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 VARIANCE MAINTAINED BY FLUCTUATING SELECTION WITH OVERLAPPING GENERATIONS: VARIANCE COMPONENTS AND COVARIANCES

The quantitative genetic variance-covariance that can be maintained in a random environment is studied, assuming overlapping generations and Gaussian stabilizing selection with a fluctuating optimum. The phenotype of an individual is assumed to be determined by additive contributions from each locus on paternal and maternal gametes (i.e., no epistasis and no dominance). Recurrent mutation is ignored, but linkage between loci is arbitrary. The genotype distribution in the evolutionarily stable population is generically discrete: only a finite number of polymorphic alleles with distinctly different effects are maintained, even though we allow a continuum of alleles with arbitrary phenotypic contributions to invade. Fluctuating selection maintains nonzero genetic variance in the evolutionarily stable population if the environmental heterogeneity is larger than a certain threshold. Explicit asymptotic expressions for the standing variance-covariance components are derived for the population near the threshold, or for large generational overlap, as a function of environmental variability and genetic parameters (i.e., number of loci, recombination rate, etc.), using the fact that the genotype distribution is discrete. Above the threshold, the population maintains considerable genetic variance in the form of positive linkage disequilibrium and positive gamete covariance (Hardy-Weinberg disequilibrium) as well as allelic variance. The relative proportion of these disequilibrium variances in the total genetic variance increases with the environmental variability.     

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.     

Evolutionary Dynamics of Sporophytic Self-Incompatibility Alleles in Plants

The stationary frequency distribution and allelic dynamics in finite populations are analyzed through stochastic simulations in three models of single-locus, multi-allelic sporophytic self-incompatibility. The models differ in the dominance relationships among alleles. In one model, alleles act codominantly in both pollen and style (SSIcod), in the second, alleles form a dominance hierarchy in pollen and style (SSIdom). In the third model, alleles interact codominantly in the style and form a dominance hierarchy in the pollen (SSIdomcod). The SSIcod model behaves similarly to the model of gametophytic self-incompatibility, but the selection intensity is stronger. With dominance, dominant alleles invade the population more easily than recessive alleles and have a lower frequency at equilibrium. In the SSIdom model, recessive alleles have both a higher allele frequency and higher expected life span. In the SSIdomcod model, however, loss due to drift occurs more easily for pollen-recessive than for pollen-dominant alleles, and therefore, dominant alleles have a higher expected life span than the more recessive alleles. The process of allelic turnover in the SSIdomcod and SSIdom models is closely approximated by a random walk on a dominance ladder. Implications of the results for experimental studies of sporophytic self-incompatibility in natural populations are discussed.     

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.     

Modeling quantitative trait Loci and interpretation of models

A quantitative genetic model relates the genotypic value of an individual to the alleles at the loci that contribute to the variation in a population in terms of additive, dominance, and epistatic effects. This partition of genetic effects is related to the partition of genetic variance. A number of models have been proposed to describe this relationship: some are based on the orthogonal partition of genetic variance in an equilibrium population. We compare a few representative models and discuss their utility and potential problems for analyzing quantitative trait loci (QTL) in a segregating population. An orthogonal model implies that estimates of the genetic effects are consistent in a full or reduced model in an equilibrium population and are directly related to the partition of the genetic variance in the population. Linkage disequilibrium does not affect the estimation of genetic effects in a full model, but would in a reduced model. Certainly linkage disequilibrium would complicate the detection of QTL and epistasis. Using different models does not influence the detection of QTL and epistasis. However, it does influence the estimation and interpretation of genetic effects.     

A general model to explore complex dominance patterns in plant sporophytic self-incompatibility systems

We developed a general model of sporophytic self-incompatibility under negative frequency-dependent selection allowing complex patterns of dominance among alleles. We used this model deterministically to investigate the effects on equilibrium allelic frequencies of the number of dominance classes, the number of alleles per dominance class, the asymmetry in dominance expression between pollen and pistil, and whether selection acts on male fitness only or both on male and on female fitnesses. We show that the so-called "recessive effect" occurs under a wide variety of situations. We found emerging properties of finite population models with several alleles per dominance class such as that higher numbers of alleles are maintained in more dominant classes and that the number of dominance classes can evolve. We also investigated the occurrence of homozygous genotypes and found that substantial proportions of those can occur for the most recessive alleles. We used the model for two species with complex dominance patterns to test whether allelic frequencies in natural populations are in agreement with the distribution predicted by our model. We suggest that the model can be used to test explicitly for additional, allele-specific, selective forces.     

