CHANGES IN VARIANCE INDUCED BY RANDOM GENETIC DRIFT

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1 3 CHANGES IN VARIANCE INDUCED BY RANDOM GENETIC DRIFT Rough Draft Version 31 July 1998 c 1998 B. Walsh and M. Lynch Please report errors to jbwalsh@u.arizona.edu We noted previously that random genetic drift leads inevitably to the loss of alleles within populations and to the fixation of alternate alleles by different populations. These conclusions extend logically to quantitative characters. We expect the within-population variance of small populations to decline and the mean phenotypes of isolated populations to diverge gradually. There are some interesting surprises however. Throughout this chapter, we will continue to assume that selection is a negligible evolutionary force. We will consider in turn the dynamics of the within- and between-population variances, initially under the assumption that these are due entirely to genetic properties of the base population. Then the covariance between relatives in inbred populations will be considered. The role of mutation will be taken up at the end of the chapter. Within-population Variance Consider a diallelic locus i with a strictly additive genetic basis such that the three genotypic values are scaled to be 0, a i,and a i. The total (and additive) genetic variance caused by this locus is, from Chapter 6, a i p i(t)[1 p i (t)] where p i (t) is the gene frequency at time t. Assuming gametic-phase equilibrium and summing over all n loci, the expected within-population genetic variance is σ A(t) = n a i E{p i (t)[1 p i (t)]} (.1) i=1 From Chapter 1, we know that the expected heterozygosity is a function of the average inbreeding coefficient. Thus, substituting from Equation 1.6, n ( σa(t) = a i p i (0)[1 p i (0)] 1 1 N i=1 e ( = σa(0) 1 1 ) t (.) N e ) t

2 30 CHAPTER 3 (Wright 1951). For a character with a purely additive genetic basis, and in the absence of any replenishing forces, the additive genetic variance within populations is expected to decline exponentially at the rate 1/N e per generation. Complications arising from nonadditive gene action. Robertson (195) first extended the theory to loci with dominance. He obtained the surprising result that rare recessive alleles can cause an initial expected increase in both the additive and dominance components of variance in an inbreeding population. A rare allele will usually be lost from a small population in which case the variance will decline, but it will sometimes increase. In the latter case, if the allele is recessive, the frequency of the extreme genotype must increase. For completely recessive alleles, a temporary inflation of the expected within-population variance will occur provided the initial frequency of the recessive genotype is less than Complete recessivity is not required for an inflation of the variance, although the relevant range of p becomes smaller in less extreme situations. In all cases, the within-population variance declines eventually as loci move towards fixation. Table 1. Factors that contribute to the additive, dominance, and additive additive components of genetic variance in finite populations, with Cockerham s notation on the right. n is the number of loci, n i the number of alleles at the ith locus, p ij the frequency of the jth allele at locus i, α ij the additive effect of the jth allele at locus i, δ ijk the dominance effect at locus i associated with genotype jk, and (αα) jk the additive additive effect of alleles j and k from different loci. All properties are defined from the standpoint of the base population. Additive variance σ A = Dominance variance σ D = n n i p ij αij i=1 j=1 n n i n i p ij p ik δijk i=1 j=1 j<k n n i n n l Epistatic variance σaa =4 p ij p lk (αα) jk i=1 j=1 l=1 k=1 n i Inbreeding depression ι i = p ij δ ijj ι = j=1 Sum of squared locus-specific n inbreeding depressions ι = ι i H i=1 Variance of dominance effects n n i in inbred individuals σdi = p ij δijj ι i D i=1 j=1 Covariance of additive and dominance effects in inbred individuals σ ADI = n i=1 ι i σ A σ D σ AA H n n i p ij α ij δ ijj D 1 i=1 j=1 Robertson considered only a single diallelic locus. Things get much messier with multiple loci and more than two alleles per locus. Only the results for the additive, dominance,

3 CHANGE IN VARIANCE WITH DRIFT 31 and additive additive epistatic components of variance are available (Cockerham 1984a,b, Cockerham and Tachida 1988, Tachida and Cockerham 1989). Even in this case, the temporal dynamics of genetic variance depend on seven properties of the base population (Table 1): the familiar parameters σa,σ D,and σ AA, the inbreeding depression ι, the sum of squared locus-specific inbreeding depressions ι, the variance of dominance effects among inbred individuals σdi, and the covariance of additive and dominance effects in inbred individuals σ ADI. Simplification is possible under certain circumstances. For example, with only two alleles per locus, ι = σd, and if all alleles have frequency 0.5, as in a cross between two pure lines, σdi = σ ADI =0. The contributions that the factors in Table 1 make to the components of genetic variance in a finite population depend upon several one- and two-locus identity coefficients (Figure 1). Of the one-locus coefficients, f is the familiar inbreeding coefficient, i.e., the probability that two gametes are identical by descent. The probabilities that the members of random groups of three and four gametes are identical by descent are γ and δ (not to be confused with the dominance effects, which are subscripted in Table 1). also involves four gametes, but it is the probability that there is identity by descent within, but not across, two pairs of gametes. For randomly mating monoecious populations, the transition equations for these coefficients are functions of N e and t, f t =1 λ t 1 γ t =1 3λt 1 +λt t =1 4λt 1 10λ t + λ t λt 1 λ t 3 5(5N e 3) (.3a) (.3b) (.3c) δ t =1 9λt 1 5λ t +λ t 3 5 3λ t 1 0(5N e 3) + λ t 1(N e 1) (8N e 3)λ t 3 30(5N e 3)(N e 1) (.3d) where λ 1 =1 (1/N e ),λ =[1 (1/N e )]λ 1, and λ 3 =[1 (3/N e )]λ (Cockerham and Weir 1983b). It should be noted that f γ δ. Figure 1. Measures of identity by descent for single loci (f, γ, δ, ) and pairs of loci ( f, γ, ). The large circles denote gametes, and the open and closed dots represent alleles from two loci. Identity by descent is indicated by a horizontal line. The two-locus coefficients (denoted by tildes in Figure 1) refer to joint identity by descent at two loci. f refers to pairs of genes on two gametes. It cannot be less than the product of the separate identity probabilities for each locus, f. γ refers to a situation in which a pair of genes from two loci is in one gamete and the other two genes are in separate

