Variances and covariances of squared linkage disequilibria in finite populations

Analysis of linkage disequilibrium D among restriction sites or bases in DNA sequences, arising from mutations in finite populations, depends on a knowledge of the variance-covariance structure of measures such as D2 between different pairs of sites. This requires evaluation of the eighth moments of gene frequencies among two, three, and four loci, and the necessary methodology is derived here and results are computed. While primary emphasis is placed on disequilibrium arising from mutation or gene conversion, the methodology also allows for the joint effects of only drift and recombination. Numerical results confirm that squared linkage disequilibria can have high variances and covariances.     

Evolution of linkage disequilibrium of the founders in exponentially growing populations

Evolution of linkage disequilibrium of the founders in exponentially growing populations was studied using a time-inhomogeneous It? process model. The model is an extension of the diffusion approximation of the Wright-Fisher model. As a measure of linkage disequilibrium, the squared standard linkage deviation, which is defined by a ratio of the moments, was considered. A system of ordinary differential equations that these moments obey was obtained. This system can be solved numerically. By simulations, it was shown that the squared standard linkage deviation gives a good approximation of the expectation of the squared correlation coefficient of gamete frequencies. In addition, a perturbative solution was obtained when the growth rate is not large. By using the perturbation, an asymptotic formula for the squared standard linkage deviation after a large number of generations was obtained. According to the formula, the squared standard linkage deviation tends to be 1/(4Nc), where N is the current size of the population and c is the recombination fraction between two loci. It is dependent on neither the initial effective size, the growth rate, nor the mutation rate. In exponentially growing populations, linkage disequilibrium will be asymptotically the same as that in a constant size population, the effective size of which is the current effective size.     

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).     

Correlation of gene frequencies between neutral linked genes in finite populations

The mean squared correlation (r²) of gene frequencies between two selectively neutral loci in segregating populations is obtained using an approximation based on expanding the formula for r² in terms up to eighth moments. The moments are evaluated for all populations, whether segregating or not, but when combined with the probability of segregation, quite accurate results are obtained. Formulae for the moments and an approximation for the probability of segregation are given for populations of large size (N) and small recombination fraction (c). The method is extended to evaluate the variance of r². The coefficient of variation of r² exceeds unity for a wide range of Nc values.     

A method for estimating the mutation,gene conversion and recombination parameters in small multigene families

A simple two-locus gene conversion model is considered to investigate the amounts of DNA variation and linkage disequilibrium in small multigene families. The exact solutions for the expectations and variances of the amounts of variation within and between two loci are obtained. It is shown that gene conversion increases the amount of variation within each locus and decreases the amount of variation between two loci. The expectation and variance of the amount of linkage disequilibrium are also obtained. Gene conversion generates positive linkage disequilibrium and the degree of linkage disequilibrium decreases as the recombination rate is increased. Using the theoretical results, a method for estimating the mutation, gene conversion, and recombination parameters is developed and applied to the data of the Amy multigene family in Drosophila melanogaster. The gene conversion rate is estimated to be approximately 60-165 times higher than the mutation rate for synonymous sites.     

Neutral two-locus multiple allele models with recombination

General formulae for the homozygosity and variance of linkage disequilibrium are derived for neutral, stationary, two-locus multiple allele models where there is a symmetric type of mutation at each locus. Particular cases examined are K allele models, the infinite alleles model, and the stepwise mutation model. The two-locus infinite allele model is examined at the molecular level and a joint probability generating function is found for the number of heterozygous sites at each locus in two randomly chosen gametes.     

On the within-sibship variance

An approximate formula for the expected within-sibship genotypic variance of a polygenic, diallelic, additive character is obtained for arbitrary recombination between the loci affecting the character. The formula is exact, when there is no recombination, or when the recombination is free. It is also shown that, if the value of $\text{1}\text{2}\text{V}{}_{\mathrm{k}}$ (one-half of the parental genotypic variance) is assigned to the within-sibship genotypic variance, as in the model of Cavalli-Sforza and Feldman (1976, Proc. Natl. Acad. Sci. USA73, 1689–1692), it implies the assumptions of random mating and of the perfect linkage. If, on the other hand, the value of $\text{1}\text{2}\text{V}{}^0$ (one-half of the linkage equilibrium genotypic variance) is assigned to the within-sibship variance, as in the model of Rice, Cloninger, and Reich (1978, Amer. J. Hum. Genet.30, 618–643), it implies the assumptions of random mating and either of the free recombination, or of the linkage equilibrium, if the recombination is not free.     

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.     

