Rates of decay of linkage disequilibrium under two-locus models of selection

The effect of selection and linkage on the decay of linkage disequilibrium, D, is investigated for a hierarchy of two-locus models. The method of analysis rests upon a qualitative classification of the dynamic of D under selection relative to the neutral dynamic. To eliminate the confounding effects of gene frequency change, the behavior of D is first studied with gene frequencies fixed at their invariant values. Second, the results are extended to certain special situations where gene frequencies are changing simultaneously.A wide variety of selection regimes can cause an acceleration of the rate of decay of D relative to the neutral rate. Specifically, the asymptotic rate of decay is always faster than the neutral rate in the neighborhood of a stable equilibrium point, when viabilities are additive or only one locus is selected. This is not necessarily the case for models in which there is nonzero additive epistasis. With multiplicative viabilities, decay is always accelerated near a stable boundary equilibrium, but decay is only faster near the stable central equilibrium (with = 0) if linkage is sufficiently loose. In the symmetric viability model, decay may even be retarded near a stable boundary equilibrium. Decay is only accelerated near a stable corner equilibrium when the double homozygote is more fit than the double heterozygotes. Decay near a stable edge equilibrium may be retarded if there is loose linkage. With symmetric viabilities there is usually an acceleration of the decay process for gene frequencies near 1/2 when the central equilibrium (with = 0) is stable. This is always the case when the sign of the epistasis is negative or zero.Conversely, the decay ofD is retarded in the neighborhood of a stable equilibrium in the multiplicative and symmetric viability models if any of the conditions above are violated. Near an unstable equilibrium of any of the models considered,D may either increase or decay at a rate slower than, equal to, or faster than the neutral rate. These analytic results are supplemented by numerical studies of the symmetric viability model.     

Linkage and inbreeding coefficients in a finite random mating population

The notion of inbreeding coefficient associated with one single locus introduced by G. Malecot can be extended to two loci. For a panmictic model with separate generation the recurrence equations are given therein allowing to calculate the coefficients in the event of migration and mutation, or loss of kinship.Hence it is derived particularly that the limit genetic distance of two groups associated with two loci is, under specific hypotheses, little different from the sum of marginal genetic distances.For an isolat this paper studies, in terms of crossing over, mutations, and population size, the evolution of the inbreading coefficients of order 2 and especially the difference of this evolution from the evolution to independence of the two loci.     

Linkage disequilibrium among multiple neutral alleles produced by mutation in finite population.

An analysis is undertaken for a finite random mating population of the linkage disequilibrium between two loci, at both of which all alleles are neutral, all mutant alleles differ from existing ones and several may be segregating at any time. Formulae are derived for the expected total squared disequilibrium, measured as the sum of squares of disequilibria between all pairs of alleles. The ratio of this quantity to the expected value of the product of the heterozygosities at the two loci is similar to that obtained previously by Ohta and Kimura for two nucleotide sites at each of which not more than two mutant types can segregate at any time.     

Convergence of two-locus gamete frequencies in random mating age-structured populations

We study two models describing infinite random mating age-structured populations. Under model I, the number of matings at any time is proportional to the number of mature females at that time, and the fecundity of a mating is assumed to be a product of factors that depend upon the ages of the mates, but not upon sex. Under model II, individuals can mate only with others of the same age group, and the number of matings of individuals of an age group is, at any time, proportional to the number of females of that age group. If model II holds, or the special case of model I in which the fecundity of all matings is the same, it is possible to show that, in the long run, allele frequencies converge and the gametic disequilibrium approaches zero at a geometric rate. A heuristic argument and a numerical example suggest that these things are also more generally true for model I. A good approximation to the time that it takes to reduce the gametic disequilibrium by a fraction equal to the recombination rate turns out to be the measure of the generation interval in Leslie's theory of population growth.     

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.     

Linkage disequilibrium under genetic hitchhiking in finite populations

The model of genetic hitchhiking predicts a reduction in sequence diversity at a neutral locus closely linked to a beneficial allele. In addition, it has been shown that the same process results in a specific pattern of correlations (linkage disequilibrium) between neutral polymorphisms along the chromosome at the time of fixation of the beneficial allele. During the hitchhiking event, linkage disequilibrium on either side of the beneficial allele is built up whereas it is destroyed across the selected site. We derive explicit formulas for the expectation of the covariance measure D and standardized linkage disequilibrium sigma 2D between a pair of polymorphic sites. For our analysis we use the approximation of a star-like genealogy at the selected site. The resulting expressions are approximately correct in the limit of large selection coefficients. Using simulations we show that the resulting pattern of linkage disequilibrium is quickly-i.e., in <0.1N generations-destroyed after the fixation of the beneficial allele for moderately distant neutral loci, where N is the diploid population size.     

