SYSTEMS OF MATING. I. THE BIOMETRIC RELATIONS BETWEEN PARENT AND OFFSPRING

De Novo Domestication: An Alternative Route toward New Crops for the Future

Current global agricultural production must feed over 7 billion people. However, productivity varies greatly across the globe and is under threat from both increased competitions for land and climate change and associated environmental deterioration. Moreover, the increase in human population size and dietary changes are putting an ever greater burden on agriculture. The majority of this burden is met by the cultivation of a very small number of species, largely in locations that differ from their origin of domestication. Recent technological advances have raised the possibility of de novo domestication of wild plants as a viable solution for designing ideal crops while maintaining food security and a more sustainable low-input agriculture. Here we discuss how the discovery of multiple key domestication genes alongside the development of technologies for accurate manipulation of several target genes simultaneously renders de novo domestication a route toward crops for the future.

Detecting Salt Tolerance in Doubled Haploid Wheat Lines

Improving salt tolerance of genotypes requires a source of genetic variation and multiple accurate selection criteria for discriminating their salt tolerance. A combination of morpho-physiological and biochemical parameters and multivariate analysis was used to detect salt tolerance variation in 15 wheat lines developed by doubled haploid (DHL) technique. They were then compared with the salt-tolerant check cultivar Sakha 93. Salinity stress was investigated at three salinity levels (0, 100, and 200 mM NaCl) for 25 days. Considerable genetic variation was observed for all traits, as was high heritability (>60%) and genetic gain (>20%). Principal component analysis indicated the ability of nine traits (root number, root length, root dry weight, shoot length, shoot dry weight, specific root length, relative water content, membrane stability index, and catalase) to identify differences in salinity tolerance among lines. Three traits (shoot length, shoot dry weight, and catalase) were indicative of salt-tolerance, indicating their importance in improving and evaluating salt tolerant genotypes for breeding programs. The salinity tolerance membership index based on these three traits classified one new line (DHL21) and the check cultivar (Sakha 93) as highly salt-tolerant, DHL25, DHL26, DHL2, DHL11, and DHL5 as tolerant, and DHL23 and DHL12 as intermediate. Discriminant function analysis and MANOVA suggested differences among the five groups of tolerance. Among the donor genotypes, Sakha 93 remained the donor of choice for improving salinity tolerance during the seedling stage. The tolerated lines (DHL21, DHL25, DHL26, DHL2, DHL11, and DHL5) could be also recommended as useful and novel genetic resources for improving salinity tolerance of wheat in breeding programs.

Breaking the curse of dimensionality to identify causal variants in Breeding 4

In the past, plant breeding has undergone three major transformations and is currently transitioning to a new technological phase, Breeding 4. This phase is characterized by the development of methods for biological design of plant varieties, including transformation and gene editing techniques directed toward causal loci. The application of such technologies will require to reliably estimate the effect of loci in plant genomes by avoiding the situation where the number of loci assayed (p) surpasses the number of plant genotypes (n). Here, we discuss approaches to avoid this curse of dimensionality (n ≪ p), which will involve analyzing intermediate phenotypes such as molecular traits and component traits related to plant morphology or physiology. Because these approaches will rely on novel data types such as DNA sequences and high-throughput phenotyping images, Breeding 4 will call for analyses that are complementary to traditional quantitative genetic studies, being based on machine learning techniques which make efficient use of sequence and image data. In this article, we will present some of these techniques and their application for prioritizing causal loci and developing improved varieties in Breeding 4.

On the Road to Breeding 4.0: Unraveling the Good, the Bad, and the Boring of Crop Quantitative Genomics

Understanding the quantitative genetics of crops has been and will continue to be central to maintaining and improving global food security. We outline four stages that plant breeding either has already achieved or will probably soon achieve. Top-of-the-line breeding programs are currently in Breeding 3.0, where inexpensive, genome-wide data coupled with powerful algorithms allow us to start breeding on predicted instead of measured phenotypes. We focus on three major questions that must be answered to move from current Breeding 3.0 practices to Breeding 4.0: ( a) How do we adapt crops to better fit agricultural environments? ( b) What is the nature of the diversity upon which breeding can act? ( c) How do we deal with deleterious variants? Answering these questions and then translating them to actual gains for farmers will be a significant part of achieving global food security in the twenty-first century.

