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Influence of Gene Interaction on Complex Trait Variation with Multilocus Models [Genetics]
[October 01, 2014]

Influence of Gene Interaction on Complex Trait Variation with Multilocus Models [Genetics]


(Genetics Via Acquire Media NewsEdge) ABSTRACT Although research effort is being expended into determining the importance of epistasis and epistatic variance for complex traits, there is considerable controversy about their importance. Here we undertake an analysis for quantitative traits utilizing a range of multilocus quantitative genetic models and gene frequency distributions, focusing on the potential magnitude of the epistatic variance. All the epistatic terms involving a particular locus appear in its average effect, with the number of two-locus interaction terms increasing in proportion to the square of the number of loci and that of third order as the cube and so on. Hence multilocus epistasis makes substantial contributions to the additive variance and does not, per se, lead to large increases in the nonadditive part of the genotypic variance. Even though this proportion can be high where epistasis is antagonistic to direct effects, it reduces with multiple loci. As the magnitude of the epistatic variance depends critically on the heterozygosity, for models where frequencies are widely dispersed, such as for selectively neutral mutations, contributions of epistatic variance are always small. Epistasis may be important in understanding the genetic architecture, for example, of function or human disease, but that does not imply that loci exhibiting it will contribute much genetic variance. Overall we conclude that theoretical predictions and experimental observations of low amounts of epistatic variance in outbred populations are concordant. It is not a likely source of missing heritability, for example, or major influence on predictions of rates of evolution.



(ProQuest: ... denotes formulae omitted.) EPISTATIC variance in quantitative traits arises from the interaction effects or epistasis between segregating genes at two or more loci that affect these complex traits. Such gene interaction is a common phenomenon because many factors have, for example, a regulatory role in a hierarchical system (Phillips 2008). The statistical theory of quantitative genetics following Fisher (1918) is based on a partition between av- erage effects across loci, which contribute to the additive genetic variance, and to interactions within loci and between loci, which contribute to the dominance and epistatic vari- ance, respectively (Cockerham 1954; Kempthorne 1954). The magnitudes of these components of the genotypic variance each depend on the frequencies, the effects, and the interac- tions among the contributing genes (see also Falconer and Mackay 1996; Lynch and Walsh 1998). The actual causal genetic factors are usually not known, but many quantitativegenetic analyses, including selection on metric traits, have been applied successfully without such knowledge.

Among quantitative geneticists, interest in epistasis con- tinues, both to understand the genetic architecture and as a potential way to improve the genomic predictions of disease and quantitative traits, utilizing some of the unexplained parts of the genetic variation (e.g., Carlborg and Haley 2004; Nelson et al. 2013; Mackay 2014). Despite the obvious interactions in the biological system, it has, however, been argued that the proportion of the genotypic variance contributed by epistatic variance expected in outbred populations is small and that data generally support this prediction (Hill et al. 2008). The theory isbasedmainlyonpopulationand statistical genetics argu- ments (the data of a more indirect form): for many traits the covariances among relatives could largely be explained by ad- ditive models, albeit estimation of epistatic variance per se with much precision is difficult in most outbred population struc- tures. Hemani et al. (2013) in contrast, taking an evolutionary perspective, have argued that much of the variation that will remain segregating long term in populations under natural section will be epistatic. It is notable, however, that the loci in the models they simulated that were still segregating all showed overdominance, for which direct data are limited (e.g., Charlesworth and Charlesworth 2009). Using models with mul- tiple loci and two-locus interactions in populations evolving with bottlenecks, Ávila et al. (2014) found only small contributions of epistatic variance. Further, it has been argued that the magni- tude of epistatic variation is essentially irrelevant in evolution (and in animal breeding) (Crow 2010), because rates of evolu- tionary change depend only on the additive variance, even in epistatic and tightly linked systems (Kimura 1965; Nagylaki 1993). Nevertheless Nelson et al. (2013) have recently argued that epistasis has been ignored in quantitative genetics and in its evolutionary studies because of convenience, and consequently statements about its insignificance are misleading.


