On the two-locus sampling distribution

Two methods are discussed for evaluating the distribution of the configuration of unlabeled gametic types in a random sample of size n from the two-locus infinitely-many-neutral-alleles diffusion model at stationarity. Both involve finding systems of linear equations satisfied by the desired probabilities. The first approach, which is due to Golding, is to include additional probabilities in the system that allow some members of the sample to be specified at only one locus. The second approach, which is new, considers the joint distribution of the sample configuration and the number of recombination events since the time of the most recent common ancestor. The first approach is used for numerical computation, whereas the second approach is used to derive a two-locus version of Hoppe's urn model. The latter permits efficient simulation of the two-locus sampling distribution, provided the recombination parameter is not too large.Supported in part by NSF grants DMS-8704369 and DMS-8902991     

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.     

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.     

Nature of allelic sequence polymorphism at the KIR3DL3 locus

KIR3DL3 is a framework gene of the Leukocyte Receptor Complex, present in all individuals and haplotypes analysed to date. We describe 17 novel KIR3DL3 alleles, including seven single nucleotide polymorphic (SNP) positions within the coding region. Sequence variation within introns included a VNTR within intron 1. As KIR3DL3 mRNA is known to be expressed in decidual NK cells, we investigated the impact of KIR3DL3 allelic variation on pre-eclampsia. No statistical difference in allele frequency or polymorphism was observed between pre-eclampsia patient and control cohorts. Linkage disequilibrium (LD) analysis of exonic SNPs suggested that recombination may be a mechanism of generating sequence diversity within KIR3DL3. A potential recombination hotspot was located within intron 5. A strong LD was detected between polymorphism in exon 6 of KIR3DL3 and the KIR gene −2DL3 or -2DS2 loci, which define the centromeric end of two main haplotypes (A and B) of the KIR cluster. Comparison of primate KIR sequences indicated that the Ig domains of KIR3DL3 are highly conserved between chimpanzee, gorilla and humans. Investigation of KIR3DL3 dN/dS ratios indicated a greater level of synonymous mutations consistent with purifying selection, although positive selection was detected acting on two sites within the stem region.Electronic supplementary material Supplementary material is available for this article at and is accessible for authorized users.     

Coexistence of competing species by the oscillation of polymorphisms

Scale-eating cichlids in Lake Tanganyika exhibit genetically determined lateral asymmetry, especially in their mouth-opening. Frequencies of the morphs oscillate due to strong frequency-dependent selection caused by the switching of prey's attention, and its delayed effect by their growth period. Two scale-eaters coexist in similar densities at south shore of the lake, with their morph frequencies oscillating in phase. We investigated the effect of the oscillation in morph frequencies to the coexistence of competing species. If the difference of two species' growth period is large, the oscillation facilitates the coexistence of the two species, while small difference of growth periods hinders their coexistence. In the latter case, the species with shorter growth period drives the other species to the extinction.     

Polymorphism in the two-locus Levene model with nonepistatic directional selection

For the Levene model with soft selection in two demes, the maintenance of polymorphism at two diallelic loci is studied. Selection is nonepistatic and dominance is intermediate. Thus, there is directional selection in every deme and at every locus. We assume that selection is in opposite directions in the two demes because otherwise no polymorphism is possible. If at one locus there is no dominance, then a complete analysis of the dynamical and equilibrium properties is performed. In particular, a simple necessary and sufficient condition for the existence of an internal equilibrium and sufficient conditions for global asymptotic stability are obtained. These results are extended to deme-independent degree of dominance at one locus. A perturbation analysis establishes structural stability within the full parameter space. In the absence of genotype-environment interaction, which requires deme-independent dominance at both loci, nongeneric equilibrium behavior occurs, and the introduction of arbitrarily small genotype-environment interaction changes the equilibrium structure and may destroy stable polymorphism. The volume of the parameter space for which a (stable) two-locus polymorphism is maintained is computed numerically. It is investigated how this volume depends on the strength of selection and on the dominance relations. If the favorable allele is (partially) dominant in its deme, more than 20% of all parameter combinations lead to a globally asymptotically stable, fully polymorphic equilibrium.     

