A symmetric two locus model with viability and fertility selection

A two locus deterministic population genetic model is analysed. One locus is under viability selection, the other under fertility selection with both forms of selection completely symmetric. It is shown that linkage equilibrium may occur at two different equilibrium points. For a two-locus polymorphism to be stable, it is necessary that the viability locus be overdominant but not necessary that the fertility locus, considered separately, be able to support a stable polymorphism. The overlaps in stability are not as complex as under two locus symmetric fertilities, but considerably more complex than with symmetric viabilities. Extensions of the analysis for the central linkage equilibrium point with multiple viability and fertility loci are indicated.Research supported in part by NIH grants GM 28106 and GM 10452     

Internode length in Pisum. A new allele at the Lh locus

A new allele at the Lh locus has been identified in Pisum sativum L. and named lhⁱ . This allele results in reduced GA levels in young shoots, and a dwarf phenotype. Gas chromatography-selected ion monitoring (GC-SIM) with dideuterated internal standards has been used to demonstrate a quantitative relationship between the level of endogenous GA₁ and internode length using the three alleles ( Lh, lh and lhⁱ ) at the Lh locus. These results are consistent with previous findings in peas (for alleles at the Le locus) and other species possessing a predominant early 13-hydroxylation pathway for GA biosynthesis and support the role of GA₁ as the major native GA in peas with biological activity in its own right. However, in contrast to alleles at the Le locus, GA₂₀ levels are also reduced in lh and lhⁱ plants. The lhⁱ allele also has possible pleiotropic effects on seed abortion, leading to a reduction in seed yield compared to plants homozygous for the previously characterised Lh or lh alleles.     

A new allele at the Pgd locus in pigs

A new electrophoretic variant of porcine 6-phosphogluconate dehydrogenase (PGD) is described. The new variant, PGD C, has been shown to be controlled by a third allele, PgdC, at the Pgd locus.     

A robust multiple-locus method for quantitative trait locus analysis of non-normally distributed multiple traits

Linear regression-based quantitative trait loci/association mapping methods such as least squares commonly assume normality of residuals. In genetics studies of plants or animals, some quantitative traits may not follow normal distribution because the data include outlying observations or data that are collected from multiple sources, and in such cases the normal regression methods may lose some statistical power to detect quantitative trait loci. In this work, we propose a robust multiple-locus regression approach for analyzing multiple quantitative traits without normality assumption. In our method, the objective function is least absolute deviation (LAD), which corresponds to the assumption of multivariate Laplace distributed residual errors. This distribution has heavier tails than the normal distribution. In addition, we adopt a group LASSO penalty to produce shrinkage estimation of the marker effects and to describe the genetic correlation among phenotypes. Our LAD-LASSO approach is less sensitive to the outliers and is more appropriate for the analysis of data with skewedly distributed phenotypes. Another application of our robust approach is on missing phenotype problem in multiple-trait analysis, where the missing phenotype items can simply be filled with some extreme values, and be treated as outliers. The efficiency of the LAD-LASSO approach is illustrated on both simulated and real data sets.     

Nonlinear frequency-dependent selection at a single locus with two alleles and two phenotypes

 The paper investigates the discrete frequency dynamics of two phenotype diploid models where genotypic fitness is an exponential function of the expected payoff in the matrix game. Phenotypic and genotypic equilibria are defined and their stability compared to frequency-dependent selection models based on linear fitness when there are two possible phenotypes in the population. In particular, it is shown that stable equilibria of both types can exist in the same nonlinear model. It is also shown that period-doubling bifurcations emerge when there is sufficient selection in favor of interactions between different phenotypes. Received: 22 October 1998     

The effect of index selection on allele frequencies and future genetic gains when traits are correlated

The application of the selection index in the case of an additive two-trait model in which the genetic effect on each trait is determined by a finite number of loci is examined. Simulation results indicate that the direction of change in the frequency of favourable alleles is not necessarily in the positive direction at all loci when index selection is used as the basis for truncation selection. When the genetic correlation was positive (or favourable with respect to the economic weights), there was little difference (<5%) in genetic gain over 20 generations and no difference in the direction of change in allele frequencies or genetic correlation whether or not updated values for the genetic (co)variances were used in constructing the selection index. However, when the genetic correlation was negative or unfavourable, the effect of using genetic parameters which were not updated had unexpected effects on the allele frequencies and genetic correlation and reduced the genetic gain by a greater amount (< 12%).     

A Bayesian method for jointly estimating allele age and selection intensity.

