Two-locus identity probabilities and identity disequilibrium in a partially selfing subdivided population

Measures of association of genes at different loci (linkage disequilibrium) are widely used to determine whether the structure of natural populations is clonal or not, to map genes from population data, or to test for the homogeneity of response of molecular markers to background selection, for example. However, the usual definitions of parameters for gametic associations may not be suitable for all these purposes. In this paper, we derive the recursion equations for one- and two-locus identity probabilities in an infinite island model. We study the role of drift, gene flow, partial selfing and mutation model on the expected association of genes across loci. We define the 'within-subpopulation identity disequilibrium' as the difference between the joint two-locus probability of identity in state and the expected product of one-locus identity probabilities. We evaluate this parameter as a function of recombination rate, effective size, gene flow and selfing rate. Within-subpopulation identity disequilibrium attains maximum values for intermediate immigration rates, whatever the selfing rate. Moreover, identity disequilibrium may be very small, even for high selfing rates. We discuss the implications of these findings for the analysis of data from natural populations.     

The effect of life-history and mode of inheritance on neutral genetic variability

Formulae for the effective population sizes of autosomal, X-linked, Y-linked and maternally transmitted loci in age-structured populations are developed. The approximations used here predict both asymptotic rates of increase in probabilities of identity, and equilibrium levels of neutral nucleotide site diversity under the infinite-sites model. The applications of the results to the interpretation of data on DNA sequence variation in Drosophila, plant, and human populations are discussed. It is concluded that sex differences in demographic parameters such as adult mortality rates generally have small effects on the relative effective population sizes of loci with different modes of inheritance, whereas differences between the sexes in variance in reproductive success can have major effects, either increasing or reducing the effective population size for X-linked loci relative to autosomal or Y-linked loci. These effects need to be accounted for when trying to understand data on patterns of sequence variation for genes with different transmission modes.     

Rate and directionality of mutations and effects of allele size constraints at anonymous, gene-associated, and disease-causing trinucleotide loci.

We studied the patterns of within- and between-population variation at 29 trinucleotide loci in a random sample of 200 healthy individuals from four diverse populations: Germans, Nigerians, Chinese, and New Guinea highlanders. The loci were grouped as disease-causing (seven loci with CAG repeats), gene-associated (seven loci with CAG/CCG repeats and eight loci with AAT repeats), or anonymous (seven loci with AAT repeats). We used heterozygosity and variance of allele size (expressed in units of repeat counts) as measures of within-population variability and GST (based on heterozygosity as well as on allele size variance) as the measure of genetic differentiation between populations. Our observations are: (1) locus type is the major significant factor for differences in within-population genetic variability; (2) the disease-causing CAG repeats (in the nondisease range of repeat counts) have the highest within-population variation, followed by the AAT-repeat anonymous loci, the AAT-repeat gene-associated loci, and the CAG/CTG-repeat gene-associated loci; (3) an imbalance index beta, the ratio of the estimates of the product of effective population size and mutation rate based on allele size variance and heterozygosity, is the largest for disease-causing loci, followed by AAT- and CAG/CCG-repeat gene-associated loci and AAT-repeat anonymous loci; (4) mean allele size correlates positively with allele size variance for AAT- and CAG/CCG-repeat gene-associated loci and negatively for anonymous loci; and (5) GST is highest for the disease-causing loci. These observations are explained by specific differences of rates and patterns of mutations in these four groups of trinucleotide loci, taking into consideration the effects of the past demographic history of the modern human population.     

Neutral two-locus multiple allele models with recombination

General formulae for the homozygosity and variance of linkage disequilibrium are derived for neutral, stationary, two-locus multiple allele models where there is a symmetric type of mutation at each locus. Particular cases examined are K allele models, the infinite alleles model, and the stepwise mutation model. The two-locus infinite allele model is examined at the molecular level and a joint probability generating function is found for the number of heterozygous sites at each locus in two randomly chosen gametes.     

Equilibrium Values of Measures of Population Subdivision for Stepwise Mutation Processes

Expected values of WRIGHT's F-statistics are functions of probabilities of identity in state. These values may be quite different under an infinite allele model and under stepwise mutation processes such as those occurring at microsatellite loci. However, a relationship between the probability of identity in state in stepwise mutation models and the distribution of coalescence times can be deduced from the relationship between probabilities of identity by descent and the distribution of coalescence times. The values of F(IS) and F(ST) can be computed using this property. Examination of the conditional probability of identity in state given some coalescence time and of the distribution of coalescence times are also useful for explaining the properties of F(IS) and F(ST) at high mutation rate loci, as shown here in an island model of population structure.     

