Analysis of genealogical structure of populations. II. The use od numerical pedigrees for calculation of inbreeding coefficient

A method for calculation of inbreeding coefficient F in a numerical pedigree with no reference to its graphic representation is suggested. For calculation of F, a formula that does not take into account inbreeding coefficients of common ancestors and admits intersections in a loop is used. An advantage of this method is that it automatically finds all loops formed by paths to common ancestors. Detecting these loops via their tracing in a graphic pedigree with intersecting lines of descent creates a possibility of errors. A criterion of existence of at least one common link for two numerical paths is presented. It enables one to exclude pairs of paths to common ancestors that do not form loops. The methods considered for computing F in a given pedigree give exact values of the inbreeding coefficient for autosomal and sex-linked loci and generalize the known approximate approaches. The methods are illustrated by examples.     

Pedigree reconstruction using identity by descent

Can we find the family trees, or pedigrees, that relate the haplotypes of a group of individuals? Collecting the genealogical information for how individuals are related is a very time-consuming and expensive process. Methods for automating the construction of pedigrees could stream-line this process. While constructing single-generation families is relatively easy given whole genome data, reconstructing multi-generational, possibly inbred, pedigrees is much more challenging. This article addresses the important question of reconstructing monogamous, regular pedigrees, where pedigrees are regular when individuals mate only with other individuals at the same generation. This article introduces two multi-generational pedigree reconstruction methods: one for inbreeding relationships and one for outbreeding relationships. In contrast to previous methods that focused on the independent estimation of relationship distances between every pair of typed individuals, here we present methods that aim at the reconstruction of the entire pedigree. We show that both our methods out-perform the state-of-the-art and that the outbreeding method is capable of reconstructing pedigrees at least six generations back in time with high accuracy. The two programs are available at http://cop.icsi.berkeley.edu/cop/.     

Huntington disease in Finland: a molecular and genealogical study

Summary Huntington disease (HD) is found at exceptionally low frequency in the Finnish population. In this population, linkage disequilibrium was earlier established with markers from the D4S10 and D4S43 loci. We now report a continuation to the restriction fragment length polymorphism haplotype analysis, in combination with a genealogical study of all the Finnish HD families. When the HD pedigrees were systematically traced to the 18th century, only one consanguinity was found, and a high percentage (28%) of the families had foreign ancestors. The majority of the Finnish ancestors were localized to border regions or trade centers of the country following the old postal routes. The observed high risk haplotypes formed with markers from the D4S10 and D4S43 loci were evenly distributed among the HD families in different geographical locations. Consequently, the HD gene(s) has most probably arrived in Finland on several occasions via foreign immigrants during the last few centuries.     

Estimation of genealogical coancestry in plant species using a pedigree reconstruction algorithm and application to an oil palm breeding population

Key message

Explicit pedigree reconstruction by simulated annealing gave reliable estimates of genealogical coancestry in plant species, especially when selfing rate was lower than 0.6, using a realistic number of markers. Genealogical coancestry information is crucial in plant breeding to estimate genetic parameters and breeding values. The approach of Fernández and Toro (Mol Ecol 15:1657–1667, 2006) to estimate genealogical coancestries from molecular data through pedigree reconstruction was limited to species with separate sexes. In this study it was extended to plants, allowing hermaphroditism and monoecy, with possible selfing. Moreover, some improvements were made to take previous knowledge on the population demographic history into account. The new method was validated using simulated and real datasets. Simulations showed that accuracy of estimates was high with 30 microsatellites, with the best results obtained for selfing rates below 0.6. In these conditions, the root mean square error (RMSE) between the true and estimated genealogical coancestry was small (<0.07), although the number of ancestors was overestimated and the selfing rate could be biased. Simulations also showed that linkage disequilibrium between markers and departure from the Hardy–Weinberg equilibrium in the founder population did not affect the efficiency of the method. Real oil palm data confirmed the simulation results, with a high correlation between the true and estimated genealogical coancestry (>0.9) and a low RMSE (<0.08) using 38 markers. The method was applied to the Deli oil palm population for which pedigree data were scarce. The estimated genealogical coancestries were highly correlated (>0.9) with the molecular coancestries using 100 markers. Reconstructed pedigrees were used to estimate effective population sizes. In conclusion, this method gave reliable genealogical coancestry estimates. The strategy was implemented in the software MOLCOANC 3.0.     

