Epistasis as a source of increased addictive genetic variance at population bottlenecks.
In these theories, epistasis is considered ubiquitous because of intense interaction among genes and their products during development. Stabilizing selection acts to build a balanced system of harmoniously interacting genes through coadaptation, or the adaptation of genes to their genetic environment (the collection of alleles present in the population at other, interacting loci). This harmonious system of epistatically interacting genes creates phenotypic uniformity at a favored, intermediate, phenotypic value despite underlying genetic diversity. The phenotypic value of an allele is not constant, but changes depending on its genetic background. Given such a balanced system of gene interaction, evolutionary change becomes difficult because alterations at any one locus will usually affect the genotype as a whole in an adverse way, hence, Mayr's (1963) unity of the genotype.
The evolutionary inertia that results from a coadapted system of interacting genes is adaptive when it is formed but needs to be dismantled if further adaptation is to occur. A potentially important means for disrupting the cohesion of the gene pool is by a founding event, where a population of small size becomes separated from a larger parental population. The founding event and subsequent genetic drift drives allele frequencies towards extreme values, exposing previously rare homozygotes to selection and, given epistasis, altering the selective values of alleles. The founding event produces a new genetic environment that will result in selection for a new system of coadapted genes. This change in genetic environments for all genes at once results in Mayr's (1963) genetic revolution in which physiologically interacting genes readapt to one another in new genetic alignments. Templeton's (1980a,b) genetic-transilience model is similar, but he restricts the revolution to a smaller set of major genes and their modifiers, which quickly develop a new coadapted system.
Although founding events and subsequent genetic drift perturb coadapted gene complexes, they are also thought to severely limit the level of additive genetic variance for quantitative traits, thereby limiting the rate of adaptation (Falconer 1989; Barton and Charlesworth 1984). In order for founder effect speciation models to produce adaptive evolutionary change and speciation, the founder event must cause a large change in the genetic environment through drift or increased inbreeding but at the same time allow sufficient additive genetic variation for a significant evolutionary response to the altered genetic and external environment (Templeton 1980a, b). This window of evolutionary opportunity is thought to be very limited (Templeton 1980b; Barton and Charlesworth 1984), although Templeton (1980a) described a series of population structure conditions that enhance the possibility of a genetic transilience while passing through a bottleneck. Prominent among these is the population flush, such that the population does not continue to lose additive genetic variance through genetic drift after the founding event. These views of speciation are by no means generally accepted, especially as they seem to contradict much of what is known from population genetic theory (Coyne 1994; Futuyma 1994).
Contrary to a commonly held view, although founder events and subsequent population bottlenecks certainly reduce genetic heterozygosity (Crow and Kimura 1970), they do not necessarily result in decreased additive genetic variance in the presence of dominance and/or epistasis. Theoretical models that include dominance and epistasis (Rose 1982; Bryant et al. 1986; Bryant and Meffert 1993; Goodnight 1987, 1988; Cockerham and Tachida 1988; Tachida and Cockerham 1989; Whitlock et al. 1993) indicate that loss of additive genetic variance as a population passes through a bottleneck can be limited or even reversed when there is a system of interacting genes. Furthermore, several experiments have noted an increase in additive genetic variance in populations passing through either one or a series of bottlenecks (Bryant et al. 1986; Bryant and Meffert 1993; Lopez-Fanjul and Villaverde 1989; Abplanalp 1988; Lints and Bourgois 1982). However, Lynch (1988) has questioned the strength of conclusions drawn from these experiments due to problems of sample size and statistical design.
We present a general two-locus genetic model describing the contribution of physiological epistasis to additive genetic variance (Cheverud and Routman 1995) and analyze the evolution of additive genetic variance during founder events and persistent population bottlenecks. We show that certain forms of epistasis can, indeed, contribute to significant increases in additive genetic variance in finite populations. However, other forms of epistasis can lead to a temporary reduction in additive genetic variance during a population bottleneck.
Epistasis occurs when phenotypic differences among genotypes at one locus depend on which genotypes are present at other loci. We defined a quantitative measure of physiological epistasis in a two-locus genetic model (Cheverud and Routman 1995), which is analogous to the familiar definition of physiological dominance (d) in a single-locus system (Falconer 1989).
