Linkage disequilibrium, selection and recombination at three Loci

Limits to the relationship among linkage disequilibrium, selection and recombination at equilibrium in three-locus, two-allele, deterministic, discrete generation models are determined using linear programming techniques. These results show that the commonly used measures of linkage disequilibrium are not appropriate for a multilocus setting. Additionally, interactions among three loci are important in reducing the strength of selection necessary to maintain a given level of disequilibrium, relative to a two-locus model.     

Three-Locus Systems Impose Additional Constraints on Pairwise Disequilibria

Combinations of allele frequencies and pairwise linkage disequilibrium terms, each of which is permissible at the two-locus level, may not always be permissible at the three-locus level. These additional constraints on the possible maximum and minimum values for the pairwise disequilibrium terms are formally determined and numerically analyzed. In some cases, the three-locus constraints on a pairwise disequilibrium (D) may be equivalent to the usual two-locus constraints, while in others, the positive or negative range may be restricted. This can result in situations where the allowable values of D are limited to only positive or only negative values up to the extreme case where there is only a single admissible value. No additional restrictions are placed on pairwise disequilibrium values when four loci are considered, other than those imposed by the three-way combinations containing the two loci of interest. A new measure of normalized pairwise linkage disequilibrium, allowing for the three-locus constraints, is defined and illustrated by an application to data from the human histocompatibility antigen (HLA) system. An analogous normalized three-way disequilibrium measure is also formulated.     

Disequilibrium among several linked neutral genes in finite population I. Mean changes in disequilibrium

Formulae are developed for computing changes in expected values in a finite population of linkage disequilibrium among neutral genes from more than two loci, although the exact analysis is taken up to only six loci. An essentially haploid model is used. As with two loci, the three-locus disequilibrium declines exponentially at all generations, but for m > 3 loci a matrix has to be constructed to give joint changes in the m-locus disequilibrium and products of disequilibria with fewer loci, for example of two m/2-locus disequilibria. The asymptotic rates of change in multilocus disequilibria depend on the arrangement of genes on the chromosome as well as its total length, but the initial rate of breakdown of disequilibrium from a line cross base is less dependent on the arrangement. With equally spaced loci the asymptotic rate of breakdown of m locus disequilibrium is roughly proportional to m. Although mutation and interference are excluded from the main analysis, it is shown how they can be incorporated.     

Selection in Complex Genetic Systems I. the Symmetric Equilibria of the Three-Locus Symmetric Viability Model

The symmetric equilibria of the three-locus symmetric viability model are determined and their stability analyzed. For tight linkage there may be four stable equilibria, each characterized by having one pair of complementary chromosomes in high frequencies, with all others low. For looser linkage the only stable symmetric equilibrium is that with complete linkage equilibrium. For intermediate recombination values both types of equilibria may be stable. A new class of equilibria with all pairwise linkage disequilibria zero, but with third order linkage disequilibrium, has been discovered. It may be stable for tight linkage.     

The hitchhiking effect on linkage disequilibrium between linked neutral loci

We analyzed a three-locus model of genetic hitchhiking with one locus experiencing positive directional selection and two partially linked neutral loci. Following the original hitchhiking approach by Maynard Smith and Haigh, our analysis is purely deterministic. In the first half of the selected phase after a favored mutation has entered the population, hitchhiking may lead to a strong increase of linkage disequilibrium (LD) between the two neutral sites if both are <0.1 s away from the selected site (where s is the selection coefficient). In the second half of the selected phase, the main effect of hitchhiking is to destroy LD. This occurs very quickly (before the end of the selected phase) when the selected site is between both neutral loci. This pattern cannot be attributed to the well-known variation-reducing effect of hitchhiking but is a consequence of secondary hitchhiking effects on the recombinants created in the selected phase. When the selected site is outside the neutral loci (which are, say, <0.1s apart), however, a fast decay of LD is observed only if the selected site is in the immediate neighborhood of one of the neutral sites (i.e., if the recombination rate r between the selected site and one of the neutral sites satisfies r<0.1 s). If the selected site is far away from the neutral sites (say, r > 0.3 s), the decay rate of LD approaches that of neutrality. Averaging over a uniform distribution of initial gamete frequencies shows that the expected LD at the end of the hitchhiking phase is driven toward zero, while the variance is increased when the selected site is well outside the two neutral sites. When the direction of LD is polarized with respect to the more common allele at each neutral site, hitchhiking creates more positive than negative linkage disequilibrium. Thus, hitchhiking may have a distinctively patterned LD-reducing effect, in particular near the target of selection.     

