributions. The process continues until the actual frequency or its expectation reaches the bound 0 or 1. A large number of mutants (e.g., 10⁸) were sampled and the averages of all those quantities were taken and then multiplied by 2λN (the expected number of new mutations each generation) to obtain expected values of V_G, V_D, m₄, Cov_p, and V_s,t at the steady state of accumulation and loss of mutations.

Individual-based Monte Carlo simulation: As a check for the above single-locus simulation, a multiple-locus and individual-based simulation procedure, modified from Keightley and Hill (1983, 1988), was used. The population was started from an isogenic state and allowed to reach equilibrium by ignoring the first 6N generations. Each generation the sequence of operations was mutation, selection, mating, and reproduction. The fitness of individual i was assigned as w_i = 1 – [Σ_js_ij + (z_i – z̄)²/2V_s,r] where z_i = Σ_ja_ij is the value of the trait and z̄, the population mean of the trait, approximates the optimum. If 0 < w_i ≤ 1, then the chance that individual i was chosen as a parent of the next generation was proportional to w_i; if w_i ≤ 0, it had no offspring. It should be noted that the population generated by this scheme of selection maintains mean fitness around a constant value from generation to generation, comparable with Haldane's (1937) law, i.e., exp(–2λ) for large populations, whereas in the model of Keightley and Hill (1983, 1988) mean fitness always keeps falling. For each generation of the equilibrium population, V_G, V_G2, and Cov(w, (z – z̄)²) were computed and averaged to estimate their means.

Diffusion approximations: Kimura's (1969) diffusion theory was applied under the infinite independent loci model. Assuming no epistasis, the density function of the stationary distribution of allele frequency at MSB, φ(x), is given by Equation 37 of Kimura (1969). In contrast to Kimura (1969), here the overall selective coefficient s̃ of a mutant depends on its frequency, dominance coefficients, the trait effect, and the pleiotropic effect on fitness (see Equation 1). The expectation I_f of an arbitrary function f(x) with respect to the equilibrium distribution φ(x) was obtained by integration of Equation 18 of Kimura (1969). As in Caballero and Keightley (1994), mutational effects were sampled from P(a, h′, s, h), and the equilibrium value of f(x) assuming no overdominance was obtained by integration of ∫−∞∞∫01∫0∞∫01IfP(a,h′,s,h)dadh′dsdh. (8) Monte Carlo integration was used to compute Equation 8.

ANALYTIC APPROXIMATIONS FOR INFINITE POPULATION

If the population size is large and mutations are not completely recessive or neutral (i.e., hs > 0 or h′ a > 0) such that 2NE(s̃) » 1, mutant frequencies then main very low and items of O(x³) or higher in Equations 1, 3, 6, and 7 can be ignored, reducing to s∼=2sh+(2h′a)2∕4Vs,r (9) VG=∑i=1n2xi(1−xi)(2hi′ai)2∕4 (10) Covp=−∑i=1n2xi(1−xi)(1−2xi)(2hisi∕2)(2hi′ai)2∕4 (11) m4=∑i=1n2xi(1−xi)(2hi′ai)4∕16. (12)