Linkage Disequilibrium and Genetic Variances under Mutation-Selection Balance

I determine the contribution of linkage disequilibrium to genetic variances using results for two loci and for induced or marginal systems. The analysis allows epistasis and dominance, but assumes that mutation is weak relative to selection. The linkage disequilibrium component of genetic variance is shown to be unimportant for unlinked loci if the gametic mutation rate divided by the harmonic mean of the pairwise recombination rates is much less than one. For tightly linked loci, linkage disequilibrium is unimportant if the gametic mutation rate divided by the (induced) per locus selection is much less than one.     

Effect of Mating Structure on Variation in Linkage Disequilibrium

Measurement of linkage disequilibrium involves two sampling processes. First, there is the sampling of gametes in the population to form successive generations, and this generates disequilibrium dependent on the effective population size (N_e) and the mating structure. Second, there is sampling of a finite number (n) of individuals to estimate the population disequilibrium.——Two-locus descent measures are used to describe the mating system and are transformed to disequilibrium moments at the final sampling. Approximate eigenvectors for the transition matrix of descent measures are used to obtain formulae for the variance of the observed disequilibria as a function of N_e, mating structure, n, and linkage or recombination parameter.——The variance of disequilibrium is the same for monoecious populations with or without random selfing and for dioecious populations with random pairing for each progeny. With monogamy, the variance is slightly higher, the proportional difference being greater for unlinked loci.     

Allele frequencies in multidimensional Wright-Fisher models with a general symmetric mutation structure

The diffusion form of a multiple-allele Wright-Fisher model of allele frequencies of types A₁,…,A_K at a neutral locus where there are general symmetric mutation rates of M_ij (=M_ji) from A_i → A_j is studied. A convenient recurrence relationship for the moments of linear forms in the allele frequencies is found. A formula is derived for the expected homozygosity in the transient and stationary models, which is applied to general stepwise mutation models where mutation over more than one step is possible. The probability that two randomly chosen genes are j steps apart at time t in the stepwise mutation model is found to have a reasonably simple form if conditioning is on the initial allele frequencies arranged in order of magnitude. Of interest is a new geometric charge state model, mutation over several steps roughly corresponding to independent charge changes. The expected homozygosity and expected distance between two randomly chosen genes is tabulated in this model.     

Dynamics of Correlated Genetic Systems. V. Rates of Decay of Linkage Disequilibria in Experimental Populations of DROSOPHILA MELANOGASTER

The dynamic behavior of four-locus gametic frequency distributions was studied in five replicate cage populations of Drosophila melanogaster for up to 50 generations. The joint frequency distributions were resolved into gene frequencies and various disequilibrium measures. In addition, F statistics for marginal single-locus genotypic frequency distributions were followed through time. The gene frequency, disequilibrium and F statistics were obtained for four chromosome 3 enzyme marker loci [isocitrate dehydrogenase (3–27.1), esterase-6 (3–36.8), phosphoglucomutase (3–43.4) and esterase-C (3–49.0)]. The initial structure of the experimental populations featured random mating proportions, and two complementary gametic types with respect to the marker loci, thus assuring complete pairwise linkage disequilibrium among the markers.——The experimental results indicate: (1) the between-replicate variance in gene frequency varied substantially among loci, with isocitrate dehydrogenase showing the greatest between-replicate variance, and esterase-C the least. (2) The F statistics initially were strongly negative but decayed to the neighborhood of zero for all marker loci except esterase-C. The rate at which the F statistics approached zero varied among the marker loci, indicating substantial differences in the distribution of selective effects along the chromosome. The centromeric region, marked by esterase-C, shows the strongest selective effects. (3) The rate of decay of linkage disequilibrium was much faster than expected for pairs of neutral loci, averaging 1.82 times the neutral rate over all replicates and pairs of loci. This acceleration, which was observed for all six pairwise combinations of loci, was interpreted as resulting from the interaction between selection and recombination. Our experimental results are consistent with many investigations of linkage disequilibrium in natural populations of Drosophila melanogaster that show little or no disequilibrium among enzyme loci. (4) A fortuitous contamination of two cages revealed an apparent regulatory interaction between the migrant and nonmigrant chromosomes at the esterase-C locus. The migrant chromosomes were very rapidly absorbed into the recipient populations, despite this interaction. This result suggests that the dynamics of migration in populations may be phenomenologically richer than anticipated by simple theory.     

