A method for mapping ancestral haplotype genes in a deep cross

Abstract

The application belongs to the technical field of biological information, and particularly relates to a method for locating ancestral haplotype genes for deep hybrid lines. In the application, the ancestral genotype of a sample to be analyzed is obtained, and then a genomic segment significantly associated with a phenotype is obtained. Then, whole genome analysis is performed. Compared with the prior art, the method for locating ancestral haplotype genes for deep hybrid lines can effectively reduce false positives, and can also be used for heterosis analysis, capture of dominant and epistatic effects between genetic markers and the like.

Description

Technical Field

[0001] This invention belongs to the field of bioinformatics technology, specifically relating to a method for ancestral haplotype gene localization for deep hybrid lines. Background Technology

[0002] Deep hybridization refers to the process of crossing two parental populations with significant differences to produce offspring, which are then continuously passed down through generations. During this continuous transmission, new recombinations are constantly generated, thereby improving the accuracy of gene mapping.

[0003] Local ancestry analysis aims to determine the ancestral origin of haplotypes in offspring samples using single nucleotide genetic markers. This analytical method is of great significance for elucidating the ancestral components, key gene localization, and the source of ancestral heterosis in offspring samples produced by hybridization breeding.

[0004] Genome-wide association studies (GWAS) can effectively capture the correlation between phenotypes and genetic markers. With advancements in sequencing technology and reduced costs, we can obtain a large number of high-density single nucleotide mutations (Snucleotide mutations). Compared to traditional methods targeting Snucleotide mutations, GWAS grouping consecutive Snucleotide mutations into haplotypes can reduce false positives and capture short-range gene interactions. However, in some cases, the false positive rate remains high, and there are issues such as low captured heritability and inability to resolve heterosis. Furthermore, GWAS can only capture additive effects between genetic markers and cannot capture dominant or epistatic effects.

[0005] Therefore, there is an urgent need for a new method for ancestral haplotype gene mapping in deep hybrid lines, which can effectively reduce false positives and can also be used for heterosis analysis, capturing dominance and epistasis effects between genetic markers, etc. Summary of the Invention

[0006] To address the aforementioned issues, the first technical solution of this application discloses a method for ancestral haplotype gene localization for deep hybrid lines, including obtaining ancestral genotypes and gene localization. The method for obtaining ancestral genotypes involves dividing the whole genome of the offspring to be analyzed into N haplotypes, inferring the origins of its two ancestors based on the haplotypes, and obtaining the whole genome ancestral genotype.

[0007] Furthermore, the gene mapping includes genome-wide association analysis, specifically: calculating the kinship matrix among the offspring of deep hybrid lines based on the ancestral genotypes of the whole genome, performing genome-wide association analysis based on the ancestral genotypes and kinship matrix, and using the false discovery rate correction method for multiple correction to obtain the genomic segments that are significantly associated with the phenotype to be analyzed.

[0008] According to the gene mapping method, the genome-wide association analysis was performed using a mixed linear model, which is as follows:

[0009] ;

[0010] Where y represents the phenotype to be analyzed in the progeny population of the deep hybrid line, Q is the covariance matrix, including sex and batch fixed effects, α is the effect vector of the covariate, X is the ancestral haplotype encoding vector of the progeny of the deep hybrid line, β is the effect size of the haplotype of the target locus, g is the kinship matrix of each pair of samples, and e is the residual vector.

[0011] Furthermore, the gene mapping includes the mapping of heterosis genomic segments of superior and middle parents, specifically: based on the whole genome ancestral genotype and corrected phenotype, the heterosis of superior and middle parents is detected using nonparametric Kruskal-Wallis and Steel-Dwass tests, respectively, to obtain heterosis genomic segments of superior and middle parents that are significantly associated with the phenotype.

[0012] Furthermore, the gene localization includes epistatic interaction genomic segment localization, specifically: based on the whole genome ancestral genotype and corrected phenotype, logistic regression is used to detect epistatic interactions, and a mixed linear model is used to perform simple effect comparisons to test the significance of interactions between genotypes, thereby obtaining epistatic interaction genomic segments that are significantly associated with the phenotype.

[0013] Furthermore, the residuals after deducting fixed effects in the phenotype to be analyzed are used as the corrected phenotype.

