Assessment of relationships between pigs based on pedigree and genomic information

Relationships play a very important role in studies on quantitative genetics. In traditional breeding, pedigree records are used to establish relationships between animals; while this kind of relationship actually represents one kind of relatedness, it cannot distinguish individual specificity, capture the variation between individuals or determine the actual genetic superiority of an animal. However, with the popularization of high-throughput genotypes, assessments of relationships among animals based on genomic information could be a better option. In this study, we compared the relationships between animals based on pedigree and genomic information from two pig breeding herds with different genetic backgrounds and a simulated dataset. Two different methods were implemented to calculate genomic relationship coefficients and genomic kinship coefficients, respectively. Our results show that, for the same kind of relative, the average genomic relationship coefficients (G matrix) were very close to the pedigree relationship coefficients (A matrix), and on average, the corresponding values were halved in genomic kinship coefficients (K matrix). However, the genomic relationship yielded a larger variation than the pedigree relationship, and the latter was similar to that expected for one relative with no or little variation. Two genomic relationship coefficients were highly correlated, for farm1, farm2 and simulated data, and the correlations for the parent-offspring, full-sib and half-sib were 0.95, 0.90 and 0.85; 0.93, 0.96 and 0.89; and 0.52, 0.85 and 0.77, respectively. When the inbreeding coefficient was measured, the genomic information also yielded a higher inbreeding coefficient and a larger variation than that yielded by the pedigree information. For the two genetically divergent Large White populations, the pedigree relationship coefficients between the individuals were 0, and 62 310 and 175 271 animal pairs in the G matrix and K matrix were greater than 0. Our results demonstrated that genomic information outperformed the pedigree information; it can more accurately reflect the relationships and capture the variation that is not detected by pedigree. This information is very helpful in the estimation of genomic breeding values or gene mapping. In addition, genomic information is useful for pedigree correction. Further, our findings also indicate that genomic information can establish the genetic connection between different groups with different genetic background. In addition, it can be used to provide a more accurate measurement of the inbreeding of an animal, which is very important for the assessment of a population structure and breeding plan. However, the approaches for measuring genomic relationships need further investigation.     

Using dominance relationship coefficients based on linkage disequilibrium and linkage with a general complex pedigree to increase mapping resolution

Dominance (intralocus allelic interactions) plays often an important role in quantitative trait variation. However, few studies about dominance in QTL mapping have been reported in outbred animal or human populations. This is because common dominance effects can be predicted mainly for many full sibs, which do not often occur in outbred or natural populations with a general pedigree. Moreover, incomplete genotypes for such a pedigree make it infeasible to estimate dominance relationship coefficients between individuals. In this study, identity-by-descent (IBD) coefficients are estimated on the basis of population-wide linkage disequilibrium (LD), which makes it possible to track dominance relationships between unrelated founders. Therefore, it is possible to use dominance effects in QTL mapping without full sibs. Incomplete genotypes with a complex pedigree and many markers can be efficiently dealt with by a Markov chain Monte Carlo method for estimating IBD and dominance relationship matrices (D(RM)). It is shown by simulation that the use of D(RM) increases the likelihood ratio at the true QTL position and the mapping accuracy and power with complete dominance, overdominance, and recessive inheritance modes when using 200 genotyped and phenotyped individuals.     

Combining microsatellite and pedigree data to estimate relationships among Skyros ponies

Relationship coefficients are particularly useful to improve genetic management of endangered populations. These coefficients are traditionally based on pedigree data, but in case of incomplete or inexistent pedigrees they are replaced by coefficients calculated from molecular data. The main objective of this study was to develop a new method to estimate relationship coefficients by combining molecular with pedigree data, which is useful for specific situations where neither pedigree nor molecular data are complete. The developed method was applied to contribute to the conservation of the Skyros pony breed, which consists of less than 200 individuals, divided into 3 main herds or subpopulations. In this study, relationships between individuals were estimated using traditional estimators as well as the newly developed method. For this purpose, 99 Skyros ponies were genotyped at 16 microsatellite loci. It appeared that the limitation of the most common molecular-based estimators is the use of weights that assume relationships equal to 0. The results showed that, as a consequence of this limitation, negative relationship values can be obtained in small inbred populations, for example. By contrast, the combined estimator gave no negative values. Using principal component analysis, the combined estimator also enabled a better graphic differentiation between the 3 subpopulations defined previously. In conclusion, this new estimator can be a promising alternative to traditionally used estimators, especially in inbred populations, with both incomplete pedigree and molecular information.     