Maximum avoidance of inbreeding in haplodiploid populations

Maximum avoidance of inbreeding (MAI) is a mating system, in which mates are as distantly related as possible. Although theoretical aspects and applications of MAI in diploid populations have been studied by many researchers, extension of MAI to haplodiploid populations is an unresolved problem. In this paper, this problem is addressed, and the following conclusions are derived. For a haplodiploid population with a Fibonacci number of females, a set of mating systems (one cycle MAI-hd) to avoid inbreeding to the maximum after one cycle practice of the set can be defined. But unlike MAI in diploid populations, repetition of one cycle MAI-hd cannot be MAI in the global range of generations. Numerical comparison with random mating and circular half-sib mating shows that as in diploid populations, repetition of one cycle MAI-hd in haplodiploid populations attains a lower inbreeding coefficient in early generations at the expense of a higher asymptotic rate of inbreeding.

Multi-objective optimized genomic breeding strategies for sustainable food improvement

The purpose of breeding programs is to obtain sustainable gains in multiple traits while controlling the loss of genetic variation. The decisions at each breeding cycle involve multiple, usually competing, objectives; these complex decisions can be supported by the insights that are gained by applying multi-objective optimization principles to breeding. The discussion in this manuscript includes the definition of several multi-objective optimized breeding approaches within the phenotypic or genomic breeding frameworks and the comparison of these approaches with the standard multi-trait breeding schemes such as tandem selection, independent culling and index selection. Proposed methods are demonstrated with two empirical data sets and simulations. In addition, we have described several graphical tools that can aid breeders in arriving at a compromise decision. The results show that the proposed methodology is a viable approach to answer several real breeding problems. In simulations, the newly proposed methods resulted in gains larger than the methods previously proposed including index selection: Compared to the best alternative breeding strategy, the gains from multi-objective optimized parental proportions approaches were about 20–30% higher at the end of long-term simulations of breeding cycles. In addition, the flexibility of the multi-objective optimized breeding strategies were displayed with methods and examples covering non-dominated selection, assignment of optimal parental proportions, using genomewide marker effects in producing optimal mating designs, and finally in selection of training populations for genomic prediction.

Relationship between Deleterious Variation, Genomic Autozygosity, and Disease Risk: Insights from The 1000 Genomes Project

Genomic regions of autozygosity (ROAs) represent segments of individual genomes that are homozygous for haplotypes inherited identical-by-descent (IBD) from a common ancestor. ROAs are nonuniformly distributed across the genome, and increased ROA levels are a reported risk factor for numerous complex diseases. Previously, we hypothesized that long ROAs are enriched for deleterious homozygotes as a result of young haplotypes with recent deleterious mutations-relatively untouched by purifying selection-being paired IBD as a consequence of recent parental relatedness, a pattern supported by ROA and whole-exome sequence data on 27 individuals. Here, we significantly bolster support for our hypothesis and expand upon our original analyses using ROA and whole-genome sequence data on 2,436 individuals from The 1000 Genomes Project. Considering CADD deleteriousness scores, we reaffirm our previous observation that long ROAs are enriched for damaging homozygotes worldwide. We show that strongly damaging homozygotes experience greater enrichment than weaker damaging homozygotes, while overall enrichment varies appreciably among populations. Mendelian disease genes and those encoding FDA-approved drug targets have significantly increased rates of gain in damaging homozygotes with increasing ROA coverage relative to all other genes. In genes implicated in eight complex phenotypes for which ROA levels have been identified as a risk factor, rates of gain in damaging homozygotes vary across phenotypes and populations but frequently differ significantly from non-disease genes. These findings highlight the potential confounding effects of population background in the assessment of associations between ROA levels and complex disease risk, which might underlie reported inconsistencies in ROA-phenotype associations.

