A method for qtl mapping in inbred multiple-parent advanced intercross lines

Abstract

The application discloses a QTL positioning method for pure-bred multi-parent high-generation intercross population, which comprises the following steps: (1) orthogonal decomposition of genetic effect; (2) construction of parent source probability matrix; (3) establishment of a statistical genetic model; (4) QTL positioning and genetic parameter estimation. Compared with the prior art, the application decomposes additive variance and superadditive epistatic variance by using the orthogonal decomposition method, improves the positioning efficiency of QTL, and reduces the false positive rate; compared with the prior art, the model does not contain the superadditive epistatic effect and the gene-environment interaction effect, the model proposed by the application contains the superadditive epistatic effect and the environment interaction effect, is more in line with the genetic characteristics of complex traits, and can effectively analyze the genetic structure of complex traits; the mixed linear model proposed by the application is high in flexibility, and the model and the analysis method can be easily extended to more parents and the MAGIC population of any generation.

Description

Technical Field

[0001] This invention relates to the fields of computational biology and crop genetics, and specifically to a QTL mapping method for high-generation intercross populations of homozygous multi-parental lines. Background Technology

[0002] Most human disease traits and crop agronomic traits are complex traits influenced by multiple genes and non-genetic factors. Quantitative trait locus (QTL) mapping can resolve the genetic variation of complex traits down to the individual locus level, facilitating the rapid discovery of functional genes controlling traits and achieving precise analysis of trait genetic structure, which is of great significance for the molecular genetic improvement of traits. Compared to traditional biparental mapping populations, multi-parent advanced generation intercross (MAGIC) populations arise from the intercrossing of multiple parents, most of which are genetic breeding materials or representative varieties with different superior traits. The combination of multiple parents increases the genetic diversity of the mapping population, which is beneficial for identifying more valuable QTLs; the large number of accumulated recombination events can improve the resolution accuracy of linkage maps, thereby accurately locating QTLs. Therefore, MAGIC populations and their QTL mapping methods have been widely used. The QTL mapping of the MAGIC population was first achieved in mice by Threadgill and Williams (Mamm. Genome 13, 175-178 (2002)). In plants, it was first achieved in Arabidopsis by Kover et al. (PLoS Genet 5, e1000551 (2009)). Subsequently, it has been gradually applied to crops, such as the QTL mapping of the wheat MAGIC population by Huang et al. (Plant Biotechnol. J. 10, 826-839 (2012)).

[0003] However, unlike QTL mapping in biparental populations, QTL mapping in MAGIC populations requires more complex QTL effect definitions and marker parental origin inference. Current MAGIC population-based QTL mapping still has several limitations: First, it cannot analyze epistasis (interactions between paired QTLs) and QTL-environment interactions, thus failing to fully utilize the advantages of MAGIC populations for effective gene mining; second, it is limited to the framework of biparental population effect definitions, still using a single fixed model to estimate the genetic parameters of QTLs in MAGIC populations. The insufficient degrees of freedom of the model gradually become apparent as the number of parents increases, making it difficult to extend to QTL mapping in more parental MAGIC populations; finally, there is still a lack of computationally efficient QTL analysis software specifically for MAGIC populations.

[0004] Specifically, at the methodological level: Wei and Xu (Genetics 202(2), 471-486(2016).) constructed a stochastic model for QTL mapping in MAGIC populations by using background control as a random effect, which to some extent solved the problem of high degree of freedom consumption in QTL mapping of MAGIC populations. However, they lacked the definition and decomposition of genetic effects in MAGIC populations, and did not further establish a model to estimate various genetic parameters. Wang et al. (Heredity 119(4), 256-264(2017).) used the Cockerham genetic model construction method to define and orthogonally decompose the additive effects of QTLs in a four-parent homozygous MAGIC population, and estimated the additive effect values ​​by fixing the model. Subsequently, the software GAPL based on this method was also successfully developed, and the maximum number of analyzable parents was extended to eight, which was successfully applied to QTL mapping analysis in homozygous MAGIC populations. Although this method has achieved the localization of additive QTLs and the estimation of additive effect values ​​in four- and eight-parent homozygous MAGIC populations, the single fixed model used in this method may have insufficient degrees of freedom when facing eight-parent MAGIC populations, especially sixteen-parent MAGIC populations.

