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.     

Genomic prediction of hybrid crops allows disentangling dominance and epistasis

We revisited, in a genomic context, the theory of hybrid genetic evaluation models of hybrid crosses of pure lines, as the current practice is largely based on infinitesimal model assumptions. Expressions for covariances between hybrids due to additive substitution effects and dominance and epistatic deviations were analytically derived. Using dense markers in a GBLUP analysis, it is possible to split specific combining ability into dominance and across-groups epistatic deviations, and to split general combining ability (GCA) into within-line additive effects and within-line additive by additive (and higher order) epistatic deviations. We analyzed a publicly available maize data set of Dent × Flint hybrids using our new model (called GCA-model) up to additive by additive epistasis. To model higher order interactions within GCAs, we also fitted “residual genetic” line effects. Our new GCA-model was compared with another genomic model which assumes a uniquely defined effect of genes across origins. Most variation in hybrids is accounted by GCA. Variances due to dominance and epistasis have similar magnitudes. Models based on defining effects either differently or identically across heterotic groups resulted in similar predictive abilities for hybrids. The currently used model inflates the estimated additive genetic variance. This is not important for hybrid predictions but has consequences for the breeding scheme—e.g. overestimation of the genetic gain within heterotic group. Therefore, we recommend using GCA-model, which is appropriate for genomic prediction and variance component estimation in hybrid crops using genomic data, and whose results can be practically interpreted and used for breeding purposes.     

Within-generation mutation variance for litter size in inbred mice

The mutational input of genetic variance per generation (sigma(m)(2)) is the lower limit of the genetic variability in inbred strains of mice, although greater values could be expected due to the accumulation of new mutations in successive generations. A mixed-model analysis using Bayesian methods was applied to estimate sigma(m)(2) and the across-generation accumulated genetic variability on litter size in 46 generations of a C57BL/6J inbred strain. This allowed for a separate inference on sigma(m)(2) and on the additive genetic variance in the base population (sigma(a)(2)). The additive genetic variance in the base generation was 0.151 and quickly decreased to almost null estimates in generation 10. On the other hand, sigma(m)(2) was moderate (0.035) and the within-generation mutational variance increased up to generation 14, then oscillating between 0.102 and 0.234 in remaining generations. This pattern suggested the existence of a continuous uploading of genetic variability for litter size (h(2)=0.045). Relevant genetic drift was not detected in this population. In conclusion, our approach allowed for separate estimation of sigma(a)(2) and sigma(m)(2) within the mixed-model framework, and the heritability obtained highlighted the significant and continuous influence of new genetic variability affecting the genetic stability of inbred strains.     

A Building Block Model for Quantitative Genetics

We introduce a quantitative genetic model for multiple alleles which permits the parameterization of the degree, D, of dominance of favorable or unfavorable alleles. We assume gene effects to be random from some distribution and independent of the D's. We then fit the usual least-squares population genetic model of additive and dominance effects in an infinite equilibrium population to determine the five genetic components--additive variance sigma 2 a, dominance variance sigma 2 d, variance of homozygous dominance effects d2, covariance of additive and homozygous dominance effects d1, and the square of the inbreeding depression h--required to treat finite populations and large populations that have been through a bottleneck or in which there is inbreeding. The effects of dominance can be summarized as functions of the average, D, and the variance, sigma 2 D. An important distinction arises between symmetrical and nonsymmetrical distributions of gene effects. With symmetrical distributions d1 = -d2/2 which is always negative, and the contribution of dominance to sigma 2 a is equal to d2/2. With nonsymmetrical distributions there is an additional contribution H to sigma 2 a and -H/2 to d1, the sign of H being determined by D and the skew of the distribution. Some numerical evaluations are presented for the normal and exponential distributions of gene effects, illustrating the effects of the number of alleles and of the variation in allelic frequencies. Random additive by additive (a*a) epistatic effects contribute to sigma 2 a and to the a*a variance, sigma 2/aa, the relative contributions depending on the number of alleles and the variation in allelic frequencies.(ABSTRACT TRUNCATED AT 250 WORDS)     

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.     

