Covariances of relatives and selection response in generations of selfing from an outcrossed base population

Summary The covariances of relatives arising under selfing from a general outbred base population in linkage equilibrium and without epistasis given by Cockerham (1983) are expressed in an alternative form which is an extension of the treatment by Mather and Jinks (1982) of the more restricted population descended from a single F₁ family. Whereas no more than two quadratic components are required to describe any covariance in the case of F₁, descendants, this more general case calls for a total of four, three of which are needed for any particular covariance. The estimation of covariances and their use for the prediction of selection response is described for breeding programs initiated by one or more cycles of intermating among a number of parental lines, as advocated by Hansel (1964) and Jensen (1970). It is pointed out that the homozygous lines descended from such a population will have up to twice as much variance as those from an F₁ between a randomly chosen pair from the same population of parents. The selection method is especially recommended for undeveloped species in which the parental lines are not well characterized and large selection responses are needed.     

CORRELATED MATINGS IN THE PARTIAL SELFER MIMULUS GUTTATUS

In partially selfing populations, siblings may be correlated for both selling and paternity. A model of the mating system based upon sampling pairs of progeny from a maternal parent is described. The model separates the correlation of selfing from the correlation of outcrossed paternal alleles and is an approach to paternity analysis suited for larger populations with fewer marker loci. Its parameters determine the components of genetic covariance between sibs and provide information about the average number of fathers in a maternal sibship. Electrophoretic markers were used to obtain estimates of correlated matings for two Mimulus guttatus populations. In both populations, about 50% selfing was observed. For two sibs randomly selected from the same capsule, the correlations of selfing between these sibs were 17% and 12% in the two populations, and the correlations of paternity (the proportion of full-sibs among outcrossed sib-pairs) were 37% and 44%. Sibs from different capsules were not correlated for selfing, and the paternity correlation dropped to near 20% in both populations. However, estimates of correlated matings have high variance, lack statistical independence, and can be difficult to obtain. The use of marker loci with many alleles can alleviate these problems.     

Variance in quantitative traits due to linked dominant genes and variance in heterozygosity in small populations

The influence of small population size (N) on the genetic variance within and between randomly bred unselected lines, with selfing permitted, is investigated for a model of a quantitative trait determined by linked genes that show dominance within loci but are additive over loci. Formulae for within-line variance include terms in linkage disequilibrum, which occurs by chance in the lines and these are evaluated in terms of N, map length and gene number.—The expected variance within lines is increased by this disequilibrium, quite substantially if there are many loci, with most of the increase being between or within full-sib families and almost no change expected between half-sib families or in the covariance of offspring and parent. If all loci are unlinked, there is no increase in variance within full-sib families. The variance between lines is little affected by disequilibrum generated by chance.—Expressions for the variance between individuals in heterozygosity over the whole genome are special cases of those for the variance due to linked dominated genes, and formulae are given and evaluated. The coefficient of variation of heterozygosity is at least (see PDF) and can be much higher for species with few chromosomes.     

The effect of genetic drift on the variance/covariance components generated by multilocus additive x additive epistatic systems

The effect of population bottlenecks on the components of the genetic variance/covariance generated by n neutral independent additive x additive loci has been studied theoretically. In its simplest version, this situation can be modelled by specifying the allele frequencies and homozygous effects at each locus, and an additional factor measuring the strength of the n-th order epistatic interaction. The variance/covariance components in an infinitely large panmictic population (ancestral components) were compared with their expected values at equilibrium over replicates randomly derived from the base population, after t bottlenecks of size N (derived components). Formulae were obtained giving the derived components (and the between-line variance) as functions of the ancestral ones (alternatively, in terms of allele frequencies and effects) and the corresponding inbreeding coefficient F(t). The n-th order derived component of the genetic variance/covariance is continuously eroded by inbreeding, but the remaining components may increase initially until a critical F(t) value is attained, which is inversely related to the order of the pertinent component, and subsequently decline to zero. These changes can be assigned to the between-line variances/covariances of gene substitution and epistatic effects induced by drift. Numerical examples indicate that: (1) the derived additive variance/covariance component will generally exceed its ancestral value unless epistasis is weak; (2) the derived epistatic variance/covariance components will generally exceed their ancestral values unless allele frequencies are extreme; (3) for systems showing equal ancestral additive and total non-additive variance/covariance components, those including a smaller number of epistatic loci may generate a larger excess in additive variance/covariance after bottlenecks than others involving a larger number of loci, provided that F(t) is low. Our results indicate that it is unlikely that the rate of evolution may be significantly accelerated after population bottlenecks, in spite of occasional increments of the derived additive variance over its ancestral value.     