Dynamics of Genetic Variability in Two-Locus Models of Stabilizing Selection

We study a two locus model, with additive contributions to the phenotype, to explore the dynamics of different phenotypic characteristics under stabilizing selection and recombination. We demonstrate that the interaction of selection and recombination results in constraints on the mode of phenotypic evolution. Let V(g) be the genic variance of the trait and C(L) be the contribution of linkage disequilibrium to the genotypic variance. We demonstrate that, independent of the initial conditions, the dynamics of the system on the plane (V(g), C(L)) are typically characterized by a quick approach to a straight line with slow evolution along this line afterward. We analyze how the mode and the rate of phenotypic evolution depend on the strength of selection relative to recombination, on the form of fitness function, and the difference in allelic effect. We argue that if selection is not extremely weak relative to recombination, linkage disequilibrium generated by stabilizing selection influences the dynamics significantly. We demonstrate that under these conditions, which are plausible in nature and certainly the case in artificial stabilizing selection experiments, the model can have a polymorphic equilibrium with positive linkage disequilibrium that is stable simultaneously with monomorphic equilibria.     

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.     

Background selection and population differentiation

A general analytical formula is derived, which predicts the effects of background selection on population differentiation at a neutral locus as a result of its linkage with selected loci of deleterious mutations. The theory is based on the assumptions of random mating, multiplicative fitness, and weak selection in hermaphrodite plants in the island model of population structure. The analytical results show that Fst at the neutral locus increases as a result of the effects of background selection, regardless of the dependence or independence among linked background selective loci. The increment in Fst is closely related to the magnitude of linkage disequilibria between the neutral locus and selected loci, and can be estimated by the ratio of Fst with background selection to Fst without background selection minus one. The steady-state linkage disequilibrium between a neutral locus and a selected locus in subpopulations, primarily attained by gene flow, decreases with the recombination rate, and can be enhanced when there are dependence among linked selected loci. Monte Carlo computer simulations with two- and three-locus models show that the analytical formulae perform well under general conditions. Application of the present theory may aid in analyzing the genome-wide mapping of the effect of background selection in terms of Fst.     

The Effects of Overdominance on Linkage in a Multilocus System

Computer simulations were performed with overdominant multiple alleles among tightly linked multiple loci under a multiplicative fitness model. The quantity X²/N(n — 1) was introduced as a new measure of linkage disequilibrium which, unlike previously available measures, can be applied to multiple allele models, where N is the sample size, and n is the number of alleles at the locus possessing fewest alleles. Simulations showed that (1) With multiple (three or four) alleles, the approach to stable disequilibrium is slower and the amount of disequilibrium established is weaker than in a two allele system. (2) The number of complementary chromosomes is a function of number of alleles and of population size. (3) As population size increases, the rate of the approach to stable disequilibrium is slower. (4) There is an optimum selection coefficient which minimizes the transient fixation probability of alleles when linkage is present. (5) The absence of linkage disequilibrium is in most cases not a practical method of testing the hypothesis of balancing selection of genetic polymorphisms because it depends strongly on population size in determining linkage disequilibria.     

