Second-order approximations for selection coefficients at polygenic loci

I determine the second-order approximation for the phenotypic distribution of an arbitrary number of quantitative traits, ignoring the effects of epistasis and linkage disequilibrium, conditioned on the presence of a specified genotype at one underlying locus of small effect. Using this approximation, I determine formulae for the effects of selection at a single locus with random mating under either Gaussian stabilizing selection, or correlated selection with truncation selection for one character. These formulae apply for arbitrary phenotypic distributions, yet even with multivariate Gaussian distributions of phenotypic effects the formula for correlated selection includes a correction to the standard formula in Falconer (1989). 1 demonstrate that this approximation has an error that is third order in the allelic or genotypic effects, independent of the form of the phenotypic distribution. I show also that the approximation of analogous form for the phenotypic distribution conditioned on the presence of a specified allele at a single locus is also correct to second order. Both approximations allow for dominance and are consistent in the sense that computing marginal fitnesses from approximations based on genotypic deviations and those based on average allelic effect yield the same answers.Supported by PHS Grant ROI GM 32130     

Second-order approximations for selection coefficients at polygenic loci

I determine the second-order approximation for the phenotypic distribution of a quantitative trait, ignoring the effects of epistasis and linkage disequilibrium, conditioned on the presence of a specified genotype at one underlying locus of small effect. I demonstrate that this approximation has an error that is third order in the allelic or genotypic effects, independent of the form of the phenotypic distribution. I also show that the approximation of analogous form for the phenotypic distribution conditioned on the presence of a specified allele at a single locus is also correct to second order. Both approximations allow for dominance and are consistent in the sense that computing marginal fitnesses from approximations based on genotypic deviations and those based on average allelic effect yield the same answers. Surprisingly, the second-order approximations derived here yield the same approximation for dynamics at a single locus as first-order approximations used earlier thus justifiying earlier stability computations based on these first-order approximations.     

A method for estimating the effective number of loci affecting a quantitative character

A likelihood method is introduced that jointly estimates the number of loci and the additive effect of alleles that account for the genetic variance of a normally distributed quantitative character in a randomly mating population. The method assumes that measurements of the character are available from one or both parents and an arbitrary number of full siblings. The method uses the fact, first recognized by Karl Pearson in 1904, that the variance of a character among offspring depends on both the parental phenotypes and on the number of loci. Simulations show that the method performs well provided that data from a sufficient number of families (on the order of thousands) are available. This method assumes that the loci are in Hardy–Weinberg and linkage equilibrium but does not assume anything about the linkage relationships. It performs equally well if all loci are on the same non-recombining chromosome provided they are in linkage equilibrium. The method can be adapted to take account of loci already identified as being associated with the character of interest. In that case, the method estimates the number of loci not already known to affect the character. The method applied to measurements of crown–rump length in 281 family trios in a captive colony of African green monkeys (Chlorocebus aethiopus sabaeus) estimates the number of loci to be 112 and the additive effect to be 0.26 cm. A parametric bootstrap analysis shows that a rough confidence interval has a lower bound of 14 loci.     

Efficiency of direct selection on quantitative trait loci for a two-trait breeding objective

Selection on known loci affecting quantitative traits (DSQ) was compared to phenotypic selection index for a single and a two-trait selection objective. Two situations were simulated; a single known quantitative locus, and ten identified loci accounting for all the additive genetic variance. Selection efficiency of DSQ relative to traitbased selection was higher for two-trait selection, than was selection on a single trait with the same heritability. The advantage of DSQ was greater when the traits were negatively correlated. Relative selection efficiency (RSE) for a single locus responsible for 0.1 of the genetic variance was 1.11 with heritabilities of 0.45 and 0.2 and zero genetic and phenotypic correlations between the traits. RSE of DSQ for ten known loci was 1.5 to 1.8 in the first 3 generations of selection, but declined in each subsequent generation. With DSQ most loci reached fixation after 7 generations. Response to trait-based selection continued through generation 15 and approached the response obtained with DSQ after 10 generations. The cumulative genetic response after 10 generations of DSQ was only 93% to 97% of the economically optimum genotype because the less favorable allele reached fixation for some loci, generally those with effects in opposite directions on the two traits.     