Statistics for Microsatellite Variation Based on Coalescence

The stepwise mutation model, which was at one time chiefly of interest in studying the evolution of protein charge-states, has recently undergone a resurgence of interest with the new popularity of microsatellites as phylogenetic markers. In this paper we describe a method which makes it possible to transfer many population genetics results from the standard infinite sites model to the stepwise mutation model. We study in detail the properties of pairwise differences in microsatellite repeat number between randomly chosen alleles. We show that the problem of finding the expected squared distance between two individuals and finding the variance of the squared distance can be reduced for a wide range of population models to finding the mean and mean square coalescence times. In many cases the distributions of coalescence times have already been studied for infinite site problems. In this study we show how to calculate these quantities for several population models. We also calculate the variance in mean squared pairwise distance (an estimator of mutation rate × population size) for samples of arbitrary size and show that this variance does not approach zero as the sample size increases. We can also use our method to study alleles at linked microsatellite loci. We suggest a metric which quantifies the level of association between loci—effectively a measure of linkage disequilibrium. It is shown that there can be linkage disequilibrium between partially linked loci at mutation–drift equilibrium.     

Interchromosomal Biased Gene Conversion, Mutation and Selection in a Multigene Family

A mathematical model of the effects of interchromosomal biased gene conversion, mutation and natural selection on a multigene family is developed and analyzed. The model assumes two allelic states at each of n loci. The effects of genetic drift are ignored. The model is developed under the assumption of no recombination, but the analysis shows that, at equilibrium, there is no linkage disequilibrium, which implies that the conclusions are valid for arbitrary recombination among loci. At equilibrium, the balance between mutation, gene conversion and selection depends on the ratio of the mutation rates to the quantity [s + g(2α - 1)/ n], where s is the increment or decrement in relative fitness with each additional copy of one of the alleles, g is the conversion rate, and α is a measure of the bias in favor of one of the alleles. When this quantity is large relative to the mutation rates, the allele that has the net advantage, combining the effects of selection and conversion, will be nearly fixed in the multigene family. A comparison of these results with those from a comparable model of intrachromosomal biased conversion shows that biased interchromosomal conversion leads to approximately the same equilibrium copy number as does intrachromosomal conversion of the same strength. Interchromosomal conversion is much more effective in causing the substitution of one allele by another. The relative frequencies of interchromosomal and intrachromosomal conversion is indicated by the extent of the linkage disequilibrium among the loci in a multigene family.     

Linkage Disequilibrium in Subdivided Populations

The linkage disequilibrium in a subdivided populaton is shown to be equal to the sum of the average linkage disequilibrium for all subpopulations and the covariance between gene frequencies of the loci concerned. Thus, in a subdivided population the linkage disequilibrium may not be 0 even if the linkage disequilibrium in each subpopulation is 0. If a population is divided into two subpopulations between which migration occurs, the asymptotic rate of approach to linkage equilibrium is equal to either r or 2(m(1) + m(2)) - (m(1) + m(2))(2), whichever is smaller, where r is the recombination value and m(1) and m(2) are the proportions of immigrants in subpopulations 1 and 2, respectively. Thus, if migration rate is high compared with recombination value, the change of linkage disequilibrium in subdivided populations is similar to that of a single random mating population. On the other hand, if migration rate is low, the approach to lnkage equilibrium may be retarded in subdivided populations. If isolated populations begin to exchange genes by migration, linkage disequilibrium may increase temporarily even for neutral loci. If overdominant selection operates and the equilibrium gene frequencies are different in the two subpopulations, a permanent linkage disequilibrium may be produced without epistasis in each subpopulation.     

Linkage Disequilibrium in Natural Populations of DROSOPHILA MELANOGASTER. Seasonal Variation

Linkage disequilibrium among ten polymorphic allozyme loci and polymorphic inversions on chromosomes 2 and 3 in a natural population of Drosophila melanogaster was examined early and late in the annual season. Similar to previous studies, little linkage disequilibrium was observed among allozymes. The two significant cases that were observed in the first sample behaved in a contradictory way. One declined much more rapidly than expected due simply to recombination; the other declined slowly as expected. There was little change in allozyme or inversion frequencies during the season.     

Quantitative genetic models for the balance between migration and stabilizing selection

The evolution of a quantitative trait subject to stabilizing selection and immigration, with the immigrants deviating from the local optimum, is considered under a number of different models of the underlying genetic basis of the trait. By comparing exact predictions under the infinitesimal model obtained using numerical methods with predictions of a simplified approximate model based on ignoring linkage disequilibrium, the increase in the expressed genetic variance as a result of linkage disequilibrium generated by migration is shown to be relatively small and negligible, provided that the genetic variance relative to the squared deviation of immigrants from the local optimum is sufficiently large or selection and migration is sufficiently weak. Deviation from normality is shown to be less important by comparing predictions of the infinitesimal model with a model presupposing normality. For a more realistic symmetric model, involving a finite number of loci only, no linkage and equal effects and frequencies across loci, additional changes in the genetic variance arise as a result of changes in underlying allele frequencies. Again, provided that the genetic variance relative to the squared deviation of the immigrants from the local optimum is small, the difference between the predictions of infinitesimal and the symmetric model are small unless the number of loci is very small. However, if the genetic variance relative to the squared deviation of the immigrants from the local optimum is large, or if selection and migration are strong, both linkage disequilibrium and changes in the genetic variance as a result of changes in underlying allele frequencies become important.     