Genomics offers tools that go much deeper, to identify the genes involved and their interactions within the system, perhaps thereby providing an understanding of the causes of specific complex diseases or traits and a route to their im- provement. There has, therefore, been a surge of interest in estimating the magnitude of epistasis and understanding its source. Despite the challenges in estimation, thousands of QTL have been found and many findings of interaction effects reported in crosses of divergent lines of chicken (Pettersson et al. 2011), yeast (Bloom et al. 2013), and Drosophila (Huang et al. 2012). Further, there is recent evidence of detection of epistatic loci for levels of gene expression in human popula- tions (Hemani et al. 2014; Brown et al. 2014).

Nevertheless the variation contributed by these epistatic loci detected in segregating populations is generally found to be small (Huang et al. 2012; Brown et al. 2014; Hemani et al. 2014), which fits with the theoretical predictions (Hill et al. 2008) that only at high heterozygosity levels do loci contrib- ute much epistatic variance; Bloom et al. (2013) note that the epistatic variance they found would have been much lower had they not been using an F1-based population.

The genetic variation accounted for by significant SNPs in genome-wide association studies (GWAS) in humans for both metric traits such as height and for multifactorial disease traits has typically been substantially less than the estimates of the additive genetic variation found in conventional pedigree- based quantitative genetic analyses. For example, the top 150 loci identified by SNPs account for only 10% of the variance in human height (Lango-Allen et al. 2010).Ifall SNPs are fitted without constraint as to statistical significance, the amount explained rises to ^40% (Yang et al. 2011) but still falls well short of the estimates of ^80% based on cova- riances among relatives. Several hypotheses for the so-called "missing heritability" (Manolio et al. 2009) have been put forward, among these that epistasis is inflating the pedigree- based estimates (Zuk et al. 2012).

We therefore have some dichotomy between, on the one hand, those who argue that, although there may be epistasis, the epistatic variance contributed is likely to be small and generally we can ignore its contribution because it is unimportant, and on theotherthosewhoarguethatthisisablinkeredview.We cannot resolve this here, but we do attempt to provide a stronger theoretical base to the arguments. The analysis and inferences of Hill et al. (2008)tookintoaccountthesizeanddirectionof interaction effects and the allele frequencies or heterozygosity at all contributing loci. Models were, however, based on pairs of loci, yet as quantitative geneticists have long assumed, have subsequently inferred from the continuing responses to selection for many traits in experiments and breeding programs, and now inferredmoredirectlyfromGWAS,verymanylociinfluence each trait. For multiple loci (n), however, the numbers of pairs with potential two-way interaction among loci increases as n2 [strictly 1/2n(n &endash; 1)], three-way interactions as n3,etc.,andsoit is possible that merely accounting for interactions among pairs of loci gives a biased impression.

In this article we therefore consider the magnitude of epistatic variance expected in outbred populations for multilocus models to check on the robustness of the argument that most variance is likely to be additive genetic. We concentrate initially on models without dominance or interactions involving domi- nance, partly for simplicity of exposition and partly because we showed previously that additive variance is also expected to contribute much more than the dominance variance (Hill et al. 2008). We consider alternative distributions of the gene fre- quency within the populations at segregating loci affecting the trait because these have an important impact on the magnitude of the genotypic variance and its components. As these distribu- tions are not static in populations evolving under selection, we also consider the impact of consequent gene frequency changes.

Analysis Genetic model with no dominance We assume that there are n loci, each of which has two alleles Ai or ai, i =1,..., n, with frequencies pi and 1 &endash; pi,respec- tively. We further assume that there is Hardy&endash;Weinberg and linkage equilibrium among all loci. Genotypic values are given in Table 1 (second column) for examples of alternative genotypes, and those for other genotypes are analogous, e.g., that for A1A1a2a2A3a3 is 2a1 + a3 +2[aa]13 (cf.fourthrow). The model readily extends to more loci and to dominance (see below and third column of Table 1) and is fully parameterized.