The genotypic probability structure of an individual at the head of a pedigree

A proper probabilistical proof of a generalization of Wright's classical formula relating the coefficient of inbreeding of an individual at the head of a pedigree to the genotypic probability structure of this individual at one gene locus is presented. It is shown that in general the knowledge of gene frequencies realized within the initial populations from which individuals entering the pedigree are selected at random is not sufficient to predict expected genotypic frequencies in the resulting inbred population. To treat any arbitrary situation concerning the choice of individuals and genotypes to enter the pedigree, it is necessary to determine an additional set of coefficients, which merely depend on the type of the pedigree. A basic method for computing these coefficients is outlined briefly.     

A novel recursive algorithm for the calculation of the detailed identity coefficients

Background

A recursive algorithm to calculate the fifteen detailed coefficients of identity is introduced. Previous recursive procedures based on the generalized coefficients of kinship provided the detailed coefficients of identity under the assumption that the two individuals were not an ancestor of each other.

Findings

By using gametic relationships to include three, four or two pairs of gametes, we can obtain these coefficients for any pair of individuals. We have developed a novel linear transformation that allows for the calculation of pairwise detailed identity coefficients for any pedigree given the gametic relationships. We illustrate the procedure using the well-known pedigree of Julio and Mencha, which contains 20 Jicaque Indians of Honduras, to calculate their detailed coefficients.

Conclusions

The proposed algorithm can be used to calculate the detailed identity coefficients of two or more individuals with any pedigree relationship.     

Improving the calculation of statistical significance in genome-wide scans

Calculations of the significance of results from linkage analysis can be performed by simulation or by theoretical approximation, with or without the assumption of perfect marker information. Here we concentrate on theoretical approximation. Our starting point is the asymptotic approximation formula presented by Lander and Kruglyak (1995, Nature Genetics, 11, 241--247), incorporating the effect of finite marker spacing as suggested by Feingold et al. (1993, American Journal of Human Genetics, 53, 234--251). We consider two distinct ways in which this formula can be improved. Firstly, we present a formula for calculating the crossover rate rho for a pedigree of general structure. For a pedigree set, these values may then be weighted into an overall crossover rate which can be used as input to the original approximation formula. Secondly, the unadjusted p -value formula is based on the assumption of a Normally distributed nonparametric linkage (NPL) score. This leads to conservative or anti-conservative p -values of varying magnitude depending on the pedigree set structure. We adjust for non-Normality by calculating the marginal distribution of the NPL score under the null hypothesis of no linkage with an arbitrarily small error. The NPL score is then transformed to have a marginal standard Normal distribution and the transformed maximal NPL score, together with a slightly corrected value of the overall crossover rate, is inserted into the original formula in order to calculate the p -value. We use pedigrees of seven different structures to compare the performance of our suggested approximation formula to the original approximation formula, with and without skewness correction, and to results found by simulation. We also apply the suggested formula to two real pedigree set structure examples. Our method generally seems to provide improved behavior, especially for pedigree sets which show clear departure from Normality, in relation to the competing approximations.     

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.     

Relatedness coefficients in pedigrees with inbred founders

We study an extension of the standard framework for pedigree analysis, in which we allow pedigree founders to be inbred. This solves a number of practical challenges in calculating coefficients of relatedness, including condensed identity coefficients. As a consequence we expand considerably the class of pedigrees for which such coefficients may be efficiently computed. An application of this is the modelling of background inbreeding as a continuous effect. We also use inbred founders to shed new light on constructibility of relatedness coefficients, i.e., the problem of finding a genealogy yielding a given set of coefficients. In particular, we show that any theoretically admissible coefficients for a pair of noninbred individuals can be produced by a finite pedigree with inbred founders. Coupled with our computational methods, implemented in the R package ribd, this allows for the first time computer analysis of general constructibility solutions, thus making them accessible for practical use.

