Journal of Heredity, 2017, 318–327 doi:10.1093/jhered/esw123 Original Article Advance Access publication January 12, 2017
Original Article
Response to Selection in Finite Locus Models with Nonadditive Effects Hadi Esfandyari, Mark Henryon, Peer Berg, Jørn Rind Thomasen, Piter Bijma, and Anders Christian Sørensen From the Center for Quantitative Genetics and Genomics, Department of Molecular Biology and Genetics, Aarhus University DK-8830 Tjele, Denmark (Esfandyari, Berg, Thomasen, and Sørensen); Seges, Danish Pig Research Centre, Copenhagen, Denmark (Henryon); School of Animal Biology, University of Western Australia, Crawley, Australia (Henryon); Nordic Genetic Resource Center, Ås, Norway (Berg); VikingGenetics, Assentoft, Denmark (Thomasen); and Animal Breeding and Genomics Centre, Wageningen University, Wageningen, The Netherlands (Bijma). Address correspondence to H. Esfandyari at the address above, or e-mail: hadi.esfandyari@ mbg.au.dk. Received May 18, 2016; First decision June 7, 2016; Accepted January 2, 2017.
Corresponding Editor: C Scott Baker
Abstract Under the finite-locus model in the absence of mutation, the additive genetic variation is expected to decrease when directional selection is acting on a population, according to quantitative-genetic theory. However, some theoretical studies of selection suggest that the level of additive variance can be sustained or even increased when nonadditive genetic effects are present. We tested the hypothesis that finite-locus models with both additive and nonadditive genetic effects maintain more additive genetic variance (VA) and realize larger medium- to long-term genetic gains than models with only additive effects when the trait under selection is subject to truncation selection. Four genetic models that included additive, dominance, and additive-by-additive epistatic effects were simulated. The simulated genome for individuals consisted of 25 chromosomes, each with a length of 1 M. One hundred bi-allelic QTL, 4 on each chromosome, were considered. In each generation, 100 sires and 100 dams were mated, producing 5 progeny per mating. The population was selected for a single trait (h2 = 0.1) for 100 discrete generations with selection on phenotype or BLUP-EBV. VA decreased with directional truncation selection even in presence of nonadditive genetic effects. Nonadditive effects influenced long-term response to selection and among genetic models additive gene action had highest response to selection. In addition, in all genetic models, BLUP-EBV resulted in a greater fixation of favorable and unfavorable alleles and higher response than phenotypic selection. In conclusion, for the schemes we simulated, the presence of nonadditive genetic effects had little effect in changes of additive variance and VA decreased by directional selection. Subject area: Quantitative genetics and Mendelian inheritance Keywords: finite locus model, nonadditive effects, selection
Dominance and epistatic effects have been detected in many experimental populations (Carlborg and Haley 2004) and these allelic interactions can influence the amount of additive genetic variance
(VA) in populations (Goodnight 1987; Cheverud and Routman 1995; Hansen and Wagner 2001; Caballero and Toro 2002; Barton and Turelli 2004). It has been argued that epistatic variance may be
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“converted” into additive variance by genetic drift when a population passes through a population bottleneck (Goodnight 1995; Cheverud and Routman 1996; Cheverud et al. 1999; López-Fanjul et al. 2002; Barton and Turelli 2004). However, this argument is not restricted to genetic drift. Changes in VA occur with variations in the genetic background, and any process that changes allele frequencies, including selection, can change VA (Hansen and Wagner 2001). Gene interactions may also affect response to selection through a build-up of linkage disequilibrium associated with favorable gene combinations, since parents transmit not only half of the additive effects to offspring, but also a quarter of the pairwise epistatic effects (A × A) and smaller fractions of higher-order interactions (Lynch and Walsh 1998). This suggests that some of the linkage disequilibrium built by epistatic selection can be converted into response to selection (Griffing 1960). The model most commonly used for genetic evaluation is the infinitesimal model. It assumes large numbers of genes affecting traits with each gene having a small additive effect (Fisher 1918). In this model, selection does not change the allele frequency significantly at any individual locus, nor does it change the genetic variance except for nonpermanent changes caused by gametic phase disequilibrium (the Bulmer effect; Bulmer 1971). The assumptions of the infinitesimal model are incorrect. In practice, there are individual genes, sometimes with large effects, and many genes showing dominance and epistasis (Mackay, 2001a, 2001b). An alternative to the infinitesimal model is a finite-locus model, which can accommodate nonadditive inheritance. Under the finite-locus model, the additive genetic variation is expected to decrease when directional selection is acting on a population, according to quantitative-genetic theory (Crow and Kimura 1970; Falconer and Mackay 1996). However, some theoretical studies of selection suggest that the level of additive variance can be sustained or even increased when nonadditive genetic effects are present, in a manner similar to the action of genetic drift (Carter et al. 2005; Fuerst et al. 1997). Experimental evidence