Simulating genealogies of selected alleles in a population of variable size.

An importance-sampling method is presented that allows the simulation of the history of a selected allele in a population of variable size. A sample path describing the number of copies of an allele that arose as a single mutant is generated by simulating backwards from the current frequency until the allele is lost. The mathematical expectation of a quantity or statistic is then estimated by taking averages over replicate simulations, weighting each replicate by the ratio of its probabilities under the Markov chains for the forward and backwards processes. This method was used to find the average age of a selected allele in an exponentially growing population. In terms of the effect on average allele age, selection in favour of an allele is not equivalent to exponential growth. To generate gene genealogies of a sample of copies of a selected allele, the neutral coalescent model is simulated for the subpopulation containing only the selected allele. From the resulting intra-allelic genealogy, it is possible to calculate the likelihood of the selection intensity as a function of the observed level of variability at marker loci closely linked to the selected allele. This method was used to estimate the intensity of selection affecting the delta 32 allele at the CCR5 locus in Europeans and a mutant at the MLH1 locus associated with colorectal cancer in the Finnish population.

The use of intraallelic variability for testing neutrality and estimating population growth rate.

To better understand the forces affecting individual alleles, we introduce a method for finding the joint distribution of the frequency of a neutral allele and the extent of variability at closely linked marker loci (the intraallelic variability). We model three types of intraallelic variability: (a) the number of nonrecombinants at a linked biallelic marker locus, (b) the length of a conserved haplotype, and (c) the number of mutations at a linked marker locus. If the population growth rate is known, the joint distribution provides the basis for a test of neutrality by testing whether the observed level of intraallelic variability is consistent with the observed allele frequency. If the population growth rate is unknown but neutrality can be assumed, the joint distribution provides the likelihood of the growth rate and leads to a maximum-likelihood estimate. We apply the method to data from published data sets for four loci in humans. We conclude that the Delta32 allele at CCR5 and a disease-associated allele at MLH1 arose recently and have been subject to strong selection. Alleles at PAH appear to be neutral and we estimate the recent growth rate of the European population to be approximately 0.027 per generation with a support interval of (0.017-0.037). Four of the relatively common alleles at CFTR also appear to be neutral but DeltaF508 appears to be significantly advantageous to heterozygous carriers.

Allele age and a test for selection on rare alleles.

An approximate expression for the probability distribution of the age of a neutral allele as a function of its frequency is derived for a population undergoing arbitrary changes in population size. A simple maximum-likelihood estimator of allele age based on frequency is also obtained. The distribution of allele age, combined with a model predicting the extent of intra-allelic variability generated by mutation and recombination, leads to a statistical test of whether a rare allele has experienced natural selection. The test is based on finding whether there is too little or too much intra-allelic variability to be consistent with the observed frequency. The test is applied to the locus, BRCA1, associated with early-onset breast cancer in humans and shows that two common disease-associated alleles (5382insC and 185delAG) appear to have been subject to natural selection.

A method for estimating the intensity of overdominant selection from the distribution of allele frequencies.

A method is proposed for estimating the intensity of overdominant selection scaled by the effective population size, S = 2Ns, from allele frequencies. The method is based on the assumption that, with strong overdominant selection, allele frequencies are nearly at their deterministic equilibrium values and that, to a first approximation, deviations depend only on S. Simulations verify that reasonably accurate estimates of S can be obtained for realistic sample sizes. The method is applied to data from several loci in the major histocompatibility complex (Mhc) in numerous human populations. For alleles distinguished by both serological typing and the sequence of the peptide-binding region, our estimates of S are comparable to those obtained by analysis of DNA sequences in showing that selection is strongest on HLA-B and weaker on HLA-A, HLA-DRB1, and HLA-DQA1. The intensity of selection on HLA-B varied considerably among populations. Two populations, Native American and Inuit, showed an excess rather than a deficiency in homozygosity. Comparable estimates of S were obtained for alleles at Mhc class II loci distinguished by serological reactions (serotyping) and by differences in the amino acid sequences of the peptide-binding region (molecular typing). A comparison of two types of data for DQA1 and DRB1 showed that serotyping led to generally lower estimates of S.

Methods for multipoint disease mapping using linkage disequilibrium.

A disease-associated mutation arises on a single chromosome such that alleles at linked markers are initially in complete linkage disequilibrium (LD) with the mutation. LD can be used as a tool for high-resolution mapping of the position of a disease mutation relative to a set of linked marker loci. When more than two linked marker loci are considered, developing a maximum likelihood approach is a challenging mathematical problem. To reduce the complexity, approximate and composite likelihood (CL) methods have been developed for multipoint LD mapping that use simplified models of population history, or of recombination, that ignore some of the statistical dependence among disease chromosomes and among marker loci. We describe the relationship among several composite likelihood methods for multipoint LD mapping, and suggest an alternative CL method that takes better account of the statistical dependence among marker loci.

Likelihood analysis of ongoing gene flow and historical association.