Non-additive Effects in Genomic Selection

In the last decade, genomic selection has become a standard in the genetic evaluation of livestock populations. However, most procedures for the implementation of genomic selection only consider the additive effects associated with SNP (Single Nucleotide Polymorphism) markers used to calculate the prediction of the breeding values of candidates for selection. Nevertheless, the availability of estimates of non-additive effects is of interest because: (i) they contribute to an increase in the accuracy of the prediction of breeding values and the genetic response; (ii) they allow the definition of mate allocation procedures between candidates for selection; and (iii) they can be used to enhance non-additive genetic variation through the definition of appropriate crossbreeding or purebred breeding schemes. This study presents a review of methods for the incorporation of non-additive genetic effects into genomic selection procedures and their potential applications in the prediction of future performance, mate allocation, crossbreeding, and purebred selection. The work concludes with a brief outline of some ideas for future lines of that may help the standard inclusion of non-additive effects in genomic selection.

Genomic Relatedness Strengthens Genetic Connectedness Across Management Units

Genetic connectedness refers to a measure of genetic relatedness across management units (e.g., herds and flocks). With the presence of high genetic connectedness in management units, best linear unbiased prediction (BLUP) is known to provide reliable comparisons between estimated genetic values. Genetic connectedness has been studied for pedigree-based BLUP; however, relatively little attention has been paid to using genomic information to measure connectedness. In this study, we assessed genome-based connectedness across management units by applying prediction error variance of difference (PEVD), coefficient of determination (CD), and prediction error correlation r to a combination of computer simulation and real data (mice and cattle). We found that genomic information (G) increased the estimate of connectedness among individuals from different management units compared to that based on pedigree (A). A disconnected design benefited the most. In both datasets, PEVD and CD statistics inferred increased connectedness across units when using G- rather than A-based relatedness, suggesting stronger connectedness. With r once using allele frequencies equal to one-half or scaling G to values between 0 and 2, which is intrinsic to A, connectedness also increased with genomic information. However, PEVD occasionally increased, and r decreased when obtained using the alternative form of G, instead suggesting less connectedness. Such inconsistencies were not found with CD. We contend that genomic relatedness strengthens measures of genetic connectedness across units and has the potential to aid genomic evaluation of livestock species.

Classification of common human diseases derived from shared genetic and environmental determinants

In this study, we used insurance claims for over one-third of the entire US population to create a subset of 128,989 families (481,657 unique individuals). We then used these data to (i) estimate the heritability and familial environmental patterns of 149 diseases and (ii) infer the genetic and environmental correlations for disease pairs from a set of 29 complex diseases. The majority (52 of 65) of our study's heritability estimates matched earlier reports, and 84 of our estimates appear to have been obtained for the first time. We used correlation matrices to compute environmental and genetic disease classifications and corresponding reliability measures. Among unexpected observations, we found that migraine, typically classified as a disease of the central nervous system, appeared to be most genetically similar to irritable bowel syndrome and most environmentally similar to cystitis and urethritis, all of which are inflammatory diseases.

Risk Prediction Modeling on Family-Based Sequencing Data Using a Random Field Method

Family-based design is one of the most popular designs in genetic studies and has many unique features for risk-prediction research. It is robust against genetic heterogeneity, and the relatedness among family members can be informative for predicting an individual's risk for disease with polygenic and shared environmental components of risk. Despite these strengths, family-based designs have been used infrequently in current risk-prediction studies, and their related statistical methods have not been well developed. In this article, we developed a generalized random field (GRF) method for family-based risk-prediction modeling on sequencing data. In GRF, subjects' phenotypes are viewed as stochastic realizations of a random field in a space, and a subject's phenotype is predicted by adjacent subjects, where adjacencies between subjects are determined by their genetic and within-family similarities. Different from existing methods that adjust for familial correlations, the GRF uses this information to form surrogates to further improve prediction accuracy. It also uses within-family information to capture predictors (e.g., rare mutations) that are homogeneous in families. Through simulations, we have demonstrated that the GRF method attained better performance than an existing method by considering additional information from family members and accounting for genetic heterogeneity. We further provided practical recommendations for designing family-based risk prediction studies. Finally, we illustrated the GRF method with an application to a whole-genome exome data set from the Michigan State University Twin Registry study.

