Estimating effects of a single gene and polygenes on quantitative traits from a diallel design

A genetic model is developed with additive and dominance effects of a single gene and polygenes as well as general and specific reciprocal effects for the progeny from a diallel mating design. The methods of ANOVA, minimum norm quadratic unbiased estimation (MINQUE), restricted maximum likelihood estimation (REML), and maximum likelihood estimation (ML) are suggested for estimating variance components, and the methods of generalized least squares (GLS) and ordinary least squares (OLS) for fixed effects, while best linear unbiased prediction, linear unbiased prediction (LUP), and adjusted unbiased prediction are suggested for analyzing random effects. Monte Carlo simulations were conducted to evaluate the unbiasedness and efficiency of statistical methods involving two diallel designs with commonly used sample sizes, 6 and 8 parents, with no and missing crosses, respectively. Simulation results show that GLS and OLS are almost equally efficient for estimation of fixed effects, while MINQUE (1) and REML are better estimators of the variance components and LUP is most practical method for prediction of random effects. Data from a Drosophila melanogaster experiment (Gilbert 1985a, Theor appl Genet 69:625–629) were used as a working example to demonstrate the statistical analysis. The new methodology is also applicable to screening candidate gene(s) and to other mating designs with multiple parents, such as nested (NC Design I) and factorial (NC Design II) designs. Moreover, this methodology can serve as a guide to develop new methods for detecting indiscernible major genes and mapping quantitative trait loci based on mixture distribution theory. The computer program for the methods suggested in this article is freely available from the authors.     

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     

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.     

Optimum spacing of genetic markers for determining linkage between marker loci and quantitative trait loci

The cost of experiments aimed at determining linkage between marker loci and quantitative trait loci (QTL) was investigated as a function of marker spacing and number of individuals scored. It was found that for a variety of experimental designs, fairly wide marker spacings (ca. 50 cM) are optimum or close to optimum for initial studies of marker-QTL linkage, in the sense of minimizing overall cost of the experiment. Thus, even when large numbers of more or less evenly spaced markers are available, it will not always be cost effective to make full utilization of this capacity. This is particularly true when costs of rearing and trait evaluation per individual scored are low, as when marker data are obtained on individuals raised and evaluated for quantitative traits as part of existing programs. When costs of rearing and trait evaluation per individual scored are high, however, as in human family data collection carried out primarily for subsequent marker — QTL analyses, or when plants or animals are raised specifically for purposes of marker — QTL linkage experiments, optimum spacing may be rather narrow. It is noteworthy that when marginal costs of additional markers or individuals are constant, total resources allocated to a given experiment will determine total number of individuals sampled, but not the optimal marker spacing.     

A note on joint versus gene-specific mixed model analysis of microarray gene expression data

Currently, linear mixed model analyses of expression microarray experiments are performed either in a gene-specific or global mode. The joint analysis provides more flexibility in terms of how parameters are fitted and estimated and tends to be more powerful than the gene-specific analysis. Here we show how to implement the gene-specific linear mixed model analysis as an exact algorithm for the joint linear mixed model analysis. The gene-specific algorithm is exact, when the mixed model equations can be partitioned into unrelated components: One for all global fixed and random effects and the others for the gene-specific fixed and random effects for each gene separately. This unrelatedness holds under three conditions: (1) any gene must have the same number of replicates or probes on all arrays, but these numbers can differ among genes; (2) the residual variance of the (transformed) expression data must be homogeneous or constant across genes (other variance components need not be homogeneous) and (3) the number of genes in the experiment is large. When these conditions are violated, the gene-specific algorithm is expected to be nearly exact.     

Optimal designs for estimating and testing interaction among multiple loci in complex traits by a Gibbs sampler

A simulation study was conducted to provide a practical guideline for experimental designs with the Bayesian approach using Gibbs sampling (BAGS), a recently developed method for estimating interaction among multiple loci. Various data sets were simulated from combinations of number of loci, within-genotype variance, sample size, and balance of design. Mean square prediction error (MSPE) and empirical statistical power were obtained from estimating and testing the posterior mean estimate of combination genotypic effect. Simultaneous use of both MSPE and power was suggested to find an optimal design because their correlation estimate (− 0.8) would not be large enough to ignore either of them. The optimal sample sizes with MSPE > 2.0 and power > 0.8 with the within-genotype variance of 30 were 135, 675, and > 8100 for 2-, 3-, and 4-locus unbalanced data. The BAGS was suggested for interaction effects among limited number (4 or less) of loci in practice. A practical guideline for determining an optimal sample size with a given power or vise versa is provided for BAGS.     

