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A quantitativelymodeled homozygosity mapping algorithm, qHomozygosityMapping, utilizing whole genome single nucleotide polymorphism genotyping data
BMC Bioinformatics volume 11, Article number: S5 (2010)
Abstract
Homozygosity mapping is a powerful procedure that is capable of detecting recessive diseasecausing genes in a few patients from families with a history of inbreeding. We report here a homozygosity mapping algorithm for highdensity single nucleotide polymorphism arrays that is able to (i) correct genotyping errors, (ii) search for autozygous segments genomewide through regions with runs of homozygous SNPs, (iii) check the validity of the inbreeding history, and (iv) calculate the probability of the diseasecausing gene being located in the regions identified. The genotyping error correction restored an average of 94.2% of the total length of all regions with run of homozygous SNPs, and 99.9% of the total length of them that were longer than 2 cM. At the end of the analysis, we would know the probability that regions identified contain a diseasecausing gene, and we would be able to determine how much effort should be devoted to scrutinizing the regions. We confirmed the power of this algorithm using 6 patients with Siiyamatype α1antitrypsin deficiency, a rare autosomal recessive disease in Japan. Our procedure will accelerate the identification of diseasecausing genes using highdensity SNP array data.
Background
Identification of the genetic factors underlying disease causation provides crucial information for disease prevention and treatment. Nevertheless, genetic factors have not yet been elucidated for many diseases [1, 2].
Homozygosity mapping [3] enables the detection of recessive diseasecausing genes in a few patients from families with a history of inbreeding; this mapping technique is especially useful for the detection of rare genes. With this technique, chromosomal segments in which all polymorphic markers are homozyogous are considered autozygous segment (AS) [4]. If a patient's coefficient of consanguinity is F, and the frequency of the diseasecausing gene in the population is p, then the chance that the recessive diseasecausing gene is located in an AS (P_{ AS }) is.
[3]
If a patient is from an inbred family (i.e., F is large) and the disease is rare (i.e., p is small), then P_{ AS }≈ 1, indicating that the gene is located in an AS. There are implementations that utilize singlenucleotide polymorphism (SNP) genotyping data obtained by highdensity arrays [5, 6]. The usable implementation should (i) correct genotyping errors because thousands of SNPs are mistyped per highdensity SNP array, adversely affecting the homozygosity mapping analysis; (ii) search for ASs genomewide; (iii) check the validity of the inbreeding history, which is vital for homozygosity mapping but is often erroneous, and (iv) calculate the probability of the diseasecausing gene being located in the regions identified. At the end of the analysis, we would know the probability that regions identified contain a diseasecausing gene, and we would be able to determine how much effort should be devoted to scrutinizing the regions.
In the current study, we present an algorithm that implements the capabilities described in the above paragraph. We confirmed the power of this algorithm using 6 patients with Siiyamatype α1antitrypsin deficiency, a rare autosomal recessive disease in Japan [7, 8]. The preliminary version of the algorithm described here has been used to prove that the SLC34A2 gene is responsible for pulmonary alveolar microlithiasis [9]; the current version has been used to show that the OPTN gene is responsible for amyotrophic lateral sclerosis [10].
Implementation
Crossover model
We used the Haldane's Poisson process model for the occurrence of crossovers and performed all calculations based on this model [11]. Information on SNPs used by Affymetrix's GenomeWide Human SNP Array 6.0 (hereafter referred to as SNP Array 6.0) was summarized in the annotation file, [12], in which the genetic distance from the telomere of the short arm of a chromosome to each SNP was obtained by interpolation using the sexaveraged data published by deCODE Genetics [13]. We restricted our analysis to a total of 890,625 autosomal SNPs with assigned dbSNP refIDs [14].
Monte Carlo simulation
The average number, the average length, and the maximal length of the ASs derived from a common ancestor were calculated for a range of m + n values (Figure 1A) using a Monte Carlo simulation. The trial was repeated until we observed 100,000 events in which at least 1 AS appeared in the autosomal region.
The length of AS
The subject is removed from the common ancestor m generations on the paternal side and n generations on the maternal side (Figure 1A). Assuming that the length of each autosome is infinite, the length of AS conforms to an exponential distribution with a probability density function of
In actuality, the autosomes have finite length; however, equation 2 provides a good approximation when the length of an AS is much shorter than the length of an autosome.