Inbreeding depression and the evolution of dispersal rates: a multilocus model

Inbreeding depression is one of the possible reasons organisms disperse. In this article, we present a two-locus model for the evolution of dispersal in the presence of inbreeding depression. The first locus codes for a modifier of the migration rate, while the second locus is a selected locus generating inbreeding depression. We express the change in frequency of the migration modifier as a function of allele frequencies and genetic associations and then use a quasi-equilibrium assumption to express genetic associations as functions of allele frequencies. Our model disentangles two effects of inbreeding depression: it gives an advantage to migrant individuals because their offspring are on average less homozygous, but it also decreases the degree of population structure, thus decreasing the strength of kin selection for dispersal. We then extend our model to include an infinite number of selected loci. When the cost of dispersal is not too high, the model predictions are confirmed by multilocus simulation results and show that inbreeding depression can have a substantial effect on the dispersal rate. For high costs of dispersal, we observe discrepancies between the model and the simulations, probably caused by associations among selected loci, which are neglected in the analysis.     

Allele Frequencies at Microsatellite Loci: The Stepwise Mutation Model Revisited

We summarize available data on the frequencies of alleles at microsatellite loci in human populations and compare observed distributions of allele frequencies to those generated by a simulation of the stepwise mutation model. We show that observed frequency distributions at 108 loci are consistent with the results of the model under the assumption that mutations cause an increase or decrease in repeat number by one and under the condition that the product Nu, where N is the effective population size and u is the mutation rate, is larger than one. We show that the variance of the distribution of allele sizes is a useful estimator of Nu and performs much better than previously suggested estimators for the stepwise mutation model. In the data, there is no correlation between the mean and variance in allele size at a locus or between the number of alleles and mean allele size, which suggests that the mutation rate at these loci is independent of allele size.     

Joint estimation of migration rate and effective population size using the island model

Using the island model of population demography, I report that the demographic parameters migration rate and effective population size can be jointly estimated with equilibrium probabilities of identity in state calculated using a sample of genotypes collected at a single point in time from a single generation. The method, which uses moment-type estimators, applies to dioecious populations in which females and males have identical demography and monoecious populations with no selfing and requires that offspring genotypes are sampled following reproduction and prior to migration. I illustrate the estimation procedure using the infinite-island model with no mutation and the finite-island model with three kinds of mutation models. In the infinite-island model with no mutation, the estimators can be expressed as simple functions of estimates of the F-statistic parameters F(IT) and F(ST). In the finite-island model with mutation among k alleles, mutation rate, migration rate, and effective population size can be simultaneously estimated. The estimates of migration rate and effective population size are somewhat robust to violations in assumptions that may arise in empirical applications such as different kinds of mutation models and deviations from temporal equilibrium.     

A multi-dimensional coalescent process applied to multi-allelic selection models and migration models

For a sample of two genes from a population divided into an arbitrary number of allele classes, a general mathematical framework is developed to address the expectation and variance of the time of the most recent common ancestor. Depending on the meaning of allele classes and the manner in which genes can change among them, this framework can be applied to a diversity of population genetic models. By adoption of the infinite sites model, the effect on heterozygosity is modelled for balancing selection among allele classes, mutation between allele classes, migration among populations, and gene conversion between loci. Most results are described for a continuous time approximation to a discrete generation model. It is also shown how the discrete generation model can be used to study the hitch-hiking effect of favorable mutations.     

Properties of a neutral allele model with intragenic recombination

An infinite-site neutral allele model with crossing-over possible at any of an infinite number of sites is studied. A formula for the variance of the number of segregating sites in a sample of gametes is obtained. An approximate expression for the expected homozygosity is also derived. Simulation results are presented to indicate the accuracy of the approximations. The results concerning the number of segregating sites and the expected homozygosity indicate that a two-locus model and the infinite-site model behave similarly for 4Nu less than or equal to 2 and r less than or equal to 5u, where N is the population size, u is the neutral mutation rate, and r is the recombination rate. Simulations of a two-locus model and a four-locus model were also carried out to determine the effect of intragenic recombination on the homozygosity test of Watterson (Genetics 85, 789-814; 88, 405-417) and on the number of unique alleles in a sample. The results indicate that for 4Nu less than or equal to 2 and r less than or equal to 10u, the effect of recombination is quite small.     

Heterogeneity of microsatellite mutations within and between loci,and implications for human demographic histories.

