Background

The definition of inbreeding coefficient (F) is the probability that two alleles in an individual are identical by descent (IBD) relative to a base population where all alleles are assumed unrelated [1]. Rates of inbreeding have increased as intensive selection was applied to the populations [2–7]. Increased levels of inbreeding result in increased probability that animals are homozygous for deleterious alleles [2, 8, 9]. Thus, inbred animals suffer from inbreeding depression with reduced fitness, and highly inbred animals may have considerably reduced lifespans [2, 6, 10–13]. Information on inbreeding is critical in the design of breeding program to control the increase in inbreeding levels and thereby controlling inbreeding depression in the progeny. Pedigree information has been used to calculate the estimated inbreeding coefficient as the expected probability that two alleles at a locus are IBD [14–16]. For example, Meuwissen and Luo proposed a method to estimate inbreeding coefficients based on pedigree data of large populations [17]. However, incomplete pedigrees result in erroneous estimates and an underestimation of levels of inbreeding [18]. VanRaden proposed a method to take into account unknown ancestors when estimating inbreeding coefficients, increasing the accuracy of inbreeding level estimates in incomplete pedigrees [19].

With the availability of Single Nucleotide Polymorphism (SNP) array genotyping technologies, long stretches of homozygous genotypes, known as runs of homozygosity (ROH) can be identified. ROH are believed to reflect an estimate of autozygosity on genomic level and generally identify genomic regions which are IBD [20, 21]. Theoretically, it is expected that ROH can be accurately estimated from the full sequence data, because these estimates do not suffer from sampling such as may be expected when subsets of loci, for instance 50k SNPs, are used [22–24]. The inbreeding coefficient can be calculated as the proportion of genome covered by ROH and has been shown to be more informative than the inbreeding coefficient estimated from pedigree data or other estimators because ROH strongly correlate with homozygous mutation load [25]. ROH have commonly been used to infer population history and to examine the effect of deleterious homozygotes caused by inbreeding in human populations [20, 26–29]. Long ROH reflect recent inbreeding, whereas short ROH reflect ancient inbreeding [26]. However, only a few studies have evaluated ROH in cattle populations. Ferenčaković et al. examined the effect of SNP density and genotyping errors when estimating autozygosity from high-throughput genomic data [24]. Estimates based on ROH also vary with different densities of genomic data. The minimum length of ROH that can be detected depends on SNP density [24, 30]. Recently, Purfield et al. detected ROH in a cattle population from SNP chip data to infer population history [31]. However, to estimate the “true” state of ROH, whole-genome sequences should be used rather than SNP chip data, but, to date, there are only few studies doing this in cattle [32]. With the advent of next-generation sequencing technology, whole-genome sequences have become available to examine the fine-scale genetic architecture of the cattle genome. It is now possible to investigate and compare how well different commonly used estimators of inbreeding level correlate with ROH estimated using next-generation sequence (NGS) data.

In recent years, widespread availability of genotype data enabled computation of inbreeding from the diagonals of genomic relationship matrices, i.e., the “GRM” method (FGRM), as a by-product of genomic selection. Similarly, using the genotypes, the inbreeding coefficient can be computed based on excess of homozygosity following Wright (1948) (FHOM) [33] and based on correlation between uniting gametes following Wright (1922) (FUNI) [1]. The objective of the present study was to compare different estimators for inbreeding coefficients calculated from pedigree, 50k SNP chip genotypes and full sequence data with estimates based on ROH, for three different dairy cattle breeds.