RHS (run of homozygous SNPs), false negative, type A false positive and type B false positive
An RHS is defined as a run of homozygous SNPs with a genetic length greater than the RHS cutoff value (Figure 1B). All SNPs in an AS are homozygous, and therefore an RHS suggests the presence of an AS. We defined 3 types of errors. False negatives are ASs that are not contained in RHSs. Type A false positives are RHSs that do not contain ASs. Type B false positives are the spaces within an RHS that do not contain an AS. The false negative rate (R_{ false negative }) is the ratio of false negatives to the total length of the AS. The false positive rate (R_{ false positive }) is the ratio of false positives (the type A false positives plus the type B false positives) to the total length of the autosomes.

(1)
R_{ false negative }, the ratio of the total length of false negatives to the total length of the AS
According to the equation 2,

(2)
R_{ Type A false positive }, the ratio of the total length of type A false positives to the total length of the autosomes
Given that N_{ SNP } is the total number of SNPs on a genotyping array, and P_{ n } and Q_{ n } are the frequencies of the major and minor alleles for the nth SNP, respectively, then the average frequencies of the major alleles ({\overline{F}}_{majorallele}) and the minor alleles ({\overline{F}}_{minorallele}) are
respectively. The numbers of homozygous SNPs (N_{ homozygousSNP }) and heterozygous SNPs (N_{ heterozygousSNP }) are approximated by
where N_{ pt } is the number of SNPs successfully genotyped. Assuming that heterozygous SNPs are randomly located, then the length between 2 heterozygous SNPs conforms to an exponential distribution with a probability density function of
where L_{ autosome } is the entire length of the autosomes. Therefore, at a cutoff value of c cM,

(3)
R_{ Type B false positive }, the ratio of the total length of type B false positives to the total length of the autosomes
R_{ Type B false positive } is not calculated mathematically but is calculated according to the actual data. An RHS containing an AS is expected to have type B false positives with an average length of \frac{1}{2}\times \frac{{L}_{autosome}}{{N}_{heterozygousSNP}} on each end. It is impossible to distinguish RHSs that contain ASs from those that do not. We calculated R_{ Type B false positive } under the assumption that every RHS contains an AS. Therefore, the R_{ Type B false positive } calculation results in an overestimation, which we consider better than an underestimation for determination of the appropriate RHS cutoff. Therefore,

(4)
R_{ false positive }, the ratio of the total length of false positives to the total length of the autosomes
\begin{array}{c}{R}_{falsepositive}={R}_{TypeAfalsepositive}+{R}_{TypeBfalsepositive}\\ =(1+\frac{{N}_{heterozygousSNP}}{{L}_{autosome}}c){e}^{\frac{{N}_{heterozygousSNP}}{{L}_{autosome}}c}+\frac{numberofRHS}{2}\times \frac{{L}_{autosome}}{{N}_{heterozygousSNP}}.\end{array}(7)
Probability that a diseasecausing gene is contained in RHSs, or the overlap of RHSs
The probability that RHSs obtained contains a diseasecausing gene is calculated using equation 1.
Here, F is the coefficient of consanguinity and is calculated by
The probability that the overlap of RHSs among multiple patients contain the gene is calculated by
Human Subjects and genotyping
This study was approved by the Institutional Review Boards of Saitama Medical University and Juntendo University. After obtaining written informed consent, DNA samples from 6 patients with α1antitrypsin deficiency were purified from peripheral blood. These patients were not related and lived in different areas of Japan. Patients 15 were from families with a history of inbreeding because their parents were first cousins. Patient 6 did not have any family history of inbreeding. These 6 patients were genotyped using the SNP Array 6.0. The genotyping data for 86 HapMap JPT were available in the HapMap3 draft release 2 http://www.hapmap.org, and were downloaded from the Wellcome Trust Sanger Institute web site http://www.sanger.ac.uk/humgen/hapmap3/. The genotyping data for NA18987, a subject in HapMap JPT, was also distributed from Affymetrix and was used in the current study.