4 3 CHAPTER 3 gametes. is the joint identity by descent of genes (two at each locus) in four different gametes. The transition equations for these double identity-by-descent measures have been derived by Weir and Cockerham (1969) for ideal monoecious populations. They depend upon the linkage parameter ρ =1 cin addition to N e and t. Letting f t = f t +f t 1, γ t = γ t +f t 1,and t = t +f t 1,the coefficients are obtained by use of Equation.3a and the following expression, f γ t+1 = (1+ρ) 4 ρ N e (N e 1)(1 ρ ) N e (N e 1)(1 ρ) 4N e 1+ρ 4N e ρ 4Ne N e 1 4N 3 e (N e 1)[N e+1+ρ(n e )] Ne (N e 1)(N e 1) Ne 3 (N e 1)(N e 3)(1 ρ) 4Ne (N e 1)(N e 1)(N e 3) 4Ne 3 f γ t (.4) The preceding formulae can accomodate changes in population density, such as a flush following a population bottleneck, by incorporating changes in N e in the appropriate generation. Table. Expected components of variance for lines derived from a base population with properties defined in Table 1. The within-population genetic variance is equal to the total of the additive (A), dominance (D), and additive additive (AA) components. Coefficients are computed with Equations.3 and.4. The two-locus terms need to be averaged over all pairs of loci. Source σ A σ D σ ADI σ DI ι ι ι σ AA Within 1 f 1 f ( δ) (f γ)f δ f f 1+f γ A 1 f [f γ ( δ)](f γ)(γ δ) (γ ) ( γ ) 4f f γ D 0 1 3f +( +γ δ) 0 f+δ γf+ γ f γ + 0 AA f f Betw. f ( δ) γ δ f f f + γ+ Total 1+f1 f f f f(1 f) f f 1+f+ f The expected dynamics of genetic variance are determined by summing the products of the seven factors listed across the top of Table and the tabulated coefficients. For example, the within-population dominance variance, expressed in terms of base-population properties is [1 3f +( +γ δ)]σd +(f+δ γ)σ DI +(f+ γ)ι +( f γ+ )(ι ι ). Although these results involve a lot of algebra, they provide a mechanistic explanation for the changes in components of genetic variance that can be induced by small population size. Thus, it can be seen that inbreeding can convert some initial dominance and additive additive variance into additive genetic variance. Although the additive genetic variance within populations must eventually decline to zero with prolonged random genetic drift, with nonadditive gene action it is difficult to predict beforehand whether a bottleneck in population size will lead to an initial flush or an erosion in the additive genetic variance. This depends critically on the relative magnitudes of σa and the six other quadratic components in the base population. The dynamics of the components contributing to the additive, dominance, and additive additive variance are shown in Figure. Note that while the contribution to the additive genetic variance from the base population σa declines each generation, all other contributions first increase before eventually decreasing to zero.

5 CHANGE IN VARIANCE WITH DRIFT 33 Figure. Dynamics of the coefficients for the terms contributing to the additive, dominance, and additive additive genetic variance within populations for a population of size 10 and freely recombining loci, obtained by use of the equations described in the text. The coefficient for the contribution of ι ι to the additive genetic variance is not visible on the scale in the graph. These results apply approximately to any other population size N, if the time scale is transformed by multiplying by N/10. To obtain the actual dynamics of the variance components, the coefficients need to be multiplied by the base-population properties. For example, the additive genetic variance in generation 50 is approximately 0.08(σA + σ ADI)+ 0.04σDI +0.01(σ D + ι )+0.8σAA, while the additive additive variance is 0, and the dominance variance is 0.04(ι + σd ). The gametic-phase disequilibrium that inevitably develops by chance in finite populations causes identity disequilibrium between loci (Weir and Cockerham 1968) individuals that are inbred at one locus are likely to be so at other linked loci. With nonadditive gene action, this contributes to an initial inflation of the within-population variance through the production of extreme phenotypes. The coefficients of the term (ι ι ) in Table are in fact equivalent to measures of identity disequilibrium (Cockerham 1984a). For example, f f is the deviation of the double identity-by-descent from that based on the assumption of independence between loci. Identity disequilibrium also contributes to the coefficients of σaa. Since (ι ι ) and σaa are functions of n(n 1) and n terms respectively, while all of the other components are functions of n terms, identity disequilibrium conceivably can have a large effect on the within-population variance. For the additive genetic variance, Figure shows that the coefficient of ι ι is always very small, but the coefficient of σaa rises to nearly 1 in a little over N generations. Therefore, a substantial additive additive variance in the base population is likely to spawn an increase in the additive genetic variance following a bottleneck. To see how this can happen, consider the following. From the standpoint of any locus, variation in the epistatic interactions with genes at other loci amounts to a reduction in the efficiency with which allelic effects are transmitted from generation to generation segregation and recombination insure that interlocus interactions in zygotes are not transmitted loyally to gametes. However, as genetic drift causes genes to move towards fixation at one or both loci, this variation in the genetic environment is reduced. In Table 3, for example, the A 1 allele is present in genetic backgrounds that lead to five distinct genotypic values in a randomly mating population. If the B allele becomes fixed then the A 1 allele can only be in two backgrounds (A 1 A 1 B B and A 1 A B B ). The epistatic interactions are still present, but they are transmitted more reliably as additive effects (the difference between adjacent pairs of A-locus genotypes is a constant a i). Table 3. A simple two-locus system with epistasis. Elements in the table are the expected genotypic values for the two-locus genotypes.