CORRELATED EVOLUTION OF MATING STRATEGY AND INBREEDING DEPRESSION WITHIN AND AMONG POPULATIONS OF THE HERMAPHRODITIC SNAIL PHYSA ACUTA

Inbreeding depression is one of the main forces opposing the evolution of self-fertilization. Of central importance is the hypothesis that inbreeding depression and selfing coevolve antagonistically, generating either low selfing rate and high inbreeding depression or vice versa. However, there is limited evidence for this coevolution within species. We investigated this topic in the hermaphroditic snail Physa acuta . In this species, isolated individuals delay the onset of egg laying compared to individuals having access to mates. Longer delays ("waiting times") indicate more intense selfing avoidance. We measured inbreeding depression and waiting time in a large quantitative-genetic experiment (281 outbred families derived from 26 natural populations). We observed large genetic variance for both traits and a strong positive genetic covariance between them, most of which resided within rather than among populations. It means that, within populations, individuals with higher mutation load avoided selfing more strongly on average. This genetic covariance may result from pleiotropy and/or linkage disequilibrium. Whatever its genetic architecture, the fact it emerges specifically when individuals are deprived of mates suggests it is not fortuitous and rather reflects the action of natural selection. We conclude that a diversity of mating strategies can arise within populations subjected to variation in inbreeding depression.     

Population differentiation in floral and life history traits ofOxalis corniculata L. (Oxalidaceae) with style length polymorphism

A quantitative genetic analysis was conducted on the amounts and distribution of variation of 20 floral, reproductive and life history traits of a self-compatible perennial,Oxalis cornlculata L. (Oxalidaceae). This species comprises two floral morphs, homostyled and long-styled, with different breeding systems. The hierarchical design of the experiments on three homostyled and three long-styled populations allowed partitioning of variation into four levels of organization using nested ANOVAs. Seven of the 20 traits examined were differentiated between homostyled and long-styled populations. Significant genetic variance components were detected in the major traits (114 of the 120 traits) examined for six populations. Average values of variance components among families within a population across 20 traits for homostyled populations were higher than those of long-styled populations. These responses likely reflect the consequences of different levels of selfing and/or mixed mating on genetic variation in the two floral morphs ofO. corniculata. Pearson product-moment correlations for family means of seven traits selected were also calculated. Two groups of trait combinations (i.e., between floral traits, and between reproductive and life history traits) showed significant family mean correlation coefficients. The origins of these varlation patterns found in different populations ofO. corniculata are discussed in terms of the underlying selective regimes operating in each population.     

The use of linear mixed model theory for the genetic analysis of repeated measures from clonal tests of forest trees. I. A focus on spatially repeated data

The paper reviews the linear mixed models (LMM) methodology that is suitable for the statistical and genetic analyses of spatially repeated measures collected from clonal progeny tests. For example, we consider a poplar clonal trial where progenies of different families are propagated by cuttings, and only one ramet per clone is planted on each block. Modeling covariance structures following the LMM theory allows improving genetic parameter estimation based on clonal testing. Besides variance components, we also obtained an estimate of the covariance between residuals (within clonal effects in two different blocks). This covariance is due to planting more than one ramet from the same genotype in the same trial, which generates correlated residual effects from different blocks. Its estimation can significantly improve the comparison among clones within a progeny test or between tests in a clonal testing network. Results indicate that the covariance is also a component of the genetic variance estimator and plays a significant role in assessing the variance of specific (micro) environmental effects. A positive covariance implies that ramets show a similar performance in more than one block. Thus, a larger and more positive covariance implies a stronger genetic effect controlling the expression of the trait in the local environment and a smaller variance of specific environmental effects. On the contrary, a negative covariance implies that the performance of individual ramets is affected by strong microenvironmental effects, specific to one or more blocks, which can also directly increase the within-clone variability.     

