A general mixture model approach for mapping quantitative trait loci from diverse cross designs involving multiple inbred lines

Most current statistical methods developed for mapping quantitative trait loci (QTL) based on inbred line designs apply to crosses from two inbred lines. Analysis of QTL in these crosses is restricted by the parental genetic differences between lines. Crosses from multiple inbred lines or multiple families are common in plant and animal breeding programmes, and can be used to increase the efficiency of a QTL mapping study. A general statistical method using mixture model procedures and the EM algorithm is developed for mapping QTL from various cross designs of multiple inbred lines. The general procedure features three cross design matrices, W, that define the contribution of parental lines to a particular cross and a genetic design matrix, D, that specifies the genetic model used in multiple line crosses. By appropriately specifying W matrices, the statistical method can be applied to various cross designs, such as diallel, factorial, cyclic, parallel or arbitrary-pattern cross designs with two or multiple parental lines. Also, with appropriate specification for the D matrix, the method can be used to analyse different kinds of cross populations, such as F2 backcross, four-way cross and mixed crosses (e.g. combining backcross and F2). Simulation studies were conducted to explore the properties of the method, and confirmed its applicability to diverse experimental designs.     

Selecting the size of a diallel cross experiment

Summary The size of a diallel cross experiment can be determined by using statistical tables and past diallel cross results; this will provide more flexibility in planning experiments and increase the opportunity to detect real differences.Contribution No. R-015, Research Program Service, Research Branch, Agriculture Canada, Ottawa, Ont. K1A OC6     

Comparative genetic analyses of metric traits using diallel and factorial mating designs in bread wheat

Summary For studying the inheritance of metric traits, diallel cross and factorial mating designs are commonly used. Since factorial mating design is less restrictive in crossing plans, the genetic information drawn from it was compared with that from a diallel cross. The comparison was made using graphical, genetic components and combining ability analyses for grain yield, grain weight and spike length in a field experiment of bread wheat (Triticum aestivum L.). Analyses were made on a nine parent diallel cross and a 4 × 5 factorial mating design which was sampled from the diallel cross. In general, there was a high degree of agreement between the results obtained from factorial mating design and diallel cross analyses showing thereby that the former provides almost equivalent genetic information to the latter.     

A comparison of four experimental designs for the estimation of heritability

Summary The partial diallel cross, the complete diallel cross, and the designs known as North Carolina Experiments 1 and 2 are compared for their usefulness in estimating heritability. It is first shown that reliable values for the sampling mean and variance of heritability estimates are obtained from approximate expressions based on the moments of the chi-square distribution. These expressions are then applied to determine the optimum experimental designs for a range of situations.The main basis for discrimination is the amount of information per unit, defined as i = 1/(N var( ²)), where ² is the estimate of the heritability h ² and N is the number of units in the experiment, either individuals or families.The two parameters considered were the heritability of individuals and the heritability of full-sib families, and for each of these the partial diallel cross was the most preferred, followed in decreasing order of preference by design NC2, the complete diallel, and design NC1.It is first shown that there is no optimum number of parents for a partial diallel cross or male parents for designs NC1 and NC2. The number of crosses per parent for a partial diallel or dams per sire for designs NC2 and NC2 should generally be six or less. Any expansion should be in the direction of using more parents in the case of the partial diallel, or more male parents in the case of designs NC1 and NC2. For the two heritability parameters considered in this study it is inefficient to increase the number of replicates beyond two.     

Additive main effect and multiplicative interaction model for a diallel-cross analysis

Diallel mating designs have proved informative in determining the inheritance of quantitative traits of interest to plant breeders. Apart from the well-established analyses of a complete diallel, the two-way factorial data structure of this design lends itself to analysis by the additive-main-effects-and-multiplicative-interaction (AMMI) model. This research article describes the joint application of the AMMI model and Griffing’s method 1, model I, to gain insight into the breeding value of inbred lines in a self-pollinated crop such as disomic, hexaploid bread wheat. Data from a multi-environment trial of a complete diallel cross between eight lines adapted to the East African highlands were analyzed to provide an example of this joint analysis. This combined approach identified not only the direction of a cross, i.e. which parent should be male or female, but also which crosses produce offspring showing F₁ heterosis. Received: 10 June 2000 / Accepted: 31 July 2000     