The population mean without dominance is ...

With H&endash;W and linkage equilibrium, average effects (also called "additive effects") and the additive variance can be found by averaging over genotypes, by regression of phenotypes on the number of increasing alleles within loci (Fisher 1918), or from the first derivative of the mean with respect to the frequency of the increasing allele (Kojima, 1959, 1961). We adopt Kojima's approach, which extends simply to epistatic effects using higher derivatives involving the corresponding loci. (In the absence of dominance higher derivatives at individual loci are zero.) For locus i, the average effect including up to third-order interaction terms is ... (1) The additive variance is obtained by summing over the n loci ...

where ... is the heterozygosity at locus i. Sim- ilarly additive 3 additive interaction effects and variances are ... (2) and so on, with the nth such partial derivatives giving n- locus epistatic effects and variances.

For three loci, for example, ...

and therefore ... (3a) ... (3b) ... (3c) The key to subsequent results is that, because two- and three- locus interactions enter the average effects and additive variance, they can make a major contribution to VA, depend- ing on their allele frequencies and sign. Similarly k-locus in- teraction effects contribute to the epistatic variances of order ,k, whereas single-locus contributions and interaction effects of order lower than k do not.

Numbers and magnitude of terms in variance components While it is assumed that many loci determine most quanti- tative traits, the number (n) is not known. To obtain some understanding of how the relative magnitudes of additive and epistatic variance depend on n, we undertake some simple calculations on the basis that the partition will depend at least to some extent on the numbers of terms contributing to each. For the additive variance there are contributions from n loci, and expansions such as in Equations 1 and 3a show that for each locus there are ...

terms in the expression for the average effects and thus n22(n 2 1) = n4n 2 1 in all when effects are squared (counting terms such as ai[aa]jk and [aa]jkai separately). There are more pairs, trios etc. of loci contributing epistatic variance, but the numbers o^f t^erms they each comprise are reduced. For VAA there are kn =1/2n(n 2 1) pairs of loci but 4n 2 2 terms in each pa^ir w^ hen squared (Equation 3b). In general, there are atotalof nk 4n 2 k terms for the kth-order epistatic variance among n loci. While terms such as [aa][aaa]thatcomprise ^pro^ducts of different interaction effects may be negative, nk 2n 2 k are squared terms and therefore nonnegative.

Intheformulaefortheaverageandothereffects,termsinL loci have a coefficient of 2L 2 k 2 1, e.g.,4forpjpk[aaa]ijk in Equation 1. But coefficients involving products of gene fre- quencies also appear in these same terms, each of which take values of 0.5 when gene frequencies are one-half. Thus if we assume gene frequencies are 0.5, which is when epistatic var- iance is typically at a maximum, this cancels the coefficients of 2 alluded to above. Hence while there are arguments for in- cluding other terms in these simple models, as a basis we re- strict them just to assuming th^ es^e factors of two cancel. Hence t^he^sum of weighted terms is nk 4n 2 k and of squared terms is nk 2n 2 k. Examples of these figures are given in Table 2, which shows how the number of contributions to the varian- ces, particularly the additive variance, can be large indeed.

Heterozygosity has a maximum of 0.5 at P =0.5,when,in the absence of epistasis, the additive variance is then at a max- imum. As the epistatic variances among k loci depend on products of k heterozygosity terms, the epistatic variances are likely to contribute correspondingly less to the genotypic variance VG than does VA, and most of the epistatic variance is likely to derive from second-order components. Thus if we include heterozygosity in the calculations, letting H be an "average" or represe^nta^tive figure, the kth component of var- iance is of order Hk nk 4n 2 k.IfH = 0.5, its m^ a^ximum, the ^we^ighted number of^ te^rms, becomes (1/2)k nk 2n 2 k = nk 4n 2 k,ofwhich kn 2n 2 2k are squared terms. Further, in all but populations derived from F1 crosses, mean heterozy- gosity will be less than one-half, and we subsequently consider the impact of different allele frequency distributions on the ex- pected proportion of epistatic va^ria^nce. According^ly, ^the weighted numbers of terms become ^ nk 4n 2 4k and kn 2n 2 4k,re- spectively, i.e., very much smaller indeed as k increases.