ESTIMATING HIERARCHICAL F-STATISTICS

This paper presents an analysis of variance (ANOVA) approach by which estimation of F-statistics can be made from data with an arbitrary s-level hierarchical population structure. Assuming a complete random-effect model, a general ANOVA procedure is developed to estimate F-statistics as ratios of different variance components for all levels of population subdivision in the hierarchy. A generalized relationship among F-statistics is also derived to extend the well-known relationship originally found by Sewall Wright. Although not entirely free from the bias particular to small number of subdivisions at each hierarchy and extreme gene frequencies, the ANOVA estimators of F-statistics consider sampling effects at each level of hierarchy, thus removing the bias incurred in the other estimators that are commonly based on direct substitution of unknown gene frequencies by their sample estimates. Therefore, the ANOVA estimation procedure presented here may become increasingly useful in analyzing complex population structure because of increasing use of the estimated hierarchical F-statistics to infer genetic and demographic structures of natural populations within and among species.     

Covariance between relatives in multibreed populations: additive model

Covariance between relatives in a multibreed population was derived for an additive model with multiple unlinked loci. An efficient algorithm to compute the inverse of the additive genetic covariance matrix is given. For an additive model, the variance for a crossbred individual is a function of the additive variances for the pure breeds, the covariance between parents, and segregation variances. Provided that the variance of a crossbred individual is computed as presented here, the covariance between crossbred relatives can be computed using formulae for purebred populations. For additive traits the inverse of the genotypic covariance matrix given here can be used both to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood in multibreed populations. For nonadditive traits, the procedure currently used to analyze multibreed data can be improved using the theory presented here to compute additive covariances together with a suitable approximation for nonadditive covariances.Supported in part by the Illinois Agricultural Experiment Station, Hatch Projects 35-0345 (RLF) and 35-0367 (MG)     

Identification of associations between SSR markers and fiber traits in an exotic germplasm derived from multiple crosses among Gossypium tetraploid species

Genetic improvement in yield and fiber quality is needed for worldwide cotton production. Identification of molecular markers associated with fiber-related traits can facilitate selection for these traits in breeding. This study was designed to identify associations between SSR markers and fiber traits using an exotic germplasm population, species polycross (SP), derived from multiple crosses among Gossypium tetraploid species. The SP population underwent 11 generations of mixed random mating and selfing followed by 12 generations of selfing. A total of 260 lines were evaluated for fiber-related traits under three environments in 2005 and 2006. Large genotypic variance components in traits were identified relative to components of genotype × environment. Eighty-six primer pairs amplified a total of 314 polymorphic fragments among 260 lines. A total of 202 fragments with above 6% allele frequency were analyzed for associations. Fifty-nine markers were found to have a significant (P < 0.05, 0.01, or 0.001) association with six fiber traits. There were six groups identified within the population using structure analysis. Allele frequency divergence among six groups ranged from 0.11 to 0.27. Of the 59 marker–trait associations, 39 remained significant after correction for population structure and kinship using a mixed linear model. The effect of population sub-structure on associations was most significant in boll weight among the traits analyzed. The sub-structure among the SP lines may be caused by natural selection, the breeding method applied during development of inbred lines, and unknown factors. The identified marker–trait associations can be useful in breeding and help determine genetic mechanisms underlying interrelationships among fiber traits. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Pollen Dispersal Between Isolated Trees in the Brazilian Savannah: A Case Study of the Neotropical Tree Hymenaea stigonocarpa

