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)     

Genetic evaluation methods for populations with dominance and inbreeding

Summary The effect of inbreeding on mean and genetic covariance matrix for a quantitative trait in a population with additive and dominance effects is shown. This genetic covariance matrix is a function of five relationship matrices and five genetic parameters describing the population. Elements of the relationship matrices are functions of Gillois (1964) identity coefficients for the four genes at a locus in two individuals. The equivalence of the path coefficient method (Jacquard 1966) and the tabular method (Smith and Mäki-Tanila 1990) to compute the covariance matrix of additive and dominance effects in a population with inbreeding is shown. The tabular method is modified to compute relationship matrices rather than the covariance matrix, which is trait dependent. Finally, approximate and exact Best Linear Unbiased Predictions (BLUP) of additive and dominance effects are compared using simulated data with inbreeding but no directional selection. The trait simulated was affected by 64 unlinked biallelic loci with equal effect and complete dominance. Simulated average inbreeding levels ranged from zero in generation one to 0.35 in generation five. The approximate method only accounted for the effect of inbreeding on mean and additive genetic covariance matrix, whereas the exact accounted for all of the changes in mean and genetic covariance matrix due to inbreeding. Approximate BLUP, which is computable for large populations where exact BLUP is not feasible, yielded unbiased predictions of additive and dominance effects in each generation with only slightly reduced accuracies relative to exact BLUP.     

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     

A study on the minimum number of loci required for genetic evaluation using a finite locus model

For a finite locus model, Markov chain Monte Carlo (MCMC) methods can be used to estimate the conditional mean of genotypic values given phenotypes, which is also known as the best predictor (BP). When computationally feasible, this type of genetic prediction provides an elegant solution to the problem of genetic evaluation under non-additive inheritance, especially for crossbred data. Successful application of MCMC methods for genetic evaluation using finite locus models depends, among other factors, on the number of loci assumed in the model. The effect of the assumed number of loci on evaluations obtained by BP was investigated using data simulated with about 100 loci. For several small pedigrees, genetic evaluations obtained by best linear prediction (BLP) were compared to genetic evaluations obtained by BP. For BLP evaluation, used here as the standard of comparison, only the first and second moments of the joint distribution of the genotypic and phenotypic values must be known. These moments were calculated from the gene frequencies and genotypic effects used in the simulation model. BP evaluation requires the complete distribution to be known. For each model used for BP evaluation, the gene frequencies and genotypic effects, which completely specify the required distribution, were derived such that the genotypic mean, the additive variance, and the dominance variance were the same as in the simulation model. For lowly heritable traits, evaluations obtained by BP under models with up to three loci closely matched the evaluations obtained by BLP for both purebred and crossbred data. For highly heritable traits, models with up to six loci were needed to match the evaluations obtained by BLP.     

Predicting response to selection on a quantitative trait: a comparison between models for mixed-mating populations

Two different theoretical frameworks have been developed to predict response to selection in a mixed mating population (in which reproduction occurs by a mixture of outcrossing and self-fertilization). The genotypic covariance model (GCM) and the structured linear model (SLM) rely on the same assumptions regarding quantitative trait inheritance, but use different genetic summary statistics. Here, we demonstrate the algebraic relationships between the various genetic metrics used in each theory. This is accomplished by reformulating the GCM in terms of the Wright-Kempthorne equation. We use stochastic simulations to investigate the relative accuracy of each theory for a range of selfing rates. The SLM is generally more accurate than the GCM, the most pronounced differences emerging in simulations with inbreeding depression for fitness. In fact, with strong inbreeding depression and high selfing rates, evolution can occur opposite the direction predicted by the GCM. The simulations also indicate that direct application of random mating models to partially selfing populations can produce very inaccurate predictions if quantitative trait loci exhibit dominance.     