Methods

SNP genotyping and sequencing

A total of 89 bulls with a high genetic contribution to current Danish dairy cattle populations were selected for whole-genome resequencing. These included 32 Holstein (HOL), 27 Jersey (JER), and 30 Danish Red Cattle (RDC) bulls. RDC cattle are a composite breed with contributions from different red breeds, including Swedish Red, Finnish Ayrshire, and Brown Swiss [34]. Only bi-allelic variants SNPs with a phred-scaled quality score [35] higher than 100 were kept for analysis to ensure the quality of variants. Genotypes were extracted from whole-genome sequence (WGS) data using GATK [36] and a perl script. The sequence variants with read depth lower than 7 or higher than 30 were filtered out. In addition, 85 of the sequenced animals were genotyped with the Illumina 50k SNP assay (BovineSNP50 BeadChip version 1 or 2, Illumina, San Diego, CA). SNP genotyping and quality control were as described by Höglund et al. [37]. Among the whole genome sequenced animals, 4 animals were not genotyped with the 50k SNP chip. Their genotypes for the SNPs on the 50k chip were extracted from their whole-genome sequences. The quality of genotype calls from SNP chips is expected to be higher than that of whole-genome sequences; therefore, only sequence variants with a high quality score (phred score > 100) were included. The corresponding corrections for reverse strand calls in the sequence data were converted to Illumina calls by correcting locus calling from reverse strands in Illumina calls to maintain consistency of allele encoding between Illumina calls and sequence data. The concordance between the SNP chip and sequence data was ~97 %.

Estimation of inbreeding

Using pedigree records (FPED)

Inbreeding coefficients for the 89 bulls were estimated using pedigree records (FPED). The average pedigree depth was ~8 generations ranging from 3 to 13. Average pedigree depth was 7, 8 and 9 for HOL, JER, and RDC, respectively. The method proposed by VanRaden [19] was used to compute inbreeding coefficients, which replaces unknown inbreeding coefficients by average inbreeding coefficients in the same generations. Inbreeding coefficients were calculated using the following formula [38]:

$$ {A}_{ii}={\displaystyle \sum_{j=1}^i}{L}_{ij}^2{D}_{jj}, $$

where \( {A}_{ii} \) is the i th diagonal element of the A matrix (pedigree relationship matrix), which is equal to the inbreeding coefficient of the i th animal plus 1. L is a lower triangular matrix containing the fraction of the genes that animals derive from their ancestors, and D is a diagonal matrix containing the within family additive genetic variances of animals [17]. The computation for matrix elements L ij and D jj follows the rule of computation of the A matrix [17]. The detailed decomposition for computing \( {A}_{ii} \) is explained by Meuwissen and Luo [17]. The analysis was conducted using Relax2 software [39].

Using genotypes (FROH, FGRM, FHOM, FUNI)

Sequence data

ROH were detected from sequence data using all bi-allelic variants according to the method of Bosse et al. [23]. This method was used to compute ROH for sequence data instead of PLINK because not all short ROH can be detected using PLINK for sequence data (the sliding window size in PLINK is fixed; therefore, ROH shorter than a certain length cannot be detected). The measure of homozygosity based on ROH (FROH) from genomic data is defined as the total length of genome covered by ROH divided by the overall length of genome covered by SNPs or sequences as follows [20]:

$$ {\mathtt{F}}_{\mathtt{ROH}}=\frac{{\mathtt{L}}_{\mathtt{ROH}}}{{\mathtt{L}}_{\mathtt{AUTO}}}, $$

where LROH is the sum of ROH lengths and LAUTO is the total length of autosomes covered by reads. The inbreeding coefficient was calculated by extracting ROH from sequence data. Three ROH estimates based on lengths were calculated from sequence data. The ROH was calculated separately by summing the ROH in different length classes: 1) based on all ROH; 2) ROH >1 Mbp; 3) ROH >3 Mbp.

In addition, three other estimates of inbreeding coefficients were calculated using sequence data (FGRM, FHOM, FUNI). The FGRM estimate was calculated following VanRaden (2008) [40] based on the variance of the additive genotypes. FGRM was derived from

$$ {\mathrm{F}}_{\mathrm{GRM}}=\frac{{\left[{x}_i-E\left({x}_i\right)\right]}^2}{h_i}-1=\frac{{\left({x}_i-2{\widehat{p}}_i\right)}^2}{h_i}-1, $$

where p i is the observed fraction of the first allele at locus i, h i  = 2p i (1 − p i ) and x i is the number of copies of the reference allele (i.e., the allele whose homozygous genotype was coded as “0”) for the ith SNP [41]. This was equivalent to estimating an individual’s relationship to itself (diagonal of the SNP-derived GRM). The FHOM estimate was calculated based on the excess of homozygosity following Wright (1948) [33]:

$$ {\mathrm{F}}_{\mathrm{HOM}} = \left[\mathrm{O}\left(\# \hom \right)\hbox{-} \mathrm{E}\left(\# \hom \right)\right]/\left[1\hbox{-} \mathrm{E}\left(\# \hom \right)\right]=1-\frac{x_i\left(2-{x}_i\right)}{h_i}, $$

where O (# hom) and E (# hom) are the observed and expected numbers of homozygous genotypes in the sample, respectively [41]. The FUNI estimate was calculated based on the correlation between uniting gametes following Wright (1922) [1]:

$$ {\mathrm{F}}_{\mathrm{UNI}}=\frac{x_i^2-\left(1+2{p}_i\right){x}_i+2{p}_i^2}{h_i}, $$

where h i and x i are the same as for FGRM [41]. The calculations for these three estimates FGRM, FHOM and FUNI were computed using the option –ibc from GCTA software [41].

50k SNP chip

ROH were detected from 50k SNP chip data using the software PLINK with adjusted parameters (–homozyg-density 1000, –homozyg-window-het 1, –homozyg-kb 10, –homozyg-window-snp 20) [23, 42]. These settings for PLINK to detect ROH in SNP data were chosen to make the detected ROH in SNP chip data and sequence data as similar as possible to enable comparisons of results when using different types of data. Genomic estimates of the inbreeding coefficient based on all ROH (FROH) were calculated using the same formula as was used for the sequence data. The other three types of estimates (FGRM, FHOM, FUNI) were also calculated for genotypes extracted from 50k SNP chip data using the same methods as for sequence data.

Pearson’s correlation coefficients were calculated between estimates of inbreeding coefficients from each of pedigree records, 50k SNP genotypes, and whole-genome sequence variants. All correlations between different inbreeding coefficient estimators were tested within breed to determine whether they were significantly different from 0 using the R (http://www.r-project.org/) cor and cor.test functions.

Impact of allele frequencies on estimators of inbreeding

As some estimators explicitly use allele frequencies to compute inbreeding coefficients, it is important to investigate how varying allele frequencies affect estimated inbreeding coefficients. Here, we investigated how the three different estimators change across the whole range of allele frequencies. For each genotype x i (homozygous for the reference allele; heterozygous for the reference and non-reference allele; homozygous for the non-reference allele), the values can be written as a function of allele frequency p i , as shown in Table 1.

Table 1 Formula for calculating three estimators (FGRM, FHOM and FUNI) for each genotype (homozygous for reference allele; heterozygous for reference and non-reference allele; homozygous for non-reference allele)
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Results

We used five different approaches (FPED, FGRM, FHOM, FUNI, FROH) to estimate inbreeding coefficients using information from three different sources: pedigree, whole genome sequence and 50k SNP chip genotype data. There were total 11 estimates of inbreeding coefficients for each animal (Table 2). The average inbreeding coefficients estimated using different approaches and different data sets are presented in Table 2. The FPED and FROH estimated from 50k data for HOL and JER are significantly higher than for RDC (p < 0.05). For inbreeding coefficients estimated from sequence data, FROH, FROH>1Mb, FROH>3Mb, FHOM and FUNI differed significantly among breeds, being highest in JER and lowest in RDC. The mean FROH for 50k SNP chip data (0.066), and sequence data (0.19) are significantly higher than FPED (0.016) (p < 0.01).