The two-locus ancestral graph in a subdivided population: convergence as the number of demes grows in the island model

We study the ancestral recombination graph for a pair of sites in a geographically structured population. In particular, we consider the limiting behavior of the graph, under Wrights island model, as the number of subpopulations, or demes, goes to infinity. After an instantaneous sample-size adjustment, the graph becomes identical to the two-locus graph in an unstructured population, but with a time scale that depends on the migration rate and the deme size. Interestingly, when migration is gametic, this rescaling of time increases the population mutation rate but does not affect the population recombination rate. We compare this to the case of a partially-selfing population, in which both mutation and recombination depend on the selfing rate. Our result for gametic migration holds both for finite-sized demes, and in the limit as the deme size goes to infinity. However, when migration occurs during the diploid phase of the life cycle and demes are finite in size, the population recombination rate does depend on the migration rate, in a way that is reminiscent of partial selfing. Simulations imply that convergence to a rescaled panmictic ancestral recombination graph occurs for any number of sites as the number of demes approaches infinity.Send offprint request to: Sabin LessardS. Lessard was supported by grants from the Natural Sciences and Research Council of Canada, the Fonds Québécois de la Recherche sur la Nature et les Technologies, and the Université de Montréal.J. Wakeley was supported by a Career Award (DEB-0133760) and by a grant (DEB-9815367) from the National Science Foundation.     

Protected polymorphism in the two-locus haploid model with unpredictable fitnesses

 In an unpredictable environment, the distributions of alleles from which polymorphism can be maintained forever belong to a certain set, the C-viability kernel. Such a set is calculated in the two-locus haploid model, as well as the corresponding fitnesses at any time which make this maintenance possible. The dependence of the C-viability kernel on the set U of admissible fitnesses and on the recombination rate r is studied. Notably, the C-viability kernel varies rapidly in the neighborhood of equal fitness of AB and ab; it becomes empty when ab has a fitness below a certain function, which is delineated, of the recombination rate. The properties of the two-locus model under constraints, out of equilibrium and with unpredictable selection are thus presented. Received: 20 May 1999     

The effects of linkage and gene flow on local adaptation: a two-locus continent-island model

Population subdivision and migration are generally considered to be important causes of linkage disequilibrium (LD). We explore the combined effects of recombination and gene flow on the amount of LD, the maintenance of polymorphism, and the degree of local adaptation in a subdivided population by analyzing a diploid, deterministic continent–island model with genic selection on two linked loci (i.e., no dominance or epistasis). For this simple model, we characterize explicitly all possible equilibrium configurations. Simple and intuitive approximations for many quantities of interest are obtained in limiting cases, such as weak migration, weak selection, weak or strong recombination. For instance, we derive explicit expressions for the measures and r² (the squared correlation in allelic state) of LD. They depend in qualitatively different ways on the migration rate. Remarkably high values of r² are maintained between weakly linked loci, especially if gene flow is low. We determine how the maximum amount of gene flow that admits preservation of the locally adapted haplotype, hence of polymorphism at both loci, depends on recombination rate and selection coefficients. We also investigate the evolution of differentiation by examining the invasion of beneficial mutants of small effect that are linked to an already present, locally adapted allele. Mutants of much smaller effect can invade successfully than predicted by naive single-locus theory provided they are at least weakly linked. Finally, the influence of linkage on the degree of local adaptation, the migration load, and the effective migration rate at a neutral locus is explored. We discuss possible consequences for the evolution of genetic architecture, in particular, for the emergence of clusters of tightly linked, slightly beneficial mutations and the evolution of recombination and chromosome inversions.     

Diffusion approximations of the two-locus Wright-Fisher model

Diffusion approximations are established for the multiallelic, two-locus Wright-Fisher model for mutation, selection, and random genetic drift in a finite, panmictic, monoecious, diploid population. All four combinations of weak or strong selection and tight or loose linkage are treated, though the proof in the case of strong selection and loose linkage is incomplete. Under certain conditions, explicit formulas are obtained for the stationary distributions of the two diffusions with loose linkage.Supported in part by NSF Grant DMS-8704369Supported in part by NSF Grant BSR-8512844     

