Genetic evaluation by best linear unbiased prediction using marker and trait information in a multibreed population.

Genetic evaluation by best linear unbiased prediction (BLUP) requires modeling genetic means, variances, and covariances. This paper presents theory to model means, variances, and covariances in a multibreed population, given marker and breed information, in the presence of gametic disequilibrium between the marker locus (ML) and linked quantitative trait locus (MQTL). Theory and algorithms are presented to construct the matrix of conditional covariances between relatives (Gv) for the MQTL effects in a multibreed population and to obtain the inverse of Gv efficiently. Theory presented here accounts for heterogeneity of variances among pure breeds and for segregation variances between pure breeds. A numerical example was used to illustrate how the theory and algorithms can be used for genetic evaluation by BLUP using marker and trait information in a multibreed population.     

Alternative partitioning of the genotype-by-environment interaction

Summary Alternative methods for partitioning the genotype-by-environment interaction, for an arbitrary number of genotypes or environments, were examined. Partitioning of the interaction is important in order to determine the nature of the interaction. Two methods of partitioning were examined; both separated the interaction into two types: (1) due to heterogeneous variances or (2) due to imperfect correlations. Method 1 was based on heterogeneity among environments in the scaling of differences among genotypes. Method 2 was based on heterogeneity among genotypes in the scaling of differences among environments. Any remaining interaction arises from deviations from the perfect positive correlation of genotypic rankings among environments (Method 1) or of environmental rankings among genotypes (Method 2). Method 1 is more appropriate for random genotypes that are to be tested in either fixed or random environments. With Method 1, the interactions that arise mainly from heterogeneity of genotypic scaling among environments are generally unimportant. However, if environments are fixed and interactions are mainly due to imperfect correlations of rankings, specialized lines may be indicated for each environment. Method 2 is more useful in evaluating fixed genotypes for sensitivity to random environments. A partitioning of the interaction into that due to the type of interaction within each genotype was shown to be useful in that situation.Journal Paper No. 123737 of the Purdue University Agricultural Experiment Station     

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)     

Developmental interactions and the constituents of quantitative variation

Development is the process by which genotypes are transformed into phenotypes. Consequently, development determines the relationship between allelic and phenotypic variation in a population and, therefore, the patterns of quantitative genetic variation and covariation of traits. Understanding the developmental basis of quantitative traits may lead to insights into the origin and evolution of quantitative genetic variation, the evolutionary fate of populations, and, more generally, the relationship between development and evolution. Herein, we assume a hierarchical, modular structure of trait development and consider how epigenetic interactions among modules during ontogeny affect patterns of phenotypic and genetic variation. We explore two developmental models, one in which the epigenetic interactions between modules result in additive effects on character expression and a second model in which these epigenetic interactions produce nonadditive effects. Using a phenotype landscape approach, we show how changes in the developmental processes underlying phenotypic expression can alter the magnitude and pattern of quantitative genetic variation. Additive epigenetic effects influence genetic variances and covariances, but allow trait means to evolve independently of the genetic variances and covariances, so that phenotypic evolution can proceed without changing the genetic covariance structure that determines future evolutionary response. Nonadditive epigenetic effects, however, can lead to evolution of genetic variances and covariances as the mean phenotype evolves. Our model suggests that an understanding of multivariate evolution can be considerably enriched by knowledge of the mechanistic basis of character development.     

On a method of estimating the genetic correlation between characters measured in different experimental units

Summary Yamada's method of estimating genetic co-variances between traits measured in different experimental units is discussed. It is shown that if the data are unbalanced, this method gives biased estimates of genetic covariances unless the traits have identical genetic and residual variances. An alternative unbiased procedure is suggested.     