Two-locus genotypic values ([G.sub.ijkl]) are defined as the average phenotype of individuals with the ij genotype at the first locus, locus A, and the kl genotype at the second locus, locus B. We consider the situation where there are two alleles, labeled 1 and 2, at each locus with allele frequencies [p.sub.1] and [p.sub.2] at locus A and [q.sub.1] and [q.sub.2] at locus B. These two-locus genotypic values are independent of allele frequencies at the loci in question and, for present purposes, also assumed independent of allele frequencies at other loci (no three-way or higher epistasis). Each locus, A and B, may have a direct effect on the phenotype in question and act epistatically to modify the effects of the other locus.
The quantitative effects of physiological epistasis are defined as the deviation of the two-locus genotypic values from the expected values of an unweighted least-squares regression of genotypic values on the number of "2" alleles and heterozygosity at both loci (see Appendix; Cheverud and Rout-man 1995). Given the definitions of genotypic values provided in the Appendix and assuming Hardy-Weinberg and linkage equilibrium, the average effect of an allele substitution ([Alpha]) at locus A is,
[[Alpha].sub.A] = [a.sub.A] + [d.sub.A]([p.sub.2] - [p.sub.1]) + [p.sub.1]([e.sub.11..] - [e.sub.12..])
+ [p.sub.2]([e.sub.12..] - [e.sub.22..]) (1)
(Cheverud and Routman 1995). A similar equation holds for locus B. These equations correspond to those given in Falconer (1989) for the average effect of an allele substitution with the addition of the epistasis terms (e) describing the effects of the modifier locus genotypes on the phenotypic effects of the locus in question. The epistatic terms in this equation are the population average epistatic values for each genotype, averaged over the alternate locus' genotypes. For example,
[e.sub.11..] = [q.sub.[1.sup.2]][e.sub.1111] + 2[q.sub.1][q.sub.2][e.sub.1112] + [q.sub.[2.sup.2]][e.sub.1122] (2)
The additive genetic variance ([V.sub.a]) is then
[V.sub.a] = 2[p.sub.1][p.sub.2][[Alpha].sub.[A.sup.2]] + 2[q.sub.1][q.sub.2][[Alpha].sub.[B.sup.2]] (3)
Because the epistatic values contribute to the average effects of alleles and allele substitutions, they also contribute to the additive genetic variance and may do so at any allele frequencies. The contributions of epistasis to dominance and interaction variances are specified in Cheverud and Routman (1995) and in the Appendix.
It is important to note that this treatment of epistasis differs from other parameterizations, such as the one described by Crow and Kimura (1970, pp. 79-81). Crucially, our parameterization defines physiological epistasis independent of allele frequencies in any specific population and independent of single-locus effects. Genotypic values are compared without regard to their population frequencies (see Appendix). This is in contrast to statistical epistasis, which refers to the population-level interaction variance produced by physiological epistasis. Physiological epistasis is a prerequisite for statistical epistasis. However, even with strong physiological epistasis, only weak statistical interaction may be present in any given population. This occurs because the contribution of physiological epistasis to all three population genetic variance components (additive genetic, dominance, and interaction variances) varies with allele frequencies (Cheverud and Routman 1995). At some allele frequencies, physiological epistasis contributes primarily to additive genetic variance, with only a small contribution to interaction variance (see below). Alternatively, at other allele frequencies physiological epistasis contributes primarily to interaction variance.
The parameters of Crow and Kimura's (1970) two-locus model cannot be used to evaluate the contribution of physiological epistasis to population genetic variance components because the epistatic and single-locus parameters are correlated with one another when estimated for genotypic values in an unweighted regression. Correlation precludes the separation of epistatic from single-locus effects on genotypic values, even though the parameters can be used to properly calculate population-level variance components. Although Crow and Kimuras (1970) parameterization serves to define statistical variance components in a population, it cannot separate the effects of physiological epistasis from independent single-locus effects due to the correlation of these parameters.
SPECIAL FORMS OF EPISTASIS
Among the myriad possible forms of epistasis, there are three special forms in which the single-locus values ([a.sub.A], [d.sub.A], [a.sub.B], [d.sub.B], see Appendix) are all zero such that the genotypic values equal the epistasis values. These are additive-by-additive (A x A), dominance-by-dominance (D x D), and additive-by-dominance (A x D or D x A) epistasis (see Table [TABULAR DATA FOR TABLE 1 OMITTED] 1). We refer to these as pure forms of epistasis because there is no overall additive or dominance genotypic value at either locus. The three epistatic patterns correspond to the three interaction variance components defined by Cockerham (1954; Crow and Kimura 1970). However, they also contribute to additive and dominance genetic variances, depending on the allele frequencies at the loci in question.
