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Increased heritable variation following population bottlenecks: the role of dominance.

Fisher (1930) believed that large populations have greater potential for adaptive evolution than small populations, because they harbor a greater variety of alleles that are not effectively neutral and because they pick up many new mutations each generation. In contrast, Wright (1977), Mayr (1954, 1963), Carson (1975), and others have argued that, because of the existence of multiple adaptive peaks and coadapted gene complexes, large random mating populations are essentially evolutionary dead ends with little capacity for change. These latter authors argue, however, that random genetic drift can disrupt these coadapted gene complexes, allowing populations to respond to selection toward new and higher adaptive peaks. An extension of this line of reasoning holds that brief but intense periods of genetic drift, such as founder events, might allow sufficient genetic change to cause reproductive isolation and speciation (Mayr 1954, 1963; Carson 1975; Templeton 1979, 1980; Carson and Templeton 1984). Several recent experiments have demonstrated that additive genetic variances and heritabilities of morphological traits and fitness components can increase following severe population bottlenecks (e.g., Bryant et al. 1986; Lopez-Fanjul and Villaverde 1989), and have therefore, lent support to the idea that greater evolutionary innovation is possible in small, rather than large, populations.

When the genetic variation underlying a trait is caused by genes that act additively within and between loci, the additive genetic variance within a population following a bottleneck event is expected to decrease in proportion to the degree of inbreeding (Crow and Kimura 1970; Lande 1980; Falconer 1981; Barton and Charlesworth 1984). The observed increases following population bottlenecks in additive genetic variation and heritabilities of eight size-related morphometric traits in the housefly (Bryant et al. 1986) and viability in Drosophila melanogaster (Lopez-Fanjul and Villaverde 1989) can therefore be attributed to nonadditive, possibly epistatic gene action. Indeed, theoretical work has shown that genetic drift can "convert" epistatic genetic variance into additive genetic variance, which can then contribute to the response to selection (Cockerham and Tachida 1988; Goodnight 1988; Tachida and Cockerham 1989).

However, Robertson (1952) also showed that when variation is caused by rare, completely recessive genes (or by overdominant loci), genetic drift can temporarily increase the additive variance, [V.sub.A]. For complete recessives, the reason for this increase is simple: as Falconer (1981, fig. 8.1b) shows, the relationship between the frequency of a recessive allele, q, and [V.sub.A] is concave upward when q is small. Slight genetic drift will cause a rare recessive allele to decrease in frequency in some replicate populations (or at some loci), leading to fairly small decreases in [V.sub.A] and increase in other populations (or at other loci), leading to disproportionately large increases in [V.sub.A]. The expected additive variance over many replicate populations (or loci) will therefore increase (Robertson 1952; Bryant et al. 1986).

In this note, we examine the effect of the degree of dominance on the expected additive genetic variance and heritability following a bottleneck. Because it is well known that many alleles affecting fitness components and other quantitative traits are partially recessive (Simmons and Crow 1977; Crow and Simmons 1983; Charlesworth and Charlesworth 1987), it is important to determine whether this large class of genes can explain the results of the experiments cited above. Although Cockerham and Tachida (Cockerham and Tachida 1988; Tachida and Cockerham 1989) present general equations relating the expected additive variance in a population following a bottleneck to the additive and dominance variance in the ancestral population, the effects of specific dominance coefficients and gene frequencies have not been examined directly. Here we show that, in the absence of epistasis, the additive variance and heritability will often increase over that in the ancestral population over a very broad range of dominance coefficients. This increase in additive variance will be accompanied by a decline in the phenotypic mean due to inbreeding depression. We discuss experimental evidence that supports a role for dominance in the observed increases in additive genetic variances and heritabilities.

Additive Variance Following a


We consider an infinitely large, randomly mating ancestral population from which a sample of N founder individuals is drawn at random to form a new population. We assume that after one generation of reproduction at size N, the founded population expands immediately to a large size with random mating for many generations, so that subsequent genetic drift is negligible and the new population is restored to Hardy-Weinberg, identity, and linkage equilibrium. We then ask how the expected additive variance of many replicate founded populations differs from that of the ancestral population.