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     

Models of Quantitative Variation of Flux in Metabolic Pathways

As a model of variation in a quantitative character, enzyme activity variation segregating in a population is assumed to affect the flux in simple metabolic pathways. The genetic variation of flux is partitioned into additive and nonadditive components. An interaction component of flux variance is present because the effect of an allelic substitution is modified by other substitutions which change the concentrations of shared metabolites. In a haploid population, the the proportion of interaction variance is a function of the gene frequencies at the loci contributing to the flux variation, enzyme activities of mutant and wild type at variable loci and activities at nonvariable loci. The proportion of interaction variance is inversely related to the ratio of mutant to wild-type activities at the loci controlling the enzyme activities. The interaction component as a function of gene frequencies is at a maximum with high mutant allele frequencies. In contrast, the dominance component which would apply to a diploid population is maximal as a proportion of the total when mutant alleles are at low frequencies. Unless there are many loci with large differences in activity between the alleles, the interaction component is a small proportion of the total variance. Data on enzyme activity variation from natural and artificial populations suggest that such variation generates little nonadditive variance despite the highly interactive nature of the underlying biochemical system.     

Distinctions among Allelic Variants Associated with Chromosome 3 Inversions in DROSOPHILA PSEUDOOBSCURA and DROSOPHILA PERSIMILIS

Efforts were made to discriminate new genetic variants among electrophoretic alleles that are associated with chromosome 3 inversions of Drosophila pseudoobscura and D. persimilis. Apparent genetic similarities for electrophoretic alleles between these two species and among the common inversions they carry were reexamined by altering gel concentration and buffer pH. At the amylase locus, the 1.09 electrophoretic allele could be further separated into two allelic classes that differentiated the WT and KL arrangements. Similarly, the 0.84 electrophoretic allele was divided into two allelic classes, one characteristic of the Santa Cruz phylad arrangements, TL and SC, and the other found in strains of the Standard phylad arrangements and CH. Uncommon amylase alleles proved to be different alleles in the two species. No new allelic variants, however, could be found among strains with the amylase 1.00 allele, the commonest allele in the Standard phylad of both species. No major new allelic variation was detected for acid phosphatase-3 and larval protein-10 that revealed any further differentiation among species or inversions. Variation at all three loci in strains of the Bogota population remained genetically similar to variation in strains of mainland D. pseudoobscura.     

Frequency Distribution of Esterase-5 Alleles in Two Populations of DROSOPHILA PSEUDOOBSCURA

Statistical tests comparing allele frequencies in natural populations with those predicted by various theories of genic variation depend critically on the accurate enumeration of alleles. This study used a series of five sequential electrophoretic conditions to characterize the allele frequency distributions of esterase-5 in two large population samples of Drosophila pseudoobscura from California. In Standard chromosome lines 12 electromorphs were discriminated using a single electrophoretic condition. When four additional criteria were used, the number of electromorphs increased to 41, 33 in one population and 22 in the other. Both populations had the same two alleles in high frequency, with other alleles present in frequencies of 6% or less. Although each population had a number of unique alleles, a χ² contingency test demonstrated no significant genetic divergence between them. A statistical comparison of allele frequencies in both populations with that predicted by neutral models suggests that the individual and combined distributions deviate from neutrality in the direction of purifying selection.—Sex-Ratio chromosomes differed markedly from Standard chromosomes in both allelic content and diversity. In 32 Sex-Ratio chromosomes from one population only three alleles were found, all of which were detected under the initial "standard" electrophoretic conditions. Moreover, none of these alleles was found in the Standard chromosome lines.     

Influence of dominance, leptokurtosis and pleiotropy of deleterious mutations on quantitative genetic variation at mutation-selection balance

In models of maintenance of genetic variance (V (G)) it has often been assumed that mutant alleles act additively. However, experimental data show that the dominance coefficient varies among mutant alleles and those of large effect tend to be recessive. On the basis of empirical knowledge of mutations, a joint-effect model of pleiotropic and real stabilizing selection that includes dominance is constructed and analyzed. It is shown that dominance can dramatically alter the prediction of equilibrium V (G). Analysis indicates that for the situations where mutations are more recessive for fitness than for a quantitative trait, as supported by the available data, the joint-effect model predicts a significantly higher V (G) than does an additive model. Importantly, for what seem to be realistic distributions of mutational effects (i.e., many mutants may not affect the quantitative trait substantially but are likely to affect fitness), the observed high levels of genetic variation in the quantitative trait under strong apparent stabilizing selection can be generated. This investigation supports the hypothesis that most V (G) comes from the alleles nearly neutral for fitness in heterozygotes while apparent stabilizing selection is contributed mainly by the alleles of large effect on the quantitative trait. Thus considerations of dominance coefficients of mutations lend further support to our previous conclusion that mutation-selection balance is a plausible mechanism of the maintenance of the genetic variance in natural populations.     

The Effect of an Experimental Bottleneck upon Quantitative Genetic Variation in the Housefly

Effects of a population bottleneck (founder-flush cycle) upon quantitative genetic variation of morphometric traits were examined in replicated experimental lines of the housefly founded with one, four or 16 pairs of flies. Heritability and additive genetic variances for eight morphometric traits generally increased as a result of the bottleneck, but the pattern of increase among bottleneck sizes differed among traits. Principal axes of the additive genetic correlation matrix for the control line yielded two suites of traits, one associated with general body size and another set largely independent of body size. In the former set containing five of the traits, additive genetic variance was greatest in the bottleneck size of four pairs, whereas in the latter set of two traits the largest additive genetic variance occurred in the smallest bottleneck size of one pair. One trait exhibited changes in additive genetic variance intermediate between these two major responses. These results were inconsistent with models of additive effects of alleles within loci or of additive effects among loci. An observed decline in viability measures and body size in the bottleneck lines also indicated that there was nonadditivity of allelic effects for these traits. Several possible nonadditive models were explored that increased additive genetic variance as a result of a bottleneck. These included a model with complete dominance, a model with overdominance and a model incorporating multiplicative epistasis.     