玉米籽粒性状的遗传模型研究

用１０个遗传上和籽粒形态性状上具有差异的玉米自交系,依多种可能的交配方法获得亲本Ｐ１、Ｐ２、Ｆ１（Ｐ１× Ｐ２）、Ｆ２、Ｂ１（Ｆ１×Ｐ１）、Ｂ２（Ｆ１× Ｐ２）及其相应反交ＲＦ１、ＲＦ２、ＲＢ１、ＲＢ２共１０个种子世代。种植２年。依广义遗传模型建立包括种子胚乳加性、胚乳显性、母体加性、母体显性和细胞质效应的遗传模型,运用种子数量性状的精细鉴别法［１］和混合模型分析法［２,３］,对粒长、粒宽、粒长宽比、粒厚及百粒重作了性状表达遗传机制的鉴别与探讨。单个组合的遗传模型精细测验表明,５个籽粒性状的遗传主要受母体显性和胚乳基因型（包括加性和灵性）的控制,一个组合的粒宽、粒厚和百粒重上还检测到细胞质效应。对２５对 Ｆ１正反交组合世代均值依ＭＩＮＱＵＥ法分析的结果表明,５个籽粒性状的遗传方差中,母体遗传方差占６０％以上,胚乳基因型方差低于４０％,粒长和百粒重还有细胞质效应,约占１０％～３０％。可见,籽粒性状的遗传特点是受多套遗传系统控制,其中以母体基因型的作用最大。     

Analysis of cytoplasmic and maternal effects. II. Genetic models for triploid endosperms

Genetic models for quantitative traits of triploid endosperms are proposed for the analysis of direct gene effects, cytoplasmic effects, and maternal gene effects. The maternal effect is partitioned into maternal additive and dominance components. In the full genetic model, the direct effect is partitioned into direct additive and dominance components and high-order dominance component, which are the cumulative effects of three-allele interactions. If the high-order dominance effects are of no importance, a reduced genetic model can be used. Monte Carlo simulations were conducted in this study for demonstrating unbiasedness of estimated variance and covariance components from the MINQUE (0/1) procedure, which is a minimum norm quadratic unbiased estimation (MINQUE) method setting 0 for all the prior covariances and 1 for all the prior variances. Robustness of estimating variance and covariance components for the genetic models was tested by simulations. Both full and reduced genetic models are shown to be robust for estimating variance and covariance components under several situations of no specific effects. Efficiency of predicting random genetic effects for the genetic models by the MINQUE (0/1) procedure was compared with the best linear unbiased prediction (BLUP). A worked example is given to illustrate the use of the reduced genetic model for kernel growth characteristics in corn (Zea mays L.).     

Estimation of deleterious genomic mutation parameters in natural populations by accounting for variable mutation effects across loci

The genomes of all organisms are subject to continuous bombardment of deleterious genomic mutations (DGM). Our ability to accurately estimate various parameters of DGM has profound significance in population and evolutionary genetics. The Deng-Lynch method can estimate the parameters of DGM in natural selfing and outcrossing populations. This method assumes constant fitness effects of DGM and hence is biased under variable fitness effects of DGM. Here, we develop a statistical method to estimate DGM parameters by considering variable mutation effects across loci. Under variable mutation effects, the mean fitness and genetic variance for fitness of parental and progeny generations across selfing/outcrossing in outcrossing/selfing populations and the covariance between mean fitness of parents and that of their progeny are functions of DGM parameters: the genomic mutation rate U, average homozygous effect s, average dominance coefficient h, and covariance of selection and dominance coefficients cov(h, s). The DGM parameters can be estimated by the algorithms we developed herein, which may yield improved estimation of DGM parameters over the Deng-Lynch method as demonstrated by our simulation studies. Importantly, this method is the first one to characterize cov(h, s) for DGM.     

