A CENTENNIAL CELEBRATION FOR QUANTITATIVE GENETICS
Quantitative genetics is at or is fast approaching its centennial. In this perspective I consider five current issues pertinent to the application of quantitative genetics to evolutionary theory. First, I discuss the utility of a quantitative genetic perspective in describing genetic variation at two very different levels of resolution, (1) in natural, free-ranging populations and (2) to describe variation at the level of DNA transcription. Whereas quantitative genetics can serve as a very useful descriptor of genetic variation, its greater usefulness is in predicting evolutionary change, particularly when used in the first instance (wild populations). Second, I review the contributions of Quantitative trait loci (QLT) analysis in determining the number of loci and distribution of their genetic effects, the possible importance of identifying specific genes, and the ability of the multivariate breeder's equation to predict the results of bivariate selection experiments. QLT analyses appear to indicate that genetic effects are skewed, that at least 20 loci are generally involved, with an unknown number of alleles, and that a few loci have major effects. However, epistatic effects are common, which means that such loci might not have population-wide major effects: this question waits upon (QTL) analyses conducted on more than a few inbred lines. Third, I examine the importance of research into the action of specific genes on traits. Although great progress has been made in identifying specific genes contributing to trait variation, the high level of gene interactions underlying quantitative traits makes it unlikely that in the near future we will have mechanistic models for such traits, or that these would have greater predictive power than quantitative genetic models. In the fourth section I present evidence that the results of bivariate selection experiments when selection is antagonistic to the genetic covariance are frequently not well predicted by the multivariate breeder's equation. Bivariate experiments that combine both selection and functional analyses are urgently needed. Finally, I discuss the importance of gaining more insight, both theoretical and empirical, on the evolution of the G and P matrices.
If we take the speculations of Yule presented at the Third International Conference of Genetics (Yule 1906) on the relationship between the biometrical and Mendelian approaches to heredity as the foundations upon which the field of quantitative genetics is based, then we have just passed the 100th birthday of quantitative genetics. On the other hand, the 1918 publication of Fisher might be taken as the real beginnings of quantitative genetics as we know it (for a detailed account of Yule's musings in relation to Fisher's contribution see Tabery 2004), in which case the centenary of quantitative genetics birth is but a few years away. No matter which date we take, the fact remains that quantitative genetics has been around for a long time, during which it has developed with a very large statistical foundation that is still in the process of being tested. Early work focused on the contribution of quantitative genetics to animal and plant breeding but the work of Russell Lande in the 1970s promoted the use of quantitative genetics by those interested in evolutionary biology. A significant difference between the interests of the breeder versus the evolutionary biologist is that whereas the breeder frequently is concerned with the improvement of a single trait (or two traits combined into a single index), the evolutionary biologist is generally faced with addressing the evolution of multiple traits simultaneously. This change in focus has raised issues of methodology and approach that are still being worked out: the purpose of this perspective is to suggest that a singular contribution that quantitative genetics can make to our understanding of organic evolution is in the area of multivariate trait evolution and that the new fields of research in genomics such as QTL and microarray analyses are contributing to, and benefiting from, a quantitative genetic perspective.
The most general form in which quantitative genetics is used in evolutionary biology is given by the equation , where is the change in trait means, G is the genetic variance–covariance matrix, P−1 is the inverse of the phenotypic variance–covariance matrix, and S is the vector of selection differentials. An alternate and equivalent formulation of this equation is , where is the response of the ith of n traits, hj is the square root of the heritability of the jth trait, rAij is the genetic correlation between traits i and j, and βj is the selection gradient on the jth trait. I shall refer to these equations as the multivariate breeder's equation. In this perspective I shall highlight five issues that bear upon the present and future contributions of quantitative genetics to our understanding of evolutionary change:
- 1The utility of a quantitative genetic approach to measuring genetic variation: In this section I outline the utility of a quantitative genetic perspective in describing and analyzing genetic variation at two very different levels of resolution, namely genetic variation in natural populations and genetic variation at the level of DNA transcription.
- 2Do the results of QTL analyses support the assumptions of quantitative genetics? As a general descriptor in the two circumstances discussed in section 1, the quantitative genetic parameters may be useful in themselves, but what we really desire is that the approach can actually be used to predict evolutionary change. In this case we need to consider whether the basic assumptions of quantitative genetics are likely to be sufficiently accurate or robust for at least short-term prediction and then whether selection on multiple traits has produced results consistent with the multivariate breeder's equation. To address the question of the number of loci and the distribution of their effects, I use information recently obtained from QTL analyses, at present possibly the premier method for investigating such questions.
- 3What is the importance to quantitative genetics of identifying specific genes? Recently there has been an enormous effort put into elucidating the molecular basis of variation with attention being given to identifying genes of major effect. Given that quantitative genetics focuses upon the totality of genomic expression of a trait as expressed in a statistical description, does such research have any messages for quantitative genetics?
- 4Testing predictions of the multivariate breeder's equation. In the fourth section I consider whether artificial selection on multiple traits or evolutionary changes in wild populations can be reasonably predicted by the multivariate breeder's equation.
- 5The evolution of the phenotypic and genetic variance covariance matrices. Application of the multivariate breeder's equation assumes that the variance–covariance matrices remain constant. In this section I examine this proposition and suggest that the present hypothesis-testing approaches should be replaced by an interval-estimation perspective.
The Utility of a Quantitative Genetic Approach to Measuring Genetic Variation
A measure of genetic variation in a population and its potential response to selection can be made by estimating G and P. The latter is readily accomplished but the estimation of G presents significant logistical problems. Three approaches can be used: (1) Bring the organism into the lab and estimate the G matrix by controlled breeding experiments, (2) use a sampling design in a wild population that matches a standard pedigree design, such as offspring on parent, (3) sample a population over a number of generations and use the animal model.