Detecting epistatic genetic variance with a clonally replicated design: models for lowvs high-order nonallelic interaction

A quantitative genetic model, that uses known family structure with clonal replicates to separate genetic variance into its additive, dominance and epistatic components, is available in the current literature. Making use of offspring testing, this model is based on the theory that components of variance from the linear model of an experimental design may be expressed in terms of expected covariances among relatives. However, if interactions between a pair of quantitative trait loci (QTLs) explain a large proportion of the total epistasis, it will seriously overestimate the additive and dominance variances but underestimate the epistatic variance. In the present paper, a new model is developed to manipulate this problem by combining parental and offspring material into the same test. Under the condition described above, the new model can provide an accurate estimate for additive x additive variances. Also, its accuracy in estimating dominance and total epistatic variances is much greater than the accuracy of the previous model. However, if there is obvious evidence showing the major contribution of high-order interactions, especially among 4QTLs, to the total epistasis, the previous model is more appropriate to partition the genetic variance for a quantitative trait. The re-analysis of an example from a factorial mating design in poplar shows large differences in estimating variance components between the new and previous models when two different assumptions (lowvs high-order epistatic interactions) are used. The new model will be an alternative to estimating the mode of quantitative inheritance for species, especially for longlived, predominantly outcrossing forest trees, that can be clonally replicated.     

Modelling the growth curve of Maine-Anjou beef cattle using heteroskedastic random coefficients models

A heteroskedastic random coefficients model was described for analyzing weight performances between the 100th and the 650th days of age of Maine-Anjou beef cattle. This model contained both fixed effects, random linear regression and heterogeneous variance components. The objective of this study was to analyze the difference of growth curves between animals born as twin and single bull calves. The method was based on log-linear models for residual and individual variances expressed as functions of explanatory variables. An expectation-maximization (EM) algorithm was proposed for calculating restricted maximum likelihood (REML) estimates of the residual and individual components of variances and covariances. Likelihood ratio tests were used to assess hypotheses about parameters of this model. Growth of Maine-Anjou cattle was described by a third order regression on age for a mean growth curve, two correlated random effects for the individual variability and independent errors. Three sources of heterogeneity of residual variances were detected. The difference of weight performance between bulls born as single and twin bull calves was estimated to be equal to about 15 kg for the growth period considered.     

Longitudinal analysis of pre- and post-treatment measurements with equal baseline assumptions in randomized trials

For continuous variables of randomized controlled trials, recently, longitudinal analysis of pre- and posttreatment measurements as bivariate responses is one of analytical methods to compare two treatment groups. Under random allocation, means and variances of pretreatment measurements are expected to be equal between groups, but covariances and posttreatment variances are not. Under random allocation with unequal covariances and posttreatment variances, we compared asymptotic variances of the treatment effect estimators in three longitudinal models. The data-generating model has equal baseline means and variances, and unequal covariances and posttreatment variances. The model with equal baseline means and unequal variance–covariance matrices has a redundant parameter. In large sample sizes, these two models keep a nominal type I error rate and have high efficiency. The model with equal baseline means and equal variance–covariance matrices wrongly assumes equal covariances and posttreatment variances. Only under equal sample sizes, this model keeps a nominal type I error rate. This model has the same high efficiency with the data-generating model under equal sample sizes. In conclusion, longitudinal analysis with equal baseline means performed well in large sample sizes. We also compared asymptotic properties of longitudinal models with those of the analysis of covariance (ANCOVA) and t-test.     

Effects of competition on means,variances and covariances in quantitative genetics with an application to general combining ability selection

Summary General expressions are established for means, variances and covariances, taking into account the effect of interactions between individuals. For simplicity, only groups of size two are considered so that interactions are restricted to pairs of individuals; genetic effects are reduced to additivity and to additive × additive interaction between the direct effect of an allele and the associate effect of another. The effect of family structure is studied, and some consequences of interest to plant breeding are discussed by an application to general combining ability selection. From a population genetic point of view, this approach reveals a new application of coefficients of kinship.     

