Quantitative Genetics of Sperm Precedence in Drosophila Melanogaster

To assess the genetic basis of sperm competition under conditions in which it occurs, I estimated additive, dominance, homozygous and environmental variance components, the effects of inbreeding, and the weighted average dominance of segregating alleles for two measures of sperm precedence in a large, outbred laboratory population. Both first and second male precedence show significant decline on inbreeding. Second male precedence demonstrates significant dominance variance and homozygous genetic variance, but the additive variance is low and not significantly different from zero. For first male precedence, the variance among homozygous lines is again significant, and dominance variance is larger than the additive variance, but is not statistically significant. In contrast, male mating success and other fitness components in Drosophila generally exhibit significant additive variance and little or no dominance variance. Other recent experiments have shown significant genotypic variation for sperm precedence and have associated it with allelic variants of accessory-gland proteins. The contrast between sperm precedence and other male fitness traits in the structure of quantitative genetic variation suggests that different mechanisms may be responsible for the maintenance of variation in these traits. The pattern of genetic variation and inbreeding decline shown in this experiment suggests that one or a few genes with major effects on sperm precedence may be segregating in this population.     

Selection theory for selfed progenies

Summary The purpose of this article was to extend the model used to predict selection response with selfed progeny from 2 alleles per locus to a model which is general for number and frequency of alleles at loci. To accomplish this, 4 areas had to be dealt with: 1) simplification of the derivation and calculation of the condensed coefficients of identity; 2) presentation of the genetic variances expressed among and within selfed progenies as linear function of 5 population parameters; 3) presentation of selection response equations for selfed progenies as functions of these 5 population parameters; and 4) to identify a set of progeny to evaluate, such that one might be able to estimate these 5 population parameters.The five population parameters used in predicting gains were the additive genetic variance, the dominance variance, the covariance of additive and homozygous dominance deviations, the variance of the homozygous dominance deviations and a squared inbreeding depression term.Contribution from the Missouri Agricultural Experiment Station. Journal Series No. 9971     

Prediction of additive and dominance effects in selected or unselected populations with inbreeding

Summary A genetic model with either 64 or 1,600 unlinked biallelic loci and complete dominance was used to study prediction of additive and dominance effects in selected or unselected populations with inbreeding. For each locus the initial frequency of the favourable allele was 0.2, 0.5, or 0.8 in different alternatives, while the initial narrow-sense heritability was fixed at 0.30. A population of size 40 (20 males and 20 females) was simulated 1,000 times for five generations. In each generation 5 males and 10 or 20 females were mated, with each mating producing four or two offspring, respectively. Breeding individuals were selected randomly, on own phenotypic performance or such yielding increased inbreeding levels in subsequent generations. A statistical model containing individual additive and dominance effects but ignoring changes in mean and genetic covariances associated with dominance due to inbreeding resulted in significantly biased predictions of both effects in generations with inbreeding. Bias, assessed as the average difference between predicted and simulated genetic effects in each generation, increased almost linearly with the inbreeding coefficient. In a second statistical model the average effect of inbreeding on the mean was accounted for by a regression of phenotypic value on the inbreeding coefficient. The total dominance effect of an individual in that case was the sum of the average effect of inbreeding and an individual effect of dominance. Despite a high mean inbreeding coefficient (up to 0.35), predictions of additive and dominance effects obtained with this model were empirically unbiased for each initial frequency in the absence of selection and 64 unlinked loci. With phenotypic selection of 5 males and only 10 females in each generation and 64 loci, however, predictions of additive and dominance effects were significantly biased. Observed biases disappeared with 1,600 loci for allelic frequencies at 0.2 and 0.5. Bias was due to a considerable change in allelic frequency with phenotypic selection. Ignoring both the covariance between additive and dominance effects with inbreeding and the change in dominance variance due to inbreeding did not significantly bias prediction of additive and dominance effects in selected or unselected populations with inbreeding.     