Analysis of Partial Diallel Crosses in Incomplete Blocks

With a large number of lines in a diallel cross experiment, the number of crosses becomes unmanageable to be accommodated in homogeneous blocks. To overcome this problem, a sample of crosses, known as partial diallel cross (PDC) is often used. The selection of a PDC is based on the criterion of high efficiency for the estimation of general combining ability (gca) effects. Even with a moderately large number of crosses, the use of incomplete blocks is necessary to obtain homogeneous experimental units. The analysis of data from a general PDC grown in general incomplete block designs is being described. An iterative scheme is being developed for obtaining a generalized inverse of the information matrix used in estimating gca effects. Properties such as connectedness and efficiency of mating designs embedded in environment designs are being examined. The paper also examines the universal optimality of some designs in a class of designs. An illustration of the numerical procedure is also presented.     

动物杂交试验的正反交循环设计法

为了提高配合力等参数估测的有效性、准确性和经济性。本文从动物自身的和试验的具体特点出发,提出了一种适于(1)客观条件仅允许试验部分组合、(2)试验目的仅为确定杂交育种的种群选配方案、(3)完全双列杂交试验分地分批进行等三种情况的新杂交试验设计法——正反交循环法,并且探讨了由这种设计法所设计的试验,即正反交循环杂交试验的最小二乘分析。     

The use of regression methods to study genotype-environment interactions: extending Griffing's model for diallel cross experiments and testing an empirical grouping method.

A model combining features of Griffing's diallel cross analysis with regression analysis for genotype-environment interactions is introduced using carp data of Moav et al. (1975) as an example. An analysis of variance based on this model provides information on the combining abilities of genetic effects and the interactions of these effects with environments from which inferences can readily be made on heterosis and heterosis-environment interactions. Applying the empirical grouping method of Lin and Thompson (1975) to these data (ignoring their diallel cross structure) established groups which were remarkably consistent with their members' crossing backgrounds.     

Juvenile body weight and gonad development in a diallel cross among lines of Japanese quail (Coturnix coturnix japonica)

Summary The influence of growth on the extent of heterosis for juvenile body weight and gonad development was studied in a diallel cross among two lines of Japanese quail differing in adult body size. A total of 1,096 birds (563 males and 533 females) was slaughtered between 25 and 49 days of age. Reciprocal cross differences were non-significant. Heterosis showed a curvilinear course with age peaking during early growth (body weight) and during sexual maturity (gonad percentage). Overall advanced physiological development of the crossbreds probably begins as early as during the embryonic stages and results in earlier sexual maturity. In females, heterosis for percentage gonads was biased strongly by the presence of a hard-shelled egg in the uterus.     

Quantitative trait mapping in a diallel cross of recombinant inbred lines

A recombinant inbred intercross (RIX) is created by generating diallel F₁ progeny from one or more panels of recombinant inbred (RI) strains. This design was originally introduced to extend the power of small RI panels for the confirmation of quantitative trait loci (QTL) provisionally detected in a parental RI set. For example, the set of 13 C × B (C57BL/6ByJ × BALB/cByJ) RI strains can, in principle, be supplemented with 156 isogenic F₁s. We describe and test a method of analysis, based on a linear mixed model, that accounts for the correlation structure of RIX populations. This model suggests a novel permutation algorithm that is needed to obtain appropriate threshold values for genome-wide scans of an RIX population. Despite the combinational multiplication of unique genotypes that can be generated using an RIX design, the effective sample size of the RIX population is limited by the number of progenitor RI genomes that are combined. When using small RI panels such as the C × B there appears to be only modest advantage of the RIX design when compared with the original RI panel for detecting QTLs with additive effects. The RIX, however, does have an inherent ability to detect dominance effects, and, unlike RI strains, the RIX progeny are genetically reproducible but are not fully inbred, providing somewhat more natural genetic context. We suggest a breeding strategy, the balanced partial RIX, that balances the advantage of RI and RIX designs. This involves the use of a partial RIX population derived from a large RI panel in which the available information is maximized by minimizing correlations among RIX progeny.     