Examples of the computed values are given in Table 2. These calculations for multiple loci are intended to serve only as a guide, but illustrate why a high proportion of epistatic variance is unlikely, because even very strong multilocus gene&endash;gene interactions do not automatically lead to a high proportion of epistatic components in the genetic variation (cf. Cheverud and Routman 1995).

Examples The relative magnitude of additive and epistatic variances for two to five loci including all possible gene&endash;gene interactions is summarized in Figure 1 for two models with the frequency of theincreasingallelethesameatallloci.Inboth,themagnitude of gene and (absolute value of) interaction effects are also assumed to be the same at all loci, and are either all synergistic ([aa]=[aaa]=...= a)orantagonistic([aa]=[aaa]=...= 2a). As expected from the above arguments, the genetic var- iance in many of these examples is seen to be mainly additive and the remainder mainly additive 3 additive, such that higher- order epistatic variances can generally be ignored. Gene inter- actions generally make the most substantial contributions to VA when the frequency of the increasing allele is very low, and only negative (antagonistic) interactions cause relatively large ratios of epistatic to additive variance because terms such as ai[aa]jk in VA are then negative.

Expected variances for different allele frequency distributions The components of genetic variance and their relative magnitudes depend on the allele frequencies in addition to the individual gene effects and their interactions. Although there is little information on allele frequencies for quantitative trait genes, it is possible to predict their frequency distribution under different assumptions about relevant evolutionary factors, such as mutation, selection, finite population size (drift), and migration. Allele frequencies of 0.5, i.e., from a re- cent inbred cross, serve as one reference point and, as evi- dence suggests that many genes influence metric traits and so there is presumably very mild natural selection on them, the- ory for neutral alleles in finite populations serves as another.

Two allele frequency distributions are considered, each with mean 0.5. One is a closed finite population under drift without mutation, when the steady-state distribution is uniform over (0, 1) (e.g., Crow and Kimura 1970; Falconer and Mackay 1996). The other is for mutation&endash;drift balance, when the frequency density of mutants is proportional to 1/p (Wright 1931); but, if increasing and decreasing mutants for the trait are assumed equally likely, the frequency distribution of segregating increasing mutants for the trait is U shaped, with f(p)}1/[p(1 &endash; p)] (Hill et al. 2008). The distributions and expected heterozygosities E(H)are ...

For the U-shaped distribution and effective population sizes N = 10, 20, 100, 1000, and 100,000, E(H)~0.167, 0.136, 0.0944, 0.0658, and 0.0410, respectively. Examples are given in Figure 2 for the partition of variance under different epistatic and frequency distribution models for three and more loci. The largest proportion of epistatic variance is expected when P = 0.5 while uniform and U-shaped allele frequency distributions yield mostly additive variance, even with antagonistic gene&endash;gene interaction. In all cases the epistatic variance is almost entirely contributed by the two- locus epistatic variance.

When n loci contribute nearly equally to the variation, for given total genetic variance, their individual effects a will tend to decline approximately in proportion to 1= n as n increases to and, similarly, epistatic interactions between pairs of loci [aa] are likely to decline roughly in proportion to 1/n and between trios of loci [aaa]by1=ðn nÞ. The dimin- ishing relative size of the interaction effects with increasing number of loci shown above for those with equal effect is therefore exacerbated as the interaction terms become indi- vidually smaller (Figure 2A vs. Figure 2B). Hence, if there is a very large number of interacting loci (say . 50) almost all the variation is expected to be additive. If their effects differ greatly, then the results become like those for fewer loci.