Effects of genetic drift on variance components under a general model of epistasis

We analyze the changes in the mean and variance components of a quantitative trait caused by changes in allele frequencies, concentrating on the effects of genetic drift. We use a general representation of epistasis and dominance that allows an arbitrary relation between genotype and phenotype for any number of diallelic loci. We assume initial and final Hardy-Weinberg and linkage equilibrium in our analyses of drift-induced changes. Random drift generates transient linkage disequilibria that cause correlations between allele frequency fluctuations at different loci. However, we show that these have negligible effects, at least for interactions among small numbers of loci. Our analyses are based on diffusion approximations that summarize the effects of drift in terms of F, the inbreeding coefficient, interpreted as the expected proportional decrease in heterozygosity at each locus. For haploids, the variance of the trait mean after a population bottleneck is var(delta(z)) = sigma(n)k=1 FkV(A(k)), where n is the number of loci contributing to the trait variance, V(A(1)) = V(A) is the additive genetic variance, and V(A(k)) is the kth-order additive epistatic variance. The expected additive genetic variance after the bottleneck, denoted (V*(A)), is closely related to var(delta(z)); (V*(A)) = (1 - F) sigma(n)k=1 kFk-1V(A(k)). Thus, epistasis inflates the expected additive variance above V(A)(1 - F), the expectation under additivity. For haploids (and diploids without dominance), the expected value of every variance component is inflated by the existence of higher order interactions (e.g., third-order epistasis inflates (V*(AA. This is not true in general with diploidy, because dominance alone can reduce (V*(A)) below V(A)(1 - F) (e.g., when dominant alleles are rare). Without dominance, diploidy produces simple expressions: var(delta(z)) = sigma(n)k=1 (2F)kV(A(k)) and (V(A)) = (1 - F) sigma(n)k=1 k(2F)k-1V(A(k)). With dominance (and even without epistasis), var(delta(z)) and (V*(A)) no longer depend solely on the variance components in the base population. For small F, the expected additive variance simplifies to (V*(A)) approximately equal to (1 - F)V(A) + 4FV(AA) + 2FV(D) + 2FC(AD), where C(AD) is a sum of two terms describing covariances between additive effects and dominance and additive X dominance interactions. Whether population bottlenecks lead to expected increases in additive variance depends primarily on the ratio of nonadditive to additive genetic variance in the base population, but dominance precludes simple predictions based solely on variance components. We illustrate these results using a model in which genotypic values are drawn at random, allowing extreme and erratic epistatic interactions. Although our analyses clarify the conditions under which drift is expected to increase V(A), we question the evolutionary importance of such increases.     

The Genetic Structure of Natural Populations of Drosophila Melanogaster. Xxiv. Effects of Hybrid Dysgenesis on the Components of Genetic Variance of Viability

Eight hundred second chromosomes were extracted from the Ishigakijima population, one of the southernmost populations of Drosophila melanogaster in Japan. Half of them were extracted in Native cytoplasm (P-type), and half in Foreign cytoplasm (M-type). Various population-genetic parameters, including the frequency of lethal-carrying second chromosomes (Q = 0.235 for the Native; 0.218 for the Foreign), the allelism rate of lethal second chromosome (Ic = 0.0217 for the Native; 0.0134 for the Foreign), the homozygous detrimental and lethal loads (D = 0.179 for the Native; 0.270 for the Foreign; L = 0.262 for the Native; 0.240 for the Foreign), the average degree of dominance of mildly deleterious mutations (?E = 0.244 for the Native; 0.208 for the Foreign), and the components of genetic variance for viability [additive (sigma A2) and dominance (sigma D2)](?igma A2 = 0.0187 for the Native; 0.0172 for the Foreign; ?igma D2 = 0.0005 for the Native; 0.0009 for the Foreign) were estimated. The data indicate that D was significantly larger and hE was significantly smaller in the Foreign cytoplasm. However, the estimates of additive and dominance variances were not significantly different between the two cytoplasms. The additive genetic variance for viability in the Ishigakijima population was greater than expected on the basis of mutation-selection balance confirming previous studies on papers of D. melanogaster in warm climates.     

