Genome sharing in large pedigrees: multiple imputation of ibd for linkage detection

Our objective is the development of robust methods for assessment of evidence for linkage of loci affecting a complex trait to a marker linkage group, using data on extended pedigrees. Using Markov chain Monte Carlo (MCMC) methods, it is possible to sample realizations from the distribution of gene identity by descent (IBD) patterns on a pedigree, conditional on observed data YM at multiple marker loci. Measures of gene IBDW which capture joint genome sharing in extended pedigrees often have unknown and highly skewed distributions, particularly when conditioned on marker data. MCMC provides a direct estimate of the distribution of such measures. Let W be the IBD measure from data YM, and W* the IBD measure from pseudo-data Y*M simulated with the same data availability and genetic marker model as the true data YM, but in the absence of linkage. Then measures of the difference in distributions of W and W* provide evidence for linkage. This approach extracts more information from the data YM than either comparison to the pedigree prior distribution of W or use of statistics that are expectations of W given the data YM. A small example is presented.     

Irreducibility and efficiency of ESIP to sample marker genotypes in large pedigrees with loops

Markov chain Monte Carlo (MCMC) methods have been proposed to overcome computational problems in linkage and segregation analyses. This approach involves sampling genotypes at the marker and trait loci. Among MCMC methods, scalar-Gibbs is the easiest to implement, and it is used in genetics. However, the Markov chain that corresponds to scalar-Gibbs may not be irreducible when the marker locus has more than two alleles, and even when the chain is irreducible, mixing has been observed to be slow. Joint sampling of genotypes has been proposed as a strategy to overcome these problems. An algorithm that combines the Elston-Stewart algorithm and iterative peeling (ESIP sampler) to sample genotypes jointly from the entire pedigree is used in this study. Here, it is shown that the ESIP sampler yields an irreducible Markov chain, regardless of the number of alleles at a locus. Further, results obtained by ESIP sampler are compared with other methods in the literature. Of the methods that are guaranteed to be irreducible, ESIP was the most efficient.     

Sibship reconstruction in hierarchical population structures using Markov chain Monte Carlo techniques

Markov chain Monte Carlo procedures allow the reconstruction of full-sibships using data from genetic marker loci only. In this study, these techniques are extended to allow the reconstruction of nested full- within half-sib families, and to present an efficient method for calculating the likelihood of the observed marker data in a nested family. Simulation is used to examine the properties of the reconstructed sibships, and of estimates of heritability and common environmental variance of quantitative traits obtained from those populations. Accuracy of reconstruction increases with increasing marker information and with increasing size of the nested full-sibships, but decreases with increasing population size. Estimates of variance component are biased, with the direction and magnitude of bias being dependent upon the underlying errors made during pedigree reconstruction.     

Inferring species trees directly from biallelic genetic markers: bypassing gene trees in a full coalescent analysis

The multispecies coalescent provides an elegant theoretical framework for estimating species trees and species demographics from genetic markers. However, practical applications of the multispecies coalescent model are limited by the need to integrate or sample over all gene trees possible for each genetic marker. Here we describe a polynomial-time algorithm that computes the likelihood of a species tree directly from the markers under a finite-sites model of mutation effectively integrating over all possible gene trees. The method applies to independent (unlinked) biallelic markers such as well-spaced single nucleotide polymorphisms, and we have implemented it in SNAPP, a Markov chain Monte Carlo sampler for inferring species trees, divergence dates, and population sizes. We report results from simulation experiments and from an analysis of 1997 amplified fragment length polymorphism loci in 69 individuals sampled from six species of Ourisia (New Zealand native foxglove).     

Characterization of 24 microsatellite markers in Primula chungensis(Primulaceae),a distylous-homostylous species,using MiSeq sequencing

Primula chungensis is a species with considerable floral and mating-system variation,including distylous(outcrossing),homostylous(selfing) and mixed populations that contain both outcrossing and selfing forms.We isolated 24 microsatellite markers from P.chungensis using Illumina Mi Seq sequencing.Polymorphism and genetic diversity were then measured based on a sample of 24 individuals from a natural population in southern Tibet.All loci were polymorphic with the number of alleles per locus ranging from 2 to 4.The observed and expected heterozygosity ranged from 0 to 1 and 0.219 to 0.708,respectively.The microsatellite markers we have identified will serve as valuable tools for the investigation of the population genetic structure and phylogeography of P.chungensis and will inform models of the evolutionary history of mating systems in the species.     

