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Heredity 84 (2000) 135±142 Received 11 October 1999, accepted 17 December 1999 Short Review The evolution of the G matrix: selection or drift? DEREK ROFF Department of Biology, McGill University, 1205 Dr Pen®eld Ave., Montreal, Quebec, Canada H3A 1B1 The evolution of quantitative characters can be described by the equation Dz GPÿ1 S where Dz is the vector of mean responses, G is the matrix of additive genetic variances and covariances, P is the matrix of phenotypic variances and covariances and S is the vector of selection dierentials. This equation can be used to predict changes in trait values or to retrospectively estimate the selection gradient and is thus a central equation of evolutionary quantitative genetics. Genetic variances and covariances will be reduced by stabilizing selection, directional selection and drift, and increased by mutation. Changes in trait values resulting from directional selection that are comparable with dierences observed among species are readily obtainable in short geological time spans (<5000 generations) with selection intensities so small that they would have an insigni®cant eect on the G matrix (of course it is possible that such changes came about by strong selection over a few genera- tions, followed by long periods of stasis; there is insucient evidence to presently distinguish these two possibilities). On the other hand, observed eective population sizes are suciently small that considerable changes in G can be expected from drift alone. The action of drift can be distinguished from selection because the former produces a proportional change in G whereas the latter, in general, will not. A survey of studies examining variation in G suggests that the null hypothesis that most of the variation can be attributed to drift rather than selection cannot be rejected. However, more research on the predicted statistical distribution of G as a result of selection and/or drift is required and further development of statistical tests to distinguish these two forces needs to be made. What is the G matrix? Of what use is G? Quantitative genetics has long made signi®cant contributions to the development of domesticated plant and animal species. In the last two decades there has been an increasing interest in the use of quantitative genetics to understand the evolution of quantitative traits in natural populations. For domestic organisms the central equation of quantitative genetics is the relationship, R h2S (response heritability ´ selection dierential), or less frequently the two-trait formulation PY CRY rA hX hY rrPX SX , where CRY is the correlated response of trait Y to direct selection on trait X, rA is the genetic correlation between X and Y, and rPi is the phenotypic standard deviation of trait i. However, natural selection does not typically act upon single or even pairs of traits, but rather the whole organism and hence it is more appropriate to consider the multivariate extension of the above equations (Lande, 1979a), Dz GPÿ1 S where Dz is the vector of mean responses (conventionally symbolized by z), G is the matrix of genetic variances and covariances, P is the matrix of phenotypic variances and covariances, and S is the vector of selection dierentials. The matrix combination GP±1 can thus be viewed as the multivariate `equivalent' of h2, with G replacing the additive genetic variance in the numerator and P±1 replacing the phenotypic variance in the denominator. The multivariate response equation has been used theoretically to explore evolutionary trajectories with particular reference to constraints on the direction of evolutionary change (e.g. Via & Lande, 1985; Arnold, 1992; Bjorklund, 1996; Schluter, 1996). With respect to the analysis of natural populations, the response equation has been used to retrospectively reconstruct the selection regime that has given rise to a change over time or among populations (Arnold, 1992), or to predict responses to selection (Morris, 1971; Grant & Grant, 1993; Ro & Fairbairn, 1999). Retrospective selection analysis assumes that both G and P remain constant and that there are no dramatic reversals in the direction of selection (Shaw et al., 1995). Depending upon the nature of the analysis, prediction of response requires either that both G and P remain constant or change in a proportional manner relative to the original matrices. There are two defences for the assumption of constancy. The ®rst is the same as applied to arti®cial selection experiments, namely that although selection will change allelic frequencies and hence genetic variance and covariances (and thus also their phenotypic counterparts), the change will be so small per generation that the equations will remain correct at least for a dozen or so generations (see Ro, 1997 for a review providing support for this assumption). This argument clearly cannot be applied to those cases in which the equations are used to predict changes taking place over hundreds or thousands of generations. For this the assumption is made that directional Keywords: G matrix, genetic covariance, genetic drift, heritability, mutation, selection *Correspondence: E-mail: dro@bio1.lan.mcgill.ca Ó 2000 The Genetical Society of Great Britain. 