Phenotypic plasticity for life-history traits in Drosophila melanogaster. III. Effect of the environment on genetic parameters.

We estimated genetic and environmental variance components for developmental time and dry weight at eclosion in Drosophila melanogaster raised in ten different environments (all combinations of 22, 25 and 28 degrees C and 0.5, 1 and 4% yeast concentration, and 0.25% yeast at 25 degrees C). We used six homozygous lines derived from a natural population for complete diallel crosses in each environment. Additive genetic variances were consistently low for both traits (h2 around 10%). The additive genetic variance of developmental time was larger at lower yeast concentrations, but the heritability did not increase because other components were also larger. The additive genetic effects of the six parental lines changed ranks across environments, suggesting a mechanism for the maintenance of genetic variation in heterogenous environments. The variance due to non-directional dominance was small in most environments. However, there was directional dominance in the form of inbreeding depression for both traits. It was pronounced at high yeast levels and temperatures but disappeared when yeast or temperature were decreased. This meant that the heterozygous flies were more sensitive to environmental differences than homozygous flies. Because dominance effects are not heritable, this suggests that the evolution of plasticity can be constrained when dominance effects are important as a mechanism for plasticity.     

Quantitative genetic variation in an ecological setting

The machinery was developed to investigate the behavior of quantitative genetic variation in an ecological model of a finite number of islands of finite size, with migration rate m and extinction rate e, for a quantitative genetic model general for numbers of alleles and loci and additive, dominance, and additive by additive epistatic effects. It was necessary to reckon with seven quadratic genetic components, whose coefficients in the genotypic variance components within demes, sigma Gw2, between demes within populations, sigma s2, and between replicate populations, sigma r2, are given by descent measures. The descent measures at any time are calculated with the use of transition equations which are determined by the parameters of the ecological model. Numerical results were obtained for the coefficients of the quadratic genetic components in each of the three genotypic variance components in the early phase of differentiation. The general effect of extinction is to speed up the time course leading to fixation, to increase sigma r2, and to decrease sigma s2 (with a few exceptions) in comparison with no extinction. The general effect of migration is to slow down the time course leading to fixation, to increase sigma Gw2, at least in the later generations, and to decrease sigma s2 (with a few exceptions) in comparison with no migration. Except for these, the effects of migration and extinction on the variance components are complex, depending on the genetic model, and sometimes involve interaction of migration and extinction. Sufficient details are given for an investigator to evaluate numerically the results for variations in the quantitative genetic and ecological models.     

Effects of bottlenecks on quantitative genetic variation in the butterfly Bicyclus anynana

The effects of a single population bottleneck of differing severity on heritability and additive genetic variance was investigated experimentally using a butterfly. An outbred laboratory stock was used to found replicate lines with one pair, three pairs and 10 pairs of adults, as well as control lines with approximately 75 effective pairs. Heritability and additive genetic variance of eight wing pattern characters and wing size were estimated using parent-offspring covariances in the base population and in all daughter lines. Individual morphological characters and principal components of the nine characters showed a consistent pattern of treatment effects in which average heritability and additive genetic variance was lower in one pair and three pair lines than in 10 pair and control lines. Observed losses in heritability and additive genetic variance were significantly greater than predicted by the neutral additive model when calculated with coefficients of inbreeding estimated from demographic parameters alone. However, use of molecular markers revealed substantially more inbreeding, generated by increased variance in family size and background selection. Conservative interpretation of a statistical analysis incorporating this previously undetected inbreeding led to the conclusion that the response to inbreeding of the morphological traits studied showed no significant departure from the neutral additive model. This result is consistent with the evidence for minimal directional dominance for these traits. In contrast, egg hatching rate in the same experimental lines showed strong inbreeding depression, increased phenotypic variance and rapid response to selection, highly indicative of an increase in additive genetic variance due to dominance variance conversion.     