Genotyping error correction
Genotyping errors may convert homozygous SNPs to heterozygous SNPs and erroneously terminate an RHS, resulting in the failure to detect a portion of an RHS. According to Affymetrix, SNP Array 6.0 has an accuracy of > 0.997, implying that the genotyping error rate (P_{ genotypingError }) may be 0.003 at maximum. A mistyped heterozygous SNP occurring in an RHS is separated by a large distance from neighboring heterozygous SNPs (Figure 1C). Therefore, if a heterozygous SNP is separated from neighboring SNPs by a distance that is rarely observed by chance, we speculated that the SNP was mistyped. Using equation 4, we calculated the probability of a heterozygous SNP being separated from neighboring SNPs at the observed distance (P_{ distanceOccurredByChance }). A SNP with P_{ distanceOccreceByChance } < 0.01 was considered a mistyped SNP and these data were removed. This algorithm may erroneously remove 20 correctly genotyped heterozygous SNPs (N_{ homozygousSNP } x P_{ genotypingError } x 0.01) from a single SNP array analysis data, which we considered acceptable.
Statistical analysis
The number of patients and controls who shared an RHS at each SNP position was compared. The assumption was made that
has a standard normal distribution, where {\widehat{p}}_{1}*=\frac{{x}_{1}+0.5}{{n}_{1}+1}, {\widehat{p}}_{2}*=\frac{{x}_{2}+0.5}{{n}_{2}+1}, {\widehat{p}}^{*}=\frac{{x}_{1}+{x}_{2}+0.5}{{n}_{1}+{n}_{2}+1}. Here, x_{1} and x_{2} represent the numbers of patients and controls sharing RHSs, respectively, and n_{1} and n_{2} represent the total numbers of patients and controls, respectively. The P value was calculated by
Computer program
The computer program was written in the ANSI standard C programming language. The program was compiled by the GNU C compiler 4.2 and run on a MacBook Pro (CPU: 2.53 GHz Intel Core 2 Duo, 4 GB RAM) computer. The command line programs and the programs equipped by graphic user interface are both available from our web site at http://www.hhanalysis.com.
Result
Strategy
Our aim was to establish an algorithm for homozygosity mapping that uses SNP genotyping data obtained by highdensity arrays, is equipped by a powerful genotyping error correction algorithm, detects ASs genomewide, allows investigation into the family inbreeding history, and is able to calculate the probability that the identified regions contain the target gene.
The algorithm searches for the ASs (Figure 1A, B(i)) through runs of homozygous SNPs, or RHSs, that are formed by consecutively homozygous SNPs and are longer than the RHS cutoff value (Figure 1B(ii)). RHSs are presumably the autozygous segments (ASs). Three types of errors were defined; false negative, type A false positive, and type B false positive (Figure 1B(iii)). The main determinants of the false negative rate (R_{ false negative }), which is the ratio of the total length of false negatives to the total length of ASs, are the number of SNPs investigated and the genotyping error rate. The main determinants of the false positive rate (R_{ false positive }), which is the ratio of the total length of type A false positives plus type B false positives to the entire length of the autosomes, are the positioning of SNPs, local haplotype block structure [15], and population substructure [16].
To attain the aims stated above while avoiding the influence of these errors, our algorithm had the following steps: Step (a) determine an appropriate RHS cutoff value based on the Haldane's recombination model; Step (b) perform genotyping error correction; Step (c) detect RHSs; Step (d) obtain the overlaps of RHSs among patients; and Step (e) correct false positives by a casecontrol approach. The validity of the family history is checked at Step (c). We used 5 patients with Siiyamatype α1antitrypsin deficiency, a rare disease in Japan, to verify our strategy. Analyses performed in the Result section can be reproduced using the program contained in additional file 1 according to the tutorial also contained in the additional file 1.
Determination of the RHS cutoff
The expected false negative and false positive rates for the SNP Array 6.0 from the Haldane's model were calculated by using equation 3 and 7 [Step (a)] (Figure 2A). We gave the priority to reducing the false positive rate than to reducing the false negative rate, because we empirically determined that it simplified the analysis. We chose 0.6 cM as the RHS cutoff value, at which the false negative rate was 0.0006 and the false positive rate was 0.0029. The probability that the RHSs contained the diseasecausing gene (P_{ GeneIsInRHS }) at this condition was calculated using equation 8 (Figure 2B).