The assumption of the first approach is that the estimate obtained from a laboratory is equal or close to that which would be obtained in the field. Because it was assumed that the environmental variance would be higher in the field than the lab, conventional wisdom suggested that heritabilities and genetic correlations in the field would be lower than in the lab. A meta-analysis of lab and field studies showed that this was not the case for heritabilities, though phenotypic variances tend to be reduced in the lab (Weigensberg and Roff 1996). An assessment of the genetic correlation or the genetic variances and covariances remains to be undertaken. The results of the analysis on heritability show that it is premature in the extreme to dismiss estimates from laboratory studies. Some studies in wild populations, most particularly those on birds, have been able to estimate genetic parameters from offspring on parent regression (e.g., Grant and Grant 1995; Réale and Festa-Bianchet 2000; Roff et al. 2004). In general, however, a study of relationships among individuals in a wild population will produce a convoluted set of relationships that cannot be analyzed using such simple methods as half-sib or offspring–parent regression. A solution to this dilemma is the animal model, which does not require any specific pedigree (Knott et al. 1995; Kruuk 2004). In addition to the important task of resolving the question of how much genetic variation is found in wild populations (e.g., Reale et al. 2003; Kruuk 2004; Charmantier et al. 2006a), use of the animal model to estimate genetic parameters in wild populations has enabled detailed investigations into the importance of maternal effects (Kruuk 2004; Wilson et al. 2005a,b; Charmantier et al. 2006b), genotype by environmental interactions (Kruuk 2003; Garant et al. 2004; Brommer et al. 2005; Nussey et al. 2005; Wilson et al. 2006), variation between ages and sexes in genetic parameters (Pettay et al. 2005; Wilson et al. 2005b; Charmantier et al. 2006c), and tests of sexual selection theory (Hadfield et al. 2006; Qvarnstrom et al. 2006). These studies have demonstrated that genetic variation is usual in wild populations and that long-term variation in trait values can only be properly understood within a quantitative genetic framework.
Quantitative genetic approaches are also making significant contributions at an entirely different level, namely variation at the level of transcription of DNA. DNA microarrays have enabled the visualization of the rates of transcription of hundreds to thousands of genes (for overviews of the techniques see Brown and Botstein 1999; Gibson 2002; Tarca et al. 2006). Early experiments showed that rates of transcription varied among genotypes and that transcription rates could themselves be viewed as heritable traits (Schadt et al. 2003; Gibson and Weir 2005; Roelofs et al. 2006). At a more general level, the analysis of microarray data can be approached using the same statistical approaches as quantitative genetics, namely mixed model analysis of variance, where both genotypic and environmental sources of variation can be resolved (Kerr et al. 2000, 2001; Wolfinger et al. 2001; Churchill 2002; Chen et al. 2004; Nettleton 2006). At the present time use of microarrays is restricted by the costs of producing arrays for the organism under study: even with model organisms the cost is sufficiently high to preclude the analysis of hundreds of microarrays, as would be necessary in a typical quantitative pedigree design. Initial studies have used inbred lines (Jin et al. 2001; Rifkin et al. 2003; Schadt et al. 2003; crosses between isogenic lines (Drnevich et al. 2004), different morphs of the same species (Derome et al. 2006), selected lines (Roberge et al. 2006), individuals from different populations (Oleksiak et al. 2002; Vuylsteke 2005), or different species (Rifkin et al. 2003). All cited studies showed genotypic differences though only in a few cases were sample sizes sufficient to attempt an estimate of variance components or a general comparison with phenotypic variation.
Jin et al. (2001) examined rates of transcription in two inbred strains of Drosophila melanogaster at ages one and six weeks and separated variance components due to sex and genotype: sex contributed most but the “genotypic contributions to transcriptional variance may be of similar magnitude to those relating to some quantitative phenotypes” (Jin et al. 2001, p. 389). In their analysis of transcriptional variation in Arabidopsis thaliana, Vuylsteke et al. (2005) found more than 30% of the genes had significant additive effects and 5–20% of genes, depending on the cross, showed nonadditive interaction. Most of the differences in transcriptional variance in the fish Fundulus heteroclitus and F. grandis were found within populations (coefficients of variation 5–15%) though significant differences among populations and species were found (Oleksiak et al. 2002).
Apart from the problem of sample size, the experiments using microarrays can suffer from a surfeit of data in the sense that hundreds of genes may show differences. One could regard the array as a variance–covariance matrix and calculate all possible variance and covariances, but this would generate enormous problems of determining statistical significance. An alternative approach is to collapse the array into a manageable number of uncorrelated variables using principal components analysis. Doing this, Bochdanovits and de Jong (2004) were able to show significant correlations between PC1 of gene expression and the two traits, body weight and larval survival in D. melanogaster. Furthermore, PC analysis showed that variation in transcription rate was, in addition to covarying with adult weight, a function of population of origin and rearing temperature (Bochdanovits et al. 2003).
Whitefish (Coregonus clupeaformis) is found within a single population in two morphs, a normal and a dwarf morph, that form reproductively isolated ecomorphs (Derome et al. 2006). Comparison of transcriptional profiles of the two morphs collected from two separate lakes, both morphs found in each lake, showed both population- and morph-specific differences (Derome et al. 2006). Variation among gene expression has also been found in the two phenotypic morphs, normal and “sneaker” (also known as a “jack”) morph, of Atlantic salmon, Salmo salar (Aubin-Horth et al. 2005a,b), and in territorial and nonterritorial males of the cichlid, Astatotilapia burtoni (Hofmann et al. 1999; Hofmann 2003). Except for the whitefish study, where different populations were raised in a common garden, separating environmental from genetic components of variation was not possible in these experiments, though the heritability of “jacking” in Chinook salmon (Onchorynchus tshawytscha) is 0.67 (Mousseau et al. 1998), suggesting that further testing for genetic differences in transcriptional rates in the two morphs of Atlantic salmon would be profitable. Genetic variation in transcription rates in Atlantic salmon has been demonstrated by comparing strains selected for rapid growth with wild strains. Five to seven generations of selection produced heritable changes in gene transcription rates (Roberge et al. 2006), but, because data on selection intensities are not available, heritabilities cannot be estimated.
