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.     

Recombination and selection at Brassica self-incompatibility loci

In Brassica species, self-incompatibility is controlled genetically by haplotypes involving two known genes, SLG and SRK, and possibly an as yet unknown gene controlling pollen incompatibility types. Alleles at the incompatibility loci are maintained by frequency-dependent selection, and diversity at SLG and SRK appears to be very ancient, with high diversity at silent and replacement sites, particularly in certain "hypervariable" portions of the genes. It is important to test whether recombination occurs in these genes before inferences about function of different parts of the genes can be made from patterns of diversity within their sequences. In addition, it has been suggested that, to maintain the relationship between alleles within a given S-haplotype, recombination is suppressed in the S-locus region. The high diversity makes many population genetic measures of recombination inapplicable. We have analyzed linkage disequilibrium within the SLG gene of two Brassica species, using published coding sequences. The results suggest that intragenic recombination has occurred in the evolutionary history of these alleles. This is supported by patterns of synonymous nucleotide diversity within both the SLG and SRK genes, and between domains of the SRK gene. Finally, clusters of linkage disequilibrium within the SLG gene suggest that hypervariable regions are under balancing selection, and are not merely regions of relaxed selective constraint.     

Social selection in human populations: fitness modification of offspring by an affected parent at two loci.

The effect of social reaction to a disease at one locus (the B locus) on the frequencies of deleterious genes at that locus and another locus (the A locus) has been studied. Social reaction is formulated so that affected parents at the B locus reduce or increase their offsprings' fitnesses. It has been shown that the genotype frequencies at the two loci depend strongly on the social reaction, but, because the magnitude of linkage disequilibrium is very small, the gene frequency at each locus can be treated as if each locus were segregating independently. Consequently, the model can be extended so that many rare diseases can be considered simultaneously.     

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.     

Polymorphism and differential selection for the sexes.

Various genetic models with different fitnesses for the sexes are investigated. Only a limited set of fitness values will result in a stable polymorphism, and the rate of approach to these equilibrium frequencies is extremely slow. These results indicate that there are problems associated with the interpretation of some human genetic traits by such models.     

Optimal selection on two quantitative trait loci with linkage

A mathematical approach to optimize selection on multiple quantitative trait loci (QTL) and an estimate of residual polygenic effects was applied to selection on two linked or unlinked additive QTL. Strategies to maximize total or cumulative discounted response over ten generations were compared to standard QTL selection on the sum of breeding values for the QTL and an estimated breeding value for polygenes, and to phenotypic selection. Optimal selection resulted in greater response to selection than standard QTL or phenotypic selection. Tight linkage between the QTL (recombination rate 0.05) resulted in a slightly lower response for standard QTL and phenotypic selection but in a greater response for optimal selection. Optimal selection capitalized on linkage by emphasizing selection on favorable haplotypes. When the objective was to maximize total response after ten generations and QTL were unlinked, optimal selection increased QTL frequencies to fixation in a near linear manner. When starting frequencies were equal for the two QTL, equal emphasis was given to each QTL, regardless of the difference in effects of the QTL and regardless of the linkage, but the emphasis given to each of the two QTL was not additive. These results demonstrate the ability of optimal selection to capitalize on information on the complex genetic basis of quantitative traits that is forthcoming.     

Contrasting effects of selection on sequence diversity and linkage disequilibrium at two phytoene synthase loci

We investigated the effects of human selection for yellow endosperm color, representing increased carotenoid content, on two maize genes, the Y1 phytoene synthase and PSY2, a putative second phytoene synthase. Multiple polymorphic sites were identified at Y1 and PSY2 in 75 white and yellow maize inbred lines. Many polymorphic sites showed strong association with the endosperm color phenotype at Y1, but no detectable association was found at PSY2. Nucleotide diversity was equivalent for whites and yellows at PSY2 but was 19-fold less in yellows than in whites at Y1, consistent with the white ancestral state of the gene. The strong sequence haplotype conservation within yellows at Y1 and a significant, negative Tajima's D both verified positive selection for yellow endosperm. We propose that two independent gain-of-function events associated with insertions into the promoter of the Y1 gene and upregulation of expression in endosperm have been incorporated into yellow maize.     

