Modifications in estimating the number of genes for a quantitative character.

In estimating the minimum number of genes contributing to a quantitative character, it is suggested that the squared difference between the means of the two parents be corrected for experimental variance and that the genetic variance stemming from differences in gene frequencies of the parents be estimated by least squares utilizing information on all entries.

Analyses of gene frequencies.

Models of variance components and their intraclass correlational equivalences are developed for genes falling into various categories of subdivisions within a population. Estimable functions are elaborated demonstrating that intraclass correlations can be estimated only relative to that for the least related genes in the informational system. The effects of different types of subdivisions-and of ignoring them-on the parameters are demonstrated. Small sample estimators are formulated for all of the parameters by three different methods, including both a weighted and an unweighted method of analysis of the variation among subpopulations. How estimators change with assumptions about the parameters is illustrated. Various tests of hypotheses are outlined in chi(2) and F-test terminology. Discussed are factors which may affect the correlations and the manner in which their effects are manifest, hopefully in clarification of some of the misconceptions that have arisen in this connection.

Analyses of gene frequencies of mates.

A genic analysis of variance of data on mate pairs for a codominant gene is developed. This analysis provides estimators of the correlation, F, of genes within individuals, of the correlation, Theta, of genes between mates, and of various variances-all relative to the correlation or variation among genes of nonmates. The data are manipulated into marginal distributions to produce another method of obtaining the same estimators. Several examples are given of how assumptions about the model and parameters modify the estimators and which were utilized in constructing chi(2) tests of hypotheses concerning F and Theta.-A recessive gene is also considered. Only the frequency of recessive genotypes and the correlation of recessive mates are estimable in this case unless one makes very demanding assumptions about the model.-Numerical examples of the analysis of variance and estimators are given for both a codominant and recessive gene.

Additive by additive variance with inbreeding and linkage.

Two-locus coancestries, which provide the coefficients of the additive by additive component in the variance and covariance of relatives for a quantitative trait, were formulated generally in terms of two-locus probabilities of identity by descent for combinations of parental and recombinant gametes. Explicit expressions, with linkage, were developed for all relatives of self-fertilization, for selfed and outbred relatives and for full and half sibs from unrelated inbred parents. The relative effect of linkage on the two-locus coancestry decreases rapidly as inbreeding and relatedness of the relatives increase. It was concluded that the error caused by ignoring linkage would probably be small in the estimation of genetic variances or in the projection of selection response with self-fertilization.

Mutation models and quantitative genetic variation.

Analyses of evolution and maintenance of quantitative genetic variation depend on the mutation models assumed. Currently two polygenic mutation models have been used in theoretical analyses. One is the random walk mutation model and the other is the house-of-cards mutation model. Although in the short term the two models give similar results for the evolution of neutral genetic variation within and between populations, the predictions of the changes of the variation are qualitatively different in the long term. In this paper a more general mutation model, called the regression mutation model, is proposed to bridge the gap of the two models. The model regards the regression coefficient, gamma, of the effect of an allele after mutation on the effect of the allele before mutation as a parameter. When gamma = 1 or 0, the model becomes the random walk model or the house-of-cards model, respectively. The additive genetic variances within and between populations are formulated for this mutation model, and some insights are gained by looking at the changes of the genetic variances as gamma changes. The effects of gamma on the statistical test of selection for quantitative characters during macroevolution are also discussed. The results suggest that the random walk mutation model should not be interpreted as a null hypothesis of neutrality for testing against alternative hypotheses of selection during macroevolution because it can potentially allocate too much variation for the change of population means under neutrality.

Variance of neutral genetic variances within and between populations for a quantitative character.

The variances of genetic variances within and between finite populations were systematically studied using a general multiple allele model with mutation in terms of identity by descent measures. We partitioned the genetic variances into components corresponding to genetic variances and covariances within and between loci. We also analyzed the sampling variance. Both transient and equilibrium results were derived exactly and the results can be used in diverse applications. For the genetic variance within populations, sigma 2 omega, the coefficient of variation can be very well approximated as [formula: see text] for a normal distribution of allelic effects, ignoring recurrent mutation in the absence of linkage, where m is the number of loci, N is the effective population size, theta 1(0) is the initial identity by descent measure of two genes within populations and t is the generation number. The first term is due to genic variance, the second due to linkage disequilibrium, and third due to sampling. In the short term, the variation is predominantly due to linkage disequilibrium and sampling; but in the long term it can be largely due to genic variance. At equilibrium with mutation [formula: see text] where u is the mutation rate. The genetic variance between populations is a parameter. Variance arises only among sample estimates due to finite sampling of populations and individuals. The coefficient of variation for sample gentic variance between populations, sigma 2b, can be generally approximated as [formula: see text] when the number of loci is large where S is the number of sampling populations.

