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C C Cockerham

Publications and source records attributed to C C Cockerham.

18 recordsLinked to original sources
PubMed1992

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)

Crosses, Genetic↗
PubMed1991

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.

Analysis of Variance↗
PubMed1990

Long-term response to artificial selection with multiple alleles--study by simulations.

The effect of multiple alleles on long-term response to selection is examined by simulations using a pseudosampling technique to simulate the multidimensional diffusion process. The effects of alleles are independently drawn from a normal distribution and the initial frequencies of alleles are assumed either to be equal or to be drawn from a neutral equilibrium population. With these two initial gene frequency distributions we examined various properties of the selection response process for the effects of number of alleles and selection intensity. For neutral initial frequencies the effects of multiple alleles compared with two alleles are minor on the ratio of final to initial response (E(R infinity/E(R1)) and the half life of response (t0.5), but are significant on the variance of response. Under certain conditions the variance of the selection limit can even increase as selection gets stronger. For equal initial frequencies the effects of multiple alleles are, however, minor on the ratio of the variance of the selection limit to the initial genetic variance, but E(R infinity/E(R1) and t0.5 increase as the number of alleles increases. The results show that for certain statistics the effects of multiple alleles can be minimized by an appropriate transformation of parameters for given initial gene frequencies, but the effects cannot, in general, be removed by any single transformation or reparameterization of parameters.

Alleles↗
PubMed1990

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.

Alleles↗
PubMed1989

Effects of identity disequilibrium and linkage on quantitative variation in finite populations.

Identity disequilibrium, ID, is the difference between joint identity by descent and the product of the separate probabilities of identity by descent for two loci. The effects of ID on the additive by additive (a*a) epistatic variance and joint dominance component between populations and in the additive, dominance and a*a variance within populations, including the effects on covariances of relatives within populations, were studied for finite monoecious populations. The effects are formulated in terms of three additive partitions, eta b, eta a and eta d, of the total ID, each of which increases from zero to a maximum at some generation dependent upon linkage and population size and decreases thereafter. eta d is about four times the magnitude of the other two but none is of any consequence except for tight linkage and very small populations. For single-generation bottleneck populations only eta d is not zero. With random mating of expanded populations eta b remains constant and eta a and eta d go to zero at a rate dependent upon linkage, very fast with free recombination. The contributions of joint dominance to the genetic components of variance within and between populations are entirely a function of the eta's while those of a*a variance to the components are functions mainly of the coancestry coefficient and only modified by the eta's. The contributions of both to the covariances of half-sibs, full-sibs and parent-offspring follow the pattern expected from their contributions to the genetic components of variance within populations except for minor terms which most likely are of little importance.

Chromosome Mapping↗
PubMed1989

Restriction map and alpha-amylase activity variation among Drosophila mutation accumulation lines.

The specific activities of alpha-amylase were measured for two sets of mutation accumulation lines, each set having originated from a different lethal-carrying second chromosome and SM1(Cy) chromosome and having been maintained by a balanced lethal system for about 300 generations. Significant variation was found to have accumulated among lines of both sets. Because of dysgenic crosses in the early generations of mutation accumulation, insertions or deletions of transposable elements in the Amy gene region were suspected of being the cause of this variation. In order to test this possibility, the structural changes in the 14 kb region of these chromosomes that includes the structural genes for alpha-amylase were investigated by restriction map analysis. We found that most part of the activity variation is due to replacements of a chromosomal region of SM1(Cy), including the structural genes for alpha-amylase, by the corresponding regions of the lethal chromosomes. One line also contained an insertion in this region but this line has an intermediate activity value. Thus, insertions of transposable elements into the Amy gene region were not found to be responsible for the new variation observed in alpha-amylase activity. If we remove those lines with structural changes from the analysis, the genetic variance of alpha-amylase specific activity among lines becomes non-significant in both sets of chromosomes.

Animals↗
PubMed1989

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)

Alleles↗
PubMed1989

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.

Alleles↗
PubMed1988

Permanency of response to selection for quantitative characters in finite populations.

To study the permanency of response to selection for a quantitative character in finite populations and the nature of the genetic effects that contribute to this response, we have used the covariance between ancestors and descendents within populations. Effects and variances are defined for an initial equilibrium random mating monoecious population that gives rise to replicate finite populations. After a prescribed history of restricted population size, the replicate populations are expanded, and the covariance between ancestors and descendents is quantified in terms of descent measures and genetic components in the initial population as a means of determining the additive variance within populations. Several dominance components including joint dominance effects of loci contribute to the additive variance, some of which can be negative. There is always a positive contribution of additive by additive variance to the additive variance within populations, which can be large. With the new definitions of components of genetic variance within populations, selection response is formulated in the same manner as for the initial random mating population, but the components have been modified considerably by the restricted population size.

Biological Evolution↗
PubMed1987

Quantitative genetic variation in an ecological setting.