Genotyping error correction
The power of the genotyping error correction algorithm was investigated using genotyping data for subject NA18987 (female) from HapMap JPT. The subject was independently genotyped in HapMap draft 3 and by Affymetrix, and data were made public from both sources. A comparison of these 2 datasets revealed that the genotyping results for 701,753 SNPs matched between these 2 sources, and they were therefore considered highly accurate. Using the matched data, RHSs were obtained with an RHS cutoff value of 0.6 cM (Figure 3A). The presence of a long RHS (36.2 cM at maximum) suggested that she had a family history of inbreeding, as described later. Considering the fact that the manufacturer (Affymetrix) claimed that the genotyping error rate for the SNP Array 6.0 is less than 0.003, we randomly introduced errors into selected 2,105 SNPs (701,753 SNPs × 0.003) and obtained RHSs. These error hampered the detection of RHSs, especially the long ones (Figure 3B). Following application of the genotyping error correction algorithm (Figure 1C), RHSs were restored (Figure 3C). The same trial repeated 100 times revealed that the genotyping error correction restored an average of 94.2% of the total length of all RHSs, and 99.9% of the total length of RHSs that were longer than 2 cM. This indicated that 99.9% of the total length of ASs resulting from first or second cousin marriages would be correctly detected as RHSs after the correction. The total length of the regions that were erroneously detected as RHSs amounted to only 0.2% of the total length of the autosomes. These results indicated that the performance of the genotyping error correction algorithm was excellent.
RHSs in the patients
We applied the genotyping error correction algorithm to the data for 5 patients with Siiyamatype α1antitrypsin deficiency [Step (b)], and then obtained RHSs [Step (c)] (Figure 4AE). All patients had long RHSs, which were likely to be the result of firstcousin marriages.
Statistics of AS
We investigated whether the RHSs obtained for each patient were consistent with family history [Step (d)]. We focused on the size of the longest AS because they are an index of the most recent occurrence of inbreeding in the patient's family (equation 2). The distribution of the length of the longest AS is calculated by a Monte Carlo simulation (Figure 4F). From this distribution we are able to say that the family history of a first cousin marriage (m + n = 6) is unlikely when the longest RHS is less than 20.9 cM. The size of the longest RHS for Patients 15 were consistent with what expected from their family histories (Table 1).
Overlap of RHSs
We then obtained the overlaps of the RHSs for Patients 15 whose parents were first cousins [Step (d)] (Figure 5A). The probability that these regions contained the diseasecausing gene (P_{ GeneIsInOverlap }) was calculated by equation 10 and is shown in Figure 5B. The prevalence of Siiyamatype α1antitrypsin deficiency is less than 1 in a million in Japan, and the frequency of the gene is suspected to be less than 0.001 in the general population, indicating that the overlaps likely contained the diseasecausing gene.
Some of the autosomal regions are prone to type A or type B false positives, and thus are likely to appear as an overlap [Step (e)]. To prioritize regions for indepth analysis, we performed a casecontrol study using 86 HapMap JPT subjects as controls. One overlap had the largest log_{10}(P) value (16.47) and was considered to be the candidate region (Figure 5C). This region (between rs10134551 and rs910349) had a genetic length of 1.44 cM, and contained 15 genes (Table 2), one of which was the diseasecausing gene for Siiyamatype α1antitrypsin deficiency, SERPIN1.
A patient without family history of inbreeding
We occasionally encounter patients who do not have a family history of inbreeding while searching for a recessive diseasecausing gene. Data from such patients are not used in the main analysis, but these data may be used for prioritizing the overlaps of RHSs as obtained in Figure 5 for an indepth search. Patient 6 had Siiyamatype α1antitrypsin deficiency but did not have a family history of inbreeding. The length of the longest RHS (6.8 cM, Figure 6A) was outside of the 95% range for the Japanese population (Figure 4F, rightmost bar and whisker). We reasoned that the patient's family might have had forgotten inbreeding history, and that the RHSs for the patient may have a high probability of containing the diseasecausing gene. This was indeed the case; addition of the data from Patient 6 excluded several overlapped regions (Figure 6B, compare with Figure 5A) and increased log_{10}(P) (Figure 6C, compare with Figure 5C), although the list of the genes was the same as Table 2. If the length of the longest RHS suggested a hidden inbreeding history, the data for subjects without an inbreeding history could be used to prioritize some RHS overlaps for an indepth search.