Testing Hypotheses about Linkage Disequilibrium with Multiple Alleles.

For loci with multiple alleles, hypotheses about linkage disequilibrium may be tested on the complete set of gametic data, or on various collapsed sets of data. Collapsing data into a few alleles at each locus can change the power of the tests, as implied in a recent paper by Zouros, Golding and Mackay (1977). We show that the nature of such changes can be found from properties of the noncentral chi-square distribution, and that the magnitude and direction of these changes depend on the levels of linkage disequilibria, allelic frequencies and degrees of freedom.

Effects of marker chromosomes on relative viability.

Viability relative to Cy/Pm as a standard was studied in Drosophila melanogaster. One experiment, E1, consisted of progeny from eleven distinct 7 x 7 factorial mating designs with reciprocals for second chromosomes extracted from a natural population. The other experiment, E2, consisted of two distinct sets of heterozygotes with reciprocals and corresponding homozygotes. It was established from E1 that there are little to no synergistic effects among different genotypes in a vial and that Cy and Pm heterozygotes vary almost as much as would be expected if one chromosome were held constant for wild-type heterozygotes. In wild-type heterozygotes, variances were estimated to be 0.0099 for average chromosomal effects, 0.0054 for interactions of chromosomes, 0.0021 for maternal effects, 0.0079 for paternal effects, and -0.0010 for the remaining interaction effects, all being significantly different from zero except the last. The variances of Cy and Pm heterozygotes, covariance of Cy and Pm heterozygotes, and covariances of Cy and Pm heterozygotes with wild-type heterozygotes, as well as the comparable statistics available in E2, all showed a large paternal component of variance and a smaller maternal component of variance, both unexpected results.-From E2 the variance of homozygotes, excluding error variance, was estimated to be 0.0149, and the covariances of homozygotes with wild-type heterozygotes to be 0.0056 for maternally derived chromosomes common and 0.0126 for paternally derived chromosomes common, again showing the larger paternal than maternal influence. The average genetic regression of heterozygotes on homozygotes of 0.61 was reduced only slightly to 0.56 by correcting for maternal and paternal variances. These genetic regressions, generally utilized as estimators of the average degree of dominance, are larger than any previously reported.-Differential meiotic drive in Cy and Pm parents was shown to be compatible with the large paternal and maternal variances, but other causes cannot be ruled out.-Approximations were developed for translating various variances, covariances, and regressions between single- and double-marker experiments, assuming that marker chromosomes behave as typical wild-type chromosomes in one case and assuming a (partially) recessive model with the population in mutation selection balance in another case. Various features, particularly the estimation of dominance, were compared and discussed between the two cases.

Design III with marker loci.

Design III is an experimental design originally proposed by R.E. COMSTOCK and H.F. ROBINSON for estimating genetic variances and the average degree of dominance for quantitative trait loci (QTL) and has recently been extended for mapping QTL. In this paper, we first extend COMSTOCK and ROBINSON's analysis of variance to include linkage, two-locus epistasis and the use of F3 parents. Then we develop the theory and statistical analysis of orthogonal contrasts and contrast x environment interaction for a single marker locus to characterize the effects of QTL. The methods are applied to the maize data of C.W. STUBER. The analyses strongly suggest that there are multiple linked QTL in many chromosomes for several traits examined. QTL effects are largely environment-independent for grain yield, ear height, plant height and ear leaf area and largely environment dependent for days to tassel, grain moisture and ear number. There is significant QTL epistasis. The results are generally in favor of the hypothesis of dominance of favorable genes to explain the observed heterosis in grain yield and other traits, although epistasis could also play an important role and overdominance at individual QTL level can not be ruled out.

Selection with partial selfing. I. Mass selection.

