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.     

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.     

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).     

Frequency spectra of neutral and deleterious alleles in a finite population

Summary One of the major goals of population genetics is to discover the nature and amount of genetic variation in natural populations. Various measures, including the population heterozygosity at any locus and the number of alleles extant at the locus, have been used for this purpose. An important task of theoretical population genetics is thus to provide expressions for the mean values of these two quantities (when calculated from a sample of genes) for various models of selection, mutation and random drift. This aim has been achieved for the selectively neutral case, where all alleles at the locus are assumed to be selectively equivalent. It is, however, generally agreed that classes of (evolutionarily unimportant) selectively deleterious alleles exist, so that the neutral theory calculations should be extended to cover this case. This has previously been done only for extremely weak selection. In this paper we obtain, via the confluent hypergeometric function and three allied functions, concise and simple exact and approximate formulae for the means of the above measures of population variation for arbitrary selective values. These all derive from the allelic frequency spectrum, which is of independent interest in assessing likely models of population variation.     

Linkage disequilibrium under skewed offspring distribution among individuals in a population

Correlations in coalescence times between two loci are derived under selectively neutral population models in which the offspring of an individual can number on the order of the population size. The correlations depend on the rates of recombination and random drift and are shown to be functions of the parameters controlling the size and frequency of these large reproduction events. Since a prediction of linkage disequilibrium can be written in terms of correlations in coalescence times, it follows that the prediction of linkage disequilibrium is a function not only of the rate of recombination but also of the reproduction parameters. Low linkage disequilibrium is predicted if the offspring of a single individual frequently replace almost the entire population. However, high linkage disequilibrium can be predicted if the offspring of a single individual replace an intermediate fraction of the population. In some cases the model reproduces the standard Wright-Fisher predictions. Contrary to common intuition, high linkage disequilibrium can be predicted despite frequent recombination, and low linkage disequilibrium under infrequent recombination. Simulations support the analytical results but show that the variance of linkage disequilibrium is very large.     

Linkage disequilibrium in two-locus,finite, random mating models without selection or mutation

Linkage disequilibrium is discussed for three two-locus, finite, random mating models in genetics, two models being Markov chains and the third a diffusion process. The expected value of the square of the disequilibrium function at any time is computed for the Markov chains. There is discussion of the relationships between the models and of the influence of finite population size on correlation between loci. It is suggested that there may be danger in assuming too simple a relationship between population size and degree of disequilibrium.     

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.     

Linkage disequilibrium among modern sugarcane cultivars

Modern sugarcane cultivars are derived from a few interspecific hybrids created early in this century. Linkage disequilibrium was investigated in a population of 59 cultivars representing the most important commercial clones bred in Mauritius as well as a few old cultivars involved in their genealogy. Thirty-eight probes scattered over the sugarcane genome map were used to reveal RFLPs. Forty-two cases of bilocus associations were observed involving a total of 33 loci. Most of them are separated by less than 10 cM. All the corresponding allele couples were found in at least 1 of the originally created cultivars, suggesting that they depict ancient associations. This global disequilibrium is interpreted as the result of the foundation bottleneck related to the first interspecific crosses; the preferential allele associations thus created have been maintained through subsequent crosses when the loci were closely linked. This phenomenon is likely also to apply to genes of agricultural interest. A practical consequence is that markers can be used to track known QTLs in modern breeding materials without the necessity to repeatedly study segregating progenies. This structure gives high value to the correlation between molecular markers and agricultural traits among cultivars. Received: 6 March 1999 / Accepted: 25 March 1999     

Linkage disequilibrium in the North American Holstein population

Linkage disequilibrium was estimated using 7119 single nucleotide polymorphism markers across the genome and 200 animals from the North American Holstein cattle population. The analysis of maternally inherited haplotypes revealed strong linkage disequilibrium ( r ²   >   0.8) in genomic regions of ∼50 kb or less. While linkage disequilibrium decays as a function of genomic distance, genomic regions within genes showed greater linkage disequilibrium and greater variation in linkage disequilibrium compared with intergenic regions. Identification of haplotype blocks could characterize the most common haplotypes. Although maximum haplotype block size was over 1 Mb, mean block size was 26–113 kb by various definitions, which was larger than that observed in humans (∼10 kb). Effective population size of the dairy cattle population was estimated from linkage disequilibrium between single nucleotide polymorphism marker pairs in various haplotype ranges. Rapid reduction of effective population size of dairy cattle was inferred from linkage disequilibrium in recent generations. This result implies a loss of genetic diversity because of the high rate of inbreeding and high selection intensity in dairy cattle. The pattern observed in this study indicated linkage disequilibrium in the current dairy cattle population could be exploited to refine mapping resolution. Changes in effective population size during past generations imply a necessity of plans to maintain polymorphism in the Holstein population.     