Mapping quantitative trait loci underlying triploid endosperm traits

Endosperm, which is derived from two polar nuclei fusing with one sperm, is a triploid tissue in cereals. Endosperm tissue determines the grain quality of cereals. Improving grain quality is one of the important breeding objectives in cereals. However, current statistical methods for mapping quantitative trait loci (QTL) under diploid genetic control have not been effective for dealing with endosperm traits because of the complexity of their triploid inheritance. In this paper, we derive for the first time the conditional probabilities of F(3) endosperm QTL genotypes given different flanking marker genotypes in F(2) plants. Using these probabilities, we develop a multiple linear regression method implemented via the iteratively reweighted least-squares (IRWLS) algorithm and a maximum likelihood method (ML) implemented via the expectation-maximization (EM) algorithm to map QTL underlying endosperm traits. We use the mean value of endosperm traits of F(3) seeds as the dependent variable and the expectations of genotypic indicators for additive and dominance effect of a putative QTL flanked by a pair of markers as independent variables for IRWLS mapping. However, if an endosperm trait is measured quantitatively using a single endosperm sample, the ML mapping method can be used to separate the two dominance effects. Efficiency of the methods is verified through extensive Monte Carlo simulation studies. Results of simulation show that the proposed methods provide accurate estimates of both the QTL effects and locations with very high statistical power. With these methods, we are now ready to map endosperm traits, as we can for regular quantitative trait under diploid control.     

The genetic basis of floral traits associated with mating system evolution in Leptosiphon (Polemoniaceae): an analysis of quantitative trait loci

Mapping of quantitative trait loci (QTL) was used to investigate the genetic architecture of divergence in floral characters associated with the mating system, an important adaptive trait in angiosperms. Two species of Leptosiphon (Polemoniaceae), one strongly self-fertilizing (L. bicolor) and the other partially outcrossing (L. jepsonii), were crossed to produce F2 and both backcross progenies. For each crossing population, a linkage map was created using amplified fragment length polymorphism markers, and QTL were identified for several dimensions of floral size. For each of the five traits examined, three to seven QTL were detected, with independent datasets yielding congruent results in some but not all cases. The phenotypic effect of individual QTL was generally moderate. We estimated that many of the QTL were additive or showed dominance toward L. bicolor, whereas comparison of mean trait values for parental and cross progenies showed apparent overall dominance of L. jepsonii traits. Colocalization of QTL for different dimensions of floral size was consistent with high phenotypic correlations between floral traits. Substantial segregation distortion was observed in marker loci, the majority favoring alleles from the large-flowered parent. A low frequency of male sterility in the F2 population is consistent with the Dobzhansky-Muller model for the evolution of reproductive isolation.     

Genotypic dispersion and covariance between relatives under linkage disequilibrium

A random mating diploid population under linkage disequilibrium is considered. In the case of two diallelic loci, the problem about condition and joint distributions of genotypes of relatives being in arbitrary genetic relations is solved. Formulae of the partitioning of genotypic variance and covariance between relatives with respect to a polygenic character are inferred (in the case of many characters - of genotypic covariance matrix).     

群体融合对遗传方差的影响

探讨了群体融合对遗传方差的影响,无显性时基因型方差对群体中的基因频率为一凸函数,群体融合将导致它的增加,完全显性时,当群体中显性基因的频率时,群体融合导致基因型方差增加;而当时,融合导致基因型方差减小。超显性时,群体融合导致基因型方差增加,对加性方差和显性方差也分别进行了探讨。     

A diffusion model for geographically structured populations

Summary A diffusion model is derived for the evolution of a diploid monoecious population under the influence of migration, mutation, selection, and random genetic drift. The population occupies an unbounded linear habitat; migration is independent of genotype, symmetric, and homogeneous. The treatment is restricted to a single diallelic locus without dominance. With the customary diffusion hypotheses for migration and the assumption that the mutation rates, selection coefficient, variance of the migrational displacement, and reciprocal of the population density are all small and of the same order of magnitude, a boundary value problem is deduced for the mean gene frequency and the covariance between the gene frequencies at any two points in the habitat. Supported by the National Science Foundation (Grant No. DEB77-21494).     

