Effect of gametic disequilibrium on means and on genetic variances of autotetraploid synthetic varieties

Summary The occurrence and effects of a gametic disequilibrium (DSE) in the first generation of a theoretical two-population synthetic variety were investigated. Theoretical development was limited to the genetics at a single locus with two alleles in an autotetraploid species with random chromosome inheritance. Algebraic expressions were developed for the differences between the mean genotypic values of the two-population synthetic variety at generation one and in random mating equilibrium (RME). For the situation where both parents of the synthetic were in RME, a numerical analysis was performed for all possible allele frequencies assuming the following types of genic action: monoplex dominance, partial monoplex dominance, duplex dominance, partial duplex dominance, and additive. The result indicated that with non-additive genic action the DSE could, in some cases, greatly depress or inflate the mean genotypic value of the first generation (Syn-1_(RME)). Thus, any change of means over advancing generations with loss of DSE could be positive or negative. When additive genic action was assumed, there was no effect associated with DSE and when both parents had the same allele frequencies there was no DSE. The DSE, with only a minor exception, decreased the genetic variance and in numerous cases forced it near zero. Expressions were developed for mean genotypic values of a first generation synthetic with DSE in one parent (Syn-1_(DSE/RME)) or both parents (Syn-1_(DSE)). The deviation of these means from those of Syn-1_(RME) was a function of digenic and quadragenic population effects. An inspection of the response equations for Syn-1_(RME) indicated that in a series of crosses with one common parent the rankings of first generation means would be the same as the ranking of populations at equilibrium though the individual means would be biased. More importantly with DSE of one or both parents there are situations when a ranking of first generation mean genotypic values would not reflect relative frequency of desirable alleles in the populations. These results indicate that statistical analyses and selections based on means of the Syn-1 generation can have an error which is not avoidable by improvement in precision of evaluation.Cooperative investigations of the Alfalfa Production Research Unit, United States Department of Agriculture, Agricultural Research Service, and the Nevada Agricultural Experiment Station, Reno, Nevada, USAPaper No. 590 Scientific Journal Series, Nevada Agricultural Experiment Station     

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

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

Theoretical value of estimates of general combining ability in the autotetraploid crop

Summary The means of half-sib progenies have been indicated as selection criteria for intra-population improvement while the average of the means of full-sib progenies in diallel analyses have been proposed as predictors, in part, of the means of untested synthetic varieties. When these measures based on progeny means are expressed as deviations from a defined greater population of crosses, they are often termed the general combining ability (GCA). In this study the GCA estimates or a facsimile were theoretically investigated for the one locus, digene, autotetraploid model to verify the genetic basis and its value for selection and prediction in the presence of a naturally occurring phenomena of autopolyploids called gametic disequilibrium with three types of non-additive inheritance. Two breeding objectives were envisioned, the selection of best parents with recurrent selection based on GCA in the continued development of elite populations and the prediction of advanced generation synthetic variety performance. The first generation means of progenies with a potential bias due to gametic disequilibrium were compared to GCA estimation of same progenies in the absence of gametic disequilibrium. The results indicated that testcrossing plants to a population without gametic disequilibrium could be used for selection of best parents. The gametic disequilibrium in the cross may increase or depress selection response dependent on the array of genotypes which happen to be evaluated, on the type of genic action at the locus, and on the frequency of the desirable allele in the testor population. The GCA estimates for prediction of synthetic performance were potentially biased by gametic disequilibrium. An assumption of pollination by the same array of gametes was made for all plants, but obviously was unrealistic for GCA estimation with partial diallels, or with no selfing, and in other situations. The GCA estimate was shown to be an unreliable predictor of synthetic variety performance. When it was assumed that different plants were pollinated by different arrays of gametes, a more realistic situation, no genetic interpretation of GCA values was possible even with purely additive gene action at the locus.Cooperative investigation of the Alfalfa Production Research Unit, United States Department of Agriculture, Agricultural Research Service, and the Nevada Agricultural Experiment Station, Reno, Nevada     

Theory for modelling means and covariances in a two-breed population with dominance inheritance