Nucleotide variation at the dopa decarboxylase (<Emphasis Type="Italic">Ddc</Emphasis>) gene in natural populations of <Emphasis Type="Italic">Drosophila melanogaster</Emphasis>

We studied nucleotide sequence variation at the gene coding for dopa decarboxylase (Ddc) in seven populations of Drosophila melanogaster. Strength and pattern of linkage disequilibrium are somewhat distinct in the extensively sampled Spanish and Raleigh populations. In the Spanish population, a few sites are in strong positive association, whereas a large number of sites in the Raleigh population are associated nonrandomly but the association is not strong. Linkage disequilibrium analysis shows presence of two groups of haplotypes in the populations, each of which is fairly diverged, suggesting epistasis or inversion polymorphism. There is evidence of two forms of natural selection acting on Ddc. The McDonald-Kreitman test indicates a deficit of fixed amino acid differences between D. melanogaster and D. simulans, which may be due to negative selection. An excess of derived alleles at high frequency, significant according to the H-test, is consistent with the effect of hitchhiking. The hitchhiking may have been caused by directional selection downstream of the locus studied, as suggested by a gradual decrease of the polymorphism-to-divergence ratio. Altogether, the Ddc locus exhibits a complicated pattern of variation apparently due to several evolutionary forces. Such a complex pattern may be a result of an unusually high density of functionally important genes.     

Distinguishing recombination and intragenic gene conversion by linkage disequilibrium patterns

Deterministic theory suggests that reciprocal recombination and intragenic, interallelic conversion have different effects on the linkage disequilibrium between a pair of genetic markers. Under a model of reciprocal recombination, the decay rate of linkage disequilibrium depends on the distance between the two markers, while under conversion the decay rate is independent of this distance, provided that conversion tracts are short. A population genetic three-locus model provides a function Q of two-locus linkage disequilibria. Viewed as a random variable, Q is the basis for a test of the relative impact of conversion and recombination. This test requires haplotype frequency data of a sufficiently variable three-locus system. One of the few examples currently available is data from the Human Leukocyte Antigen (HLA) class I genes of three Amerindian populations. We find that conversion may have played a dominant role in shaping haplotype patterns over short stretches of DNA, whereas reciprocal recombination may have played a greater role over longer stretches of DNA. However, in order to draw firm conclusions more independent data are necessary.     

Cline coupling and uncoupling in a stickleback hybrid zone

Strong ecological selection on a genetic locus can maintain allele frequency differences between populations in different environments, even in the face of hybridization. When alleles at divergent loci come into tight linkage disequilibrium, selection acts on them as a unit and can significantly reduce gene flow. For populations interbreeding across a hybrid zone, linkage disequilibria between loci can force clines to share the same slopes and centers. However, strong ecological selection on a locus can also pull its cline away from the others, reducing linkage disequilibrium and weakening the barrier to gene flow. We looked for this “cline uncoupling” effect in a hybrid zone between stream resident and anadromous sticklebacks at two genes known to be under divergent natural selection (Eda and ATP1a1) and five morphological traits that repeatedly evolve in freshwater stickleback. These clines were all steep and located together at the top of the estuary, such that we found no evidence for cline uncoupling. However, we did not observe the stepped shape normally associated with steep concordant clines. It thus remains possible that these clines cluster together because their individual selection regimes are identical, but this would be very surprising given their diverse roles in osmoregulation, body armor, and swimming performance.     

New properties of the two-locus partial selfing model with selection

Analytic solutions are obtained for the equilibria of a simple two-locus, heterotic selection model with mixed selfing and random outcrossing. Two general phenomena are possible, depending upon the viabilities and the degree of selfing: (1) Negative disequilibrium potential, under which only gametic disequilibrium is possible; and (2) positive disequilibrium potential, which can result in permanent gametic disequilibrium provided that linkage is sufficiently tight. Under random mating (s = 0), these two situations correspond to negative and positive additive epistasis, respectively. With partial self-fertilization, however, this is no longer true, and a more appropriate measure of gametc disequilibrium potential, Δ(s), is introduced. A numerically aided examination of the model results in the discovery of two new properties of partial selfing with selection: (1) With negative disequilibrium potential (Δ(s) < 0), the equilibrium mean fitness increases with increasing recombination. With positive disequilibrium potential (Δ(s) > 0), the opposite is true. (2) Gametic disequilibrium can increase or decrease as the degree of selfing is increased. Therefore, it is apparent that partial selfing and linkage are not analogous as regards the maintenance of disequilibrium.     

