Drawing inferences about the coancestry coefficient

The coancestry coefficient, also known as the population structure parameter, is of great interest in population genetics. It can be thought of as the intraclass correlation of pairs of alleles within populations and it can serve as a measure of genetic distance between populations. For a general class of evolutionary models it determines the distribution of allele frequencies among populations. Under more restrictive models it can be regarded as the probability of identity by descent of any pair of alleles at a locus within a random mating population. In this paper we review estimation procedures that use the method of moments or are maximum likelihood under the assumption of normally distributed allele frequencies. We then consider the problem of testing hypotheses about this parameter. In addition to parametric and non-parametric bootstrap tests we present an asymptotically-distributed chi-square test. This test reduces to the contingency-table test for equal sample sizes across populations. Our new test appears to be more powerful than previous tests, especially for loci with multiple alleles. We apply our methods to HapMap SNP data to confirm that the coancestry coefficient for humans is strictly positive.     

Defining the assumptions underlying modeling of epistatic QTL using variance component methods

Variance component models are commonly used to detect quantitative trait loci (QTL) in general pedigrees. The variance-covariance structure of the random QTL effect is given by the identity by descent (IBD) between genotypes. Epistatic effects have previously been modeled, both for unlinked and linked loci, as a random effect with a variance-covariance structure given by the Hadamard product between the IBD matrices of the direct QTL effects. In the original papers, the model was given but not derived. Here, we identify the underlying assumptions of this previously proposed model. It assumes that either an unlinked QTL or a fully informative marker (i.e., all marker alleles are unique in the base generation) is located between the loci. We discuss the need of developing a general algorithm to estimate the variance-covariance structure of the random epistatic effect for linked loci.     

Descent measures for two loci with some applications

For any four genes, two at each of two loci, in a population, a 15 component descent measure has been introduced. These components are the probabilities of the 15 possible arrangements on a set of initial gametes of those genes of which the four of interest are copies. Since identity by descent of genes is equivalent to their being copies of a single gene on an initial gamete, descent measures have inbreeding coefficients as special cases. The individual descent measure, defined for four genes on two uniting gametes can be evaluated for any pedigree by means of an algorithm developed here. If initial gametic frequencies are specified, descent measures allow genotypic frequencies and disequilibria functions at one and two loci to be found. The procedures are illustrated for selfing and for sib mating. Several applications of the descent measures are discussed.     

Behavior of pairs of loci in finite monoecious populations

Transition equations are established for descent measures for pairs of loci in finite randomly mating monoecious populations. The two special cases of equal chance gamete formation and two gametes per parent are considered in detail. The descent measures allow genotypic frequencies to be found but are used mainly to evaluate three quadrigenic moments, including the variance of the linkage disequilibrium. Numerical properties of these moments are compared to previously reported values.     

Genomic prediction based on runs of homozygosity

Background

Genomic prediction is based on the accurate estimation of the genomic relationships among and between training animals and selection candidates in order to obtain accurate estimates of the genomic estimated breeding values (GEBV). Various methods have been used to predict GEBV based on population-wide linkage disequilibrium relationships (G_IBS) or sometimes on linkage analysis relationships (G_LA). Here, we propose a novel method to predict GEBV based on a genomic relationship matrix using runs of homozygosity (G_ROH). Runs of homozygosity were used to derive probabilities of multi-locus identity by descent chromosome segments. The accuracy and bias of the prediction of GEBV using G_ROH were compared to those using G_IBS and G_LA. Comparisons were performed using simulated datasets derived from a random pedigree and a real pedigree of Italian Brown Swiss bulls. The comparison of accuracies of GEBV was also performed on data from 1086 Italian Brown Swiss dairy cattle.

Results

Simulations with various thresholds of minor allele frequency for markers and quantitative trait loci showed that G_ROH achieved consistently more accurate GEBV (0 to 4% points higher) than G_IBS and G_LA. The bias of GEBV prediction for simulated data was higher based on the real pedigree than based on a random pedigree. In the analyses with real data, G_ROH and G_LA had similar accuracies. However, G_LA achieved a higher accuracy when the prediction was done on the youngest animals. The G_IBS matrices calculated with and without standardized marker genotypes resulted in similar accuracies.

Conclusions

The present study proposes G_ROH as a novel method to estimate genomic relationship matrices and predict GEBV based on runs of homozygosity and shows that it can result in higher or similar accuracies of GEBV prediction than G_LA, except for the real data analysis with validation of young animals. Compared to G_IBS, G_ROH resulted in more accurate GEBV predictions.     

The IBD process along four chromosomes

In this note, we present the continuous-time Markov rate matrix that models identity by descent (ibd) patterns among four chromosomes in a population. The equilibrium distribution of this Markov process along a chromosome is the set of 4-gene state probabilities given by the Ewens sampling formula. This model will facilitate inference of identity by descent among the four chromosomes of a pair of individuals, using data at dense SNP loci among which there may be linkage disequilibrium.     