Impact of interpopulation divergence on additive and dominance variance in hybrid populations

We present a theoretical proof that the ratio of the dominance vs. the additive variance decreases with increasing genetic divergence between two populations. While the dominance variance is the major component of the variance due to specific combining ability (sigma(SCA)(2)), the additive variance is the major component of the variance due to general combining ability (sigma(GCA)(2)). Therefore, we conclude that interpopulation improvement becomes more efficient with divergent than with genetically similar heterotic groups, because performance of superior hybrids can be predicted on the basis of general combining ability effects.     

A classical setting for associations between markers and loci affecting quantitative traits

We examine the relationships between a genetic marker and a locus affecting a quantitative trait by decomposing the genetic effects of the marker locus into additive and dominance effects under a classical genetic model. We discuss the structure of the associations between the marker and the trait locus, paying attention to non-random union of gametes, multiple alleles at the marker and trait loci, and non-additivity of allelic effects at the trait locus. We consider that this greater-than-usual level of generality leads to additional insights, in a way reminiscent of Cockerham's decomposition of genetic variance into five terms: three terms in addition to the usual additive and dominance terms. Using our framework, we examine several common tests of association between a marker and a trait.     

Combined genetic analysis of partial blast resistance in an upland rice population and recurrent selection for line and hybrid values

The CNA-IRAT 5 upland rice population has been improved for 4 years by recurrent selection for blast resistance in Brazil. In order to predict the efficiency of recurrent selection in different test systems and to compare the relative advantage of hybrids versus pure line breeding, a combined genetic analysis of partial blast resistance in the CNA-IRAT 5 population was undertaken. A three-level hierarchical design in inbreeding and a factorial design were derived from the base population. Partial blast resistance of lines and hybrids was evaluated in the greenhouse and in the field by inoculation with one virulent blast isolate. The means and genetic variances of the hybrids and lines were estimated. Genetic advance by recurrent selection was predicted from estimates of variance components. The inheritance of partial blast resistance was mainly additive but non-additive effects were detected at both levels of means and variances. Mean heterosis ranged from 4%–8% for lesion size and lesion density to 10–12% for leaf and panicle resistance. High dominance or homozygous dominance variances relative to additive variance and negative covariance between additive and homozygous dominance effects were estimated. A low frequency of favourable alleles for partial resistance would explain the observed organisation of genetic variability in the base population. Recurrent selection will efficiently improve partial blast resistance of the CNA-IRAT 5 population. Genetic advance for line or hybrid values was expected to be higher testing doubled haploid lines than S1 lines, or than general combining ability. Two components of partial resistance assessed in the greenhouse, lesion size and lesion density, could be used as indirect selection criteria to improve field resistance. On the whole, hybrid breeding for partial blast resistance appeared to be slightly more advantageous than pure line breeding.     

Extending the definition of additive genetic variance to more than one gene-a viewpoint

Following the Fisherian approach, the expression for additive genetic variance is derived in a single gene system through a regression equation in two variables which are used to obtain the additive and dominance variances. The approach is extended to two genes with restricted linkage and inbreeding. It was brought out that additive genetic variance defined essentially for one gene does not extendper se to multi-gene systems.     