To illustrate the effect of increasing the number of loci, we include interactions only up to second or third order with positive (or negative) sign consistent over loci. Again, unless interaction terms are such that, for example, [aa] is negative and its products in the expressions for variance balance the positive a2 and [aa]2 term (Figure 2), the general conclusion is that, with a very large number of loci, only in special cases are the positive and negative terms likely to counter each other and the epistatic variance to be larger than the addi- tive variance. Similarly, as the number of loci become very large, epistatic variance of higher order also become trivial compared to VAA.

Dominance The diploid models can easily be extended to incorporate dominance effects and Kojima's (1959) method can be used to compute the different components of genetic variance. This has been done to illustrate the close correspondence with ad- ditive models (Appendix 1). The inclusion of dominance intro- duces three additional two-locus interaction terms [ad], [da], and [dd].Similartothewayinwhichtheadditiveinteractions contribute to the additive variance, the dominance interactions contribute to both the additive and dominance variance. The number of additional terms in the formulae for the variances increases rapidly, so for clarity, results are given only for two loci with positive interactions (Figure 3).

As in the case of additive interactions, the proportion of epistatic variance is small, in particular when allele frequen- cies are not concentrated at intermediate values. The pro- portion of dominance variance stays the same or is slightly increased when adding the dominance interaction while the ratio between additive and dominance variance is not changed, implying that the interaction terms contribute to these variances in the same way. When the allele frequencies are 0.5 there is most epistatic variance, but its contribution rapidly decreases for a higher number of loci, with the same pattern as with only additive effects and interactions.

Effect of selection on the expected variances For traits under stabilizing selection with intermediate opti- mum, frequencies of increasing and decreasing alleles will also tend to be distributed around 0.5 as in the neutral case. In a population newly under directional selection, the mean fre- quency of increasing alleles will increase .0.5, but under long continued selection stabilizeasnewmutantscomein.Hence we consider the results for neutral distributions to be still relevant. We examine some limited examples, however.

In a population under selection the contribution of epistatic to total variance changes as frequencies change. Thus as Hansen (2013), for example, emphasizes, although selection response at any given time depends almost entirely on the additive variance (Kimura 1965; Crow 2010), the long-term trajectory depends on the nature of the interactions among the loci. Further, allele frequency distributions can no longer remain symmetric around P = 0.5 under directional selection. In a small investigation of the robustness of some of our con- clusions under selection, for example, that a high proportion of VG is contributed by VA, we used transition probability matrices to obtain the distributions for pairs of unlinked loci, modeling with quite small population size values. For sim- plicity a haploid model with population size Nh was assumed. Details of the model are given in Appendix 2.

Several simple situations were modeled using two loci with synergistic or antagonistic epistatic effects ([aa]=+a or &endash;a). These included cases where (i) the increasing alleles at each locus were mutants at initial frequency 1/Nh, (ii) the frequency at one locus was at steady state under neutrality (i.e., U shaped) and at the other locus a mutant allele of either increasing or decreasing effect, and (iii) initial fre- quencies at the two loci were set as either 1/Nh or 1 &endash; 1/ Nh with equal probability. Components of variance were obtained by averaging over 100 generations.

Detailed findings are given in Appendix 2. Although lim- itedinscope,theresultsareclear:theasymmetryinthe allele frequency distributions caused by selection does not seem to influence the relative proportions of additive and nonadditive variance and thus substantially alter the gen- eral expectation that the proportion of epistatic variance is likely to be small.

Discussion Main findings When the genetic variation is caused by segregation at many loci, the magnitude of the epistatic variance is almost certain to be trivial relative to the additive variance. This result is nearly invariant to the order of the interaction, allele fre- quencies, and type and magnitude of interaction effects. The models show how the gene-interaction effects appear in the average effects and contribute to the additive variance and produce more positive terms there than in the epistatic components. The epistatic components depend on the magni- tude of heterozygosity and are generally small for low hetero- zygosity. For the same reasons, under most of the assumptions the majority of the epistatic variance is due to the two-locus epistatic component.