On the prediction of simultaneous inbreeding coefficients at multiple loci

A new deterministic method for predicting simultaneous inbreeding coefficients at three and four loci is presented. The method involves calculating the conditional probability of IBD (identical by descent) at one locus given IBD at other loci, and multiplying this probability by the prior probability of the latter loci being simultaneously IBD. The conditional probability is obtained applying a novel regression model, and the prior probability from the theory of digenic measures of Weir and Cockerham. The model was validated for a finite monoecious population mating at random, with a constant effective population size, and with or without selfing, and also for an infinite population with a constant intermediate proportion of selfing. We assumed discrete generations. Deterministic predictions were very accurate when compared with simulation results, and robust to alternative forms of implementation. These simultaneous inbreeding coefficients were more sensitive to changes in effective population size than in marker spacing. Extensions to predict simultaneous inbreeding coefficients at more than four loci are now possible.     

Covariance between relatives for X-chromosomal loci in a population in disequilibrium

Summary According to Hardy-Weinberg, for a single autosomal locus, a population achieves equilibrium in one generation of random mating if allelic frequency is the same in the sexes, or in two generations if the frequency is not. For a single X-chromosomal locus, however, the approach to equilibrium oscillates and is gradual. Covariances between relatives for autosomal and for X-chromosomal loci are in the literature for a random mating population in equilibrium. Although assumption of equilibrium is defensible for an autosomal locus, it is less defensible for an X-chromosomal locus. Covariances between collateral and between lineal relatives are derived for X-chromosomal loci in a random mating population not in equilibrium. Collateral relatives such as sibs are of the same generation, and lineal relatives such as parent-offspring are of different generations. Coefficient of co-ancestry between relatives, based on identity by descent, was used in this development. Results are applicable to crossbreeding in livestock and poultry, and also to haplo-diploid organisms, such as the honeybee, in which the entire genome is equivalent to being X-chromosomal.Supported in part by the Illinois Agricultural Experiment Station, Hatch Project 35-0367     

Estimation of relatedness by DNA fingerprinting

The recent discovery of hypervariable VNTR (variable number of tandem repeat) loci has led to much excitement among population biologists regarding the feasibility of deriving individual estimates of relatedness in field populations by DNA fingerprinting. It is shown that unbiased estimates of relatedness cannot be obtained at the individual level without knowledge of the allelic distributions in both the individuals of interest and the base population unless the proportion of shared marker alleles between unrelated individuals is essentially zero. Since the latter is usually on the order of 0.1-0.5 and since there are enormous practical difficulties in obtaining the former, only an approximate estimator for the relatedness can be given. The bias of this estimator is individual specific and inversely related to the number of marker loci and frequencies of marker alleles. Substantial sampling variance in estimates of relatedness arises from variation in identity by descent within and between loci and, with finite numbers of alleles, from variation in identity in state between genes that are not identical by descent. In the extreme case of 25 assayed loci, each with an effectively infinite number of alleles, the standard error of a relatedness estimate is no less than 14%, 20%, 35%, and 53% of the expectation for full sibs and second-, third-, and fourth-order relationships, respectively. Attempts to ascertain relatedness by means of DNA fingerprinting should proceed with caution.      

Effect of Mating Structure on Variation in Linkage Disequilibrium

Measurement of linkage disequilibrium involves two sampling processes. First, there is the sampling of gametes in the population to form successive generations, and this generates disequilibrium dependent on the effective population size (N_e) and the mating structure. Second, there is sampling of a finite number (n) of individuals to estimate the population disequilibrium.——Two-locus descent measures are used to describe the mating system and are transformed to disequilibrium moments at the final sampling. Approximate eigenvectors for the transition matrix of descent measures are used to obtain formulae for the variance of the observed disequilibria as a function of N_e, mating structure, n, and linkage or recombination parameter.——The variance of disequilibrium is the same for monoecious populations with or without random selfing and for dioecious populations with random pairing for each progeny. With monogamy, the variance is slightly higher, the proportional difference being greater for unlinked loci.     

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.     

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.     

SINEs of speciation: tracking lineages with retroposons

The value of short interspersed elements (SINEs) for diagnosing common ancestry is being expanded to examine the differential sorting of lineages through the course of speciation events. Because most SINEs are neutral markers of identical descent, are not precisely excised from the genome and have a known ancestral condition, they are advantageous for reconciling gene trees and species trees with minimal phylogenetic error. A population perspective on SINE evolution combined with coalescence theory provides a context for investigating the phenomenon of ancestral polymorphism and its role in producing incongruent SINE insertion patterns among multiple loci. Studies of human Alu repeats demonstrate the value of young polymorphic SINEs for assessing human genomic diversity and tracking ancient demographics of human populations, whereas incongruent insertion patterns revealed by older fixed SINE loci, such as those in African cichlid fishes, contain information that might help identify ancient radiations that are otherwise obscured by accumulated mutations in sequence data. Here, we review the utility of retroposons for inferring common ancestry, discuss limits to the method, and clarify confusion by providing examples from the literature that illustrate how discordant multi-locus insertion patterns of retroelements can indicate lineage-sorting events that should not be misinterpreted as phylogenetic noise.     