Table 2 Estimated mean (min-max) of pedigree-based inbreeding coefficient (FPED), GRM-based inbreeding coefficient (FGRM), inbreeding coefficients based on excess of homozygosity (FHOM), inbreeding coefficients based on correlation between uniting gametes (FUNI), ROH-based inbreeding coefficients (FROH). FROH greater than 1 Mb, 3 Mb derived from sequence data were reported
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FROH estimated from sequence data is a direct and accurate estimate of the levels of homozygosity. It mostly reflects regions which were IBD on the genome; therefore, we limited our comparisons to comparing between FROH from sequence data with other estimates of F. High correlations were observed between FROH estimated from the 50k and sequence data with FROH>1Mb and FROH>3Mb from the sequence data for all three breeds (Tables 3, 4 and 5). The correlation between FROH estimated from 50k data and FROH>3Mb was higher than FROH estimated from 50k data and FROH>1Mb in JER and RDC (Tables 4 and 5). FROH was consistently positively correlated with FHOM and FUNI, when both were computed from either 50k or sequence data in all three breeds (Tables 3, 4 and 5). A high correlation was found between FROH and FUNI, when both were computed from either 50k or sequence data in all three breeds (Tables 3, 4 and 5). However, for different breeds, FHOM and FUNI were correlated differently across different densities of genomic data. For HOL and RDC, the higher the density of genomic data used for FUNI, the higher the correlation was between FUNI and FROH from sequence data (Tables 3 and 5). For HOL, the correlation between FUNI and FROH from sequence data (0.95) was still higher than the correlation between FROH estimated from 50k SNP chip data and sequence data (0.87) (Table 3). In contrast to JER, FHOM and FUNI were most highly correlated with FROH estimated from sequence data (Table 5).

Table 3 Correlation coefficients between different estimates for inbreeding from different data sets for HOL
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Table 4 Correlation coefficients between different estimates for inbreeding from different data sets for JER
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Table 5 Correlation coefficients between different estimates for inbreeding from different data sets for RDC
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FPED was mostly intermediately correlated with FHOM and FROH estimated from 50k and sequence data. The highest correlation between FPED and FROH estimated from 50k and sequence data was found in HOL (Table 3). The strongest correlation among estimators of FROH (FROH from 50k or sequence data or FROH>3Mb or FROH>1Mb from sequence data) and FPED was observed between FPED and FROH>3Mb from sequence data in HOL (Table 3). A moderate correlation was found between FPED and FROH estimated from 50k and sequence data for JER and RDC (Tables 4 and 5).

The estimate FGRM from both 50k and sequence data and FPED had a correlation close to zero in all three breeds and the values were often negative (Tables 3, 4 and 5). At the same time, FGRM estimated from 50k and sequence data generally showed a low correlation with other estimates except between two estimates FGRM estimated from 50k and sequence data in HOL and JER, and between FGRM and FUNI estimated from 50k data (Tables 3 and 4).

Discussion

Pedigree has been used to estimate inbreeding coefficients in animal breeding for over 50 years [1, 17]. Recently, researchers have utilized runs of homozygosity (ROH) estimated from medium density genotype data such as 50k SNP chip data to estimate inbreeding coefficients in livestock populations [22–24, 30]. ROH were initially used to explore regions of inbreeding in the genome and further investigate the fitness effect of these regions on different traits [2, 9, 11, 43]. Population subdivision and either inbreeding or inbreeding avoidance affects the whole genome composition, whereas selection and assortative mating will affect only those loci associated with particular phenotypes. However, we observed that inbreeding coefficient FROH estimated from sequence data were relatively higher for chromosome 1 and 10 for all four breeds (Fig. 1). This is most likely because the local recombination rate is relatively lower than average, which results in high levels of homozygosity on average [23, 44].

Fig. 1
figure 1

Distribution of inbreeding coefficients FROH estimated from sequence data using ROH for each chromosome in three breeds. Inbreeding coefficients FROH estimated from sequence data versus chromosomes 1–29 in HOL. JER and RDC. Standard error bars were computed among individuals within HOL, JER and RDC

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Our study is the first to calculate inbreeding coefficient based on ROH from full sequence data in cattle. The objective of this study was to compare estimates of inbreeding calculated from different methods and different data sources (pedigree, 50k SNP chip genotypes and full sequence data).

The pedigree-based inbreeding coefficient, FPED, was moderately correlated with FHOM and FROH in all breeds. These moderate correlations (~0.47 to 0.56) may be partly explained by the relatively shallow depth of the pedigree records (~8–9) for these bulls. Another difference between FROH and FPED is that short ROH capture ancient inbreeding while long ROH capture recent inbreeding whereas pedigree captures only relatively recent inbreeding. Pedigree accounts only for inbreeding that occurred since pedigree recording began. Therefore, after excluding ROH smaller than 1 or 3 Mbp, the correlation between FPED and FROH from sequence data increased slightly for all breeds. We should also point out that a very long stretch of homozygosity using marker data might not actually be completely homozygous and therefore, higher density data was suggested to be used to detect selective sweeps through runs of homozygosity [45]. Sørensen et al. [7] has estimated inbreeding in Danish Dairy Cattle Breeds and our estimates FPED are lower than theirs. This is because our sampled animals for sequencing are founder and older animals compare to the other study where they used all animals [7].