[0014] The present invention has the following beneficial effects: By obtaining the ancestral genotype of the sample to be analyzed, this application further obtains genomic segments with significant phenotypic association; then whole-genome analysis is performed. Compared with existing methods, the ancestral haplotype gene localization method for deep hybrid lines provided by this application can effectively reduce false positives, and can also be used for heterosis analysis, capturing the dominance and epistatic effects between genetic markers, etc. Attached Figure Description

[0015] Figure 1 Manhattan plot for genome-wide association analysis of ancestral haplotypes;

[0016] Figure 2 Corrected phenotypic and genotypic distribution maps for superior parent hybrid vigor and mid-parent hybrid vigor;

[0017] Figure 3 This refers to the interaction between two ancestral genotypes. Detailed Implementation

[0018] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] Unless otherwise specifically stated, the numerical expressions and values ​​of the steps set forth in these embodiments do not limit the scope of this application. Techniques and methods known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques and methods should be considered part of the specification. In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values.

[0020] The first embodiment of this application discloses a method for ancestral haplotype gene mapping for deep hybrid lines, including obtaining ancestral genotypes and gene mapping. The method for obtaining ancestral genotypes is as follows: the whole genome of the offspring to be analyzed is divided into N haplotypes, and the origins of its two ancestors are inferred based on the haplotypes to obtain the whole genome ancestral genotype.

[0021] In this embodiment, based on haplotype data composed of consecutive single nucleotide mutations, a local ancestry inference tool is used to obtain the two ancestral origins of the haplotypes of the offspring samples to be analyzed, and the ancestral haplotype information is encoded to obtain the ancestral genotype. Specifically, through the local ancestry inference method, all offspring samples to be analyzed are divided into several haplotypes across their entire genome using the same fixed number of consecutive single nucleotide mutation sites.

[0022] The haplotypes of the offspring samples to be analyzed from diploid deep hybrid lines are used to infer the origins of their two ancestors. Specifically, ancestor 1 and ancestor 2 are represented by 0 and 1, respectively.

[0023] The ancestral origin of each haplotype in each offspring sample to be analyzed can be represented by 0, 1, 2, also known as the ancestral genotype, indicating the number of copies of the haplotype in the diploid plant / animal offspring sample from ancestor 1. Therefore, each offspring sample can form an ancestral haplotype vector according to the haplotype order, and several offspring samples form an ancestral haplotype matrix. Among them, haplotypes 0 and 2 in the offspring to be analyzed represent ancestral homozygotes, and 1 represents ancestral heterozygotes.

[0024] It is understandable that when classifying haplotypes, it is only necessary to classify them according to the fixed frame length of the whole genome. For example, in Example 1 below, the chicken whole genome is divided into fixed frame lengths of 5 single nucleotides. In actual application, the fixed frame lengths can also be divided into 6, 10, or other single nucleotides, depending on the actual situation, so as to classify N haplotypes (N = number of single nucleotides in the whole genome / number of single nucleotides in the fixed frame length).

[0025] It should be noted that after obtaining the whole-genome ancestral genotype, genome-wide association studies (GWAS), heterosis genomic region mapping for superior and mid-parents, and epistatic interaction genomic region mapping can be performed. Compared with existing base-based genotype mapping methods, the genome-wide ancestral genotype-based mapping method of this application can effectively reduce false positives for GWAS, capture dominant effects based on heterosis analysis for heterosis genomic regions for superior and mid-parents, and capture epistatic effects between genetic markers for epistatic interaction genomic region mapping. Therefore, the genome-wide ancestral genotype-based mapping method of this application has a wider range of applications and higher accuracy.

[0026] In one specific implementation, the gene localization is a genome-wide association analysis, which specifically involves: calculating the kinship matrix among the offspring of deep hybrid lines based on the ancestral genotypes of the whole genome, and performing a genome-wide association analysis based on the ancestral genotypes and the kinship matrix to obtain genomic segments that are significantly associated with the phenotype to be analyzed.