Methods for linkage analysis of quantitative trait loci in humans

This paper reviews linkage analysis methods for detecting loci associated with quantitative traits in humans. All such methods are based on the underlying principle that family members who have similar trait values should have higher than expected levels of sharing of genetic material (identity by descent) near the genes that influence those traits. A number of different statistical methods for testing that association between shared trait values and shared identity by descent have been developed over the past 30 or more years. These different types of tests are reviewed here, with emphasis on their theory and derivations. Robustness and power are also discussed.     

Inference on the strength of balancing selection for epistatically interacting loci

Existing inference methods for estimating the strength of balancing selection in multi-locus genotypes rely on the assumption that there are no epistatic interactions between loci. Complex systems in which balancing selection is prevalent, such as sets of human immune system genes, are known to contain components that interact epistatically. Therefore, current methods may not produce reliable inference on the strength of selection at these loci. In this paper, we address this problem by presenting statistical methods that can account for epistatic interactions in making inference about balancing selection. A theoretical result due to Fearnhead (2006) is used to build a multi-locus Wright-Fisher model of balancing selection, allowing for epistatic interactions among loci. Antagonistic and synergistic types of interactions are examined. The joint posterior distribution of the selection and mutation parameters is sampled by Markov chain Monte Carlo methods, and the plausibility of models is assessed via Bayes factors. As a component of the inference process, an algorithm to generate multi-locus allele frequencies under balancing selection models with epistasis is also presented. Recent evidence on interactions among a set of human immune system genes is introduced as a motivating biological system for the epistatic model, and data on these genes are used to demonstrate the methods.     

Molecular identification of individual oak and fir trees from maternal tissues of their fruits or seeds

The applicability of DNA markers to purely maternal tissues has been scarcely addressed in trees. We have focused on non-parenchymatic maternal tissues of the fruits and seeds of pedunculate oak (Quercus robur L.) and silver fir (Abies alba Mill.) and investigated whether they can be used for a direct molecular identification of the mother trees. Total DNA with sufficient quantity and quality was extracted from single woody pericarps of acorns as well as from single dry wings of silver fir seeds. The DNA was analysed by PCR at highly polymorphic microsatellite loci. A comparison of the multi-locus genotypes from pericarps and wings with those of the respective mother trees revealed absolute identity. Thus, mother trees could be identified by genotyping their fruits or seeds. The results demonstrate the applicability of DNA fingerprinting to woody and/or dry seed tissues without the destruction of embryos and endosperm or a significant contamination. Progress is now expected in dispersal biology as well as in forensics and forest management.     

An Approximate EM Algorithm for Nonlinear Mixed Effects Models

In this article, we construct an approximate EM algorithm to estimate the parameters of a nonlinear mixed effects model. The iterative procedure can be viewed as an iterative method of moments procedure for estimating the variance components and an iterative reweighted least squares estimates for estimating the fixed effects. Therefore, it is valid without the normality assumptions on the random components. A computationally simple method of moments estimates of the model parameters are used as the starting values for our iterative procedure. A simulation study was conducted to compare the performances of the proposed procedure with the procedure proposed by Lindstrom and Bates (1990) for some normal models and nonnormal models.     

Sampling Issues Affecting Accuracy of Likelihood-based Classification Using Genetical Data

We demonstrate the effectiveness of a genetic algorithm for discovering multi-locus combinations that provide accurate individual assignment decisions and estimates of mixture composition based on likelihood classification. Using simulated data representing different levels of inter-population differentiation (F_st～ 0.01 and 0.10), genetic diversities (four or eight alleles per locus), and population sizes (20, 40, 100 individuals in baseline populations), we show that subsets of loci can be identified that provide comparable levels of accuracy in classification decisions relative to entire multi-locus data sets, where 5, 10, or 20 loci were considered. Microsatellite data sets from hatchery strains of lake trout, Salvelinus namaycush, representing a comparable range of inter-population levels of differentiation in allele frequencies confirmed simulation results. For both simulated and empirical data sets, assignment accuracy was achieved using fewer loci (e.g., three or four loci out of eight for empirical lake trout studies). Simulation results were used to investigate properties of the ‘leave-one-out’ (L1O) method for estimating assignment error rates. Accuracy of population assignments based on L1O methods should be viewed with caution under certain conditions, particularly when baseline population sample sizes are low (<50).     

One- and multi-locus multi-allele selection models in a random environment

Summary We deduce conditions for stochastic local stability of general perturbed linear stochastic difference equations widely applicable in population genetics. The findings are adapted to evaluate the stability properties of equilibria in classical one- and multi-locus multi-allele selection models influenced by random temporal variation in selection intensities. As an example of some conclusions and biological interpretations we analyse a special one-locus multi-allele model in more detail.This work was supported in part by Stiftung Volkswagenwerk.