The problem of jointly estimating the intensity of past selection affecting an allele and the allele's age is formulated in a Bayesian framework. The prior distribution of allele age given its frequency is obtained from existing population genetics theory. The prior distribution of selection intensity is assumed to reflect the fact that positive selection on a new mutant is more likely to be weak than strong. The general approach is illustrated by the development of an importance sampling method applicable to low-frequency alleles. This method can be used either when the haplotypes of closely linked marker loci are known or when the lengths of linked ancestral chromosomal segments can be inferred. The method is illustrated with an application to the A-allele of G6PD in Africa. Because changes in allele frequency and recombination are both intrinsically stochastic, there are limits to the accuracy achievable with any method.     

Viability-fecundity selection at a single multiallelic locus in random mating populations: Protectedness of an allele

To simplify comprehension of the effect of combined viability and fecundity selection on the genetic structure, emphasis is put on the definition of those population parameters, such as average fitness of genotypes or alleles, which provide insight into the principles which govern the system's dynamics. With the help of these parameters, it is possible to point out the resemblance of the model presented here to the known models of pure viability selection. Protectedness of alleles is considered in two steps: (1) treatment of the possibility that alleles become extinct in finite time, and (2) provided sudden extinction cannot occur, derivation of conditions for the initial increase of a new allele. The range of relevance of these results is demonstrated by applying them to the most frequently used factorizations of fecundity into the male and female contributions, namely the multiplicative and the additive.     

Variation,selection and evolution of function-valued traits

We describe an emerging framework for understanding variation, selection and evolution of phenotypic traits that are mathematical functions. We use one specific empirical example – thermal performance curves (TPCs) for growth rates of caterpillars – to demonstrate how models for function-valued traits are natural extensions of more familiar, multivariate models for correlated, quantitative traits. We emphasize three main points. First, because function-valued traits are continuous functions, there are important constraints on their patterns of variation that are not captured by multivariate models. Phenotypic and genetic variation in function-valued traits can be quantified in terms of variance-covariance functions and their associated eigenfunctions: we illustrate how these are estimated as well as their biological interpretations for TPCs. Second, selection on a function-valued trait is itself a function, defined in terms of selection gradient functions. For TPCs, the selection gradient describes how the relationship between an organism's performance and its fitness varies as a function of its temperature. We show how the form of the selection gradient function for TPCs relates to the frequency distribution of environmental states (caterpillar temperatures) during selection. Third, we can predict evolutionary responses of function-valued traits in terms of the genetic variance-covariance and the selection gradient functions. We illustrate how non-linear evolutionary responses of TPCs may occur even when the mean phenotype and the selection gradient are themselves linear functions of temperature. Finally, we discuss some of the methodological and empirical challenges for future studies of the evolution of function-valued traits.     

Serum esterase genetics in rabbits. III. A third allele on the Est-2 locus

A fast-migrating F' zone of the prealbumin serum esterase system of rabbits is demonstrated in low frequency in the breed Vienna White (stock Cpb:VW). Evidence is given that this zone is controlled by a third allele of the Est-2 locus. The zymotypic expression of this allele (Est-2-F') shows codominance in combination with the Est-2-F allele and complete dominance in combination with the Est-2-F' allele. In contradistinction to the F zones of the Est-2-F allele, the F' zone possesses no atropinesterase activity.     

Experimental comparison of methods for simultaneous selection of two correlated traits in Tribolium

Summary Two lines of Tribolium castaneum were selected in each of three replicates for egg laying between 7 and 11 days after adult emergency and for adult weight at 12 days, using theoretical (IT) and empirical (IP) index selection methods. Index coefficients were given empirically in the IP line and they were adjusted in the successive generations of selection according to the results obtained in the previous ones. Highly repeatable selection responses in all replicates occurred in both lines for the aggregate genotype (egg laying plus adult weight) and for each individual trait. The IP line tended to increase slightly more than the IT line for aggregate genotype and egg laying, while the highest response in adult weight was obtained with the IT method. The two methods gave consistently different responses in each replicate. The expected results were that IT selection should not exceed IP selection for the aggregate genotype and egg laying while theoretically the IT method should have been superior for increase adult weight. Theoretical expectations for adult weight have been fulfilled in practice. The IP method would be preferred in a practical sense because of its simplicity and freedom from need of parameter estimation.     

Selection in complex genetic systems. VI. Equilibrium properties of two locus selection models with partial selfing

The results of a combined analytical and numerical study of two locus selection models with partial selfing indicate that several commonly held opinions about the effects of partial self-fertilization do not hold in general. For example, the heterozygosity of a population may actually increase as the selfing rate is increased. Similarly, selection strong enough to guarantee a two locus polymorphism with complete selfing does not necessarily guarantee a two locus polymorphism with intermediate amounts of self-fertilization. The results presented here and a brief review of previously existing results indicate that the predictions of population genetic models based on the assumption of random mating will not be greatly altered by a small amount of self-fertilization, unless the loci involved are tightly linked. On the other hand, the results presented indicate that a very small amount of outcrossing may lead to marked differences from the expectation based on complete self-fertilization.