Multipoint quantitative-trait linkage analysis in general pedigrees.

Multipoint linkage analysis of quantitative-trait loci (QTLs) has previously been restricted to sibships and small pedigrees. In this article, we show how variance-component linkage methods can be used in pedigrees of arbitrary size and complexity, and we develop a general framework for multipoint identity-by-descent (IBD) probability calculations. We extend the sib-pair multipoint mapping approach of Fulker et al. to general relative pairs. This multipoint IBD method uses the proportion of alleles shared identical by descent at genotyped loci to estimate IBD sharing at arbitrary points along a chromosome for each relative pair. We have derived correlations in IBD sharing as a function of chromosomal distance for relative pairs in general pedigrees and provide a simple framework whereby these correlations can be easily obtained for any relative pair related by a single line of descent or by multiple independent lines of descent. Once calculated, the multipoint relative-pair IBDs can be utilized in variance-component linkage analysis, which considers the likelihood of the entire pedigree jointly. Examples are given that use simulated data, demonstrating both the accuracy of QTL localization and the increase in power provided by multipoint analysis with 5-, 10-, and 20-cM marker maps. The general pedigree variance component and IBD estimation methods have been implemented in the SOLAR (Sequential Oligogenic Linkage Analysis Routines) computer package.     

The age of human mutation: genealogical and linkage disequilibrium analysis of the CLN5 mutation in the Finnish population.

Variant late infantile neuronal ceroid lipofuscinosis (vLINCL) is an autosomal recessive progressive encephalopathy of childhood enriched in the western part of Finland, with a local incidence of 1 in 1500. We recently assigned the locus for vLINCL, CLN5, to 13q21.1-q32. In the present study, the haplotype analysis of Finnish CLN5 chromosomes provides evidence that one single mutation causes vLINCL in the Finnish population. Eight microsatellite markers closely linked to the CLN5 gene on chromosome 13q were analyzed, to study identity by descent by shared haplotype analysis. One single haplotype formed by flanking markers D13S160 and D13S162 in strong linkage disequilibrium (P < .0001) was present in 81% of disease-bearing chromosomes. Allele 4 at the marker locus D13S162 was detected in 94% of disease-bearing chromosomes. To evaluate the age of the CLN5 mutation by virtue of its restricted geographical distribution, church records were used to identify the common ancestors for 18 vLINCL families diagnosed in Finland. The pedigrees of the vLINCL ancestors merged on many occasions, which also supports a single founder mutation that obviously happened 20 to 30 generations ago (i.e., approximately 500 years ago) in this isolated population. Linkage disequilibrium was detected with seven markers covering an extended genetic distance of 11 cM, which further supports the young age of the CLN5 mutation. When the results of genealogical and linkage disequilibrium studies were combined, the CLN5 gene was predicted to lie approximately 200 - 400 kb (total range 30 - 1360 kb) from the closest marker D13S162.     

A resolution of the ascertainment sampling problem. III. Pedigrees.

When nuclear families are sampled by an ascertainment procedure whose properties are not known, biased estimates of genetic parameters will arise if an incorrect specification of the ascertainment procedure is made. Elsewhere we have put forward a resolution of this problem by introducing an ascertainment-assumption-free (AAF) method, for nuclear family data, which gives asymptotically unbiased estimators no matter what the true nature of the ascertainment process. In the present paper we extend this method to cover pedigree data. Problems that arise with pedigrees but not with families--for example, the question of which families in a pedigree are "ascertainable"--are also considered. Comparisons of numerical results for pedigrees and nuclear families are also made.     