Evaluation of inheritance to leaf rust in wheat using area under disease progress curve

Six varieties, Kundan (K), Galvez-87 (G), Trap (T), Chris (C), Mango (M) and PBW-348 (P) along with fast ruster, Agra Local (AL), were screened for seedling reaction and adult pant response to leaf rust. Seedlings of all six varieties were susceptible while adult plants showed lower susceptability response than Agra Local. The F1s among the varieties, and also with Agra Local, showed the values lesser than the respective mid parental values for AUDPC suggesting a polygenic mode of inheritance. ANOVA for combining ability effects indicated variation due to the GCA and SCA effects, which indicated that both additive as well as non-additive type of genetic variances, govern AUDPC. The higher values for the GCA variance over the SCA variance indicated the predominance of an additive component over the dominance component for AUDPC. Significant values for GCA effects indicated that Kundan, Galvez-87 and Trap can be used as good general combiners for AUDPC. The crosses, KxAL, GxAL and TxAL showed significant sca effects for AUDPC, which indicated the predominance of non-additive gene effects in these crosses. Additive x additive and dominance x dominance components of the 5- parameter model were highly significant and contributed maximum extent compared to the additive and dominance components in the cross KxG, while dominance and dominance x dominance components contributed maximum in the remaining crosses. Under such a situation, improvement in the character may be expected through standard selection procedure, which may first exploit the additive gene effects and simultaneously care should be taken to see that the dominance effects are not dissipated, but rather they should be concentrated.     

Genetic analyses for certain plant and ear characters in pearl millet top-crosses

Summary Combining ability and the genetics of tiller number, days taken to flower and ear thickness were studied in top-cross progenies of pearl millet. General combining ability seemed to be more important for all the characters. The prevalence of epistatic variation, presumably of the type additive x additive, additive x dominance and dominance x dominance gene effects, with a non-significant contribution of additive and dominance components of genetic variance, was observed for tiller number. For days taken to flower, the importance of additive genetic variance was greater than that of the dominance component with directional dominance towards the recessive allele. However, for ear thickness, the existence of additive genetic variability together with the additive x additive type of genic interaction was suggested.An appreciable effect of epistasis on and ₁ components was observed for tiller number, whereas this effect was not so marked for other characters.The author is grateful to Dr. D. S. Athwal, formerly Professor and Head (now Assistant Director, The International Rice Research Institute, Los Banõs, Laguna, Philippines), Department of Plant Breeding, Punjab Agricultural University, Ludhiana, for providing the facilities.     

A comparison of different methods of half-diallel analysis

Summary A comparison among various forms of half-diallel analysis was made. The different half-diallel techniques used were: Griffing's model I, method 2 and 4, Morley-Jones' model; Walters and Morton's model, and Gardner and Eberhart's model. All these methods of diallel analysis were found to be interrelated. However, as the Gardner and Eberhart's model partitioned heterosis into different components as well as gave information about combining ability, this method had certainly some advantages over the others. The results further indicated the possibility of dominance variance being confounded with the additive variance of general combining ability.     

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.     

Neutral additive genetic variance in a metapopulation

For neutral, additive quantitative characters, the amount of additive genetic variance within and among populations is predictable from Wright's FST, the effective population size and the mutational variance. The structure of quantitative genetic variance in a subdivided metapopulation can be predicted from results from coalescent theory, thereby allowing single-locus results to predict quantitative genetic processes. The expected total amount of additive genetic variance in a metapopulation of diploid individual is given by 2Ne sigma m2 (1 + FST), where FST is Wright's among-population fixation index, Ne is the eigenvalue effective size of the metapopulation, and sigma m2 is the mutational variance. The expected additive genetic variance within populations is given by 2Ne sigma e2(1-FST), and the variance among demes is given by 4FSTNe sigma m2. These results are general with respect to the types of population structure involved. Furthermore, the dimensionless measure of the quantitative genetic variance among populations, QST, is shown to be generally equal to FST for the neutral additive model. Thus, for all population structures, a value of QST greater than FST for neutral loci is evidence for spatially divergent evolution by natural selection.     

A general approach for the study of a population of testcross progenies and consequences for recurrent selection

Summary A model to study genetic effects at the level of a population of testcross progenies is presented. As there is no dominance for the testcross value, with the restriction of epistasis to pairs of loci, only additive x additive epistasis can contribute to the variance among progenies. To estimate the variance among progenies due to epistasis, it is necessary to have the population structured in families of full sibs, half sibs or S₁, with only a few plants per family tested in combination with the tester. Using a two-way mating design to produce the families, it is possible to estimate the variance due to additive x additive epistasis. The consequence of the presence of epistasis is studied at the level of recurrent selection for combining ability with the tester. It seems that epistasis itself does not change the efficiency of the breeding methods considered. However, when the population from intercrossing is structured in families, it could be efficient to use a combined selection when the heritability is very low. In this case it would be efficient to produce full-sib families (by single-pair matings) at the level of intercrossing. The best procedure is to produce such families at the same time as crossing with the tester. In comparison to the classical scheme of selection for combining ability with a tester, such a modification increases the efficiency of selection 41.1% if an off-season generation can be used.     