Linkage disequilibrium mapping of Arabidopsis CRY2 flowering time alleles

The selfing plant Arabidopsis thaliana has been proposed to be well suited for linkage disequilibrium (LD) mapping as a means of identifying genes underlying natural trait variation. Here we apply LD mapping to examine haplotype variation in the genomic region of the photoperiod receptor CRYPTOCHROME2 and associated flowering time variation. CRY2 DNA sequences reveal strong LD and the existence of two highly differentiated haplogroups (A and B) across the gene; in addition, a haplotype possessing a radical glutamine-to-serine replacement (AS) occurs within the more common haplogroup. Growth chamber and field experiments using an unstratified population of 95 ecotypes indicate that under short-day photoperiod, the AS and B haplogroups are both highly significantly associated with early flowering. Data from six genes flanking CRY2 indicate that these haplogroups are limited to an approximately 65-kb genomic region around CRY2. Whereas the B haplogroup cannot be delimited to <16 kb around CRY2, the AS haplogroup is characterized almost exclusively by the nucleotide polymorphisms directly associated with the serine replacement in CRY2; this finding strongly suggests that the serine substitution is directly responsible for the AS early flowering phenotype. This study demonstrates the utility of LD mapping for elucidating the genetic basis of natural, ecologically relevant variation in Arabidopsis.     

Linkage disequilibrium between mutation and RFLP haplotype at the phenylalanine hydroxylase locus in the German population

Summary Restriction fragment length polymorphism (RFLP) haplotypes at the phenylalanine hydroxylase (PAH) locus have been determined in 60 German families with PAH deficiency. Similar to the Danish population, about 90% of the mutant alleles are confined to four distinct haplotypes. There are however, differences in the frequency distributiion of these haplotypes among the mutant alleles between the two populations. Using an oligonucleotide probe for the splicing mutation associated with mutant haplotype 3 in the Danish population, a tight association between the mutation and the RFLP haplotype has also been observed in Germany. The results provide strong evidence that the splicing mutation occurred on a haplotype 3 chromosome and that the mutant allele has spread into different populations smong Caucasians.     

Correlation-based inference for linkage disequilibrium with multiple alleles

The correlation between alleles at a pair of genetic loci is a measure of linkage disequilibrium. The square of the sample correlation multiplied by sample size provides the usual test statistic for the hypothesis of no disequilibrium for loci with two alleles and this relation has proved useful for study design and marker selection. Nevertheless, this relation holds only in a diallelic case, and an extension to multiple alleles has not been made. Here we introduce a similar statistic, R(2), which leads to a correlation-based test for loci with multiple alleles: for a pair of loci with k and m alleles, and a sample of n individuals, the approximate distribution of n(k - 1)(m - 1)/(km)R(2) under independence between loci is chi((k-1)(m-1))(2). One advantage of this statistic is that it can be interpreted as the total correlation between a pair of loci. When the phase of two-locus genotypes is known, the approach is equivalent to a test for the overall correlation between rows and columns in a contingency table. In the phase-known case, R(2) is the sum of the squared sample correlations for all km 2 x 2 subtables formed by collapsing to one allele vs. the rest at each locus. We examine the approximate distribution under the null of independence for R(2) and report its close agreement with the exact distribution obtained by permutation. The test for independence using R(2) is a strong competitor to approaches such as Pearson's chi square, Fisher's exact test, and a test based on Cressie and Read's power divergence statistic. We combine this approach with our previous composite-disequilibrium measures to address the case when the genotypic phase is unknown. Calculation of the new multiallele test statistic and its P-value is very simple and utilizes the approximate distribution of R(2). We provide a computer program that evaluates approximate as well as "exact" permutational P-values.     

