Epistasis in the multiple locus symmetric viability model

The n-locus two-allele symmetric viability model is considered in terms of the parameters measuring the additive epistasis in fitness. The dynamics is analysed using a simple linear transformation of the gametic frequencies, and then the recurrence equations depend on the epistatic parameters and Geiringer's recombination distribution only. The model exhibits an equilibrium, the central equilibrium, where the 2 ⁿ gametes are equally frequent. The transformation simplifies the stability analysis of the central point, and provides the stability conditions in terms of the existence conditions of other equilibria. For total negative epistasis (all epistatic parameters are negative) the central point is stable for all recombination distributions. For free recombination either a central point (segregating one, two, ... or n loci) or the n-locus fixation states are stable. For no recombination and some epistatic parameters positive the central point is unstable and several boundary equilibria may be locally stable. The sign structure of the additive epistasis is therefore an important determinant of the dynamics of the n-locus symmetric viability model. The non-symmetric multiple locus models previously analysed are dynamically related, and they all have an epistatic sign structure that resembles that of the multiplicative viability model. A non-symmetric model with total negative epistasis which share dynamical properties with the similar symmetric model is suggested.Supported in part by NIH grant GM 28016, and by grant 81-5458 from the Danish Natural Science Research Council     

Global convergence properties in multilocus viability selection models: the additive model and the Hardy-Weinberg law

A natural coordinate system is introduced for the analysis of the global stability of the Hardy-Weinberg (HW) polymorphism under the general multilocus additive viability model. A global convergence criterion is developed and used to prove that the HW polymorphism is globally stable when each of the loci is diallelic, provided the loci are overdominant and the multilocus recombination is positive. As a corollary the multilocus Hardy-Weinberg law for neutral selection is derived.Research supported in part by NIH grants GM 39907-01, GM 10452-26 and NSF Grant DMS 86-06244Research supported in part by a US-Israel Binational Science Foundation grant 85-00021 and NIH grant GM 28016     

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 model for marker-assisted selection among single crosses with multiple genetic markers

 Trait means of marker genotypes are often inconsistent across experiments, thereby hindering the use of regression techniques in marker-assisted selection. Best linear unbiased prediction based on trait and marker data (TM-BLUP) does not require prior information on the mean effects associated with specific marker genotypes and, consequently, may be useful in applied breeding programs. The objective of this paper is to present a flanking-marker, TM-BLUP model that is applicable to interpopulation single crosses that characterize maize (Zea mays L.) breeding programs. The performance of a single cross is modeled as the sum of testcross additive and dominance effects at unmarked quantitative trait loci (QTL) and at marked QTL (MQTL). The TM-BLUP model requires information on the recombination frequencies between flanking markers and the MQTL and on MQTL variances. A tabular method is presented for calculating the conditional probability that MQTL alleles in two inbreds are identical by descent given the observed marker genotypes (G ^k _obs) at the kth MQTL. Information on identity by descent of MQTL alleles can then be used to calculate the conditional covariance of MQTL effects between single crosses given G ^k _obs. The inverse of the covariance matrix for dominance effects at unmarked QTL and MQTL can be written directly from the inverse of the covariance matrices of the corresponding testcross additive effects. In practice, the computations required in TM-BLUP may be prohibitive. The computational requirements may be reduced with simplified TM-BLUP models wherein dominance effects at MQTL are excluded, only the single crosses that have been tested are included, or information is pooled across several MQTL. Received: 22 June 1997 / Accepted: 25 February 1998     

The two-locus model of Gaussian stabilizing selection

We study the equilibrium structure of a well-known two-locus model in which two diallelic loci contribute additively to a quantitative trait that is under Gaussian stabilizing selection. The population is assumed to be infinitely large, randomly mating, and having discrete generations. The two loci may have arbitrary effects on the trait, the strength of selection and the recombination rate may also be arbitrary. We find that 16 different equilibrium patterns exist, having up to 11 equilibria; up to seven interior equilibria may coexist, and up to four interior equilibria, three in negative and one in positive linkage disequilibrium, may be simultaneously stable. Also, two monomorphic and two fully polymorphic equilibria may be simultaneously stable. Therefore, the result of evolution may be highly sensitive to perturbations in the initial conditions or in the underlying genetic parameters. For the special case of equal effects, global stability results are proved. In the general case, we rely in part on numerical computations. The results are compared with previous analyses of the special case of extremely strong selection, of an approximate model that assumes linkage equilibrium, and of the much simpler quadratic optimum model.     

The stationary distribution of allele frequencies when selection acts at unlinked loci

We consider population genetics models where selection acts at a set of unlinked loci. It is known that if the fitness of an individual is multiplicative across loci, then these loci are independent. We consider general selection models, but assume parent-independent mutation at each locus. For such a model, the joint stationary distribution of allele frequencies is proportional to the stationary distribution under neutrality multiplied by a known function of the mean fitness of the population. We further show how knowledge of this stationary distribution enables direct simulation of the genealogy of a sample at a single-locus. For a specific selection model appropriate for complex disease genes, we use simulation to determine what features of the genealogy differ between our general selection model and a multiplicative model.     