First consider the case of variation caused by segregation of two alleles, A and a, at a single locus. Following the notation of Falconer (1981), the phenotypic values of the three diploid genotypes are AA = 1, Aa = d, and aa = -1, where the range of 2 is an arbitrary scale. When the dominance coefficient, d, is 0, the genes act additively and when d = 1, a is completely recessive to A. If the frequency of A in the ancestral population is p, then the additive variance before the bottleneck is [V.sub.A] = 2pq[1 + d(q - p.sup.2)],

(1) where q = (1 - p) gives the frequency of a (Falconer 1981). Expansion of (1) yields

[V.sub.A] = 2p(1 + 2d + [d.sup.2]) - [2p.sup.2](1 + 6d + [5d.sup.2] + [8p.sup.3](d + [2d.sup.2) - [8p.sup.4.d.sup.2]. (2)

Random genetic drift during a bottleneck will cause the gene frequencies in the replicate founder populations to diverge from the frequencies in the ancestral population. Following Crow and Kimura (1970, pp. 335-344) and Bulmer (1980, pp. 218-226), the expected additive variance for arbitrary degrees of dominance and initial gene frequencies of many replicate founder populations can be obtained by substituting the first four moments of the gene-frequency distribution with binomial sampling for p, [p.sup.2], [p.sup.3], and [p.sup.4] in (2). This method can of course be used to obtain the expected additive variance in founder populations that have had any history of inbreeding, but here we illustrate the results with a single generation bottleneck of small size. Because the number of diploid adults during the bottleneck, N, can be sufficiently small that terms of order 1/[N.sup.2], and so forth, cannot be ignored, we use the exact gene frequency moments found in Crow and Kimura (1970, p. 335, correcting for the typographical error in the fourth moment, in which the second term should read [(18N - 11)/ (10N - 6)]p(1 - p)(1 - [1/2N.sup.t]). When the bottleneck event lasts only one generation, it can be shown that the expected additive variance, E([V.sup.A]), is

E([V.sub.A]) = [p(1 - p)(2N - 1)/[N.sup.3]] .{(dN + N - [d.sup.2]) - 2dp(N - 1) .[2N(d + 1) - 3d - d(2N - 3)p]},

(3) where E represents the expectation taken over many replicate founder populations. Identical results can be obtained using the least-squares approach of Cockerham and Tachida (Cockerham and Tachida 1988; Tachida and Cockerham 1989). Notice that when d = 0, E([V.sub.A]) = [p(1 - p) (2N - 1)]/N and E(V.sub.A])/[V.sub.A] = 1 - 1/2N, as expected. In the case of a very rare partially recessive allele (p [is nearly equal to] 1), equation (3) is approximately

E([V.sub.A]) [is nearly equal to] (1 - p)(2N - 1) .(dN - d - [N.sup.2])/[N.sup.3]. (4)

Figure 1 shows the ratio of the expected additive variance among many replicate bottleneck populations of size N = 2 to the additive variance in the ancestral population for a range of dominance coefficients (0 [less than or equal to] d [less than or equal to] 1), as a function of q in the ancestral population. As expected, there is a 25% loss of additive genetic variance when the alleles act additively (d = 0), regardless of initial allele frequency. With dominance, the change in [V.sub.A] depends on the initial allele frequency as well as on the dominance coefficient. However, when the partially recessive allele is very rare, as would be expected with deleterious alleles, the expected additive variance always increases if d is greater than about 0.25, with the greatest increase occurring with complete recessivity (d = 1). Indeed, from equations 1 and 4, it can be shown that, when variation is due to rare deleterious alleles, a bottleneck of size N always increases E([V.sub.A]) if d > N[3N - 2N(1 - [1/2N.sup.1/2]) - 1]/(1 - 4N + [5N.sup.2]). (5)

This critical value of d is remarkably insensitive to the size of the bottleneck. When N = 1, E([V.sub.A]) increases if d > 0.29. As N increases, the critical value of d quickly reaches an asymptote at 0.20.

The critical value of the frequency of the partially recessive allele, (1 - p), below which [V.sub.A] is expected to increase with drift, increases with d, so that even moderate frequencies of a completely recessive allele will cause an increase in the expected additive variance. The results presented above were derived for a single locus, but extension to multiple loci will not greatly alter the results. In the simplest case, where all segregating loci in the ancestral population have identical allele frequencies and dominance coefficients, the results in figure 1 are independent of the number of loci. Indeed, an increase in the number of loci segregating for recessives will simply decrease the variation of replicate founder populations about the expected additive variance (Robertson 1952).

We chose to illustrate the effect of inbreeding on additive variance with partial dominance with a single generation bottleneck of N = 2. Less extreme inbreeding will cause the expected additive variance to show less change after a bottleneck event. The critical value of (1 - p), below which the expected additive variance will increase, will be slightly greater for a given degree of dominance with less extreme inbreeding than with a bottleneck of N = 2. For example, the critical value of (1 - p) approaches 0.5, when d = 1, as the degree of inbreeding approaches zero (Bulmer 1980, p. 226, compare with fig. 1).