135 136 D. ROFF or stabilizing selection is suciently weak that any erosion of variance is countered by new variation added by mutation (Lande, 1976a). How weak is weak selection? We can examine this question by asking what selection is necessary to produce a change in mean trait value that is characteristic of some given taxonomic level. A survey of tests of variation in G matrices at dierent taxonomic levels suggests that statistically signi®cant dierences are found at least at the level of species (Ro & Mousseau, 1999). Most studies have focused upon morphological traits and thus we must ask how much variation is characteristically found among species within any given genus. An analysis of variation among species within 32 dierent genera covering a wide range of animals suggests that a typical range is a two- to four-fold dierence in linear dimensions, though dierences larger than tenfold also occur (Fig. 1). For illustration, suppose that h2 0.5 (a typical heritability for morphological traits), VA 25 and the initial trait value is 50. When the top 99.9% of the population is selected each generation then after 5000 generations the trait value will be 50 + (5000)(0.00337) (Ö[0.5][25]) 110, a two-fold increase, while if the top 99% is selected the trait value will be 50 + (5000)(0.02692)(Ö[0.5][25]) 526, a 10-fold increase. Both of these changes are within the range of interspeci®c variation in morphological traits and the former is probably most typical (see above). On a geological time-scale 5000 generations is short and changes of the above magnitude would be considered rapid. A cull of 0.1% of the population per generation is extremely weak (certainly below present levels of detection for most studies). Lande (1976b) estimated the required cull rate per generation to account for change in the tooth dimensions during the evolution of the horse: to obtain the observed changes requires selective deaths of the order of 2 per million individuals, which is considerably weaker than the above rates (99.9998% survive), but the timespan is in the millions rather than thousands of generations. Of course, it is unlikely that selection acts at a constant, minute rate each generation but the important point is that even very low rates of selection can produce very large phenotypic changes in what is no more than a geological `instant'. Thus the hypothesis of weak selection is at least tenable, but requires empirical support (Turelli, 1988). Endler (1986; p. 210) surveyed a large number of studies in which estimates of selection intensities in natural populations were made and found that `natural selection is often as strong as arti®cial selection'. This result does not, however, demonstrate that the assumption of weak selection is invalid. First, it remains to be shown if these are typical or simply those that Fig. 1 Distribution of the ratio R2 among 32 dierent genera (lower panel) with two examples of the distribution of sizes within genera (upper panels). Data sources: birds (Dunning, 1984: note that weight was the trait given, and to make it comparable to a length measurement I used the cube root); mammals (Barbour & Davis, 1969; Schober & Grimmberger, 1987; Kingdon, 1997); reptiles (Conant, 1975); amphibia (Barker et al., 1995); ®sh (Scott & Crossman, 1973); beetles (Lindroth, 1966); leafhoppers (Beirne, 1956); grasshoppers (Otte, 1984); spiders (Locket & Millidge, 1953); cone shells (Wagner & Abbott, 1977). Ó The Genetical Society of Great Britain, Heredity, 84, 135±142. THE EVOLUTION OF THE G MATRIX 137 have been selected a priori because the researchers suspected that the traits in question would be under strong selection. There is little value in selecting traits for which selection intensities cannot be measured although these may represent the majority of traits. Secondly, single or short-term episodes of strong selection will not greatly aect the additive genetic variance (Ro, 1997). What we need to know is the frequency with which episodes of strong selection occur. additive variance, r2A , after one generation to (Keightley & Hill, 1987) How does selection and/or drift alter G? where h2t is the heritability at time t, which changes because r2A changes, k i(i ) x), where i is the intensity of selection and x is the standardized deviation of the truncation point of selection and r2g;t is the genic variance in the population, which is the genetic variance that would be present under linkage and Hardy±Weinberg equilibria. Assuming this to be true in the initial, unselected population we have r2A;0 r2g;0 . The genetic variance introduced by mutation is r2M . At equilibrium the additive genetic variance maintained by selection, drift and mutation, r2A , is obtained from the solution to the quadratic When genetic correlations arise principally from pleiotropy, populations are large and mating is random, the G matrix is determined by the joint eects of pleiotropic mutation and multivariate selection (Lande, 1980). Changes in the selective regime will produce changes in G but there is no a priori general prediction that can be made (or