When the variation is due to only a few interacting loci or there are major genes with substantial gene&endash;gene interac- tion effects segregating in the population, there are no gen- eral patterns in the proportion of epistatic variance, but the following are the main observations: high heterozygosity (or allele frequency 0.5) and/or negative interaction can gener- ate high proportions of epistatic variance; the proportion of the variance that is epistatic variance is lowest at low het- erozygosity, notably for the U-shaped allele frequency distri- bution but even for the uniform distribution compared to P = 0.5; and deviations from symmetry of the distribution caused by mild selection do not increase the low proportion of epistatic variance.

Models and assumptions Only biallelic loci were analyzed but we cannot see that the general results would be fundamentally affected if loci are multiallelic. Each of the m two-way interactions for m alleles at each locus with no dominance appear in the average effects for the m alleles at the m loci, giving m3 terms after squaring effects, whereas the number of additive 3 additive effects and variance components increases only in propor- tion to the number of allelic pairs, m2, as each has only one two-way interaction. Although we focused on models with- out dominance, as the interaction effects containing domi- nance contribute to the dominance variance in the same way as do the additive 3 additive effects to the additive variance, the qualitative finding remains that the proportion of epi- static variance in the total genotypic variance is small.

Epistasis also arises, even with infinitesimal additive effects on some underlying variable, if there is a nonlinear relation- ship between genotypic and observed phenotypic values, such as with a multiplicative, optimum, or threshold model (e.g., Wright 1931; Dempster and Lerner 1950; Cockerham 1959). Kojima's (1959, 1961) methodology provides a simple way to partition the genotypic variance and can also be applied to these models. Basically, the proportion of epistatic variance is high only when the transformation departs far from linearity: ahighcoefficient of variation in the multiplicative model, the population mean near the optimum value in the optimum model,andwithaproportiontruncatednear0or1inthe threshold model (see also Mäki-Tanila and Hill 2014).

In an attempt to explain some of the missing heritability, especially that estimated from human twin studies, Zuk et al. (2012) suggested that the additive component was overestimated, biased inter alia by epistatic variance. They proposed an extreme type of threshold model, with the all- or-none output dependent on whether, in some underlying system of multiple identically independently (i.i.d.) nor- mally distributed variables, the lowest exceeded the cutoff. This model undoubtedly generates substantial epistatic var- iance, but has been queried both because it is sensitive to assumptions, such as i.i.d. and is biologically implausible (Stringer et al. 2013). Indeed there is little evidence of such limiting pathways, and models based on flux through sys- tems without "limiting steps" have firmer foundation (Kacser and Burns 1973). Analysis of genetic models of flux in pathways shows that largely additive variance is to be expected (Keightley 1989; Hill et al. 2008). Much of the missing heritability can be accounted for by fitting all SNPs without regard to statistical significance (Purcell et al. 2009; Yang et al. 2011), showing that multiple sites are involved. This observation that multiple loci influence the traits coupled with our demonstration that this lends support to additive rather than nonepistatic models argues against the belief that much of the missing heritability is due to epistatic variation.

In the examples studied, we set the gene&endash;gene interac- tions [aa], [aaa], ..., to be of the same magnitude as the single-locus effects a. This is important in assessing the pro- portion of epistatic variance for models with very few loci but ceases to do so as the number of loci increase. It seems biologically reasonable to assume that the interaction effects are smaller, however, in which case a higher proportion of additive variance is seen with even a few loci and strong antagonistic pleiotropy. In the models studied we assumed that all loci were interacting with each other, but most inter- actions would not involve them all and consequently there would be relatively more additive variance.