This case study examines the pollen dispersal distance, pollen dispersal patterns and intra‐family genetic structure for isolated trees in pastures of the bat‐pollinated Neotropical tree species Hymenaea stigonocarpa using six microsatellite loci and parentage analysis. The sampling included 28 grouped trees (referred to as the population) and six isolated trees in pastureland along a highway in Mato Grosso do Sul State, Brazil. From the population, we sampled 137 seeds from 12 seed‐trees, and from the isolated trees, we sampled 34 seeds from two seed‐trees. The results showed that pollen was dispersed over long distances (reaching 7353 m) and therefore the spatially isolated trees were not reproductively isolated. The pollen immigration rate in the population was also high (31%). Isolated trees presented a higher selfing rate (s=26%) than trees in the population (s=12%), suggesting that the spatial isolation of the trees increased selfing. However, selfing was responsible for only 30 percent of the inbreeding in offspring and mating among relatives was 70 percent. In the population, excluding selfing, ca 72 percent of the pollen was dispersed over distances <1000 m (average: 860 m). For the two isolated seed‐trees, excluding selfing, the average pollen dispersal distance was 5229 m. The results demonstrate that although pollen can be dispersed over long distances for H. stigonocarpa isolated trees, a high percentage of pollen comes from the same tree (selfing) and mating was correlated. Consequently, seeds must be collected from a large number of seed‐trees for conservation purposes.     

POPULATION GENETIC STRUCTURE IN CHEILANTHES GRACILLIMA

Population genetic structure was examined in five populations of the xerically adapted homosporous fern Cheilanthes gracillima. F statistics using allozymic data indicated substantial genetic structure in all populations. To determine the factors responsible for genetic structure, we calculated levels of intragametophytic selfing and the fixation index for each subpopulation of each population and estimated levels of intrapopulational gene flow in each population. These analyses indicated that each subpopulation was a panmictic unit; thus, population genetic structure is not due to family structure, arising via matings between relatives. Intrapopulational gene flow was surprisingly low, given the typically high dispersibility of fern spores. However, it seems unlikely that spore dispersal in C. gracillima is significantly reduced relative to other homosporous ferns. Instead, we propose that the low rates of intrapopulational gene flow reflect limited availability of safesites for spore germination and gametophyte establishment. This ecological factor may play a primary role in generating and/or maintaining population genetic structure in C. gracillima.     

A Markov chain Monte Carlo approach for joint inference of population structure and inbreeding rates from multilocus genotype data

Nonrandom mating induces correlations in allelic states within and among loci that can be exploited to understand the genetic structure of natural populations (Wright 1965). For many species, it is of considerable interest to quantify the contribution of two forms of nonrandom mating to patterns of standing genetic variation: inbreeding (mating among relatives) and population substructure (limited dispersal of gametes). Here, we extend the popular Bayesian clustering approach STRUCTURE (Pritchard et al. 2000) for simultaneous inference of inbreeding or selfing rates and population-of-origin classification using multilocus genetic markers. This is accomplished by eliminating the assumption of Hardy-Weinberg equilibrium within clusters and, instead, calculating expected genotype frequencies on the basis of inbreeding or selfing rates. We demonstrate the need for such an extension by showing that selfing leads to spurious signals of population substructure using the standard STRUCTURE algorithm with a bias toward spurious signals of admixture. We gauge the performance of our method using extensive coalescent simulations and demonstrate that our approach can correct for this bias. We also apply our approach to understanding the population structure of the wild relative of domesticated rice, Oryza rufipogon, an important partially selfing grass species. Using a sample of n = 16 individuals sequenced at 111 random loci, we find strong evidence for existence of two subpopulations, which correlates well with geographic location of sampling, and estimate selfing rates for both groups that are consistent with estimates from experimental data (s approximately 0.48-0.70).     