This paper presents theory and methods to compute genotypic means and covariances in a two breed population under dominance inheritance, assuming multiple unlinked loci. It is shown that the genotypic mean is a linear function of five location parameters and that the genotypic covariance between relatives is a linear function of 25 dispersion parameters. Recursive procedures are given to compute the necessary identity coefficients. In the absence of inbreeding, the number of parameters for the mean is reduced from five to three and the number for the covariance is reduced from 25 to 12. In a two-breed population, for traits exhibiting dominance, the theory presented here can be used to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood.Supported in part by the Illinois Agricultural Experiment Station, Hatch Projects 35-0345 (R.L.F.) and 35-0367 (M.G.). A computer program implementing the methods described here is available upon request to R.L.F.     

Selection theory for selfed progenies

Summary The purpose of this article was to extend the model used to predict selection response with selfed progeny from 2 alleles per locus to a model which is general for number and frequency of alleles at loci. To accomplish this, 4 areas had to be dealt with: 1) simplification of the derivation and calculation of the condensed coefficients of identity; 2) presentation of the genetic variances expressed among and within selfed progenies as linear function of 5 population parameters; 3) presentation of selection response equations for selfed progenies as functions of these 5 population parameters; and 4) to identify a set of progeny to evaluate, such that one might be able to estimate these 5 population parameters.The five population parameters used in predicting gains were the additive genetic variance, the dominance variance, the covariance of additive and homozygous dominance deviations, the variance of the homozygous dominance deviations and a squared inbreeding depression term.Contribution from the Missouri Agricultural Experiment Station. Journal Series No. 9971     

Is Fisher's model necessary for the theory of population improvement?

Summary It is shown here that genetic advance in one cycle of recurrent selection can be formulated directly in terms of covariances between relatives by application of the general statistical principle of linear prediction. For practical use of such formulae it is necessary to estimate the corresponding covariance between relatives from the mating design used. With General Combining Ability selection such estimation is direct. For other types of selection, it is necessary to derive associated covariances from other types of covariances but it is not necessary to use classical results of covariances between relatives in terms of genetic effects. Indeed, covariances can be derived without factorial decomposition of the genetic effects at one locus, i.e., without the concept of additivity and dominance. This approach allows a simple derivation of the genetic advance after n cycles of selection, followed by m generations of intercrossing, with a minimum of assumptions.     

Covariance between relatives in multibreed populations: additive model

Covariance between relatives in a multibreed population was derived for an additive model with multiple unlinked loci. An efficient algorithm to compute the inverse of the additive genetic covariance matrix is given. For an additive model, the variance for a crossbred individual is a function of the additive variances for the pure breeds, the covariance between parents, and segregation variances. Provided that the variance of a crossbred individual is computed as presented here, the covariance between crossbred relatives can be computed using formulae for purebred populations. For additive traits the inverse of the genotypic covariance matrix given here can be used both to obtain genetic evaluations by best linear unbiased prediction and to estimate genetic parameters by maximum likelihood in multibreed populations. For nonadditive traits, the procedure currently used to analyze multibreed data can be improved using the theory presented here to compute additive covariances together with a suitable approximation for nonadditive covariances.Supported in part by the Illinois Agricultural Experiment Station, Hatch Projects 35-0345 (RLF) and 35-0367 (MG)     

Genetic correlation and response to selection in simulated populations

Summary Effects of truncation selection of a primary trait upon genetic correlation between the primary trait and an unselected secondary trait were observed during 30 generations. Populations were 24 male and 24 female parents per generation randomly mated with replacement, the number of offspring set by intensity of selection. Each trait was controlled by genes with equal effects and complete dominance segregating independently from starting frequencies of 0.5 at each of 48 loci. Three levels each of genetic correlation, selection, and environmental variation were simulated.Genetic correlation decreased faster under more intense selection by lower than by upper truncation but behaved similarly in both by remaining near initial level when as many as one-half of the offspring were saved for parents. Truncation selection decreased genetic correlation in the offspring selected to be parents whether selection was by upper or lower truncation. Estimates of genetic correlation from covariances between phenotypes of parent and offspring were erratic for both directions of selection.Michigan Agricultural Experiment Station Journal Article4841. Part of North Central Regional Project NC-2.     