for this phenomenon was found by Martinez et al. (2000), when they selected mice for body fat, and by Sorensen and Hill (1982), who selected D. melanogaster for abdominal bristle number. Furthermore, Carlborg et al. (2006) showed that epistatic interactions between 4 loci mediated a considerably higher response to selection for growth in chicken than predicted by a single-locus model. In a simulation study, Hallander and Waldmann (2007) investigated changes inVA in the presence of nonadditive effects in a trait subjected to directional selection. They showed that by including dominance and epistatic effects, VA was increased during the initial generations of selection. Fuerst et al. (1997) showed similar results by using a 2-locus genetic model to simulate a trait with different levels of additive and nonadditive genetic effects. In these simulation studies VA increased by including nonadditive effects, but it seems considering 2–4 loci with equal additive and dominance effects across all loci with initial frequency of 0.5 is not a realistic model of the underlying genes. In fact, finite-locus models used in these studies to test quantitative theory are too restrictive. Indeed, it is possible to fit less restrictive models with many genes each having a unique effect, thus allowing a range from genes of large to zero effect. These genes could display nonadditive effects such as dominance or epistasis. In theory, such a model seems to agree more closely with what we know about the genetics of quantitative traits than the simple models (Goddard 2001). In animal breeding, response to short-term selection of a few generations depends on the intensity and accuracy of selection. Response to long-term selection depends on how much genetic variability, in particular, additive variance is maintained and how much inbreeding depression occurs over generations, and therefore on effects effective
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population size (Robertson 1960). In the absence of new mutations, the total response to selection is limited by the initial standing variation. Selection acts on the variation present in the population, and will typically act to reduce it; the extent to which it does so depends on the allele frequencies, and on the relation between genotype and phenotype (i.e., the “genetic architecture”). Interactions between alleles (epistasis) are a key component of this genetic architecture, and their effect on the response to selection has been controversial. On the one hand, artificially selected populations are usually well approximated by the infinitesimal model, which, in its simplest form, assumes infinitely many loci of small and additive effect (Hill et al. 2008). This suggests that gene interactions do not play an important role. On the other hand, biology can hardly be additive, and genomes are finite. If the trait depended on a small number of additive alleles, these would quickly fix contradicting the robust observation of sustained response to selection. One possibility is that epistasis sustains additive genetic variance for longer: Alleles that were initially deleterious or near-neutral may acquire favorable effects as the genetic background changes, “converting” epistatic variance into additive, and so prolonging the response to selection. The question of whether selection can generate additive genetic variance is an important one for a number of reasons, including the more nuanced understanding we are developing of the relationship between molecular variation (“QTL”) and phenotypic variation, and the results of long-term selection experiments that can show an approximately linear response to selection for a hundred generations or more (Goodnight 2015). Previous studies for investigating effect of selection on additive genetic variance in presence of nonadditive effects have focused on phenotypic selection (Fuerst et al. 1997; Carter et al. 2005; Hallander and Waldmann 2007). An alternative to selection based only on the phenotypic record of the individual is selection based on best linear unbiased predictor (BLUP) of breeding value (Henderson 1975), which uses records on all relatives, in addition to the individual’s own record, in genetic evaluation. The BLUP theory was first proposed by Henderson in 1949 to describe the environmental and genetic trend estimates in cattle and was subsequently developed (Henderson 1975). The main advantage of the BLUP methodology is the high level of accuracy of the breeding (or genetic evaluations) and has the property to account for selection of parents in a breeding population. Hence, it fairly accounts for the fact that some animals are from better parents than other animals. Many simulation studies have compared selection responses based on BLUP with those based on either phenotype or family index. In 2 simulation studies of a pig population, phenotypic selection led to response rates of 64% and 91% of those for BLUP for heritabilities of 0.1 and 0.6, respectively (Belonsky and Kennedy 1988). However, for a trait under selection, there is no theory to quantify the differences between phenotypic selection or BLUP-EBV selection when the trait is controlled both by additive and nonadditive genetic effects. This work explores the effect of presence of nonadditive effects in genetic models and assesses their importance in medium- to long-term selection experiments. For this objective, we examine how the genetic variance and genetic gain are affected by the presence of nonadditive genetic effects, using BLUP-EBV and phenotypes as selection criteria.