We develop a Monte Carlo-based likelihood method for estimating migration rates and population divergence times from data at unlinked loci at which mutation rates are sufficiently low that, in the recent past, the effects of mutation can be ignored. The method is applicable to restriction fragment length polymorphisms (RFLPs) and single nucleotide polymorphisms (SNPs) sampled from a subdivided population. The method produces joint maximum-likelihood estimates of the migration rate and the time of population divergence, both scaled by population size, and provides a framework in which to test either for no ongoing gene flow or for population divergence in the distant past. We show the method performs well and provides reasonably accurate estimates of parameters even when the assumptions under which those estimates are obtained are not completely satisfied. Furthermore, we show that, provided that the number of polymorphic loci is sufficiently large, there is some power to distinguish between ongoing gene flow and historical association as causes of genetic similarity between pairs of populations.

Balancing selection at closely linked, overdominant loci in a finite population.

High levels of allelic diversity and strong linkage disequilibrium are found in the major histocompatibility (MHC) system in humans and other vertebrates. This article proposes several descriptive statistics that quantify the extent and pattern of strong linkage disequilibrium between pairs of highly polymorphic loci. It also develops an approximate analytic theory incorporating the effects of balancing selection, mutation, recombination, and genetic drift at two closely linked loci and compares the theoretical predictions with published surveys of the MHC class II loci, DQA1 and DQB1, in humans and nonhuman primates. The descriptive statistics proposed include the fraction of complementary haplotypes (haplotypes with D' = 1), the fraction of excess haplotypes, and the numbers of alleles at each locus in complementary haplotypes with one or more alleles at the other locus. The model assumes the infinite alleles model of mutation and the symmetric overdominance model of selection. Analytic approximations in some cases are obtained in the strong selection, weak mutation (SSWM) limit introduced by J. Gillespie. The predictions of the approximate analysis are confirmed by simulation. Both the analytic theory and simulations show that relatively few haplotypes will be found when selection is strong and recombination is weak relative to genetic drift. The model can reproduce many of the observed patterns at DQA1 and DQB1 provided that the recombination rate is assumed to be very small.

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.

Estimating allele age.

The age of an allele can be estimated both from genetic variation among different copies (intra-allelic variation) and from its frequency. Estimates based on intra-allelic variation follow from the exponential decay of linkage disequilibrium because of recombination and mutation. The confidence interval depends both on the uncertainty in recombination and mutation rates and on randomness of the genealogy of chromosomes that carry the allele (the intra-allelic genealogy). Several approximate methods to account for variation in the intra-allelic genealogy have been derived. Allele frequency alone also provides an estimate of age. Estimates based on frequency and on intra-allelic variability can be combined to provide a more accurate estimate or can be contrasted to show that an allele has been subject to natural selection. These methods have been applied to numerous cases, including alleles associated with cystic fibrosis, idiopathic torsion dystonia, and resistance to infection by HIV. We emphasize that estimates of allele age depend on assumptions about demographic history and natural selection.

Using maximum likelihood to estimate population size from temporal changes in allele frequencies.

We develop a maximum-likelihood framework for using temporal changes in allele frequencies to estimate the number of breeding individuals in a population. We use simulations to compare the performance of this estimator to an F-statistic estimator of variance effective population size. The maximum-likelihood estimator had a lower variance and smaller bias. Taking advantage of the likelihood framework, we extend the model to include exponential growth and show that temporal allele frequency data from three or more sampling events can be used to test for population growth.

Overdominant alleles in a population of variable size.

An approximate method is developed to predict the number of strongly overdominant alleles in a population of which the size varies with time. The approximation relies on the strong-selection weak-mutation (SSWM) method introduced by J. H. Gillespie and leads to a Markov chain model that describes the number of common alleles in the population. The parameters of the transition matrix of the Markov chain depend in a simple way on the population size. For a population of constant size, the Markov chain leads to results that are nearly the same as those of N. Takahata. The Markov chain allows the prediction of the numbers of common alleles during and after a population bottleneck and the numbers of alleles surviving from before a bottleneck. This method is also adapted to modeling the case in which there are two classes of alleles, with one class causing a reduction in fitness relative to the other class. Very slight selection against one class can strongly affect the relative frequencies of the two classes and the relative ages of alleles in each class.

Disequilibrium mapping of a quantitative-trait locus in an expanding population.

Linkage disequilibrium (LD) mapping can be successful if there is strong nonrandom association between marker alleles and an allele affecting a trait of interest. The principles of LD mapping of dichotomous traits are well understood, but less is known about LD mapping of a quantitative-trait locus (QTL). It is shown in this report that selective genotyping can increase the power to detect and map a rare allele of large effect at a QTL. Two statistical tests of the association between an allele and a quantitative character are proposed. These tests are approximately independent, so information from them can be combined. Analytic theory is developed to show that these two tests are effective in detecting the presence of a low-frequency allele with a relatively large effect on the character when the QTL is either already a candidate locus or closely linked to a marker locus that is in strong LD with the QTL. The latter situation is expected in a rapidly growing population in which the allele of large effect was present initially in one copy. Therefore, the proposed tests are useful under the same conditions as those for successful LD mapping of a dichotomous trait or disease. Simulations show that, for detection of the presence of a QTL, these tests are more powerful than a simple t-test. The tests also provide a basis for defining a measure of association, gamma, between a low-frequency allele at a putative QTL and a low-frequency allele at a marker locus.