6 34 CHAPTER 3 A 1 A 1 A 1 A A A B 1 B 1 4a + i 3a a i B 1 B 3a a a B B a i a i These results are highly relevant to the rather intense controversy that exists over the importance of population bottlenecks (founder effects) for the speciation process (Mayr 1954, Templeton 1980, Carson and Templeton 1984, Barton and Charlesworth 1984). Much of the debate derives from verbal arguments, regarding additive and epistatic gene action, that sometimes appear rather intuitive on the surface. However, the preceding examples illustrate that intuition can be quite misleading in this case. It is certainly not true that temporary declines in population size always lead to an erosion of additive genetic variance as emphasized in Mayr s (1954) genetic revolution arguments. In Carson s (1968, 1975) founder-flush theory it is assumed that during a period of population expansion following a bottleneck and under relaxed selection pressures, various types of epistatic interactions (coadapted gene complexes) will be converted into additive genetic variance. Similar arguments are made by Templeton (1980) in his hypothesis of speciation via genetic transilience. The theory indicates that this can indeed happen. Further theoretical work in the vein of Goodnight (1987, 1988) and Tachida and Cockerham (1989) could provide important insights to these issues. No attempts have yet been made to derive formulations to describe the dynamics of variance due to higher order epistatic interactions. We can anticipate that the algebra will be extremely messy here since it would involve descent measures involving three and more loci. For heuristic purposes, we simply point out the approximate case for epistasis involving only additive effects. Assuming freely recombining loci, the contribution of n- locus interactions to the total within-population variance is [(1 + f) n (f) n ]σa n, where σa refers to additive epistatic effects involving n loci, i.e., (1 n f)σ A for additive effects, and (1 + f 3f )σaa for additive additive effects, etc. A simple way to obtain this result is to recall that the expected covariance between relatives x and y is σ G (x, y) = θ xy σa +(θ xy) σaa + +(θ xy) n σa n, where θ xy is the coefficient of coancestry. The total genetic variance is equivalent to the covariance of individuals with themselves, which is obtained by letting θ xy =(1+f)/.The variance between isolated subpopulations is equivalent to the covariance of random members within the same subpopulation, which is obtained by letting θ xy = f. Thus, for any n-locus interaction, the contribution to the total genetic variance is (1 + f) n σa and to the between-population component of variance is n (f) n σa n. For n =, 3, and 4, the contributions to the within-population variance are maximized when f is approximately 0.33, 0.55, and 0.66, which for randomly mating populations occur at 0.8N e, 1.6N e, and.n e generations. For these same n, the peak contributions to the total within-population genetic variance are magnified by factors of 1.3,.4, and 4.4. Thus, even if levels of higher-order epistatic genetic variance are relatively low in a base population, they may have a substantial impact on the within-population variance under inbreeding, and this may continue to increase for many generations. To this point we have emphasized ideal monoecious populations that become inbred via random genetic drift. In fact, the system of equations given in Table applies to any mating system, provided the formulae for the identity coefficients are modified appropriately. For monoecy with the avoidance of selfing and for separate sexes, see Weir et al. (1980) and Weir and Hill (1980). Explicit formulae for obligate self-fertilization, full-sib mating, and other special systems of mating are given in Cockerham and Weir (1968, 1973) and Weir and Cockerham (1968). A useful review is provided in Cockerham and Weir (1977).