Methods Procedure We used stochastic simulation to estimate genetic gain and monitor changes in genetic-variance components generated by 4 genetic models, with 2 different selection criteria and 2 distributions of QTL
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effects. The models were applied to a population undergoing directional truncation selection for a single trait over 100 discrete generations. During each generation, the genetic gain and changes in additive, dominance, and epistasis variances were calculated. The simulations were carried out using a modified version of ADAM (Pedersen et al. 2009). For each scenario, 100 replicates were performed.
Genetic Models The 4 genetic models, 2 selection criteria, and 2 distributions of QTL effects are presented in Table 1. The first genetic model (A) assumed that the trait is controlled by additive-gene actions. In the second model (A/D), the trait was controlled by additive and dominance effects. In the third model (A/AA), additive and additive × additive epistatic effects were included. In the final model, a full-genetic model (A/D/AA) consisted of additive, dominance, and additive × additive epistatic effects.
Selection Criteria Truncation selection was applied in each generation and the criterion for truncation selection was either the phenotypic observation of the individual or estimated breeding values (EBV) obtained from standard BLUP evaluations. The BLUP-EBVs were obtained by Henderson’s (1975) mixed linear model. Two pieces of information were used: phenotypic records and pedigree data. The following model was used to estimate breeding values: y = µ + Zu + e, where y is a vector of phenotypic values, µ is the overall mean Z is known design matrices relating observations to the random effects u, u is the vector of breeding values, and e is the vector of random residual effects. The numerator relationship matrix (A) was used in the following mixed model equations to derive BLUP of random additive effects (breeding values):
why selection response does not decline in the first few generations, because selection increases the frequency of rare favorable alleles and, hence, increases the genetic variance due to these loci, which compensates for the loss of variance caused by selection for common favorable alleles (Goddard 2001). Two distributions were fitted for QTL effects (ai); either a gamma distribution (0.4, 1.66) or a mixture of a double exponential distribution and a normal distribution, that is,
(
ai ~ 0.95 . L (0, u2 ) + 0.05.N 0, (5u)
2
)
(Figure 1). With appropriate u > 0. These distributions resulted in many QTL with small effects and few QTL with large effect (Bennewitz and Meuwissen 2010). Dominance effects (di) were generated between alleles at the same locus and interactions between loci (aaij) only occurred for each pair of neighboring loci, and a locus had only interaction with one other locus. Dominance degrees (hi) were normally distributed with mean µh = 0.5, variance Vh = 1, and they were independent of the additive effects. Dominance effects were then calculated as di = hi ai so they became dependent to the additive effects (Wellmann and Bennewitz 2011). First-order (additiveby-additive) epistatic degrees (kij) also followed a normal distribution (0, 1), and the epistatic effects were calculated as aaij = kij ai ⋅ aj , where aaij is the epistatic effect of the 2 adjacent loci, ki is the epistatic degree, ai and aj are the additive effects at the first and second locus. The total genotypic value of an individual was obtained by summing the genotypic contribution of each locus pair (Table 2).
σ e2 −1 Z ′Z + A σ 2 [aˆ ] = [Z ′y] a where σ e2 is the residual variance and σ a2 is the additive genetic variance.
Simulated Genome and Distributions of QTL Effects The simulated genome consisted of 25 chromosomes. One hundred QTL, 4 on each chromosome, were considered. All chromosomes had a length of 1 M and the QTL were assumed to be positioned randomly on each chromosome following a uniform distribution. All QTL were bi-allelic and initial frequencies of the alleles followed a U-shaped distribution as suggested by Crow and Kimura (1970). The U-shaped distribution of gene frequencies explains
Table 1. Values of the initial ratio of nonadditive variances on additive variance, QTL effects distribution and selection criteria in simulated scenarios Tested parameters
Values
Distribution of QTL effects
Mixture, Gamma P, BLUP VD/VA 0 1/2 0 1/2
Selection criteria Genetic model A A/D A/AA A/D/AA P stands for phenotypic selection.