has so far been published). If selection is strong enough to signi®cantly alter the gene frequencies of the traits subject to selection there will be a change in that portion of the G matrix in which the genes have in¯uence, i.e. the additive genetic variance of the target trait plus the additive genetic covariances between the target trait and other nontarget traits. Thus, in general, the change in the G matrix will show no overall general pattern, but will depend upon the selection regime and the traits included in the matrix. In contrast, genetic drift will cause G matrices to diverge but remain proportional to each other (Lande, 1979a). Population bottlenecks can convert nonadditive genetic (co)variance into additive genetic (co)variance (Ro, 1997; chapter 8) but the general eect of genetic drift is to erode additive genetic variance. This should be particularly true for morphological traits in which non additive eects are typically small (Mousseau & Ro, 1987; Crnokrak & Ro, 1995; Derose & Ro, 1999). Non-additive eects are present in life history traits (Crnokrak & Ro, 1995; DeRose & Ro, 1999) and hence changes in the G matrix due to short episodes of genetic drift could be unpredictable. To illustrate the potential importance of genetic drift when there is only additive genetic variance, consider the change in the additive genetic variance of a single trait as a result of drift alone. The proportional decline in the additive genetic variance after t generations is approximately e)t/2Ne (Robertson, 1960). After 5000 generations the proportional decline for an eective population size of 5000 will be 0.61, which is a substantial reduction in a very short geological period. Thus even very large populations can be subject to signi®cant erosion in additive genetic variance due to genetic drift. A review of estimated eective population sizes produced less than 20% greater than 1000 and over 70% in the range 10±500 (see table 8.3 and ®gs 8.2 in Ro, 1997). The populations studied do not represent a random sample (there are few invertebrates) but do suggest that random genetic drift could play a very important role in evolutionary changes in many populations (see also Lande, 1979b). The erosion of additive genetic (co)variance resulting from the combined eects of selection (directional or stabilizing) and drift will be compensated in part or whole by new variation added by mutation. For the in®nitesimal model, truncation selection, drift and mutation will change the Ó The Genetical Society of Great Britain, Heredity, 84, 135±142. r2A;t r2g;t 1 r2M 1ÿ 1 ÿ h2t k 2 2 Ne 1 r2g;tÿ1 1 ÿ ; 2Ne r2A;t1 r2g;t r4A 1 k r2A r2E ÿ 2Ne r2M ÿ 2Ne r2M r2E 0 1 2 which is r2A 2Ne r2M ÿ r2E q 2Ne r2M ÿ r2E 2 8Ne r2M r2E 1 k 2 1 k 3 From the above it is evident that the equilibrium additive genetic variance is approximately proportional to the eective population size. For example, suppose the proportion culled per generation is 0.1%, and the mutational variance is 0.002, which is within the observed range (see tables 9.7 in Ro, 1997), then for eective population sizes of 10, 100, 1000 the equilibrium additive genetic variance is 0.04, 0.40 and 4.0, respectively. Even if we make selection very strong by culling 50% of the population per generation, the respective additive genetic variances are still approximately proportional to Ne (0.04, 0.34, 2.73, respectively). BuÈrger (1993), assuming weak selection, obtained the simple result that at equilibrium under directional selection the additive genetic variance would be equal to 2Ne r2M , which is the equilibrium variance under the neutral mutation±drift balance as derived by Lynch & Hill (1986). What statistical methods are presently available to test for variation among G matrices? Given the G matrix of two or more populations consisting of either the same or dierent species we are in the ®rst instance interested in two questions: ®rst, `can we reject the hypothesis that the matrices came from the same statistical population?'. This null hypothesis supposes that the populations do not dier in population size, selection history or mutational variances. We might be inclined to entertain such a hypothesis when comparing several populations of the same species but it : 138 D. ROFF seems an unlikely scenario when considering dierent species. The second question is, `can we reject the hypothesis that dierences among the G matrices can be accounted for on the basis of drift alone?' This hypothesis does not preclude the possibility that there are dierences in selection history or mutational variances among the populations, but postulates that genetic drift will be relatively so large that it overwhelms such eects. It is important to remember that proportionality of the G matrices does not imply that the mean phenotypes have not been drastically altered by selection. If equality and proportionality can be rejected we accept the hypothesis that dierences are, at least in part, a consequence of variation in selection history and/or mutational eects. The statistical methods proposed to address the above hypotheses can be classi®ed into four groups: element-byelement comparisons, correlation between