The expected responses to mass selection carried out before or after reproduction in a population whose members all have a fixed probability of self pollination (s) are formulated using covariances of relatives and their component quadratic functions for a model with arbitrary additive and dominance effects. The response measured in the first generation offspring after selection (immediate gain) can differ from that retained when the population has regained equilibrium (permanent gain). The population mean behaves in a predictable manner during the return to equilibrium, and its value at any time can be predicted from earlier generations. The permanent gain from selection after reproduction is always (1 + s)/2 times as large as that from selection before reproduction, but the relationship of the immediate gains depends on the genetic model assumed. Numerical analysis applied to a model with two alleles per locus and varying allele frequencies, dominance ratios and numbers of loci showed that the proportion of the immediate gain retained at equilibrium was reduced with the large inbreeding depression associated with increasing dominance levels and numbers of loci and was generally lower for selection after reproduction than before. In the absence of information as to the magnitude of genetic variances and inbreeding depression in species reproducing by partial selfing, the importance of this phenomenon is unknown.

Use of the multinomial Dirichlet model for analysis of subdivided genetic populations.

The distribution found by compounding the multinomial distribution with the Dirichlet distribution has been suggested as a basis for the estimation of parameters in subdivided populations, in particular of the "correlation between genotypes" within subpopulations. It is shown that the estimators deriving from these procedures perform poorly when the data are generated by the classical Wright drift model of subdivided populations. This conclusion suggests that the compound distribution estimation approach does not provide a good estimation procedure for real populations which are reasonably described by the Wright model.

A building block model for quantitative genetics.

We introduce a quantitative genetic model for multiple alleles which permits the parameterization of the degree, D, of dominance of favorable or unfavorable alleles. We assume gene effects to be random from some distribution and independent of the D's. We then fit the usual least-squares population genetic model of additive and dominance effects in an infinite equilibrium population to determine the five genetic components--additive variance sigma 2 a, dominance variance sigma 2 d, variance of homozygous dominance effects d2, covariance of additive and homozygous dominance effects d1, and the square of the inbreeding depression h--required to treat finite populations and large populations that have been through a bottleneck or in which there is inbreeding. The effects of dominance can be summarized as functions of the average, D, and the variance, sigma 2 D. An important distinction arises between symmetrical and nonsymmetrical distributions of gene effects. With symmetrical distributions d1 = -d2/2 which is always negative, and the contribution of dominance to sigma 2 a is equal to d2/2. With nonsymmetrical distributions there is an additional contribution H to sigma 2 a and -H/2 to d1, the sign of H being determined by D and the skew of the distribution. Some numerical evaluations are presented for the normal and exponential distributions of gene effects, illustrating the effects of the number of alleles and of the variation in allelic frequencies. Random additive by additive (a*a) epistatic effects contribute to sigma 2 a and to the a*a variance, sigma 2/aa, the relative contributions depending on the number of alleles and the variation in allelic frequencies.(ABSTRACT TRUNCATED AT 250 WORDS)

Multiplicative vs. arbitrary gene action in heterosis.

In this article we investigate multiplicative effects between genes in relation to heterosis. The extensive literature on heterosis due to multiplicative effects between characters is reviewed, as is earlier work on the genetic description of heterosis. A two-locus diallelic model of arbitrary gene action is used to derive linear parameters for two multiplicative models. With multiplicative action between loci, epistatic effects are nonlinear functions of one-locus effects and the mean. With completely multiplicative action, the mean and additive effects form similar restrictions for all the rest of the effects. Extensions to more than two loci are indicated. The linear parameters of various models are then used to describe heterosis, which is taken as the difference between respective averages of a cross (F1) and its two parent populations (P). The difference (F2 - P) is also discussed. Two parts of heterosis are distinguished: part I arising from dominance, and part II due to additive x additive (a x a)-epistasis. Heterosis with multiplicative action between loci implies multiplicative accumulation of heterosis present at individual loci in part I, in addition to multiplicative (a x a)-interaction in part II. Heterosis with completely multiplicative action can only be negative (i.e., the F1 values must be less than the midparent), but the difference (F2 - P) can be positive under certain conditions. Heterosis without dominance can arise from multiplicative as well as any other nonadditive action between loci, as is exemplified by diminishing return interaction. The discussion enlarges the scope in various directions: the genetic significance of multiplicative models is considered.(ABSTRACT TRUNCATED AT 250 WORDS)

How informative is Wright's estimator of the number of genes affecting a quantitative character?