Heritability Following a Bottleneck

It is more difficult to assess the effect of a founder event on the expected heritability than on the expected additive variance, because allele-frequency changes caused by genetic drift will alter both the total genetic variance and the additive variance. Unfortunately, although it is possible to obtain the expected total phenotypic variance, [V.sub.P] as well as the expected additive variance, following a bottleneck using the moments of the gene frequency distribution and making assumptions about the environmental variance, [V.sub.E], the expected heritability will not in general equal the ratio of these two quantities. This inequality of the expected heritability to the ratio of the expected variances is primarily due to the covariance between the additive variance and the total phenotypic variance, which is caused by the mathematical relationship between them. The expected heritability is

E([V.sub.A]/[V.sub.P]) [is nearly equal to] E([V.sub.A])/E([V.sub.P) - Cov([V.sub.A], [V.sub.P])/[E([V.sub.P.sup.2])] + E([V.sub.A])Var([V.sub.P])/[E([V.sub.P.sup.3])]. (6)

The situation is simplified considerably when numerous unlinked, identical loci contribute to the variability in the trait of interest (i.e., all loci are identical with respect to d and the initial p). In this simple case, each founded population will be nearly identical in terms of the distribution of gene frequencies across loci and therefore replicate populations will not vary greatly in components of variance or in heritability. The second and third terms of the right-hand side of (6) would therefore be approximately zero, and the expected heritability (mean of the ratio of variances) will be nearly equal to the ratio of the expected variances. This usually occurs when the number of loci is larger than about ten (unpublished simulation results; P. Phillips and M. Wade, unpublished results).

In order to calculate the expected heritability following a bottleneck, one must calculate the expected total genetic variance using the gene-frequency moments. In the absence of epistasis, the genetic variance, [V.sub.G] is simply equal to the sum of the additive and dominance variances (Falconer 1981): [V.sub.G] = 2pq[1 + d(q - [p.sup.2])] + [2pqd.sup.2]. (7)

This can be expanded to give

[V.sub.G] = 2p(1 + 2d + [d.sup.2] - [2p.sup.2](1 + 6d + [3d.sup.2]) + [8p.sup.3](d + [d.sup.2]) - [4p.sup.4.d.sup.2], (8) and we can, once again, substitute the first four moments of the gene-frequency distribution with binomial sampling for p, [p.sup.2], [p.sup.3], and [p.sup.4].

The expected heritability following a founder event, making the assumptions listed above, is equal to the expected [V.sub.A], divided by the sum of the expected [V.sub.G] and [V.sub.E]:

E([h.sup.2]) = E([V.sub.A])/[E([V.sub.G]) + [V.sub.E]. (9)

A difficulty in assessing the expected change in the heritability following a bottleneck is in deciding what [V.sub.E] should be. We choose to illustrate the situation by making three different assumptions about [V.sub.E]. In all three cases, we assume that [V.sub.E] is unchanged by a bottleneck event. First, we assume [V.sub.E] = 0. Although biologically unrealistic, this case has the advantage of revealing the complex ways that the expected heritability is influenced by changes in the additive variance and the total genetic variance. The maximum heritability in this case is unity. Second, we examine the case in which [V.sub.E] is assumed to equal [V.sub.G] in the ancestral population, such that the maximum heritability before the bottleneck is 0.5. This range of heritabilities might be relevant to traits not directly associated with fitness (Mousseau and Roff 1987; Roff and Mousseau 1987; Houle 1992). In our third case, [V.sub.E] is assumed to equal nine times the [V.sub.G] in the ancestral population. Here the maximum [h.sup.2] before the bottleneck is 0.1, and this range of values may approximate the heritabilities of many fitness components (Mousseau and Roff 1987; Roff and Mousseau 1987; Houle 1992), and is roughly the heritability for viability in the non-inbred lines of Lopez-Fanjul and Villaverde (1989).

Figure 2 shows the expected changes in [h.sup.2] when N = 2. Figures 2A and 2B illustrate the case of [V.sub.E] = 0. The relationship between the change in heritability after a bottleneck and the initial gene frequency is clearly more complex than with the change in additive variance. With complete recessivity (d = 1), the heritability is expected to increase when the recessive allele is rare, and the absolute magnitude of this increase can approach 0.4 (fig. 2B). This increase in expected [h.sup.2] at small (1 - p) is extremely sensitive to the degree of dominance, so that [h.sup.2] is expected to decrease with d < 1 when (1 - p) is very small. However, when the partially recessive allele is moderately common [0.2 > (1 - p) < 0.6, approximately], there can be small increases in the expected heritability with intermediate degrees of dominance.