[0027] In this embodiment, whole-genome ancestral genotypes are used to calculate the phylogenetic matrix of the samples to be analyzed. For the phenotypic data to be analyzed, a mixed linear model is used to perform genome-wide association analysis based on whole-genome ancestral genotypes and the phylogenetic matrix. Multiple correction is performed using the false discovery rate correction method to calculate the genomic regions significantly associated with the phenotype. The mixed linear model is as follows:

[0028] ;

[0029] Where y represents the phenotype to be analyzed in the progeny population of the deep hybrid line, Q is the covariance matrix, including fixed effects such as sex and batch, α is the effect vector of the covariates, X is the ancestral haplotype coding matrix of the progeny of the deep hybrid line, β is the effect size of the haplotype at the target locus, g is the kinship matrix of each pair of samples, and e is the residual vector.

[0030] It should be noted that in the method of genome-wide association analysis based on ancestral genotype in this embodiment, the kinship matrix g calculated based on the ancestral genotype is first used. Compared with the prior art, the calculation model of this application can effectively reduce false positives.

[0031] In one specific implementation, the gene mapping includes the mapping of heterotic genomic segments of superior and middle parents, specifically: based on the whole genome ancestral genotype and corrected phenotype, the heterotic genomic segments of superior and middle parents are detected using nonparametric Kruskal-Wallis and Steel-Dwass tests, respectively, to obtain heterotic genomic segments of superior and middle parents that are significantly associated with the phenotype.

[0032] In this embodiment, the corrected phenotype is the residual after deducting fixed effects from the phenotype to be analyzed. For example, a linear regression model is established with the phenotype to be analyzed as the dependent variable and fixed effects such as sex and batch number as independent variables, and the residual of the regression model is extracted as a new phenotype to become the corrected phenotype. Each ancestral haplotype of the sample to be analyzed is encoded into three types: 0, 1, and 2, where 0 represents a homozygous ancestral haplotype of ancestor 1, 1 represents a heterozygous ancestral haplotype of ancestor 1 and ancestor 2, and 2 represents a homozygous ancestral haplotype of ancestor 2. The mean of the corrected phenotype for each haplotype and each ancestral genotype is calculated at the progeny population level. If the mean of the corrected phenotype of ancestral genotype 1 is higher than that of ancestral genotypes 0 and 2, the haplotype is considered a potential heterozygous haplotype of superior parent; if the mean of the corrected phenotype of ancestral genotype 1 is higher than that of ancestral genotypes 0 and 2, the haplotype is considered a potential heterozygous haplotype of middle parent. The Kruskal-Wallis test was used to examine whether there were significant differences between different genotypes and the corrected phenotypes in the overall population for potential superior heterotic haplotypes and potential mid-parent heterotic haplotypes, respectively. The Steel-Dwass test was used to examine which specific groups had differences.

[0033] In one specific implementation, the gene localization includes epistatic interaction genomic segment localization, specifically: based on the whole genome ancestral genotype and corrected phenotype, logistic regression is used to detect epistatic interactions, and a mixed linear model is used to perform simple effect comparisons to test the significance of interactions between genotypes, thereby obtaining epistatic interaction genomic segments that are significantly associated with the phenotype.

[0034] In this embodiment, the corrected phenotype is consistent with the above, and the residual after deducting fixed effects in the phenotype to be analyzed is used as the corrected phenotype. Using the ancestral genotypes and phenotypes of all samples to be analyzed, with fixed effects such as sex and batch number as covariates, gene-gene interaction analysis is performed using the MatrixEpistasis R package. Significant (P<0.05) interacting ancestral genotype pairs are output, denoted as ancestral genotype 1 and ancestral genotype 2. For each significant ancestral genotype pair, with the corrected phenotype as the dependent variable, ancestral genotype 1, ancestral genotype 2, and the interaction between ancestral genotypes 1 and 2 are referred to as different genotype groups, and different genotype groups are the independent variables, a linear model is established. The least squares mean is calculated, and slice analysis is performed to compare which levels have significant differences, obtaining the epistatic interaction dominant genomic regions significantly associated with the phenotype.

[0035] It should be noted that existing technologies based on base genotype analysis can only analyze the genetic information of the organism itself. However, the applicant has found that heterosis, epistasis, etc., may have different ancestral origins. Therefore, this application uses ancestral genotype to analyze heterosis and epistasis, which can effectively improve accuracy.

[0036] The following specific embodiments will provide a more detailed explanation of the method for locating ancestral haplotype genes obtained from deep hybridization.

