Testing for increased mutation rate for neutral alleles

Methods are given to test whether, at a selectively neutral locus, one can reasonably assume a recent increase in mutation rate to new alleles. The tests use the number of singletons, doubletons, etc. occurring at this locus and also the frequency of the most frequent allele.     

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.     

The population genealogy of the infinitely-many neutral alleles model

A process analogous to Kingman's coalescent is introduced to describe the genealogy of populations evolving according to the infinitelymany neutral alleles model. The process records population frequencies in old and new classes, and labels the new classes in order of decreasing age. Its marginal distribution is characterized in a form which is amenable to explicit calculations and the transition densities of the associated K-allele models follow readily from this representation.     

Tetrasomic segregation for multiple alleles in alfalfa

Evidence of tetrasomic inheritance in alfalfa, Medicago sativa L. and M. falcata L., for multiple codominant alleles at three isozymic loci is reported in this study. The locus Prx-1 governing anodal peroxidase and the loci Lap-1 and Lap-2 governing anodal leucine-aminopeptidase were studied by starch gel electrophoresis in seedling root tissue or seeds. The progenies from several di-, tri- or tetra-allelic plants belong to the species M. sativa and M. falcata and their hybrids were studied for the segregation of the three genes. In all cases, tetrasomic inheritance of chromosomal-type segregation was observed. In another progeny resulting from the crossing of two plants involving four different alleles at locus Lap-2, tetrasomic segregation with the possible occurrence of double reduction was observed. This study presents direct evidence of autotetraploidy and the existence of tetra-allelic loci in alfalfa. It also supports the concept that the species M. sativa and M. falcata are genetically close enough to be considered biotypes of a common species.     

The inheritance of resistance alleles in multiple sclerosis

Multiple sclerosis (MS) is a complex trait in which alleles at or near the class II loci HLA-DRB1 and HLA-DQB1 contribute significantly to genetic risk. HLA-DRB1*15 and HLA-DRB1*17-bearing haplotypes and interactions at the HLA-DRB1 locus increase risk of MS but it has taken large samples to identify resistance HLA-DRB1 alleles. In this investigation of 7,093 individuals from 1,432 MS families, we have assessed the validity, mode of inheritance, associated genotypes, and the interactions of HLA-DRB1 resistance alleles. HLA-DRB1*14-, HLA-DRB1*11-, HLA-DRB1*01-, and HLA-DRB1*10-bearing haplotypes are protective overall but they appear to operate by different mechanisms. The first type of resistance allele is characterised by HLA-DRB1*14 and HLA-DRB1*11. Each shows a multiplicative mode of inheritance indicating a broadly acting suppression of risk, but a different degree of protection. In contrast, a second type is exemplified by HLA-DRB1*10 and HLA-DRB1*01. These alleles are significantly protective when they interact specifically in trans with HLA-DRB1*15-bearing haplotypes. HLA-DRB1*01 and HLA-DRB1*10 do not interact with HLA-DRB1*17, implying that several mechanisms may be operative in major histocompatibility complex-associated MS susceptibility, perhaps analogous to the resistance alleles. There are major practical implications for risk and for the exploration of mechanisms in animal models. Restriction of antigen presentation by HLA-DRB1*15 seems an improbably simple mechanism of major histocompatibility complex-associated susceptibility.     

A class of models of selectively neutral alleles

Using exchangeability as a statistical analog of neutrality, we derive a generalized sampling distribution for neutral alleles. The distribution depends upon a parameter that determines the underlying marginal distribution of the number of copies of a neutral allele and that can range from zero to infinity. The sampling model of Ewens (1972) is a special case characterized by an extreme value (0) of this parameter. Two other special cases are considered, one of which seems to be applicable to populations with a structure like that of the Yanomama Indians of South America. We then investigate the expected frequency spectra under these three special cases and discover that all three models yield a broad range of possible spectra with overlap between the special cases. We finally show that Ewens' sampling model cannot be used to construct tests of neutrality versus selection tending to maintain polymorphisms, but it can be used to construct tests of directional selection versus neutrality plus selection tending to yield polymorphic states.     

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.     