Do the Results of QTL Analyses Support the Assumptions of Quantitative Genetics?
Prior to QTL analysis the estimation of the number of loci was problematic and very uncertain. The advent of the use of molecular markers to divide the chromosome into many segments has enabled QTL analysis, first conceived in 1923 (Sax 1923), to make great strides in enunciating the number of chromosomal segments involved in quantitative genetic variation. Quantitative trait loci analyses most frequently employ crosses between inbred lines, though methods exist for the analysis of outbred populations (Lynch and Walsh 1998; for briefer, less mathematical descriptions see Mackay 2001a or Mauricio 2001). Although the statistical power of QTL analyses using inbred lines is generally greater than those using outbred pedigrees (Erickson et al. 2004), their weakness lies in their inability to detect more than two alleles at any locus. Thus even if a QTL analysis picks up relatively few regions with significant effects we cannot immediately conclude that the assumption of a large number of genes of small effect is violated, because there may still be numerous alleles at each detected QTL or allelic variation at QTLs that happen to be homozygous for the two inbred lines chosen. However, if QTL analyses consistently found few regions of major effect we might certainly begin to entertain the hypothesis that the assumption of many genes of small effect may be incorrect, presuming that the QTL segments are sufficiently small that they contain only a single or possibly a few actual loci that affect the trait under study.
Early QTL analyses did indeed produce results that pointed to relatively few regions of major effect. One would expect that as sample size increased the number of QTLs detected would increase, the percentage of phenotypic variance explained per QTL would decrease, and the total variance explained would go up. Rather surprisingly, though the first two predictions were upheld, the third was not (see fig. 1.10 in Roff 1997). Using simulation, Beavis (1994, 1998) showed that with small sample sizes QTL effects are grossly inflated and hence the effective number of QTLs contributing to a trait is underestimated. This phenomenon has become known as the Beavis effect and its theoretical basis investigated in detail by Xu (2003), though the general nature of the problem was forecasted in the early 1990s both by the work of Beavis and others (Kearsey and Farquhar 1998). To obtain reasonably accurate estimates of effects sample size must exceed 500, whereas sample sizes much before the year 2000 were more typically in the range of 200 individuals (Roff 1997). It is still not clear how many loci generally contribute to a trait or the distribution of effects. In a review of QTL analyses in plants, Remington and Purugganan (2003, p. S13) concluded “that individual genes with relatively large effects on trait variation are probably important in evolution,” but did not provide a meta-analysis to support this statement. A meta-analysis of the distribution of QTL effects in pigs and dairy did indicate that effects were skewed with a few QTL of large effect, the total number of QTL for the dairy data being estimated to lie between 50 and 100, depending on population size (Hayes and Goddard 2001). Mackay (2004, p. 254) in reviewing QTL analyses in Drosophila drew the conclusion that “the distribution of homozygous QTL effects is exponential, with a large number of QTLs with small effects, and a smaller number with large effects; the latter contribute most of the variation between parental lines.” To date I know of no study that has carried out a meta-analysis of QTL effects from which general statements about the number of QTLs and the distribution of their effects can be made with statistical precision: such a study is greatly needed.
Given that we know that there is not an infinite number of loci, how many is enough to satisfy the assumption of normality if each did indeed have a similar effect? The well-known answer is that very few are required: four or five loci with two alleles per locus will provide a distribution that is statistically very close to normal. The problem lies in the fact that in such a situation selection can very rapidly erode genetic variation. A typical number of QTL detected appears to be about 20 (Cheverud et al. 1996; Vaughn et al. 1999; Mackay 2001a; Rocha et al. 2004; Nuzhdin et al. 2005), though Walsh (2001, citing personal communication with Mackay) gives an estimate of 130 genes for sternopleural bristle number in D. melanogaster based on mutational effects. Given the limitations of QTL detection, the number of loci might well be typically in the dozens if not hundreds. What we do not know, and need to know, is how many alleles typically segregate at each locus (defining what is a locus at the molecular level itself presents difficulties, but here I refer only to QTL variation, which admittedly is at best only a crude measure of a locus). The number of QTL is probably sufficient to satisfy the assumption of normality provided that the distribution of effects is itself close to normal. The present QTL results, as expounded in the above-cited reviews, suggest that the QTL effects are highly skewed. On the other hand, this is not necessarily important, because the multivariate breeder's equation focuses upon the change in mean trait values and hence even if the distribution of genetic effects is not normal, by the central limit theorem, the distribution of mean effects will be normal. Lack of normality is certainly observed in many phenotypic distributions but these can be brought into line by a suitable transformation. Theoretical and numerical analyses have shown that the Gaussian approximation is satisfactory even under strong selection, which produces large deviations from normality (Turelli and Barton 1994). Thus, for single traits the question of normality is probably not critical, at least for the case of short-term (approximately 10 generations) selection. It is not clear how important the assumption of multivariate normality will be, and this question needs to be addressed via analytical, or more likely, numerical analysis. If phenotypic effects mirror the underlying additive genetic effects, then the question of multivariate normality may be not be particularly significant because one can find a suitable transformation. However, more troublesome is the possibility that the distribution of additive genetic effects is not the same as the composite phenotypic effects and hence a transformation applied to the phenotypic scale does little or nothing to improve the distribution on the additive genetic scale.