[0037] Example 1

[0038] Using the fast-growing and large-scale Lingnan Yellow Chicken A03 strain and the local Guangdong breed Huiyang Bearded Chicken as parents, a distant deep hybrid line was constructed using the reciprocal cross method. The F9 offspring of the deep hybrid line were used as the sample to be analyzed, and the body weight at week 8 was used as the phenotype to be analyzed (original phenotype) for ancestral haplotype gene mapping analysis of the deep hybrid line.

[0039] S1. Obtaining ancestral genotypes: Divide the whole genome of the offspring to be analyzed into N haplotypes, infer the origin of its two ancestors based on the haplotypes, and obtain the whole genome ancestral genotypes.

[0040] Using RFMix software, based on the A03 strain and Huiyang Bearded Chicken as the two parental populations, haplotypes were segmented using 5 single nucleotides as a fixed frame length, and the ancestral origin of each haplotype in each sample of the F9 progeny was calculated. In this example, the ancestral population of Huiyang Bearded Chicken is represented by 0, and the ancestral population of A03 strain is represented by 1. The ancestral origin of each haplotype in the F9 progeny can be represented by 0, 1, and 2, referred to as the ancestral genotype. 0 indicates that both haplotypes of a sample originated from Huiyang Bearded Chicken; 1 indicates that one haplotype came from Huiyang Bearded Chicken and the other haplotype came from A03 strain; 2 indicates that both haplotypes came from A03 strain.

[0041] S2. Obtaining associated genomic segments: Calculate the kinship matrix among the offspring of deep hybrids based on the ancestral genotypes of the whole genome, and perform whole genome association analysis based on the ancestral genotypes and kinship matrix. Use the false discovery rate correction method for multiple correction to obtain genomic segments with significant phenotypic association.

[0042] Based on the ancestral genotypes of the F9 progeny population, a kinship matrix was calculated using GCTA software. For the phenotypic information regarding weight at week 8, a genome-wide association analysis (GWA) was performed using a mixed linear model within GCTA software, based on the F9 progeny genotypes and the kinship matrix. Multiple corrections were applied using the false discovery rate correction method. Specifically, Figure 1The Manhattan plot for genome-wide association analysis of ancestral haplotypes (FDR ≤ 0.05) is shown, where the horizontal axis represents genomic location and the vertical axis represents significance -log10(p). Haplotype segments associated with phenotypically significant (adjusted p-value < 0.05) were further extracted. Specifically, Table 1 shows the significance of genome-wide association analysis on some segments located on chromosome GGA12, representing some significant genomic segments for this study. In the table, the first column is the chromosome number, the second column is the haplotype number, the third column is the genomic segment containing the corresponding haplotype, the fourth column is the p-value hypothesis probability obtained from the mixed dominant model in GCTA software, the fifth column represents the effect value of the ancestral haplotype segment, and the sixth column is the false discovery rate (FDR) obtained after multiple corrections for false discovery rate.

[0043] In this embodiment, the hybrid linear model used is:

[0044] ;

[0045] Where y represents the phenotype to be analyzed in the F9 population, Q is the covariance matrix, including sex and batch fixed effects, α is the effect vector of the covariates, X is the ancestral genotype matrix of the progeny of the deep hybrid line, β is the effect size of the haplotype at the target locus, g is the kinship matrix of each pair of samples, and e is the residual vector.

[0046] Table 1. Significant genomic regions (partial list)

[0047] .

[0048] Furthermore, using the weight data of the week 8 to be analyzed as the dependent variable, and gender and batch fixed effects as independent variables, the lm function in R language was used to fit a linear model, and the residuals function was used to obtain the residuals of the linear model. The residuals were used as a new phenotype, called the corrected phenotype.

