Individual and group selection with competition

Summary The response of a randomly mating population which is expected to follow selection of phenotypic units, comprising individuals or groups whose members have an arbitrary degree of relatedness, was formulated using a model which included additive and dominance competition effects. The derivation involved three steps. Twenty-two quadratic components were defined, six describing individual (direct) and neighbor (associate) effects, and 16 describing direct by associate interactions for different loci, for single loci with different alleles, and for identical alleles. Six covariances between pairs of individual phenotypes and three of individuals with their offspring were defined according to whether or not their direct or associate genotypes are common, and expressed in terms of the quadratic components. Finally, variances of selection units of different types and their covariance with their offspring were expressed as compounds of these individual covariances. Explicit formulations for mass, clonal and full-sib selection show that without constraints on the quadratic components, and hence on the magnitude and type of competition operative, no predictions as to the relative efficiencies of these three methods can be made.     

Influence of mom and dad: quantitative genetic models for maternal effects and genomic imprinting

The expression of an imprinted gene is dependent on the sex of the parent it was inherited from, and as a result reciprocal heterozygotes may display different phenotypes. In contrast, maternal genetic terms arise when the phenotype of an offspring is influenced by the phenotype of its mother beyond the direct inheritance of alleles. Both maternal effects and imprinting may contribute to resemblance between offspring of the same mother. We demonstrate that two standard quantitative genetic models for deriving breeding values, population variances and covariances between relatives, are not equivalent when maternal genetic effects and imprinting are acting. Maternal and imprinting effects introduce both sex-dependent and generation-dependent effects that result in differences in the way additive and dominance effects are defined for the two approaches. We use a simple example to demonstrate that both imprinting and maternal genetic effects add extra terms to covariances between relatives and that model misspecification may over- or underestimate true covariances or lead to extremely variable parameter estimation. Thus, an understanding of various forms of parental effects is essential in correctly estimating quantitative genetic variance components.     

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.     

Analysis of some nonrandom mating models

In this paper a few asymmetric models are presented taking account of the effects of assortative mating on an autosomal trait controlled by a single locus possibly with multiple alleles. The models are developed by specifying the intensities for preference mating for various phenotypes. The analysis is confined to the case in which preference is exercised by the individuals of one sex only. It is assumed that males possess unlimited fertility.The dynamics of the population and its equilibrium distribution are discussed. The gene frequency usually changes with time and equilibrium distribution in most cases depends only on the assortment parameters. Expressions are obtained giving the additive and dominance components of variance, and covariances for relatives for populations in equilibrium for some of the models.     

Prediction of effects of genetic drift on variance components under a general model of epistasis

For a model of diallelic loci with arbitrary epistasis, Barton and Turelli [2004. Effects of genetic drift on variance components under a general model of epistasis. Evolution 58, 2111-2132] gave results for variances among and within replicate lines obtained by inbreeding without selection. Here, we discuss the relation between their population genetic methods and classical quantitative genetic arguments. In particular, we consider the case of no dominance using classical identity by descent arguments, which generalizes their results from two alleles to multiple alleles. To clarify the connections between the alternative methods, we obtain the same results using an intermediate method, which explicitly identifies the statistical effects of sets of loci. We also discuss the effects of population bottlenecks on covariances among relatives.     

Estimation aux moindres carrés des effets géniques chez un organisme diploïde—Un locus à deux allèles—Application àNicotiana tabacum

When the genotypes of the individuals in a population can be identified and thereafter quantified, direct estimates of the genetic parameters and of the genetic variability components can be obtained from a sample proportional to the population genotypic frequencies. In such a situation, covariances among relatives are not needed. This paper provides the complete least squares computational procedure for one locus and two alleles, with frequenciesp and 1-p, in a population in Hardy-Weinberg equilibrium. Data onN. tabacum are throughly treated.     

Theory of inbreeding and covariances between relatives under full-sib mating in diploids.