Linkage disequilibrium An important assumption made is that there is linkage equilib- rium (no LD) among all pairs of loci comprising the analysis, indeed contributing to the trait. In higher organisms most pairs of loci are on different chromosomes or far apart on the same chromosome. Further, even for closely linked loci, LD is likely to be small in populations of large effective size, such as for humans and Drosophila melanogaster, although not so for closely linked loci in domestic livestock (e.g., de Roos et al. 2008, Veroneze et al. 2013) and laboratory populations of experimental species. A theoretical analysis of variance parti- tion in the presence of LD is problematic because trait effects of different loci are not orthogonal; i.e., fitting locus B after A removes less variance than fitting B alone, and epistatic var- iance may be confounded with that of average effects. An alternative approach is to be pragmatic and consider just the quantities that can readily be estimated from the cova- riances among relatives such as half-sibs, full-sibs, parent&endash; offspring, and grandparent&endash;offspring. We have analyzed this complete LD scenario for pairs of epistatic loci including only additive effects and interactions. We made a further assumption that, as substantial LD occurs only among very tightly linked loci, recombination between close relatives can be ignored. Our preliminary analysis shows that the magnitude of epistatic variance, as judged say by the sire 3 dam interac- tion component or the difference between the full-sib and offspring&endash;parent covariance, remains small relative to the 'additive' component (e.g.,43 covariance of half sibs), as in similar models examined here for linkage equilibrium (Mäki-Tanila and Hill 2014).

Prediction of selection responses due to epistasis The success of a selection scheme within populations depends on utilizing the additive variance. Griffing (1960) showed with theoretical analyses how the genetic changes obtained with mass selection that derive from the epistatic variation are only temporary, as selected haplotypes break up. Similar predictions were made by Bulmer (1980) by comparing the offspring&endash;parent and grandoffspring&endash;grandparent regres- sions. Estimates of VA based on four times the half-sib covari- ance, which has expectation 1/4VA +(1/16)VAA + ...,are, however, unlikely to be substantially biased by epistatic vari- ance because, for typical quantitative VAA,theyareexpectedto be small compared to VA. While the rate of recombination loss is lower with linkage, as shown by Kimura (1965) and Crow (2010), only the additive variance contributes to the steady- state response even in the presence of linkage.

Hansen (2013) has argued that epistatic effects must be included in predicting long-term responses to natural selec- tion. While short-term response to selection can be predicted from the additive variance, changes over many generations depend on genetic architecture, including the distribution of gene effects and frequencies at loci that do not exhibit epistasis and therefore on many unknowns. Thus, under most circumstances, Hansen's objective is not achievable in practice, and adding another level of complexity would only exacerbate the uncertainty. Further, the changes ob- served in plant and animal breeding have continued for very many generations, often with little or no sign of slowing down even in very intensive selection schemes (Dudley and Lambert 2004; Hill 2010). The achieved changes have been far beyond what could be foreseen in the early stage of selection.

Nelson et al. (2013, p. 671) find classical quantitative genetics incapable of giving "insight to infer the underlying, functional architecture of the traits required to predict phe- notypes of individuals in other populations, as desired in medical diagnostics, or long-term changes in populations studied in evolutionary biology." The incorporation of epis- tasis in predicting breeding values or individual risks would, for example, also depend on the distribution of gene effects across loci and the genome. Although they consider the existing research insufficient for characterizing epistatic effects needed in predicting individual phenotypes, their objective, like that of Hansen (2013), is laudable. The incorporation of epistasis in predicting breeding values or individual risks woulddependondistributionofgeneeffectsacrossloci and genome. The prediction of response to directional selection over multiple generations, however, already contains many unknown even without epistasis, not least the consequences of mutations. Adding another level of complexity would only exacerbate problems.