Genetic parameters and direct,maternal and heterosis effects on litter size in a diallel cross among three commercial varieties of Iberian pig

The Iberian pig is one of the pig breeds that has the highest meat quality. Traditionally, producers have bred one of the available varieties, exclusively, and have not used crosses between them, which has contrasted sharply with other populations of commercial pigs for which crossbreeding has been a standard procedure. The objective of this study was to perform an experiment under full diallel design among three contemporary commercial varieties of Iberian pig and estimate the additive genetic variation and the crossbreeding effects (direct, maternal and heterosis) for prolificacy. The data set comprised 18 193 records for total number born and number born alive from 3800 sows of three varieties of the Iberian breed (Retinto, Torbiscal and Entrepelado) and their reciprocal crosses (Retinto × Torbiscal, Torbiscal × Retinto, Retinto × Entrepelado, Entrepelado × Retinto, Torbiscal × Entrepelado and Entrepelado × Torbiscal), and a pedigree of 4609 individuals. The analysis was based on a multiple population repeatability model, and we developed a model comparison test that indicated the presence of direct line, maternal and heterosis effects. The results indicated the superiorities of the direct line effect of the Retinto and the maternal effect of the Entrepelado populations. All of the potential crosses produced significant heterosis, and additive genetic variation was higher in the Entrepelado than it was in the other two populations. The recommended cross for the highest yield in prolificacy is a Retinto father and an Entrepelado mother to generate a hybrid commercial sow.     

Good genes, genetic compatibility and the evolution of polyandry: use of the diallel cross to address competing hypotheses

Genetic benefits can enhance the fitness of polyandrous females through the high intrinsic genetic quality of females' mates or through the interaction between female and male genes. I used a full diallel cross, a quantitative genetics design that involves all possible crosses among a set of genetically homogeneous lines, to determine the mechanism through which polyandrous female decorated crickets (Gryllodes sigillatus) obtain genetic benefits. I measured several traits related to fitness and partitioned the phenotypic variance into components representing the contribution of additive genetic variance ('good genes'), nonadditive genetic variance (genetic compatibility), as well as maternal and paternal effects. The results reveal a significant variance attributable to both nonadditive and additive sources in the measured traits, and their influence depended on which trait was considered. The lack of congruence in sources of phenotypic variance among these fitness-related traits suggests that the evolution and maintenance of polyandry are unlikely to have resulted from one selective influence, but rather are the result of the collective effects of a number of factors.     

Dual Designs and Their Application in Genetical Experiments

The dual of incomplete block designs has been studied with th́eir applications in genetical experiments. Partial diallel crosses (PDC) of type I have been constructed using balanced incomplete block (BIB) designs, partially balanced incomplete block (PBIB) designs and their dual designs. Simplified analysis of PDC has been presented using the dual property of these designs. List of optimal PDC having simple analysis has been given.     

Impact of mating design on selection response in Brassica rapa L.

The impact of four mating designs on selection response for leaf area was assessed at four different population sizes, using fast-cycling Brassica rapa L. Mating designs were either balanced (partial diallel or pair mating) or unbalanced (factorial mating designs with either one or two testers). When balanced, the mating designs required different numbers of crossings for the same number of parents: the partial diallel design, in the configuration retained here, required three times as many crossings as pair mating. Population sizes were 4, 8, 16, and 32. The percentage of selected individuals was kept constant at 25%. Despite an average estimated heritability around 0.4, the overall response to selection after five generations was fairly weak in all three replicates. For a given population size, selection response was larger under balanced mating designs than under unbalanced ones. There was no difference among balanced mating designs. Both results indicate that effective population size is more important than population size or the number of crossings in maintaining genetic gain.     