A general mixture model for mapping quantitative trait loci by using molecular markers

Summary In a segregating population a quantitative trait may be considered to follow a mixture of (normal) distributions, the mixing proportions being based on Mendelian segregation rules. A general and flexible mixture model is proposed for mapping quantitative trait loci (QTLs) by using molecular markers. A method is discribed to fit the model to data. The model makes it possible to (1) analyse non-normally distributed traits such as lifetimes, counts or percentages in addition to normally distributed traits, (2) reduce environmental variation by taking into account the effects of experimental design factors and interaction between genotype and environment, (3) reduce genotypic variation by taking into account the effects of two or more QTLs simultaneously, (4) carry out a (combined) analysis of different population types, (5) estimate recombination frequencies between markers or use known marker distances, (6) cope with missing marker observations, (7) use markers as covariables in detection and mapping of QTLs, and finally to (8) implement the mapping in standard statistical packages.     

Heterosis for chiasma frequency and quantitative traits in common beans (Phaseolus vulgaris L.)

Summary Ten genotypes, including inbreds, hybrids, and advanced populations, were examined in order to elucidate the relationship between position and frequency distribution of chiasmata and quantitative traits, including yield heterosis in common beans. The hybrid and advanced population groups were determined to possess 83% and 54% increased chiasma frequency, respectively in contrast to inbred lines. The increase in chiasma frequency of these populations was further manifested in a high number of interstitial chiasmata. The regular and superior chromosome behaviour of the hybrids was found to be positively associated with quantitative measures on bean yield, harvest index and bean yield efficiency. The results were discussed from the point of view that: a) increased interstitial chiasmata may provide an effective mechanism for maintaining genetic diversity and heterosis in hybrid populations; and b) heterosis for chiasma frequency and quantitative traits may be due to dispersed genes on the chromosomes having combined intra-and interallelic interactions. The data provide evidence for the existence of positive associations between interstitially localized chiasmata with its recombination potential and regular chromosome behaviour to bean yield heterosis. The role of enhanced interstitial chiasmata to promote higher levels of genetic variation and heterozygous advantage is discussed.     

The effects of selective genotyping on estimates of proportion of recombination between linked quantitative trait loci

Selective genotyping is the marker assay of only the more extreme phenotypes for a quantitative trait and is intended to increase the efficiency of quantitative trait loci (QTL) mapping. We show that selective genotyping can bias estimates of the recombination frequency between linked QTLs — upwardly when QTLs are in repulsion phase, and downwardly when QTLs are in coupling phase. We examined these biases under simple models involving two QTLs segregating in a backcross or F₂ population, using both analytical models and computer simulations. We found that bias is a function of the proportion selected, the magnitude of QTL effects, distance between QTLs and the dominance of QTLs. Selective genotyping thus may decrease the power of mapping multiple linked QTLs and bias the construction of a marker map. We suggest a large proportion than previously suggested (50%) or the entire population be genotyped if linked QTLs of large effects (explain > 10% phenotypic variance) are evident. New models need to be developed to explicitly incorporate selection into QTL map construction.     

A model for marker-assisted selection among single crosses with multiple genetic markers

 Trait means of marker genotypes are often inconsistent across experiments, thereby hindering the use of regression techniques in marker-assisted selection. Best linear unbiased prediction based on trait and marker data (TM-BLUP) does not require prior information on the mean effects associated with specific marker genotypes and, consequently, may be useful in applied breeding programs. The objective of this paper is to present a flanking-marker, TM-BLUP model that is applicable to interpopulation single crosses that characterize maize (Zea mays L.) breeding programs. The performance of a single cross is modeled as the sum of testcross additive and dominance effects at unmarked quantitative trait loci (QTL) and at marked QTL (MQTL). The TM-BLUP model requires information on the recombination frequencies between flanking markers and the MQTL and on MQTL variances. A tabular method is presented for calculating the conditional probability that MQTL alleles in two inbreds are identical by descent given the observed marker genotypes (G ^k _obs) at the kth MQTL. Information on identity by descent of MQTL alleles can then be used to calculate the conditional covariance of MQTL effects between single crosses given G ^k _obs. The inverse of the covariance matrix for dominance effects at unmarked QTL and MQTL can be written directly from the inverse of the covariance matrices of the corresponding testcross additive effects. In practice, the computations required in TM-BLUP may be prohibitive. The computational requirements may be reduced with simplified TM-BLUP models wherein dominance effects at MQTL are excluded, only the single crosses that have been tested are included, or information is pooled across several MQTL. Received: 22 June 1997 / Accepted: 25 February 1998     

A general model for detecting genetic determinants underlying longitudinal traits with unequally spaced measurements and nonstationary covariance structure

A mixture model for determining quantitative trait loci (QTL) affecting growth trajectories has been proposed in the literature. In this article, we extend this model to a more general situation in which longitudinal traits for each subject are measured at unequally spaced time intervals, different subjects have different measurement patterns, and the residual correlation within subjects is nonstationary. We derive an EM-simplex hybrid algorithm to estimate the allele frequencies, Hardy-Weinberg disequilibrium, and linkage disequilibrium between QTL in the original population and parameters contained in the growth equation and in the covariance structure. A worked example of head circumference growth in 145 children is used to validate our extended model. A simulation study is performed to examine the statistical properties of the parameter estimation obtained from this example. Finally, we discuss the implications and extensions of our model for detecting QTL that affect growth trajectories.     