Linkage disequilibrium analysis in the genetically isolated Norfolk Island population

Norfolk Island is a human genetic isolate, possessing unique population characteristics that could be utilized for complex disease gene localization. Our intention was to evaluate the extent and strength of linkage disequilibrium (LD) in the Norfolk isolate by investigating markers within Xq13.3 and the NOS2A gene encoding the inducible nitric oxide synthase. A total of six microsatellite markers spanning approximately 11 Mb were assessed on chromosome Xq13.3 in a group of 56 men from Norfolk Island. Additionally, three single nucleotide polymorphisms (SNPs) localizing to the NOS2A gene were analyzed in a subset of the complex Norfolk pedigree. With the exception of two of the marker pairs, one of which is the most distantly spaced marker, all the Xq13.3 marker pairs were found to be in significant LD indicating that LD extends up to 9.5-11.5 Mb in the Norfolk Island population. Also, all SNPs studied showed significant LD in both Norfolk Islanders and Australian Caucasians, with two of the marker pairs in complete LD in the Norfolk population only. The Norfolk Island study population possesses a unique set of characteristics including founder effect, geographical isolation, exhaustive genealogical information and phenotypic data of use to cardiovascular disease risk traits. With LD extending up to 9.5-11 Mb, the Norfolk isolate should be a powerful resource for the localization of complex disease genes.     

Linkage disequilibrium in a population undergoing periodic fragmentation and admixture

Glacial and interglacial cycles are considered to have caused the fragmentation and admixture of populations in many organisms. A simple model incorporating such periodic changes of the population structure is analysed in order to investigate the behaviour of neutral genetic variation at one and two loci. The equilibrium is reached very quickly in terms of cycles if the length of a cycle is long, as would be expected of the glaciation cycles. Heterozygosity and linkage disequilibrium are shown to depend on the length of time of the fragmented and admixed phases, population sizes, and number (n) of subpopulations in the fragmented phase. If the population size is small in the fragmented phase and its duration is long, the squared correlation coefficient of two loci (a measure of linkage disequilibrium) just after the admixture is approximated by 1/(n-1) for n > 1. After admixture, the correlation decays at a rate of approximately twice the recombination rate. Therefore, if post-glaciation admixture created linkage disequilibrium, we expect to observe linkage disequilibrium even between moderately linked loci, and its decay pattern along the chromosome is very different from that in a random mating population at equilibrium. This is especially true in organisms with long generation times such as trees.     

Power studies for the transmission/disequilibrium tests with multiple alleles.

Case-control studies compare marker-allele distributions in affected and unaffected individuals, and significant results suggest linkage but may simply reflect population structure. For markers with m alleles (m > or = 2), a McNemar-like statistic, I, estimates the level of population association between marker and disease loci. To test for linkage after significant case-control tests, within-family tests are performed. These operate on the contingency table, with i, jth element equal to the number of parents that transmit marker allele Mi and do not transmit marker allele Mi to an affected offspring. The dimension of the table is the number of alleles at the marker locus. Three test statistics have recently been proposed in the literature: Tc compares symmetric pairs of cells (i, j) and (j, i), Tm compares row and column totals for the same marker allele, and a likelihood ratio statistic Tl uses all the cells in the table. In addition, we consider a new statistic, Tmhet, that uses only the heterozygous parents and is approximately chi2 with (m - 1) df. We use a Monte Carlo test to guarantee valid tests and to demonstrate the inferiority of Tc and the equality of Tm and Tl in terms of power. The power of the Tmhet test is close but not always equal to the power of the Tm test. We also show that under the alternative hypothesis of linkage, Tm is approximately noncentral chi2 with (m - 1) df and noncentrality parameter 2NT(1 - 2theta)2I*, when data on single affecteds in NT families are used. If the disease has a low population frequency, then I* is estimated using the case-control statistic I. This offers a basis for choosing sample size, or choosing a marker system.     

The transition density for multiple neutral alleles

For a single genetic locus with multiple alleles, Littler and Fackerall (1975, Biometrics 31, 117–123) found the transition density of the allelic proportions in the case of no mutation and no selection. Their method is found to be suitable for a model in which symmetric mutation is allowed between alleles. The sampling probabilities and moments are found. Watterson's (1977, Genetics 85, 789–814) test for selection is found to be non-robust for non-stationary populations.