Estimating pairwise relatedness in a small sample of individuals

The genetic relatedness between individuals because of their recent common ancestry is now routinely estimated from marker genotype data in molecular ecology, evolutionary biology and conservation studies. The estimators developed for this purpose assume that marker allele freque218 in a population are known without errors. Unfortunately, however, these frequencies, upon which both the definition and the estimation of relatedness are based, are rarely known in reality. Frequently, the only data available in a relatedness analysis are a sample of multilocus genotypes from which both allele frequencies and relatedness must be deduced. Furthermore, because of various constraints, sample sizes of individuals can be quite small (say <50 individuals) in practice. This study shows, for the first time, that the widely used relatedness estimators become severely biased when they use allele frequencies calculated from an extremely small sample (say <10 individuals). The extent of bias depends on the sample size, the (unknown) population allele frequencies, the actual relatedness and the estimators. It also shows that relatedness estimators become even more biased when they use allele frequencies calculated from a sample by omitting a focal pair of individuals whose relatedness is being estimated. This study modifies two estimators to suit small samples and shows, both analytically and by analysing simulated and empirical data, that the two modified estimators are much less biased, more precise and more accurate than the original estimators. These performance advantages of the modified estimators are shown to increase with a decreasing sample size of individuals and with an increasing value of actual relatedness.

INBREEDING EFFECTS IN CLARKIA TEMBLORIENSIS (ONAGRACEAE) POPULATIONS WITH DIFFERENT NATURAL OUTCROSSING RATES

Inbreeding depression is commonly observed in natural populations. The deleterious effects of forced inbreeding are often thought to be less pronounced in populations with self-pollinating mating systems than in primarily outcrossing populations. We tested this hypothesis by comparing the performance of plants produced by artificial self- and cross-pollination from three populations whose outcrossing rate estimates were 0.03, 0.26, and 0.58. Outcrossing rates and inbreeding coefficients were estimated using isozyme polymorphisms as genetic markers. Analysis of F statistics suggests that biparental inbreeding as well as self-fertilization contribute to the level of homozygosity in the seed crop. Biparental inbreeding will reduce the heterozygosity of progeny produced by outcrossing, relative to random outcrossing expectations, and hence will reduce the effects of outcrossing versus self-fertilization. Heterotic selection may increase the average heterozygosity during the life history. Selfed and outcrossed seeds from all three populations were equally likely to germinate and survive to reproduce. However, inbreeding depression was observed in fecundity traits of plants surviving to reproduction in all three populations. Even in the population whose natural self-fertilization rate was 97%, plants grown from seed produced by self-pollination produced fewer fruits and less total seed weight than plants grown from outcrossed seed. There was no detectable inbreeding depression in estimated lifetime fitness. Inbreeding effects for all reproductive yield characters were most severe in the accession from the most outcrossing population and least severe in the accession from the most self-fertilizing population.

Potential emigration of Siberian cattle germplasm on Chirikof Island, Alaska

Feral cattle residing in Chirikof Island, Alaska, are relatively distinct from breeds used in commercial production in North America. However, preliminary evidence suggested that they exhibit substantial genetic relationship with cattle from Yakutian region of Siberia. Thus, our objective was to further elucidate quantify the origins, admixture and divergence of the Chirikof Island cattle relative to cattle from Siberia and USA. Subject animals were genotyped at 15 microsatellite loci. Compared with Turano-Mongolian and North American cattle, Chirikof Island cattle had similar variation, with slightly less observed heterozygosity, fewer alleles per locus and a positive fixation index. Analysis of the genetic distances revealed two primary clusters; one that contained the North American breeds and the Kazakh White head, and a second that contained the Yakutian and Kalmyk breeds, and the Chirikof population. Thus, it is suggested that Chirikof Island cattle may be a composite of British breeds emanating from North America and Turano-Mongolian cattle. A potential founder effect, consistent with historical records of the Russian-American period, may contribute to the adaptation of the Chirikof Island cattle to their harsh high-latitude environment. Further study of adaptive mechanisms manifest by these cattle is warranted.

What does the ‘closed herd’ really mean for Australian breeding companies and their customers?