With additive-by-additive epistasis, genotypic differences at each locus are additive within modifying locus genotypes but the additive effects differ in sign depending on which modifying locus genotype is considered (see Table 1). This results in an overall simple average homozygote difference (a) of zero at each locus. Physiological dominance (d) is also zero both within modifying genotypes and overall. The A x A epistasis pattern described here corresponds to the special case analyzed by Goodnight (1988) and Whitlock et al. (1993).
The additive genetic variance with A x A epistasis is plotted against two-locus allele frequencies in Figure 1A. For A x A epistasis, the minimum additive genetic variance ([V.sub.a] = 0.00) occurs when alleles at both loci are at intermediate frequency ([p.sub.1] = 0.50; [q.sub.1] = 0.50). This is also the point of maximum interaction variance ([V.sub.i] = 0.0625). The additive genetic variance maxima ([V.sub.a] = 0.125) occur with one locus fixed and the second locus with alleles at intermediate frequency [ILLUSTRATION FOR FIGURE 1A OMITTED]. These maxima correspond to those expected for additive effects at single loci (Falconer 1989). There is no interaction variance when the additive variance is at a maximum.
With dominance by dominance epistasis (see Table 1), all of the two-locus homozygotes have the same phenotype; thus, there is no homozygous difference (a) among genotypes. However, heterozygote values vary such that genotypic values display underdominance or overdominance depending on the modifying locus genotype. In this case, the overall simple average dominance (d) is zero at both loci.
The level of additive genetic variance with D x D epistasis is plotted against allele frequencies in Figure 1B. In this case, there are three classes of minimum additive variance ([V.sub.a] = 0.000). First, an additive genetic variance minimum occurs when alleles at both loci are at intermediate frequency ([V.sub.a] = 0.000; [V.sub.d] = 0.031; [V.sub.i] = 0.141). This is also the set of allele frequencies with maximum interaction variance. A second class of minima occurs when an allele is fixed at one locus and alleles at the other locus are at intermediate frequencies ([V.sub.a] = 0.000; [V.sub.d] = 0.063; [V.sub.i] = 0.000). In this case, dominance variance is at a maximum. A third class of minima occurs when allele frequencies are either 0.2 or 0.8 at both loci ([V.sub.a] = 0.000; [V.sub.d] = 0.000; [V.sub.i] = 0.107). Additive genetic variance maxima ([V.sub.a] = 0.031; [V.sub.d] = 0.016; [V.sub.i] = 0.000) occur when an allele is fixed at one locus and alleles at the other locus have a frequency of 0.15 or 0.85. These maxima correspond to those expected under "pure" overdominance at single loci (Falconer 1989).
With additive-by-dominance epistasis (see Table 1), phenotypic differences among genotypes are additive at one locus (locus B) but differ in direction depending on the genotype at the second locus. At the second locus (locus A), genotypic values display underdominance or overdominance depending on the first locus' genotype. This results in the overall simple average homozygote difference (a) equaling zero at the B locus while dominance (d) is zero within each of the modifying locus genotypes. Moreover, the overall simple average dominance (d) is zero at the A locus while the homozygote difference (a) is zero within each of the modifying locus genotypes.
The level of additive genetic variance for A x D epistasis is plotted against the two-locus allele frequencies in Figure 1C. Minima occur when the alleles are fixed at the locus displaying varying additivity (locus B) and at intermediate values at the locus displaying varying dominance (locus A). At these minima, there is no additive genetic or interaction variance, but dominance variance is at a maximum ([V.sub.d] = 0.187). Maxima ([V.sub.a] = 0.094; [V.sub.d] = 0.049; [V.sub.i] = 0.000) occur when an allele at the B locus is fixed and alleles at the A locus are at frequencies of 0.15 and 0.85. There is also a minor peak of relatively low additive genetic variance ([V.sub.a] = 0.010) when alleles are at intermediate frequency at both loci. This local maximum also has the highest level of interaction variance ([V.sub.i] = 0.094) but no dominance variance. Local maxima also occur when an allele is fixed at locus A and at intermediate frequencies at locus B ([V.sub.a] = 0.042; [V.sub.d] = 0.000; [V.sub.i] = 0.000).