Efficient path-based computations on pedigree graphs with compact encodings

A pedigree is a diagram of family relationships, and it is often used to determine the mode of inheritance (dominant, recessive, etc.) of genetic diseases. Along with rapidly growing knowledge of genetics and accumulation of genealogy information, pedigree data is becoming increasingly important. In large pedigree graphs, path-based methods for efficiently computing genealogical measurements, such as inbreeding and kinship coefficients of individuals, depend on efficient identification and processing of paths. In this paper, we propose a new compact path encoding scheme on large pedigrees, accompanied by an efficient algorithm for identifying paths. We demonstrate the utilization of our proposed method by applying it to the inbreeding coefficient computation. We present time and space complexity analysis, and also manifest the efficiency of our method for evaluating inbreeding coefficients as compared to previous methods by experimental results using pedigree graphs with real and synthetic data. Both theoretical and experimental results demonstrate that our method is more scalable and efficient than previous methods in terms of time and space requirements.     

Estimating the power of a proposed linkage study: a practical computer simulation approach.

I describe a simulation method to estimate the power to detect linkage given a set of pedigrees of known structure and for which family history data may be available. This method can be applied to autosomal and X-linked dominant diseases; depending on the pedigrees under consideration, it will often be applicable for autosomal and X-linked recessive diseases. This power calculation can most usefully be undertaken after family history data are gathered, but prior to examination and testing of pedigree members to obtain marker information. Of key importance, the power calculation is straightforward to carry out and not too time-consuming; it is practical even on a microcomputer. The result of the power calculation is an objective answer to the question: Will my families be sufficient to demonstrate linkage?     

Inheritance of total serum IgE in the isolated Tangier Island population from Virginia: complexities associated with genealogical depth of pedigrees in segregation analyses

OBJECTIVES: This study was aimed at performing a segregation analysis of total serum immunoglobulin E (tIgE) in an isolated population using maximal genealogical information permitted by current software and computer capacities, while assessing the reliability of the best-fitting model of inheritance for tIgE through simulations. METHODS: All current Tangier Island, VA, residents (n = 664) belonged to one large extended pedigree (n = 3,501) spanning 13 generations, with an average inbreeding coefficient of 0.009. Phenotype data were obtained on 453 (68.2%) of the residents using a population-based recruitment scheme. Due to computational limitations resulting from the extremely complex pedigree structure, analysis on only two pedigree reconstructions was feasible: a reduced pedigree retaining all phenotyped individuals and their parents as 57 distinct families, and 922 nuclear families. RESULTS: Familial correlations and heritability calculations reveal a significant genetic component to tIgE in these data (heritability = 26%). The most parsimonious model to explain tIgE distribution indicated by the reduced pedigree structure was a two-distribution Mendelian model. However, larger and non-genetic models could not be rejected. Simulations over 200 replicates performed to evaluate the reliability of this model, indicated that using restricted genealogical information had minimal impact on results of segregation analyses performed here.     

Genetic genealogical studies of 20 north Swedish families with the rare blood group p

In the county of V?sterbotten in northern Sweden, a large number of individuals with the rare blood group p have been found. The ancestors of all known 31 cases were studied genealogically, and the data showed that about one half of all cases (15 out of 31 in 9 out of 20 families) could be traced to one gene source in the north-eastern part of the county in the middle of the 17th century. In two of the families the parents were first cousins and in three they were second cousins. In the 18th and 17th centuries genealogical connections between the parents were found in another 10 of the families.     