The covariance between relatives conditional on genetic markers

The development of molecular genotyping techniques makes it possible to analyze quantitative traits on the basis of individual loci. With marker information, the classical theory of estimating the genetic covariance between relatives can be reformulated to improve the accuracy of estimation. In this study, an algorithm was derived for computing the conditional covariance between relatives given genetic markers. Procedures for calculating the conditional relationship coefficients for additive, dominance, additive by additive, additive by dominance, dominance by additive and dominance by dominance effects were developed. The relationship coefficients were computed based on conditional QTL allelic transmission probabilities, which were inferred from the marker allelic transmission probabilities. An example data set with pedigree and linked markers was used to demonstrate the methods developed. Although this study dealt with two QTLs coupled with linked markers, the same principle can be readily extended to the situation of multiple QTL. The treatment of missing marker information and unknown linkage phase between markers for calculating the covariance between relatives was discussed.     

Effects of random and non-random errors on phenotypic selection in autotetraploids

Summary A theoretical investigation was made to ascertain the effects of random and non-random deviations, called errors, of phenotypic from genotypic values on population means and on the response to phenotypic recurrent selection. The study was motivated as a selection experiment for disease resistance where there was either variability in the inoculation or environment (the random errors) or where the inoculation was above or below the the optimum rate where genetic differences in resistance are maximized (the non-random errors). The study was limited to the genetics at a diallelic locus (alleles B and b) in an autotetraploid population in random mating equilibrium. The response to selection was measured as the covariance of selection and compared to the exact covariance which was the covariance of selection without errors in phenotype. The random errors were modeled by assuming that a given percentage () of the population was uniformly distributed among the five possible genotype classes independent of their true genotypes. This model was analyzed numerically for a theoretical population with the frequency of the B allele (p) ranging from 0.0 to 1.0 and assumed errors of=0.1 and 0.5 for the following six types of genic action of the B allele: additive, monoplex dominance, partial monoplex dominance, duplex dominance, partial duplex dominance, and recessive. The effect of random error was to consistently reduce the response to selection by a percentage independent of the type of genic action at the locus. The effect on the population mean was an upward bias when p was low and a downward bias when p approached unity. In the non-random error model below optimum inoculations altered the phenotypes by systematically including percentage of susceptible genotypes into one or more other genotype classes with more genetic resistance (a positive shift). With above optimum inoculations, some resistant genotypes are classed with the non-resistant genotypes (a negative shift). The effects on the covariance of selection were found by numerical analysis for the same types of genic action and's as investigated for random error. With a negative shift and a low p, the covariance of selection was always reduced, but for an increasing p the covariance approached and exceeded the exact covariance for all types of genic action except additive. With a positive shift and a low p, response to selection was greatly improved for three types of genic action: duplex dominance, partial duplex dominance, and recessive. The effect of a non-random error on population means was to greatly bias the means upwards for a low p and positive shift, but with increasing p the bias decreased. A relatively slight decrease in the mean occurred with a negative shift. This study indicated check varieties commonly used to monitor selection pressures in screening programs are very responsive to positive non-random shifts, but are relatively unresponsive to negative shifts. The interaction of selection pressure, types of genic action, and genotypes in the class shift models was suggested as a partial explanation for the lack of response to increasing selection pressures observed in some breeding programs.Cooperative investigations of the Alfalfa Production Research Unit, United States Department of Agriculture, Agricultural Research Service, and the Nevada Agricultural Experiment Station, Reno, Nevada. Paper No. 404 Scientific Journal Series. Nevada Agricultural Experiment Station     

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.     