Figure 1. Distribution of QTL effects.
Table 2. Genetic effects of genotypes for a pair of loci based on Fuerst et al. (1997) VAA/VA 0 0 1/4 1/4
Genotype at locus 1
AA Aa aa
Genotype at locus 2 BB
Bb
bb
a1 + a2 + aa12 d1 + a2 −a1 + a2 − aa12
a1 + d2 d1 + d2 −a1 + d2
a1 − a2 − aa12 d1 − a2 −a1 − a2 + aa12
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The genetic variance components depend on gene frequencies and values of different genetic effects. Following Fuerst et al. (1997), they were computed as: np
VA = ∑{2 p1q1 a1 + (q1 − p1 ) d1 + ( p2 − q2 ) aa12
2
l
+ 2 p2 q2 a2 + (q2 − p2 ) d2 + ( p1 − q1 ) aa12 }l 2
VD =
np
∑{4p q d
2 2 2 1 1 1
+ 4p22 q22 d22 }l
l
VAA =
(1)
(2)
np
∑{4p q p q aa 1 1 2
l
2
}
2 12 l
(3)
where VA is the additive variance (variance of breeding values), np is the total number of pairs of loci, VD is the dominance variance (variance of dominance deviations), VAA is the additive-by-additive variance, a1 and a2 are the additive effect at loci 1 and 2, d1 and d2 are the dominance effects, and aa12 is the additive-by-additive effect at the pair (1 and 2). The gene frequencies of alleles A, B, a, and b are p1, p2, q1, and q2.
Population Structure One hundred sires and 100 dams were in the base population (generation 0) of each scenario and were mated randomly to produce the first generation of offspring. Offspring produced by the base population were selected in generation 1 and the first generation of offspring from selected parents was produced in generation 2. Offspring produced in generation 100 were the result of 99 generations of selection. In generation 1 through 100, 200 animals (the top 100 males and top 100 females) were selected from the 500 available candidates in each generation, based on their phenotype or BLUP-EBVs. Thus, the selected proportions were 40% (100 out of 250) in males and 40% in females (100 out of 250). Each sire was mated with 1 dam, and 5 full-sib offspring were produced per mating, resulting in 500 offspring in each generation.
Trait The trait under selection had a narrow-sense heritability of 0.10 in the base population. The environmental values were sampled from a
normal distribution for each individual with mean zero and variance VE = Vp (VA + VD + VAA).
Results Additive Variance In the following, results for the Mixture distribution of QTL effects are not presented as we found similar and consistent results for the 2 distributions (gamma vs. mixture) fitted for QTL effects. In order to compare trends of VA across scenarios, additive genetic variance in each scenario was scaled by dividing all values to initial additive variance of each scenario. Figure 2 shows changes of VA over generations in 4 genetic models. There was a systematic pattern in the loss of VA when comparing the 4 genetic models, and differences between genetic models were relatively small. In the A/AA model, the loss of VA was faster than in the other genetic models. Initially, the decline in VA was smallest in the A-model. In the long-term, however, the amount of retained VA was highest in full genetic model (A/D/ AA). Two distinct selection criteria showed differences in reduction of VA over time, as the BLUP-EBV selection criteria accelerated the reduction in VA.
Dominance Variance In the 2 genetic models, A/D/AA and A/D, where dominance effects were included, VD increased or was constant in the initial generations (Figure 3). Afterwards VD decreased in the A/D model. Changes in VD over time differed for the 2 selection criteria. When selection was based on BLUP-EBV, the reduction of VD over time was faster than with phenotypic selection. A striking result was that phenotypic selection hardly affected VD in the A/D/AA model over the entire period of 100 generations.
Additive-by-Additive Genetic Variance The epistatic variance decreased in a similar way over the 100 generations of selection in both genetic models and both selection criteria (Figure 4). The decrease was fastest for selection on BLUPEBV, where VAA approached its lowest level (~0) at generation 60. For phenotypic selection, longer time was needed for VAA to vanish completely.
Figure 2. Changes in additive genetic variance (VA) in 4 genetic models separated by selection criteria. The plotted additive variances for each scenario are means from 100 replicates, standardized by the initial additive variance of each scenario. (a) Selection method: phenotypic and (b) selection method: BLUP-EBV.