The perception that the genetic background of the Australian pig population is limiting for genetic improvement of commercial pigs in Australia is considered in the context of well established theory combined with practical evidence. The diversity of pig breeds used in modern commercial pig-breeding programs is diminished worldwide relative to all the pig breeds available. Australia is no different in this respect. The use of predominantly three main breeds (Large White, Landrace, Duroc) and synthetic lines, with contributions from other minor breeds to form the basis of a cross-breeding system for commercial pig production is well established internationally. The Australian concern of relatively small founder populations is potentially of relevance, from a theoretical perspective, for (1) the prevalence of defects or the presence of desirable alleles, and (2) the loss of genetic variation or increase in inbreeding depression resulting from increased inbreeding in closed nucleus lines, potentially reducing response to selection. However, rates of response achieved in Australian herds are generally commensurate with the performance recording and selection emphasis applied, and do not appear to be unduly restricted. Moreover, favourable alleles present in unrepresented breeds are frequently present in the three major breeds elsewhere, and therefore would be expected to be present within the Australian populations. Wider testing would provide confirmation of this. Comparison of estimates of effective population size of Australian populations with experimental selection lines overseas (e.g. INRA) or other intensely selected species (e.g. Holstein cattle) suggest adequate genetic diversity to achieve ongoing genetic improvement in the Australian pig industry. However, fitness traits should be included in breeding goals. What remains to be seen is whether novel phenotypes or genotypes are required to meet future challenges, which might be imposed by changes in the environment (e.g. climate change, disease) or market needs. Given probable overlap in genetic merit across Australian and foreign populations for unselected attributes, we suggest that sufficient genetic resources are already present in Australian herds to continue commercial progress within existing Australian populations that have adapted to Australian conditions.

Additive genetic risk from five serotonin system polymorphisms interacts with interpersonal stress to predict depression

Behavioral genetic research supports polygenic models of depression in which many genetic variations each contribute a small amount of risk, and prevailing diathesis-stress models suggest gene-environment interactions (G×E). Multilocus profile scores of additive risk offer an approach that is consistent with polygenic models of depression risk. In a first demonstration of this approach in a G×E predicting depression, we created an additive multilocus profile score from 5 serotonin system polymorphisms (1 each in the genes HTR1A, HTR2A, HTR2C, and 2 in TPH2). Analyses focused on 2 forms of interpersonal stress as environmental risk factors. Using 5 years of longitudinal diagnostic and life stress interviews from 387 emerging young adults in the Youth Emotion Project, survival analyses show that this multilocus profile score interacts with major interpersonal stressful life events to predict major depressive episode onsets (hazard ratio [HR] = 1.815, p = .007). Simultaneously, there was a significant protective effect of the profile score without a recent event (HR = 0.83, p = .030). The G×E effect with interpersonal chronic stress was not significant (HR = 1.15, p = .165). Finally, effect sizes for genetic factors examined ignoring stress suggested such an approach could lead to overlooking or misinterpreting genetic effects. Both the G×E effect and the protective simple main effect were replicated in a sample of early adolescent girls (N = 105). We discuss potential benefits of the multilocus genetic profile score approach and caveats for future research.

Prediction and estimation of effective population size

Effective population size (Ne) is a key parameter in population genetics. It has important applications in evolutionary biology, conservation genetics and plant and animal breeding, because it measures the rates of genetic drift and inbreeding and affects the efficacy of systematic evolutionary forces, such as mutation, selection and migration. We review the developments in predictive equations and estimation methodologies of effective size. In the prediction part, we focus on the equations for populations with different modes of reproduction, for populations under selection for unlinked or linked loci and for the specific applications to conservation genetics. In the estimation part, we focus on methods developed for estimating the current or recent effective size from molecular marker or sequence data. We discuss some underdeveloped areas in predicting and estimating Ne for future research.

Mating strategies with genomic information reduce rates of inbreeding in animal breeding schemes without compromising genetic gain

We tested the hypothesis that mating strategies with genomic information realise lower rates of inbreeding (∆F) than with pedigree information without compromising rates of genetic gain (∆G). We used stochastic simulation to compare ∆F and ∆G realised by two mating strategies with pedigree and genomic information in five breeding schemes. The two mating strategies were minimum-coancestry mating (MC) and minimising the covariance between ancestral genetic contributions (MCAC). We also simulated random mating (RAND) as a reference point. Generations were discrete. Animals were truncation-selected for a single trait that was controlled by 2000 quantitative trait loci, and the trait was observed for all selection candidates before selection. The criterion for selection was genomic-breeding values predicted by a ridge-regression model. Our results showed that MC and MCAC with genomic information realised 6% to 22% less ∆F than MC and MCAC with pedigree information without compromising ∆G across breeding schemes. MC and MCAC realised similar ∆F and ∆G. In turn, MC and MCAC with genomic information realised 28% to 44% less ∆F and up to 14% higher ∆G than RAND. These results indicated that MC and MCAC with genomic information are more effective than with pedigree information in controlling rates of inbreeding. This implies that genomic information should be applied to more than just prediction of breeding values in breeding schemes with truncation selection.

