The loss of statistical power to distinguish populations when certain samples are ambiguous

Case-control studies are used to map loci associated with a genetic disease. The usual case-control study tests for significant differences in frequencies of alleles at marker loci. In this paper, we consider the problem of comparing two or more marker loci simultaneously and testing for significant differences in haplotype rather than allele frequencies. We consider two situations. In the first, genotypes at marker loci are resolved into haplotypes by making use of biochemical methods or by genotyping family members. In the second, genotypes at marker loci are not resolved into haplotypes, but, by assuming random mating, haplotypes can be inferred using a likelihood method such as the expectation-maximization (EM) algorithm. We assume that a causative locus has two alleles with a multiplicative effect on the penetrance of a disease, with one allele increasing the penetrance by a factor pi. We find, for small values of pi-1 and large sample sizes, asymptotic results that predict the statistical power of a test for significant differences in haplotype frequencies between cases and a random sample of the population, both when haplotypes can be resolved and when haplotypes have to be inferred. The increase in power when haplotypes can be resolved can be expressed as a ratio R, which is the increase in sample size needed to achieve the same power when haplotypes are resolved over when they are not resolved. In general, R depends on the pattern of linkage disequilibrium between the causative allele and the marker haplotypes but is independent of the frequency of the causative allele and, to a first approximation, is independent of pi. For the special situation of two di-allelic marker loci, we obtain a simple expression for R and its upper bound.     

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.     

Exploring variation in the d(N)/d(S) ratio among sites and lineages using mutational mappings: applications to the influenza virus

We use a likelihood-based method for mapping mutations on a phylogeny in a way that allows for both site-specific and lineage-specific variation in selection intensity. The method accounts for many of the potential sources of bias encountered in mapping of mutations on trees while still being computationally efficient. We apply the method to a previously published influenza data set to investigate hypotheses about changes in selection intensity in influenza strains. Influenza virus is sometimes propagated in chicken cells for several generations before sequencing, a process that has been hypothesized to induce mutations adapting the virus to the lab medium. Our analysis suggests that there are approximately twice as many replacement substitutions in lineages propagated in chicken eggs as in lineages that are not. Previous studies have attempted to predict which viral strains future epidemics may arise from using inferences regarding positive selection. The assumption is that future epidemics are more likely to arise from the strains in which positive selection on the so-called “trunk lineages” of the evolutionary tree is most pervasive. However, we find no difference in the strength of selection in the trunk lineages versus other evolutionary lineages. Our results suggest that it may be more difficult to use inferences regarding the strength of selection on mutations to make predictions regarding viral epidemics than previously thought. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. Reviewing Editor: Dr. Willie Swanson     

Non-equilibrium theory of the allele frequency spectrum

A forward diffusion equation describing the evolution of the allele frequency spectrum is presented. The influx of mutations is accounted for by imposing a suitable boundary condition. For a Wright-Fisher diffusion with or without selection and varying population size, the boundary condition is lim(x downward arrow0)xf(x,t)=thetarho(t), where f(.,t) is the frequency spectrum of derived alleles at independent loci at time t and rho(t) is the relative population size at time t. When population size and selection intensity are independent of time, the forward equation is equivalent to the backwards diffusion usually used to derive the frequency spectrum, but this approach allows computation of the time dependence of the spectrum both before an equilibrium is attained and when population size and selection intensity vary with time. From the diffusion equation, a set of ordinary differential equations for the moments of f(.,t) is derived and the expected spectrum of a finite sample is expressed in terms of those moments. The use of the forward equation is illustrated by considering neutral and selected alleles in a highly simplified model of human history. For example, it is shown that approximately 30% of the expected total heterozygosity of neutral loci is attributable to mutations that arose since the onset of population growth in roughly the last 150,000 years.     

The Quantitative Genetic Consequences of Pleiotropy under Stabilizing and Directional Selection

The independence of two phenotypic characters affected by both pleiotropic and nonpleiotropic mutations is investigated using a generalization of M. Slatkin's stepwise mutation model of 1987. The model is used to determine whether predictions of either the multivariate normal model introduced in 1980 by R. Lande or the house-of-cards model introduced in 1985 by M. Turelli can be regarded as typical of models that are intermediate between them. We found that, under stabilizing selection, the variance of one character at equilibrium may depend on the strength of stabilizing selection on the other character (as in the house-of-cards model) or not (as in the multivariate normal model) depending on the types of mutations that can occur. Similarly, under directional selection, the genetic covariance between two characters may increase substantially (as in the house-of-cards model) or not (as in the multivariate normal model) depending on the kinds of mutations that are assumed to occur. Hence, even for the simple model we consider, neither the house-of-cards nor the multivariate normal model can be used to make predictions, making it unlikely that either could be used to draw general conclusions about more complex and realistic models.     