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Figure 3. Changes in dominance genetic variance (VD) in 2 genetic models (A/D/AA and A/D) separated by selection criteria. The plotted dominance variances for each scenario are means from 100 replicates, standardized by the initial dominance variance of each scenario. (a) Selection method: phenotypic and (b) selection method: BLUP-EBV.
Figure 4. Changes in additive-by-additive genetic variance (VAA) in 2 genetic models (A/D/AA and A/AA) separated by selection criteria. The plotted additive by additive variances for each scenario are means from 100 replicates, standardized by the initial additive by additive variance of each scenario. (a) Selection method: phenotypic and (b) selection method: BLUP-EBV.
Response to Selection The mean observed genotypic value of individuals in each generation, expressed in initial additive genetic standard deviations and as a deviation from the initial mean, is plotted in Figure 5. As selection proceeds, the difference between genetic models became larger. In generation 100, the additive model had the highest cumulative response to selection, and the full genetic model (A/D/AA) had the lowest cumulative response in long-term. When selection was based on BLUP-EBV, for all genetic models response plateaued earlier than for phenotypic selection.
Fixation and Loss of Favorable QTL The additive genetic model (A) had the highest percentage of fixation of favorable alleles at generation 100 (Figure 6). This result agrees with the highest response found for the additive model in Figure 5. The percentage of fixed favorable alleles decreased when more nonadditive effects were added to the model. In all genetic models, the
percentage of favorable alleles that became fixed was higher than percentage of favorable alleles that were lost. Of the total fraction of fixed alleles, however, the additive model had a greater proportion of loci fixed for the favorable allele. BLUP-EBV generated higher levels of fixation and loss of favorable alleles than phenotypic selection, which agrees with the earlier plateau of response seen with BLUP selection in Figure 5. In addition, the ratio of favorable fixed over total allele fixation was higher in BLUP-EBV than phenotypic selection.
Discussion Our findings did not support the hypothesis that finite-locus models with both additive and nonadditive genetic effects maintain more VA and realize larger medium- to long-term genetic gains than models with only additive effects when the trait under selection is subject to truncation selection. We used 4 genetic models to simulate a population undergoing directional truncation selection. In all 4 models, VA
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Figure 5. Response to selection over 100 generations of selection separated by selection criteria. The plotted mean genotypic values for each scenario are means from 100 replicates, standardized by initial additive genetic standard deviations of each scenario. (a) Selection method: phenotypic and (b) selection method: BLUP-EBV.
Figure 6. Percentage of allele fixation and lost at generation 100 in 4 genetic models separated by selection criteria. Fav. fix: percentage of QTL in which frequency of favorable allele is 1 in generation 100, Fav. lost: percentage of QTL in which frequency of favorable allele is 0 in generation 100.
decreased by directional selection, also in the presence of nonadditive effects, but the rate at which variation decreased varied among genetic models and selection criteria. The main reason that we did not found any enhancement of the additive genetic variance due to nonadditive effects may be explained as the following. First, the ratio of VAA/VA used in this study was 1/4, however, Goodnight (1987) showed that the additive genetic variance was not expected to increase unless the VAA was at least 1/3 of the VA. Also, epistatic effects in our simulation were nondirectional. It has been shown that pure nondirectional epistasis has no significant effects on the response and additive variance (Carter et al. 2005). Furthermore, in our simulation the allele frequencies were drawn from a U-shaped frequency distribution. Epistatic variance will only show up as a measurable variance component when the allele frequencies are near 0.5. In summary, we believe that discrepancy of our results compared to those of the previous models (Fuerst et al. 1997; Hallander and Waldmann 2007) investigating the effect of nonadditive effects on response to selection was due to less restrictive assumptions in our simulated models.