A generation matrix theory of full-sib mating is developed in which 13 mating "classes" are distinguished according to identity of genes in individuals mated and identity of genotypes as belonging to homozygous, parental, or offspring sets. The 13 times 13 matrix reveals some properties of the full-sib mating system not shown by previous work. The eigenvalues and a set of eigenvectors for the generation matrix, and the general solution for the frequencies of mating classes among descendants of an original mating of genotypes ab times cd, are given. The genotypic array of descendants in an arbitrary generation is also given. A new formula is derived for the coefficient of inbreeding in generation n + m in terms of coefficients of inbreeding in earlier generations. An algorithm is presented for calculating the probability of a given situation of identity of alleles carried by two individuals given only the indices of their own respective generations and the generation of their most recent common ancestor. The application of such probabilities to obtaining covariances between relatives in a full-sib mating system, under the assumptions of independence and non-interaction among loci, is illustrated. All results are shown to agree with previous work in special cases. All possible full sib, generation n - 1 parent-generation n + m offspring, and generation n uncle-generation n + m nephew covariances for 1 less than n + m less than or equal to 8 are obtained using the given algorithm.     

The effect of cultural transmission on continuous variation.

Cultural transmission may depend on the non-genetic transfer of information from parent to offspring. The consequences of such cultural transmission for continuous variation are investigated theoretically for randomly mating populations. Cultural inheritance may act on genetical and environmental differences between individuals. The consequences for cultural inheritance of polygenic variation and variation due to chance environmental factors are considered. An equilibrium may occur in which the population variance and the covariances between relatives can be expressed as functions of estimable parameters of genetical and environmental variation. Whatever the ultimate origin of culturally inherited differences they are expected to lead to environmental differences between families ("E2" variation). In addition, if cultural transmission maintains differences due ultimately to segregation at many gene loci we may find genotype-environmental covariation is generated.     

Covariances between Arbitrary Relatives in Autotetraploids with Panmictic Disequilibrium

In random mating autopolyploid populations which have not reached equilibrium, two alleles may be interdependent as a result of the phenomenon of gametic recombination, i.e. the maintenance through successive gamete generations of an association of two alleles from the same gamete in a source reference generation. Any two alleles are dependent by this relationship if they derive by descent from the same ancestral gamete in the source population. Applied with the classical notion of identity by descent, the concept thus defined identifies new coefficients of dependence between arbitrary relatives. Coefficients of dependence are probabilities attached to the drawing of genes from two zygotes such that there are certain relationships amongst them. Applying the concept to autotetraploids, consideration of the states of dependence between the genes of a zygote or of a pair of zygotes leads to the definition of new parameters bearing on population means, variances and covariances, for arbitrary inbreeding. The absence of epistasis is supposed. Some applications of interest in artificial selection are briefly envisaged, with simplifying restrictions on genetic effects. The particular case of diallelism is also considered.     

Inbreeding, coancestry, and covariance between relatives for X-chromosomal loci

Knowledge about the relationships between relatives for X-chromosomal loci is necessary to compute genetic variances and covariances for the genetic evaluation of individuals for economically important traits in livestock and poultry. Using a unified approach, we derived explicitly the coefficient of inbreeding for individuals and the coefficient of coancestry between collateral and lineal relatives of the same or different sex, assuming that the male is heterogametic and the female is homogametic. Collateral relatives include full sibs, paternal and maternal half-sibs, paternal and maternal single first cousins, and double first cousins. Lineal relatives include parent-offspring, paternal and maternal grandparent-grandoffspring, and aunt- or uncle-niece or -nephew. We also defined additive and dominance relationships to compute genetic covariance between relatives, assuming random mating equilibrium, and clarified misinterpretations and corrected errors in the literature. Our results are also applicable to organisms that have few autosomal loci, such as Drosophila, in which X-chromosomal loci can account for a large amount of genetic variance, and to haplodiploid organisms, such as the honeybee, in which the entire genome is equivalent to being X-chromosomal.     

Aspects of variance and covariance analysis with cultural inheritance

Linear Gaussian models of genotypic, phenotypic, and environmental transmission are studied. The nature of equilibrium assumptions under various modes of assortative mating are discussed with particular emphasis on expected correlations between relatives. Assorting models based only on phenotype, or only on environment, are compared with those in which the mating correlation structure is more complex. Explicit values in terms of transmission parameters and within individual covariances are given for the usual correlations between relatives. The possibility of decomposing these in terms of correlations involving adoptive families is indicated.     

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.     