7 CHANGE IN VARIANCE WITH DRIFT 35 Sampling error. It cannot be emphasized too strongly that Equation. gives only the expected change of the within-population variance for a neutral quantitative character. Due to the stochastic nature of random genetic drift, departures from the expectation can be expected to arise in any individual population. Thus, we denote the realized additive genetic variance for any particular population by σ A (t). Estimation error on the part of the investigator aside, three sources contribute to the variation in σ A (t): 1) variation in the genetic variance among founder populations caused by sampling, ) subsequent random departures of the within-population heterozygosity from its expectation caused by drift, and 3) deviations from Hardy-Weinberg and gametic-phase equilibrium. The quantification of these sources of variation is difficult, but some general conclusions have emerged. The additive genetic variance within a particular finite population can be written as σ A(t) = σ A(t)+ σ HW (t)+ σ L (t) (.5) where σ A (t) is the variance due to the true gene effects expected if the line were expanded into an infinitely large, randomly mating population, while σ HW (t) and σ L (t) are transient covariances of genic effects within and between loci caused by deviations from Hardy- Weinberg and gametic-phase equilibria. The expected value of σ A (t) is σ A (t), and the disequilibria are equally likely to occur in positive and negative directions in the absence of selection. Thus, the expected value of σ A (t) is also equal to σ A (t). Each of the terms on the right of Equation.5 has a variance associated with it. The variance of the true additive genetic variance may be written σ [ σ A(t)] = n a 4 i σh i (t) (.6) where σ H i (t) is the variance in heterozygosity, H i (t) =p i (t)[1 p i (t)], at locus i among replicate populations t generations after divergence. For loci with two alleles, σ H i (t) =H i (0) [ i=1 λ t 1 5H i (0)/ 1 5H i(0)/ 5H i (0)/ ( 1 3 ) ] t λ t 1 N e where λ 1 =1 (1/N e ) (Bulmer 1980). While the dynamics of σ [ σ A (t)] will depend on the initial gene frequencies at all loci, which are generally unknown, a useful qualitative statement can be made. For fixed initial genetic variance in the base population, a i must scale inversely with the number of loci. Thus, since σ [ σ A (t)] is the sum of n terms, each a function of a 4 i n, it must be inversely proportional to n. Therefore, for characters with large effective numbers of loci, deviations from the true additive genetic variance caused by variance in heterozygosity are potentially of minimum importance. The expected variance of the within-population variance resulting from Hardy-Weinberg deviations is σ [ σ HW (t)] σa 4 (t)/n e (Bulmer 1976a, 1980). The variation due to gameticphase disequilibrium is more substantial and its computation is quite laborious. The general details appear in Avery and Hill (1977) and Bulmer (1980). Regardless of the degree of linkage, σ [ σ L (1)] σa 4 (0)/N e initially. Thereafter, for the special case of unlinked loci, σ [ σ L (t)] 5σA 4 (t)/3n e. With linkage σ [ σ L (t)] is necessarily larger, but for most cases it will not be substantially so (Avery and Hill 1977). Regardless of the state of disequilibrium in the base population, the expected value of σ [ σ L (t)] is almost always attained within five generations. The expressions for the variance of the components of the within-population genetic variance are in terms of measureable quantities, but to achieve this useful property, several assumptions (ideal population structure, no association between map distances and effects (.7)

8 36 CHAPTER 3 of genes, additivity of gene effects) were made. Violations of any these will tend to inflate the variance of σ A (t). Thus, summing over the two disequilibrium sources, we find that σ [ σ A (t)] must be at least 8σ4 A (t)/3n e. These theoretical results have important implications for the interpretation of observed changes of genetic variance in small populations. Estimates of σ A (t) derived from a small number of replicate populations, even over several generations, provide unreliable assessments of the expected dynamics of σa (t). Averaging over L independent lines, the sampling variance of the mean genetic variance within lines is approximately 8σA 4 (t)/3ln e. Therefore, if it is desirable to keep the standard error at a level of 10% of the expectation, the design must be such that N e L 70, i.e., approximately 70 lines of N e =4or 17 of N e =16. For self-fertilizing lines, the sampling variance is closer to 7σA 4 (t)/l over the first five generations of inbreeding (Lynch 1988c), so at least 700 lines would have to be monitored. In practice, it is useful to set the target number of lines even higher than these estimates, since the additional variation due to parameter estimation [the deviation of the observation Var(A, t) from the realized parameter σ A (t)], which may be considerable, has been ignored. There is another problem. The σ A (t) observed in successive generations are not independent. The minimum correlation is one-half for unlinked loci. Thus, if the genetic variation within a particular population exceeds the expectation due to chance in one generation, it is likely to remain in excess for several consecutive generations. When this problem is confounded with the sampling variance described above, there is a substantial possibility that σ A (t) for a particular replicate may on occasion increase for several generations, contrary to the expected trend. See Avery and Hill (1977) and Bulmer (1980) for further results. In summary, even in the case of purely additive gene action, a reliable tracking of the expected dynamics of the additive genetic variance requires a large number of replicate populations. There are three levels at which sampling error plays a role. In any single replicate, the variance observed by the investigator, Var(A), is likely to deviate substantially from the parametric value σ A. In turn, the realized variance σ A may deviate considerably from that expected if all genes in the replicate were in equilibrium within and between loci, σ A. Finally, sampling error will also cause σ A to deviate from the global expectation σ A. One can expect the situation to get even messier in the presence of nonadditive gene action. The details of the sampling theory have not been worked out for this case, although considerable attention has been given to the expected dynamics of the variance components. The data. Despite the central importance of the theory of small populations for practical issues (such as genetic conservation), there are remarkably few empirical studies on the consequences of inbreeding for the phenotypic variance within or between populations. Most studies have yielded essentially linear declines in the phenotypic variance with f or such a noisy response that no general conclusion can be drawn (Figure 3). Only a few of the existing studies have employed controls, and even these have not used the controls in the formal analysis. [As in the case of inbreeding depression analysis, control lines can be treated as a covariate to remove temporal trends in the variance unassociated with inbreeding (Lynch 1988c)].