THE MIXED-MODEL ANALYSIS OF VARIANCE APPLIED TO QUANTITATIVE GENETICS: BIOLOGICAL MEANING OF THE PARAMETERS

The mixed-model factorial analysis of variance has been used in many recent studies in evolutionary quantitative genetics. Two competing formulations of the mixed-model ANOVA are commonly used, the “Scheffe” model and the “SAS” model; these models differ in both their assumptions and in the way in which variance components due to the main effect of random factors are defined. The biological meanings of the two variance component definitions have often been unappreciated, however. A full understanding of these meanings leads to the conclusion that the mixed-model ANOVA could have been used to much greater effect by many recent authors. The variance component due to the random main effect under the two-way SAS model is the covariance in true means associated with a level of the random factor (e.g., families) across levels of the fixed factor (e.g., environments). Therefore the SAS model has a natural application for estimating the genetic correlation between a character expressed in different environments and testing whether it differs from zero. The variance component due to the random main effect under the two-way Scheffe model is the variance in marginal means (i.e., means over levels of the fixed factor) among levels of the random factor. Therefore the Scheffe model has a natural application for estimating genetic variances and heritabilities in populations using a defined mixture of environments. Procedures and assumptions necessary for these applications of the models are discussed. While exact significance tests under the SAS model require balanced data and the assumptions that family effects are normally distributed with equal variances in the different environments, the model can be useful even when these conditions are not met (e.g., for providing an unbiased estimate of the across-environment genetic covariance). Contrary to statements in a recent paper, exact significance tests regarding the variance in marginal means as well as unbiased estimates can be readily obtained from unbalanced designs with no restrictive assumptions about the distributions or variance-covariance structure of family effects.     

Ontogeny of additive and maternal genetic effects: lessons from domestic mammals

Evolution of size and growth depends on heritable variation arising from additive and maternal genetic effects. Levels of heritable (and nonheritable) variation might change over ontogeny, increasing through "variance compounding" or decreasing through "compensatory growth." We test for these processes using a meta-analysis of age-specific weight traits in domestic ungulates. Generally, mean standardized variance components decrease with age, consistent with compensatory growth. Phenotypic convergence among adult sheep occurs through decreasing environmental and maternal genetic variation. Maternal variation similarly declines in cattle. Maternal genetic effects are thus reduced with age (both in absolute and relative terms). Significant trends in heritability (decreasing in cattle, increasing in sheep) result from declining maternal and environmental components rather than from changing additive variation. There was no evidence for increasing standardized variance components. Any compounding must therefore be masked by more important compensatory processes. While extrapolation of these patterns to processes in natural population is difficult, our results highlight the inadequacy of assuming constancy in genetic parameters over ontogeny. Negative covariance between direct and maternal genetic effects was common. Negative correlations with additive and maternal genetic variances indicate that antagonistic pleiotropy (between additive and maternal genetic effects) may maintain genetic variance and limit responses to selection.     

COVARIATION OF SELFING RATES WITH PARENTAL GENE FIXATION INDICES WITHIN POPULATIONS OF MIMULUS GUTTATUS

Wright's gene fixation index F and two single-locus effective selfing rates—the selfing rate at loci with fixed alleles, and the selfing rate at loci without fixed alleles—were estimated in five populations of Mimulus guttatus. These two effective selfing rates describe the inbreeding observed at a single locus when both uniparental and biparental inbreeding are practiced. Estimates were made using progeny arrays assayed for six allozyme loci and two morphological loci exhibiting dominance. The average of the two selfing rates computed for subpopulations (ca. 10 m diameter) ranged from 24% to 59%, with a mean of 37%. When computed for populations (ca. 1 km diameter), average selfing rates were about 10% higher. In four populations, the selfing rate at loci with fixed alleles was higher than the selfing rate at loci without fixed alleles. Thus, the covariance of selfing with parental gene fixation was positive. In one of the populations, estimates for individual plants sampled along a transect gave positive correlations for selfing rates and for gene-fixation indices between adjacent plants. A highly positive correlation between selfing rate and gene fixation of individual plants was also observed. In another population, the covariance of selfing with gene fixation was higher for a locus causing leaf spots than for allozyme loci. This covariance is partially caused by 1) variation in homozygosity among neighborhoods and 2) biparental inbreeding within neighborhoods. The consequences of this covariance are discussed.     