But can we trust the distribution of effects as presently suggested by the QTL analyses? Here we must consider the issue of epistatic effects. First, it is very evident from analyses of line crosses between inbred lines, different populations or different species that both dominance and epistatic effects are exceedingly common in both morphological and life-history traits, dominance effects being demonstrated in more than 95% of cases and epistatic effects demonstrated in more than 65% of cases (Roff and Emerson 2006). Quantitative trait loci analyses have made similar findings (e.g., Cheverud and Routman 1995; Shook and Johnson 1999; Mackay 2001b, 2004; Leips and Mackay 2002; Alonso-Blanco et al. 2003; Carlborg et al. 2003; Caicedo et al. 2004; Carlborg and Haley 2004; Weinig and Schmitt 2004; Malmber et al. 2005; Mackay and Lyman 2005; Carlborg et al. 2006; Yi et al. 2006) despite the fact that the power to detect epistatic interactions is low (Flint and Mott 2001; Mackay 2001b; Carlborg and Haley 2004; Erickson et al. 2004). If epistatic effects are so pervasive, the presence of alleles of large effect may be contingent on the particular genetic background used in the QTL analysis, in which case the effect of this “major” QTL may disappear when measured over the entire complement of genomes present in a wild, outbreeding population. In this scenario, epistatic effects could contribute significantly to additive genetic variance but not epistatic variance. This is the argument that Fisher himself used to dismiss the importance of epistatic variance in large populations (Whitlock et al. 1995; Brodie 2000). In small populations, epistatic effects may come to the fore and contribute to nonadditive genetic variance, a consequence of which would be that the selection coefficient on a locus in local populations may differ as a consequence of epistatic effects, which is the argument Wright used in championing his model of evolution (Hill 1989; Wade and Goodnight 1998). As with QTL analysis, the detection of epistatic effects within populations using traditional quantitative genetics approaches is extremely difficult, but this is itself no reason to ignore their possible effects. In the light of the consistent evidence for epistatic effects, we are in need of research both in the empirical investigation of epistatic effects within populations, which will rely upon further QTL analyses, and theoretical work on whether the effects as presently observed could influence evolutionary trajectories.
At present, the conclusions on the importance of epistatic variance, as opposed to epistatic effects, in modulating evolutionary responses is mixed. Keightley (1996) showed theoretically that epistatic effects generated in metabolic pathways could cause asymmetric responses to selection but empirical demonstration is as yet absent. Other theoretical models have also shown that epistatic effects could be important (Fuerst et al. 1997; Soriano 2000), but much remains to be done in working out the circumstances under which epistatic variance should be incorporated into quantitative genetic models and analyses. Carlborg et al. (2006) present an interesting case in which a single major QTL for growth in chickens is actually composed of four interacting loci, the effect of which is to produce a higher response to selection than would be predicted by a single locus. Theoretical models incorporating epistatic effects have also been studied in regards to mutation-selection balance, the general finding being that, as with the additive model, the final balance is sensitive to parameter values (Hermisson et al. 2003; Hansen et al. 2006), about which we know very little. Epistasis underlies canalization and may under particular conditions be a source of cryptic variation (Hansen 2006). Several studies have shown that, as a result of the population passing through a bottleneck, epistatic effects can be converted into significant additive genetic variance (Goodnight 1988, 2000; Cheverud and Routman. 1996; Cheverud et al. 1999; Wade 2002) though its importance has been questioned by other studies (Lopez-Fanjul et al. 2002, 2006; Hill et al. 2006; Turelli and Barton 2006).
Finding considerable epistasis at the molecular level does not mean that epistatic variance will be significant at the phenotypic level at which most traits forming the focus of quantitative genetic analysis are found. As discussed in the next section, epistatic interactions are expected in such complex traits as morphology, fecundity, or longevity but this may be irrelevant to quantitative genetics if such effects are manifested as additive variance.
What is the Importance to Quantitative Genetics of Identifying Specific Genes?
Quantitative genetics is primarily a statistical description of the action of genes and does not, in principle, concern itself with the details of the genetic mechanism except in as much as such mechanisms result in additive and nonadditive genetic variance. On the other hand, such a broad brush approach to evolutionary responses to selection or random sampling through drift may not capture evolutionary responses in some instances. It is thus important to continually assess how the increasing information on genetic mechanisms impacts quantitative genetic predictions. Recent advances in genomics has led to the identification of specific genes involved in traits of obvious evolutionary significance such as morphology, life span, and resistance to pesticides (Table 1). How do such findings impinge upon quantitative genetics?
Table 1. Examples of quantitative traits in which single genes have major effects and the molecular basis of regulation has been studied.