Genetic covariances within and between species: indirect selection for hybrid inviability

Genotype-environment interactions and natural selection can result in local specialization when different genotypes are favored in different environments. Restricted gene flow or genetic subdivision enhances local genetic diversification across a species when natural selection acts on such variation. The indirect evolution of reproductive isolation and the restriction of gene flow between species in statu nascendi may provide a central role for genotype-environment interactions in speciation genetics. We derive the expected genetic covariance between heterospecific and conspecific viability fitness under several different models of selection, dominance, and breeding structure. Standard quantitative genetic methods can be used to estimate these covariances in experimental studies. These genetic covariances permit us to evaluate in a formal way the indirect effects of selection within a species on the evolution of hybrid inviability between species. We find that, for autosomal loci and random mating, the genetic covariance across species is equal to the product of three quantities: (1) the viability of the best hybrid genotype; (2) the viability effect of an allele in hybrids; and, (3) the change in allele frequency due to selection in the conspecific population. Inbreeding within the conspecific population, expressed as Wright's coefficient, F, increases the genetic covariance by a factor (1 + F). In all cases, a negative genetic covariance across species is evidence for hybrid inviability evolving as an indirect effect of selection within species for adaptive (as opposed to neutral) genetic change. “It is an irony of evolutionary genetics, that although it is a fusion of Mendelism and Darwinism, it has made no direct contribution to what Darwin obviously saw as the fundamental problem: the origin of species…. While it is a question of elementary population genetics to state how many generations will be required for the frequency of an allele to change from q₁ to q₂, we do not know how to incorporate such a statement into speciation theory, in large part because we know virtually nothing about the genetic changes that occur in species formation.” (Lewontin 1974, p. 159)     

Combined S2 and crossbred family selection in full-sib reciprocal recurrent selection

Summary A method (CRRS) that combines S₂ and crossbred family selection in full-sib reciprocal recurrent selection (FSRRS) is proposed. The method requires four generations per cycle in single-eared maize populations. Selection is based on performance of S₂ and full-sib families by applying selection index theory. Equations to estimate the coefficients included in the index are given. These estimates are functions of the genetic and phenotypic variances and covariances among and between the two kinds of families. Comparisons of FSRRS and CRRS under equivalent amount of effort show that CRRS has some advantage over FSRRS for low heritability of the trait being selected (e.g., maize yield) and when only one or two locations with two replications are involved in the selection experiment.Joint contribution: Institute Nacional de Investigaciones Agrarias, La Coruna, Spain; and Agricultural Research, Science and Education Administration, U.S. Department of Agriculture, and Journal Paper No. J-10118 of the Iowa Agriculture and Home Economics Experiment Station, Ames, IA 50011. Project 2194     

RESISTANCE OF GENETIC CORRELATION STRUCTURE TO DIRECTIONAL SELECTION IN DROSOPHILA MELANOGASTER

The genetic covariance and correlation matrices for five morphological traits were estimated from four populations of fruit flies, Drosophila melanogaster, to measure the extent of change in genetic covariances as a result of directional selection. Two of the populations were derived from lines that had undergone selection for large or small thorax length over the preceding 23 generations. A third population was constituted using flies from control lines that were maintained with equivalent population sizes as the selected lines. The fourth population contained flies from the original cage population from which the selected and control lines had been started. Tests of the homogeneity of covariance matrices using maximum likelihood techniques revealed significant changes in covariance structure among the selected lines. Prediction of base population trait means from selected line means under the assumption of constant genetic covariances indicated that genetic covariances for the small population differed more from the base population than did the covariances for the large population. The predicted small population means diverged farther from the expected means because the additive genetic variance associated with several traits increased in value and most of the genetic covariances associated with one trait changed in sign. These results illustrate that genetic covariances may remain nearly constant in some situations while changing markedly in others. Possible developmental reasons for the genetic changes are discussed.     