Equilibrium behavior of population genetic models with non-random mating. Part II: Pedigrees, Homozygosity and Stochastic Models

Wright (1921) computed various correlations of relatives by a rather cumbersome procedure called the “method of path coefficients”. Wright's method is basically a disguised form of the use of Bayes' rule and the law of total probabilities. Malecot (1948) reorganized Wright's calculations by introducing the fundamental concept of identity by descent and exploiting its properties. The method of identity by descent has been perfected and developed by Malécot and his students, especially Gillois, Jauquard and Bouffette. Kempthorne (1957) has applied the concept of identity by descent to the study of quantitative inheritance. Kimura (1963) elegantly employed the ideas of identity by descent in determining rates of approach to homozygosity in certain mating situations with finite population size. Later in this chapter we will extend and refine the results of Kimura (1963) to give a more complete study of rates of approach to homozygosity. Ellison (1966) established several important limit theorems corresponding to polyploid, multi-locus random mating infinite populations by judicious enlargement of the concepts of identity by descent. Kesten (unpublished)) has recently refined the technique of Ellison.

Another approach to some Markov chain models in population genetics

This paper treats the well known model of sib mating of individuals with autosomal genes; a single locus is studied. The general derivations are valid for any number of alleles. At the end of the paper, some simple examples are included; here several familiar results concerning models for two alleles are obtained as special cases of the general theory.

Another approach to some Markov chain models in population genetics

This paper treats the well known model of sib mating of individuals with autosomal genes; a single locus is studied. The general derivations are valid for any number of alleles. At the end of the paper, some simple examples are included; here several familiar results concerning models for two alleles are obtained as special cases of the general theory.

Equilibrium behavior of population genetic models with non-random mating. Part II: Pedigrees, Homozygosity and Stochastic Models

Wright (1921) computed various correlations of relatives by a rather cumbersome procedure called the “method of path coefficients”. Wright's method is basically a disguised form of the use of Bayes' rule and the law of total probabilities. Malecot (1948) reorganized Wright's calculations by introducing the fundamental concept of identity by descent and exploiting its properties. The method of identity by descent has been perfected and developed by Malécot and his students, especially Gillois, Jauquard and Bouffette. Kempthorne (1957) has applied the concept of identity by descent to the study of quantitative inheritance. Kimura (1963) elegantly employed the ideas of identity by descent in determining rates of approach to homozygosity in certain mating situations with finite population size. Later in this chapter we will extend and refine the results of Kimura (1963) to give a more complete study of rates of approach to homozygosity. Ellison (1966) established several important limit theorems corresponding to polyploid, multi-locus random mating infinite populations by judicious enlargement of the concepts of identity by descent. Kesten (unpublished)) has recently refined the technique of Ellison.

Evolution in diploid populations by continuity of gametic types

This work studies the long-term effects of mutation and selection pressures on a diploid population embracing many genetic types. A number of results previously established for the simpler asexual case (see [4]) are extended to the cases of random mating and complete inbreeding (Theorem 1), and then, under particular conditions, to certain circumstances of mixed random mating and inbreeding (Theorem 3 and Corollary 1). Several implications for sex and diploidity are drawn from Theorem 2 and its corollaries. Further biological interpretations of these findings, especially of Theorem 2, are given in [3].