Polymorphism at the ovine major histocompatibility complex class II loci

Southern hybridization analysis of the ovine major histocompatibility complex (MHC) ( MhcOvar ) class II region, using sheep-specific probes for the DQA1, DQA2, DQB and DRA loci, has revealed extensive polymorphism. DQA1 and DQAP had eight and 16 alleles respectively, DQB had six and DRA had three alleles. Little information was derived from the DRB locus owing to extensive cross-hybridization between the DRB probe and the DQB locus. Differences in allele frequency between breeds were revealed. At the DQA1 locus a null allele (DQA1-N) was observed with a frequency of between 27% and 45%, making this the most common DQA1 allele in all breeds examined. The frequency of DQA1-N homozygotes was between 11% and 18%, raising questions as to the functional significance of the DQA1 gene. Linkage analysis between the DQA1, DQA2, DQB and DRA loci did not reveal any recombination.     

Analysis of metabolic networks using a pathway distance metric through linear programming

The solution of the shortest path problem in biochemical systems constitutes an important step for studies of their evolution. In this paper, a linear programming (LP) algorithm for calculating minimal pathway distances in metabolic networks is studied. Minimal pathway distances are identified as the smallest number of metabolic steps separating two enzymes in metabolic pathways. The algorithm deals effectively with circularity and reaction directionality. The applicability of the algorithm is illustrated by calculating the minimal pathway distances for Escherichia coli small molecule metabolism enzymes, and then considering their correlations with genome distance (distance separating two genes on a chromosome) and enzyme function (as characterised by enzyme commission number). The results illustrate the effectiveness of the LP model. In addition, the data confirm that propinquity of genes on the genome implies similarity in function (as determined by co-involvement in the same region of the metabolic network), but suggest that no correlation exists between pathway distance and enzyme function. These findings offer insight into the probable mechanism of pathway evolution.     

Evaluating human autosomal loci for sexually antagonistic viability selection in two large biobanks

Sex and sexual differentiation are pervasive across the tree of life. Because females and males often have substantially different functional requirements, we expect selection to differ between the sexes. Recent studies in diverse species, including humans, suggest that sexually antagonistic viability selection creates allele frequency differences between the sexes at many different loci. However, theory and population-level simulations indicate that sex-specific differences in viability would need to be very large to produce and maintain reported levels of between-sex allelic differentiation. We address this contradiction between theoretical predictions and empirical observations by evaluating evidence for sexually antagonistic viability selection on autosomal loci in humans using the largest cohort to date (UK Biobank, n = 487,999) along with a second large, independent cohort (BioVU, n = 93,864). We performed association tests between genetically ascertained sex and autosomal loci. Although we found dozens of genome-wide significant associations, none replicated across cohorts. Moreover, closer inspection revealed that all associations are likely due to cross-hybridization with sex chromosome regions during genotyping. We report loci with potential for mis-hybridization found on commonly used genotyping platforms that should be carefully considered in future genetic studies of sex-specific differences. Despite being well powered to detect allele frequency differences of up to 0.8% between the sexes, we do not detect clear evidence for this signature of sexually antagonistic viability selection on autosomal variation. These findings suggest a lack of strong ongoing sexually antagonistic viability selection acting on single locus autosomal variation in humans.     

The signature of positive selection at randomly chosen loci

In Drosophila and humans, there are accumulating examples of loci with a significant excess of high-frequency-derived alleles or high levels of linkage disequilibrium, relative to a neutral model of a random-mating population of constant size. These are features expected after a recent selective sweep. Their prevalence suggests that positive directional selection may be widespread in both species. However, as I show here, these features do not persist long after the sweep ends: The high-frequency alleles drift to fixation and no longer contribute to polymorphism, while linkage disequilibrium is broken down by recombination. As a result, loci chosen without independent evidence of recent selection are not expected to exhibit either of these features, even if they have been affected by numerous sweeps in their genealogical history. How then can we explain the patterns in the data? One possibility is population structure, with unequal sampling from different subpopulations. Alternatively, positive selection may not operate as is commonly modeled. In particular, the rate of fixation of advantageous mutations may have increased in the recent past.