In each case of pure epistasis, the additive genetic variance is lower and the interaction variance higher at intermediate allele frequencies than at extreme allele frequencies. This tendency is especially strong for A x A epistasis. The maximum additive genetic variance produced by D x D and A x D epistasis is much lower, 25% and 75% of the A x A epistasis maximum, respectively.
INCREASE IN ADDITIVE GENETIC VARIANCE AT A POPULATION BOTTLENECK
The dynamics of additive genetic variance during a period of restricted population size was modeled by calculating the average additive genetic variance of populations of constant finite size drawn from a parental population with intermediate allele frequencies at both loci. The frequency distribution of populations displaying various combinations of allele frequencies was obtained using the Markov chain model (Crow and Kimura 1970). The average additive genetic variance was then obtained by multiplying the frequency of populations with a specified set of two-locus allele frequencies by its corresponding additive genetic variance and summing over all possible allele frequency combinations. The additive genetic variance is calculated from the allele frequencies and from genotypic values corresponding to a specific form of epistasis.
The increase in additive genetic variance with succeeding generations of constant finite population size is presented in Figure 2 for population sizes of 2, 8, 16, 32, and 64. Each curve is drawn until 1 generation after the maximum average additive genetic variance or to 50 generations. After the maximum is reached, the average additive genetic variance decreases due to fixation at both loci.
With each form of pure epistasis, there is a dramatic rise in additive genetic variance as the populations pass through the bottleneck. The increase is most immediate and dramatic at the smallest population sizes (N = 2). The maximum average additive genetic variance reached does not seem to depend heavily on population size. An exception is that at N = 2 a significant fraction of populations are fixed at both loci after the founding event, thus limiting the maximum additive genetic variance. For each form of epistasis, larger population sizes continue to increase their level of additive genetic variance over a considerable number of generations, up to 100 generations at population sizes of 64.
The dynamics of increase in average additive genetic variance is relatively simple in the case of A x A epistasis [ILLUSTRATION FOR FIGURE 2A OMITTED]. The maximum average additive genetic variance reached ([V.sub.a] = 0.0625) is one-half the maximum variance for any single population. This maximum is reached at 2 generations at N = 2, 11 generations at N = 8, 22 generations at N = 16, 44 generations at N = 32, and 88 generations at N = 64.
The evolution of additive genetic variance with D x D epistasis takes on a more complex double sigmoidal shape [ILLUSTRATION FOR FIGURE 2B OMITTED]. This dynamic is caused by the nonintermediate minima in additive genetic variance at various combinations of allele frequencies of 0.20 and 0.80. The maximum average additive genetic variance in these finite populations is one-third of the maximum additive genetic variance for any single population. Maximum average additive variance is reached at 3 generations with N = 2, 6 generations with N = 8, 13 generations at N = 16, 56 generations at N = 32, and more than 100 generations at N = 64.
With A x D epistasis, the average additive genetic variance declines precipitously immediately after a bottleneck and then, after a few generations, increases to its maximum average value [ILLUSTRATION FOR FIGURE 2C OMITTED]. The initial decline is more longlasting at larger population sizes but results in a maximum decrease of only 20% of the original additive genetic variance ([V.sub.a] = 0.010 versus 0.008). The rate of variance loss during the first few generations is faster than would be predicted by a purely additive single-locus model. The decrease occurs because additive genetic variance is at a minor local maximum in the additive genetic variance surface in the initial population [ILLUSTRATION FOR FIGURE 1C OMITTED]. In the first few generations after a bottleneck, most populations will deviate slightly from intermediate allele frequencies and thus have a lower additive genetic variance than the parental population. However, as genetic drift continues and allele frequencies become more extreme, average additive genetic variance increases. The maximum average additive genetic variance is approximately 0.027, only a little more than one-quarter of the maximum variance for any single population. Maximum average additive variance is reached at 3 generations with N = 2, 12 generations with N = 8, 25 generations at N = 16, 50 generations at N = 32, and more than 100 generations at N = 64.