S. Wright suggested an estimator, m, of the number of loci, m, contributing to the difference in a quantitative character between two differentiated populations, which is calculated from the phenotypic means and variances in the two parental populations and their F1 and F2 hybrids. The same method can also be used to estimate m contributing to the genetic variance within a single population, by using divergent selection to create differentiated lines from the base population. In this paper we systematically examine the utility and problems of this technique under the influences of unequal allelic effects and initial allele frequencies, and linkage, which are known to lead m to underestimate m. In addition, we examine the effects of population size and selection intensity during the generations of selection. During selection, the estimator m rapidly approaches its expected value at the selection limit. With reasonable assumptions about unequal allelic effects and initial allele frequencies, the expected value of m without linkage is likely to be on the order of one-third of the number of genes. The estimates suffer most seriously from linkage. The practical maximum expectation of m is just about the number of chromosomes, considerably less than the "recombination index" which has been assumed to be the upper limit. The estimates are also associated with large sampling variances. An estimator of the variance of m derived by R. Lande substantially underestimates the actual variance. Modifications to the method can ameliorate some of the problems. These include using F3 or later generation variances or the genetic variance in the base population, and replicating the experiments and estimation procedure. However, even in the best of circumstances, information from m is very limited and can be misleading.

Effects of mutation on selection limits in finite populations with multiple alleles.

The ultimate response to directional selection (i.e., the selection limit) under recurrent mutation is analyzed by a diffusion approximation for a population in which there are k possible alleles at a locus. The limit mainly depends on two scaled parameters S (= 4Ns sigma a) and theta (= 4Nu) and k, the number of alleles, where N is the effective population size, u is the mutation rate, s is the selection coefficient, and sigma 2a is the variance of allelic effects. When the selection pressure is weak (S less than or equal to 0.5), the limit is given approximately by 2S sigma a[1 - (1 + c2)/k]/(theta + 1) for additive effects of alleles, where c is the coefficient of variation of the mutation rates among alleles. For strong selection, other approximations are devised to analyze the limit in different parameter regions. The effect of mutation on selection limits largely relies on the potential of mutation to introduce new and better alleles into the population. This effect is, however, bounded under the present model. Unequal mutation rates among alleles tend to reduce the selection limit, and can have a substantial effect only for small numbers of alleles and weak selection. The selection limit decreases as the mutation rate increases.

Estimation of the coancestry coefficient: basis for a short-term genetic distance.

A distance measure for populations diverging by drift only is based on the coancestry coefficient theta, and three estimators of the distance D = -ln(1 - theta) are constructed for multiallelic, multilocus data. Simulations of a monoecious population mating at random showed that a weighted ratio of single-locus estimators performed better than an unweighted average or a least squares estimator. Jackknifing over loci provided satisfactory variance estimates of distance values. In the drift situation, in which mutation is excluded, the weighted estimator of D appears to be a better measure of distance than others that have appeared in the literature.

Maternal and Reciprocal Effects on Seedling Characters in ARABIDOPSIS THALIANA (L.) Heynh.

Five early growth characters were examined in six races of Arabidopsis thaliana (L.) Heynh, their reciprocal F(1) hybrids (1974) and F(1) by tester hybrids, using a seventh race as a paternal tester. Three of the five characters were also examined at two nutrient levels in reciprocal F(1) hybrids (1972) of all seven races. Analyses of F(1) and F(1) by tester hybrids revealed significant maternal effects in all characters examined in F(1) hybrids (1972) and in root length and plant weight of F(1) (1974) and F(1) by tester hybrids. Significant reciprocal effects were found for plant weight in F(1) by tester hybrids and for seed weight, percentage of germination and root length in F(1) (1974) and F(1) by tester hybrids. The presence of significant maternal and/or reciprocal components in both F(1) (1974) and F(1) by tester diallels suggests that differences in maternal cytoplasm rather than maternal genotype per se were responsible for much of the variation resulting from these non-direct genetic effects.

Inversions fail to account for allozyme clines.

Allozyme and inversion data from natural populations of Drosophila melanogaster from the eastern United States were analyzed to determine whether the clines at allozyme loci are due to nonrandom associations with common cosmopolitan inversions. All inversions show strong clines. Clines were large and significant for half of the eight allozyme loci. An analysis of the contribution of inversions to clines of allozyme genes revealed three outcomes: the inversion cline (1) enhanced the allozyme cline, but was only partly responsible, (2) reduced the allozyme cline, and (3) had no effect. The allozyme clines were mainly determined by the pattern of allele frequencies within the chromosomal arrangements. Consequently, it was concluded that allozyme clines would exist in the absence of inversion clines.