Efficient Markov chain Monte Carlo implementation of Bayesian analysis of additive and dominance genetic variances in noninbred pedigrees

Accurate and fast computation of quantitative genetic variance parameters is of great importance in both natural and breeding populations. For experimental designs with complex relationship structures it can be important to include both additive and dominance variance components in the statistical model. In this study, we introduce a Bayesian Gibbs sampling approach for estimation of additive and dominance genetic variances in the traditional infinitesimal model. The method can handle general pedigrees without inbreeding. To optimize between computational time and good mixing of the Markov chain Monte Carlo (MCMC) chains, we used a hybrid Gibbs sampler that combines a single site and a blocked Gibbs sampler. The speed of the hybrid sampler and the mixing of the single-site sampler were further improved by the use of pretransformed variables. Two traits (height and trunk diameter) from a previously published diallel progeny test of Scots pine (Pinus sylvestris L.) and two large simulated data sets with different levels of dominance variance were analyzed. We also performed Bayesian model comparison on the basis of the posterior predictive loss approach. Results showed that models with both additive and dominance components had the best fit for both height and diameter and for the simulated data with high dominance. For the simulated data with low dominance, we needed an informative prior to avoid the dominance variance component becoming overestimated. The narrow-sense heritability estimates in the Scots pine data were lower compared to the earlier results, which is not surprising because the level of dominance variance was rather high, especially for diameter. In general, the hybrid sampler was considerably faster than the blocked sampler and displayed better mixing properties than the single-site sampler.     

油用向日葵主要农艺性状的遗传效应及相关性研究

根据加性-显性与环境互作的遗传模型,对6个油用向日葵自交系及其配制的9个杂交组合在2个环境下的7个农艺性状表现进行遗传分析,揭示油用向日葵主要农艺性状遗传性质、规律以及主要农艺性状对含油率的贡献率。结果表明:株高、茎粗、盘径、百粒重、籽仁率和单盘粒重等6个遗传性状主要受加性和显性共同控制,结实率的遗传以加性、显性×环境互作效应为主,籽仁率、单盘粒重以加性、显性、显性×环境互作效应为主;性状间的各项遗传相关性多以加性遗传相关为主。百粒重的净效应对籽实含油率的加性遗传方差贡献率最高,结实率的净效应对籽实含油率的显性遗传方差贡献率最高,单盘粒重对籽实含油率的加性×环境互作遗传方差的贡献率最高。     

Combining ability analysis over F1-F5 generations in diallel crosses of bread wheat

Summary Combining ability studies for grain yield and its primary component traits in diallel crosses involving seven diverse wheat cultivars of bread wheat (Triticum aestivum L.) over generations F₁-F₅ are reported. The general and specific combining ability variances were significant in all generations for all the traits except specific combining ability variance for number of spikes per plant in the F₅. The ratio of general to specific combining ability variances was significant for all the traits except grain yield in all the generations. This indicated an equal role of additive and non-additive gene effects in the inheritance of grain yield, and the predominance of the former for its component traits. The presence of significant specific combining ability variances in even the advanced generations may be the result of an additive x additive type of epistasis or evolutionary divergence among progenies in the same parental array. The relative breeding values of the parental varieties, as indicated by their general combining ability effects, did not vary much over the generations. The cheap and reliable procedure observed for making the choice of parents, selecting hybrids and predicting advanced generation (F₅) bulk hybrid performance was the determination of breeding values of the parents on the relative performance of their F₂ progeny bulks.     

Analysis of diallel cross in Phaseolus aureus roxb

Summary In a diallel cross of Phaseolus aureus involving five varieties, combining ability and gene action for grain yield, grains per pod and pods per plant were estimated. The study indicated that both general combining ability and specific combining ability effects were significant and important for all three traits. Partial dominance for grain yield and partial to over-dominance for grains per pod and pods per plant were observed. Dominant genes seem to govern the inheritance of all three characters. Combining ability, and graphical and component variance analyses indicated that the grain yield and two of its components are influenced by both additive and non-additive gene action.     

Genetic Analysis of the Latent Period of Stripe Rust in Wheat Seedlings

Genetics of slow‐rusting resistance to yellow rust (Puccinia striiformis f.sp. tritici) was studied by a half‐diallel design using six wheat varieties, Tiritea (susceptible), Tancred, Kotare, Otane, Karamu, and Briscard. The parents and 15 F₁ progenies were evaluated in the greenhouse by three pathotypes 7E18A^?, 38E0A⁺, and 134E134A⁺. The latent period was measured as the number of days from inoculation to the appearance of the first pustule. For each pathotype a randomized complete block design was used and data were analysed by methods of Griffing and Hayman. The range of average degree of dominance was from complete dominance to over‐dominance. Positive and negative degrees of dominance were observed for each pathotype that showed the reversal of dominance. Analysis of variance showed the importance of both additive and dominance effects in controlling the latent period. Broad‐sense heritabilities were 0.99 and narrow‐sense heritabilities ranged from 0.85 to 0.94. Briscard and Karamu for the pathotypes 38E0A⁺ and 134E134A⁺, Kotare for the pathotype 7E18A^? and Tancred for the pathotype 38E0A⁺ had significant and positive general combining ability (GCA) (more resistance) for latent period. The crosses of Kotare with Tancred, Briscard and Karamu indicated the highest and positive specific combining ability (SCA) for the pathotype 7E18A^?. Significant additive genetic component and moderate narrow‐sense heritability indicate the possibility of improving for longer latent period of stripe rust in breeding programmes.     