With this rare allele assumption, only the heterozygous effects of mutants, hs and h′a, are relevant to the genetic processes that control the equilibrium genetic variance. If 2NE(s̃) » 1, diffusion theory shows that the asymptotic expectations of the functions 2x(1 – x) and 2x(1 – x)(1 – 2x) approach 4λ/s̃ (Kimura 1969; Zhanget al. 2002). Thus the genetic variance V_G, the covariance of relative fitness and squared deviation due to pleiotropic effect Cov_p, and the fourth moment under selection m₄ can be calculated through expression (8) using I_f = 4λV_s,r[(2h′a)²/4V_s,r]/s̃, I_f = 4λV_s,r(2hs/2)[(2h′a)²/4V_s,r]/s̃ and I_f = 4λ[(2h′a)⁴/16]/s̃, respectively. These quantities are evaluated by Monte Carlo integration (e.g., Keightley and Hill 1990). Compared with the additive model (Zhang and Hill 2002), all else being the same, the equilibrium genetic variance V_G decreases if h > h′ and increases if h < h′, as expected intuitively. In fact, the strength of apparent stabilizing selection can be simplified as Vs,t=Vs,r[1−Covp∕(VG2∕Vs,r+(2h′)Vm)] (13) or Vs,t−1=Vs,r−1+Vs,p−1 (14) (cf. Zhang and Hill 2002, Appendix A), where the strength of apparent stabilizing selection due to pleiotropic effects is Vs,p=[VG2+((2h′)2VM−Covp)Vs,r]∕Covp. (15) Since Cov_p < (2h′)²V_M (Zhang and Hill 2002), V_s,p > V²_G/(2h′)²V_M, suggesting that whenever the genetic variance is close to the observed levels, the stabilizing selection due to pleiotropic effects of mutations is quite weak, in agreement with the pure pleiotropic model (Barton 1990; Zhanget al. 2002). As the asymptotic expectation of [x(1 – x)]² vanishes when N → ∞ (Kimura 1969), the dominance variance vanishes due to the fact that almost all segregating mutant alleles are in heterozygotes. Thus additive variance comprises most of V_G within a large population.

If dominance coefficients are constant across loci, algebraic expressions for V_G and V_s,t can be obtained for some distributions of mutational effects (cf. Table 1 of Zhang and Hill 2002). For instance, if the squared effect on the trait (a²) and the pleiotropic effect on fitness (s) of mutations follow independent gamma (β₁) and gamma (β₂ = 1 –β₁) distributions, respectively, then VG=4λVs,r(1−θ1−β2)∕(1−θ), (16) Vs,t=Vs,r{1−(2h′)2Vm(β2+β1θ2−θ2−β2)∕[β1(1−θ)2×(VG2∕Vs,r+(2h′)2Vm)]}. (17) (see Appendix). In Equations 16 and 17 θ≡(2hs¯p∕β2)∕((2h′)2−sr∕β1) denotes the mean homozygous pleiotropic effect on fitness, and s_r ≡ E(a²/4V_s,r) = 2V_m/(4λV_s,r) represents the mean selection coefficient arising from real stabilizing selection on homozygous mutational effects on the trait. If the two dominance coefficients h and h′ are the same, the genetic variance V_G increases with dominance coefficients whereas V_s,t decreases. However, if all mutations are additive for the trait (i.e., h′ = 0.5), both V_G and V_s,t increase as h decreases, and at the extreme situation of h = 0, the house-of-cards approximations hold: V_s,t = V_s,r and V_G = 4λV_s,r(Turelli 1984). This shows that even though pleiotropic effects on the carriers of the mutations when homozygous occur in principle, the population experiences little pleiotropic selection as real stabilizing selection is effective in keeping recessive mutants rare and thus heterozygous. In the other extreme case where pleiotropic effects on fitness follow the geometric distribution (Hill 1982b; i.e., β₂ → 0), Equations 16 and 17 also return to the house-of-cards approximations, reflecting the fact that most mutants have little effect on fitness. It should be noted that varying dominance coefficients across loci as described above increases the predictions of V_G and V_s,t, especially for more recessive mutants for fitness (cf. Table 1 and Figure 3 below).

NUMERICAL RESULTS

Monte Carlo simulation methods are employed to illustrate the dependence of both V_G and V_s,t on the population size, to verify the above analytical approximations in a large population when 2NE(s̃) » 1 and, using single-locus simulation, to show how the dependence of V_G on the degree of dominance of genes is affected by population size. For other cases, diffusion approximations are used to save computation time. Our aim is to display quantitatively what differences the variable dominance and the exact shape of distributions of mutational effects cause to the predictions of V_G and V_s,t.