Admixture, Population Structure, and F-Statistics

Many questions about human genetic history can be addressed by examining the patterns of shared genetic variation between sets of populations. A useful methodological framework for this purpose isF-statistics that measure shared genetic drift between sets of two, three, and four populations and can be used to test simple and complex hypotheses about admixture between populations. This article provides context from phylogenetic and population genetic theory. I review how F-statistics can be interpreted as branch lengths or paths and derive new interpretations, using coalescent theory. I further show that the admixture tests can be interpreted as testing general properties of phylogenies, allowing extension of some ideas applications to arbitrary phylogenetic trees. The new results are used to investigate the behavior of the statistics under different models of population structure and show how population substructure complicates inference. The results lead to simplified estimators in many cases, and I recommend to replace F3 with the average number of pairwise differences for estimating population divergence.

Likelihood Ratio Test for Excess Homozygosity at Marker Loci on X Chromosome

The assumption of Hardy-Weinberg equilibrium (HWE) is generally required for association analysis using case-control design on autosomes; otherwise, the size may be inflated. There has been an increasing interest of exploring the association between diseases and markers on X chromosome and the effect of the departure from HWE on association analysis on X chromosome. Note that there are two hypotheses of interest regarding the X chromosome: (i) the frequencies of the same allele at a locus in males and females are equal and (ii) the inbreeding coefficient in females is zero (without excess homozygosity). Thus, excess homozygosity and significantly different minor allele frequencies between males and females are used to filter X-linked variants. There are two existing methods to test for (i) and (ii), respectively. However, their size and powers have not been studied yet. Further, there is no existing method to simultaneously detect both hypotheses till now. Therefore, in this article, we propose a novel likelihood ratio test for both (i) and (ii) on X chromosome. To further investigate the underlying reason why the null hypothesis is statistically rejected, we also develop two likelihood ratio tests for detecting (i) and (ii), respectively. Moreover, we explore the effect of population stratification on the proposed tests. From our simulation study, the size of the test for (i) is close to the nominal significance level. However, the size of the excess homozygosity test and the test for both (i) and (ii) is conservative. So, we propose parametric bootstrap techniques to evaluate their validity and performance. Simulation results show that the proposed methods with bootstrap techniques control the size well under the respective null hypothesis. Power comparison demonstrates that the methods with bootstrap techniques are more powerful than those without bootstrap procedure and the existing methods. The application of the proposed methods to a rheumatoid arthritis dataset indicates their utility.

A Bayesian Approach to Inferring Rates of Selfing and Locus-Specific Mutation

We present a Bayesian method for characterizing the mating system of populations reproducing through a mixture of self-fertilization and random outcrossing. Our method uses patterns of genetic variation across the genome as a basis for inference about reproduction under pure hermaphroditism, gynodioecy, and a model developed to describe the self-fertilizing killifish Kryptolebias marmoratus. We extend the standard coalescence model to accommodate these mating systems, accounting explicitly for multilocus identity disequilibrium, inbreeding depression, and variation in fertility among mating types. We incorporate the Ewens sampling formula (ESF) under the infinite-alleles model of mutation to obtain a novel expression for the likelihood of mating system parameters. Our Markov chain Monte Carlo (MCMC) algorithm assigns locus-specific mutation rates, drawn from a common mutation rate distribution that is itself estimated from the data using a Dirichlet process prior model. Our sampler is designed to accommodate additional information, including observations pertaining to the sex ratio, the intensity of inbreeding depression, and other aspects of reproduction. It can provide joint posterior distributions for the population-wide proportion of uniparental individuals, locus-specific mutation rates, and the number of generations since the most recent outcrossing event for each sampled individual. Further, estimation of all basic parameters of a given model permits estimation of functions of those parameters, including the proportion of the gene pool contributed by each sex and relative effective numbers.

Estimating Relatedness in the Presence of Null Alleles

Studies of genetics and ecology often require estimates of relatedness coefficients based on genetic marker data. However, with the presence of null alleles, an observed genotype can represent one of several possible true genotypes. This results in biased estimates of relatedness. As the numbers of marker loci are often limited, loci with null alleles cannot be abandoned without substantial loss of statistical power. Here, we show how loci with null alleles can be incorporated into six estimators of relatedness (two novel). We evaluate the performance of various estimators before and after correction for null alleles. If the frequency of a null allele is <0.1, some estimators can be used directly without adjustment; if it is >0.5, the potency of estimation is too low and such a locus should be excluded. We make available a software package entitled PolyRelatedness v1.6, which enables researchers to optimize these estimators to best fit a particular data set.