Dynamics of a quantitative character subject only to stabilising selection

The implications of stabilising selection on a quantitative trait, in the absence of other evolutionary forces, are theoretically investigated in a randomly mating population. The dynamics of various statistics that describe the alleles contributing to the trait are determined and used to infer the behaviour of the trait. Dynamical solutions of the distribution of allelic effects and the distribution of the trait are found when all initial distributions of allelic effects are Gaussian and linkage disequilibria are neglected. Some results for the behaviour of the mean and the variance of genotypic effects of the population, when subject to a moving optimum, are derived. When the initial distributions of allelic effects are not Gaussian, but possess a small asymmetry, the mean and the variance of the allelic effects differ only slightly from the Gaussian results. By contrast, the third central moments of allelic effects, are, at all loci, strictly zero in the Gaussian case but are generally non-zero for non-symmetric initial distributions. To leading order in a quantitative measure of the asymmetry of the distribution, we determine the third central moment of allelic effects.     

Assortative mating for a quantitative character

Phenotypic assortative mating is investigated for a character determined by additive loci without dominance and a stochastically independent environment. Conditional-expectation arguments are used to calculate the equilibrium values of the phenotypic variance and the correlation between sundry relatives. For the latter, it suffices to suppose that the regressions of an individual's genotype on his phenotype and of his phenotype on that of his mate are linear. For the former, linearity of the regression of the allelic effects on the phenotype is also posited. The biological implications of these assumptions are discussed.Supported by National Science Foundation Grant DEB81-03530     

High level of functional polymorphism indicates a unique role of natural selection at human immune system loci

Several studies have shown that immune system proteins have on average a higher rate of amino acid evolution between different species of mammals than do most other proteins. To test whether immune-system-expressed loci show a correspondingly elevated rate of within-species nonsynonymous (amino acid altering) polymorphism, we examined gene diversity (heterozygosity) at 4,911 single nucleotide polymorphism (SNP) sites at 481 protein-coding loci. At loci with nonimmune functions, gene diversity at nonsynonymous SNP sites was typically lower than that at silent SNP sites (those not altering the amino acid sequence) in the same gene, a pattern that is an evidence of purifying selection acting to eliminate slightly deleterious variants. However, this pattern was not seen at nonsynonymous SNPs causing conservative amino acid replacements in immune system proteins, indicating that the latter are subject to a reduced level of functional constraint. Similarly, immune system genes showed higher gene diversities in their 5′ noncoding regions than did other proteins. These results identified certain immune system loci that are likely to be subject to balancing selection that acts to maintain polymorphism in either coding or regulatory regions. Electronic Supplementary Material Supplementary material is available for this article at .     

The generalized multiplicative model for viability selection at multiple loci

Selection due to differential viability is studied in an n-locus two-allele model using a set indexation that allows the simplicity of the one-locus two-allele model to be carried to multi-locus models. The existence condition is analyzed for polymorphic equilibria with linkage equilibrium: Robbins' equilibria. The local stability condition is given for the Robbins' equilibria on the boundaries in the generalized non-epistatic selection regimes of Karlin and Liberman (1979). These generalized non-epistatic regimes include the additive selection model, the multiplicative selection model and the multiplicative interaction model, and their symmetric versions cover all the symmetric viability models.Research supported by grant no. 11-7805 from the Danish Natural Science Research Council, by NIH grant GM 28016, by a fellowship from the Research Foundation of Aarhus University, and by a visiting fellowship from the University of New England, N.S.W.     

Comparing measures of breeding inequality and opportunity for selection with sexual selection on a quantitative character in bighorn rams

The reliability and consistency of the many measures proposed to quantify sexual selection have been questioned for decades. Realized selection on quantitative characters measured by the selection differential i was approximated by metrics based on variance in breeding success, using either the opportunity for sexual selection I_s or indices of inequality. There is no consensus about which metric best approximates realized selection on sexual characters. Recently, the opportunity for selection on character mean OSM was proposed to quantify the maximum potential selection on characters. Using 21 years of data on bighorn sheep (Ovis canadensis), we investigated the correlations between seven indices of inequality, I_s, OSM and i on horn length of males. Bighorn sheep are ideal for this comparison because they are highly polygynous and sexually dimorphic, ram horn length is under strong sexual selection, and we have detailed knowledge of individual breeding success. Different metrics provided conflicting information, potentially leading to spurious conclusions about selection patterns. I_δ, an index of breeding inequality, and, to a lesser extent, I_s showed the highest correlation with i on horn length, suggesting that these indices document breeding inequality in a selection context. OSM on horn length was strongly correlated with i, I_s and indices of inequality. By integrating information on both realized sexual selection and breeding inequality, OSM appeared to be the best proxy of sexual selection and may be best suited to explore its ecological bases.     