Mathematical consequences of the genealogical species concept

A genealogical species is defined as a basal group of organisms whose members are all more closely related to each other than they are to any organisms outside the group ("exclusivity"), and which contains no exclusive group within it. In practice, a pair of species is so defined when phylogenies of alleles from a sample of loci shows them to be reciprocally monophyletic at all or some specified fraction of the loci. We investigate the length of time it takes to attain this status when an ancestral population divides into two descendant populations of equal size with no gene exchange, and when genetic drift and mutation are the only evolutionary forces operating. The number of loci used has a substantial effect on the probability of observing reciprocal monophyly at different times after population separation, with very long times needed to observe complete reciprocal monophyly for a large number of loci. In contrast, the number of alleles sampled per locus has a relatively small effect on the probability of reciprocal monophyly. Because a single mitochondrial or chloroplast locus becomes reciprocally monophyletic much faster than does a single nuclear locus, it is not advisable to use mitochondrial and chloroplast DNA to recognize genealogical species for long periods after population divergence. Using a weaker criterion of assigning genealogical species status when more than 50% of sampled nuclear loci show reciprocal monophyly, genealogical species status depends much less on the number of sampled loci, and is attained at roughly 4-7 N generations after populations are isolated, where N is the historically effective population size of each descendant. If genealogical species status is defined as more than 95% of sampled nuclear loci showing reciprocal monophyly, this status is attained after roughly 9-12 N generations.     

SAMPLING PROPERTIES OF GENEALOGICAL PATHWAYS UNDERLYING POPULATION PEDIGREES

In sexual species, autosomal alleles are transmitted through multigeneration organismal pedigrees via pathways of descent involving both genders. Here, models assess the sampling properties of these gender-described transmission pathways. An isolation-by-distance model of mating was used to construct a series of computer population pedigrees by systematically varying neighborhood size and the timing of isolation events in sundered populations. For each known pedigree, a matrix of true coancestry coefficients between all individuals in the final generation was calculated and compared (using cophenetic correlations) to mean pairwise times to common ancestry as estimated by sampling varying numbers of gender-defined lineage routes available to individual alleles through that pedigree. When few lineage routes were sampled, agreement between the estimated and the true pedigree was poor and showed a large variance. Agreement improved as more lineage routes were incorporated and asymptotically approached plateau levels predictably relatable to the magnitude of population structure. Results underscore a distinction between the composite genealogical information in a population pedigree and the subsets of that information registered in allelic lineage pathways.     

A test for linkage and association in general pedigrees: the pedigree disequilibrium test

Family-based tests of linkage disequilibrium typically are based on nuclear-family data including affected individuals and their parents or their unaffected siblings. A limitation of such tests is that they generally are not valid tests of association when data from related nuclear families from larger pedigrees are used. Standard methods require selection of a single nuclear family from any extended pedigrees when testing for linkage disequilibrium. Often data are available for larger pedigrees, and it would be desirable to have a valid test of linkage disequilibrium that can use all potentially informative data. In this study, we present the pedigree disequilibrium test (PDT) for analysis of linkage disequilibrium in general pedigrees. The PDT can use data from related nuclear families from extended pedigrees and is valid even when there is population substructure. Using computer simulations, we demonstrated validity of the test when the asymptotic distribution is used to assess the significance, and examined statistical power. Power simulations demonstrate that, when extended pedigree data are available, substantial gains in power can be attained by use of the PDT rather than existing methods that use only a subset of the data. Furthermore, the PDT remains more powerful even when there is misclassification of unaffected individuals. Our simulations suggest that there may be advantages to using the PDT even if the data consist of independent families without extended family information. Thus, the PDT provides a general test of linkage disequilibrium that can be widely applied to different data structures.     

A note on algorithms for genotype and allele elimination in complex pedigrees with incomplete genotype data

Elimination of genotypes or alleles for each individual or meiosis, which are inconsistent with observed genotypes, is a component of various genetic analyses of complex pedigrees. Computational efficiency of the elimination algorithm is critical in some applications such as genotype sampling via descent graph Markov chains. We present an allele elimination algorithm and two genotype elimination algorithms for complex pedigrees with incomplete genotype data. We modify all three algorithms to incorporate inheritance restrictions imposed by a complete or incomplete descent graph such that every inconsistent complete descent graph is detected in any pedigree, and every inconsistent incomplete descent graph is detected in any pedigree without loops with the genotype elimination algorithms. Allele elimination requires less CPU time and memory, but does not always eliminate all inconsistent alleles, even in pedigrees without loops. The first genotype algorithm produces genotype lists for each individual, which are identical to those obtained from the Lange-Goradia algorithm, but exploits the half-sib structure of some populations and reduces CPU time. The second genotype elimination algorithm deletes more inconsistent genotypes in pedigrees with loops and detects more illegal, incomplete descent graphs in such pedigrees.     