Genetic hitch-hiking in a subdivided population.

The problem of genetic hitch-hiking in a geographically subdivided population is analysed under the assumption that migration rates among populations are relatively small compared with the selection coefficient for a newly arising advantageous allele. The approximate method used in the paper is valid when the number of emigrants per generation (Nm) is less than one. The approximate analysis shows that hitch-hiking can result in substantial differences among populations in the frequencies of neutral alleles closely linked to the advantageous allele. Thus, in cases for which genetic hitch-hiking is thought to be responsible for low levels of genetic variability in regions of the genome with restricted crossing over, it might be possible to find confirmatory evidence for that hypothesis by finding unusual patterns of geographic differentiation in the same regions of the genome.

Epistatic selection in a multi-locus Levene model and implications for linkage disequilibrium.

We analyze a multiple-locus extension of the Levene (1953) model of population subdivision. We show that stable or quasistable linkage disequilibrium between two selected loci can be maintained even with free recombination, provided that there is a strong enough epistatic interaction. We then consider the dynamics of a third neutral locus and show that its approach to linkage equilibrium depends on the recombination rates and the selection intensities. There is an embedding or hitchhiking effect that extends the time during which a neutral locus which is closely linked to one of the selected loci remains in disequilibrium with both selected loci. Therefore, strong disequilibrium between two loci does not necessarily indicate that those loci are themselves selected, but it does indicate that there is strong selection acting at least on nearby loci. This property implies a warning that screening for linkage disequilibrium as a tool to identify functionally important sites in a genome can be misleading.

Likelihood analysis of disequilibrium mapping, and related problems.

In this paper a theory is developed that provides the sampling distribution of alleles at a diallelic marker locus closely linked to a low-frequency allele that arose as a single mutant. The sampling distribution provides a basis for maximum-likelihood estimation of either the recombination rate, the mutation rate, or the age of the allele, provided that the two other parameters are known. This theory is applied to (1) the data of Hästbacka et al., to estimate the recombination rate between a locus associated with diastrophic dysplasia and a linked RFLP marker; (2) the data of Risch et al., to estimate the age of a presumptive allele causing idiopathic distortion dystonia in Ashkenazi jews; and (3) the data of Tishkoff et al., to estimate the date at which, at the CD4 locus, non-African lineages diverged from African lineages. We conclude that the extent of linkage disequilibrium can lead to relatively accurate estimates of recombination and mutation rates and that those estimates are not very sensitive to parameters, such as the population age, whose values are not known with certainty. In contrast, we also conclude that, in many cases, linkage disequilibrium may not lead to useful estimates of allele age, because of the relatively large degree of uncertainly in those estimates.

Incorporating genotypes of relatives into a test of linkage disequilibrium.

Genetic data from autosomal loci in diploids generally consist of genotype data for which no phase information is available, making it difficult to implement a test of linkage disequilibrium. In this paper, we describe a test of linkage disequilibrium based on an empirical null distribution of the likelihood of a sample. Information on the genotypes of related individuals is explicitly used to help reconstruct the gametic phase of the independent individuals. Simulation studies show that the present approach improves on estimates of linkage disequilibrium gathered from samples of completely independent individuals but only if some offspring are sampled together with their parents. The failure to incorporate some parents sharply decreases the sensitivity and accuracy of the test. Simulations also show that for multiallelic data (more than two alleles) our testing procedure is not as powerful as an exact test based on known haplotype frequencies, owing to the interaction between departure from Hardy-Weinberg equilibrium and linkage disequilibrium.

The sampling distribution of disease-associated alleles.

A theory is developed that provides the sampling distribution of low frequency alleles at a single locus under the assumption that each allele is the result of a unique mutation. The numbers of copies of each allele is assumed to follow a linear birth-death process with sampling. If the population is of constant size, standard results from theory of birth-death processes show that the distribution of numbers of copies of each allele is logarithmic and that the joint distribution of numbers of copies of k alleles found in a sample of size n follows the Ewens sampling distribution. If the population from which the sample was obtained was increasing in size, if there are different selective classes of alleles, or if there are differences in penetrance among alleles, the Ewens distribution no longer applies. Likelihood functions for a given set of observations are obtained under different alternative hypotheses. These results are applied to published data from the BRCA1 locus (associated with early onset breast cancer) and the factor VIII locus (associated with hemophilia A) in humans. In both cases, the sampling distribution of alleles allows rejection of the null hypothesis, but relatively small deviations from the null model can account for the data. In particular, roughly the same population growth rate appears consistent with both data sets.