Random mating and random union of gametes models for finite dioecious populations

This paper compares a random mating model for independent trials with two random union of gametes models in the case when the population is finite and dioecious. The first random union of gametes model is examined as a two-locus dioecious model with unequal recombination values in each sex, which is a generalization of previous work. The second model can be of some biological use under certain circumstances, and it is easier to analyse (when appropriate) than the first. It is examined with and without mutation to obtain both old and new results about the fixation probabilities (in the multilocus case) and the related rates in the autosomal locus case.     

Linkage disequilibrium in a finite population that is partially selfing

The linkage disequilibrium expected in a finite, partially selfing population is analyzed, assuming the infinite allele model. Formulas for the expected sum of squares of the linkage disequilibria and the squared standard linkage disequilibrium are derived from the equilibrium values of sixteen inbreeding coefficients required to describe the behavior of the system. These formulas are identical to those obtained with random mating if the effective population size N_e = (1-½S)N and the effective recombination value r_e = (1-S)r/(1-½S), where S is the proportion of selfing, are substituted for the population size and the recombination value. Therefore, the effect of partial selfing at equilibrium is to reduce the population size by a factor 1-½S and the recombination value by a factor (1-S)/(1-½S).     

Linkage disequilibrium mapping of quantitative trait loci under truncation selection

As a dense map of single nucleotide polymorphism (SNP) markers are available, population-based linkage disequilibrium (LD) mapping or association study is becoming one of the major tools for identifying quantitative trait loci (QTL) and for fine gene mapping. However, in many cases, LD between the marker and trait locus is not very strong. Approaches that maximize the potential of detecting LD will be essential for the success of LD mapping of QTL. In this paper, we propose two strategies for increasing the probability of detecting LD: (1) phenotypic selection and (2) haplotype LD mapping. To provide the foundations for LD mapping of QTL under selection, we develop analytic tools for assessing the impact of phenotypic selection on allele and haplotype frequencies, and LD under three trait models: single trait locus, two unlinked trait loci, and two linked trait loci with or without epistasis. In addition to a traditional chi(2) test, which compares the difference in allele or haplotype frequencies in the selected sample and population sample, we present multiple regression methods for LD mapping of QTL, and investigate which methods are effective in employing phenotypic selection for QTL mapping. We also develop a statistical framework for investigating and comparing the power of the single marker and multilocus haplotype test for LD mapping of QTL. Finally, the proposed methods are applied to mapping QTL influencing variation in systolic blood pressure in an isolated Chinese population.     

Linkage disequilibrium between GABRB3 gene and nonsyndromic familial cleft lip with or without cleft palate

The malformation of nonsyndromic cleft lip with or without cleft palate (CL/P) is a common congenital disease that affects approximately 1/1000 newborns in Caucasian populations. Genetic studies indicate that CL/P has the characteristics of a complex genetic trait. Linkage analysis and mouse-model knockout studies have suggested several candidate genes mapping in different chromosome regions for CL/P malformation. On these grounds, we have investigated, by linkage disequilibrium (LD) and parametric and nonparametric linkage analyses, five different candidate genes, including those for the beta3 subunit of the gamma-aminobutyric acid receptor (GABRB3), glutamic acid decarboxylase 1 (GAD1), retinoic acid receptor alpha (RARA), transforming growth factor beta3 (TGFB3), and msh ( Drosophila) homeobox homolog 1 (MSX1). Interestingly, a significant LD between GABRB3 and CL/P was obtained ( P-value=0.008 in the allele-wise analysis for multiallelic markers), suggesting that the GABRB3 gene is involved in this congenital disease. This new finding in humans is in agreement with previously reported data obtained with the murine model. Indeed, mouse studies indicate a role for gamma-aminobutyric acid (GABA) and its receptor in normal palate development. Exclusion of the GAD1 gene, which encodes the GABA-producing enzyme, in CL/P pathogenesis was obtained in our study. Moreover, we were unable to confirm the involvement of the MSX1 gene in nonsyndromic CL/P. Modest evidence of LD between marker alleles and CL/P was found at the RARA and TGFB3 loci suggesting a minor role for these genes in our family set of nonsyndromic CL/P.     

Linkage disequilibrium in Bougainville Island

Linkage disequilibria are estimated for three 2-locus systems in 18 samples from Bougainville Island, Solomon Islands. The systems are haptoglobin, acid phosphatase and MN blood group. The disequilibria are estimated two ways: by maximum likelihood (ML) and by the covariance between the non-alleles. Though seven of the 52 ML estimates are statistically different than zero, none of the covariance estimates are significant. We conclude that because linkage disequilibrium for loosely linked loci is a small quantity and because the sample sizes for most populations studied by anthropologists are small, linkage disequilibrium will not be a useful parameter for the study of natural selection in these populations.