|Wing polyphenism||Ants||Multiple||Interruption of developmental network occurs at different places in different species|| Abouheif and Wray (2002) |
| Growth of insect hind legs || Numerus species || Ubx and abd-A expression || Regulator of growth but precise molecular mechanism not elucidated || Mahfooz et al. (2004) |
|Beak size||Chickens, ducks, Darwin's finches|| Bmp4 ||Differential expression in beak morphogenesis|| Abzhanov et al. (2004); Wu et al. (2004)|
| Heavy metal tolerance || Orchesella cincta || Mt || Transcription rate regulates Cd excretion || Roelofs et al. (2006) |
|Bolting|| A. thaliana || FRI gene encodes a protein that represses transition to flowering||Epistatic interaction with FLC genes produces clinal variation|| Sheldon et al. (2000); Caicedo et al (2004)|
| Life span || D. melanogaster || Multiple || Multiple genes individually have major effects (8–85% increase) || Table 1 in Hughes and Reynolds (2005) |
|Life span|| C. elegans ||Cell-signaling pathway||Up to 3× increase||p. 432 in Hughes and Reynolds (2005). See also Leroi (2001)|
| Resistance || A. thaliana || Plasma membrane protein that confers ability to recognize pathogen || Not driven to fixation because of a trade-off with seed production || Tian et al. (2003) |
|Resistance|| A. thaliana ||Auxin response||3 mutations affecting same pathway|| Roux and Reboud (2005) |
| Resistance || D. melanogaster || JH receptor || Mutant Met alleles significantly affect resistance and have statistically significant pleiotropic effects on life-history traits || Flatt and Kawecki (2004) |
|Resistance|| D. melanogaster ||Overtranscription||Different paths can be affected (e.g., cytochrome P450s, glutathione-S-transferases, esterases)|| Le Goff et al. (2003); Festucci-Buselli et al. (2005)|
| Resistance || Culex pipiens || Esterase overproduction || Two loci involved, with two nonexclusive mechanisms of operation || Berticat et al. (2002) |
|Resistance|| Myzus persicae, Musca domestica ||Overproduction of carboxylases or sodium-channel modulation||Pleiotropic effects on behavior|| Foster et al. (2003) |
| Resistance || Saccharomyces cerevisiae || Transcriptional regulators || Two separate mutations, which singly increase resistance but together have negative effects on fitness || Anderson et al. (2006) |
|Resistance|| Candida albicans ||Overexpression of drug-resistance determinant||3 different patterns evolved, one involving a single major gene, the other two, which were more common, involving multiple genes|| Cowen et al. (2002) |
Wing polyphenism is found in a large number of ant species and is completely determinate in that particular castes, but not all castes (queens in some species may be dimorphic for wings), always exhibit the same phenotype. Although the same network of gene interactions appear to be involved in the expression of morphology, the point at which the network is interrupted differs among species (Abouheif and Wray 2002). This information is important for an understanding of the genetic and physiological regulation of caste development but does not affect the general genetic model of caste determination nor does it imply that wing development, or lack of, is a consequence of a single gene or a single mutation. Caste determination in hymenoptera is generally determined by environmental conditions during rearing, including maternal inputs to the egg (Suzzoni et al. 1980; Wheeler 1986, 1991), although there are cases of genetic differences among castes (Fraser 2000; Julian et al. 2002; Cahan and Keller 2003). Genomic comparisons among castes have shown that there is differential expression of numerous genes (Evans and Wheeler 1999; Pereboom et al. 2005; Sumner et al. 2006), though which ones are specifically required for determination is not known. A general model for this mechanism of determination is the threshold model in which it is envisaged that at some point in development the subsequent developmental trajectory is determined by the value of some trait, called the liability, and a threshold: values of the liability above the threshold shift development into one trajectory whereas values below the threshold shift development into the alternate path (Falconer 1965; Wright 1977; Roff 1996a). The liability could indeed be equated to a single gene that up- or downregulates some product, such as a hormone, that controls future development. On the other hand, the liability may be a function of a large number of factors, in which case the liability may show a continuous distribution. In the former case, evolution can be modeled using simple Mendelian population genetics, whereas in the latter a quantitative genetic approach is appropriate. With respect to wing dimorphism, it is intriguing to find that in holometabolous development, as found in the hymenoptera, the vast majority of cases can be modeled using a single locus model with winglessness being dominant, but in those insects with hemimetabolous development a polygenic model best fits the data (Roff and Fairbairn 1991).
Insects show enormous variation in the relative size of their hind legs, which can be related to variation in the pattern of expression of the Ubx and abd-A genes (the method of detection could not distinguish between these two and so their relative contributions are not known; Mahfooz et al. 2004). Similarly, the activity of the Bmp4 gene correlates with adult beak size in Darwin's ground finches (Abzhanov et al. 2004). Does this mean that leg length or beak size is determined by the action of only one or two genes? It is readily observed that there is continuous phenotypic variation within populations and estimates of heritability of morphological traits are relatively large (approximately 0.4, Mousseau and Roff 1987), which is inconsistent with single gene action. Morphology is a result of a sequence of developmental processes, which both affect and are affected by the expression of particular genes such as Ubx, abd-A, and Bmp4. Even if we knew all the genes involved in the developmental process, it is unlikely that we could predict the results of selection on a single trait better than that done by quantitative genetics (for reasons discussed below, the jury is still out on whether the prediction of multivariate evolution will typically require more than a traditional quantitative genetic model).
An excellent illustration of the relationship between single gene action and quantitative genetics is given by the study of Roelofs et al. (2006) on additive genetic variation in metallothionen expression in the collembolan Orchesella cincta. Tolerance to heavy metals in this species is mediated by cadmium (Cd) excretion, which has been shown to have a heritability between 0.33 and 0.48 (Posthuma et al. 1993). It is known that the gene mt plays a significant role in Cd excretion. Using quantitative polymerase chain reaction (PCR), Roelofs et al. (2006) determined variation in transcription rate of this gene and from parent–offspring regression estimated the heritability of transcription rate to be between 0.36 and 0.46, which agrees very well with the previously estimated heritability for Cd excretion. Cadmium excretion is largely under the control of a single gene (mt) but the expression of this gene is itself modified by the action of other genes, thereby producing continuous genotypic and phenotypic distributions.
Another example in which a single gene plays a key role but its effect may be modeled by a quantitative genetic approach is the gene FLC, which controls the onset of flowering in A. thaliana (Sheldon et al. 2000). Flowering is controlled by the induced transcription rate of FLC, variation in which is a function of both allelic variation at the FLC locus and the action of other genes (Sheldon et al. 2000; Caicedo et al. 2004; see Remington and Purugganan 2003 for a review of other genes affecting flowering time in plants). Epistatic interaction between the genes FRI and FLC is responsible for clinal variation in flowering time (Caicedo et al. 2004). It would be extremely interesting to know if this epistatic interaction resulted in significant epistatic variance.