When [V.sub.E] > 0, expected changes in the heritability more closely resemble the expected changes in the additive genetic variance. Figures 2C and 2D illustrate the expected changes in [h.sup.2] when [V.sub.E] = [V.sub.G]. Clearly, increases in heritability following a bottleneck are expected in this case when genetic variation is caused by rare recessives or partial recessives. As with [V.sub.A], a broad range of gene frequencies and dominance coefficients cause an increase in the expected heritability. The expected absolute changes (fig. 2D) are also quite large, with increases of greater than about 0.1 when d > 0.6 and (1 - p) is small. Figure 2E and 2F illustrate the case in which [V.sub.E] = 9([V.sub.G]). The relative changes in [h.sup.2] in this case (fig. 2E) are quite similar to the relative changes in [V.sub.A] seen in figure 1, and substantial increases are to be expected when (1 - p) is small for a broad range of dominance coefficients. Here the expected absolute changes in [h.sup.2] (fig. 2F) are not quite as large as when [V.sub.E] = [V.sub.G], yet even in this case increases greater than about 0.1 are expected when d > 0.6 and (1 - p) is small.

In sum, when [V.sub.E] is small, expected changes in the heritability will resemble the case illustrated in figures 2A and 2B, in which [V.sub.E] = 0. When [V.sub.E] is large, changes in the heritability will be expected to resemble the changes in [V.sub.A], as illustrated in figure 1. The reason the environmental variance plays such an important role in the expected changes in the [h.sup.2] is that it determines the degree to which the phenotypic variance reflects the total genetic variance. In the extreme case of [V.sub.E] [much greater than] [V.sub.G], the total phenotypic variance will be roughly constant before and after the bottleneck, and changes in the heritability will primarily reflect changes in the additive genetic variance.


The results presented here show that unless alleles are nearly completely additive in their heterozygous effects (d < 0.25) or the partially recessive alleles are common in the ancestral population, brief population bottlenecks will increase the expected additive genetic variance and heritability. Although Robertson (1952) pointed out that genetic drift could increase the additive variance with completely recessive genes, it has not been generally appreciated just how broad a range of dominance coefficients will cause an increase in additive genetic variance and heritability. Furthermore, the expected absolute increases in heritability can be substantial (greater than 0.1 when d > 0.6 and (1 - p) < 0.01, approximately, see fig. 2) and should be detectable in experiments of reasonable size. Although recent theoretical work has emphasized the potential for epistasis to cause temporary increases in additive variances caused by genetic drift (Goodnight 1987, 1988; Cockerham and Tachida 1988; Tachida and Cockerham 1989), it appears, for the reasons discussed below, that partial dominance could be the primary cause of the increases in genic variance for fitness and size related traits seen in several experiments (e.g., Bryant et al. 1986; Lopez-Fanjul and Villaverde 1989).

Although it is not usually possible to distinguish experimentally between dominance and epistasis as causes of increases in additive variance (particularly because epistatic interactions can involve dominance), there is compelling evidence that dominance is involved in at least some of the increases seen in Bryant et al.'s (1986) and Lopez-Fanjul and Villaverde's (1989) experiments. In the housefly work, for instance, substantial inbreeding depression for body size was observed; body size was highly correlated with the morphological traits studied (Bryant et al. 1986, p. 1198). Similarly, Lopez-Fanjul and Villaverde (1989) observed a considerable decline in mean as well as an increases in additive genetic variance in their study of viability in bottleneck populations of D. melanogaster. Although dominance must be present to cause inbreeding depression (Crow and Kimura 1970), and the authors of these studies carefully considered the role of dominance in their results, they argued, for various reasons, that at least some of the increases in additive variance were caused by epistasis.