[0005] Currently, QTL mapping methods based on MAGIC populations and QTL mapping software for homozygous MAGIC populations, such as GAPL (Zhang et al., The Crop Journal, 7(3), 283-293.(2019)) and R / qtl2 (Broman et al., Genetics, 211(2), 495-502.(2019)), still have many limitations: First, they cannot analyze epistasis (interactions between paired QTLs) and QTL-environment interactions, thus failing to fully utilize the rich genetic variation of MAGIC populations for effective gene mining; second, they are limited to the definition framework of additive effects in biparental populations, still using fixed models to estimate QTL effects in MAGIC populations, which consumes a large number of degrees of freedom, making it difficult to extend the models and methods to QTL mapping analysis of MAGIC populations with more parents. Furthermore, they neglect the genetic characteristics of complex traits involving gene-gene interactions and gene-environment interactions. Summary of the Invention

[0006] This application addresses the aforementioned shortcomings in the prior art by providing a QTL localization method for homozygous MAGIC populations.

[0007] The specific technical solution of the present invention is as follows:

[0008] This invention provides a QTL localization method for homozygous MAGIC populations, comprising the following steps:

[0009] (1) Orthogonal decomposition of genetic effects:

[0010] Based on the Cockerham genetic model, the total genetic variance is decomposed into the variance components of each genetic effect. Then, combined with the constraints satisfied by each effect, pairwise orthogonal coefficient vectors are constructed to obtain the orthogonal coefficient matrix of the homozygous MAGIC population.

[0011] The genetic effects mentioned are additive and additive in situ genetic effects;

[0012] This step aims to decompose the additive variance components into the sum of additive and central variance components, thereby ensuring the accuracy of subsequent QTL localization and the unbiasedness of effect estimation.

[0013] (2) Construction of the parental origin probability matrix:

[0014] Unlike biparental populations, the parental origin of each molecular marker and the QTL between markers in the MAGIC population needs to be inferred. Therefore, the probability that each marker and the QTL between markers originate from each parent can be obtained by using the Markov chain algorithm based on the recombination information between markers, and a parental origin probability matrix can be constructed. Combined with the corresponding orthogonal coefficient matrix in step (1), an effect coefficient matrix can be constructed.

[0015] (3) Establishment of statistical genetic models:

[0016] A homozygous MAGIC population of size n is subjected to genetic experiments under p different environments. The variation of the target trait is controlled by s QTLs, abbreviated as Q1, Q2, ..., Qt. s There exist t pairs of QTLs interacting. Combining the orthogonal decomposition of QTL genetic effects and the Markov chain algorithm, the coefficient matrix is ​​obtained, and the following mixed linear model is constructed:

[0017]

[0018] Where y ij Let μ be the phenotypic value of the i-th individual in the j-th environment, and μ be the population mean; k For Q k The additive effect vector after orthogonal decomposition is taken as a fixed effect, x ki For the i-th individual located in Q k The coefficient vector x of the additive effect at point hi For the i-th individual located in Q h The coefficient vector of the additive effect at point aa kh For Q k With Q h After orthogonal decomposition, the summative effect vector is used as a fixed effect, e j Let ae represent the main effect of the j-th environment as a random effect.kj For Q k The additive effect between the environment and the environment j, and the vector of the interaction effect with the environment, are treated as random effects, aae khj For Q k With Q h The summation of the interaction vector between the intrinsic and environmental j is taken as a random effect, ε ij For residuals, The Kronecker product represents the vector;

[0019] Obtain the corresponding coefficient matrix;

[0020] The homozygous MAGIC population can be four-parent, eight-parent, sixteen-parent, etc.

[0021] (4) QTL mapping and genetic parameter estimation

[0022] Based on the corresponding coefficient matrix, one-dimensional and two-dimensional scans of QTLs are performed across the entire genome to obtain QTLs and their genetic parameters.

[0023] In some embodiments of the present invention, when one of the QTLs in a four-parent homozygous MAGIC population has multiple different genotypes sequentially from the four parents, the effect value of each genotype can be represented by the following unit point model:

[0024]

[0025] Where g i Let μ be the homozygous genotype value of the i-th parent, μ be the population mean, specifically the mean of the effect values ​​of the four homozygous genotypes, and a be the mean of the effect values ​​of the four homozygous genotypes. i Let be the additive effect value of the homozygous genotype of the i-th parent. Let α1, α2, and α3 be the coefficients corresponding to the additive effects α1, α2, and α3 after orthogonal decomposition, respectively; satisfying the constraint condition ∑ i a i =0;

[0026] When two QTLs in a four-parent homozygous MAGIC population have four different genotypes from each of the four parents, the genotype value for each pair of genotype combinations can be represented by the following two-locus model:

[0027] g ij =μ+a i +a′ j +aa ij

[0028] Where g ij For each of the two loci, there are four homozygous genotype combinations, resulting in a total of 16 effect values. μ is the population mean, specifically the mean of these 16 genotype combination effect values. i a j′、aa ij Let i, j = 1, 2, 3, 4, be the additive effect of the first QTL, the additive effect of the second QTL, and the sum of the secondary effects of the two QTLs, respectively, satisfying the constraint ∑ i a i =∑ j a j ′=∑ i，j aai j =∑ i aa ij =∑ j aa ij =0.