Discussion
In the current report, we described the quantitativelymodeled homozygosity mapping algorithm that uses high density array SNP genotyping data.
Homozygosity mapping is simple in principle, but many pitfalls were discovered when it was actually applied. Problems that included (i) unexpected allelic heterogeneity, (ii) identification of a homozygous identicalbydescent (IBD) region to the disease locus, (iii) underestimation of the extent of inbreeding, were pointed out in the analyses using microsatellite markers [17] and are still observed in the analyses using SNPs. Moreover, use of highdensity SNP arrays introduced a novel problem, (iv) a large number of mistyped SNPs. Although the genotyping error rate is low for highdensity arrays, the huge number of SNPs in these arrays inevitably produces a large number of mistyped SNPs. Even a single mistyped SNP erroneously terminates an RHS, making the detection of large RHSs difficult. Our algorithm has overcome all these problems: problem (i) is solved by using highdensity SNP arrays, problem (ii) by casecontrol approach, problem (iii) by identifying ASs as RHSs and calculating F by the total length of RHSs divided by the total length of the autosomes, and problem (iv) by applying genotyping error correction algorithm.
As stated as Problem (ii) above, we observed some autosomal regions had a high probability of having RHSs. This may be caused by SNP positioning, local haplotype block structure, or population substructure. The effect of them was eliminated by using a casecontrol approach, which is performed in the order that (a) obtain overlap of RHS among patients, and (b) perform a casecontrol analysis targeting obtained overlaps.
Homozygosity mapping has power to identify a diseasecausing gene in as few as 3 patients, and we have indeed identified the SLC34A2 gene in pulmonary alveolar microlithiasis and the OPTN gene in the amyotrophic lateral sclerosis both in 3 patients [9, 10]. Amyotrophic lateral sclerosis has multiple causative genes. In the latter report, we were able to identify one of the genes by investigating each combination of 3 patients from 7 patients with a history of inbreeding, seeking for 3 patients harboring the same diseasecausing gene. Our algorithm worked fine in this approach. During the process, it was quite helpful that the algorithm provided the probability that the identified regions contain the diseasecausing gene, which determined how much effort should be further devoted. To our knowledge, the algorithm presented in the current study is the first to provide this information.
Conclusions
We described an algorithm that enables homozygosity mapping to be performed based on a quantitative model using SNP genotyping data. Our procedure will accelerate the identification of diseasecausing genes using highdensity SNP array data.
Availability and requirements
Project name: qHomozygosityMapping
Project home page: http://www.hhanalysis.com
Operating system(s): Mac, Linux and Windows.
Programming language: C
License: GNU GPL.
Any restrictions to use by nonacademics: The software is for academic purpose only.
Funding
This work is supported in part by the grantinaid for scientific research (No. 18390242) from the Japan Society of Promotion of Science, and in part by the grantsinaid for Health and Labor Science (Nos. H22NanchiIppan005 and H20NanchiIppan023) from the Ministry of Health, labor and Welfare, Japan.
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Acknowledgements
The authors thank Ms. Tomoko Hirata for her technical assistance.
This article has been published as part of BMC Bioinformatics Volume 11 Supplement 7, 2010: Ninth International Conference on Bioinformatics (InCoB2010): Bioinformatics. The full contents of the supplement are available online at http://www.biomedcentral.com/14712105/11?issue=S7.
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Authors' contribution
Huqun, S.F., H.M., T.T., T.S., M.K., H.K., Y.O., and K.S. tested the programs, did genetic analyses and provided ideas to improve the program. K.S. collected the patients' samples. K.H. provided basic ideas, wrote the program, and prepared manuscript.
Shunichiro Fukuyama and Koichi Hagiwara contributed equally to this work.
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Huqun*, Fukuyama, Si., Morino, H. et al. A quantitativelymodeled homozygosity mapping algorithm, qHomozygosityMapping, utilizing whole genome single nucleotide polymorphism genotyping data. BMC Bioinformatics 11 (Suppl 7), S5 (2010). https://doi.org/10.1186/1471210511S7S5
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DOI: https://doi.org/10.1186/1471210511S7S5
Keywords
 Amyotrophic Lateral Sclerosis
 Homozygosity Mapping
 Genotyping Error Rate
 Heterozygous SNPs
 Cousin Marriage