9 CHANGE IN VARIANCE WITH DRIFT 37 Figure 3. Response of the average within-line phenotypic variance and the between-line phenotypic variance to inbreeding. References and system of mating from top to bottom: a) Horner and Weber (1956), selfing; b) López-Fanjul and Jódar (1977), full-sib mating, control-corrected; c) Kidwell and Kidwell (1966), full-sib mating, control-corrected; d) Bateman and Mather (1951), selfing. Fitted lines are the expectations based on the reported estimates of σ A and σ E, where available. The two plant studies were initiated with a cross between two varieties. Some empirical evidence has emerged for an increase in additive genetic variance with inbreeding. López-Fanjul and Villaverde (1989) took 16 replicate populations of Drosophila melanogaster through single generations of full-sib mating and assayed them, along with parallel controls, for egg-to-pupa viability. The average additive genetic variance in the control lines, 19.8±11.8, was not significantly different from zero, whereas that in the inbred lines (f =0.5), 98.1±36.7, was quite substantial. The character exhibited significant inbreeding depression, consistent with the existence of dominance genetic variance in the base population. Bryant et al. (1986a) put populations of houseflies through single-generation bottlenecks of 1, 4, and 16 pairs, and then rapidly expanded them for several generations prior to the measurement of the additive genetic variance. Such random mating should have reduced the variation in the within-line variance caused by gametic-phase disequilibrium. For several morphological characters, their analysis suggested an increase in σa in the bottlenecked lines relative to a control (Figure 4). The authors suggested that this was a consequence of the conversion of epistatic to additive genetic variance. Several potential problems with this experiment highlight the extreme difficulties that exist in tracking the dynamics of genetic variance. First, only L =4replicate populations were maintained at each population density. The minimum coefficients of variation of the additive genetic variance, caused by the drift process alone under an additive model, are 8/3N e L =0.59, 0.30, and 0.15 for the three bottleneck sizes, and this does not include sampling error on the part of the investigator. There is therefore a substantial chance that the within-line variance may have increased by chance even if the genetic variance was purely additive. Second, since the eight characters studied were correlated genetically, they should not be interpreted as independent assays. Third, some of the characters exhibited a five-fold inflation in the additive genetic variance over the control. This is hard to accept as a real consequence of inbreeding in the 16-pair lines, which were essentially non-inbred (f 0.03). If one regards the latter lines as the control and takes the previous two problems into account, the evidence that inbreeding created additive genetic variance in these lines is weakened substantially.

10 38 CHAPTER 3 Figure 4. Additive genetic variances for eight morphometric traits averaged over four replicate lines of bottlenecked housefly populations. Horizontal lines connect variances that were not significantly different at the 0.05 level. C denotes a large randomly mating control population. WLdenotes wing length, WW wing length, HW head width, SL scutellum length, IE inner eye separation, SW scutellum width, ML metafemur length, and TS thoracic suture length. From Bryant et al. (1986a). The importance of the problem of the variance of the within-population variance is illustrated by a large experiment performed by López-Fanjul et al. (1989). Starting from a large random-bred base population of D. melanogaster, they produced 75 lines, which were maintained by full-sib mating for four generations and then expanded for an additional six generations. The components of variance for abdominal bristle number were evaluated for the initial lines (f =0)and for the fourth and tenth generations (both f =0.5) by several techniques including sib analysis. Consistent with the view that this character has a strictly additive genetic basis, no change in the mean was induced by the inbreeding (Table 4). Moreover, there was an approximately 50% reduction in the additive genetic variance in the inbred lines, as the additive theory predicts. The data from this experiment are in excellent accord with the sampling theory presented above. Summing the expected variances contributed by Hardy-Weinberg and gametic-phase disequilibria, the expected value of the coefficient of variation for the additive genetic variance at time t =1,which is based on random populations with N e 8, is (/N e ) 1/ = This is reasonably close to the observed value For both of the inbred generations, the observed values of CV[Var(A)] are close to the theoretical minimum 8/3N e =1.04 (using N e =.5for full-sib mating). Several generations of random mating were expected to cause a reduction in CV[Var(A)], through an elimination of gametic-phase disequilibrium, but this was not observed, perhaps because most of the constituent loci are in tightly linked chromosomal regions. Table 4. Changes in mean phenotype and within-line variance components for abdominal bristle number in Drosophila melanogaster with inbreeding. From López-Fanjul et al. (1989). Generation f z Var(A) Var(E) CV[Var(A)] Despite the theoretical importance of the subject, it is still unclear whether the special mixes of gene frequencies and genetic effects required for flushes of additive genetic variance under inbreeding are common in natural populations. It should also be emphasized that an enhancement in the level of additive genetic variance upon inbreeding will not necessarily have beneficial evolutionary effects. If directional dominance is important, such an increase