The minor intermediate additive genetic variance maximum with A x D epistasis is due to the modifying locus heterozygote displaying a larger additive genotypic value (a) at the focal locus than either of the modifying locus homo-zygotes (see Table 1; 0.578 [greater than] 0.289). Therefore, modifying locus heterozygosity enhances additive genetic variance at the focal locus. This enhancing form of epistasis leads to a dramatic reduction in [V.sub.a] during the first few generations of limited population size as modifying locus heterozygotes become rare. Average [V.sub.a] subsequently increases again due to the contributions of epistasis for dominance values.
These results clearly demonstrate that with epistasis, additive genetic variance can increase as populations pass through bottlenecks. In the long term this is true for all three forms of pure epistasis considered, although a transient decline in additive genetic variance may occur in the first few generations with A x D epistasis. The increase in additive genetic variance is most pronounced for A x A epistasis. The maximum average additive genetic variance with D x D and A x D epistasis is much lower than the maximum average under A x A epistasis.
With pure epistasis, the increase in additive genetic variance is not restricted to the founder event but continues through succeeding generations until populations fixed at both loci become common. The increase continues longer with larger population sizes, indicating that epistasis may contribute to additive genetic variance for quantitative characters even after a long period of finite population size.
We provide two empirical examples of two-locus genotypic values exhibiting epistasis for 10-wk body weight and predict their effects on additive genetic variance when populations pass through a period of finite size. The data are derived from 534 [F.sub.2] mice formed from an intercross of the LG/J and SM/J strains (Cheverud et al. 1996). The locus pairs described here [TABULAR DATA FOR TABLE 2 OMITTED] exhibit extreme levels of epistasis. Body weight was analyzed on its natural scale (grams). It should be kept in mind that the presence or absence and form of epistasis can be profoundly affected by the scale of measurement (Falconer 1989).
Examining the role of epistasis in additive variance evolution is more complex for these empirical examples than for cases of "pure" epistasis because the single-locus genotypic values ([a.sub.A], [d.sub.A], [a.sub.B], and [d.sub.B]) are not zero and also contribute to additive genetic variance. If locus pairs exhibit only homozygous differences (a), genetic drift will always result in a decline in additive genetic variance as allele frequencies deviate from intermediate values (Falconer 1989). However, with dominance the maximum additive variance may occur at more extreme allele frequencies, such as 0.25 or 0.75 with complete dominance or 0.15 or 0.85 with overdominance (Falconer 1989). Therefore, when the parental population has intermediate allele frequencies, dominance can cause increased additive genetic variance during a founding event even without epistasis (Robertson 1952; Willis and Orr 1993). Thus, we must separate the variance effects of epistasis from those of the single-locus, nonepistasis values (see Appendix).
This separation was accomplished by calculating the additive genetic variance separately with and without epistasis. First, the additive variance is calculated using only the nonepistatic values. Then the additive variance was calculated using the total genotypic values. The ratio of the total additive genetic variance to additive genetic variance calculated from the nonepistatic values alone is an index of the effect of epistasis on additive genetic variance, the additive variance ratio (AVR).
D10Mit10 and D11Mit64 are a pair of microsatellite loci developed by Dietrich et al. (1992). These loci show a significant association with adult body weight indicating linkage to a quantitative trait locus (Cheverud et al. 1996). The two-locus genotypic, nonepistatic, and epistatic values are presented in Table 2, along with the homozygous difference (a) and dominance values (d) at each locus for each alternate locus genotype. Epistasis is due to the relatively large weight of the LG double homozygote. This pattern is similar to the example given in Crow and Kimura (1970, p. 126, table 4.1.3). It is clear that in this example the "a" and "d" values for any given locus vary substantially depending on the genotype at the second locus. For the LG homozygous genotypes ([A.sub.2][A.sub.2] and [B.sub.2][B.sub.2]) at each locus, the additive genotypic value (a) is much increased whereas the SM alleles are partially dominant or underdominant. Because the alternate locus heterozygotes have relatively small additive genotypic values (a), this pair of loci display suppressing epistasis in that epistasis results in diminished additive genetic variance at intermediate allele frequencies. Therefore, this pair of loci should show increased additive genetic variance when a population passes through a bottleneck.
The distribution of additive genetic variance at various allele frequencies is shown in Figure 3. The largest additive genetic variances occur when the LG double homozygote represents slightly more than one-half of the population. The additive genetic variance at these allele frequencies is more than 3 1/2 times larger than when allele frequencies are intermediate.
