Maintenance of Genetic Variability under the Pressure of Neutral and Deleterious Mutations in a Finite Population

In order to assess the effect of deleterious mutations on various measures of genic variation, approximate formulas have been developed for the frequency spectrum, the mean number of alleles in a sample, and the mean homozygosity; in some particular cases, exact formulas have been obtained. The assumptions made are that two classes of mutations exist, neutral and deleterious, and that selection is strong enough to keep deleterious alleles in low frequencies, the mode of selection being either genic or recessive. The main findings are: (1) If the expected value (q) of the sum of the frequencies of deleterious alleles is about 10% or less, then the presence of deleterious alleles causes only a minor reduction in the mean number of neutral alleles in a sample, as compared to the case of q = 0. Also, the low- and intermediate-frequency parts of the frequency spectrum of neutral alleles are little affected by the presence of deleterious alleles, though the high-frequency part may be changed drastically. (2) The contribution of deleterious mutations to the expected total number of alleles in a sample can be quite large even if q is only 1 or 2%. (3) The mean homozygosity is roughly equal to (1--2q)/(1 + theta 1), where theta 1 is twice the number of new neutral mutations occurring in each generation in the total population. Thus, deleterious mutations increase the mean heterozygosity by about 2q/(1 + theta 1). The present results have been applied to study the controversial problem of how deleterious mutations may affect the testing of the neutral mutation hypothesis.     

Maintenance of Genetic Variability under the Joint Effect of Mutation, Selection and Random Drift

Formulae are developed for the distribution of allele frequencies (the frequency spectrum), the mean number of alleles in a sample, and the mean and variance of heterozygosity under mutation pressure and under either genic or recessive selection. Numerical computations are carried out by using these formulae and Watterson's (1977) formula for the distribution of allele frequencies under overdominant selection. The following properties are observed: (1) The effect of selection on the distribution of allele frequencies is slight when 4Ns 相似文献   

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.     

Human neutral alpha-glucosidase C: genetic polymorphism including a "null" allele.

We describe a genetic polymorphism of human neutral alpha-glucosidase C, detected in lymphoid cells by a combination of starch gel electrophoresis and isoelectric focusing. The seven phenotypes observed appear to result from the expression of four different alleles. The distribution of the observed phenotypes fits the expected distribution predicted from calculated gene frequencies in Hardy-Weinberg equilibrium. Family studies are consistent with autosomal inheritance of the gene. The product of one of the alleles is unusual in that it is "silent," with an estimated gene frequency of .174 in an outbred white population. Approximately one-third of the population is heterozygous "null." Homozygosity for the allele has not been associated with any obvious disease state. This is the third example of a "null" allele which has a substantial gene frequency in an outbred population but does not appear to result in disease in the homozygous state.     

On the persistence and pervasiveness of a new mutation

It has frequently been assumed that the persistence of a deleterious mutation (the average number of generations before its loss) and its pervasiveness (the average number of individuals carrying the gene before its loss) are equal. This is true for a particular simple, widely used infinite model, but this agreement is not general. If hs > 1/(4N(e)), where hs is the selective disadvantage of mutant heterozygotes and N(e) is the effective population number, the contribution of homozygous mutants can be neglected and the simple approximate formula 1/hs gives the mean pervasiveness. But the expected persistence is usually much smaller, 2(log(e)(1/2hs) + 1 - gamma) where gamma = 0.5772. For neutral mutations, the total number of heterozygotes until fixation or loss is often the quantity of interest, and its expected value is 2N(e), with remarkable generality for various population structures. In contrast, the number of generations until fixation or loss, 2(N(e)/N)(1 + log(e)2N), is much smaller than the total number of heterozygotes. In general the number of generations is less than the number of individuals.     