     

A mathematical approach to multiple genetic relationships

A fundamental concept in the treatment of genetic relationships is that of gene identity which first was introduced by Cotterman (1940). Based on this notion several measures of relationship evolved such as the inbreeding coefficient, the coefficient of kinship, and the identity coefficients; by means of these quantities joint and conditional phenotype probabilities could be derived. This paper is an attempt at a general mathematical treatment of genetic relationships: Identity states are defined for any number of individuals, a method is given for the calculation of the corresponding identity coefficients by means of generalized coefficients of kinship, and applications are emphasized.     

一个可用于复杂大群体的可视化系谱绘制和分析软件

对于在遗传研究和家系研究中大的系谱结构图还很难分析。系谱的绘制通常是遗传性状的分析研究的第一步。系图可以反映整个群体的结构、每个个体之间的相互关系以及基因流的走向,便于理解遗传性状的本质。因为所用家系数目的增大和复杂性的增加,绘制1个清晰的系谱有时变得十分困难。因此开发了1种名为Pedigraph软件,可以解决这个问题。Pedigraph能够完成对于大的复杂的群体的系谱绘制工作,并能进行相应的系谱分析。初步的测试表明这个软件在研究动植物的遗传育种中是1个有用的工具,同时它也可以用于人类的群体和历史等方面的研究。     

Comparison of genetic parameters from marker-based relationship, sibship, and combined models in Scots pine multi-site open-pollinated tests

Nine microsatellite DNA markers (simple sequence repeats, SSRs) were used to estimate pairwise relationships among 597 Scots pine (Pinus sylvestris) trees as well as to generate a sibship structure for quantitative genetic parameters’ estimation comparison. The studied trees were part of an open-pollinated progeny test of 102 first-generation parents. Three methods were used to estimate variance components and heritabilities, namely, structured pedigree (half- and full-sib), marker-based pairwise relationships (four pairwise estimators), and a combined pedigree and marker-based relationship. In each of the three methods, the same animal model was used to compute variances except when marker-based relationship was used wherein we substituted the average numerator relationship matrix (i.e., pedigree-based matrix) with that computed based on markers’ pairwise relationships. Our results showed a high correlation in estimated breeding values between the pedigree (full-sib) and the combined marker-pedigree estimates. The marker-based relationship method produced high correlations when individual site data were analyzed. In contrast, the marker-based relationship method resulted in a significant decrease in both variance estimation and their standard errors which were in concordance with earlier published results; however, no estimates were produced when across-site analyses were attempted. We concluded that the combined pedigree method is the best approach as it represents the historical (pairwise) and contemporary (pedigree) relationships among the tested individuals, a situation that cannot be attained by any of the used methods individually. This method is dependent on the number and informativeness of the markers used.     

Comparisons of three probability formulae for parentage exclusion

Three general formulae calibrate the average capability of marker systems to dispute falsely reported pedigree records in uniparous species. The most familiar exclusion formula applies to paternity, although the same formula applies equally to maternity. Another formula faults the relationship of a single offspring with its putative parent; for example, where the genotype of the other parent is not available. The remaining formula excludes both of the falsely recorded parents of a substituted offspring. Simplified forms of the three general formulae facilitate the calculation of maximal average exclusion values over a range of hypothetical markers. Allele frequency data on eight marker systems in horses provide practical examples. The exclusion values of the three formulae are compared.     