Given a parental population with intermediate allele frequencies, the average increase in additive genetic variance during a population bottleneck for this pair of loci is shown in Figure 4A. Additive genetic variance increases at a substantial rate at all population sizes. This occurs because the increased additive genetic variance due to epistasis [ILLUSTRATION FOR FIGURE 4B OMITTED] overwhelms the loss due to additive genotypic differences. As in the cases of pure epistasis, this increase lasts for a few generations at small population sizes but continues for many more generations as the population size increases. Unlike the cases of pure epistasis, the largest maximum average additive genetic variance occurs at the smallest population size (N = 2) and decreases as population size increases.
[TABULAR DATA FOR TABLE 3 OMITTED]
The additive genetic variance ratio (AVR; [ILLUSTRATION FOR FIGURE 4C OMITTED]) for this pair of loci indicates that additive variance was suppressed at intermediate allele frequencies (AVR = 0.75 at generation zero). Subsequent genetic drift leads to additive genetic variances two or three times higher than if there were no epistasis [ILLUSTRATION FOR FIGURE 4C OMITTED].
The increase in additive genetic variance at these two loci is due entirely to the effects of epistasis, not dominance. At the generation displaying the peak additive genetic variance [ILLUSTRATION FOR FIGURE 4A OMITTED], the additive genetic variance of the nonepistatic values is approximately 50% to 55% of its original value, only 5% larger than expected assuming no dominance or epistasis. This indicates that epistasis itself is responsible for a more than 150% increase in additive genetic variance compared with a situation with no epistasis. Clearly, epistatic interactions between quantitative trait loci marked by D10Mit10 and D11Mit64 can be a potent source of increased additive genetic variance during population bottlenecks.
The two locus genotypic values for the second locus pair, D4Mit17 and D13Mit1, are provided in Table 3. The epistasis at this pair of loci is due largely to the especially small value of the D4Mit17 SM homozygote and the especially large value of the D4Mit17 LG homozygote among D13Mit1 heterozygotes. This results in an especially large additive genotypic value (a) for the D4Mit17 locus within the D13Mit1 heterozygote and contrasting over- and underdominance at the D13Mit1 locus within the LG and SM homozygotes at the D4Mit17 locus. Therefore, this locus pair displays A x D epistasis and should show an initial enhanced rate of decrease in additive genetic variance as the population passes through a bottleneck.
The additive genetic variances at various allele frequencies for this locus pair are displayed in Figure 5. This surface has the appearance of a ridge running from a local maximum of [p.sub.LG] = 0.8, [q.sub.LG] = 0.0 to the global maximum of [p.sub.LG] = 0.25, [q.sub.LG] = 0.95. The maximum additive genetic variance is approximately 1 1/2 times larger than the additive genetic variance at intermediate allele frequencies.
Again, given a parental population with intermediate allele frequencies, the change in additive genetic variance over the generations of genetic drift at varied population sizes is presented in Figure 6A. For this locus pair, additive genetic variance declines at a slightly faster rate than expected given no dominance or epistasis. The enhanced rate of additive genetic variance loss is due to epistasis. The nonepistatic contribution to additive genetic variance declines at approximately the same rate as predicted for a population with little or no dominance. However, immediately after the founding event the epistatic contribution to additive genetic variance declined at an accelerated rate [ILLUSTRATION FOR FIGURE 6B OMITTED] attaining only 86% of its expected nonepistatic value at N = 2, 94% at N = 8, and 96% at N = 16. This represents a 4% to 14% decrease relative to the loss of variance expected with no dominance or epistasis.
The additive variance ratio [ILLUSTRATION FOR FIGURE 6C OMITTED] indicates that epistasis has enhanced additive genetic variance at this pair of loci at intermediate allele frequencies (AVR = 1.40 at generation zero). Subsequent genetic drift leads to a depression of additive genetic variance relative to that expected from single-locus values alone. At smaller population sizes the additive variance is only 40% to 60% of the value it would attain in the absence of epistasis.
We have shown that epistasis can lead to an increase in additive genetic variance at population bottlenecks. This result is general for all forms of epistasis, although pure A x D epistasis does result in an initial, transient decrease in variance prior to a later increase. This theoretical result suggests that models of founder effect speciation (Mayr 1963; Carson 1968, 1982; Templeton 1980a,b) may not be limited by loss of additive genetic variance due to the founding event and subsequent finite population size. Instead, with the kind of epistasis envisioned by Schmalhausen (1949), Waddington (1957), Mayr (1963), and Templeton (1980a), founder events may actually release previously suppressed additive genetic variance allowing rapid response to new genetic and external environments.