Influence of the chain length of ubiquinones on their interaction with DPPC in mixed monolayers

The thermodynamic behavior of representative short (UQ2), middle (UQ4 and UQ6) and long-chain (UQ10) ubiquinones (UQ) mixed with dipalmitoyl-phosphatidylcholine (DPPC) was studied in monolayers at the air-water interface. The influence of isoprenoid chain-length of UQ on miscibility of both lipids was investigated by analysis of surface pressure-area isotherms and using fluorescence microscopy. Analysis of excess areas (A_ex) and free energies of mixing (ΔG_m), calculated from compression isotherms in the full range of ubiquinones concentrations, has given evidences for UQ-rich constant-size (UQ6, UQ10) or less growth limited (UQ2, UQ4) microdomains formation within mixed films. Fluorescence microscopy observation revealed that ubiquinones are preferentially soluble in the expanded phase. When lateral pressure increased, concomitant evolutions of A_ex and ΔG_m parameters, and composition dependence of collapse surface pressures, argue for an evolution towards a total segregation, never reached due to expulsion of ubiquinones from the film. The possible significance of these observations is discussed in relation to ubiquinones organization and similar chain length effects in membranes.     

Identification and estimation of a QTL model and its effects

 A joint segregation analysis of a genetic system and the effects of QTLs based on the six populations P₁, F₁, P₂, B₁, B₂ and F₂ is proposed in this paper. The major steps were as follows. Firstly, under the supposition that the segregating population was composed of component distributions controlled by a major gene(s) and modified by both polygenes and environments, four groups and 17 types of genetic models, including a one major-gene model, a two major-gene model, a polygene model, and a mixed one-major gene and polygene model, were set up. Secondly, the joint maximum-likelihood function was constructed from the six generations so as to estimate the parameters of component distributions through an EM algorithm. Thirdly, the best-fitting genetic model was chosen according to Akaike’s information criterion, a likelihood-ratio test, and tests for goodness of fit. Fourthly, the related genetic parameters, including gene effects, as well as the genetic variances of major genes and polygenes, were obtained from the estimates of component distributions. Finally, the individuals in segregating populations were classified into their major-gene genotypes according to their posterior probabilities. An example of the genetic analysis of plant height of a rice cross between Nanjing No. 6 and Guangcong was used to illustrate the above procedure. The method was especially appropriate to those crops with easy to obtain hybrid seeds. Received: 11 February 1998 / Accepted: 28 May 1998     

The finite polygenic mixed model: An alternative formulation for the mixed model of inheritance

This paper presents a mixed model of inheritance with a finite number of polygenic loci. This model leads to a likelihood that can be calculated using efficient algorithms developed for oligogenic models. For comparison, likelihood profiles were obtained for the finite polygenic mixed model, the usual mixed model, with exact and approximate calculations, and for a class D regressive model. The profiles for the finite polygenic mixed model were closest to the profiles for the usual mixed model with exact calculations.     

Linkage and association studies of QTL for nuclear families by mixed models

The transmission disequilibrium test (TDT) has been utilized to test the linkage and association between a genetic trait locus and a marker. Spielman et al. (1993) introduced TDT to test linkage between a qualitative trait and a marker in the presence of association. In the presence of linkage, TDT can be applied to test for association for fine mapping (Martin et al., 1997; Spielman and Ewens, 1996). In recent years, extensive research has been carried out on the TDT between a quantitative trait and a marker locus (Allison, 1997; Fan et al., 2002; George et al., 1999; Rabinowitz, 1997; Xiong et al., 1998; Zhu and Elston, 2000, 2001). The original TDT for both qualitative and quantitative traits requires unrelated offspring of heterozygous parents for analysis, and much research has been carried out to extend it to fit for different settings. For nuclear families with multiple offspring, one approach is to treat each child independently for analysis. Obviously, this may not be a valid method since offspring of one family are related to each other. Another approach is to select one offspring randomly from each family for analysis. However, with this method much information may be lost. Martin et al. (1997, 2000) constructed useful statistical tests to analyse the data for qualitative traits. In this paper, we propose to use mixed models to analyse sample data of nuclear families with multiple offspring for quantitative traits according to the models in Amos (1994). The method uses data of all offspring by taking into account their trait mean and variance-covariance structures, which contain all the effects of major gene locus, polygenic loci and environment. A test statistic based on mixed models is shown to be more powerful than the test statistic proposed by George et al. (1999) under moderate disequilibrium for nuclear families. Moreover, it has higher power than the TDT statistic which is constructed by randomly choosing a single offspring from each nuclear family.     