Inheritance of body size in the Barnacle Goose under different environmental conditions

Heritabilities, genetic variances and covariances for body size traits, i.e. tarsus length, head length and body mass, were estimated under different environmental conditions in a Barnacle Goose (Branta leucopsis) population. Under poor growth conditions, that is, when average body size of fully grown offspring in a given cohort was small, the offspring-parent regressions and full-sib analyses yielded heritability estimates not significantly different from zero. By contrast, when growth conditions were normal or good the heritability estimates were generally significantly positive. Comparisons of genetic covariance estimates indicated that they also differed across the analysed environmental conditions. This result, together with similar results obtained in studies of passerine birds, suggests that genotype-environment interactions might be frequent within the range of environments normally encountered by birds in natural populations. If general, such results might question the validity of assuming approximate constancy of additive genetic variances and covariances over time and environments in evolutionary models.     

Coevolutionary Constraints? The Environment Alters Tripartite Interaction Traits in a Legume

Third party species, which interact with one or both partners of a pairwise species interaction, can shift the ecological costs and the evolutionary trajectory of the focal interaction. Shared genes that mediate a host's interactions with multiple partners have the potential to generate evolutionary constraints, making multi-player interactions critical to our understanding of the evolution of key interaction traits. Using a field quantitative genetics approach, we studied phenotypic and genetic correlations among legume traits for rhizobium and herbivore interactions in two light environments. Shifts in plant biomass allocation mediated negative phenotypic correlations between symbiotic nodule number and herbivory in the field, whereas positive genetic covariances suggested shared genetic pathways between nodulation and herbivory response. Trait variance-covariance (G) matrices were not equal in sun and shade, but nevertheless responses to independent and correlated selection are expected to be similar in both environments. Interactions between plants and aboveground antagonists might alter the evolutionary potential of traits mediating belowground mutualisms (and vice versa). Thus our understanding of legume-rhizobium genetics and coevolution may be incomplete without a grasp of how these networks overlap with other plant interactions.     

Genotype x environment interaction of crossover type: detecting its presence and estimating the crossover point

Genotype-environment interaction (GEI) introduces inconsistency in the relative rating of genotypes across environments and plays a key role in formulating strategies for crop improvement. GEI can be either qualitative (i.e., crossover type) or only quantitative (i.e., non-crossover type). Since the presence of crossover-type interaction has a strong implication for breeding for specific adaptation, it is important to assess the frequency of crossover interactions. This paper presents a test for detecting the presence of crossover-type interaction using the response-environment relationship and enumerates the frequency of crossovers and estimation of the crossover point (CP) on the environment axis, which serves as a cut-off point for the two environments groups where different/specific selections can be made. Sixty-four barley lines with various selection histories were grown in northern Syria and Lebanon giving a total of 21 environments (location-year combinations). Linear regression of the genotypic response on the environmental index represented a satisfactory model, and heterogeneity among regressions was significant. At a 5% level of significance, 38% and 19% of the pairs showed crossover interactions when the error variances were considered heterogeneous and homogeneous, respectively, implying that an appreciable number of crossovers took place in the case of barley lines responding to their environments. The CP of 1.64 t/ha, obtained as the CP of regression lines between the genotype numbers 19 and 31, provided maximum genotype x environment-group interaction. Across all environments, genotype nos. 59 and 12 stood first and second for high yield, respectively. The changes in the ranks of genotypes under the groups of environments can be used for selecting specifically adapted genotypes. Received: 25 January 1999 / Accepted: 16 March 1999     

Nonlinear selection and the evolution of variances and covariances for continuous characters in an anole

The pattern of genetic variances and covariances among characters, summarized in the additive genetic variance‐covariance matrix, G , determines how a population will respond to linear natural selection. However, G itself also evolves in response to selection. In particular, we expect that, over time, G will evolve correspondence with the pattern of multivariate nonlinear natural selection. In this study, we substitute the phenotypic variance‐covariance matrix ( P ) for G to determine if the pattern of multivariate nonlinear selection in a natural population of Anolis cristatellus, an arboreal lizard from Puerto Rico, has influenced the evolution of genetic variances and covariances in this species. Although results varied among our estimates of P and fitness, and among our analytic techniques, we find significant evidence for congruence between nonlinear selection and P , suggesting that natural selection may have influenced the evolution of genetic constraint in this species.     