Some proponents of founder effect speciation models hypothesized that a period of dramatic population growth must follow the founder event, in part to prevent further loss of additive genetic variance (Carson 1968, 1982; Templeton 1980a,b; Carson and Templeton 1984). Our models, like those of Goodnight (1988), indicate that the gain in additive genetic variance can continue over an extended number of generations. An immediate population flush would tend to limit this further by fixing the variance level at that achieved before the flush. A delayed flush, however, could overcome the loss of heterozygosity and additive genetic variance caused by an extended period of low population size. Contrary to the expectations of an additive model, this indicates that species with slow-growth life-history strategies may be more, rather than less, likely to undergo founder effect speciation.
Our results also confirm and generalize those of previous theoretical work (Bryant et al. 1986; Bryant and Meffert 1993; Goodnight 1987, 1988; Cockerham and Tachida 1988; Tachida and Cockerham 1989: Whitlock et al. 1993; Willis and Orr 1993). These models dealt primarily with A x A epistasis whereas the approach taken here allowed us to also consider arbitrary epistasis patterns and those involving dominance.
Goodnight (1988) describes A x A epistatic variance in the parental population as contributing to additive genetic variance in the finite-sized daughter populations. With our two-locus genetic model of this process, such a transfer is not described because the contributions of physiological epistasis to additive genetic variance are considered directly, not in relation to its contributions to other genetic variance components in the parent population. It is crucially important to distinguish between the genetic phenomenon of epistasis and its contributions to population genetic variance components. Although epistasis is the sole contributor to interaction variance, it also contributes to additive and dominance variance components (Cheverud and Routman 1995). Genotypic values are constant while changes in allele frequency due to drift alter their contribution to the genetic variance components in a complex fashion.
Our results also lend support to Bryant and colleagues' (Bryant et al. 1986; Bryant and Meffert 1993) interpretation of their empirical findings in the housefly. Epistasis for morphological characters may have contributed to the observed increases in additive genetic variances after a population bottleneck. However, it is likely that dominance also contributed to these increases (Willis and Orr 1993).
One problem with previous theoretical and empirical studies of the role of epistasis in additive genetic variance evolution was the difficulty in measuring epistasis in terms of the models. A full evaluation of Goodnight's (1988) or Whitlock et al's (1993) models, requires measuring interaction variance at the population level, a difficult task using the covariance among relatives (Falconer 1989). Bryant and colleagues (Bryant et al. 1986; Bryant and Meffert 1993) were unable to measure epistasis or its contribution to genetic variance components in their experiments.
In contrast, it is possible to measure the extent and pattern of epistasis and its influence on the various genetic variance components with measured genotypes at quantitative trait loci (Cheverud and Routman 1995). Although our models relax the conditions for founder effect speciation, whether such speciation occurs depends on the existence of substantial epistasis for quantitative traits. The recent and continuing molecular and quantitative advances allowing detection of the effects of specified loci on quantitative traits make epistasis measurement and empirical evaluation of its evolutionary role possible (Cheverud and Routman 1993, 1995).
Our results are also important for the maintenance of additive genetic variance in endangered populations. Genetic variation in such populations is often evaluated using the heterozygosity of molecular genetic markers. However, in the presence of epistasis, we have shown that additive genetic variance for quantitative phenotypes can be maintained or even increase as heterozygosity declines. Of course, extended periods of small population size will eventually lead to a loss of both herterozygosity and additive genetic variance. Loss of additive genetic variance due to fixation occurs in a few generations for N = 2 but comes much later for population sizes greater than 16. One possible example of low heterozygosity combined with normal levels of additive genetic variance is the work of Cheverud and colleagues (Cheverud 1996; Cheverud et al. 1994) who found normal levels of heritable variance for body weight and morphological characters in a restricted population of cotton-top tamarins, despite very low levels of allozyme heterozygosity. Perhaps evaluations of endangered species should also include surveys for additive genetic variation in quantitative characters in order to determine whether the extent or duration of the bottleneck has been sufficient to reduce additive genetic variance.