微卫星跨物种交叉PCR扩增的一个新问题:扩增非同源产物

微卫星已被广泛应用于群体遗传学、生态学和进化生物学研究。然而,一些物种微卫星尚未克隆。为了节省时间和经费,研究人员往往使用一个物种已发表的微卫星引物扩增其近缘物种的微卫星。该研究对属于3个不同科(Clariidae、Heteropneustidae 和Pimelodidae)的7个鲶鱼物种的微卫星跨物种PCR扩增产物进行了序列分析,研究发现扩增非同源（non-orthologous）产物是微卫星跨物种PCR扩增的一个新问题。该研究共采用4对胡子鲶微卫星座位引物对7个鲶鱼物种进行了跨物种PCR扩增。对获得的204个PCR产物的序列分析结果表明,两对微卫星座位引物扩增了所有7个物种的同源特异产物。而其他两个座位的引物扩增了特异但非同源的多态产物,对近缘物种的扩增也获得类似结果。另外,除胡子鲶等位基因大小异源同型(size homoplasy)的特征不明显外,其他物种在3个微卫星座位都具有这一非常明显的特征。这些数据表明,微卫星跨物种间交叉扩增能产生非同源产物;等位基因大小异源同型与微卫星座位本身有关,而与物种间的亲缘关系无明显的相关性。微卫星跨物种扩增产生的非同源产物和等位基因大小异源同型将使系统发育、群体遗传学和进化研究明显复杂化。因此,在应用微卫星跨物种交叉扩增数据以前,最好对跨物种交叉扩增产物进行测序验证。     

The effects of selective genotyping on estimates of proportion of recombination between linked quantitative trait loci

Selective genotyping is the marker assay of only the more extreme phenotypes for a quantitative trait and is intended to increase the efficiency of quantitative trait loci (QTL) mapping. We show that selective genotyping can bias estimates of the recombination frequency between linked QTLs — upwardly when QTLs are in repulsion phase, and downwardly when QTLs are in coupling phase. We examined these biases under simple models involving two QTLs segregating in a backcross or F₂ population, using both analytical models and computer simulations. We found that bias is a function of the proportion selected, the magnitude of QTL effects, distance between QTLs and the dominance of QTLs. Selective genotyping thus may decrease the power of mapping multiple linked QTLs and bias the construction of a marker map. We suggest a large proportion than previously suggested (50%) or the entire population be genotyped if linked QTLs of large effects (explain > 10% phenotypic variance) are evident. New models need to be developed to explicitly incorporate selection into QTL map construction.     

Some additional results about polymorphisms in models of an additive quantitative trait under stabilizing selection

 The existence of two stable, symmetric (allelic frequency 0.5 in each locus) polymorphic states is demonstrated for a two-locus model of an additive quantitative trait under strong Gaussian selection. Linkage disequilibrium at one of the states is negative whereas it is positive at the other state. For a three-locus model, it is shown that in order to maintain a stable polymorphism in all three loci, selection must be sufficiently but not exces- sively strong relative to recombination. Also, positive linkage disequilibrium can be maintained in a three-locus model under stabilizing selection that is not very strong. Received 15 July 1995     

Herbicide response polymorphisms in wild emmer wheat: ecological and isozyme correlations

Summary We demonstrate that the scores and frequencies of chlortoluron (CT) and metoxuron (MX) resistance and susceptible phenotypes of wild emmer wheat, Triticum dicoccoides, are correlated with ecological factors and allozyme markers. Some isozyme markers located on chromosome 6B (e.g. Adh,Est-4 and Got), which also harbours the CT and MX resistance gene, provide good genetic markers for herbicide resistance breeding. Significant correlations between herbicide and photosynthetic characters suggest that the evolution of herbicide resistance polymorphisms may be related to the process of photosynthesis in nature and predated domestication of cultivated wheat.     

Selection by humans searching for computer- generated prey images: the effect of prey density

We investigated the effect of 'prey' density on frequency-dependent selection by human 'predators'. Eighty subjects were presented with computer-generated populations of two cryptically coloured characters on the screen of a colour monitor. Each subject was given the prey at a single combination of one of two frequencies and one of five densities, and was instructed to delete the prey as fast as possible with a light-pen. The results suggested that the degree of selection was inversely proportional to prey density, but there was no evidence that any frequency-dependent component of selection was affected.     

The evolutionary consequences of niche construction: a theoretical investigation using two-locus theory

This paper addresses the joint evolution of environment-altering (niche constructing) traits, and traits whose fitness depends on alterable sources of natural selection in environments. We explore the evolutionary consequences of this niche construction using a two-locus population genetic model. The novel conclusions are that niche construction can (1) cause evolutionary inertia and momentum, (2) lead to the fixation of otherwise deleterious alleles, (3) support stable polymorphisms where none are expected, (4) eliminate what would otherwise be stable polymorphisms, and (5) influence disequilibrium. The results suggest that the changes that organisms bring about in their niche can themselves be an important source of natural selection pressures, and imply that evolution may proceed in cycles of selection and niche construction.     

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.     

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.