Microsatellites have been widely used to reconstruct human evolution. However, the efficient use of these markers relies on information regarding the process producing the observed variation. Here, we present a novel approach to the locus-by-locus characterization of this process. By analyzing somatic mutations in cancer patients, we estimated the distributions of mutation size for each of 20 loci. The same loci were then typed in three ethnically diverse population samples. The generalized stepwise mutation model was used to test the predicted relationship between population and mutation parameters under two demographic scenarios: constant population size and rapid expansion. The agreement between the observed and expected relationship between population and mutation parameters, even when the latter are estimated in cancer patients, confirms that somatic mutations may be useful for investigating the process underlying population variation. Estimated distributions of mutation size differ substantially amongst loci, and mutations of more than one repeat unit are common. A new statistic, the normalized population variance, is introduced for multilocus estimation of demographic parameters, and for testing demographic scenarios. The observed population variation is not consistent with a constant population size. Time estimates of the putative population expansion are in agreement with those obtained by other methods.     

Measures of Variation at DNA Repeat Loci under a General Stepwise Mutation Model

Polymorphisms at tandem repeat loci are caused by mutations with allele sizes occasionally altered by more than one repeat unit in both forward and backward directions. Such mutational changes may occur with asymmetric probabilities. Therefore, a one-step symmetric stepwise mutation model may not be appropriate for studying the population dynamics at all repeat loci. In this work, we evaluated the expectation and variance of the within-population variance of the allele size distribution in a finite population, and the expected homozygosity at a locus by the coalescence approach under a general stepwise mutation model, where mutational transitions of allele sizes can be arbitrary, including being asymmetric. Under the special cases of symmetric one-step, two-step, and multi-step geometric distributions of mutations, our general results reduce to the corresponding results obtained by earlier investigators. The general results indicate that in a finite population, which has reached a steady state under the (general stepwise) mutation and drift balance, the within-population variance of allele sizes has a simple expectation (i.e., proportional toNν, the product of the mutation rate,ν, and effective population size,N). However, its stochastic variance is a quadratic function of this composite parameter,Nν. Furthermore, this second-order variance does not decay with the number of alleles sampled from a population. Application of this theory to data on allele size distributions in unrelated Caucasians from the CEPH pedigree (obtained from the Genome Data Base) shows that the relationship of the variance and mean of within-population variance of allele sizes at tandem repeat loci, grouped by their chromosomal assignment, has a trend compatible with the theory. However, there is an indication that the second-order variance is generally underestimated. One reason for this departure might be that the CEPH sample may not represent a single homogeneous population that reached equilibrium at all tandem repeat loci.     

Maximum-likelihood estimation of migration rates and effective population numbers in two populations using a coalescent approach.

A new method for the estimation of migration rates and effective population sizes is described. It uses a maximum-likelihood framework based on coalescence theory. The parameters are estimated by Metropolis-Hastings importance sampling. In a two-population model this method estimates four parameters: the effective population size and the immigration rate for each population relative to the mutation rate. Summarizing over loci can be done by assuming either that the mutation rate is the same for all loci or that the mutation rates are gamma distributed among loci but the same for all sites of a locus. The estimates are as good as or better than those from an optimized FST-based measure. The program is available on the World Wide Web at http://evolution.genetics. washington.edu/lamarc.html/.     

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.     

Multilocus estimation of selfing and its heritability

We describe a new method of estimating the selfing rate (S) in a mixed mating population based on a population structure approach that accounts for possible intergenerational correlation in selfing rate, giving rise to an estimate of the upper limit for heritability of selfing rate (h(2)). A correlation between generations in selfing rate is shown to affect one- and two-locus probabilities of identity by descent. Conventional estimates of selfing rate based on a population structure approach are positively biased by intergenerational correlation in selfing. Multilocus genotypes of individuals are used to give maximum-likelihood estimates of S and h(2) in the presence of scoring artifacts. Our multilocus estimation of selfing rate and its heritability (MESH) method was tested with simulated data for a range of conditions. Selfing rate estimates from MESH have low bias and root mean squared error, whereas estimates of the heritability of selfing rate have more uncertainty. Increasing the number of individuals in a sample helps to reduce bias and root mean squared error more than increasing the number of loci of sampled individuals. Improved estimates of selfing rate, as well as estimates of its heritability, can be obtained with this method, although a large number of loci and individuals are needed to achieve best results.     