Evaluating pedigree data. II. Identifying the cause of error in families with inconsistencies

Pedigree data can be evaluated, and subsequently corrected, by analysis of the distribution of genetic markers, taking account of the possibility of mistyping . Using a model of pedigree error developed previously, we obtained the maximum likelihood estimates of error parameters in pedigree data from Tokelau. Posterior probabilities for the possible true relationships in each family are conditional on the putative relationships and the marker data are calculated using the parameter estimates. These probabilities are used as a basis for discriminating between pedigree error and genetic marker errors in families where inconsistencies have been observed. When applied to the Tokelau data and compared with the results of retyping inconsistent families, these statistical procedures are able to discriminate between pedigree and marker error, with approximately 90% accuracy, for families with two or more offspring. The large proportion of inconsistencies inferred to be due to marker error (61%) indicates the importance of discriminating between error sources when judging the reliability of putative relationship data. Application of our model of pedigree error has proved to be an efficient way of determining and subsequently correcting sources of error in extensive pedigree data collected in large surveys.     

Application of DNA fingerprinting to parentage and extended family relationship testing

For the purpose of applying DNA fingerprinting to parentage or other extended blood relationship testing, we have developed a general system for estimating blood relationship between a set of individuals. This system can be used for any blood relationship testing, even if critical family members are unavailable. A general formula for calculating the probability of such relationships based on Bayes' theorem has been devised and the ability of DNA fingerprinting produced by Jeffrey's polycore probe to determine the relationships has been evaluated.     

Analyse des pedigrees et calcul des coefficients d'identité par les arbres géniques

Two genes in a pedigree are identical by descent if they are two copies of a common ancestor gene. To obtain an unambiguous definition of the set of genes, at some autosomal locus, any gene is defined as an ordered pair of zygotes: the zygote who carries the gene, and the parent who transmitted it. The natural ordered structure on the set of zygotes yields an ordered structure upon the set of genes. Any event of the mendelian segregation splits down the set of genes into non-overlaping classes of identical genes: when considered as an ordered sub-set of genes, each class is shown to have the algebraic properties of a tree. Given a sub-set ?? of genes, a family of exclusive events ensuring identity between all genes of ?? is identified as a family of genic trees with some property. This relationship between segregational events and genic trees is extended to the case where two sub-sets ?? and ??′ of genes are considered together. As a consequence, a general method is obtained to compute either identity coefficients involving any number of genes splitted into one or two identity classes, or the fifteen coefficients defined among four genes, whichever the relationships between zygotes and genes might be. Using this approach to deal with the allelic structure in a set of genes carried by related zygotes is suggested.     

Pedigrees and microsatellites among endangered ungulates: what do they tell us?

Relationships between pedigree coefficients of inbreeding and molecular metrics are generally weak, suggesting that measures of heterozygosity estimated using microsatellites may be poor surrogates of genome-wide inbreeding. We compare three endangered species of gazelles ( Gazella ) with different degrees of threat in their natural habitats, for which captive breeding programmes exist. For G. dorcas, the species with the largest founding population, the highest and most recent number of founding events, the correlation between pedigree coefficient of inbreeding and molecular metrics was higher than for outbred populations of mammals, probably because it has both higher mean f and variance. For the two species with smaller founding populations, conventional assumptions about founders, i.e. outbred and unrelated, are unrealistic. When realistic assumptions about the founders were made, clear relationships between pedigree coefficients of inbreeding and molecular metrics were revealed for G. cuvieri. This population had a small founding population, but it did experience admixture years later; thus, the relationship between inbreeding and molecular metrics in G. cuvieri is very similar to the expected values but lower than in G. dorcas . In contrast, no relationship was found for G. dama mhorr which had a much smaller founding population than had been previously assumed, which probably had high levels of inbreeding and low levels of genetic variability, and no admixture. In conclusion, the strength of the association between pedigree coefficient of inbreeding and molecular metrics among endangered species depends on the level of inbreeding and genetic variability present in the founding population, its size and its history.     

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.     