The resolution of sexual antagonism by gene duplication

Disruptive selection between males and females can generate sexual antagonism, where alleles improving fitness in one sex reduce fitness in the other. This type of genetic conflict arises because males and females carry nearly identical sets of genes: opposing selection, followed by genetic mixing during reproduction, generates a population genetic "tug-of-war" that constrains adaptation in either sex. Recent verbal models suggest that gene duplication and sex-specific cooption of paralogs might resolve sexual antagonism and facilitate evolutionary divergence between the sexes. However, this intuitive proximal solution for sexual dimorphism potentially belies a complex interaction between mutation, genetic drift, and positive selection during duplicate fixation and sex-specific paralog differentiation. The interaction of these processes--within the explicit context of duplication and sexual antagonism--has yet to be formally described by population genetics theory. Here, we develop and analyze models of gene duplication and sex-specific differentiation between paralogs. We show that sexual antagonism can favor the fixation and maintenance of gene duplicates, eventually leading to the evolution of sexually dimorphic genetic architectures for male and female traits. The timescale for these evolutionary transitions is sensitive to a suite of genetic and demographic variables, including allelic dominance, recombination, sex linkage, and population size. Interestingly, we find that female-beneficial duplicates preferentially accumulate on the X chromosome, whereas male-beneficial duplicates are biased toward autosomes, independent of the dominance parameters of sexually antagonistic alleles. Although this result differs from previous models of sexual antagonism, it is consistent with several findings from the empirical genomics literature.     

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.     

Inheritance and dominance of self-incompatibility alleles in polyploid Arabidopsis lyrata

Natural populations of diploid Arabidopsis lyrata exhibit the sporophytic type of self-incompatibility system characteristic of Brassicaceae, in which complicated dominance interactions among alleles in the diploid parent determine self-recognition phenotypes of both pollen and stigma. The purpose of this study was to investigate how polyploidy affects this already complex system. One tetraploid population (Arabidopsis lyrata ssp kawasakiana from Japan) showed complete self-compatibility and produced viable selfed progeny for at least three generations subsequent to field collection. In contrast, individuals from a second tetraploid population (A. lyrata ssp petraea from Austria) were strongly self-incompatible (SI). Segregation of SI genotypes in this population followed Mendelian patterns based on a tetrasomic model of inheritance, with two to four alleles per individual, independent segregation of alleles, and little evidence of dosage effects of alleles found in multiple copies. Similar to results from diploids, anomalous compatibility patterns involving particular combinations of individuals occurred at a low frequency in the tetraploids, suggesting altered dominance in certain genetic backgrounds that could be due to the influence of a modifier locus. Overall, dominance relationships among S-alleles in self-incompatible tetraploid families were remarkably similar to those in related diploids, suggesting that this very important and complicated locus has not undergone extensive modification subsequent to polyploidization.     

Comparison of selection methods for improvement of the population hybrid

Summary The objective of this study was to compare several selection procedures with respect to expected genetic gain in the population hybrid across a range of initial allelic frequencies, degrees of dominance, and environmental variances. The methods compared were intrapopulation recurrent selection using full-sib or S₁ families, full-sib and two half-sib reciprocal recurrent selection procedures, and convergent improvement applied to populations. Comparisons were made by calculating expected allelic frequency changes for each method. The optimal selection method for a given set of allelic frequencies and degree of dominance depended little on the environmental variance. Partly because of its short cycle, full-sib intrapopulation selection was the most effective method for the majority of allelic frequency combinations when the degree of dominance was small and an off-season nursery could be used to make recombinations. With larger values for the degree of dominance, S₁ and reciprocal full-sib methods became optimal, the former method especially when favorable alleles had a high frequency and the latter when populations were highly divergent. When off-season nursery use was restricted to making self-pollinations or was absent, S₁ selection was optimal for the majority of allelic frequency combinations. Convergent improvement was superior only for extremely divergent allelic frequencies and then only when the degree of dominance was less than 0.10. Half-sib reciprocal methods were never optimal, although the gain for the standard half-sib reciprocal procedure differed little from that of full-sib reciprocal selection when the degree of dominance was 0.75.     

More about quantitative trait locus mapping with diallel designs

We present a general regression-based method for mapping quantitative trait loci (QTL) by combining different populations derived from diallel designs. The model expresses, at any map position, the phenotypic value of each individual as a function of the specific-mean of the population to which the individual belongs, the additive and dominance effects of the alleles carried by the parents of that population and the probabilities of QTL genotypes conditional on those of neighbouring markers. Standard linear model procedures (ordinary or iteratively reweighted least-squares) are used for estimation and test of the parameters.