Dependence of V_G and V_s,t on the population size: As natural populations may not be sufficiently large and the distribution of mutational effects is highly leptokurtic such that 2NE(s̃) » 1 may not hold for some mutant genes, it is relevant to investigate the influence of N on the predictions of V_G, V_D, and V_s,t. Diffusion approximations with single-locus and individual-based Monte Carlo simulation results are shown in Figure 1, where the results from different methods are in good agreement. As the population size N increases, V_G and V_s,t increase and approach asymptotic values predicted by analytical approximations, while V_D increases to a peak and then decreases and vanishes, although larger population sizes are required to approach infinite approximations when mutational effects become more leptokurtic. The trend of dependence of V_G and V_s,t on N is similar to that for the additive model (Figure 4 of Zhang and Hill 2002); but, as expected, it is quite different from that of the purely pleiotropic model (cf. Figure 4 of Caballero and Keightley 1994). In the following, except where stated otherwise, we consider large populations such that 2NE(s̃) » 1 holds and thus analytical approximations apply.

Influence of dominance on V_G and V_s,t: This is shown in the following three different situations.

Dominance coefficients equal and constant: Consider the situation where dominance coefficients for the pleiotropic effect on fitness and for the trait are the same, i.e., h = h′, and remain constant across loci. The results in Figure 2 show that the genetic variance V_G increases with the dominance coefficients, but that the increase in V_G is not large, in agreement with the prediction of Caballero and Keightley (1994). The value of V_s,t decreases as the degree of dominance of mutants increases; i.e., the apparent stabilizing selection becomes stronger, but more slowly so as the kurtosis of s increases. The results indicate that with dominant mutations, only a slightly higher V_G is maintained under relatively strong apparent stabilizing selection, compared to additive mutant genes.

Dominance coefficients different and constant: Results are shown in Figure 3, a and b (dotted curves), for the case where mutant genes are assumed to be additive for the trait (h′ = 0.5). Both V_G and V_s,t decrease with increasing degree of dominance of the pleiotropic effect (h). Intuitively, this is because, with increasing h, selection becomes more effective in removing mutant genes from the population, thus reducing mutant frequencies. As the rate of mutation decreases, real stabilizing selection becomes significant in relation to pleiotropic selection (cf. Figure 3 of Zhang and Hill 2002) and V_G becomes nearly independent of h over a larger range. With an increasing rate of mutation, however, real stabilizing selection becomes weak and V_G becomes increasingly affected by h. Thus high levels of V_G can be maintained and apparent stabilizing selection is due mainly to real stabilizing selection, as mutants for pleiotropic effect on fitness become more recessive.

Variable dominance coefficients: For the case where dominance coefficients h and h′ vary across loci as described in model and methods, the results are shown in Figure 3, a and b (solid curves), and Table 1. Comparison with constant dominance coefficients (dotted curves, fixing h at h̄ and h′ at 0.5) displays that, all else being the same, varying dominance coefficients increases V_G and V_s,t (i.e., reduces apparent stabilizing selection). This is due to the assumption that the dominance coefficients are inversely correlated with mutational effects so that the heterozygous effects of mutations on fitness (hs) always remain small, resulting in a relatively weak selection. Moreover, all mutant genes become mildly deleterious when heterozygous and thus segregate for a long time in the population and reach higher frequencies, so that the genetic variance increases. Dominance variance appears as h′ fluctuates around the mean 0.5, but the value of V_D is very small, <1% of V_G. Given h¯′=0.5 , V_G increases rapidly and approaches the house-of-cards approximation V_G = 4λV_s,r (Turelli 1984) as h̄ reduces to nil. Consider typical estimates of parameters s̄_P = 0.1, λ = 0.1, V_m = 10^–3V_E, and h̄ = 0.1 (Lynchet al. 1999; Garcia-Dorado and Caballero 2000; Carr and Dudash 2003; García-Doradoet al. 2003) with a and s following a reflected gamma (0.0847) and gamma (½) distributions, respectively. Variable dominance coefficients result in V_G = 1.89V_E, over 15 times that for the additive model (0.12V_E) (Zhang and Hill 2002), while the strength of apparent stabilizing selection is reduced by only 50% (from 12.1V_E to 19.9V_E, see Table 1). However, as h̄ increases and exceeds 0.3, V_G and V_s,t become roughly independent of h̄, which partly reflects the constraint h ≤ h′ used in the calculations.

