Computational procedure for a weighted diallel analysis

Summary A diallel analysis is described for the case in which the values of the characteristic used in an inheritance study have unequal variances. Such a characteristic can be a mean, a slope of a regression line, or the estimates of some parameter of a linear or nonlinear model. Computational formulae are presented which incorporate the necessary weighting along with the statistics of the Hayman-Jinks method for diallel analysis. The method described can also be used to perform a weighted diallel analysis based on means when there are unequal numbers of replications per cross. A simple example demonstrates the computations necessary to complete a weighted diallel analysis.Journal paper no. 87-117-J of the Kansas Agricultural Experiment Station. This study was conducted by the senior author in partial fulfillment for the PhD degree in genetics     

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.     

Evaluation of circulant partial diallel crosses in maize

Summary The present study was conducted in maize (Zea mays L.) on crosses among 20 diverse parents. The materials were evaluated in four different environments for eight characters. Combining ability analysis was carried out following diallel and partial diallel crosses. The number of crosses per parent (s) varied from 3 to 19 and the results were studied to identify the critical value of s that would provide an adequate information with minimum resources. The S₅ partial diallel was as good as the S₁₉ for the detection of differences among general combining ability (GCA) effects. Even the S₃ gave adequate information in the case of characters with high heritability. However, partial diallel analysis was less efficient in detecting the differences due to specific combining ability (SCA) effects. These results varied with environments, and characters with low heritability were more prone to misinterpretation. GCA effects showed fluctuations in partial diallel analysis which were more pronounced in S₅ and S₃, particularly for characters with low heritability. The average standard error of difference between GCA effects increased with a decrease in s, with a steep slope for s < 7. The partial diallel analysis was more efficient for the estimation of the variance component of GCA than for SCA, as the estimate of SCA was biased upwards. Estimates of broad sense heritability obtained from the partial diallels agreed with the full diallel analysis better than the narrow sense estimates. Smaller partial diallels gave erratic estimates of heritability, particularly for the characters with low heritability.     

An allelism test for quantitative trait genes

Analytical modeling has been used to test assumptions on the mode of inheritance of a quantitative trait in the course of diallel crossing between pure lines that are sufficient for adequacy of a simple regression model. This model frequently proved to be adequate in analysis of numerous data on diallel crossings of wheat and maize. An allelism test for quantitative trait genes has been suggested. Computer simulation has been used to estimate the effect of random experimental errors and deviations from the model assumptions.     

Using mating designs to uncover QTL and the genetic architecture of complex traits

Analysis of quantitative trait loci (QTL) affecting complex traits is often pursued in single-cross experiments. For most purposes, including breeding, some assessment is desired of the generalizability of the QTL findings and of the overall genetic architecture of the trait. Single-cross experiments provide a poor basis for these purposes, as comparison across experiments is hampered by segregation of different allelic combinations among different parents and by context-dependent effects of QTL. To overcome this problem, we combined the benefits of QTL analysis (to identify genomic regions affecting trait variation) and classic diallel analysis (to obtain insight into the general inheritance of the trait) by analyzing multiple mapping families that are connected via shared parents. We first provide a theoretical derivation of main (general combining ability (GCA)) and interaction (specific combining ability (SCA)) effects on F(2) family means relative to variance components in a randomly mating reference population. Then, using computer simulations to generate F(2) families derived from 10 inbred parents in different partial-diallel designs, we show that QTL can be detected and that the residual among-family variance can be analyzed. Standard diallel analysis methods are applied in order to reveal the presence and mode of action (in terms of GCA and SCA) of undetected polygenes. Given a fixed experiment size (total number of individuals), we demonstrate that QTL detection and estimation of the genetic architecture of polygenic effects are competing goals, which should be explicitly accounted for in the experimental design. Our approach provides a general strategy for exploring the genetic architecture, as well as the QTL underlying variation in quantitative traits.     

A general Bayesian approach to analyzing diallel crosses of inbred strains

The classic diallel takes a set of parents and produces offspring from all possible mating pairs. Phenotype values among the offspring can then be related back to their respective parentage. When the parents are diploid, sexed, and inbred, the diallel can characterize aggregate effects of genetic background on a phenotype, revealing effects of strain dosage, heterosis, parent of origin, epistasis, and sex-specific versions thereof. However, its analysis is traditionally intricate, unforgiving of unplanned missing information, and highly sensitive to imbalance, making the diallel unapproachable to many geneticists. Nonetheless, imbalanced and incomplete diallels arise frequently, albeit unintentionally, as by-products of larger-scale experiments that collect F(1) data, for example, pilot studies or multiparent breeding efforts such as the Collaborative Cross or the Arabidopsis MAGIC lines. We present a general Bayesian model for analyzing diallel data on dioecious diploid inbred strains that cleanly decomposes the observed patterns of variation into biologically intuitive components, simultaneously models and accommodates outliers, and provides shrinkage estimates of effects that automatically incorporate uncertainty due to imbalance, missing data, and small sample size. We further present a model selection procedure for weighing evidence for or against the inclusion of those components in a predictive model. We evaluate our method through simulation and apply it to incomplete diallel data on the founders and F(1)'s of the Collaborative Cross, robustly characterizing the genetic architecture of 48 phenotypes.     