Characterization of 24 microsatellite markers in Primula chungensis (Primulaceae), a distylous homostylous species,using MiSeq sequencing

Primula chungensis is a species with considerable floral and mating-system variation, including distylous (outcrossing), homostylous (selfing) and mixed populations that contain both outcrossing and selfing forms. We isolated 24 microsatellite markers from P.chungensis using Illumina MiSeq sequencing. Polymorphism and genetic diversity were then measured based on a sample of 24 individuals from a natural population in southern Tibet. All loci were polymorphic with the number of alleles per locus ranging from 2 to4. The observed and expected heterozygosity ranged from 0 to 1 and 0.219 to 0.708, respectively. The microsatellite markers we have identified will serve as valuable tools for the investigation of the population genetic structure and phylogeography of P.chungensis and will inform models of the evolutionary history of mating systems in the species.     

Finding noncommunicating sets for Markov chain Monte Carlo estimations on pedigrees.

Markov chain Monte Carlo (MCMC) has recently gained use as a method of estimating required probability and likelihood functions in pedigree analysis, when exact computation is impractical. However, when a multiallelic locus is involved, irreducibility of the constructed Markov chain, an essential requirement of the MCMC method, may fail. Solutions proposed by several researchers, which do not identify all the noncommunicating sets of genotypic configurations, are inefficient with highly polymorphic loci. This is a particularly serious problem in linkage analysis, because highly polymorphic markers are much more informative and thus are preferred. In the present paper, we describe an algorithm that finds all the noncommunicating classes of genotypic configurations on any pedigree. This leads to a more efficient method of defining an irreducible Markov chain. Examples, including a pedigree from a genetic study of familial Alzheimer disease, are used to illustrate how the algorithm works and how penetrances are modified for specific individuals to ensure irreducibility.     

Markov chain Monte Carlo segregation and linkage analysis for oligogenic models.

A new method for segregation and linkage analysis, with pedigree data, is described. Reversible jump Markov chain Monte Carlo methods are used to implement a sampling scheme in which the Markov chain can jump between parameter subspaces corresponding to models with different numbers of quantitative-trait loci (QTL's). Joint estimation of QTL number, position, and effects is possible, avoiding the problems that can arise from misspecification of the number of QTL's in a linkage analysis. The method is illustrated by use of a data set simulated for the 9th Genetic Analysis Workshop; this data set had several oligogenic traits, generated by use of a 1,497-member pedigree. The mixing characteristics of the method appear to be good, and the method correctly recovers the simulated model from the test data set. The approach appears to have great potential both for robust linkage analysis and for the answering of more general questions regarding the genetic control of complex traits.     

A study of deleterious gene structure in plants using Markov chain Monte Carlo

The characteristics of deleterious genes have been of great interest in both theory and practice in genetics. Because of the complex genetic mechanism of these deleterious genes, most current studies try to estimate the overall magnitude of mortality effects on a population, which is characterized classically by the number of lethal equivalents. This number is a combination of several parameters, each of which has a distinct biological effect on genetic mortality. In conservation and breeding programs, it is important to be able to distinguish among different combinations of these parameters that lead to the same number of lethal equivalents, such as a large number of mildly deleterious genes or a few lethal genes, The ability to distinguish such parameter combinations requires more than one generation of mating. We propose a model for survival data from a two-generation mating experiment on the plant species Brassica rapa, and we enable inference with Markov chain Monte Carlo. This computational strategy is effective because a vast amount of missing genotype information must be accounted for. In addition to the lethal equivalents, the two-generation data provide separate information on the average intensity of mortality and the average number of deleterious genes carried by an individual. In our Markov chain Monte Carlo algorithm, we use a vector proposal distribution to overcome inefficiency of a single-site Gibbs sampler. Information about environmental effects is obtained from an outcrossing experiment conducted in parallel with the two-generation mating experiments.     