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.     

Combining ability analysis over environments in diallel crosses of linseed (Linum usitatissimum L.)

Summary This paper reports on combining ability studies for yield and its component traits in diallel crosses involving ten ecogeographically and genetically diverse linseed (Linum usitatissimum L.) cultivars in the F₂ generation over three locations. The general combining ability (GCA) and specific combining ability (SCA) mean squares were significant at all three locations for all traits. Combined analysis over locations showed the same trend of significance. The ratio of GCA to SCA mean squares was significant for all the traits in individual location analysis as well as in combined analysis. This indicated the predominant role of additive gene effects in the inheritance of these characters. The GCA mean squares were several times larger than SCA mean squares for all the traits, indicating the presence of considerable magnitude of additive genetic variance and the additive x additive components of the epistatic variance. Consequently, effective selection should be possible within these F₂ populations for all characters. Significant genotype-location and GCA-location interactions indicated that more than one test location is required to obtain reliable information. The inexpensive and reliable procedure used for making the choice of parents was the determination of breeding values of the parents on the relative performance of their F₂ progeny bulks.Part of the thesis submitted by the senior author in partial fulfillment of the requirements for the Ph. D. degree of the Marathwada Agricultural University, Parbhani, 431 402, India     

Including Dominance Effects in the Genomic BLUP Method for Genomic Evaluation

We evaluated the performance of GBLUP including dominance genetic effect (GBLUP-D) by estimating variances and predicting genetic merits in a computer simulation and 2 actual traits (T4 and T5) in pigs. In simulation data, GBLUP-D explained more than 50% of dominance genetic variance. Moreover, GBLUP-D yielded estimated total genetic effects over 1.2% more accurate than those yielded by GBLUP. In particular, when the dominance genetic variance was large, the accuracy could be substantially improved by increasing the number of markers. The dominance genetic variances in T4 and T5 accounted for 9.6% and 6.3% of the phenotypic variances, respectively. Estimates of such small dominance genetic variances contributed little to the improvement of the accuracies of estimated total genetic effects. In both simulation and pig data, there were nearly no differences in the estimates of additive genetic effects or their variance between GBLUP-D and GBLUP. Therefore, we conclude GBLUP-D is a feasible approach to improve genetic performance in crossbred populations with large dominance genetic variation and identify mating systems with good combining ability.     

Genetic Architecture of Two Fitness-related Traits in Drosophila melanogaster: Ovariole Number and Thorax Length

In Drosophila melanogaster, ovariole number and thorax length are morphological characters thought to be associated with fitness. Maximum daily egg production in females is positively correlated with ovariole number, while thorax length is correlated with male reproductive success and female fecundity. Though both traits are related to fitness, ovariole number is likely to be under stabilizing selection, while thorax length appears to be under directional selection. Current research has focused on examining the sources of variation for ovariole number in relation to fitness, with a view towards elucidating how segregating variation is maintained in natural populations. Here, we utilize a diallel design to explore the genetic architecture of ovariole number and thorax length in nine isogenic lines derived from a natural population. The full diallel design allows the estimation of general combining ability (GCA), specific combining ability (SCA), and also describes variation due to reciprocal effects (RGCA and RSCA). Ovariole number and thorax length differed with respect to their genetic architecture, reflective of the independent selective forces acting on the traits. For ovariole number, GCA accounted for the majority (67.3%) of variation segregating between the lines, with no evidence of reciprocal effects or inbreeding depression; SCA accounted for a small percentage (3.9%) of the variance, suggesting dominance variation; no reciprocal effects were observed. In contrast, for thorax length, the majority of the non-error variance was accounted for by SCA (17.9%), with only one third as much variance (6.2%) due to GCA. Interestingly, RSCA (nuclear–extranuclear interactions) accounted for slightly more variation (7.5%) than GCA in these data. Thus, genetic variation for thorax length is largely in accord with predictions for a fitness trait under directional selection: little additive genetic variation and substantial dominance variation (including a suggestion of inbreeding depression); while the mechanisms underlying the maintenance of variation for ovariole number are more complex.