Genetic correlation and response to selection in simulated populations

Summary Effects of truncation selection of a primary trait upon genetic correlation with a secondary trait were examined over 30 generations in genetic populations simulated by computer. Populations were 24 males and 24 females mated randomly with replacement; number of offspring was determined by intensity of selection. Each trait was controlled by 48 loci segregating independently, effects were equal at every locus, and gene frequency was arbitrarily set at 0.5 at each locus in the initial generation. All combinations of three genetic correlations, three intensities of selection, and three environmental variances were simulated. Gene action was additive. Genetic correlation was set by number of loci which affected both traits and was measured each generation as the product-moment correlation of genotypic values and estimated by two methods of combining phenotypic covariances between parent and offspring.Genetic correlations in each offspring generation remained consistently near initial correlations for all environmental variances when fraction of offspring saved as parents was as large as one-half. When the fraction of offspring saved was as small as one-fifth, genetic correlations decreased but most rapidly with heritability high and after the 15th generation of selection. Truncation selection caused genetic correlation to decrease in those offspring selected to become parents of the next generation. Amount of reduction depended on heritability of the selected trait rather than on degree of truncation selection. Estimates of genetic correlation from phenotypic covariances between parent and offspring fluctuated markedly from real correlations in the small populations simulated.Michigan Agricultural Experiment Station Journal Article 4836. Part of North Central Regional Project NC-2.     

Gamete selection with an inbred tester

Summary A common inbred tester was used to evaluate gametes selected from three complex hybrid populations using two different inbreds as elite lines. The results support Stadler'S contention that gamete selection is an efficient method for extracting superior gene combinations from hybrid populations. The inbred tester increases the resolving power of the method by eliminating extraneous genetic variability and by providing a homogeneous check population for comparative purposes.Technical Article No. TA7973, Texas Agricultural Experiment Station, College Station, Texas. Project No. 1280. Parts of dissertations submitted by the first two authors in partial fulfillment of requirements for Ph. D. degrees.     

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.     

Errors in genetic theory equations

Summary This study addresses the consequences of eliminating terms such as x² and x³ from genetic equations when the variable x is known to be small. This paper indicates logically that to assign such terms a value of 0.0 requires knowing the magnitude of the coefficients for each of these terms as well as the magnitude of all other terms in a given expression. Since most genetic expressions of interest involve several unknowns, the elimination of these terms appears difficult to justify in most situations. The effects of the elimination of a single term from an expression in a classical plant breeding paper were investigated as a simple exemplifying case. In the example, the simplified equation for change in population mean with selection sometimes greatly overestimated the response to selection and in some cases also altered conclusions as to best procedure. Though simplified equations are usually much more tractable and interpretable, the bias which is introduced into the research results and the potential for propagation of such biases in subsequent studies indicates that no term can be uncritically ignored in a genetic equation. The obvious alternatives are (1) do not simplify by eliminating terms, (2) perform a complete error analysis, or (3) restrict the range of values for variables so that terms can be justifiably eliminated in the error analysis.Cooperative investigations of the Alfalfa Production Research Unit, United States Department of Agriculture, Agricultural Research Service, and the Nevada Agricultural Experiment Station, Reno, NV     

Prediction of additive and dominance effects in selected or unselected populations with inbreeding

Summary A genetic model with either 64 or 1,600 unlinked biallelic loci and complete dominance was used to study prediction of additive and dominance effects in selected or unselected populations with inbreeding. For each locus the initial frequency of the favourable allele was 0.2, 0.5, or 0.8 in different alternatives, while the initial narrow-sense heritability was fixed at 0.30. A population of size 40 (20 males and 20 females) was simulated 1,000 times for five generations. In each generation 5 males and 10 or 20 females were mated, with each mating producing four or two offspring, respectively. Breeding individuals were selected randomly, on own phenotypic performance or such yielding increased inbreeding levels in subsequent generations. A statistical model containing individual additive and dominance effects but ignoring changes in mean and genetic covariances associated with dominance due to inbreeding resulted in significantly biased predictions of both effects in generations with inbreeding. Bias, assessed as the average difference between predicted and simulated genetic effects in each generation, increased almost linearly with the inbreeding coefficient. In a second statistical model the average effect of inbreeding on the mean was accounted for by a regression of phenotypic value on the inbreeding coefficient. The total dominance effect of an individual in that case was the sum of the average effect of inbreeding and an individual effect of dominance. Despite a high mean inbreeding coefficient (up to 0.35), predictions of additive and dominance effects obtained with this model were empirically unbiased for each initial frequency in the absence of selection and 64 unlinked loci. With phenotypic selection of 5 males and only 10 females in each generation and 64 loci, however, predictions of additive and dominance effects were significantly biased. Observed biases disappeared with 1,600 loci for allelic frequencies at 0.2 and 0.5. Bias was due to a considerable change in allelic frequency with phenotypic selection. Ignoring both the covariance between additive and dominance effects with inbreeding and the change in dominance variance due to inbreeding did not significantly bias prediction of additive and dominance effects in selected or unselected populations with inbreeding.     