The Coalescent Process in Models with Selection

Statistical properties of the process describing the genealogical history of a random sample of genes are obtained for a class of population genetics models with selection. For models with selection, in contrast to models without selection, the distribution of this process, the coalescent process, depends on the distribution of the frequencies of alleles in the ancestral generations. If the ancestral frequency process can be approximated by a diffusion, then the mean and the variance of the number of segregating sites due to selectively neutral mutations in random samples can be numerically calculated. The calculations are greatly simplified if the frequencies of the alleles are tightly regulated. If the mutation rates between alleles maintained by balancing selection are low, then the number of selectively neutral segregating sites in a random sample of genes is expected to substantially exceed the number predicted under a neutral model.     

Premeiotic clusters of mutation and the cost of natural selection

Haldane stated that there is a cost of natural selection for new beneficial alleles to be substituted over time. Most of this cost, which leads to "genetic deaths," is in the early generations of the substitution process when the new allele is low in frequency. It depends on the initial frequency and dominance value, but not the selection coefficient, of the advantageous allele. There have been numerous suggestions on how to reduce the cost for preexisting genetic variation that goes from disadvantageous, or neutral, to advantageous with a change in the environment. However, the cost of natural selection for new alleles that arise by mutation is assumed to be high, based on the assumption that new mutant alleles arise in natural populations as single events [1/(2N) of the total alleles]. However, not all mutant alleles arise as single events. Premeiotic mutations occur frequently in individuals (germinal mosaics), giving rise to multiple copies of identical mutant alleles called a "cluster" (C) with an initial allele frequency of C/(2N) instead of 1/(2N). These clusters of new mutant alleles reduce the cost of natural selection in direct proportion to the relative size of the cluster. Hence new advantageous alleles that arise by mutation have the greatest chance of going to fixation if they occur in large clusters in small populations.     

On the distribution of allele frequencies in a diffusion model

An expression is derived and values tabulated for the expected allele frequencies and their variances, arranged in decreasing order in a population, from the finite and infinite alleles diffusion model in Watterson (1976). The neutral model and also a model with heterozygote selection are considered. Some observed ABO blood group allele frequencies are compared with the tabulated expected frequencies in the neutral three allele model. This extends the results of Watterson and Guess (1977) who tabulate the expected value of the most common allele. One test of neutrality previously advocated is to consider the distribution of F, the population homozygosity, conditional on G, the product of allele frequencies. However it is shown here that for a large number of alleles, F and G are asymptotically independent, the test would not be a good one in this case. A limit theorem is derived for the distribution of allele frequencies in the neutral model when the mutation rate is large. In this case F is shown to be asymptotically normal. An inequality is derived for the probability that the oldest allele in a population is amongst the r most frequent types. An inequality is also found for the probability that a sample will only contain representatives of the r most frequent allele types in the population.     

A Markov Process of Gene Frequency Change in a Geographically Structured Population

A Markov process (chain) of gene frequency change is derived for a geographically-structured model of a population. The population consists of colonies which are connected by migration. Selection operates in each colony independently. It is shown that there exists a stochastic clock that transforms the originally complicated process of gene frequency change to a random walk which is independent of the geographical structure of the population. The time parameter is a local random time that is dependent on the sample path. In fact, if the alleles are selectively neutral, the time parameter is exactly equal to the sum of the average local genetic variation appearing in the population, and otherwise they are approximately equal. The Kolmogorov forward and backward equations of the process are obtained. As a limit of large population size, a diffusion process is derived. The transition probabilities of the Markov chain and of the diffusion process are obtained explicitly. Certain quantities of biological interest are shown to be independent of the population structure. The quantities are the fixation probability of a mutant, the sum of the average local genetic variation and the variation summed over the generations in which the gene frequency in the whole population assumes a specified value.     

Conditions under Which the Mean Fertility Is Maximized When a Population Is at a Stable Equilibrium

It is assumed that there is a population with two alleles at one locus, random mating of adults and selection only involving differential fertilities. By making use of the Kuhn-Tucker theory of optimization under constraints, conditions are derived under which stable equilibrium frequencies x, y and z of the three genotypes are the same as those that maximize the mean fertility of the population. We derive all sets of frequencies of this type for the Hadeler-Liberman symmetric fertility model and all such sets for which at least one genotype is missing for the general model. If the population has frequencies that are initially near those at which there is both a stable equilibrium and maximization of the mean fertility, then the mean fertility (t) at time t is nondecreasing with t as t -> &. It is found that it is possible for the stable equilibrium maximum points (x, y, z) to be one or two points on a ridge on which the mean fertility is maximized or the entire set of points on the ridge. Furthermore, may be smaller on this ridge than at another stable equilibrium point at which is not even locally maximized.     