Maintenance of Genetic Variability under Strong Stabilizing Selection: A Two-Locus Model

We study a two locus model with additive contributions to the phenotype to explore the relationship between stabilizing selection and recombination. We show that if the double heterozygote has the optimum phenotype and the contributions of the loci to the trait are different, then any symmetric stabilizing selection fitness function can maintain genetic variability provided selection is sufficiently strong relative to linkage. We present results of a detailed analysis of the quadratic fitness function which show that selection need not be extremely strong relative to recombination for the polymorphic equilibria to be stable. At these polymorphic equilibria the mean value of the trait, in general, is not equal to the optimum phenotype, there exists a large level of negative linkage disequilibrium which ``hides' additive genetic variance, and different equilibria can be stable simultaneously. We analyze dependence of different characteristics of these equilibria on the location of optimum phenotype, on the difference in allelic effect, and on the strength of selection relative to recombination. Our overall result that stabilizing selection does not necessarily eliminate genetic variability is compatible with some experimental results where the lines subject to strong stabilizing selection did not have significant reductions in genetic variability.     

Strong linkage disequilibrium near the selected Yr17 resistance gene in a wheat experimental population

Dynamic management (DM) is a method of genetic resources conservation that aims at maintaining evolutionary process in subdivided populations cultivated in contrasted environments. Such populations are often submitted to strong natural selection as it was the case for experimental wheat populations maintained under DM. Understanding impacts of selection on genetic diversity around selected genes is necessary for the middle-term maintenance of genetic variability in DM populations. Evolution of diversity at six neutral markers located near the yellow rust resistance gene Yr17 has been studied for the parental lines and for generations 1, 5, 10 and 17 in one of the DM populations. Yr17 provided complete resistance to yellow rust in France until 1997 and thus was suspected to be under strong selection. The gene is located on a fragment introgressed in winter wheat from a wild species. The presence of the gene was estimated using a marker closely related to the gene. We showed that the Yr17 gene has been selected between generations 5 and 10. Generally, selection tends to reduce diversity around selected genes, generating linkage disequilibrium (LD) between a gene and adjacent markers. Here, the major effect of the Yr17 gene selection was a reduction of multilocus diversity and the maintenance of strong pre-existing LD in the zone surrounding the gene for a distance of 20 cM. As expected, the presence of the exogenous introgression was responsible for restrictions to recombination which contributed to the maintenance of strong correlations between loci. However, we found a noticeable number of recombinations around the gene indicating a progressive incorporation of the fragment into the wheat genome.     

Disequilibrium among several linked neutral genes in finite population 1. mean changes in disequilibrium

Formulae are developed for computing changes in expected values in a finite population of linkage disequilibrium among neutral genes from more than two loci, although the exact analysis is taken up to only six loci. An essentially haploid model is used. As with two loci, the three-locus disequilibrium declines exponentially at all generations, but for m > 3 loci a matrix has to be constructed to give joint changes in the m-locus disequilibrium and products of disequilibria with fewer loci, for example of two $\text{m}\text{2}\text{-}\text{locus}$ disequilibria. The asymptotic rates of change in multilocus disequilibria depend on the arrangement of genes on the chromosome as well as its total length, but the initial rate of breakdown of disequilibrium from a line cross base is less dependent on the arrangement. With equally spaced loci the asymptotic rate of breakdown of m locus disequilibrium is roughly proportional to m. Although mutation and interference are excluded from the main analysis, it is shown how they can be incorporated.     

On Models of Quantitative Genetic Variability: A Stabilizing Selection-Balance Model

A model of stabilizing selection on a multilocus character is proposed that allows the maintenance of stable allelic polymorphism and linkage disequilibrium. The model is a generalization of Lerner's model of homeostasis in which heterozygotes are less susceptible to environmental variation and hence are superior to homozygotes under phenotypic stabilizing selection. The analysis is carried out for weak selection with a quadratic-deviation model for the stabilizing selection. The stationary state is characterized by unequal allele frequencies, unequal proportions of complementary gametes, and a reduction of the genetic (and phenotypic) variance by the linkage disequilibrium. The model is compared with Mather's polygenic balance theory, with models that include mutation-selection balance, and others that have been proposed to study the role of linkage disequilibrium in quantitative inheritance.     