Changes in Variance In our finite locus model, VA decreased by selection, which is in agreement with results from other studies (Villanueva and Kennedy
1990; Fuerst et al. 1997). Directional selection changes the mean of a trait and it can also change the variance. First, directional selection decreases VA due to the generation of negative gametic phase disequilibrium (Bulmer 1971). In the intermediate term, regardless of selection criteria and QTL distribution, genetic models with epistatic terms (A/AA and A/D/AA), showed faster reduction in VA compared to the purely additive model (Figure 2). One explanation for this could be that the double homozygote genotype is more favored by selection and that reduces VA by inducing negative linkage disequilibrium among selected genes. If the selected genes are linked, the decay of linkage disequilibrium is delayed, and the reduction of VA is enhanced (Nomura 2005). However, it has been shown in several studies as well as experimental results that epistatic variance and to some extent dominance variance might convert to additive genetic variance (Fuerst et al. 1997; Hallander and Waldmann 2007). In addition, any changes in VA might depend on the ratio of VAA to VA. If VAA is smaller than VA, as in our case, then epistatic values for a pair of loci might be small and this will cause more decrease in VA by selection in comparison of purely additive gene action (Mueller and James 1983). In order to figure out the impact of initial ratio of VAA over VA on both response to selection and changes in VA we did extra analysis. We run Model A/AA with varying the ratio of VAA to VA in the model. The VAA was considered to be half, equal and twice of the VA (Supplementary Figure 1). The results confirmed the impact of initial ratio on response to selection. Increasing the VAA/VA ratio, resulted in higher response to selection, both in short- and long-term, however, VA decreased in the all ratios considered, nonetheless the reduction in VA was faster when VAA was twice the additive variance. Second, a more important cause for changes in VA is due to changes in allele frequency. If all alleles are at intermediate frequencies, it can be expected that variance will decline monotonically with time (assuming additivity), whereas if some are at low frequency, an initial increase in variance might be observed. If the distribution of frequencies is U shaped, as in this study, then the increase in variance due to alleles at low frequency might be expected to outweigh the decrease from those at high frequency (Hill and Bünger 2010). However, analyses undertaken by Hill and Rasbash (1986) for finite populations indicate that the pattern of response and change in VA is somewhat robust to the gene
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frequency distribution and VA decreases eventually as favorable alleles are moved to fixation. Third, in finite populations, VA also declines due to drift and, in the absence of selection, this decline can be predicted as VAt = VA0 (1 − 1/2Ne)t when assuming additive-gene action (Robertson 1960), where VAt is additive variance at generation t, VA0 is initial additive variance, and Ne is effective population size. We applied the approach developed by PerezEnciso (1995) to estimate Ne. By regressing ln(1 − Ft ) on generation tt, Ne is estimated from Ne = ½(1 − eβ), where β is the slope from the regression of ln(1 − Ft ) on generation number and Ft is the average inbreeding coefficients at generation t (PerezEnciso 1995). We plotted VA predicted by this formula versus observed VA for the additive model (A) to see how they would differ (Figure 7). The large difference between predicted and observed VA in Figure 7 is due to the fixation of favorable alleles by selection, as the formula based on effective population size (or inbreeding) assumes changes the allele frequency due to drift only. The lines for predicted VA in Figure 7 show that, as expected, selection based on BLUP-EBV generates more fixation due to drift than phenotypic selection and as pointed out earlier fraction of favorable fixed over total fixation was higher in BLUP-EBV. In contrast to VA, which decreased by selection, VD was preserved especially by phenotypic selection over generations. This maintenance of dominance variation can be explained by the occurrence of overdominance at some loci. In the A/D/AA genetic model, around 25% and 20% of loci showed overdominance for the mixture and gamma distribution, respectively. It has been shown that overdominance results in stabilizing selection, maintaining heterozygosity in the population rather than driving one allele to fixation. In order to investigate the effect of amount of overdominance on the preservation of dominance variance over generations, we run model A/D with varying amount of loci that showed over dominance. So, 4 cases of A/D model was considered in which 0 (overdominance was not allowed), 10%, 20%, and 30% of loci were allowed to have over dominance. All other parameters in the model such as ratio of dominance variance to additive variance were kept constant and similar to the original model described in the Methods section. The results showed clear effect of amount of overdominance on the changes of dominance variance (Supplementary Figure 2). When there was
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no overdominance or its percentage was small (10%), dominance variance decreased over generations, while increasing amount of overdominance in the model resulted in sustained dominance variation over generations. This result implies that occurrence of overdominance can serve as an explanation for the maintenance of VD in our study. Similar to VA, epistatic variance decreased by directional selection in this study. Contrary to VD that its changes over time was affected by the amount of overdominance, changes in VAA was not affected neither by the type of epistasis effects (e.g., directional or nondirectional epistasis) nor the ratio of VAA/VA (results not shown). In all the models including epistasis effects, we found a reduction in epistatic variance that is probably due to the fixation of the loci over the course of selection.