Analysis of genetic effects of major genes and polygenes on quantitative traits I. Genetic model for diploid plants and animals

A genetic model was proposed to simultaneously investigate genetic effects of both polygenes and several single genes for quantitative traits of diploid plants and animals. Mixed linear model approaches were employed for statistical analysis. Based on two mating designs, a full diallel cross and a modified diallel cross including F₂, Monte Carlo simulations were conducted to evaluate the unbiasedness and efficiency of the estimation of generalized least squares (GLS) and ordinary least squares (OLS) for fixed effects and of minimum norm quadratic unbiased estimation (MINQUE) and Henderson III for variance components. Estimates of MINQUE (1) were unbiased and efficient in both reduced and full genetic models. Henderson III could have a large bias when used to analyze the full genetic model. Simulation results also showed that GLS and OLS were good methods to estimate fixed effects in the genetic models. Data on Drosophila melanogaster from Gilbert were used as a worked example to demonstrate the parameter estimation. Received: 11 November 2000 / Accepted: 2 May 2001     

A note on the application of diallel crosses for the analysis of genetic variation in natural populations

Summary The complete diallel cross among homozygous lines can be a useful tool to analyze the genetic architecture of natural populations. However, it represents the natural population only approximately, in particular if the number of lines is small and the analyzed traits exhibit inbreeding depression or other forms of directional dominance. Some incorrect expected mean squares that can be found in the literature suggest tests for genetic variance components that can be misleading under such circumstances. Expected mean squares for a factorial analysis and for a modified Hayman analysis are presented and the effect of the number of lines and directional dominance is discussed.     

Mapping quantitative trait loci controlling silking date in a diallel cross among four lines of maize

 We describe and apply an interval mapping method for quantitative trait locus (QTL) detection using F₃ and testcross progenies derived from F₂ populations obtained from a diallel cross among four elite lines of maize. Linear model-based procedures were used for the test and estimation of putative QTL effects together with genetic interactions including epistasis. We mapped QTL associated with silking date and explored their genetic effects. Ten QTL were detected, and these explained more than 40% of the phenotypic variance. Most of these QTL had consistent and stable effects among genetic backgrounds and did not show significant epistasis. QTL-by-environment interaction was important for four QTL and was essentially due to changes in magnitude of allelic effects. These results show the efficiency of our method in several genetic situations as well as the power of the diallel design in detecting QTL simultaneously over several populations. Received: 2 September 1996 / Accepted: 20 December 1996     

Estimating single gene effects on quantitative traits

Summary Experimental designs for measuring the effects of single loci on quantitative traits are compared for statistical properties. The designs tested are single population, combined strains, multiple strains, diallel of strains, and co-isogenic strains. Testing was done by simulating population genotypic and phenotypic arrays. Statistical properties measured are type I error, power, bias and efficiency. The relative ranking of designs is consistent for all properties and over eight conditions examined. The co-isogenic design is superior, followed closely by the single population method. The other three designs are similar in ability, with the diallel design somewhat superior. Based on its good statistical performance and wide feasibility, the single population method is recommended. The diallel method provides the most information on genetic components of variation.     

Estimation of genetic parameters in barley (Hordeum vulgare L.)

Summary Four different sets of partial diallels were analysed for their relative efficiencies for estimating the genetic parameters in barley: (1) partial diallel with 12 parents, each involved in only 5 crosses; (2) partial diallel with 12 parents, each involved in only 3 crosses; (3) partial diallel with 8 parents, each involved in only 5 crosses; and (4) partial diallel with 8 parents, each involved in only 3 crosses. In partial diallel experiments, the estimates of gca effects were higher than in those of full diallel. Ranking pattern of the parents on the basis of gca effects in partial diallels deviated considerably from the ranking in full diallel. With decreasing s per parent, the deviation in ranking was also more. This clearly suggests the unsuitability of partial diallel analysis for screening high general combiners. Selection of best cross combinations is also not possible because only a sample of crosses (s out of n) is analysed under partial diallel so that there is every possibility of the best cross being excluded from the sample. In general, overdominance was exhibited, indicating that there is ample scope for heterosis breeding in barley.     

A diallel cross analysis of gum content in barley (Hordeum vulgare)

Summary A diallel cross analysis of gum content in barley (Hordeum vulgare) was made using six cultivars of two-rowed spring barley as parents. A Jinks-Hayman analysis of F2 progeny means showed that gum content was controlled by a simple additive-dominance genetic system and that low gum content was strongly dominant. The analysis suggested that gum content was principally controlled by two or three genes showing a high degree of dominance. Some genotype-environment interaction was detected in a comparison between the F2 and F3 generations which were grown in different years and locations. However, the character was found to be highly heritable both within and between generations, suggesting that the selection and breeding of barleys of reduced gum content should not be difficult.