ACROSS‐ENVIRONMENT GENETIC CORRELATIONS AND THE FREQUENCY OF SELECTIVE ENVIRONMENTS SHAPE THE EVOLUTIONARY DYNAMICS OF GROWTH RATE IN IMPATIENS CAPENSIS

Trade‐offs can exist within and across environments, and constrain evolutionary trajectories. To examine the effects of competition and resource availability on trade‐offs, we grew individuals of recombinant inbred lines of Impatiens capensis in a factorial combination of five densities with two light environments (full light and neutral shade) and used a Bayesian logistic growth analysis to estimate intrinsic growth rates. To estimate across‐environment constraints, we developed a variance decomposition approach to principal components analysis, which accounted for sample size, model‐fitting, and within‐RIL variation prior to eigenanalysis. We detected negative across‐environment genetic covariances in intrinsic growth rates, although only under full‐light. To evaluate the potential importance of these covariances, we surveyed natural populations of I. capensis to measure the frequency of different density environments across space and time. We combined our empirical estimates of across‐environment genetic variance–covariance matrices and frequency of selective environments with hypothetical (yet realistic) selection gradients to project evolutionary responses in multiple density environments. Selection in common environments can lead to correlated responses to selection in rare environments that oppose and counteract direct selection in those rare environments. Our results highlight the importance of considering both the frequency of selective environments and the across‐environment genetic covariances in traits simultaneously.     

Choice of environments in reciprocal recurrent selection programs

Summary An alternative method of reciprocal recurrent selection (RRS) in which populations A and B are each evaluated in a different environment is proposed. This method is called dual-environment reciprocal recurrent selection (DERRS). Two genetic models are considered in the theoretical study. A comparison of selection methods shows that genetic gain is larger in DERRS than in RRS for the two models. The difference grows greater as the dominance effects operating in the two environments are more divergent and as the number of selection cycles increases. A greater gain is obtained when the genetic covariances between crosses in the two chosen environments are lower.     

Correlation and path-coefficient analyses of seed-yield components in pearl millet x elephantgrass hybrids

Correlation and path-coefficient analyses have been successful tools in developing selection criteria. Since increased seed yield is an important goal in our pearl millet x elephantgrass [Pennisetum glaucum (L.) R.Br. x P. purpureum Schum.] hexaploid breeding program, we used correlation and path-coefficient analyses on seed data. This study was conducted to develop appropriate selection criteria by determining the direct and indirect effects of seed-yield components on seed yield plant^-1. Number of tillers plant^-1, panicles tiller^-1, seeds panicle^-1, 100-seed weight, and seed yield plant^-1, were estimated for individual plants in seven families. Phenotypic (r_p) and genetic correlations (r_g) were calculated, and path analyses (phenotypic and genetic) were carried out according to predetermined causal relationships. Phenotypic and genetic correlations differed in several cases due to large environmental variance and covariance. Phenotypically, all components were positively and significantly associated with seed yield plant^-1. Genotypically, only seeds panicle^-1 and 100-seed weight were significantly correlated. These two components were also positively correlated (r _p=0.55, r _g=0.63), so simultaneous improvement for both components would be feasible. Panicles tiller^-1 and seeds panicle^-1 were negatively correlated (r _g=-0.97). In the path analyses, all direct effects of the components on seed yield plant^-1 were positive. Phenotypic indirect effects were not as important as genetic indirect effects. The components seeds panicle^-1 and 100-seed weight influenced seed yield plant^-1 the greatest, both directly and indirectly.Florida Agricultural Experimental Station Journal Series No. R-03339     

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

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