A note on the application of diallel crosses for the analysis of genetic variation in natural populations

Summary The complete diallel cross among homozygous lines can be a useful tool to analyze the genetic architecture of natural populations. However, it represents the natural population only approximately, in particular if the number of lines is small and the analyzed traits exhibit inbreeding depression or other forms of directional dominance. Some incorrect expected mean squares that can be found in the literature suggest tests for genetic variance components that can be misleading under such circumstances. Expected mean squares for a factorial analysis and for a modified Hayman analysis are presented and the effect of the number of lines and directional dominance is discussed.     

Effect on linkage disequilibrium of selection for a quantitative character with epistasis

Summary Selection for a character controlled by additive genes induces linkage disequilibrium which reduces the additive genetic variance usable for further selective gains. Additive x additive epistasis contributes to selection response through development of linkage disequilibrium between interacting loci. To investigate the relative importance of the two effects of linkage disequilibrium, formulae are presented and results are reported of simulations using models involving additive, additive x additive and dominance components. The results suggest that so long as epistatic effects are not large relative to additive effects, and the proportion of pairs of loci which show epistasis is not very high, the predominant effect of linkage disequilibrium will be to reduce the rate of selection response.     

Model selection for quantitative trait loci mapping in a full-sib family

Statistical methods for mapping quantitative trait loci (QTLs) in full-sib forest trees, in which the number of alleles and linkage phase can vary from locus to locus, are still not well established. Previous studies assumed that the QTL segregation pattern was fixed throughout the genome in a full-sib family, despite the fact that this pattern can vary among regions of the genome. In this paper, we propose a method for selecting the appropriate model for QTL mapping based on the segregation of different types of markers and QTLs in a full-sib family. The QTL segregation patterns were classified into three types: test cross (1:1 segregation), F₂ cross (1:2:1 segregation) and full cross (1:1:1:1 segregation). Akaike’s information criterion (AIC), the Bayesian information criterion (BIC) and the Laplace-empirical criterion (LEC) were used to select the most likely QTL segregation pattern. Simulations were used to evaluate the power of these criteria and the precision of parameter estimates. A Windows-based software was developed to run the selected QTL mapping method. A real example is presented to illustrate QTL mapping in forest trees based on an integrated linkage map with various segregation markers. The implications of this method for accurate QTL mapping in outbred species are discussed.     

A new test for detecting ongoing selection

Much effort has been made to search for signatures of past natural selection in DNA sequences. However, currently acting selection is rarely detected in natural populations because of its rarity, low detection power of available methods, or both. Here, we develop a new test to detect viability selection over a single generation. In this test, one specific type of chromosomes is chosen as a reference, while all other chromosomes are designated as "focal". The test compares measures of variation between two groups of "focal" chromosomes: those found in reference/focal heterozygous individuals and those found in focal/focal homozygous individuals. In the absence of selection, we do not expect differences between these two groups as long as mating is random. On the other hand, currently acting selection can cause differences in some measures of variation. We applied this test to typing data for In(2L)t inversion polymorphism in a Drosophila melanogaster population, using "standard" (non-inverted) chromosomes as the focal class. Although the frequencies of In(2L)t and standard chromosomes did not deviate from the Hardy-Weinberg equilibrium, we found differences in allele frequency and the number of haplotypes between the two groups of standard chromosomes. This new test, in conjunction with the Hardy-Weinberg test, may shed light on how often strong selection is operating in extant populations.     