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Figure 1.

—Equilibrium genetic (V_G) (a) and dominance variance (V_D) (b) and the strength of apparent stabilizing selection (V_s,t) (c), plotted against population size (N) for mutational effects of variable dominance coefficients with means h̄ = 0.2 and . Mutational effects on both the trait and fitness are independent, homozygous pleiotropic effects on fitness follow a gamma (0.125) distribution (kurtosis = 47.2), and homozygous effects on the trait a reflected gamma (0.0846) distribution (kurtosis = 70). Typical estimates of mutation and selection parameters are assumed: the mutation rate λ = 0.1 per haploid genome per generation, mean pleiotropic effect on fitness s̄_p = 0.1, mutational variance V_m = 10^–3V_E, and real stabilizing selection of strength V_s,r = 20V_E. Single-locus simulation results (triangle) are obtained by averaging over 10⁸ mutation events, and individual-based simulations (circles) are obtained assuming 10³ mutable loci and averaging over 10³ equilibrium generations. Diffusion results (diamond) are obtained by Monte Carlo integration over 10⁵ samples; the curves are the solid lines through the diffusion data. The dash at the end of curves represents the infinite population approximation, which is approached on the diffusion results (e.g., V_G = 0.9V_E for N = 10¹³). The dashed curves are obtained by diffusion approximation for a and s following independent reflected gamma (0.196) and gamma (0.25) distributions, respectively.

If mutations for the trait are exclusively additive, a negligible increase in V_G and a slight increase in V_s,t occur, compared to the above where h′ varies. Further, removing the constraint h ≤ h′ = 0.5 on variable dominance coefficients results in a minor decrease in V_s,t and a slight decrease in V_G. If h¯′<0.5 , all else being the same, the prediction of V_G is smaller and that of V_s,t is larger than those for h¯′=0.5 . However, such changes in V_G and V_s,t due to a decrease in h′ are not large if the deviation of h¯′ from 0.5 is small (cf. Caballero and Keightley 1994). For example, with h¯′=0.4 , the decrease in V_G is <10% when h̄ = 0.1 and shrinks rapidly as h̄ becomes smaller, while the increase in V_s,t is <1% (see Figure 3, a and b).

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Figure 2.

—Influence of dominance coefficients of mutations on the genetic variance (a) and the strength of apparent stabilizing selection (b). Dominance coefficients for fitness and for the trait are assumed to be the same and constant across loci. Mutational effects on the trait follow a reflected gamma (0.091) distribution (kurtosis = 65), while pleiotropic effects of mutations are distributed as gamma (⅙), gamma (¼), or gamma (½) with kurtosis 35.3, 23.4, 11.7, respectively. Other parameters of mutation and selection are as in Figure 1 (i.e., λ = 0.1, V_m = 10^–3 V_E, s̄_p = 0.1, and V_s,r = 20V_E).