Estimating selfing rates from reconstructed pedigrees using multilocus genotype data

Several methods have been developed to estimate the selfing rate of a population from a sample of individuals genotyped for several marker loci. These methods can be based on homozygosity excess (or inbreeding), identity disequilibrium, progeny array (PA) segregation or population assignment incorporating partial selfing. Progeny array-based method is generally the best because it is not subject to some assumptions made by other methods (such as lack of misgenotyping, absence of biparental inbreeding and presence of inbreeding equilibrium), and it can reveal other facets of a mixed-mating system such as patterns of shared paternity. However, in practice, it is often difficult to obtain PAs, especially for animal species. In this study, we propose a method to reconstruct the pedigree of a sample of individuals taken from a monoecious diploid population practicing mixed mating, using multilocus genotypic data. Selfing and outcrossing events are then detected when an individual derives from identical parents and from two distinct parents, respectively. Selfing rate is estimated by the proportion of selfed offspring in the reconstructed pedigree of a sample of individuals. The method enjoys many advantages of the PA method, but without the need of a priori family structure, although such information, if available, can be utilized to improve the inference. Furthermore, the new method accommodates genotyping errors, estimates allele frequencies jointly and is robust to the presence of biparental inbreeding and inbreeding disequilibrium. Both simulated and empirical data were analysed by the new and previous methods to compare their statistical properties and accuracies.     

Sampling genotypes in large pedigrees with loops

Markov chain Monte Carlo (MCMC) methods have been proposed to overcome computational problems in linkage and segregation analyses. This approach involves sampling genotypes at the marker and trait loci. Scalar-Gibbs is easy to implement, and it is widely used in genetics. However, the Markov chain that corresponds to scalar-Gibbs may not be irreducible when the marker locus has more than two alleles, and even when the chain is irreducible, mixing has been observed to be slow. These problems do not arise if the genotypes are sampled jointly from the entire pedigree. This paper proposes a method to jointly sample genotypes. The method combines the Elston-Stewart algorithm and iterative peeling, and is called the ESIP sampler. For a hypothetical pedigree, genotype probabilities are estimated from samples obtained using ESIP and also scalar-Gibbs. Approximate probabilities were also obtained by iterative peeling. Comparisons of these with exact genotypic probabilities obtained by the Elston-Stewart algorithm showed that ESIP and iterative peeling yielded genotypic probabilities that were very close to the exact values. Nevertheless, estimated probabilities from scalar-Gibbs with a chain of length 235 000, including a burn-in of 200 000 steps, were less accurate than probabilities estimated using ESIP with a chain of length 10 000, with a burn-in of 5 000 steps. The effective chain size (ECS) was estimated from the last 25 000 elements of the chain of length 125 000. For one of the ESIP samplers, the ECS ranged from 21 579 to 22 741, while for the scalar-Gibbs sampler, the ECS ranged from 64 to 671. Genotype probabilities were also estimated for a large real pedigree consisting of 3 223 individuals. For this pedigree, it is not feasible to obtain exact genotype probabilities by the Elston-Stewart algorithm. ESIP and iterative peeling yielded very similar results. However, results from scalar-Gibbs were less accurate.     

Mixed linear model approach for mapping quantitative trait loci underlying crop seed traits

The crop seed is a complex organ that may be composed of the diploid embryo, the triploid endosperm and the diploid maternal tissues. According to the genetic features of seed characters, two genetic models for mapping quantitative trait loci (QTLs) of crop seed traits are proposed, with inclusion of maternal effects, embryo or endosperm effects of QTL, environmental effects and QTL-by-environment (QE) interactions. The mapping population can be generated either from double back-cross of immortalized F₂ (IF₂) to the two parents, from random-cross of IF₂ or from selfing of IF₂ population. Candidate marker intervals potentially harboring QTLs are first selected through one-dimensional scanning across the whole genome. The selected candidate marker intervals are then included in the model as cofactors to control background genetic effects on the putative QTL(s). Finally, a QTL full model is constructed and model selection is conducted to eliminate false positive QTLs. The genetic main effects of QTLs, QE interaction effects and the corresponding P-values are computed by Markov chain Monte Carlo algorithm for Gaussian mixed linear model via Gibbs sampling. Monte Carlo simulations were performed to investigate the reliability and efficiency of the proposed method. The simulation results showed that the proposed method had higher power to accurately detect simulated QTLs and properly estimated effect of these QTLs. To demonstrate the usefulness, the proposed method was used to identify the QTLs underlying fiber percentage in an upland cotton IF₂ population. A computer software, QTLNetwork-Seed, was developed for QTL analysis of seed traits.     