We thank G. P. Wagner, A. Templeton, and C. Goodnight for helpful discussions of the material presented here and S. Beyene, M. Butler, E. Cheverud, K. Cothran, F. M. Duarte, D. Irshick, C. Perel, B. van Swinderen, and N. Vessey for laboratory help. This work was supported by National Science Foundation grant BSR-9106565.
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Genotypic values and population genic values and variances for a two-locus model with epistasis (Cheverud and Routman 1995). Two loci, A and B, are considered with two alleles, 1 and 2, each. Allele frequencies are [p.sub.1]([p.sub.2] = 1 - [p.sub.1]) and [q.sub.1][q.sub.2] = 1 - [q.sub.1]) at locus A and B respectively. When equations are given for locus A only, similar equations can be defined for locus B. Genotypic values are defined without regard to their frequencies.
(1) Two-locus genotypic value ([G.sub.ijkl]) = Average phenotypic value for individuals with genotypes ij at locus A and kl at locus B.
(2) Unweighted marginal genotypic values for locus A and locus B
[G.sub.ij..] = ([G.sub.ij11] + [G.sub.ij12] + [G.sub.ij22])/3
[G.sub...kl] = ([G.sub.11kl] + [G.sub.12kl] + [G.sub.22kl])/3.
(3) Non-epistatic values
[n.sub.ijkl] = [G.sub.ij..] + [G.sub..kl] - [G.sub.....],
where [G.sub.....] is the unweighted average of the [G.sub.ijkl] values.
(4) Epistasis values
[e.sub.ijkl] = [G.sub.ijkl] - [n.sub.ijkl].
(5) Additive genotypic value
[a.sub.A] = ([G.sub.11..] - [G.sub.22..])/2.
(6) Dominance genotypic value
[d.sub.A] = [G.sub.12..] - ([G.sub.11..] + [G.sub.22..])/2.
Population means, values and variances (assuming H - W and linkage equilibrium).
(7) Mean ([Mu]) = [a.sub.A]([p.sub.1] - [p.sub.2]) + 2[p.sub.1][p.sub.2][d.sub.A] + [a.sub.B]([q.sub.1] - [q.sub.2]) + 2[q.sub.1][q.sub.2]B + [e.sub.....], where [e.sub.....] is the population average of the epistasis values.
(8) Population genotypic values
[a[prime].sub.A] = [a.sub.A] + ([e.sub.11..] - [e.sub.22..])/2
[d[prime].sub.A] = [d.sub.A] - ([e.sub.11..] - 2[e.sub.12..] + [e.sub.22..])/2,
where [e.sub.ij..] is the population average epistasis value for the ij genotype,
[e.sub.ij..] = [[q.sub.1].sup.2][e.sub.ij11] + 2[q.sub.1][q.sub.2][e.sub.ij12] + [[q.sub.2].sup.2][e.sub.ij22].
(9) Average effects of alleles ([[Alpha].sub.i]) and allele substitutions ([Alpha])
[Mathematical Expression Omitted],
[Mathematical Expression Omitted],
[[Alpha].sub.A] = [a.sub.A] + [d.sub.A]([p.sub.2] - [p.sub.1])
+ [p.sub.1]([e.sub.11..] - [e.sub.12..]) + [p.sub.2]([e.sub.12..] - [e.sub.22..]).
(10) Additive genetic variance
[Mathematical Expression Omitted].
(11) Dominance deviations ([[Delta].sub.ij])
[Mathematical Expression Omitted],
[[Delta].sub.A12] = 2[p.sub.1][p.sub.2][d.sub.A] - [p.sub.1][p.sub.2]([e.sub.11..] - 2[e.sub.12..] + [e.sub.22..]),
[Mathematical Expression Omitted].
(12) Dominance variance
[V.sub.d] = [[[p.sub.1][p.sub.2](2[d.sub.A] - [e.sub.11..] + [2e.sub.12..] - [e.sub.22..])].sup.2]
+ [[[q.sub.1][q.sub.2](2[d.sub.B] - [e.sub...11] + [2e.sub...12] + [e.sub...22])].sup.2].
(13) Interaction deviation
[I.sub.ijkl] = [e.sub.ijkl] - [e.sub.ij..] - [e.sub..kl] + [e.sub.....].
(14) Interaction variance
[Mathematical Expression Omitted],
where f([G.sub.ijkl]) is the frequency of the ijkl genotype.
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|Author:||Cheverud, James M.; Routman, Eric J.|
|Date:||Jun 1, 1996|
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