Impact of distributions of mutational effects on V_G and V_s,t: The results listed in Table 1 show that under the constraint k₄(a) > k₄(s), even the additive model can generate abundant genetic variance with highly leptokurtic mutational effects. Here values of the kurtosis k₄(a) Ξ E(a⁴)/E(a²)² and k₄(s) Ξ E(s⁴)/E(s²)² are intended to describe the shape of distributions of mutational effects. If mutational effects on the trait follow a reflected squared exponential distribution with kurtosis k₄(a) = 70, for example, V_G can increase dramatically from 0.19 to 0.69 when the distribution of pleiotropic effects changes from gamma (½) to gamma (¼). Data in Table 1 further show how different assumptions for distributions of mutational effects change the predictions of V_G and V_s,t. The three distributions of mutational effects on the trait, reflected square-root gamma (0.0145), reflected gamma (0.0847), and reflected squared exponential, for example, have the same kurtosis k₄(a) = 70. However, they give very different predicted values of V_G, 0.22, 0.38, and 0.69 when h = 0.5, and 3.67, 4.70, and 5.79 when h = 0.1, if the pleiotropic effects on fitness are assumed to have a gamma (¼) distribution. The change in V_s,t is comparatively small, however. This implies that the assumption of the squared gamma distribution actually allows more mutants of large effect on the trait than do the other two distributions, and different distributions of a give different predictions of V_G. Therefore, a description of the distribution solely in terms of kurtosis is not sufficient.

If the population size is not sufficiently large to retain 2NE(s̃) » 1 for most genes, predictions of the genetic variance maintained, which are calculated using formulas for infinite population size, blow up. Considering a population size of 10⁵, the predicted value of V_G is the same as or slightly smaller than the infinite approximation if constant dominance coefficients are assumed (Figure 2 and Table 1). For the variable dominance case, the predicted value of V_G decreases rapidly as h̄ is reduced, but that of V_s,t remains roughly the same (see Figure 3, c and d, and Table 1). For a population of this size, the increase in V_G relative to the additive case, >60% for h̄ = 0.2 and >90% for h̄ = 0.1, is substantial (see Table 1), and the dependence of V_G on dominance still holds, although it is somewhat weaker (see also Figure 3c). Results in Figure 3c and Table 1 for finite population sizes show that for typical estimates of mutation and selection parameters, wherever k₄(s) < k₄(a) < 10k₄(s), variable dominance coefficients increase the prediction of V_G to the levels observed in natural populations.

Impact of the correlation between |a| and s on V_G and V_s,t: It is biologically plausible that mutational effects on the trait and fitness are correlated (see Keightley and Hill 1990). If the marginal distributions of a and s have quite different degrees of kurtosis, effects of mutations on the trait and on fitness can be only partially correlated (Whittaker 1974). For a fitness-related trait such as life history, the difference between k₄(a) and k₄(s) is likely to be smaller than that for a trait less directly related to fitness, e.g., morphology. Figure 4 shows the influence of the correlation coefficient, ρ=cov(∣a∣,s)∕V[∣a∣]V[s] , between |a| and s on both V_G and V decreases or the difference between k₄(a) and k₄(s) increases, the impact of the correlation on V_G becomes large while its impact on V_s,t becomes small. However, it can still be concluded, as in the additive model (Zhang and Hill 2002), that if the correlation between |a| and s is at most intermediate, its influence on V_G and V_s,t is not large.

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Figure 3.

—Influence of dominance coefficients of mutations on the genetic variance (a and c) and the strength of apparent stabilizing selection (b and d). The distributions of homozygous mutational effects are assumed to be reflected squared exponential for a and gamma (½) for s. The x-axis is the average dominance coefficient for pleiotropic effects of mutations on fitness. (a and b) Results are shown assuming infinite population size for three genome-wide mutation rates. Dotted curves are results for constant dominance coefficients (i.e., h̄ = h, and h′ = 0.5); solid thick curves are for variable dominance coefficients as described in the text with means ; and solid thin curves are for variable dominance coefficients with means and mutation rate λ = 0.1. Other parameters of mutation and selection are as in Figure 1 (i.e., , and V_s,r = 20V_E). (c and d) Comparison of results between infinite populations (solid curves) and finite populations of N = 10⁶,10⁵, and 10⁴ (solid thin, dashed, and dotted curves, respectively, obtained by single-locus Monte Carlo simulation) for , λ = 0.1.