11 CHANGE IN VARIANCE WITH DRIFT 39 will be accompanied by inbreeding depression, which may offset the population s ability to respond to directional selection. Much more empirical work is needed in this area. Little work has been done on the consequences of population bottlenecks for the covariance between characters. Lande (1979) has shown that, in the absence of selection and nonadditive gene action, the expected change of genetic covariance within populations follows the same dynamics as the variance. Consequently, the expected genetic correlation is not expected to be altered much by random genetic drift. However, the variance in realized outcomes is a difficult theoretical problem (Avery and Hill 1977). Bryant and Meffert (1988) have demonstrated substantial changes in the degree of morphological integration in bottlenecked housefly populations (as measured by genetic correlations), but it remains to be seen whether these differences exceed the expectations for neutral additive characters. Between-population Variance While random genetic drift ultimately leads to a reduction in genetic variance within populations, it encourages diversification of the mean phenotypes of isolated populations. One measure of the divergence of population means is the between-population variance. In principle, observed changes in the variance between populations can be used to test hypotheses regarding evolutionary mechanisms. For example, an observed divergence of isolated lines that is significantly less than the neutral expectation provides evidence of stabilizing selection. The reverse suggests the possibility of diversifying selection. In this section, we will take as a null model the special case in which all gene action is additive and random genetic drift is the only evolutionary mechanism. Because many of the predictions of this neutral model can be expressed in terms of observable quantities (additive genetic variance in the base population, effective population sizes of lines), it provides a useful reference point. Other null hypotheses are possible, but the need to incorporate more parameters makes their implementation much more difficult. The expected between-population genetic variance, σḡ(t), under the null model is obtained by noting that the mean genotypic value at locus i is a i p i. The variance between populations for this locus is then E{[a i p i (t)] } {E[a i p i (t)]}, which simplifies to 4a i σ p i (t) where σp i (t) is the expected between-population variance in gene frequency. Summing over all loci, assuming negligible gametic-phase disequilibrium, and substituting from Equation 1.9, [ n ( σḡ(t) =4 a i p i (0)[1 p i (0)] ) ] t N e i=1 =f t σ A(0) (.8) Thus, the expected variance between the genotypic means of isolated populations increases linearly with the inbreeding coefficient, asymptotically approaching a limit equal to twice the additive genetic variance in the base population (Wright 1951). This limiting result may be obtained in a simpler manner. When the process of random drift is allowed to go to completion, a proportion p i (0) of the populations will have genotypic value a i, and a proportion 1 p i (0) will have genotypic value 0. The mean genotypic value is therefore a i p i (0) and the mean square is 4a i p i(0), which yields the between-population variance 4a i p i(0)[1 p i (0)] = σ A i (0). The above results hold regardless of the number of alleles at a locus. A few words should be said about the potential importance of nonadditive gene action. From Table, it can be seen that with dominance and additive additive epistasis, the

12 40 CHAPTER 3 between-population variance asymptotes eventually at σḡ =σa +σ ADI + σdi +4σ AA. With only two alleles per locus, the terms involving dominance drop out. With more than two alleles per locus, dominance can magnify or reduce the between-population variance depending upon the signs and magnitudes of σdi and σ ADI, since the latter can be negative. As noted above, a more general expression for the asymptotic contribution of additive epistasic effects involving sets of n loci, assumed to be unlinked, is n σa n. Thus, epistasis involving large numbers of loci can greatly magnify the development of between-population variance, even if it appears to be of relatively minor importance within populations. Sampling error. We will now consider the sampling properties of the between-population variance by reference to a particular type of experiment, assuming a character with a strictly additive basis (Hill 197b, Lynch 1988c). Starting from a base population with additive genetic variance σa (0), Lreplicate lines are extracted and maintained with N/ random monogamous matings occuring in each line each generation. Due to the fact that only a finite number of lines is studied, the between-population variance that actually develops in any particular experiment, σ ḡ(t), will deviate from the expectation σḡ(t) somewhat. Moreover, due to finite sample sizes within populations, the between-population variance estimated by the investigator, Var( z,t), will deviate from σ ḡ(t). The first source of variation (the realization variance) is a function of population genetic structure and, for a fixed system of mating, is largely beyond the control of the investigator. The second source of variation (the sampling variance) can be minimized by the use of large sample sizes. In an experiment of this nature, we make inferences about the genetic divergence of lines from observations on the mean phenotypes of the lines, the expected variance of which is σ z(0) = σ z(0) N = σ A (0) + σ E g + σe s (.9a) N σ z(t) ( ) 1 = N +f t σa(0) + 1 ] [(1 f t 1 )σ Nn A(0)+nσ Eg +σ Es for t>1 (.9b) where n is the number of offspring measured from each full-sib family. The variance in generation 0, which has been ignored in Equation.8, is due entirely to the sampling of N individuals for each founding population. The first term in Equation.9b represents the true dispersion of genotypic means, σḡ(t), and is cumulative over generations. The second term refers to the variance due entirely to the limitations of finite sample size. It contains the segregational variance within full-sib families, (1 f t 1 )σa (0)/, plus the variance caused by maternal and special environmental effects, σe g and σe s. The exact structure of this term depends on the family structure. In this case, where there are N/ full-sib families, the segregational and special environmental effects variance are discounted by the total sample size (Nn/), whereas the contribution from maternal effects is reduced by the number of mothers (N/). Due to inheritance, there is also a covariance between the mean phenotypes of lines in different generations, the expectation of which is σ z (0,t)= σ A (0) for t>0 (.10a) σ z (t, t )= N ( 1 N +f t ) σ A(0) for 0 <t<t (.10b) Ideally, the environmental sources of variance, which contribute only to the withinpopulation variance, should be eliminated from the observed variance of mean phenotypes