Effect of gametic disequilibrium on selection in an autotetraploid population

Summary The effects of a gametic disequilibrium (DSE) in an autotetraploid population on response to selection as measured by the covariance of selection were investigated. The theoretical responses were calculated for mass selection [Mass (1)] and half-sib progeny test selection (HSPT) in a two-allele (B and b), single locus, autotetraploid population. The complexity of calculations precluded analytical expressions for the covariances so numerical analysis was used assuming the following genetic models: monoplex dominance, partial monoplex dominance, duplex dominance, partial duplex dominance, and additive gene action.The results indicated the DSE could greatly affect the covariance of selection. For a constant allele frequency the DSE might double the covariance expected with selection in a population at random mating equilibrium (RME) of gametes, but in other instances approach zero. For all genetic models and the two breeding methods investigated the covariance of selection was always increased when the frequency of BB gamete exceeded p² (where p is frequency of allele B) and decreased when the frequency of BB gamete was less than p². The possible incorporation of this information into a long term breeding program and some other ramifications were briefly discussed.With the DSE the covariances of selection with HSPT and Mass (1) had a proportionality of 1:2, respectively, with the additive genetic model, but this relationship rarely occurred for other genetic models. The deviations from this ratio were not large in comparison to differences between selection in populations in DSE and RME.Cooperative investigations of the Alfalfa Production Research Unit, United State Department of Agriculture, Agricultural Research Service, and the Nevada Agricultural Experiment Station, Reno, Nevada. Paper No. 512. Scientific Journal Series, Nevada Agricultural Experiment Station     

Bias in the prediction of genetic gain due to mass and half-sib selection in random mating populations

The prediction of gains from selection allows the comparison of breeding methods and selection strategies, although these estimates may be biased. The objective of this study was to investigate the extent of such bias in predicting genetic gain. For this, we simulated 10 cycles of a hypothetical breeding program that involved seven traits, three population classes, three experimental conditions and two breeding methods (mass and half-sib selection). Each combination of trait, population, heritability, method and cycle was repeated 10 times. The predicted gains were biased, even when the genetic parameters were estimated without error. Gain from selection in both genders is twice the gain from selection in a single gender only in the absence of dominance. The use of genotypic variance or broad sense heritability in the predictions represented an additional source of bias. Predictions based on additive variance and narrow sense heritability were equivalent, as were predictions based on genotypic variance and broad sense heritability. The predictions based on mass and family selection were suitable for comparing selection strategies, whereas those based on selection within progenies showed the largest bias and lower association with the realized gain.     

Mixed Model Methods for Genomic Prediction and Variance Component Estimation of Additive and Dominance Effects Using SNP Markers

We established a genomic model of quantitative trait with genomic additive and dominance relationships that parallels the traditional quantitative genetics model, which partitions a genotypic value as breeding value plus dominance deviation and calculates additive and dominance relationships using pedigree information. Based on this genomic model, two sets of computationally complementary but mathematically identical mixed model methods were developed for genomic best linear unbiased prediction (GBLUP) and genomic restricted maximum likelihood estimation (GREML) of additive and dominance effects using SNP markers. These two sets are referred to as the CE and QM sets, where the CE set was designed for large numbers of markers and the QM set was designed for large numbers of individuals. GBLUP and associated accuracy formulations for individuals in training and validation data sets were derived for breeding values, dominance deviations and genotypic values. Simulation study showed that GREML and GBLUP generally were able to capture small additive and dominance effects that each accounted for 0.00005–0.0003 of the phenotypic variance and GREML was able to differentiate true additive and dominance heritability levels. GBLUP of the total genetic value as the summation of additive and dominance effects had higher prediction accuracy than either additive or dominance GBLUP, causal variants had the highest accuracy of GREML and GBLUP, and predicted accuracies were in agreement with observed accuracies. Genomic additive and dominance relationship matrices using SNP markers were consistent with theoretical expectations. The GREML and GBLUP methods can be an effective tool for assessing the type and magnitude of genetic effects affecting a phenotype and for predicting the total genetic value at the whole genome level.