Numerous mutations affecting life span in D. melanogaster and C. elegans have been isolated (Table 1). Life span in D. melanogaster has been the subject of intense study, both using quantitative and genomic approaches. Life span responds readily to selection with attendant correlated responses in physiology and life-history traits, the mechanisms for which are still unresolved (Valenzuela et al. 2004; Rose et al. 2005; Vermeulen and Bijlsma 2006). Single gene manipulations have shown that life span can be substantially increased (up to 85%) and that no single gene determines this trait or even a unique mechanism by which longevity is altered (see Table 1 in Hughes and Reynolds 2005). Genomic analyses have identified up to 25 QTL affecting longevity (Nuzhdin et al. 2005). Eleven QTL were identified on chromosome 3 containing from 12 to 170 positional candidate genes (Wilson et al. 2006). Quantitative trait loci analyses have also shown significant dominance, epistatic and genotype by environment interactions (Leips and Mackay 2002; Forbes et al. 2004). It is evident that longevity is a highly complex trait likely involving a large array of different physiological components. Although single gene mutations can have significant effects on longevity, the antagonistic effects on other fitness components may exclude their invasion into most natural populations: in the absence of such effects we would expect these genes to be generally favored and spread to fixation at which point they no longer contribute to variation (of course high extrinsic mortality will greatly reduce the selective advantage of longevity genes). The presence of dominance and epistatic effects is not unexpected: life-history traits typically show directional dominance (Roff 1997) and the complexity of physiological components of longevity would argue for interactions among loci. Whereas dominance effects may be revealed by contributions to genetic variance, epistatic effects, even of large effect, may contribute little to genetic variance, as illustrated, for example by the lack of epistatic variance in Drosophila bristle number but significant epistatic interaction revealed by QTL analysis (Mackay and Lyman 2005). It remains to be shown if the epistatic effects in longevity could play a significant role in evolutionary change and thus need to be incorporated into a quantitative genetic model of the evolution of life span.
Major gene effects conferring increased resistance to pesticides, herbicides, and drugs have frequently been observed in natural populations of animals and plants (Table 1; Jasieniuk et al. 1996; ffrench-Constant et al. 2004). In many cases the molecular mechanism underlying this resistance is an upregulation of transcription (Table 1; Taylor and Feyereisen 1996). Major gene action is common in the evolution of insecticide resistance in natural populations but artificial selection appears to act on polygenic variation (McKenzie and Batterham 1994). The precise reasons for this are still debated (McKenzie and Batterham 1998; McKenzie 2000) but certainly a contributing factor is that the relevant mutations are likely to be absent in laboratory populations but available in the much larger natural populations. An understanding of the potential genetic mechanisms available for resistance is, in this case, important for predicting the circumstances under which particular evolutionary trajectories will be taken.
In summary, detailing the genetic mechanisms is an important enterprise but, with relatively few exceptions, knowledge of those mechanisms does not contribute to the prediction of the evolutionary trajectory of traits that fall within the purview of quantitative genetics. At present, although it is possible to simulate simple gene networks and follow their evolutionary change (Frank 1999; Omholt et al. 2000; Hasty 2001; de la Fuente et al. 2002; Nijhout 2002; Bergman and Siegal 2003), we are nowhere close to simulating the complexity of genetic interactions involved in the determination of such traits as fecundity, longevity, or body size.
An important area of research that is highlighted by the search for specific genetic mechanisms is that of determining the number of alternate paths to a single phenotype. Selection in the “classical” quantitative genetic model assumes equality of all loci, with the result that the change in allelic frequencies during selection may differ due to chance. Under this scenario two lines selected in the same direction could display the same phenotype but differ at particular loci. Because of nonadditive genetic interactions, a cross between two such selected lines would not then necessarily produce the same phenotype. For example, crosses between lines of mice selected for growth rate revealed extensive nonadditive interactions (Mohamed et al. 2001). Differences in genetic mechanism have also been demonstrated in mice selected for thermoregulatory nest-building behavior (Bult and Lynch 1996). Perhaps, even more intriguing is the possibility of different morphological, physiological, or behavioral pathways leading to the same selection response. Examples in this category include selection on competitive ability in Drosophila (Joshi and Thompson 1995), growth rate patterns in mice (Rhees and Atchley 2000), wheel running in mice (Garland et al. 2002), and adaptive evolution to growth media in Escherichia coli (Fong et al. 2005). Different pathways to the same phenotype have also been found in natural populations of D. subobscura. Near identical clines in wing size is found in D. subobscura populations in Europe, South America, and North America, but the components of the wing show striking differences (Gilchrist et al. 2004). Crosses among populations of D. melanogaster also suggest that different clines in wing size have different genetic bases (Gilchrist and Partridge 1999). Such results do suggest that research incorporating quantitative genetic and mechanistic approaches are fundamental to the prediction of evolutionary change and that we need to develop models that combine these two components.
Testing Predictions of the Multivariate Breeder's Equation
Whereas QTL analysis can provide direct evidence for or against the assumptions underlying the quantitative genetic framework, indirect tests are provided by the ability of quantitative genetic models to predict the rate and direction of evolutionary change. Perhaps more importantly, the results of such experimental comparisons inform us on the robustness of the models to the underlying assumptions. In this respect the record on predictions for single trait responses to artificial selection is very good, excellent predictions being made for the first 10–15 generations of selection, followed generally by the predicted decline in response as variation is eroded (Hill and Caballero 1992; fig. 2.5 in Roff 2002). Artificial selection experiments also tend to support the results from QTL analyses that genes of large effect frequently occur (Hill and Caballero 1992). The analysis of trait variation in wild populations, particularly responses following changes in biotic or abiotic conditions, depends generally upon being able to predict multivariate responses, and here even the results of artificial selection experiments are somewhat discouraging.
The general finding of artificial bivariate selection experiments is that antagonistic selection, meaning selection opposite to the sign of the genetic correlation, frequently does not accord with prediction (Table 2). Various reasons have been advanced for the poor response: incorrect initial estimates, maternal effects, genetic drift, asymmetry in gene frequencies, type of index selection applied, and functional constraints. However, in no case has the cause of the irregularity in response been adequately analyzed. If quantitative genetic theory cannot account for response to artificial bivariate selection then it will be severely limited in how useful it can be in understanding short-term evolutionary change in wild populations.