Electrophoretically silent alleles in a finite population

Summary The expected number of silent alleles in an electromorph is computed for various values of population size (N), mutation rate (u), and sample size (s) under the assumption of no selection. The proportion of alleles undetectable by electrophoresis is higher when Nu is large than when this is small. It is shown that an electromorph of high population frequency has more silent alleles than an electromorph of low frequency if the sample size is the same.     

An Approach to Population and Evolutionary Genetic Theory for Genes in Mitochondria and Chloroplasts, and Some Results

We developed population genetic theory for organelle genes, using an infinite alleles model appropriate for molecular genetic data, and considering the effects of mutation and random drift on the frequencies of selectively neutral alleles. The effects of maternal inheritance and vegetative segregation of organelle genes are dealt with by defining new effective gene numbers, and substituting these for 2N(e) in classical theory of nuclear genes for diploid organisms. We define three different effective gene numbers. The most general is N(lambda), defined as a function of population size, number of organelle genomes per cell, and proportions of genes contributed by male and female gametes to the zygote. In many organisms, vegetative segregation of organelle genomes and intracellular random drift of organelle gene frequencies combine to produce a predominance of homoplasmic cells within individuals in the population. Then, the effective number of organelle genes is N(eo), a simple function of the numbers of males and females and of the maternal and paternal contributions to the zygote. Finally, when the paternal contribution is very small, N( eo) is closely approximated by the number of females, N( f). Then if the sex ratio is 1, the mean time to fixation or loss of new mutations is approximately two times longer for nuclear genes than for organelle genes, and gene diversity is approximately four times greater. The difference between nuclear and organelle genes disappears or is reversed in animals in which males have large harems. The differences between nuclear and organelle gene behavior caused by maternal inheritance and vegetative segregation are generally small and may be overshadowed by differences in mutation rates to neutral alleles. For monoecious organisms, the effective number of organelle genes is approximately equal to the total population size N. We also show that a population can be effectively subdivided for organelle genes at migration rates which result in panmixis for nuclear genes, especially if males migrate more than females.     

The effect of recurrent mutation on the frequency spectrum of a segregating site and the age of an allele

The sample frequency spectrum of a segregating site is the probability distribution of a sample of alleles from a genetic locus, conditional on observing the sample to be polymorphic. This distribution is widely used in population genetic inferences, including statistical tests of neutrality in which a skew in the observed frequency spectrum across independent sites is taken as a signature of departure from neutral evolution. Theoretical aspects of the frequency spectrum have been well studied and several interesting results are available, but they are usually under the assumption that a site has undergone at most one mutation event in the history of the sample. Here, we extend previous theoretical results by allowing for at most two mutation events per site, under a general finite allele model in which the mutation rate is independent of current allelic state but the transition matrix is otherwise completely arbitrary. Our results apply to both nested and nonnested mutations. Only the former has been addressed previously, whereas here we show it is the latter that is more likely to be observed except for very small sample sizes. Further, for any mutation transition matrix, we obtain the joint sample frequency spectrum of the two mutant alleles at a triallelic site, and derive a closed-form formula for the expected age of the younger of the two mutations given their frequencies in the population. Several large-scale resequencing projects for various species are presently under way and the resulting data will include some triallelic polymorphisms. The theoretical results described in this paper should prove useful in population genomic analyses of such data.     

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.      

Mutant alleles of small effect are primarily responsible for the loss of fitness with slow inbreeding in Drosophila melanogaster.