Effective size of harvested ungulate populations

The harvest of ungulate populations is often directed against certain sex or age classes to maximize the yield in terms of biomass, number of shot animals or number of trophies. Here we examine how such directional harvest affects the effective size of the population. We parameterize an age-specific model assumed to describe the dynamics of Fennoscandian moose. Based on expressions for the demographic variance     for a small subpopulation of heterozygotes Aa bearing a rare neutral allele a , we use this model to calculate how different harvest strategies influence the effective size of the population, given that the population remains stable after harvest. We show that the annual genetic drift, determined by     , increases with decreasing harvest rate of calves and increasing sex bias in the harvest towards bulls 1 year or older. The effective population size per generation decreased with reduced harvest of calves and increased harvest of bulls 1 year or older. The magnitude of these effects depends on the age-specific pattern of variation in reproductive success, which influences the demographic variance. This shows that the choice of harvest strategy strongly affects the genetic dynamics of harvested ungulate populations.     

Variance of Neutral Genetic Variances within and between Populations for a Quantitative Character

The variances of genetic variances within and between finite populations were systematically studied using a general multiple allele model with mutation in terms of identity by descent measures. We partitioned the genetic variances into components corresponding to genetic variances and covariances within and between loci. We also analyzed the sampling variance. Both transient and equilibrium results were derived exactly and the results can be used in diverse applications. For the genetic variance within populations, sigma 2 omega, the coefficient of variation can be very well approximated as [formula: see text] for a normal distribution of allelic effects, ignoring recurrent mutation in the absence of linkage, where m is the number of loci, N is the effective population size, theta 1(0) is the initial identity by descent measure of two genes within populations and t is the generation number. The first term is due to genic variance, the second due to linkage disequilibrium, and third due to sampling. In the short term, the variation is predominantly due to linkage disequilibrium and sampling; but in the long term it can be largely due to genic variance. At equilibrium with mutation [formula: see text] where u is the mutation rate. The genetic variance between populations is a parameter. Variance arises only among sample estimates due to finite sampling of populations and individuals. The coefficient of variation for sample gentic variance between populations, sigma 2b, can be generally approximated as [formula: see text] when the number of loci is large where S is the number of sampling populations.     

Fractional populations in multiple gene inheritance

MOTIVATION: With complete knowledge of the human genome sequence, one of the most interesting tasks remaining is to understand the functions of individual genes and how they communicate. Using the information about genes (locus, allele, mutation rate, fitness, etc.), we attempt to explain population demographic data. This population evolution study could complement and enhance biologists' understanding about genes. RESULTS: We present a general approach to study population genetics in complex situations. In the present approach, multiple allele inheritance, multiple loci inheritance, natural selection and mutations are allowed simultaneously in order to consider a more realistic situation. A simulation program is presented so that readers can readily carry out studies with their own parameters. It is shown that the multiplicity of the loci greatly affects the demographic results of fractional population ratios. Furthermore, the study indicates that some high infant mortality rates due to congenital anomalies can be attributed to multiple loci inheritance. AVAILABILITY: The simulation program can be downloaded from http://won.hongik.ac.kr/~mhchung/index_files/yapop.htm. In order to run this program, one needs Visual Studio.NET platform, which can be downloaded from http://msdn.microsoft.com/netframework/downloads/default.asp.     

Genetic Differentiation in Populations with Different Classes of Individuals

I formulate and analyse a model of population structure with different classes of individuals. These different classes may be age classes, other demographic classes, or different types of habitats homogeneously distributed over a geographical area. The value of population differentiation under an island model of dispersal and the increase of differentiation with geographical distance in one- and two-dimensional "isolation by distance" models are then obtained for a generalization of the FST measure of population structure, as a function of "effective" mutation, migration, and population size parameters. The relevant effective subpopulation size is related to the "mutation effective population size" of a single isolated subpopulation and, in models of age-structured populations, to the inbreeding effective population size.     

Recombination,selection, and the evolution of tandem gene arrays

Multigene families—immunity genes or sensory receptors, for instance—are often subject to diversifying selection. Allelic diversity may be favored not only through balancing or frequency-dependent selection at individual loci but also by associating different alleles in multicopy gene families. Using a combination of analytical calculations and simulations, we explored a population genetic model of epistatic selection and unequal recombination, where a trade-off exists between the benefit of allelic diversity and the cost of copy abundance. Starting from the neutral case, where we showed that gene copy number is Gamma distributed at equilibrium, we derived also the mean and shape of the limiting distribution under selection. Considering a more general model, which includes variable population size and population substructure, we explored by simulations mean fitness and some summary statistics of the copy number distribution. We determined the relative effects of selection, recombination, and demographic parameters in maintaining allelic diversity and shaping the mean fitness of a population. One way to control the variance of copy number is by lowering the rate of unequal recombination. Indeed, when encoding recombination by a rate modifier locus, we observe exactly this prediction. Finally, we analyzed the empirical copy number distribution of 3 genes in human and estimated recombination and selection parameters of our model.