matrices, maximum likelihood, and the Flury hierarchical comparison. Most analyses have involved only two populations and for simplicity I shall assume that only two matrices are being compared and that the elements of the matrices are `lined up' in two vectors. Element-by-element comparison entails doing a separate test on each element of the matrices (Brodie, 1993; Paulsen, 1996; Ro & Mousseau, 1999; Ro et al., 1999). The simplest such test is to check whether the dierence between the elements is greater than expected by chance, which may be done using a bootstrap approach (Paulsen, 1996), randomization (Ro et al., 1999), or a t-test where the standard errors are derived by using the jackknife (Brodie, 1993; Ro, 1997). The problem with the method is that the elements within a matrix are not independent of each other and there are likely to be so many tests that after correction for multiple tests there is too little statistical power to be useful. However, while the method may not serve as a means of testing for overall dierences it is useful as a `data-exploration' technique, serving to indicate where large dierences might lie. The most commonly used test is to compare the correlation between the matrices. To circumvent the problem of nonindependence, Lofsvold (1986) suggested the randomization test known as Mantel's test. This test has been quite widely used although its statistical validity has been questioned (Shaw, 1992). My objection to the method, apart from those enunciated by Shaw (1992) is that it cannot distinguish the hypotheses presented above. A highly signi®cant correlation between matrices could indicate equality (i.e. yi xi, where yi and xi are the corresponding elements in the two `vectorized' matrices), proportionality (i.e. yi bxi, where b ¹ 1), or the joint eects of drift and selection/mutation (i.e. yi a + bxi, where a ¹ 0). The three hypotheses can be distinguished by a regression approach using the reduced major axis method and randomization to estimate the required probabilities (Ro et al., 1999). Maximum likelihood methods have become particularly popular in quantitative genetic analyses in general; they have been used to address the question of similarity between matrices (e.g. Shaw, 1991; Holloway et al., 1993), but have not been used to distinguish equality from proportionality vs. other types of dierences. A method that shows considerable promise and is itself based on maximum likelihood is the Flury approach which considers a hierarchy of possible dierences. This method looks at the matrices from the perspective of their common principal components: starting from the `top' of the hierarchy the matrices can be (a) identical, (b) proportional, in which case the matrices share identical principal components but their eigenvalues dier by a proportional constant, (c) they can have one or more, but not all, principal components in common, or (d) they can have completely unrelated structures (Phillips & Arnold, 1999; Fig. 2). The argument for analysing the principle components is that evolutionary change will be in¯uenced by the size of the eigenvalues corresponding to the principal components (Bjorklund, 1996; Schluter, 1996). Eigenvalues that are zero indicate directions in which there is no genetic variation and hence in which there cannot be evolutionary change. Very large eigenvalues indicate highly correlated traits, and thus selection will be strongly in¯uenced by the associated principal component. Determining that two matrices dier does not itself tell us which process (drift vs. selection) is the more signi®cant. For this purpose we require some method of partitioning the eect that can be attributed to drift vs. that which can be attributed to selection. The following approach based on mean-square deviations is one possible method. First, we compute the mean-square error for the three models Model 1, Equal matrices MSE1 1X xi ÿ yi 2 c Model 2, Proportional matrices 1X 1X MSE2 xi ÿ b0 yi 2 yi ÿ B0 xi 2 4 2c 2c Model 3, Different matrices 1X 1X MSE3 xi ÿ a byi 2 yi ÿ A Bxi 2 ; 2c 2c where c is the number of variances and covariances, b0 and B0 ( 1/b0) are the slopes of the reduced major axis regression forced through the origin, and a, b, A and B are the parameters of the reduced major axis regression with the intercept included. The MSE is calculated using both `x on y' and `y on x', because these do not each give the same value. We can now ask how much the MSE is reduced by the assumption of a model that only assumes drift (model 2) vs. one that assumes both drift and selection (model 3). To illustrate this I compare the G matrices for two populations of garter snake studied by Arnold & Phillips (1999). Using the Flury hierarchical method the G matrices for females from a coastal population were found to have common principal components but not to be proportional (Arnold & Phillips, 1999). However, it is visually evident that the elements of the matrices do not dier much from proportionality (Fig. 3). This is con®rmed by the comparison of the percentage reduction: model 2 accounts for a 17.1% reduction in the mean-square error whereas model 3 accounts for a 18.3% reduction. Thus 17.1% reduction