Response to Selection Our results demonstrate that nonadditive effects may affect response to selection in long-term. Comparing genetic models, A and A/AA models had higher response in long-term than the models having a dominance component (Figure 5). The reason for the greater response is probably due to the constellation of the genotypic values, where one double homozygote pair of loci has higher value. Hansen and Wagner (2001) argued that nonadditive effect such as directional epistasis will affect the response to selection due to systematic changes in the effects of alleles as their genetic background changes. On the other hand, if the epistatic interactions are random and nondirectional as in this simulation, these effects will tend to cancel out or add random noise. However, over many generations, the dynamics of gene effect reinforcement and competition can become very complex, and may lead to substantial departures from simple additive response to selection (Carter et al. 2005). We run model A/D/AA when epistatic effects in the model were allowed to be both directional and nondirectional (i.e., positive and negative). We found that directional epistasis enhanced the response to selection in long-term (Supplementary Figure 6). One deduction of these results could be that estimates of classical epistatic variance components are of little value in predicting response in short term, as these estimation do not distinguish between
Figure 7. Comparison of changes in observed VA in pure additive genetic model with VA based on inbreeding calculated as VAt = VA0 (1 − 1/2Ne). (a) Selection method: phenotypic and (b) selection method: BLUP-EBV.
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directional and nondirectional forms of functional epistasis. Griffing (1960) showed when directional epistasis is present, gametic-phase disequilibrium increases the response to directional selection, with 2 the response augmented by sσ AA / 2σ p2 , where S is intensity of selec2 tion σ AA is additive by additive variance and σ p2 is phenotypic variance. This increase in rate of response has been termed the “Griffing effect.” Thus, in the presence of directional epistasis, disequilibrium is on the one hand expected to increase the rate of response, while it is also expected to decrease the rate of response by decreasing additive genetic variance (the Bulmer effect). Based on a small simulation study, Mueller and James (1983) concluded that if epistatic variance is small relative to additive variance and the proportion of pairs showing epistasis is also small, the Bulmer effect dominates the Griffing effect, and disequilibrium reduces the response to selection (Walsh and Lynch 2010). We did not observe difference between 2 distributions fitted for QTL effects. Assuming that effects of mutant genes follow a gamma distribution but their frequencies are independent of their effects (i.e., a neutral model), Hill and Rasbash (1986) examined the influence of number and effects of mutant genes on response to selection and variance in response among replicates and found that the shape of the distribution of effects of mutant genes on the quantitative trait is not usually important, which is in agreement with our findings. In long-term, models including dominance (A/D and A/D/AA) had lower amount of response. This can be due to the rather frequent overdominance in this simulation, so loci with positive overdominance get stuck at intermediate frequencies. To investigate whether overdominance may be a causal factor limiting long-term response to selection, we run model A/D/AA with reduced proportion of overdominant loci (Supplementary Figure 5). We compared response in model A/D/AA when overdominance was not allowed with a scenario in which 20% of loci showed overdominance. When overdominance was not allowed, we found slightly higher response to selection in long-term compared to a model with overdominant loci. So, the presence of overdominance might have some effect on the cumulated response to selection. Gill (1965a, 1965b) in a simulation study showed that over dominance (positive dominance effect) reduces selection advance.
(Woolliams 1989; Wray and Hill 1989; Verrier et al. 1993). We also found that BLUP-EBV generated higher levels of fixation and loss of favorable alleles than phenotypic selection. In fact, fixation probabilities of favorable mutations depend on the population size; the effective size of a population under selection depends on the number of parents selected each generation and the method by which they are selected. Since effective population size is reduced more rapidly with BLUP-based selection, the loss of favorable mutations in the long-term is a concern in breeding programs (Hill 2000).