Drift and selection influence geographic variation at immune loci of prairie-chickens

Previous studies of immunity in wild populations have focused primarily on genes of the major histocompatibility complex (MHC); however, studies of model species have identified additional immune-related genes that also affect fitness. In this study, we sequenced five non-MHC immune genes in six greater prairie-chicken (Tympanuchus cupido) populations that have experienced varying degrees of genetic drift as a consequence of population bottlenecks and fragmentation. We compared patterns of geographic variation at the immune genes with six neutral microsatellite markers to investigate the relative effects of selection and genetic drift. Global F(ST) outlier tests identified positive selection on just one of five immune genes (IAP-1) in one population. In contrast, at other immune genes, standardized G'(ST) values were lower than those at microsatellites for a majority of pairwise population comparisons, consistent with balancing selection or with species-wide positive or purifying selection resulting in similar haplotype frequencies across populations. The effects of genetic drift were also evident as summary statistics (e.g., Tajima's D) did not differ from neutrality for the majority of cases, and immune gene diversity (number of haplotypes per gene) was correlated positively with population size. In summary, we found that both genetic drift and selection shaped variation at the five immune genes, and the strength and type of selection varied among genes. Our results caution that neutral forces, such as drift, can make it difficult to detect current selection on genes.     

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 effect of variable environments on polymorphism at loci with several alleles I. A symmetric model

Much of the extant polymorphism has been attributed to spatial and temporal variation among selection regimes. Analysis of models entailing two alleles at a single locus has demonstrated that polymorphism may result from variation among selection regimes which prescribe monomorphism if constant. This relationship is studied in the context of several alleles at a locus.One result which is not valid with only two alleles is that variation among selection regimes which specify polymorphic equilibria may lead to a stable monomorphic equilibrium. The analyses of temporal variation and total panmixia spatial variation among environments show that temporal variation allows the simultaneous stability of equilibrium configurations which cannot be simultaneously stable under total panmixia spatial variation (hard or soft selection). The principle that polymorphism is more readily maintained with spatial than temporal variation is invalidated.Supported in part by Purdue Research Foundation and National Science Foundation (USA) grant MCS-8002227     

Perturbation analysis of a two-locus model with directional selection and recombination

A population genetic two-locus model with additive, directional selection and recombination is considered. It is assumed that recombination is weaker than selection; i.e., the recombination parameter r is smaller than the selection coefficients. This assumption is appropriate for describing the effects of two-locus selection at the molecular level. The model is formulated in terms of ordinary differential equations (ODES) for the gamete frequencies x = (x ₁, x ₂, x ₃, x ₄), defined on the simplex S ₄. The ODEs are analyzed using first a regular pertubation technique. However, this approach yields satisfactory results only if r is very small relative to the selection coefficients and if the initial values x(0) are in the interior part of S ₄. To cope with this problem, a novel two-scale perturbation method is proposed which rests on the theory of averaging of vectorfields. It is demonstrated that the zeroth-order solution of this two-scale approach approximates the numerical solution of the model well, even if recombination rate is on the order of the selection coefficients.     

Variable selection for the single-index model

We consider variable selection in the single-index model. Weprove that the popular leave-m-out crossvalidation method hasdifferent behaviour in the single-index model from that in linearregression models or nonparametric regression models. A newconsistent variable selection method, called separated crossvalidation,is proposed. Further analysis suggests that the method has betterfinite-sample performance and is computationally easier thanleave-m-out crossvalidation. Separated crossvalidation, appliedto the Swiss banknotes data and the ozone concentration data,leads to single-index models with selected variables that havebetter prediction capability than models based on all the covariates.     

Divergent patterns of selection on the DAB and DXB MHC class II loci in Xiphophorus fishes

Two MHC class II loci, DAB (a classical class II locus) and DXB (putatively a non-classical class II locus), were sequenced in samples of individuals from two populations of swordtail fish, Xiphophorus multilineatus and X. pygmaeus. The DAB locus showed higher levels of genetic variation in the B1-encoding region, (putative binding region) than the DXB locus. We used two methods to investigate d_N/d_S ratios. The results from a maximum likelihood method based on phylogenetic relationships indicated positive selection on the B1 region of DAB (this method could not be used on DXB). Results from a coalescent-based method also showed evidence for positive selection in the B1 region of DAB, but only weak evidence for selection on the DXB. Further analyses indicated that recombination is an important source of variation in the B1 region of DAB, but has a relatively small effect on DXB. Overall, our results were consistent with the hypothesis that the DAB locus is under positive selection driven by antagonistic coevolution, and that the DXB locus plays the role of a non-classical MHC II locus. We also used simulations to investigate the presence of an elevated synonymous substitution rate in the binding region. The simulations revealed that the elevated rate could be caused by an interaction between positive selection and codon bias. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Haplotype structure and evidence for positive selection at the human IL13 locus