A quantitative genetic model for mixed diploid and triploid hybrid progenies in tree breeding and evolution

Interspecific hybridization has played a critical role in tree evolution and breeding. The findings of triploidy in forest trees stimulate the development of a quantitative genetic model to estimate the nature of gene action. The model is based on clonally replicated triploid progenies derived from a two-level population and individual-within-population mating design in which offspring have a double dose of alleles from the parent and a single dose of alleles from the other parent. With the same genetic assumptions of a diploid model, except non-Mendelian behavior at meiosis, and the experimental variances estimated from a linear statistical model, total genetic variances in the triploid progenies are separated into additive, dominance, and epistatic components. In addition, by combining the new model with the already existing model based on disomic expression, the partitioning of additive, dominant, and epistatic variances can be obtained for a mixed diploid/triploid F₁ progeny population. This paper provides an alternative technique to study the modes of quantitative inheritance for outcrossing, long-lived forest trees in which inbred lines cannot be easily generated. The accuracy for estimating gene action using this technique is discussed.     

The distribution of the effects of genes affecting quantitative traits in livestock

Meta-analysis of information from quantitative trait loci (QTL) mapping experiments was used to derive distributions of the effects of genes affecting quantitative traits. The two limitations of such information, that QTL effects as reported include experimental error, and that mapping experiments can only detect QTL above a certain size, were accounted for. Data from pig and dairy mapping experiments were used. Gamma distributions of QTL effects were fitted with maximum likelihood. The derived distributions were moderately leptokurtic, consistent with many genes of small effect and few of large effect. Seventeen percent and 35% of the leading QTL explained 90% of the genetic variance for the dairy and pig distributions respectively. The number of segregating genes affecting a quantitative trait in dairy populations was predicted assuming genes affecting a quantitative trait were neutral with respect to fitness. Between 50 and 100 genes were predicted, depending on the effective population size assumed. As data for the analysis included no QTL of small effect, the ability to estimate the number of QTL of small effect must inevitably be weak. It may be that there are more QTL of small effect than predicted by our gamma distributions. Nevertheless, the distributions have important implications for QTL mapping experiments and Marker Assisted Selection (MAS). Powerful mapping experiments, able to detect QTL of 0.1σ_p, will be required to detect enough QTL to explain 90% the genetic variance for a quantitative trait.     

Use of the additive main effects and multiplicative interaction model in QTL mapping for adaptation in barley

The additive main effects and multiplicative interaction (AMMI) model has emerged as a powerful analytical tool for genotype x environment studies. The objective of the present study was to assess its value in quantitative trait locus (QTL) mapping. This was done through the analysis of a large two-way table of genotype-by-environment data of barley (Hordeum vulgare L.) grain yields, where the genotypes constituted a genetic population suitable for mapping studies. Grain yield data of 150 doubled haploid lines derived from the Steptoe x Morex cross, and the two parental lines, were taken by the North American Barley Genome Mapping Project (NABGMP) at 16 environments throughout the barley production areas of the USA and Canada. Four regions of the genome were responsible for most of the differential genotypic expression across environments. They accounted for approximately 50% of the genotypic main effect and 30% of the genotype x environment interaction (GE) sums of squares. The magnitude and sign of AMMI scores for genotypes and sites facilitate inferences about specific interactions. The parallel use of classification (cluster analysis of environments) and ordination (principal component analysis of GE matrix) techniques allowed most of the variation present in the genotype x environment matrix to be summarized in just a few dimensions, specifically four QTLs showing differential adaptation to four clusters of environments. Thus, AMMI genotypic scores, when the genotypes constituted a population suitable for QTL mapping, could provide an adequate way of resolving the magnitude and nature of QTL x environment interactions.Ignacio Romagosa was on sabbatical leave from the University of Lleida and the Institut de Recerca i Tecnologia Agroalimentàries, Lleida, Spain, when this study was conducted     

A test of linkage for complex discrete and continuous traits in nuclear families

This article presents a score test for genetic linkage in nuclear families which applies to any trait having a distribution belonging to the exponential family, which includes binary and normal distributions, and distributions which are skewed or have nonnormal kurtosis. The specific distribution need not be specified and the method applies to sibships of arbitrary size. Tests of complex genetic effects are given, including unspecified mode of inheritance or additive, dominant, overdominant, and recessive modes of inheritance, covariates, multiple-locus models, including gene-gene interactions, and gene-environment interactions. The relation of our method to the Haseman-Elston methods is studied theoretically and by simulation.