Genetic polymorphism at two linked loci,Sod and Est-6, in Drosophila melanogaster

We have examined the patterns of polymorphism at two linked loci, Sod and Est-6, separated by nearly 1000 kb on the left arm of chromosome 3 of Drosophila melanogaster. The evidence suggests that natural selection has been involved in shaping the polymorphisms. At the Sod locus, a fairly strong (s>0.01) selective sweep, started ≥2600 years ago, increased the frequency of a rare haplotype, F(A), to about 50% frequency in populations of Europe, Asia, and the Americas. More recently, an F(A) allele mutated to an S allele, which has increased to frequencies 5–15% in populations of Europe, Asia and North America. All S alleles are identical (or very nearly) in sequence and differ by one nucleotide substitution (which accounts for the F→S electrophoretic difference) from F(A) alleles. At the Est-6 locus, the evidence indicates both directional and balancing selection impacting separately the promoter and the coding regions of the gene, with linkage disequilibrium occurring within each region. Some linkage disequilibrium also exists between the two genes.     

Evidence for selection at cytokine loci in a natural population of field voles (Microtus agrestis)

Individuals in natural populations are frequently exposed to a wide range of pathogens. Given the diverse profile of gene products involved in responses to different types of pathogen, this potentially results in complex pathogen-specific selection pressures acting on a broad spectrum of immune system genes in wild animals. Thus far, studies into the evolution of immune genes in natural populations have focused almost exclusively on the Major Histocompatibility Complex (MHC). However, the MHC represents only a fraction of the immune system and there is a need to broaden research in wild species to include other immune genes. Here, we examine the evidence for natural selection in a range of non-MHC genes in a natural population of field voles (Microtus agrestis). We concentrate primarily on genes encoding cytokines, signalling molecules critical in eliciting and mediating immune responses and identify signatures of natural selection acting on several of these genes. In particular, genetic diversity within Interleukin 1 beta and Interleukin 2 appears to have been maintained through balancing selection. Taken together with previous findings that polymorphism within these genes is associated with variation in resistance to multiple pathogens, this suggests that pathogen-mediated selection may be an important force driving genetic diversity at cytokine loci in voles and other natural populations. These results also suggest that, along with the MHC, preservation of genetic variation within cytokine genes should be a priority for the conservation genetics of threatened wildlife populations.     

The genetic consequences of selection in natural populations

The selection coefficient, s, quantifies the strength of selection acting on a genetic variant. Despite this parameter's central importance to population genetic models, until recently we have known relatively little about the value of s in natural populations. With the development of molecular genetic techniques in the late 20th century and the sequencing technologies that followed, biologists are now able to identify genetic variants and directly relate them to organismal fitness. We reviewed the literature for published estimates of natural selection acting at the genetic level and found over 3000 estimates of selection coefficients from 79 studies. Selection coefficients were roughly exponentially distributed, suggesting that the impact of selection at the genetic level is generally weak but can occasionally be quite strong. We used both nonparametric statistics and formal random‐effects meta‐analysis to determine how selection varies across biological and methodological categories. Selection was stronger when measured over shorter timescales, with the mean magnitude of s greatest for studies that measured selection within a single generation. Our analyses found conflicting trends when considering how selection varies with the genetic scale (e.g., SNPs or haplotypes) at which it is measured, suggesting a need for further research. Besides these quantitative conclusions, we highlight key issues in the calculation, interpretation, and reporting of selection coefficients and provide recommendations for future research.     

Action of natural selection in lines of Drosophila selected for a new level of canalization

Summary Four lines of Drosophila melanogaster previously selected for a stabilized phenotype of two extra dorsocentral bristles were examined for 20 generations of canalizing selection and relaxation of selection. A substantial frequency of flies with either two anterior or two posterior extra bristles was maintained in the relaxed lines. These patterns were the only ones tolerated by natural selection, i.e., the only symmetric ones. It was concluded that anterior and posterior dorsocentral bristles are two independent development structures, and the results are discussed in relation to two proposed genetic systems for bristle determination.     

Artificial selection for 18-day pupa weight and opposing simulated naturel selection in Tribolium castaneum

Summary The effect of simulated opposing natural selection on the response to mass selection for 18-day pupa weight of Tribolium castaneum was studied for 10 generations of selection. Natural selection was simulated in replicated treatment lines by imposing a negative relationship between mid-parent genetic value for pupa weight and fertility. Responses to selection and realized heritabilities were smaller (P < 0.05 and P < 0.10, respectively) for the treatment lines than for control lines under selection for pupa weight only. One treatment, line E3G1, reached an intermediate selection plateau by generation 10, and responded linearly to 4 generations of artificial selection after natural selection had been discontinued. Possible explanations for the different behaviors of the replicate lines E3G1 and E3G2 were also discussed.