Combining the meiosis Gibbs sampler with the random walk approach for linkage and association studies with a general complex pedigree and multimarker loci

A linkage analysis for finding inheritance states and haplotype configurations is an essential process for linkage and association mapping. The linkage analysis is routinely based upon observed pedigree information and marker genotypes for individuals in the pedigree. It is not feasible for exact methods to use all such information for a large complex pedigree especially when there are many missing genotypic data. Proposed Markov chain Monte Carlo approaches such as a single-site Gibbs sampler or the meiosis Gibbs sampler are able to handle a complex pedigree with sparse genotypic data; however, they often have reducibility problems, causing biased estimates. We present a combined method, applying the random walk approach to the reducible sites in the meiosis sampler. Therefore, one can efficiently obtain reliable estimates such as identity-by-descent coefficients between individuals based on inheritance states or haplotype configurations, and a wider range of data can be used for mapping of quantitative trait loci within a reasonable time.     

Inference of population structure under a Dirichlet process model

Inferring population structure from genetic data sampled from some number of individuals is a formidable statistical problem. One widely used approach considers the number of populations to be fixed and calculates the posterior probability of assigning individuals to each population. More recently, the assignment of individuals to populations and the number of populations have both been considered random variables that follow a Dirichlet process prior. We examined the statistical behavior of assignment of individuals to populations under a Dirichlet process prior. First, we examined a best-case scenario, in which all of the assumptions of the Dirichlet process prior were satisfied, by generating data under a Dirichlet process prior. Second, we examined the performance of the method when the genetic data were generated under a population genetics model with symmetric migration between populations. We examined the accuracy of population assignment using a distance on partitions. The method can be quite accurate with a moderate number of loci. As expected, inferences on the number of populations are more accurate when theta = 4N(e)u is large and when the migration rate (4N(e)m) is low. We also examined the sensitivity of inferences of population structure to choice of the parameter of the Dirichlet process model. Although inferences could be sensitive to the choice of the prior on the number of populations, this sensitivity occurred when the number of loci sampled was small; inferences are more robust to the prior on the number of populations when the number of sampled loci is large. Finally, we discuss several methods for summarizing the results of a Bayesian Markov chain Monte Carlo (MCMC) analysis of population structure. We develop the notion of the mean population partition, which is the partition of individuals to populations that minimizes the squared partition distance to the partitions sampled by the MCMC algorithm.     

The use of marker‐based relationship information to estimate the heritability of body weight in a natural population: a cautionary tale

A number of procedures have been developed that allow the genetic parameters of natural populations to be estimated using relationship information inferred from marker data rather than known pedigrees. Three published approaches are available; the regression, pair‐wise likelihood and Markov Chain Monte Carlo (MCMC) sib‐ship reconstruction methods. These were applied to body weight and molecular data collected from the Soay sheep population of St. Kilda, which has a previously determined pedigree. The regression and pair‐wise likelihood approaches do not specify an exact pedigree and yielded unreliable heritability estimates, that were sensitive to alteration of the fixed effects. The MCMC method, which specifies a pedigree prior to heritability estimation, yielded results closer to those determined using the known pedigree. In populations of low average relationship, such as the Soay sheep population, determination of a reliable pedigree is more useful than indirect approaches that do not specify a pedigree.     

Genetic architecture and adaptive significance of the selfing syndrome in Capsella

The transition from outcrossing to predominant self-fertilization is one of the most common evolutionary transitions in flowering plants. This shift is often accompanied by a suite of changes in floral and reproductive characters termed the selfing syndrome. Here, we characterize the genetic architecture and evolutionary forces underlying evolution of the selfing syndrome in Capsella rubella following its recent divergence from the outcrossing ancestor C. grandiflora. We conduct genotyping by multiplexed shotgun sequencing and map floral and reproductive traits in a large (N= 550) F2 population. Our results suggest that in contrast to previous studies of the selfing syndrome, changes at a few loci, some with major effects, have shaped the evolution of the selfing syndrome in Capsella. The directionality of QTL effects, as well as population genetic patterns of polymorphism and divergence at 318 loci, is consistent with a history of directional selection on the selfing syndrome. Our study is an important step toward characterizing the genetic basis and evolutionary forces underlying the evolution of the selfing syndrome in a genetically accessible model system.     