[0049] S3. Genomic segment mapping of superior and mid-parent hybrids

[0050] Based on the corrected phenotype obtained using the same method in S2, the mean of the corrected phenotype for each ancestral genotype is calculated at the F9 offspring population level. If the mean of the corrected phenotype of ancestral genotype 1 is higher than that of ancestral genotypes 0 and 2, then ancestral genotype 1 is considered a potential superior hybrid genotype. If the mean of the corrected phenotype of ancestral genotype 1 is higher than that of ancestral genotypes 0 and 2, then ancestral genotype 1 is considered a potential mid-parent hybrid genotype, and the location information of the above genotypes is recorded. For each potential superior parent hybrid and mid-parent hybrid vigor, the corrected phenotype and genotype at the corresponding positions in the F9 population were extracted. The kruskal.test function in R language was used with the corrected phenotype as the dependent variable and the genotype as the independent variable to test whether there was a significant difference in the population. If there was a significant difference, the Steel.Dwass function in the steel.Dwass.test package in R language was used with the corrected phenotype as the dependent variable and the genotype as the grouping variable to test and compare the differences among the three genotypes. Specifically, Table 2 shows some segments with significant superior parent and mid-parent hybrid vigor in this study (in the table, the first column represents the chromosome number, the second column represents the haplotype number, the third column corresponds to the genomic segment where the haplotype is located, and the fourth column (type) indicates whether the haplotype belongs to superior parent hybrid vigor or mid-parent hybrid vigor after identification by the above method). Figure 2 This study presents the corrected phenotypic and genotypic distribution maps of heterosis in superior and middle parents, respectively. The left map shows the segment identified as heterosis in the GGA2-H2082 haplotype (GGA2: 47,996,576-48,032,258 bp). Red indicates ancestral genotype 2 (two ancestral haplotypes from the A03 population), blue indicates ancestral genotype 0 (two ancestral genotypes from the Huiyang Bearded Chicken population), and orange indicates ancestral genotype 1 (one ancestral haplotype from the A03 population and the other from the Huiyang Bearded Chicken population). It can be seen that the phenotypic value of ancestral genotype 1 is significantly higher than that of ancestral genotypes 1 and 2, respectively. The right map shows the GGA4-H132 haplotype (GGA4: 74,902,003-74,907,736 bp). The above data shows hybrid vigor in the middle-parent population. Orange represents ancestral genotype 1 (one ancestral haplotype comes from the A03 population, and the other ancestral haplotype comes from the Huiyang Bearded Chicken population). Blue represents ancestral genotypes 0 and 2 (two ancestral haplotypes come from the same ancestral population). The vertical axis represents the corrected body weight; it can be seen that the phenotypic value of ancestral genotype 1 is significantly higher than the phenotypic mean of ancestral genotypes 1 and 2.

[0051] Table 2 shows the regions with significant heterosis between superior and mid-parental offspring (partial list).

[0052] .

[0053] S4. Location of epistatic interaction genomic segments

[0054] Furthermore, based on the F9 progeny population, gene-gene interaction analysis was performed using the `MatrixEpistasis_main` function in the `MatrixEpistasis` package of the R language. Interacting genes A and B were whole-genome ancestral genotypes, and the phenotype was the original weight phenotype at week 8. The covariance matrix was for sex and batch, outputting ancestral genotype pairs with significant interactions (P < 0.05). For each significant ancestral genotype pair, ancestral genotype A, ancestral genotype B, and the interaction between ancestral genotypes A and B were termed different genotype groups. Using the `glimmix` generalized linear mixed model procedure in the SAS language, with corrected weight at week 8 as the dependent variable and the three different genotype groups as independent variables, a regression model was established, and the parameter estimates for each effect (genotype group) were displayed. Subsequently, the least squares mean was calculated, and slice analysis was performed to compare which genotype groups showed significant differences. Figure 3 This study demonstrates the interaction between two ancestral genotypes at the interaction site between haplotypes GGA4-H3634 and GGA15-H112 (where the ordinate represents corrected body weight, 0 indicates that both haplotypes are homozygous from the Huiyang Bearded Chicken population, 2 indicates that both haplotypes are homozygous from the A03 population, and 1 indicates heterozygous ancestry, with one haplotype from the Huiyang Bearded Chicken population and the other from the A03 population; the red bars represent the epistatic interaction site between genotypes 0 and 2 (p < 0.01). When GGA4-HAP3634 is a homozygous ancestor of the Huiyang Bearded Chicken (ancestral genotype 0) and GGA15-HAP11 is a homozygous ancestor of the A03 population (ancestral genotype 2), Alternatively, when GGA4-HAP3634 is a homozygous ancestor of the A03 population (ancestral genotype 2) and GGA15-HAP112 is a homozygous ancestor of the Huiyang Bearded Chicken (ancestral genotype 0), these individuals show a significant increase in body weight.