The machinery was developed to investigate the behavior of quantitative genetic variation in an ecological model of a finite number of islands of finite size, with migration rate m and extinction rate e, for a quantitative genetic model general for numbers of alleles and loci and additive, dominance, and additive by additive epistatic effects. It was necessary to reckon with seven quadratic genetic components, whose coefficients in the genotypic variance components within demes, sigma Gw2, between demes within populations, sigma s2, and between replicate populations, sigma r2, are given by descent measures. The descent measures at any time are calculated with the use of transition equations which are determined by the parameters of the ecological model. Numerical results were obtained for the coefficients of the quadratic genetic components in each of the three genotypic variance components in the early phase of differentiation. The general effect of extinction is to speed up the time course leading to fixation, to increase sigma r2, and to decrease sigma s2 (with a few exceptions) in comparison with no extinction. The general effect of migration is to slow down the time course leading to fixation, to increase sigma Gw2, at least in the later generations, and to decrease sigma s2 (with a few exceptions) in comparison with no migration. Except for these, the effects of migration and extinction on the variance components are complex, depending on the genetic model, and sometimes involve interaction of migration and extinction. Sufficient details are given for an investigator to evaluate numerically the results for variations in the quantitative genetic and ecological models.

Alleles↗
PubMed1987

Evolution and maintenance of quantitative genetic variation by mutations.

The genotypic variance within, sigma 2w, and between, sigma 2b, random mating populations and rates and times for convergence to equilibrium values from different founder populations are formulated for an additive genetic model with an arbitrary number of alleles k, number of loci m, population size N, and mutation rate u, with unequal mutation rates for alleles. As a base of reference, the additive variance sigma 2a in an infinite equilibrium population is used. sigma 2a increases as k increases and decreases with variation in the mutation rates. Both transitional and equilibrium values of the variance within populations could be expressed as sigma 2w = (1 - theta)sigma 2a, where theta is the coancestry with mutations of individuals within populations. Thus, rates of convergence and evolutionary times are a function of those for theta, which involves both N and u. When the founder population is fixed, very long times are required to obtain a perceptible increase in sigma 2w and equilibrium values of sigma 2w are very small when 4Nu less than or equal to 10(-1). The variance between populations can be expressed as sigma 2b = 2 theta sigma 2a when the founder population is an infinite equilibrium population, and as sigma 2b = 2(theta - alpha)sigma 2a when the founder population is fixed, where alpha is a function only of u. In this latter case, rates of divergence, while affected by both N and u, are dominated by u and asymptotically a function of u only. With u = 10(-5), very long times (10(3) generations) are required for any perceptible divergence, even for N = 1-10. At equilibrium, most of the variance is between small populations and within very large populations. Migration increases the variance within populations and decreases the variance between populations.

Alleles↗
PubMed1987

Correlations, descent measures: drift with migration and mutation.

The analysis of gene frequencies for a nested structure of genes within individuals, individuals within subpopulations, and subpopulations within populations is considered. Alternative parameterizations are provided by measures of correlation and of identity by descent, but the latter parameters provide more flexibility. The effects of population size, mating system, mutation, and migration can be incorporated into transition equations for identity measures and the structure of equilibrium populations can be determined; the procedures are illustrated for a finite island model. With parameters defined before estimation procedures are developed, problems of estimates depending on the numbers of sampled subpopulations are avoided, while the descent measures also avoid the approximations found in other treatments.

Emigration and Immigration↗
PubMed1987

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.

Gene Frequency↗
PubMed1986

Estimation of inbreeding parameters in stratified populations.

A clarification is given of the differences in approaches to the estimation of F-statistics of Nei & Chesser [Ann. Hum. Genet. (1983) 47, 253-259] and Cockerham [Genetics (1973) 74, 679-700]. The principal difference is that Nei & Chesser define quantities with respect to fixed extant populations, while Cockerham allows for evolutionary variation between populations. Weighted and unweighted analyses are compared, and a numerical example given.

Alleles↗
PubMed1977

Spontaneous mutation rates at enzyme loci in Drosophila melanogaster.

In a marked-inversion-balanced lethal system mutations were accumulated at a minimum pressure of natural selection on 2000 second chromosomes of Drosophila melanogaster that originated from 4 stem chromosomes. Five enzyme loci were tested: alpha-glycerol-3-phosphate dehydrogenase (EC 1.1.1.8), malate dehydrogenase (Mdh, EC 1.1.1.37), alcohol dehydrogenase (EC 1.1.1.1), hexokinase-C (Hex-C tentative name), and alpha-amylase (Amy, EC 3.2.1.1). Three band-morph mutants, one at the Mdh locus, one at the Hex-C locus, and one at the Amy locus, were detected out of 1,658,308 allele replications. In addition, 17 null mutants were found. Accepting that the number of structural genes is the same as that of bands in the salivary gland chromosomes, the total mutation rate per generation for all the structural genes in the second chromosomes is estimated to be 0.008-0.040, which is much smaller than that estimated for viability polygenes (0.12-0.17). Thus, it is speculated that most viability and other fitness polygenes are located in controlling regions outside the structural genes.

Alcohol Oxidoreductases↗
PubMed1977

Quadratic analyses of reciprocal crosses.

Three different models, a two-way factorial model for familiarity, an orthogonalizing transform of this model to a diallel model, and a bio model more representative of the biological situation, are interrelated in terms of their components of variance and covariance. It is clarified that there are five components that can be reckoned with in the analysis of reciprocal crosses, including distinct maternal and paternal variances. Estimation of the components and tests of hypotheses concerning them are outlined for two types of mating designs with reciprocals. One deisgn involves a factorial mating design between two distinct sets of parents or parental lines and the other a diallel of all crosses from a single set of parents or parental lines. Both designs provide the same types of information and similar tests of hypotheses. At least some parts of the analyses corresponding to the factorial model are required to separate the maternal and paternal variances. A least squares partitioning of the sums of squares according to the diallel model, but with expectations expressed in terms of the bio model, provides most of the tests of hypotheses of interest. Worked examples are given.

Crosses, Genetic↗