Dissecting total genetic variance into additive and dominance components of purebred and crossbred pig traits

The partition of the total genetic variance into its additive and non-additive components can differ from trait to trait, and between purebred and crossbred populations. A quantification of these genetic variance components will determine the extent to which it would be of interest to account for dominance in genomic evaluations or to establish mate allocation strategies along different populations and traits. This study aims at assessing the contribution of the additive and dominance genomic variances to the phenotype expression of several purebred Piétrain and crossbred (Piétrain × Large White) pig performances. A total of 636 purebred and 720 crossbred male piglets were phenotyped for 22 traits that can be classified into six groups of traits: growth rate and feed efficiency, carcass composition, meat quality, behaviour, boar taint and puberty. Additive and dominance variances estimated in univariate genotypic models, including additive and dominance genotypic effects, and a genomic inbreeding covariate allowed to retrieve the additive and dominance single nucleotide polymorphism variances for purebred and crossbred performances. These estimated variances were used, together with the allelic frequencies of the parental populations, to obtain additive and dominance variances in terms of genetic breeding values and dominance deviations. Estimates of the Piétrain and Large White allelic contributions to the crossbred variance were of about the same magnitude in all the traits. Estimates of additive genetic variances were similar regardless of the inclusion of dominance. Some traits showed relevant amount of dominance genetic variance with respect to phenotypic variance in both populations (i.e. growth rate 8%, feed conversion ratio 9% to 12%, backfat thickness 14% to 12%, purebreds-crossbreds). Other traits showed higher amount in crossbreds (i.e. ham cut 8% to 13%, loin 7% to 16%, pH semimembranosus 13% to 18%, pH longissimus dorsi 9% to 14%, androstenone 5% to 13% and estradiol 6% to 11%, purebreds-crossbreds). It was not encountered a clear common pattern of dominance expression between groups of analysed traits and between populations. These estimates give initial hints regarding which traits could benefit from accounting for dominance for example to improve genomic estimated breeding value accuracy in genetic evaluations or to boost the total genetic value of progeny by means of assortative mating.     

The consequences of including non-additive effects on the genetic evaluation of harvest body weight in Coho salmon (Oncorhynchus kisutch)

Background

In this study, we used different animal models to estimate genetic and environmental variance components on harvest weight in two populations of Oncorhynchus kisutch, forming two classes i.e. odd- and even-year spawners.

Methods

The models used were: additive, with and without inbreeding as a covariable (A + F and A respectively); additive plus common environmental due to full-sib families and inbreeding (A + C + F); additive plus parental dominance and inbreeding (A + D + F); and a full model (A + C + D + F). Genetic parameters and breeding values obtained by different models were compared to evaluate the consequences of including non-additive effects on genetic evaluation.

Results

Including inbreeding as a covariable did not affect the estimation of genetic parameters, but heritability was reduced when dominance or common environmental effects were included. A high heritability for harvest weight was estimated in both populations (even = 0.46 and odd = 0.50) when simple additive models (A + F and A) were used. Heritabilities decreased to 0.21 (even) and 0.37 (odd) when the full model was used (A + C + D + F). In this full model, the magnitude of the dominance variance was 0.19 (even) and 0.06 (odd), while the magnitude of the common environmental effect was lower than 0.01 in both populations. The correlation between breeding values estimated with different models was very high in all cases (i.e. higher than 0.98). However, ranking of the 30 best males and the 100 best females per generation changed when a high dominance variance was estimated, as was the case in one of the two populations (even).

Conclusions

Dominance and common environmental variance may be important components of variance in harvest weight in O. kisutch, thus not including them may produce an overestimation of the predicted response; furthermore, genetic evaluation was seen to be partially affected, since the ranking of selected animals changed with the inclusion of non-additive effects in the animal model.     

On the Additive and Dominant Variance and Covariance of Individuals Within the Genomic Selection Scope