13 CHANGE IN VARIANCE WITH DRIFT 41 in order to obtain an estimate of the divergence caused solely by genetic causes. This is readily accomplished by a nested analysis of variance, the observed mean squares being those for populations, families within populations, and individuals within families. The within- and between-population components of variance can be separated by equating the observed mean squares to their expectations, as we assume in the following. Suppose that the same experiment has been replicated many different times, each time taking L lines from the same base population. Due to the variation in the drift process and the finite number of observed lines, each experimental set of lines develops its own temporal pattern of realized between-population variance. The theoretical variation in the realized variance among replicate experiments provides a measure of confidence that one can have in the results of any single experiment. Let σ ḡ(t) be the realized between-population variance at generation t for a particular experiment. The expected variance of this quantity among experiments is σ [ σ ḡ(0)] σ4 A (0) (L 1)N (.11a) σ [ σ ḡ(t)] A 1 L 1 N ft +σ N f(t) 4σ4 (0) [ ( ) ] for t>0 (.11b) The term σf (t), which will be considered below, represents the between-population variance in the realized inbreeding coefficient. The sampling covariance among estimates of the realized variance obtained in different generations provides a measure of the degree of nonindependence of variance estimates derived from the same set of lines in different generations, σ[ σ ḡ(t), σ ḡ(t )] 4σ4 A (0) L 1 [ ( 1 N ) ] f t f t +λ t t 1 σ N f(t) for t<t (.1) The variation described by these formulae is an inevitable consequence of the randomness of the drift process. Since the expected divergence is f t σa (0), and since Equations.11 1 do not include sampling error on the part of the investigator, the coefficient of variation of the between-population variance can be no less than /L. This implies that studies of phenotypic divergence need to be very large to be statistically reliable. For example, if it is desirable to reduce the standard error of the between-line variance to 10% of the expectation under the null hypothesis, approximately 00 lines need to be sampled (closer to 300 and 400 for the cases of full-sib mating and selfing, Lynch 1988c). The additional problem of variance in inbreeding has been covered in Lynch (1988c), drawing heavily from the results of Weir et al. (1980) and Cockerham and Weir (1983). Under many mating schemes, some individuals mate by chance with closer relatives than do others. This results in variation in f among members of the same population, and because of sampling, accumulates as between-population variance in f. The expected value of σf (t) under different systems of mating is of special interest since empirical studies usually do not record the essential pedigree information for its computation. For freely recombining loci, σf is zero when the pedigree structure is fixed obligate selfing, full-sib mating, the maximum avoidance systems of Wright (191b) and the circular systems of Kimura and Crow (1963). Even with fairly tightly linked loci, there is little reason for concern with σf (t) in any generation of self-fertilization or full-sib mating. However, with larger N, especially if the sexes are separate and matings are monogamous, the squared coefficient of variation σf (t)/f t can attain values of 0.1 to 1.0 in the first to 4 generations of isolation, even with unlinked loci. This is enough to contribute significantly

14 4 CHAPTER 3 to σ [ σ ḡ(t)]. After six or so generations have passed, σf (t) can be safely ignored regardless of the population size, even with tightly linked loci. When a temporal sequence of estimates of the between-population variance is available, these may be regressed on f t to test the null hypothesis of neutral additive genes. The expected slope of such a regression is σa (0), and the intercept estimates the baseline environmental contribution to population divergence. However, since consecutive estimates obtained from the same lines are nonindependent, a fundamental assumption of least-squares analysis is violated. For example, once the lines have become completely inbred, then all future values of σ ḡ(t) are fixed; they should not be given equal weight in the regression analysis. Lynch (1988c) has given approximate expressions for the standard errors of the slope and intercept that account for the intrinsic correlations in the data. The variance of the regression coefficient increases with the duration of the experiment, but is essentially constant after the fourth generation of inbreeding. At that point, the standard error ranges from approximately 4σA (0)/ L under obligate self-fertilization to 3σA (0)/ L with larger N. Thus, conservatively speaking, to reduce the standard error to 10% of the expectation σa (0), approximately 360 lines should be monitored under self-fertilization and 00 with larger N. For the case in which one has only a single estimate of the between-population divergence, Var( z,t), Lande (1977b) suggested the statistic F = Var( z,t) t Var(A, 0)/N e (.13) as a test for neutrality. The denominator of this expression is an estimate of the expected between-population variance, which is obtained from f t σ A (0) tσ A (0)/N e for t<<n e. Under the assumption of a normal sampling distribution of population mean phenotypes, F is expected to be F -distributed. This requires that the numerator is χ -distributed with variance equal to [f t σ A (0)] /(L 1). Ignoring the added contribution from sampling error, this can be seen to be true for large N by reference to Equation.11b. However, with selfing and full-sib mating, the expected variance is about twice and 1.5 times too high respectively. Thus, Lande s approach should be restricted to lines with at least moderate effective size. As we will see below, all of the formulae in this section become questionable for t>n e,since they ignore the contribution from new mutations. Example 1. Lande (1977b) used the preceding test statistic to evaluate the results of a 1- year divergence experiment involving five populations of Drosophila pseudoobscura (Anderson 1973). Two of the populations had been maintained at 5 C, two at 7 C, and one at 16 C. They were then raised in two common environments (16 and 5 C) and measured for wing length. Estimates of the additive genetic variance for these two environments were 0.88 and 0.77, while the between-population variances were approximately 6.6 and 4.37 respectively. An approximate upper bound for the number of generations of divergence is t = 150, whereas the effective population size probably always exceeded N e = 1000; the use of these extreme bounds gives conservative estimates of F, making it more difficult to demonstrate diversifying selection on wing length. Even so, the ratios of observed to expected between-population variance are 50 and 38, both of which are highly significant (comparing these with the critical F ratio with four degrees of freedom in the numerator, and infinite degrees of freedom in the denominator). Thus, the hypothesis that the observed line divergence is solely attributable to random genetic drift can be rejected confidently. More likely, the different thermal conditions resulted in selection for different wing lengths.