An exact test for Hardy-Weinberg and multiple alleles

Algorithms for generating the exact distribution of a finite sample drawn from a population in Hardy-Weinberg equilibrium are given for multiple alleles. The finite sampling distribution is derived analogously to Fisher's 2 X 2 exact distribution and is equivalent to Levene's conditional finite sampling distribution for Hardy-Weinberg populations. The algorithms presented are fast computationally and allow for quick alternatives to standard methods requiring corrections and approximations. Computation time is on the order of a few seconds for three-allele examples and up to 2 minutes for four-allele examples on an IBM 3081 machine.     

The number of distinguishable alleles according to the Ohta-Kimura model of neutral mutation

We calculate how many alleles one can expect to distinguish in a large sample from a large population which develops according to the Ohta-Kimura model. This number tends to infinity with the sample size, but so slowly that it is bounded for all practical purposes.Research supported by the NSF through a grant to Cornell University     

The sampling theory of neutral alleles and an urn model in population genetics

The behaviour of a Pólya-like urn which generates Ewens' sampling formula in population genetics is investigated. Connections are made with work of Watterson and Kingman and to the Poisson-Dirichlet distribution. The order in which novel types occur in the urn is shown to parallel the age distribution of the infinitely many alleles diffusion model and consequences of this property are explored. Finally the urn process is related to Kingman's coalescent with mutation to provide a rigorous basis for this parallel.This research was partially supported by the Sloan Foundation under Grant 85-6-14 and by the National Science Foundation     

A discussion of simulation results for various aspects of the neutral alleles model

In this paper a number of simulation results relating to the theory of neutral alleles are discussed. In particular, results derived from the distribution of Karlin, McGregor, and Ewens for the number and configuration of alleles in a sample of genes from a selectively neutral locus are considered. The problems discussed concern efficiencies of estimation, an approximation to the distribution of a test-statistic for neutrality, and the effect of changes in population size on an index of neutrality.     

Rare variant alleles in the light of the neutral theory

Based on the neutral theory of molecular evolution and polymorphism, and particularly assuming "the model of infinite alleles," a method is proposed which enables us to estimate the fraction of selectively neutral alleles (denoted by Pneut) among newly arisen mutations. It makes use of data on the distribution of rare variant alleles in large samples together with information on the average heterozygosity. The formula proposed is Pneut = [He/(1-He)] [loge(2nq)/n alpha (x less than q)], where n alpha(x less than q) is the average number of rare alleles per locus whose frequency, x, is less than q; n is the average sample size used to count rare alleles; He is the average heterozygosity per locus; and q is a small preassigned number such as q = 0.01. The method was applied to observations on enzyme and other protein loci in plaice, humans (European and Amerindian), Japanese monkeys, and fruit flies. Estimates obtained for them range from 0.064 to 0.21 with the mean and standard error Pneut = 0.14 +/- 0.06. It was pointed out that these estimates are consistent with the corresponding estimate Pneut(Hb) = 0.14 obtained independently based on the neutral theory and using data on the evolutionary rate of nucleotide substitutions in globin pseudogenes together with those in the normal globins.      

On the genealogy of a sample of neutral rare alleles

This paper concerns the genealogical structure of a sample of chromosomes sharing a neutral rare allele. We suppose that the mutation giving rise to the allele has only happened once in the history of the entire population, and that the allele is of known frequency q in the population. Within a coalescent framework C. Wiuf and P. Donnelly (1999, Theor. Popul. Biol. 56, 183-201) derived an exact analysis of the conditional genealogy but it is inconvenient for applications. Here, we develop an approximation to the exact distribution of the conditional genealogy, including an approximation to the distribution of the time at which the mutation arose. The approximations are accurate for frequencies q<5-10%. In addition, a simple and fast simulation scheme is constructed. We consider a demography parameterized by a d-dimensional vector alpha=(alpha(1), em leader, alpha(d)). It is shown that the conditional genealogy and the age of the mutation have distributions that depend on a=qalpha and q only, and that the effect of q is a linear scaling of times in the genealogy; if q is doubled, the lengths of all branches in the genealogy are doubled. The theory is exemplified in two different demographies of some interest in the study of human evolution: (1) a population of constant size and (2) a population of exponentially decreasing size (going backward in time).     

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.