Genomic evaluation models can fit additive and dominant SNP effects. Under quantitative genetics theory, additive or “breeding” values of individuals are generated by substitution effects, which involve both “biological” additive and dominant effects of the markers. Dominance deviations include only a portion of the biological dominant effects of the markers. Additive variance includes variation due to the additive and dominant effects of the markers. We describe a matrix of dominant genomic relationships across individuals, D, which is similar to the G matrix used in genomic best linear unbiased prediction. This matrix can be used in a mixed-model context for genomic evaluations or to estimate dominant and additive variances in the population. From the “genotypic” value of individuals, an alternative parameterization defines additive and dominance as the parts attributable to the additive and dominant effect of the markers. This approach underestimates the additive genetic variance and overestimates the dominance variance. Transforming the variances from one model into the other is trivial if the distribution of allelic frequencies is known. We illustrate these results with mouse data (four traits, 1884 mice, and 10,946 markers) and simulated data (2100 individuals and 10,000 markers). Variance components were estimated correctly in the model, considering breeding values and dominance deviations. For the model considering genotypic values, the inclusion of dominant effects biased the estimate of additive variance. Genomic models were more accurate for the estimation of variance components than their pedigree-based counterparts.     

An Analysis of Heterosis VS. Inbreeding Effects with an Autotetraploid Cross-Fertilized Plant: MEDICAGO SATIVA L

Self-fertilization and crossing were combined to produce a large number of levels of inbreeding and of degrees of kinship. The inbreeding effect increases with the complexity of the character and with its supposed relationship with fitness. A certain amount of heterozygosity appears to be necessary for the expression of variability. With crossing of unrelated noninbred plants, genetic variance is mainly additive, but with inbreeding its major part is nonadditive. High additivity in crossing, therefore, coexists with strong inbreeding depression. However, even in inbreeding the genetic coefficient of covariation among relatives appears to be strongly and linearly related to the classical coefficient of kinship. This means that deviations from the additive model with inbreeding could be partly due to an effect of inbreeding on variances through an effect on means. An attempt to analyze genetic effects from a theoretical model, based upon the identity by descent relationship at the level of means and of covariances between relatives, tends to show that allelic interactions are more important and nonallelic interactions are less important for a character closely related to fitness. For a complex character, these results lead to the conception of a genome organized in polygenic complementary blocks integrating epistasis and dominance. Some consequences for plant breeding are also discussed.     

Effects of identity disequilibrium and linkage on quantitative variation in finite populations

Identity disequilibrium, ID, is the difference between joint identity by descent and the product of the separate probabilities of identity by descent for two loci. The effects of ID on the additive by additive (a*a) epistatic variance and joint dominance component between populations and in the additive, dominance and a*a variance within populations, including the effects on covariances of relatives within populations, were studied for finite monoecious populations. The effects are formulated in terms of three additive partitions, eta b, eta a and eta d, of the total ID, each of which increases from zero to a maximum at some generation dependent upon linkage and population size and decreases thereafter. eta d is about four times the magnitude of the other two but none is of any consequence except for tight linkage and very small populations. For single-generation bottleneck populations only eta d is not zero. With random mating of expanded populations eta b remains constant and eta a and eta d go to zero at a rate dependent upon linkage, very fast with free recombination. The contributions of joint dominance to the genetic components of variance within and between populations are entirely a function of the eta's while those of a*a variance to the components are functions mainly of the coancestry coefficient and only modified by the eta's. The contributions of both to the covariances of half-sibs, full-sibs and parent-offspring follow the pattern expected from their contributions to the genetic components of variance within populations except for minor terms which most likely are of little importance.     

Maintenance of genetic variation in quantitative traits of a woodland rodent during generations of captive breeding

Breeding programs to conserve diversity are predicated on the assumption that genetic variation in adaptively important traits will be lost in parallel to the loss of variation at neutral loci. To test this assumption, we monitored quantitative traits across 18 generations of Peromyscus leucopus mice propagated with protocols that mirror breeding programs for threatened species. Ears, hind feet, and tails became shorter, but changes were reversible by outcrossing and therefore were due to accumulated inbreeding. Heritability of ear length decreased, because of an increase in phenotypic variance rather than the expected decrease in additive genetic variance. Additive genetic variance in hind foot length increased. This trait initially had low heritability but large dominance or common environmental variance contributing to resemblance among full-sibs. The increase in the additive component indicates that there was conversion of interaction variances to additive variance. For no trait did additive genetic variation decrease significantly across generations. These findings indicate that the restructuring of genetic variance that occurs with genetic drift and novel selection in captivity can prevent or delay the loss of phenotypic and heritable variation, providing variation on which selection can act to adapt populations to captivity and perhaps later to readapt to more natural habitats after release. Therefore, the importance of minimizing loss of gene diversity from conservation breeding programs for threatened wildlife species might lie in preventing immediate reduction in individual fitness due to inbreeding and protecting allelic diversity for long-term evolutionary change, more so than in protecting variation in quantitative traits for rapid re-adaptation to wild environments.     