Genealogy of neutral genes in two partially isolated populations

Gene genealogy in two partially isolated populations which diverged at a given time t in the past and have since been exchanging individuals at a constant rate m is studied based upon an analytic method for large t and a simulation method for any t. Particular attention is paid to the conditions under which neutral genes sampled from populations are mono-, para-, and polyphyletic in terms of coalescence (divergence) times of genes. It is shown tha the probability of monophyly is high if M = 2Nm less than 0.5 and T = t/(2N) greater than 1, where N is the size of ancestral and descendant haploid populations, in which case most gene genealogies are likely to be concordant with the population relatedness. This probbility decreases as the sample size of genes increases. On the other hand, the case where the probability of monophyly is low will be either that of M greater than 1 and any T or that of M less than 1 and T less than 1, but the clear distinction between these conditions appears very difficult to make. These results are also examined if the gene genealogy is reconstructed from nucleotide differences. It is then shown that the results based upon coalescence times remain valid if the number of nucleotide differences between any pair of genes is not much smaller than 10. To observe such large nucleotide differences in small populations and therefore infer a reliable gene genealogy, we must examine a fairly long stretch of DNA sequences.     

New insights into the history of the C-14010 lactase persistence variant in Eastern and Southern Africa

Lactase persistence (LP), the ability to digest lactose into adulthood, is strongly associated with the cultural traits of pastoralism and milk-drinking among human populations, and several different genetic variants are known that confer LP. Recent studies of LP variants in Southern African populations, with a focus on Khoisan-speaking groups, found high frequencies of an LP variant (the C-14010 allele) that also occurs in Eastern Africa, and concluded that the C-14010 allele was brought to Southern Africa via a migration of pastoralists from Eastern Africa. However, this conclusion was based on indirect evidence; to date no study has jointly analyzed data on the C-14010 allele from both Southern African Khoisan-speaking groups and Eastern Africa. Here, we combine and analyze published data on the C-14010 allele in Southern and Eastern African populations, consisting of haplotypes with the C-14010 allele and four closely-linked short tandem repeat loci. Our results provide direct evidence for the previously-hypothesized Eastern African origin of the C-14010 allele in Southern African Khoisan-speaking groups. In addition, we find evidence for a separate introduction of the C-14010 allele into the Bantu-speaking Xhosa. The estimated selection intensity on the C-14010 allele in Eastern Africa is lower than that in Southern Africa, which suggests that in Eastern Africa the dietary changes conferring the fitness advantage associated with LP occurred some time after the origin of the C-14010 allele. Conversely, in Southern Africa the fitness advantage was present when the allele was introduced, as would be expected if pastoralism was introduced concomitantly. Am J Phys Anthropol 156:661–664, 2015. © 2014 Wiley Periodicals, Inc.     

Inbreeding coefficients and coalescence times.

This paper describes the relationship between probabilities of identity by descent and the distribution of coalescence times. By using the relationship between coalescence times and identity probabilities, it is possible to extend existing results for inbreeding coefficients in regular systems of mating to find the distribution of coalescence times and the mean coalescence times. It is also possible to express Sewall Wright's FST as the ratio of average coalescence times of different pairs of genes. That simplifies the analysis of models of subdivided populations because the average coalescence time can be found by computing separately the time it takes for two genes to enter a single subpopulation and time it takes for two genes in the same subpopulation to coalesce. The first time depends only on the migration matrix and the second time depends only on the total number of individuals in the population. This approach is used to find FST in the finite island model and in one- and two-dimensional stepping-stone models. It is also used to find the rate of approach of FST to its equilibrium value. These results are discussed in terms of different measures of genetic distance. It is proposed that, for the purposes of describing the amount of gene flow among local populations, the effective migration rate between pairs of local populations, M, which is the migration rate that would be estimated for those two populations if they were actually in an island model, provides a simple and useful measure of genetic similarity that can be defined for either allozyme or DNA sequence data.