Estimates of inbreeding coefficients differed with methods. Inbreeding coefficients estimates from methods using allele frequencies, i.e., FGRM, FHOM and FUNI, showed considerable variation across data type and breeds. These estimators were sensitive to allele frequencies compared to ROH estimators, especially for populations with divergent allele frequencies (e.g., Fig. 2; RDC population). The estimates of genomic inbreeding coefficients are dependent on the allele frequencies in the base population [40].

Fig. 2
figure 2

Minor allele frequency distribution for HOL, JER, and RDC bulls from sequence data. Minor allele frequency in HOL (yellow), JER (blue), and RDC (green) bulls against the minor allele frequency among all loci

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In order to explore the reasons about the various correlations between inbreeding coefficients estimates using allele frequencies, FGRM, FHOM and FUNI were plotted against the allele frequency changing from 0 to 1 when the number of copies of reference alleles for ith SNP is 0, 1 or 2 (Figs. 3, 4 and 5). When a locus is homozygous for either the reference alleles or the non-reference alleles with the allele frequency ranging from 0 to 1, FGRM ranged from -1 to infinity, FHOM has a constant value of 1 and FUNI ranged from 0 to infinity (Figs. 3 and 5). FHOM gave constant estimates for homozygous genotypes, regardless of the allele frequency (Figs. 3 and 5). When the allele frequency of the non-reference alleles is smaller than 0.2 or larger than 0.8, FGRM was less than 0 (Figs. 3 and 5). When the allele frequency of the non-reference allele was between 0.2 and 0.5 or when the allele frequency of the reference allele was between 0.5 and 0.8, FGRM become positive and ranges from 0 to 1 (Figs. 3 and 5).

Fig. 3
figure 3

Inbreeding coefficients FGRM, FHOM and FUNI against the reference allele frequency changing from 0 to 1 when the number of copies of reference alleles for the ith SNP is 0. Black line represents FGRM; red line represents FHOM and blue line represents FUNI

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Fig. 4
figure 4

Inbreeding coefficients FGRM, FHOM and FUNI against the reference allele frequency changing from 0 to 1 when the number of copies of reference alleles for the ith SNP is 1. Black line represents FGRM; red line represents FHOM and blue line represents FUNI

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Fig. 5
figure 5

Inbreeding coefficients FGRM, FHOM and FUNI against the reference allele frequency changing from 0 to 1 when the number of copies of reference alleles for the ith SNP is 2. Black line represents FGRM; red line represents FHOM and blue line represents FUNI

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For a heterozygous locus with an allele frequency ranging from 0 to 1, FGRM and FHOM ranged from minus infinity to plus infinity, and FUNI has a constant value of 0 (Fig. 4). If the allele frequency was smaller than 0.2 or larger than 0.8 FGRM become very large positive whereas FHOM become a large negative. FHOM was always negative, and FGRM was always positive (Fig. 4). Thus, when a population has a high level of heterozygosity and some rare alleles with small frequency, FGRM would yield large positive inbreeding coefficients, which can be misleading. This result explains why FGRM was positive in the RDC breed (Table 2): this population had a higher level of heterozygosity than HOL and JER. FUNI gave a stable value of 0 when the locus was heterozygous and therefore was robust to allele frequency (Fig. 4).

The correlation between the three estimators FGRM, FHOM and FUNI was computed for each of the three genotypes (i.e., homozygotes for allele 1, homozygotes for allele 2 and heterozygotes) for comparison between FGRM, FHOM and FUNI when the allele frequency was varied between 0 and 1 (Fig. 6). Correlations reached the maximal value (i.e., 1) when the allele frequencies were 0.5. When the allele frequencies were extremely high or low, correlations between estimators became low, especially the correlation between FGRM and FHOM (0.27). The correlation plot (Fig. 6) reflected a similar result as those in Figs. 3, 4 and 5. Therefore, when computing inbreeding coefficients using allele frequencies, populations with different allele frequencies might have very different inbreeding coefficients and the correlations between those inbreeding coefficients might be very low, with different allele frequencies.