Multilocus simulation is used to identify genetic models that can account for the observed rates of inbreeding and fitness decline in laboratory populations of Drosophila melanogaster. The experimental populations were maintained under crowded conditions for approximately 200 generations at a harmonic mean population size of Nh approximately 65-70. With a simulated population size of N = 50, and a mean selective disadvantage of homozygotes at individual loci approximately 1-2% or less, it is demonstrated that the mean effective population size over a 200-generation period may be considerably greater than N, with a ratio matching the experimental estimate of Ne/Nh approximately 1.4. The buildup of associative overdominance at electrophoretic marker loci is largely responsible for the stability of gene frequencies and the observed reduction in the rate of inbreeding, with apparent selection coefficients in favor of the heterozygote at neutral marker loci increasing rapidly over the first N generations of inbreeding to values approximately 5-10%. The observed decline in fitness under competitive conditions in populations of size approximately 50 in D. melanogaster therefore primarily results from mutant alleles with mean effects on fitness as homozygotes of sm < or = 0.02. Models with deleterious recessive mutants at the background loci require that the mean selection coefficient against heterozygotes is at most hsm approximately 0.002, with a minimum mutation rate for a single Drosophila autosome 100 cM in length estimated to be in the range 0.05-0.25, assuming an exponential distribution of s. A typical chromosome would be expected to carry at least 100-200 such mutant alleles contributing to the decline in competitive fitness with slow inbreeding.     

A strategy to discover genes that carry multi-allelic or mono-allelic risk for common diseases: a cohort allelic sums test (CAST)

A method is described to discover if a gene carries one or more allelic mutations that confer risk for any specified common disease. The method does not depend upon genetic linkage of risk-conferring mutations to high frequency genetic markers such as single nucleotide polymorphisms. Instead, the sums of allelic mutation frequencies in case and control cohorts are determined and a statistical test is applied to discover if the difference in these sums is greater than would be expected by chance. A statistical model is presented that defines the ability of such tests to detect significant gene-disease relationships as a function of case and control cohort sizes and key confounding variables: zygosity and genicity, environmental risk factors, errors in diagnosis, limits to mutant detection, linkage of neutral and risk-conferring mutations, ethnic diversity in the general population and the expectation that among all exonic mutants in the human genome greater than 90% will be neutral with regard to any effect on disease risk. Means to test the null hypothesis for, and determine the statistical power of, each test are provided. For this "cohort allelic sums test" or "CAST", the statistical model and test are provided as an Excel program, CASTAT(c) at . Based on genetics, technology and statistics, a strategy of enumerating the mutant alleles carried in the exons and splice sites of the estimated approximately 25,000 human genes in case cohort samples of 10,000 persons for each of 100 common diseases is proposed and evaluated: A wide range of possible conditions of multi-allelic or mono-allelic and monogenic, multigenic or polygenic (including epistatic) risk are found to be detectable using the statistical criteria of 1 or 10 "false positive" gene associations approximately 25,000 gene-disease pair-wise trials and a statistical power of >0.8. Using estimates of the distribution of both neutral and gene-inactivating nondeleterious mutations in humans and the sensitivity of the test to multigenic or multicausal risk, it is estimated that about 80% of nullizygous, heterozygous and functionally dominant gene-common disease associations may be discovered. Limitations include relative insensitivity of CAST to about 60% of possible associations given homozygous (wild type) risk and, more rarely, other stochastic limits when the frequency of mutations in the case cohort approaches that of the control cohort and biases such as absence of genetic risk masked by risk derived from a shared cultural environment.     