Quantitative studies on the mating system of opium poppy (Papaver somniferum L.)

Summary Nearly 400 individuals at two locations and over a number of years were crossed and subsequently scored for selfing versus outcrossing in eight monohybrid populations of opium poppy (Papaver somniferum). Two different marker loci, petal colour (R/r) and capsule size (B/b) were used to determine the male gametes that had effected fertilizations in F₂ recessives (rr and bb). The estimates of the outcrossing parameter were found to vary with year, location and for the marker locus used ( range: 0.0988–0.3704). Study of two dihybrid crosses involving the two loci simultaneously, further confirmed that outcrossing at the R/r locus was significantly greater than that at the B/b locus. The nature of the outcrossing was, in general, nonrandom. Selfmg predominated in this species; however, there was a high frequency of natural outcrossing for generating variations in P. somniferum.CIMAP publication No. 1086     

The genetic variance for multiple linked quantitative trait loci conditional on marker information in a crossed population

In the prediction of genetic values and quantitative trait loci (QTLs) mapping via the mixed model method incorporating marker information in animal populations, it is important to model the genetic variance for individuals with an arbitrary pedigree structure. In this study, for a crossed population originated from different genetic groups such as breeds or outbred strains, the variance of additive genetic values for multiple linked QTLs that are contained in a chromosome segment, especially the segregation variance, is investigated assuming the use of marker data. The variance for a finite number of QTLs in one chromosomal segment is first examined for the crossed population with the general pedigree. Then, applying the concept of the expectation of identity-by-descent proportion, an approximation to the mean of the conditional probabilities for the linked QTLs over all loci is obtained, and using it an expression for the variance in the case of an infinite number of linked QTLs marked by flanking markers is derived. It appears that the approach presented can be useful in the segment mapping using, and in the genetic evaluation of, crosses with general pedigrees in the population of concern. The calculation of the segregation variance through the current approach is illustrated numerically, using a small data-set.     

Gene frequency-based estimation of natural outcrossing in opium poppy (<Emphasis Type="Italic">Papaver somniferum</Emphasis> L.)

Mating systems are generally thought to be important factors in determining the amount and nature of genetic variability in a population. Nearly 1,000 individuals at a single location (Lucknow) and over two years were crossed and subsequently scored for selfing versus outcrossing in 9–10 monohybrid populations of opium poppy (Papaver somniferum). Three different alleles of two marker loci (two—R/r and P/p—for anthocyanin locus and B/b for capsule size locus) were used to determine the male gametes that had effected fertilizations in F₂ recessives (rr, pp and bb). The estimates of the gene frequency-based outcrossing parameter (α) were found to vary with year, cross and marker locus used (α range: 7.21–71.03%). Study of the two dihybrid crosses concerning the two marker loci simultaneously, further confirmed that outcrossing at the R/r or P/p locus was significantly greater than that at the B/b locus. The nature of the outcrossing was, in general, nonrandom. In this species, in general, selfing predominated, with one exception in respect of monohybrid crosses involving the purple form of anthocyanin locus, in which outcrossing predominated.     

Marker-assisted estimation of quantitative genetic parameters in rainbow trout,Oncorhynchus mykiss

Estimation of quantitative genetic parameters conventionally requires known pedigree structure. However, several methods have recently been developed to circumvent this requirement by inferring relationship structure from molecular marker data. Here, two such marker-assisted methodologies were used and compared in an aquaculture population of rainbow trout (Oncorhynchus mykiss). Firstly a regression-based model employing estimates of pairwise relatedness was applied, and secondly a Markov Chain Monte Carlo (MCMC) procedure was employed to reconstruct full-sibships and hence an explicit pedigree. While both methods were effective in detecting significant components of genetic variance and covariance for size and spawning time traits, the regression model resulted in estimates that were quantitatively unreliable, having both significant bias and low precision. This result can be largely attributed to poor performance of the pairwise relatedness estimator. In contrast, genetic parameters estimated from the reconstructed pedigree showed close agreement with ideal values obtained from the true pedigree. Although not significantly biased, parameters based on the reconstructed pedigree were underestimated relative to ideal values. This was due to the complex structure of the true pedigree in which high numbers of half-sibling relationships resulted in inaccurate partitioning of full-sibships, and additional unrecognized relatedness between families.