Multifactorial genetic models for quantitative traits in humans.

Quantitative traits measured in human families can be analyzed to partition the total population variance into genetic and environmental components, or to elucidate the genetic mechanism involved. We review the estimation of variance components directly from human pedigree data, or in the form of path coefficients from correlations between pairs of relatives. To elucidate genetic mechanisms, a mixed model that allows for segregation at a major locus, a polygenic effect and a sibling environmental correlation is described for nuclear families. In each case appropriate likelihoods are derived as a basis, using numerical maximum likelihood methods, for parameter estimation and hypothesis testing. A general model is then described that allows for several familial sources of environmental variation, assortative mating, and both major gene and polygenic effects; and an algorithm for calculating the likelihood of a pedigree under this model is indicated. Finally, some of the remaining problems in this area of biometric analysis are pointed out.     

Estimation of heritability from limited family data using genome-wide identity-by-descent sharing

Background

In classical pedigree-based analysis, additive genetic variance is estimated from between-family variation, which requires the existence of larger phenotyped and pedigreed populations involving numerous families (parents). However, estimation is often complicated by confounding of genetic and environmental family effects, with the latter typically occurring among full-sibs. For this reason, genetic variance is often inferred based on covariance among more distant relatives, which reduces the power of the analysis. This simulation study shows that genome-wide identity-by-descent sharing among close relatives can be used to quantify additive genetic variance solely from within-family variation using data on extremely small family samples.

Methods

Identity-by-descent relationships among full-sibs were simulated assuming a genome size similar to that of humans (effective number of loci ~80). Genetic variance was estimated from phenotypic data assuming that genomic identity-by-descent relationships could be accurately re-created using information from genome-wide markers. The results were compared with standard pedigree-based genetic analysis.

Results

For a polygenic trait and a given number of phenotypes, the most accurate estimates of genetic variance were based on data from a single large full-sib family only. Compared with classical pedigree-based analysis, the proposed method is more robust to selection among parents and for confounding of environmental and genetic effects. Furthermore, in some cases, satisfactory results can be achieved even with less ideal data structures, i.e., for selectively genotyped data and for traits for which the genetic variance is largely under the control of a few major genes.

Conclusions

Estimation of genetic variance using genomic identity-by-descent relationships is especially useful for studies aiming at estimating additive genetic variance of highly fecund species, using data from small populations with limited pedigree information and/or few available parents, i.e., parents originating from non-pedigreed or even wild populations.     

Impact of nonrandom mating on genetic variance and gene flow in populations with mass selection

The mechanisms by which nonrandom mating affects selected populations are not completely understood and remain a subject of scientific debate in the development of tractable predictors of population characteristics. The main objective of this study was to provide a predictive model for the genetic variance and covariance among mates for traits subjected to directional selection in populations with nonrandom mating based on the pedigree. Stochastic simulations were used to check the validity of this model. Our predictions indicate that the positive covariance among mates that is expected to result with preferential mating of relatives can be severely overpredicted from neutral expectations. The covariance expected from neutral theory is offset by an opposing covariance between the genetic mean of an individual's family and the Mendelian sampling term of its mate. This mechanism was able to predict the reduction in covariance among mates that we observed in the simulated populations and, in consequence, the equilibrium genetic variance and expected long-term genetic contributions. Additionally, this study provided confirmatory evidence on the postulated relationships of long-term genetic contributions with both the rate of genetic gain and the rate of inbreeding (deltaF) with nonrandom mating. The coefficient of variation of the expected gene flow among individuals and deltaF was sensitive to nonrandom mating when heritability was low, but less so as heritability increased, and the theory developed in the study was sufficient to explain this phenomenon.