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

Summary Four exotic and four indigenous strains of barley were used for making diallel crosses. The sets of parents and crosses making full, half and quarter diallel were analysed in a randomized block design for plant height, number of effective tillers, ear length, grain yield per plant, 100 grain weight and number of grains per ear.The three alternatives of diallel were similar with respect to the estimates of degree of dominance, general combining ability and specific combining ability, indicating that all these three methods of diallel were equally efficient. However, as the number of entries are minimum in quarter diallel, it would be economical in terms of cost, time and labour to estimate genetic parameters by this method. Average degree of dominance was found in the range of overdominance. The ranking of parents on the basis of their array mean was similar to the ranking based on gca effects. Similarly, the ranking of crosses on the basis of per se performance was similar to the ranking based on sca effects. This suggests that the selection of best general combiner or best cross combinations may be easier and more effective through array mean for per se performance rather than through high gca and sea effects, respectively. From among 56 crosses, IB-226 X X C-164 was the one which showed superiority for maximum number of characters followed by AB-12/59 X PTS-57. High sea effect for plant height, ear length, grain yield, 100 grain weight and number of grains per ear was the result of cross between parents having high X low general combining ability, indicating additive X dominance type of gene interaction. For number of effective tillers, high sca was produced by low x low general combiners, indicating dominance x dominance gene interaction.Part of a Ph. D. thesis submitted by senior author to Haryana Agricultural University, Hissar.     

Genetic analysis of phytic acid content in pearl millet

Summary The genetics of phytic acid content in pearl millet (Pennisettum typhoides (Burm) S & H) was studied using a 12 parent diallel. The analysis of variance of diallel progenies exhibited significant genotypic differences. Different analyses, i.e., combining ability analysis, analysis of variance of diallel tables and genetic component analysis, showed that both additive and non additive gene effects were significant. It is suggested that a population improvement is possible by breeding for low phytic acid cultivars of pearl millet.     

A comparison between the full diallel cross and the simplified triple-test cross

Summary The simplified triple-test cross (sTTC) is a mating design that, because of its economic use of the experimental material as compared with other designs, seems very attractive. In theory, its power is almost equal to that of more elaborate designs such as the diallel cross. To evaluate the merits of both designs in a genetic analysis of mouse behavior, the results of a previous replicated 4×4 diallel cross (Crusio and van Abeelen 1986) were reanalyzed as a sTTC. We found that, at least with the fairly low number of strains employed, the sTTC analysis is clearly inferior to the diallel cross. This finding, in combination with some theoretical considerations, leads to the conclusion that the sTTC design is not a very useful one for such studies.     

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.     

Analysis of Triangular Diallel Table for Experiments in Block Designs

In this paper the analysis of genotypes obtained in diallel crossing system, including one set of F₁′s and parents, arranged in triangular table is given. This analysis is presented for data obtained from the experiments laid out in block designs. The analysis of variance, estimators of general and specific combining abilities as well as statistics for testing of hypotheses concerning those parameters are given.     

Inheritance of seed yield and quality traits in peas (Pisum sativum L.)

Summary A half diallel analysis involving nine cultivars showed that additive as well as non-additive gene effects were important for the inheritance of seed yield per plant, 100-seed weights, protein content and potassium per cent. For remaining traits non-additive genetic components were important. Overdominance was observed for all traits except for 100-seed weight, which expressed partial dominance. Parents PMR-T10, EC21857, EC109182, T163 and EC109189 were good general combiners for seed yield, seed weight and quality traits. In general there was a good relationship between per se performance and the gca effects of the parents for all traits. Cross combinations such as LMR8 x EC109182,LMR8 x PMR-T10,LMR8 x EC21857,PMRT10 x EC21857 and P23 x EC21857 were found promising. The seed yield was positively correlated with other quality traits. Protein had a positive correlation with methionine and phosphorus. All the values of correlation co-efficients were non-significant except for yield with potassium, 100-seed weight and protein with methionine, indicating that yield and quality attributes can be improved simultaneously by simple selection procedures.     