RAPD and pedigree-based genetic diversity estimates in cultivated diploid potato hybrids

In this study, RAPD and pedigree data were used to investigate the genetic relationships in a group of 45 diploid hybrid potato clones used in the breeding and genetics program of the Agriculture and Agri-Food Canada Potato Research Centre in Fredericton, New Brunswick, and used for the potato after-cooking darkness program at the Nova Scotia Agricultural College. These hybrids were derived from crossing primitive cultivated South American diploid species such as Solanum phureja or Solanum stenotomum and wild diploid species such as Solanum chacoense and other wild Argentine species with haploids of Solanum tuberosum. These hybrids have subsequently undergone up to 30 years of breeding and selection, for adaptation to local growing and storage conditions, processing traits and pest resistances. The objectives of this study were to estimate the level of genetic similarity (GS) among these sets of clones and to investigate the correlation between RAPD-based GS and f, based on pedigree information. Genetic similarity coefficients varied from 0.29 to 0.90 with a mean of 0.65 when based on the RAPD data, whereas the coefficient of parentage varied from zero to 0.75 with a mean of 0.11. The degree of relationship between the similarity matrices based on RAPD and pedigree was measured by comparing the similarity matrices with the normalized Mantel test. A low positive correlation (R = 0.104, p = 0.999) between the two matrices was observed. Cluster analysis using GS divided the clones into many subgroups that did not correspond well with the grouping based on pedigree. The level of genetic variation present in this set of potato clones is very high. Rigorous selection pressure aimed at different breeding purposes may result in the genetic differentiation of the clones from the same origin.     

Discovery of hidden pedigree errors combining genomic information with the genomic relationship matrix in Texel sheep

Genomic variants such as Single Nucleotide Polymorphisms and animal pedigree are now used widely in routine genetic evaluations of livestock in many countries. The use of genomic information not only can be used to enhance the accuracy of prediction but also to verify pedigrees for animals that are extensively managed using natural mating and enabling multiple-sire mating groups to be used. By so doing, the rate of genetic gain is enhanced, and any bias associated with incorrect pedigrees is removed. This study used a set of 8 764 sheep genotypes to verify the pedigree based on both the conventional opposing homozygote method as well as a novel method when combined with the inclusion of the genomic relationship matrix (GRM). The genomic relationship coefficients between verified pairs of animals showed on average a relationship of 0.50 with parent, 0.25 with grandparent, 0.13 with great grandparent, 0.50 with full-sibling and 0.27 with half-sibling. Minimum obtained values from these verified pairs were then used as thresholds to determine the pedigree for unverified pairs of animals, to detect potential errors in the pedigree. Using a case study from a population partially genotyped UK sheep, the results from this study illustrate a powerful way to resolve parentage inconsistencies, when combining the conventional ‘opposing homozygote’ method using genomic information together with GRM for pedigree checking. In this way, previously undetected pedigree errors can be resolved.     

The importance of identity-by-state information for the accuracy of genomic selection

Background

It is commonly assumed that prediction of genome-wide breeding values in genomic selection is achieved by capitalizing on linkage disequilibrium between markers and QTL but also on genetic relationships. Here, we investigated the reliability of predicting genome-wide breeding values based on population-wide linkage disequilibrium information, based on identity-by-descent relationships within the known pedigree, and to what extent linkage disequilibrium information improves predictions based on identity-by-descent genomic relationship information.

Methods

The study was performed on milk, fat, and protein yield, using genotype data on 35 706 SNP and deregressed proofs of 1086 Italian Brown Swiss bulls. Genome-wide breeding values were predicted using a genomic identity-by-state relationship matrix and a genomic identity-by-descent relationship matrix (averaged over all marker loci). The identity-by-descent matrix was calculated by linkage analysis using one to five generations of pedigree data.

Results

We showed that genome-wide breeding values prediction based only on identity-by-descent genomic relationships within the known pedigree was as or more reliable than that based on identity-by-state, which implicitly also accounts for genomic relationships that occurred before the known pedigree. Furthermore, combining the two matrices did not improve the prediction compared to using identity-by-descent alone. Including different numbers of generations in the pedigree showed that most of the information in genome-wide breeding values prediction comes from animals with known common ancestors less than four generations back in the pedigree.

Conclusions

Our results show that, in pedigreed breeding populations, the accuracy of genome-wide breeding values obtained by identity-by-descent relationships was not improved by identity-by-state information. Although, in principle, genomic selection based on identity-by-state does not require pedigree data, it does use the available pedigree structure. Our findings may explain why the prediction equations derived for one breed may not predict accurate genome-wide breeding values when applied to other breeds, since family structures differ among breeds.