Phenotype Versus BLUP Selection
Comparison of Simulated Models with Hallander and Waldmann (2007)
Across all simulated genetic models, compared to phenotypic selection, selection based on BLUP-EBV accelerated the reduction in VA over generations. In fact, compared to phenotypic selection, use of family information in BLUP increases the correlation between predicted breeding values of relatives and therefore the probability of co-selection of close relatives, which in turn leads to increase in amount of inbreeding and reduction in the effective population size and additive genetic variance (Robertson 1960). This increase in amount of inbreeding and reduction in additive genetic variance explains the observed earlier plateau of response for BLUP than phenotypic selection in our results. Also, we found that, phenotypic selection realized higher response in long-term than selection based on BLUP-EBV (results not shown). Compared to long-term response, response to short-term selection of a few generations depends on the intensity and accuracy of selection in these generations, however, in long-term selection depends on how much genetic variability is maintained and how much inbreeding depression occurs over generations, and therefore on effective population size (Robertson 1960). As BLUP uses family information to increase the accuracy of selection, so BLUP-based selection is expected to give larger shortterm but smaller long-term responses than phenotypic selection
Fixation and Loss of Favorable QTL Favorable allele in our study was defined based on the sign of the additive effect of an allele. However, in models with gene interaction favorability can change if the favorability of an allele is determined by its average effect (or average excess) at the time the measurement is made. In fact, in systems with genetic interactions, selection has effects that are qualitatively different from selection acting in additive systems. Selection leads to regular changes in phenotype regardless of the underlying genetic system. When there is dominance or epistasis, however, the average effects of alleles are changing randomly as allele frequencies change. Some alleles may increase in frequency, only to subsequently decrease in frequency as selection proceeds. So, selection in systems with gene interaction leads to random shifts in the average effects of alleles (Goodnight 2010). This will make it difficult to clearly label an allele as “beneficial” as the actual effect of the allele will depend on the genetic background, and will change as selection proceeds. We compared percentage of favorable allele fixation and lost for model A/D/AA with 2 definitions of favorable allele, that is, favorability based on the main effect of an allele or its average effect. We found that if favorable allele was defined as its average effect, then percentage of favorable allele fixation in the last generation was 9% less than when favorable allele was defined as its main effect. In the same manner, percentage of favorable allele lost was also 20% higher when average effect was considered as favorability. Supplementary Figure 3 shows the changes in average effect of 8 randomly selected alleles over generations for A/D/AA model (Supplementary Figure 3).
To compare our results with the findings of Hallander and Waldmann (2007), we simulated their genetic model. When the initial allele frequency and additive effects were same across all loci, as in their analysis, we obtained similar results and including nonadditive effects increased VA in the initial generations (Supplementary Figure 4). However, in our study, VA decreased in all genetic models and including nonadditive effects did not have effect on the amount of VA at least in short-term. In order to figure out the factors that contributed to the observed differences between our results and those of Hallander and Waldmann (2007), we run their model by changing the parameters assumed in their simulation. The parameters being tested were number of loci, initial distribution of allele frequencies, and distribution of allelic effects. We run 4 different models and in each model one parameter was changed compared to Hallander and Waldmann (2007) keeping other parameters constant. For example, in Model 1, the number of loci affecting the trait of interest was increased to 100 rather than 4 loci considered in Hallander and Waldmann (2007). However, similar to Hallander and Waldmann (2007), the initial allele frequency and loci effects were the same across all loci (for more details see footnote in Supplementary
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Figure 4). The results showed that any change in the assumptions of Hallander and Waldmann (2007) would result in depletion of additive variance by directional selection except changing the number of loci affecting the trait. We found that by increasing number of the loci affecting the trait and keeping other parameters constant, additive genetic variance increased in initial generations similar to Hallander and Waldmann (2007). However, changes in other parameters such as distribution of allelic effects and frequencies resulted in reduction of additive variance (Supplementary Figure 4).
Conclusion In the schemes we simulated, additive genetic variance decreased by directional truncation selection, also in presence of nonadditive genetic effects. The distribution of QTL effects underlying the trait and the presence of nonadditive genetic effects had relatively small effects on the changes in additive variance. Response was relatively robust to nonadditive genetic effects in short term, but dominance decreased long-term response to selection.
Supplementary Material Supplementary data are available at Journal of Heredity online.
Funding First author benefited from a joint grant from the European Commission and Aarhus University within the framework of the Erasmus-Mundus joint doctorate “EGS-ABG.” This research was supported in part by the Danish Strategic Research Council (GenSAP: Centre for Genomic Selection in Animals and Plants, contract no. 12-132452).
Acknowledgments We thank anonymous reviewer and Charles Goodnight for useful comments and suggestions that significantly improved the article.
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