Table 2. A survey of bivariate artificial selection experiments.
| Selection in all four directions|
| Bicyclus anyana || Anterior eyespot || Posterior eyespot || No || Both traits responded || Beldade et al. (2002) |
| Bicyclus anyana||Forewing area||Body weight||No||Both traits responded|| Frankino et al. (2005) |
| Mouse || 12 to 21-day weight gain || 51 day weight || Yes || Good in two directions (H–H; H–L), poor in the other two (L–L; L–H) || Berger and Harvey (1975) |
| D. melanogaster||Coxals||Sternopleural bristles||Yes||Good fit in early generation (10) poor fit in later generation (22)|| Sheridan and Barker (1974) |
| Tribolium casteneum || 13-day larval weight || Pupal weight || Yes || Poor, particularly for antagonistic selection || Bell and Burris (1973) |
| Tribolium casteneum||14-day larval weight||30-day larval weight||Yes||Poor, particularly for antagonistic selection||Okada and Hardin (1967)|
| Tribolium casteneum || Pupal weight || Egg laying || Yes || Good, consistent with estimates || Campo and de la Fuente (1991) |
| One direction of selection reinforcing and a second antagonistic|
| Onthophagus acuminatus || Horn length || Body size || No || Response same in both directions || Emlen (1996) |
| Reinforcing selection|
| Mouse || Post weaning weight gain || Litter size at birth || No || Both traits responded || Doolittle et al. (1972) |
| D. melanogaster||Abdominal bristles||Sternopleural bristles||No||Both traits responded|| Sen and Robertson (1964) |
| Tribolium casteneum || Pupal weight || Family size || Yes || Poor, but good for single traits || Berger (1977) |
| Plant||Height||Number of leaves||Yes||Good|| Matzinger et al. (1977) |
| Antagonistic selection|
| Mouse || Tail length || Body weight || No || Both traits responded || Cockrem (1959) |
| Mouse || Food intake, or gonadal fat pad, or fat pad || Body weight || No || Both traits responded || Sharp et al. (1984) |
| Brassica rapa||Filament length||Corolla length||No||Both traits responded|| Conner (2003) |
| Boar || Daily weight gain between 30–80 kg || Back fat index at 80 kg || No || Both traits responded || Ollivier (1980) |
| Mouse||Tail length||Body weight||Yes||Poor|| Rutledge et al (1973) |
| Mouse || Weight at 5 weeks || Weight at 10 weeks || Yes || Poor || McCarthy and Doolittle (1977) |
| Mouse||6-week body weight||Litter size at birth||Yes||Poor|| Eisen (1978) |
| Mouse || 12-week fat weight || Constant 12-week body weight || Yes || Fair in one direction, poor in the other || Eisen (1992) |
| Mouse||8-week body weight||3–5 week weight gain||Yes||Poor|| von Butler et al. (1986) |
| Mouse || Weight gain, birth to 10 days || Weight gain, 28 days to 56 days || Yes || Good in one regime, poor in another || Atchley et al. (1997) |
| Tribolium casteneum||Body size||Pupal weight||Yes||Poor|| Campo and Raya (1986) |
| Tribolium casteneum || Larval weight, or development time, or pupal weight || Index of other two || Yes || Generally poor || Scheinberg et al. (1967) |
| Tribolium casteneum||Pupal weight at 21 days||Adult body weight at 21 days||Yes||Good in one direction, poor in other|| Campo and Velasco (1989) |
| Chicken || Egg weight || Body weight || Yes || Poor || Festing and Nordskog (1967); Nordskog (1977) |
| Turkey||Days tested||Egg weight and rate of lay||Yes||Accurate for the one generation tested|| Garwood et al. (1978) |
| Turkey || 8-week body weight || 24-week body weight || Yes || Moderate || Abplanalp et al. (1963) |
| Maize||Yield||Ear height||Yes||Poor||Moll et al. (1975)|
| Soybean || Protein concentration || Oil content || Yes || Good in one direction, poor in other || Openshaw and Hadley (1984) |
The breeder's equation has been applied to explain selection response in three natural populations: changes in a diapause component, KP, in the lepidopteran, Hyphantria cunea (Morris 1971); the evolution of body components in Darwin's medium ground finch, Geospiza fortis (Grant and Grant 1995); the evolution of juvenile hormone esterase (JHE) activity in the Bermuda population of the sand cricket, Gryllus firmus (Roff and Fairbairn 1999). For H. cunea, Morris (1971) successfully used the single trait breeder's equation to predict the change in Kp in three populations over 12 years (for a summary and discussion see Roff 1997, p. 157–159). Roff and Fairbairn (1999) successfully predicted the correlated response of JHE activity to changes in proportion macropterous (long-winged and capable of flight) female G. firmus using genetical parameters derived from laboratory rearings. Finally, Grant and Grant (1995) obtained good agreement between observed and predicted response to natural selection in G. fortis using genetical parameters estimated from offspring–parent pedigrees in the same population. Multivariate selection was overwhelmingly reinforcing on G. fortis, meaning that selection was in the direction of the genetic correlations, which is the case most likely in theory to produce predictable responses (this also appears to be supported by the empirical findings in Table 2, though the number of cases I have been able to locate is surprisingly few).
To address the utility of the multivariate breeder's equation we need bivariate selection experiments in which both functional and genetic factors that could restrict the evolutionary trajectory can be explored. Such experiments will most likely be achievable using an invertebrate or plant system, though a fast growing vertebrate such as guppies might also be useful. Possible candidates for which we have considerable information on functional and genetic parameters are D. melanogaster (e.g., Roff and Mousseau 1987; Flatt et al. 2005; Chippindale et al. 2003; Rose et al. 2005), Arabidopsis (e.g., Pigliucci 1998; Ungerer and Rieseberg 2003; Ungerer et al. 2003; Koornneef et al. 2004), Manduca sexta (e.g., D'Amico et al. 2001; Davidowitz et al. 2005; Nijout et al. 2006, 2007), Bicyclus anynana (e.g., Beldade et al. 2002; Brakefield et al. 2003; Zijlstra et al. 2003; Frankino et al. 2005), Onthophagus sp. (Emlen 1996, 2000, 2001; House and Simmons 2005; Emlen et al. 2006; Moczek 2006), and G. firmus (e.g., Roff and Fairbairn 1999, 2001, 2006; Roff et al., 2002; Zera and Harshman 2001).