Multiple feedbacks and the prevalence of alternate stable states on coral reefs

The prevalence of alternate stable states on coral reefs has been disputed, although there is universal agreement that many reefs have experienced substantial losses of coral cover. Alternate stable states require a strong positive feedback that causes self-reinforcing runaway change when a threshold is passed. Here we use a simple model of the dynamics of corals, macroalgae and herbivores to illustrate that even weak positive feedbacks that individually cannot lead to alternate stable states can nonetheless do so if they act in concert and reinforce each other. Since the strength of feedbacks varies over time and space, our results imply that we should not reject or accept the general hypothesis that alternate stable states occur in coral reefs. Instead, it is plausible that shifts between alternate stable states can occur sporadically, or on some reefs but not others depending on local conditions. Therefore, we should aim at a better mechanistic understanding of when and why alternate stable states may occur. Our modelling results point to an urgent need to recognize, quantify, and understand feedbacks, and to reorient management interventions to focus more on the mechanisms that cause abrupt transitions between alternate states.     

Positive Selection in the Chromosome 16 VKORC1 Genomic Region Has Contributed to the Variability of Anticoagulant Response in Humans

VKORC1 (vitamin K epoxide reductase complex subunit 1, 16p11.2) is the main genetic determinant of human response to oral anticoagulants of antivitamin K type (AVK). This gene was recently suggested to be a putative target of positive selection in East Asian populations. In this study, we genotyped the HGDP-CEPH Panel for six VKORC1 SNPs and downloaded chromosome 16 genotypes from the HGDP-CEPH database in order to characterize the geographic distribution of footprints of positive selection within and around this locus. A unique VKORC1 haplotype carrying the promoter mutation associated with AVK sensitivity showed especially high frequencies in all the 17 HGDP-CEPH East Asian population samples. VKORC1 and 24 neighboring genes were found to lie in a 505 kb region of strong linkage disequilibrium in these populations. Patterns of allele frequency differentiation and haplotype structure suggest that this genomic region has been submitted to a near complete selective sweep in all East Asian populations and only in this geographic area. The most extreme scores of the different selection tests are found within a smaller 45 kb region that contains VKORC1 and three other genes (BCKDK, MYST1 (KAT8), and PRSS8) with different functions. Because of the strong linkage disequilibrium, it is not possible to determine if VKORC1 or one of the three other genes is the target of this strong positive selection that could explain present-day differences among human populations in AVK dose requirement. Our results show that the extended region surrounding a presumable single target of positive selection should be analyzed for genetic variation in a wide range of genetically diverse populations in order to account for other neighboring and confounding selective events and the hitchhiking effect.     

Genomic islands of speciation or genomic islands and speciation?

Populations of the malaria mosquito, Anopheles gambiae, are comprised of at least two reproductively isolated, sympatric populations. In this issue, White et al. (2010) use extensive sampling, high‐density tiling microarrays, and an updated reference genome to clarify and expand our knowledge of genomic differentiation between these populations. It is now clear that DNA near the centromeres of all three chromosomes are in near‐perfect disequilibrium with each other. This is in stark contrast to the remaining 97% of the assembled genome, where fixed differences between populations have not been found, and many polymorphisms are shared. This pattern, coupled with direct evidence of hybridization in nature, supports models of “mosaic” speciation, where ongoing hybridization homogenizes variation in most of the genome while loci under strong selection remain in disequilibrium with each other. However, unambiguously demonstrating that selection maintains the association of these pericentric “speciation islands” in the face of gene flow is difficult. Low recombination at all three loci complicates the issue, and increases the probability that selection unrelated to the speciation process alters patterns of variation in these loci. Here, we discuss these different scenarios in light of this new data.