[0055] Table 3 further presents some of the results of the upper-level interaction detection.

[0056] Table 3. Results of Upper-Level Interaction Detection (Partial)

[0057] .

[0058] The first column represents the chromosome number of interacting gene A, the second column represents the haplotype number of interacting gene A, the third column represents the chromosome number of interacting gene B, the fourth column represents the haplotype number of interacting gene B, the fifth column represents the significance of the interaction, and the sixth column represents which two genotypes have a significant interaction. Interacting genotype pairs are separated by "||".

[0059] The above embodiments demonstrate that obtaining the whole genome ancestral genotype and corrected phenotype of the deep hybrid line through the technical means of this application can not only significantly reduce false positives that are prone to occur based on base genotype analysis, but also be used for heterosis analysis, capturing the dominance and epistatic effects between genetic markers, etc., confirming that heterosis, epistatic effects, etc. may have different ancestral origins, and therefore has higher accuracy than the existing technology based on base genotype analysis.

[0060] The above embodiments are only used to illustrate the technical solutions of the present invention, but the protection scope of the present invention is not limited thereto. The embodiments and features in the embodiments of the present invention can be arbitrarily combined with each other without conflict. Any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be covered within the protection scope of the present invention. Therefore, the protection scope of the present invention should be determined by the scope of the claims.

Claims

1. A method for ancestral haplotype gene mapping in deep hybrid lines, characterized in that, This includes obtaining ancestral genotypes and gene mapping. Obtaining ancestral genotypes involves using a local ancestry inference tool to divide the genome of the offspring to be analyzed into N haplotypes based on a fixed frame length. The haplotypes are then encoded to infer their two ancestral origins, thus obtaining the whole-genome ancestral genotype. Specifically, encoding the haplotypes to infer their two ancestral origins involves using 0 and 1 to represent ancestor 1 and ancestor 2, respectively. Each haplotype in each offspring sample can be represented by 0, 1, or 2, also known as the ancestral genotype, indicating the number of copies of the haplotype from ancestor 1 in the diploid plant / animal offspring sample. Therefore, each offspring sample can form an ancestral haplotype vector according to the haplotype order, and several offspring samples form an ancestral haplotype matrix. Here, haplotypes 0 and 2 in the offspring to be analyzed represent homozygous ancestry, and 1 represents heterozygous ancestry. The gene mapping includes the mapping of heterotic genomic segments of superior and intermediate parents. Specifically, based on the whole genome ancestral genotype and corrected phenotype, the heterotic genomic segments of superior and intermediate parents are detected by nonparametric Kruskal-Wallis and Steel-Dwass tests, respectively, to obtain heterotic genomic segments of superior and intermediate parents that are significantly associated with the phenotype. The gene mapping includes epistatic interaction genomic segment mapping, specifically: based on the whole genome ancestral genotype and corrected phenotype, logistic regression is used to detect epistatic interactions, and a mixed linear model is used to perform simple effect comparisons to test the significance of interactions between genotypes, thereby obtaining epistatic interaction genomic segments that are significantly associated with the phenotype. The residuals after deducting fixed effects from the phenotype to be analyzed are used as the corrected phenotype.

2. The gene localization method according to claim 1, characterized in that, The gene mapping includes genome-wide association analysis, specifically: calculating the kinship matrix among the offspring of deep hybrid lines based on the ancestral genotypes of the whole genome, and performing genome-wide association analysis based on the ancestral genotypes and kinship matrix, and using the false discovery rate correction method for multiple correction to obtain the genomic segments that are significantly associated with the phenotype to be analyzed.

3. The gene localization method according to claim 1, characterized in that, The genome-wide association analysis was performed using a mixed linear model, which is as follows: ； Where y represents the phenotype to be analyzed in the progeny population of the deep hybrid line, Q is the covariance matrix, including sex and batch fixed effects, α is the effect vector of the covariate, X is the ancestral haplotype encoding vector of the progeny of the deep hybrid line, β is the effect size of the haplotype of the target locus, g is the kinship matrix of each pair of samples, and e is the residual vector.

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