The Use of Combining Ability Analysis to Identify Elite Parents for Artemisia annua F1 Hybrid Production

Artemisia annua is an important medicinal crop used for the production of the anti-malarial compound artemisinin. In order to assist in the production of affordable high quality artemisinin we have carried out an A. annua breeding programme aimed at improving artemisinin concentration and biomass. Here we report on a combining ability analysis of a diallel cross to identify robust parental lines for hybrid breeding. The parental lines were selected based on a range of phenotypic traits to encourage heterosis. The general combining ability (GCA) values for the diallel parental lines correlated to the positive alleles of quantitative trait loci (QTL) in the same parents indicating the presence of beneficial alleles that contribute to parental performance. Hybrids generated from crossing specific parental lines with good GCA were identified as having an increase in both artemisinin concentration and biomass when grown either in glasshouse or experimental field trials and compared to controls. This study demonstrates that combining ability as determined by a diallel cross can be used to identify elite parents for the production of improved A. annua hybrids. Furthermore, the selection of material for breeding using this approach was found to be consistent with our QTL-based molecular breeding approach.     

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.     

Combining ability of tropical maize lines for seed quality and agronomic traits

Studies of genetic effects of early selection of maize based on seed quality traits are rare, especially those that use materials from different heterotic groups. These studies are also useful in tropical environments and for the advancement of sustainable agriculture with cropping during seasons not commonly used for cultivation. We estimated, through diallel crosses, the predominant genetic effects on the expression of agronomic traits and seed quality and on the general combining ability of nine maize lines from commercial hybrids and the specific combining ability of hybrid combinations among them. In the evaluation of seed quality, seven tests were used: first count and final count of seed germination, seedling vigor classification, cold tolerance, seedling emergence rate in a sand seedbed, speed of emergence in a sand seedbed, and speed of emergence index. Plant height, first ear height and grain yield were the estimated agronomic traits. In the diallel analysis, method 3 (model I) proposed by Griffing was used. There was a greater significance of non-additive genetic effects in the genetic control of seed quality of the various lines. The Flash, Dekalb 350 and P 30F80 lines combined high seed quality and high grain yield. For growth during the normal planting season, the combinations CD 3121-1 x P 30F80, Speed x CD 3121-2, Dow 8330 x AG 8080 and Dekalb 350 x CD 3121-2 were the most promising for both seed quality and agronomic traits.     

作物杂种后代基因型值和杂种优势的预测方法

本文提出了利用作物亲本和F_1预测杂种后代基因型值和杂种优势的统计分析方法.该方法运用加性-显性遗传模型,分析双列杂交试验资料,用MINQUE(1)法估算方差分量以及预测遗传效应值.由加性和显性效应预测值可进一步预测F_1,F_2,BC_1,BC_2,等不同世代的基因型值,在预测F_1群体平均优势和群体超亲优势的基础上,可以推导出其它各世代的杂种优势.提出了预测杂种后代保持超亲优势世代数的简单公式,根据杂交组合F_1群体平均优势和双亲相对遗传差异,便可预测该组合能在生产上直接利用的世代数.以棉花六个品种完全双列杂交试验资料为例,分析了各组合F_1和F_2的基因型值、超亲优势和保持5％超亲优势的世代数.     

Analysis of Diallel Table for Experiments Carried Out in BIB Designs-Mixed Model

In this paper the analysis of genotypes obtained in a diallel crossing system, including parents, one set of F₁′s and reciprocal F₁′s, is given. This analysis is presented for data obtained from the experiments carried out in a block design in which block effects are random. The analysis of variance, estimators of reciprocal effects, estimators of general and specific combining abilities are considered. Also statistics for testing hypotheses concerning some genetical parameters are given.     

Mathematical expressions useful in the construction, description and evaluation of protein libraries

The creation of protein libraries by random mutagenesis and cassette mutagenesis has proven to be a successful method of protein engineering. Appropriate statistical analysis is important for the proper construction of these libraries and even more important for the interpretation of data from these libraries. We present simple mathematical expressions useful in the creation and evaluation of such libraries. These equations are useful in estimating the distribution of mutations, the degeneracy of the library and the frequency of a particular clone in the library. In addition, general equations addressing the probability that a particular clone is in a library, the probability that a library is complete, and as the consequences of retransformation of the library on these probabilities are presented.     

Analysis of Diallel Table for Experiments in Block Designs

In this paper the analysis of genotypes obtained in diallel crossing system, including parents, one set of F₁s and reciprocal F₁'s, is given. This analysis is presented for data obtained from the experiments laid out in block designs. The analysis of variance, estimator of reciprocal effects, estimators of general and specific combining abilities as well as statistics for testing of hypotheses concerning those parameters are given.     

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.