Epistatic variance and findings on functional epistasis Experiments from crosses among lines have revealed many clear cases of epistatic QTL, by Bloom et al. (2013) in yeast, Huang et al. (2012) in Drosophila, and Pettersson et al. (2011) in chickens. These contributed a sizable proportion of the variance. As we have shown, however, the contribu- tion of epistasis is likely to be largest with genes at frequency one-half, and so these findings do not imply substantial con- tributions of epistatic variance in natural or domesticated populations maintained segregating with large effective pop- ulation size. In contrast to such studies using defined crosses e.g., in which gene frequencies are all 0.5, in (outbred) hu- man and livestock populations minor allele frequencies are likely to be diverse and typically low. There are hundreds of loci affecting both continuous and disease traits, hence vast numbers of potential locus combinations, most of which are not present in the populations analyzed. This is in line with the findings by Huang et al. (2012) and Hemani et al. (2014) of several epistatic QTL but nevertheless these made very little overall contribution to the genetic variances. In an anal- ysis of gene expression, for loci found to be interacting by variance heterogeneity, Brown et al. (2014) detected on av- erage 4.3% of the variance to be epistatic, with the highest up to 16%. This represented two successive selection processes in the analysis, however.

Mackay (2014) argues for the importance of epistasis, and while acknowledging the fact that substantial amounts of epistasis do not in themselves produce much epistatic variance, provides experimental data on behavioral traits in D. melanogaster. For Drosophila inbred lines that were de- rived from an outbred population the variance among lines was much more than twice the additive variance within populations expected under an additive model. This com- parison is, however, subject to potential biases due to non- additive gene action within loci: notably low-frequency recessives contribute little variance within lines but a lot between lines. Her group also concluded from comparisons of QTL detected from GWAS and from selection within a population that there was little correspondence in the estimates of effects (regions of large effect) between them, interpreting this as epistasis. In such an experiment, al- though lack of power in either analysis leads to noncorres- pondence and apparent interaction, this was minimized by focusing on variants with highly significant marginal effects.

The best-known biochemical networks of up to a few dozen nodes in laboratory organism exhibit much interac- tion (Phillips 2008). Therefore it is important that research on functional epistasis of individual genes is pursued to gain more knowledge about the nature of quantitative trait var- iation. The new findings can be then incorporated in the statistical models elucidating the state and potential in the variation. Quantitative geneticists would like to exploit all the information underlying genotypic values. The challenges in detecting interaction effects are great for a number of reasons: multilocus genotypic classes have lower frequency, epistatic effects are likely to be smaller than main effects, power of detection in GWAS depends on there being LD between two pairs of trait genes and markers, and their detection has to pass more stringent statistical thresholds because so many tests are undertaken. Therefore it is not yet feasible to obtain information on epistasis among indi- vidual loci for quantitative traits pending as-yet-unforeseen advances in technology or ideas are achieved.

Conclusions We show that substantial epistasis among multiple loci is likely to lead to mostly additive genetic variance in outbred populations. The common error is, however, to assume a lot of one implies a lot of the other. Perhaps the controversies arise because it is not fully appreciated or scientists choose not to acknowledge this rather mundane theoretical finding. An understanding of the biology, however, requires that the epistasis be identified. A pragmatic approach is likely to be more successful in areas such as livestock improvement or understanding the variation of most traits in humans. Only when at the individual gene or gene pair do we need to concern ourselves greatly with epistasis.

Acknowledgments We thank Nick Barton for helpful comments on the manu- script. A.M.T. acknowledges the receipt of a fellowship from the OECD (The Organisation for Economic Co-operation and Development) Co-operative Research Programme: Biological Resource Management for Sustainable Agricultural Systems in 2013 for the visit to Edinburgh.

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Communicating editor: S. Sen Asko Mäki-Tanila*,1 and William G. Hill[dagger] *Biotechnology and Food Research, MTT Agrifood Research Finland, 31600 Jokioinen, Finland, and [dagger]Institute of Evolutionary Biology, School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JT, United Kingdom Copyright © 2014 by the Genetics Society of America doi: 10.1534/genetics.114.165282 Manuscript received April 14, 2014; accepted for publication June 19, 2014; published Early Online July 1, 2014.

Available freely online through the author-supported open access option.

1Corresponding author: Biotechnology and Food Research, MTT Agrifood Research Finland, 31600 Jokioinen, Finland. E-mail: [email protected] (ProQuest: Appendix omitted.) (c) 2014 Genetics Society of America

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