Bryant et al. (1986) concluded that epistasis was probably involved because they observed that, for some traits, bottlenecks of intermediate size produced greater increases in additive variances and heritabilities than either smaller or larger bottleneck sizes. This seems to contradict the predictions of the dominance model, where smaller bottlenecks are expected to yield greater increases in variance (Robertson 1952; present results). Bryant et al. (1986) showed, however, that their observation was consistent with predictions from a model in which effects across loci were multiplicative. Unfortunately, a couple of factors make it difficult to interpret Bryant et al.'s observation. First, estimates of additive variances and heritabilities following drift are subject to many sources of error variance, and it is entirely possible that standard errors incorporating these sources of variation might be large enough to render the pattern nonsignificant (Lynch 1988). Also, Goodnight (1988) has pointed out that the greatest increases in additive variance are not expected to occur at intermediate bottleneck sizes with all types of epistasis. Indeed, Goodnight's models of genetic drift with additive by additive epistasis yield the same qualitative prediction as dominance (Goodnight 1987, 1988). In any case, more recent experiments have failed to confirm the pattern of greater increases in additive variance with intermediate bottleneck sizes (Bryant and Meffert 1993).

It seems reasonable to expect that much of the genetic variation for major components of fitness and for morphological traits closely associated with body size will be caused by the segregation of rare partially recessive alleles that may be unconditionally deleterious. Careful studies of the genetic basis of variation for viability in natural populations of D. melanogaster have revealed that the amount of additive genetic variation is often at least an order of magnitude greater than the residual genetic variance, which includes dominance or epistatic components of variance (Mukai 1985). This pattern is expected if most of the genetic variation in fitness is caused by the mutation-selection balance of partially recessive alleles (Mukai 1985; Charlesworth 1987). Furthermore, the magnitude of additive genetic variation estimated in these studies is consistent with that expected under mutation-selection balance at many genes of minor effect (except in southern populations where there is circumstantial evidence for diversifying selection at a minority of loci) (Mukai et al. 1974; Charlesworth 1987; Mukai 1988). Studies of the dominance coefficients of deleterious alleles in Drosophila have shown that d is about 0.96 for lethals and 0.6 for mildly deleterious mutations segregating in wild populations [Simmons and Crow 1977; Crow and Simmons 1983; these data were originally reported in terms of the dominance coefficient, h, where h = (1 - d)/2]. Both of these values are well within the range of dominance coefficients expected to produce an increase in additive genetic variance and heritability following a bottleneck event (figs. 1 and 2). For example, at mutation-selection balance with mildly deleterious mutations (d = 0.6), the expected additive variance after a bottleneck of N = 2 is about 2.3 times that in the ancestral population [assuming (1 - p) [is nearly equal to] [micro]/hs, where [micro] = [10.sup.-5], h = 0.2, and s = 0.03; Crow and Simmons 1983]. Although direct estimates of d are not available for other organisms, the ubiquity of inbreeding depression for many fitness components in a wide variety of animal and plant species (reviewed by Charlesworth and Charlesworth 1987) shows that deleterious partial recessives are common.

There are less data on the dominance coefficients for genes underlying variation in quantitative traits that are not major components of fitness. Some traits, like bristle number in D. melanogaster (Kidwell and Kidwell 1966), show no inbreeding depression while others like leaf size in Papaver dubium (Thomas and Gale 1977) show substantial inbreeding depression. It remains to be seen how common inbreeding depression is in quantitative traits of ecological and evolutionary interest. We suggest, however, that any study that examines the effect of drift on the genetic variance of traits should also report the effect of drift on the phenotypic means. Only in this way can one determine how likely a role dominance plays in changes in [V.sub.A] and [h.sup.2]. We also note that a lack of inbreeding depression does not by itself rule out the possible contribution of dominance to changes in genetic variance, as inbreeding depression results only from directional dominance or overdominance.

It seems clear then that much, if not all, of the increase in additive genetic variances and heritabilities observed in experiments like those described by Bryant et al. (1986) and Lopez-Fanjul and Villaverde (1989) might be due to chance increases in the frequencies of rare partially recessive alleles. If an increase in the additive variance of a trait related to fitness is invariably accompanied by inbreeding depression, as is expected with directional dominance or overdominance, then it seems unlikely that a population following a bottleneck event will have a greater evolutionary potential than the ancestral population. It is of course possible that a bottleneck event will be followed by a change in the selective environment (Barton 1989), such that formerly deleterious alleles are suddenly favored, but it is far from clear that a population that had passed through a bottleneck would be more able to respond over a long period of time to that selective change than the ancestral population. After all, although some rare alleles will increase in frequency caused by the bottleneck, many more will be lost. The ancestral population would therefore be expected to be polymorphic at many more loci and so might achieve a greater response in the long run to altered selective pressures. In any case, mutation must produce many unconditionally deleterious alleles and any increase in their frequencies during a population bottleneck could only be harmful.

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Author:Willis, John H.; Orr, H. Allen
Date:Jun 1, 1993
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