PRIMUS: Rapid Reconstruction of Pedigrees from Genome-wide Estimates of Identity by Descent

Understanding and correctly utilizing relatedness among samples is essential for genetic analysis; however, managing sample records and pedigrees can often be error prone and incomplete. Data sets ascertained by random sampling often harbor cryptic relatedness that can be leveraged in genetic analyses for maximizing power. We have developed a method that uses genome-wide estimates of pairwise identity by descent to identify families and quickly reconstruct and score all possible pedigrees that fit the genetic data by using up to third-degree relatives, and we have included it in the software package PRIMUS (Pedigree Reconstruction and Identification of the Maximally Unrelated Set). Here, we validate its performance on simulated, clinical, and HapMap pedigrees. Among these samples, we demonstrate that PRIMUS can verify reported pedigree structures and identify cryptic relationships. Finally, we show that PRIMUS reconstructed pedigrees, all of which were previously unknown, for 203 families from a cohort collected in Starr County, TX (1,890 samples).     

Inherent intractability of the ascertainment problem for pedigree data: a general likelihood framework.

The problem of ascertainment in segregation analysis arises when families are selected for study through ascertainment of affected individuals. In this case, ascertainment must be corrected for in data analysis. However, methods for ascertainment correction are not available for many common sampling schemes, e.g., sequential sampling of extended pedigrees (except in the case of "single" selection). Concerns about whether ascertainment correction is even required for large pedigrees, about whether and how multiple probands in the same pedigree can be taken into account properly, and about how to apply sequential sampling strategies have occupied many investigators in recent years. We address these concerns by reconsidering a central issue, namely, how to handle pedigree structure (including size). We introduce a new distinction, between sampling in such a way that observed pedigree structure does not depend on which pedigree members are probands (proband-independent [PI] sampling) and sampling in such a way that observed pedigree structure does depend on who are the probands (proband-dependent [PD] sampling). This distinction corresponds roughly (but not exactly) to the distinction between fixed-structure and sequential sampling. We show that conditioning on observed pedigree structure in ascertained data sets obtained under PD sampling is not in general correct (with the exception of "single" selection), while PI sampling of pedigree structures larger than simple sibships is generally not possible. Yet, in practice one has little choice but to condition on observed pedigree structure. We conclude that the problem of genetic modeling in ascertained data sets is, in most situations, literally intractable. We recommend that future efforts focus on the development of robust approximate approaches to the problem.     

A note on the rationale for estimating genealogical coancestry from molecular markers

Background

Genetic relatedness or similarity between individuals is a key concept in population, quantitative and conservation genetics. When the pedigree of a population is available and assuming a founder population from which the genealogical records start, genetic relatedness between individuals can be estimated by the coancestry coefficient. If pedigree data is lacking or incomplete, estimation of the genetic similarity between individuals relies on molecular markers, using either molecular coancestry or molecular covariance. Some relationships between genealogical and molecular coancestries and covariances have already been described in the literature.

Methods

We show how the expected values of the empirical measures of similarity based on molecular marker data are functions of the genealogical coancestry. From these formulas, it is easy to derive estimators of genealogical coancestry from molecular data. We include variation of allelic frequencies in the estimators.

Results

The estimators are illustrated with simulated examples and with a real dataset from dairy cattle. In general, estimators are accurate and only slightly biased. From the real data set, estimators based on covariances are more compatible with genealogical coancestries than those based on molecular coancestries. A frequently used estimator based on the average of estimated coancestries produced inflated coancestries and numerical instability. The consequences of unknown gene frequencies in the founder population are briefly discussed, along with alternatives to overcome this limitation.

Conclusions

Estimators of genealogical coancestry based on molecular data are easy to derive. Estimators based on molecular covariance are more accurate than those based on identity by state. A correction considering the random distribution of allelic frequencies improves accuracy of these estimators, especially for populations with very strong drift.