Efficiency of selection, as measured by single nucleotide polymorphism variation, is dependent on inbreeding rate in Drosophila melanogaster

It is often hypothesized that slow inbreeding causes less inbreeding depression than fast inbreeding at the same absolute level of inbreeding. Possible explanations for this phenomenon include the more efficient purging of deleterious alleles and more efficient selection for heterozygote individuals during slow, when compared with fast, inbreeding. We studied the impact of inbreeding rate on the loss of heterozygosity and on morphological traits in Drosophila melanogaster. We analysed five noninbred control lines, 10 fast inbred lines and 10 slow inbred lines; the inbred lines all had an expected inbreeding coefficient of approximately 0.25. Forty single nucleotide polymorphisms in DNA coding regions were genotyped, and we measured the size and shape of wings and counted the number of sternopleural bristles on the genotyped individuals. We found a significantly higher level of genetic variation in the slow inbred lines than in the fast inbred lines. This higher genetic variation was resulting from a large contribution from a few loci and a smaller effect from several loci. We attributed the increased heterozygosity in the slow inbred lines to the favouring of heterozygous individuals over homozygous individuals by natural selection, either by associative over‐dominance or balancing selection, or a combination of both. Furthermore, we found a significant polynomial correlation between genetic variance and wing size and shape in the fast inbred lines. This was caused by a greater number of homozygous individuals among the fast inbred lines with small, narrow wings, which indicated inbreeding depression. Our results demonstrated that the same amount of inbreeding can have different effects on genetic variance depending on the inbreeding rate, with slow inbreeding leading to higher genetic variance than fast inbreeding. These results increase our understanding of the genetic basis of the common observation that slow inbred lines express less inbreeding depression than fast inbred lines. In addition, this has more general implications for the importance of selection in maintaining genetic variation.     

Effects on phenotypic variability of directional selection arising through genetic differences in residual variability

In standard models of quantitative traits, genotypes are assumed to differ in mean but not variance of the trait. Here we consider directional selection for a quantitative trait for which genotypes also confer differences in variability, viewed either as differences in residual phenotypic variance when individual loci are concerned or as differences in environmental variability when the whole genome is considered. At an individual locus with additive effects, the selective value of the increasing allele is given by ia/sigma + 1/2 ixb/sigma2, where i is the selection intensity, x is the standardized truncation point, sigma2 is the phenotypic variance, and a/sigma and b/sigma2 are the standardized differences in mean and variance respectively between genotypes at the locus. Assuming additive effects on mean and variance across loci, the response to selection on phenotype in mean is isigma2(Am)/sigma + 1/2 ixcov(Amv)/sigma2 and in variance is icov(Amv)/sigma + 1/2 ixsigma2(Av)/sigma2, where sigma2(Am) is the (usual) additive genetic variance of effects of genes on the mean, sigma2(Av) is the corresponding additive genetic variance of their effects on the variance, and cov(Amv) is the additive genetic covariance of their effects. Changes in variance also have to be corrected for any changes due to gene frequency change and for the Bulmer effect, and relevant formulae are given. It is shown that effects on variance are likely to be greatest when selection is intense and when selection is on individual phenotype or within family deviation rather than on family mean performance. The evidence for and implications of such variability in variance are discussed.