Disequilibrium among several linked neutral genes in finite population. II. Variances and covariances of disequilibria

A method is derived for computing the variances and covariances of linkage disequilibria between neutral genes in finite populations, which is based on a linear transformation of results given previously for the mean values of disequilibria. The formulae obtained are limited to moments of sixth order or less, such as the variance of the three-locus disequilibrium. It is shown that there is no covariance between any pair of disequilibria in populations starting equilibrium. The pattern of change with time in variance of the three-locus disequilibrium from populations initially in equilibrium is similar to that for two loci, except that the highest values are achieved rather earlier and are smaller.     

Heterogeneity of the blood pressure distribution among Solomon Islands societies with increasing acculturation

This study illustrates the very complex nature of gene by environmental interactions influencing the blood pressure (BP) distribution in a series of genetically distinctive populations undergoing rapid acculturation. We report the results of two BP and anthropometric surveys on Solomon Islands societies separated by an interval of 14 to 19 years. While differences in acculturation existed at the time of the initial survey, the interval between surveys was marked by rapid acculturation in almost all societies. Seven of the eight societies originally covered were included in the resurvey, and a large but variable proportion of the original sample subjects was recovered in the follow-up. Because the genetic relationships of the societies have been described, we were able to establish the following points concerning the role of genetic differences in determining the distribution of BP among these populations and, more important, the interaction of these genetic differences with changes associated with increasing acculturation: 1) In the initial survey, mean adjusted systolic and diastolic BPs were significantly heterogeneous among societies within and among genetically related clusters of societies (genetic clusters) and sexes. At the same time, rank differences in these means were not associated with rank differences in acculturation status among societies ignoring cluster membership. 2) Importantly, in the follow-up survey increasing acculturation resulted in the disappearance of significant differences in mean systolic and diastolic BP among genetic clusters in males, despite continued significant heterogeneity among societies within genetic clusters. In females, differences among genetic clusters persisted, but the degree of significance was substantially less with increasing acculturation. We interpret these changes as evidence for genotype by environment interaction. 3) There were significant differences in interindividual variances of both systolic and diastolic BPs among genetic clusters in the first survey. Ranks of these variances were not significantly associated with acculturation rank. In the follow-up survey, however, most societies showed striking increases in the variance of both systolic and diastolic BPs with increased acculturation. These increases in variance of both systolic and diastolic BPs may be related to a) shifts in demography and/or anthropometry of some societies; b) increased range and intensity of environmental factors affecting BP and associated with increased acculturation; and/or c) genotype by environmental interactions. 4) The correlation between systolic and diastolic BP decreased over the interval for all societies within and among genetic clusters. This trend was partly the result of larger changes in variances for systolic than diastolic BP in the resurveys.(ABSTRACT TRUNCATED AT 400 WORDS)     

Protein deprivation facilitates the independent evolution of behavior and morphology

Ecological conditions such as nutrition can change genetic covariances between traits and accelerate or slow down trait evolution. As adaptive trait correlations can become maladaptive following rapid environmental change, poor or stressful environments are expected to weaken genetic covariances, thereby increasing the opportunity for independent evolution of traits. Here, we demonstrate the differences in genetic covariance among multiple behavioral and morphological traits (exploration, aggression, and body weight) between southern field crickets (Gryllus bimaculatus) raised in favorable (free‐choice) versus stressful (protein‐deprived) nutritional environments. We also quantify the extent to which differences in genetic covariance structures contribute to the potential for the independent evolution of these traits. We demonstrate that protein‐deprived environments tend to increase the potential for traits to evolve independently, which is caused by genetic covariances that are significantly weaker for crickets raised on protein‐deprived versus free‐choice diets. The weakening effects of stressful environments on genetic covariances tended to be stronger in males than in females. The weakening of the genetic covariance between traits under stressful nutritional environments was expected to facilitate the opportunity for adaptive evolution across generations. Therefore, the multivariate gene‐by‐environment interactions revealed here may facilitate behavioral and morphological adaptations to rapid environmental change.     