Fig. 6
figure 6

Correlations between FGRM and FHOM, FGRM and FUNI, and FHOM and FUNI when reference allele frequency changes between 0 and 1. Black line represents correlation between FGRM and FHOM against reference allele frequencies; red line represents correlation between FGRM and FUNI against reference allele frequencies and blue line represents correlation between FHOM and FUNI against reference allele frequencies

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The comparison between FGRM and other estimators showed a very low correlation and FGRM was mostly negatively correlated with other estimators. FHOM based on excess of homozygosity was positively correlated with other estimators and was relatively highly correlated with FROH detected from 50k and sequence data. FUNI based on correlations between uniting gametes estimated from 50k data generally was negatively correlated with other estimators. However, with increasing marker density, the correlation between FUNI and other estimators became positive for the HOL and RDC populations. Surprisingly, when using sequence data, FUNI was highly correlated with other estimators, especially FROH, detected from sequence data (~0.95) for HOL. This correlation may have resulted from the nature of the estimators: FROH uses only runs of homozygosity, whereas the other estimators (to some extent) capture all of the homozygosity. This high correlation for FUNI and FROH compared with low correlation between FGRM and FROH might also be explained by the algorithms: FGRM = (1 + F)-1 and F is the correlation between uniting gametes. This estimator has only sampling on the F-term, whereas in the FGRM estimator there is also sampling variance on the “1”, which creates additional sampling variance.

It is known that RDC is an admixed breed with introgressed haplotypes from Old Danish Red, Holstein and Brown Swiss breeds. HOL and JER are relatively pure breeds and more inbred than RDC (Zhang Q, Guldbrandtsen B, Bosse M, Lund MS, Sahana G. Runs of homozygosity and distribution of functional variants in the cattle genome. BMC Genomics (in press)). Therefore, minor allele frequencies tend to be lower in HOL and JER breeds than in RDC. FGRM is negatively correlated with other estimators for all three breeds. FHOM becomes negative for RDC, which is likely due to the admixture present in RDC. Therefore, it appears that FGRM tends to be less accurate for populations with a low minor allele frequency and that FHOM tends to be less accurate for populations with a higher level of heterozygosity. This argument is supported by our results that the three inbreeding estimators FGRM, FHOM and FUNI were most closely correlated with each other when the allele frequency is approximately 0.5 (Figs. 3, 4 and 5). Therefore, the three estimators FGRM, FHOM and FUNI depend strongly on the estimation of allele frequencies in the population, unlike FROH. However, here we only took one locus as an example to study the impact of allele frequencies on three estimators FGRM, FHOM and FUNI.

Conclusion

In this study, we compared different estimators of inbreeding coefficient with different types of data (pedigree, 50k SNP chip genotypes and full sequence data). Methods based on GRM, excess of homozygosity and the correlation between uniting gametes were observed to be sensitive to allele frequencies in the base population. The estimator based on pedigree data was moderately correlated with estimators based on ROH when a pedigree is relatively complete. Estimators based on ROH from SNP chip genotypes and full sequence directly reflect homozygosity on the genome, and have the advantage of not being affected by estimates of allele frequency or incompleteness of the pedigree. Inbreeding estimated from ROH was shown to be affected by the marker density used. Using sequence data, we obtained a full picture of the distribution of ROH on the genome, including short and medium length ROH that reflect ancient inbreeding regions which are possibly IBD. Detecting ROH based on high-density or 50k chip data was shown to give estimates most closely related to ROH from sequence data. However, more than 50k genotypes are required to accurately detect short ROH that are most likely identical by descent (IBD).

Availability of supporting data

Data used in this study are from the 1000 Bull Genome Project (Daetwyler et al. 2014 Nature Genet. 46:858–865). Whole genome sequence data of individual bulls of the 1000 Bull Genomes Project are already available at NCBI using SRA no. SRP039339 (http://www.ncbi.nlm.nih.gov/bioproject/PRJNA238491).