15 CHANGE IN VARIANCE WITH DRIFT 43 The data. As an example of the application of the preceding formulae, consider the results of a large drift experiment with laboratory cultures of Tribolium castaneum (Rich et al. 1984). The authors followed 1 replicate populations at four population sizes (1:1 sex ratio, random mating) over 0 consecutive generations. Each generation, the mean pupal weight (in µg) of each population was obtained from a bulk sample of 100 random individuals. The additive genetic variance was estimated to be 460 in the base population. The observed Var( z,t) are plotted as a function of f t in Figure 5, along with the expected divergence 90f t (solid lines). The authors argued that the striking downward trend in Var( z,t) in the last few generations of three of the four treatments was due to the suppression of random drift and the operation of stabilizing selection. However, this could also have been a response to a shift in the environment that influenced the expression of variation. A control would have been useful. Figure 5. Observed and expected levels of the between-population variance for pupal weight in a divergence experiment with the flour beetle Tribolium. The lines are described in the text. Data from Rich et al. (1984). The dashed lines in Figure 5, obtained by use of Equation.11, give the limits of the between-population variance beyond which there is less than a 5% chance for the realization of the drift process in either direction. Since these bounds are based on a χ distribution and also ignore measurement error, they may be regarded as conservative confidence limits. Nevertheless, almost all of the observations, with the exception of the clusters of the late generations at N =10and 0, lie within these limits. The least-squares regressions of the data are given by the dotted lines. The slope of each regression is less than the expected 90, but all are within two standard errors of the expectation. Thus, this fairly conservative analysis indicates that the observed patterns, even in the absence of a control, are consistent with a hypothesis of random drift of neutral additive genes. There is a significant probability that the observed declines in Var( z,t) late in the experiment at the two smallest population sizes arose by chance, and due to inter-generational correlations remained there. The results of some other short-term divergence experiments are given in Figure. With one possible exception, there is no evidence for nonlinear increases in the betweenpopulation variance with inbreeding. Eisen and Hanrahan (1974) have argued that the divergence of inbred lines of mice is more rapid than can be accounted for by the additive genetic variance in the base population (Figure 6), and Bryant et al. (1986b) suggested the same for their bottlenecked housefly lines. In neither case, however, has it been verified that the departures from expectations are significant.

16 44 CHAPTER 3 Figure 6. Change in the between-line variance for life-history characters in mice as a function of inbreeding, compared to the expectations (solid lines) based on the additive model. Succesive points are based on two 8-pair lines, two 4-pair lines, four first cousin-mated lines, and eight full-sib mated lines. From Eisen and Hanrahan (1974). Covariance Between Inbred Relatives When individuals are inbred with respect to the base population, the expressions for the genetic covariance between relatives can become quite complicated if there is any nonadditive gene action. We will assume an absence of epistasis initially, and even in this simple case all of the new terms introduced in Table 1 must be considered. There are three new issues to consider. First, inbreeding causes a statistical dependence between alleles within individuals, and this creates a covariance between the additive effects in one relative and the dominance effects in the other. Second, if two individuals have identical genotypes by descent, their dominance covariance will differ depending on whether they are inbred since inbred individuals cannot be heterozygous. Third, with dominance, the mean phenotype of inbred individuals will differ from that of noninbred individuals, and this can inflate the covariance between certain types of relatives by breaking the population up into classes of inbred vs. noninbred individuals. Harris (1964) and Gillois (1965) first derived an expression for the covariance between inbred relatives assuming an absence of both epistasis and gametic-phase disequilibria. Cockerham (1984a) extended their analyses to allow for gametic-phase disequilibrium. The genetic covariance between individuals x and y is σ G (x, y) =Θ xy σ A + 7 σ D + 1 σ DI +( )σ ADI +( f x f y )ι +( f x f y )(ι ι ) (.14) where the coefficients have been defined in Figure (Chapter 11), and the variance components are defined from the standpoint of the noninbred base population. is the average joint probability of identity by descent at one locus in individual x and at a second locus in individual y. It depends upon the linkage relationships between all pairs of loci (Weir and Cockerham 1968, 1969), and in most cases is likely to be approximately equal to f x f y, so the last term in Equation.14 may be of negligible importance. One conclusion that can be drawn immediately from Equation.14 is that with inbreeding, dominance can contribute to the covariance between all types of relatives. All of the terms in this equation are necessarily positive except σ ADI, which can be positive or negative. Thus, while it is likely that inbreeding will inflate the covariance between relatives, this cannot be stated with certainty.