can be attributed to drift while only 1.3% can be attributed to selection. This does not mean that drift actually caused the divergence, for it is possible that selection caused a proportional change in the matrices. However, in general, Ó The Genetical Society of Great Britain, Heredity, 84, 135±142. THE EVOLUTION OF THE G MATRIX 139 Fig. 2 Schematic view of the Flury hierarchy for two covariance matrices consisting of two traits. The ellipses represent the covariance structure with the axis orientation denoting the principal components and the spread of the ellipses along each axis the eigenvalues. The analysis proceeds from the top to the bottom. selection will cause nonproportional changes. Thus, in the absence of evidence to the contrary, the hypothesis that the major force causing divergence was drift cannot be rejected. What have the analyses of G matrices told us to date? Almost all studies have compared either dierent populations of the same species or dierent species within the same genus. Most have not distinguished between the eects of drift vs. the eects of selection. More importantly many used a correlational test, which as discussed above, cannot distinguish between equality, proportionality and a general linear relationship. The majority of studies comparing populations have found Ó The Genetical Society of Great Britain, Heredity, 84, 135±142. either no signi®cant dierences or signi®cant correlations, whereas comparisons between species have most frequently found dierences (Fig. 4). Of course this does not mean that there are no dierences between populations, simply that the dierence is smaller than can typically be detected given the sample sizes typically used. Another way of approaching the problem is to ask how much of the dierence at the various taxonomic levels is attributable to drift vs. selection. Where signi®cant variation between groups has been found it can almost all be accounted for by drift (Fig. 5). Interestingly, the points showing the greatest `eect' of drift and selection (as indicated by the deviation from the line of equality) are all nonsigni®cant indicating that analyses should consider both the results of hypothesis testing and interval estimation. Also included in 140 D. ROFF Fig. 3 Plot of genetic variances and covariances of two populations of garter snake (data from Arnold & Phillips (1999). Fig. 5 are the results from a selection experiment on body size in D. melanogaster. Comparison of the G matrices of the base and control populations showed signi®cant dierences (Shaw et al., 1995), which, as shown in the plot, can be attributed to drift, as would be expected. Comparisons of the control population with those selected for large or small size suggests that drift has caused even greater deviation of the selected lines from the control. Further, there appears to have been no eect of selection on G in the population selected for small size but an eect in the population selected for large size (Fig. 5). However, these conclusions are suspect because in neither case did the G matrix of the control population dier from that of the selected population (Shaw et al., 1995). Where do we go from here? There is considerable scope for further theoretical analysis of the evolution of the G matrix. In particular, what is the relationship between selection and the deviation from proportionality expected under drift alone? There is also need for more research into both the statistical properties of the proposed tests and further development of methods for estimating the dierence between matrices. I have presented one measure, the reduction in mean-squared deviation, which suggests that most of the dierentiation observed between populations and species is due to drift rather than selection. Drift is a consequence of dierences in eective population Fig. 4 A survey of analyses comparing genetic covariance matrices (data from table 7 of Ro & Mousseau, 1999; plus the garter snake data from Arnold & Phillips, 1999). NSD, Not Signi®cantly Dierent; SC, Signi®cantly Correlated; SD, SC, Signi®cantly Dierent and Signi®cantly Correlated; SD, Signi®cantly Dierent; NSC, Not Signi®cantly Correlated. Ó The Genetical Society of Great Britain, Heredity, 84, 135±142. THE EVOLUTION OF THE G MATRIX 141 Fig. 5 A comparison of percentage reduction in mean-square error attributable to Drift vs. that attributable to Drift and Selection. The data were taken from those studies in table 7 of Ro & Mousseau (1999) in which the necessary variances and covariances were presented, Arnold & Phillips (1999) and Shaw et al. (1995). Sig; matrices were found to be signi®cantly dierent from each other. NS; the null hypothesis of equality could not be rejected. size, which are likely to be a consequence of ecological factors: thus we might expect that variation in G should correlate with both taxonomic and ecological variables. Support for the latter is provided by the study of the genetic correlations between morphological characters in the amphipod Gammarus minus (Jernigan et al., 1994). The G matrix is central to the quantitative genetics of evolutionary processes. 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