Variance in quantitative traits due to linked dominant genes and variance in heterozygosity in small populations

The influence of small population size (N) on the genetic variance within and between randomly bred unselected lines, with selfing permitted, is investigated for a model of a quantitative trait determined by linked genes that show dominance within loci but are additive over loci. Formulae for within-line variance include terms in linkage disequilibrum, which occurs by chance in the lines and these are evaluated in terms of N, map length and gene number.—The expected variance within lines is increased by this disequilibrium, quite substantially if there are many loci, with most of the increase being between or within full-sib families and almost no change expected between half-sib families or in the covariance of offspring and parent. If all loci are unlinked, there is no increase in variance within full-sib families. The variance between lines is little affected by disequilibrum generated by chance.—Expressions for the variance between individuals in heterozygosity over the whole genome are special cases of those for the variance due to linked dominated genes, and formulae are given and evaluated. The coefficient of variation of heterozygosity is at least (see PDF) and can be much higher for species with few chromosomes.     

Polymorphic DNA haplotypes at the human phenylalanine hydroxylase locus and their relationship with phenylketonuria

Summary Eight polymorphic restriction enzyme sites at the phenylalanine hydroxylase (PAH) locus were analyzed from the parental chromosomes in 33 Danish nuclear families with at least one phenylketonuric (PKU) child. Determination of haplotypes of 66 normal chromosomes and 66 chromosomes bearing mutant allele (S) demonstrated that there are at least two haplotypes which occur predominantly on PKU chromosomes and rarely otherwise. Overall, the relative frequencies of the various haplotypes are significantly different on PKU-and normal-allele bearing chromosomes, even though there is no predominantly occurring unique haplotype which can characterize the PKU chromosomes. In addition, no significant association (linkage disequilibrium) between any single polymorphic site and the mutant allele (s) was observed. The results suggest that either the phenylketonuric mutation was very ancient so that the polymorphic sites and the mutation have reached linkage equilibrium or the mutant allele (s) are the results of multiple mutations in the phenylalanine hydroxylase gene in man. Furthermore, a crude relationship between standardized linkage disequilibria and physical map distances of the polymorphic sites indicates that there is no apparent recombination hot-spot in the human phenylalanine hydroxylase gene, since the recombination rate within the locus apears to be uniform and likely to be occurring at a rate similar to that within the HLA gene cluster. The limitations of this later analysis are discussed in view of the sampling errors of disequilibrium measure used, and the potential untility of the PAH haplotypes for prenatal diagnosis and detection of PKU carriers is established.     

Linkage disequilibrium analyses and restriction mapping of four RFLPs at the proα2(I) collagen locus: lack of correlation between linkage disequilibrium and physical distance

Summary Restriction fragment length polymorphisms (RRLPs) located at short distances may demonstrate linkage disequilibrium. Under the assumption that the distances between the loci of the RFLPs are inversely related to the linkage disequilibria, gene order may be deduced. However, if the assumption is invalid, the results may be incorrect. We have studied four different DNA polymorphisms at the COLIA2 locus in 180 unrelated Norwegian individuals. Observed frequencies (presence/absence) for the different polymorphic sites were as follows: site A (EcoRI) 0.30/0.70, site B (MspI) 0.83/0.16, site C (StuI) 0.86/0.14, and site D (RsaI) 0.66/0.34. Of 16 possible haplotypes 12 were demonstrated, and 2 additional were deduced to be present. Restriction mapping of the four polymorphic sites gave the following order of the sites from the 5 to the 3 of the gene: A-D-B-C. Linkage disequilibrium was not found between the sites A and D; strong disequilibrium was found between sites A and C, and B and C; and less strong, between A and B, B and D, and C and D. Analysis of linkage disequilibrium coefficients between all pairs of loci demonstrated that there is no consistent relationship between linkage disequilibrium and physical distance (=-0.07). These results suggest that for a small region of the genome, factors such as deviating mutation rate and gene conversion may add significantly to rearrangements by recombination. Thus, a deduced gene order from linkage disequilibrium data has to be regarded with great caution.