Covariances of relatives and selection response in generations of selfing from an outcrossed base population

Summary The covariances of relatives arising under selfing from a general outbred base population in linkage equilibrium and without epistasis given by Cockerham (1983) are expressed in an alternative form which is an extension of the treatment by Mather and Jinks (1982) of the more restricted population descended from a single F₁ family. Whereas no more than two quadratic components are required to describe any covariance in the case of F₁, descendants, this more general case calls for a total of four, three of which are needed for any particular covariance. The estimation of covariances and their use for the prediction of selection response is described for breeding programs initiated by one or more cycles of intermating among a number of parental lines, as advocated by Hansel (1964) and Jensen (1970). It is pointed out that the homozygous lines descended from such a population will have up to twice as much variance as those from an F₁ between a randomly chosen pair from the same population of parents. The selection method is especially recommended for undeveloped species in which the parental lines are not well characterized and large selection responses are needed.     

An estimator for pairwise relatedness using molecular markers

I propose a new estimator for jointly estimating two-gene and four-gene coefficients of relatedness between individuals from an outbreeding population with data on codominant genetic markers and compare it, by Monte Carlo simulations, to previous ones in precision and accuracy for different distributions of population allele frequencies, numbers of alleles per locus, actual relationships, sample sizes, and proportions of relatives included in samples. In contrast to several previous estimators, the new estimator is well behaved and applies to any number of alleles per locus and any allele frequency distribution. The estimates for two- and four-gene coefficients of relatedness from the new estimator are unbiased irrespective of the sample size and have sampling variances decreasing consistently with an increasing number of alleles per locus to the minimum asymptotic values determined by the variation in identity-by-descent among loci per se, regardless of the actual relationship. The new estimator is also robust for small sample sizes and for unknown relatives being included in samples for estimating allele frequencies. Compared to previous estimators, the new one is generally advantageous, especially for highly polymorphic loci and/or small sample sizes.     

Effects of epistasis on tests for linkage in self-pollinated species

Summary Tests for linkage based on covariances among relatives in self-pollinated species are usually based upon an assumption that epistasis is not important. This study was conducted to determine the impact of epistasis on, and to investigate the sensitivity of, such tests. Thirty covariances were calculated for each of ten non-epistatic and ten epistatic genetic models with varying probabilities of recombination between two coupling or repulsion loci. Each set of covariances was tested for linkage by comparing covariances calculated for the model with those expected for an additive-dominance model with no linkage. Results showed that the test for linkage is quite insensitive to the effects of linkage due to the disproportionate influence of inbreeding. Repulsion linkages should be easier to detect than coupling linkages for all models. Epistasis was found to mimic or counteract the effects of linkage. Tests for linkage based on covariances within a hierarchical mating design appear to be insensitive to linkage and may confuse the effects of linkage and epistasis.     

Epistatic selection opposed by immigration in multiple locus genetic systems

A model of “complete” epistatis is considered in which all “plus” alleles must be present in an individual before the adaptive phenotype is expressed. The conditions under which the plus alleles and hence the adaptive phenotype can increase and reach a stable equilibrium in the presence of immigration of gametes carrying minus alleles are found. In haploids and diploids in which the plus alleles are recessive, frequencies of the plus alleles are the same at all loci, regardless of the linkage relationships. Tight linkage favors the existence of a locally stable polymorphic equilibrium, but the equilibrium with only minus alleles is locally stable unless there is very tight linkage or very strong selection. Thus, this kind of epistasis, which provides a simple model for a character that requires several components to be present at the same time, is very sensitive to even a small amount of immigration. Hence, the evolution of such characters is likely only in completely rather than partially isolated populations.     

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.     

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.     

Exchangeable models of complex inherited diseases

A model of unlinked diallelic loci affecting the risk of a complex inherited disease is explored. The loci are equivalent in their effect on disease risk and are in Hardy-Weinberg and linkage equilibrium. The goal is to determine what assumptions about dependence of disease risk on genotype are consistent with data for diseases such as schizophrenia, bipolar disorder, autism, and multiple sclerosis that are relatively common (0.1-2% prevalence) and that have high concordance rates for monozygotic twins (30-50%) and high risks to first-degree relatives of affected individuals (risk ratios exceeding 4). These observations are consistent with a variety of models, including generalized additive, multiplicative, and threshold models, provided that disease risk increases rapidly for a narrow range of numbers of causative alleles. If causative alleles are in relatively high frequency, then the combined effects of numerous causative loci are necessary to substantially elevate disease risk.