Effect of gene conversion on variances of digenic identity measures

The variances and covariances of digenic descent measures are studied for a two-locus model incorporating mutation, gene conversion, recombination, drift, and finite sampling. Gene conversion can occur between allelic pairs of genes or between non-allelic pairs on the same or different gametes within individuals. Most interest therefore centers on pairs of genes, and five digenic identity measures are required. The behavior over time of these measures is studied, with an emphasis on the effects of gene conversion. Because of the stochastic nature of the forces of drift, recombination, mutation, and conversion, the actual identity status of gene pairs can vary from expectation among replicate populations. To study this variation we compute the expected variances and covariances of the measures, and show that this requires the introduction of trigenic and quadrigenic measures. Allowing for conversion between genes on different gametes requires a large number of these higher-order measures.     

Evolutionary potential of a widespread clonal grass under changing climate

Adaptive responses are probably the most effective long‐term responses of populations to climate change, but they require sufficient evolutionary potential upon which selection can act. This requires high genetic variance for the traits under selection and low antagonizing genetic covariances between the different traits. Evolutionary potential estimates are still scarce for long‐lived, clonal plants, although these species are predicted to dominate the landscape with climate change. We studied the evolutionary potential of a perennial grass, Festuca rubra, in western Norway, in two controlled environments corresponding to extreme environments in natural populations: cold–dry and warm–wet, the latter being consistent with the climatic predictions for the country. We estimated genetic variances, covariances, selection gradients and response to selection for a wide range of growth, resource acquisition and physiological traits, and compared their estimates between the environments. We showed that the evolutionary potential of F. rubra is high in both environments, and genetic covariances define one main direction along which selection can act with relatively few constraints to selection. The observed response to selection at present is not sufficient to produce genotypes adapted to the predicted climate change under a simple, space for time substitution model. However, the current populations contain genotypes which are pre‐adapted to the new climate, especially for growth and resource acquisition traits. Overall, these results suggest that the present populations of the long‐lived clonal plant may have sufficient evolutionary potential to withstand long‐term climate changes through adaptive responses.     

On selection criteria and estimation of parameters when the variance is heterogeneous

Summary Procedures for ranking candidates for selection and for estimating genetic and environmental parameters when variances are heterogeneous are discussed. The best linear unbiased predictor (BLUP) accounts automatically for heterogeneous variance provided that the covariance structure is known and that the assumptions of the model hold. Under multivariate normality BLUP allowing for heterogeneous variance maximizes expected genetic progress. Examples of application of BLUP to selection when residual or genetic variances are heterogeneous are given. Restricted maximum likelihood estimation of heterogeneous variances and covariances via the expectation-maximization algorithm is presented.     

The calculation and significance testing of genetic correlations across environments

Genetic correlations within a trait across environments (r_g) are important in the analysis of phenotypic plasticity. Not all methods are, however, equally reliable. An overview of all different methods for estimation of r_g with one generation data sets is given. Formulae for the relationship between causal variance components and family means are derived. When these formulae are used covariances derived from family means, thought to be incorrect, are exactly the same as those derived with the ANOVA method. The bias, precision and power of the different methods are compared with Monte Carlo simulations. For all methods bias is small and precision is high for the large balanced data sets analyzed here, except when the variance in one or both of the environments is close to 0. Significance testing causes more problems. Confidence intervals with or without z-transformation are not suitable for testing, nor is testing for g*e interaction in an ANOVA suitable for testing whether the r_g is different from 1. The F-test in a mixed model ANOVA and a likelihood ratio test in a REML-analysis can be used for testing a difference from 0 but not from 1 or other values. Jackknife and Bootstrap, however, are suitable tests both for differences with 0,1 and other values, though negative variances can make these tests difficult to apply.