Interleukin-13 (IL13) is believed to play an important role in the pathogenesis of atopy and allergic asthma. To better understand genetic variation at the IL13 locus, we resequenced a 5.1-kb genomic region spanning the entire locus and identified 26 single-nucleotide polymorphisms (SNPs) in 74 individuals from three major populations-Chinese, Caucasian, and African. Our survey suggests exceptionally high and significant geographic structure at the IL13 locus between African and outside Africa populations. This unusual pattern suggests that positive selection that acts in some local populations may have played a role on the IL13 locus. In support of this suggestion, we found a significant excess of high frequency-derived SNPs in the Chinese population and Caucasian population, respectively, as expected after a recent episode of positive selection. Further, the unusual haplotype structure indicates that different scenarios of the action of positive selection on the IL13 locus in different populations may exist. In the Caucasian population, the skewed haplotype distribution dominated by one common haplotype supports the hypothesis of simple directional selection. Whereas, in the Chinese population, the two-round hitchhiking hypothesis may explain the skewed haplotype structure with three dominant ones. These findings may provide insight into the likely relative roles of selection and population history in establishing present-day variation at the IL13 locus, and, motivate further studies of this locus as an important candidate in common diseases association studies.     

Mutation loci and intragenic selection marker of the granule-bound starch synthase gene in waxy maize

Four pairs of specific PCR primers have been designed on the basis of the sequence of the granule-bound starch synthase gene (GBSS; dominant non-waxy gene Wx) and used to amplify its homologous sequence from thirteen waxy and two non-waxy inbred lines. Results from electrophoresis indicated that the recessive waxy gene was wx, derived from the dominant non-waxy gene Wx by mutation at its 3′ end. The sequence of the mutated 3′ end was amplified by the TAIL-PCR technique. Sequence alignment showed that the mutation of the wx gene was caused by transposition of the aldehyde dehydrogenase gene rf2. Two pairs of specific primers were designed on the basis of the sequence difference between the dominant gene Wx and its mutated recessive allele wx and used as intragenic selection markers to identify individual plants of genotypes WxWx, Wxwx, and wxwx by PCR amplification from the segregating population of the F₂ generation crossed between waxy and non-waxy inbred lines. Iodine solution staining and starch component assay showed that all the 35 F₂ plants identified as genotype WxWx produced non-waxy kernels of the F₃ generation and that all 33 F₂ plants identified as genotype wxwx produced waxy kernels of the F₃ generation. This result can be used to improve the selection efficiency of waxy maize breeding and for selection of other single genes and major polygenes.     

Footprints of intragenic recombination at HLA loci

 To evaluate the effect of balancing selection and intragenic recombination (or gene conversion) at six individual HLA loci, synonymous nucleotide diversity in different exon groups is examined within (π_w) and between (π_b) allelic lineages that may be defined by either serological or DNA sequence differences. Both π values are high in exons which encode for the peptide binding region (PBR) and tend to decrease in other exons. The value of π_w is significantly smaller than that of π_b in any exon of any locus. However, even π_w is much greater than nucleotide diversity at non-HLA loci. These observations provide additional strong evidence for the operation of balancing selection in PBR-encoding exons and its indirect effects on polymorphism at linked neighboring regions. It appears that allelic lineages have generally evolved in isolation but the linkage relationships within and between exons are incomplete throughout the long evolutionary history. To quantify intragenic recombination and account for the large discrepancy between the HLA and non-HLA diversity, a population genetics model is analyzed with special reference to the evolution of modern humans. The analysis suggests that the recombination rate between two sites 1000 base pairs apart is about 10^–5 per generation and that the effective size of human populations (equivalent roughly to the number of breeding individuals in a randomly mating population) has dropped from 10⁵ to 10⁴ in most of the Quaternary. One possibility for this reduction is discussed. Received: 11 August 1997 / Revised: 8 October 1997     

Robustness of statistical tests for multiplicative terms in the additive main effects and multiplicative interaction model for cultivar trials

The additive main effects multiplicative interaction model is frequently used in the analysis of multilocation trials. In the analysis of such data it is of interest to decide how many of the multiplicative interaction terms are significant. Several tests for this task are available, all of which assume that errors are normally distributed with a common variance. This paper investigates the robustness of several tests (Gollob, F _GH1, F_GH2, F_R)to departures from these assumptions. It is concluded that, because of its better robustness, the F _Rtest is preferable. If the other tests are to be used, preliminary tests for the validity of assumptions should be performed.