A Study on a Nearly Neutral Mutation Model in Finite Populations

As a nearly neutral mutation model, the house-of-cards model is studied in finite populations using computer simulations. The distribution of the mutant effect is assumed to be normal. The behavior is mainly determined by the product of the population size, N, and the standard deviation, sigma, of the distribution of the mutant effect. If 4N sigma is large compared to one, a few advantageous mutants are quickly fixed in early generations. Then most mutation becomes deleterious and very slow increase of the average selection coefficient follows. It takes very long for the population to reach the equilibrium state. Substitutions of alleles occur very infrequently in the later stage. If 4N sigma is the order of one or less, the behavior is qualitatively similar to that of the strict neutral case. Gradual increase of the average selection coefficient occurs and in generations of several times the inverse of the mutation rate the population almost reaches the equilibrium state. Both advantageous and neutral (including slightly deleterious) mutations are fixed. Except in the early stage, an increase of the standard deviation of the distribution of the mutant effect decreases the average heterozygosity. The substitution rate is reduced as 4N sigma is increased. Three tests of neutrality, one using the relationship between the average and the variance of heterozygosity, another using the relationship between the average heterozygosity and the average number of substitutions and Watterson's homozygosity test are applied to the consequences of the present model. It is found that deviation from the neutral expectation becomes apparent only when 4N sigma is more than two. Also a simple approximation for the model is developed which works well when the mutation rate is very small.     

Population estimators or progeny tests: what is the best method to assess null allele frequencies at SSR loci?

Nuclear SSRs are notorious for having relatively high frequencies of null alleles, i.e. alleles that fail to amplify and are thus recessive and undetected in heterozygotes. In this paper, we compare two kinds of approaches for estimating null allele frequencies at seven nuclear microsatellite markers in three French Fagus sylvatica populations: (1) maximum likelihood methods that compare observed and expected homozygote frequencies in the population under the assumption of Hardy-Weinberg equilibrium and (2) direct null allele frequency estimates from progeny where parent genotypes are known. We show that null allele frequencies are high in F. sylvatica (7.0% on average with the population method, 5.1% with the progeny method), and that estimates are consistent between the two approaches, especially when the number of sampled maternal half-sib progeny arrays is large. With null allele frequencies ranging between 5% and 8% on average across loci, population genetic parameters such as genetic differentiation (F _ST) may be mostly unbiased. However, using markers with such average prevalence of null alleles (up to 15% for some loci) can be seriously misleading in fine scale population studies and parentage analysis.     

The divergence of a polygenic system subject to stabilizing selection, mutation and drift

Polygenic variation can be maintained by a balance between mutation and stabilizing selection. When the alleles responsible for variation are rare, many classes of equilibria may be stable. The rate at which drift causes shifts between equilibria is investigated by integrating the gene frequency distribution W2N II (pq)4N mu-1. This integral can be found exactly, by numerical integration, or can be approximated by assuming that the full distribution of allele frequencies is approximately Gaussian. These methods are checked against simulations. Over a wide range of population sizes, drift will keep the population near an equilibrium which minimizes the genetic variance and the deviation from the selective optimum. Shifts between equilibria in this class occur at an appreciable rate if the product of population size and selection on each locus is small (Ns alpha 2 less than 10). The Gaussian approximation is accurate even when the underlying distribution is strongly skewed. Reproductive isolation evolves as populations shift to new combinations of alleles: however, this process is slow, approaching the neutral rate (approximately mu) in small populations.     

The Sampling Distribution of Linkage Disequilibrium under an Infinite Allele Model without Selection

The sampling distributions of several statistics that measure the association of alleles on gametes (linkage disequilibrium) are estimated under a two-locus neutral infinite allele model using an efficient Monte Carlo method. An often used approximation for the mean squared linkage disequilibrium is shown to be inaccurate unless the proper statistical conditioning is used. The joint distribution of linkage disequilibrium and the allele frequencies in the sample is studied. This estimated joint distribution is sufficient for obtaining an approximate maximum likelihood estimate of C = 4Nc, where N is the population size and c is the recombination rate. It has been suggested that observations of high linkage disequilibrium might be a good basis for rejecting a neutral model in favor of a model in which natural selection maintains genetic variation. It is found that a single sample of chromosomes, examined at two loci cannot provide sufficient information for such a test if C less than 10, because with C this small, very high levels of linkage disequilibrium are not unexpected under the neutral model. In samples of size 50, it is found that, even when C is as large as 50, the distribution of linkage disequilibrium conditional on the allele frequencies is substantially different from the distribution when there is no linkage between the loci. When conditioned on the number of alleles at each locus in the sample, all of the sample statistics examined are nearly independent of theta = 4N mu, where mu is the neutral mutation rate.