The Evolution of the Phenotypic and Genetic Variance–Covariance Matrices
A fundamental assumption of the foregoing discussion is that the variance–covariance matrices do not themselves evolve. This is clearly not the case but we presently lack a detailed theory on how the matrix will change over time (Arnold et al. 2001; McGuigan 2006; Phillips and McGuigan 2006). Schluter (1996), assuming that the G matrix remains constant, pointed out that evolution would tend to follow the trajectory in which additive genetic variances are maximal: thus, for example, with two traits the evolutionary path would at least initially tend to be in the same direction as the major axis of the bivariate distribution. On the other hand, we would also expect that variances and covariances would be selected such that the resulting major axis of the bivariate distribution would be in the direction of selection, that is, the G matrix aligns itself to selection (Lande 1980; Cheverud 1984; Arnold 1992; Arnold et al. 2001). Some understanding of the evolution of the G matrix under different types of selection has been achieved by simulation (Jones et al. 2003, 2004) but empirical descriptions of the manner of variation in the G matrix are still rather few and general patterns have yet to emerge with which theoretical analysis can be compared.
To date most empirical studies of the G matrix have focused upon the question of whether G matrices vary among populations or species (Roff 2000; Steppan et al. 2002). It comes as no surprise that given sufficiently large sample sizes statistically significant variation among G matrices is very common, if not ubiquitous. It is time to depart from this hypothesis-testing approach and consider the question of interval-estimation, that is, asking not whether two matrices differ but what is the amount and pattern of differences (a point also pressed by Phillips and McGuigan 2006). A potential stumbling block is the logistical difficulty of determining G matrices. One solution may be to use the P matrix as a surrogate, which for morphological traits can be justified by the close correspondence between the phenotypic and genetic correlations (Cheverud 1988; Roff 1996b). However, a comparison of how the P and G matrices interpreted variation among species of crickets suggested that using P as a surrogate for G can be misleading even for morphological traits (Begin and Roff 2004). Because it is an important component of the breeder's equation, a study of variation in the P matrix is also of importance in its own right, regardless of its possible use in place of G (Roff and Mousseau 2005).
Quantitative genetics has shown itself to be an extremely fruitful approach to the analysis of quantitative variation. The intellectual achievement of bringing together Mendelian genetics and statistics was a landmark in evolutionary biology. The breeder's equation, in either its singular or multivariate formulation, is elegant in its simplicity yet has proven to be of high explanatory ability, though the predictive ability of the multivariate version needs much more testing.
As a descriptor of genetic variation in a population the quantitative genetic perspective has been extremely important. Analyses of wild populations have shown that genetic variation is rampant and generally ample for rapid evolutionary responses. At the level of the genome the quantitative genetics perspective has played a major role in bringing order to the vast amount of data extracted from microarrays. The potential of this combination has yet to be realized, but surely will be as the cost of microarray analysis drops to a point at which it can be applied to standard pedigree designs.
The underlying assumptions of the breeder's equation, in either its singular or multivariate forms, have always been known to be mathematical approximations of what is actually happening at the level of the gene. The important question is simply how good are these approximations? The results of single trait selection suggest that they are reasonable for short-term selection but predictable deviations occur over the long term. Quantitative trait loci analyses have shed more light on the number of loci and the distribution of genetic effects but much remains to be learned about the number of alleles and the distribution of genetic effects in a large outbreeding population. In particular, do QTLs of major effect show such effects across the large array of genetic backgrounds expected in an outbreeding population?
The considerable research done at the level of molecular genetics has revealed extensive networks of interacting genes. Frequently, genes of large effect are found and these may indeed be common, though research tends to be directed towards detecting such genes. While it is important to elucidate the genetic mechanisms underlying traits, those of interest to quantitative geneticists will generally be composed of such a large array on interacting components that at this stage there is no possibility of creating a useful mechanistic model for such traits. Quantitative genetics cuts through this Gordian knot to provide a convenient and powerful summary tool of these interactions. Further developments in the mathematics of gene networks may provide a means of connecting mechanism and statistical description, but it remains to be seen whether such would provide greater explanatory power than presently provided by the quantitative genetic approach.
An area in which quantitative genetics has been surprisingly unsuccessful is that of the quantitative prediction of multivariate evolution when selection is antagonistic to the genetic covariances. In large part this failing is due to a paucity of bivariate selection experiments with follow-up analyses of the mechanistic causes for the deviations from expectation. The recent acknowledgment of the utility of selection experiments (Gibbs 1999; Brakefield et al. 2003; Conner 2003; Garland 2003; Fuller et al. 2005; Swallow and Garland 2005) and of experimental evolution approaches (Bennett and Lenski 1999; Stearns et al. 2000; Mery and Kawecki 2002; Rose et al. 2005; Roff and Fairbairn 2007) will likely remedy this situation.
Just as genetic variances are expected to evolve under the force of selection so too are the G and P matrices expected to evolve. Theoretical and numerical analyses of such evolution is beginning, as is the description of variation of the matrices in wild populations in relation to possible factors of selection, but we are still at the early stages and this area is ripe for further study.
Quantitative genetics has proven itself of great utility over the last 100 years. The advent of massive computing power and the ability to delve into the molecular foundation of genetic variation promises to contribute to an increasing refinement of the multivariate breeder's equation: there is no reason to expect that the contribution of quantitative genetics to our understanding of the basis and evolution of trait variation will diminish in the near future.
Associate Editor: M. Rausher
This work is supported by National Science Foundation (NSF) grant DEB-0445140. I am very grateful for the constructive criticisms of the two reviewers.