[0029] Furthermore, when the homozygous MAGIC population is an eight-parent homozygous MAGIC population or a sixteen-parent homozygous MAGIC population, the corresponding orthogonal coefficient matrix is ​​recursively generated using the Hadamard matrix.

[0030] Optionally, in step (2), the marker may further include a marker for genotype deletion, and the QTL between the markers may further include a QTL to be tested between the two markers.

[0031] In some embodiments of the present invention, in step (4), the expression for the corresponding coefficient matrix is ​​as follows:

[0032]

[0033] Where y is an n×1 phenotypic value vector; μ is the population mean vector; and b is the effect vector. A =[a1 T a2 T , ..., a s T ] T b AA =[aa1 T aa2 T , ..., aa t T ] T , and The corresponding coefficient matrices are X A X AA U E , and Let I(U) be an n×1 residual effect vector; r+1 ) is an n×n identity matrix.

[0034] Furthermore, when constructing the statistical genetic model corresponding to the sixteen-parent homozygous MAGIC population, the additive effect aa will be added. vAs a random effect, X is represented by the corresponding coefficient matrix expression. AA b AA Replace with

[0035] In some embodiments of the present invention, in step (4), in order to estimate the genetic parameters of the detected QTL, the least second-order unbiased estimation method is used to estimate the variance components of each random effect in the mixed linear full model, and the adjusted unbiased prediction method is used to predict the random effect value based on the variance component estimate; the generalized least squares method is used to estimate the fixed effects in the model.

[0036] Specifically, the obtained variance component estimates are used as prior values ​​in the Markov chain Monte Carlo process. Based on the Gibbs sampling technique, distribution samples of each parameter are drawn, and the parameters are estimated and significance tests are performed to make corresponding statistical inferences.

[0037] Furthermore, step (4) also includes using stepwise regression to select the model and eliminate false positive QTLs.

[0038] This invention also provides a statistical genetic model for QTL mapping in homozygous MAGIC populations, the statistical genetic model being as follows:

[0039]

[0040] Where y ij Let μ be the phenotypic value of the i-th individual in the j-th environment, and μ be the population mean; k For Q k The additive effect vector after orthogonal decomposition is taken as a fixed effect, x ki For the i-th individual located in Q k The coefficient vector x of the additive effect at point hi For the i-th individual located in Q h The coefficient vector of the additive effect at point aa kh For Q k With Q h After orthogonal decomposition, the summative effect vector is used as a fixed effect, e j Let ae represent the main effect of the j-th environment as a random effect. kj For Q k The additive effect between the environment and the environment j, and the vector of the interaction effect with the environment, are treated as random effects, aae khj For Q k With Q h The summation of the interaction vector between the intrinsic and environmental j is taken as a random effect, ε ij For residuals, The Kronecker product represents the vector;

[0041] Based on the statistical genetic model, the corresponding coefficient matrix is ​​obtained.

[0042] The innovation of this invention compared with the prior art is as follows:

[0043] a) The addition variance and the addition of central variance were decomposed using orthogonal decomposition, which improved the localization power of QTL and reduced the false positive rate.

[0044] b) Compared with existing MAGIC population QTL mapping and genetic parameter estimation models that do not include epistasis and gene-environment interaction effects, the model proposed in this method includes epistasis and environmental interaction effects, which is more consistent with the genetic characteristics of complex traits and can more effectively resolve the genetic structure of complex traits.

[0045] c) The model proposed in this invention is a hybrid linear model, which has great flexibility and can be easily extended to more parents and MAGIC populations of any generation. Attached Figure Description

[0046] Figure 1 This diagram illustrates the genetic distance and parameter settings for simulating QTLs. Detailed Implementation

[0047] This study argues that the rich genetic variation in MAGIC populations should be fully utilized, and a new definition and estimation method for additive and additive-additive effects of QTLs, consistent with the multiple allele characteristics of MAGIC populations, should be adopted: In MAGIC populations with a typical number of parents (such as four-parent and eight-parent MAGIC populations), the genetic components of detected QTLs (including additive and additive-additive effects) and their interactions with the environment should be treated as fixed and random effects, respectively; In MAGIC populations with a larger number of parents (such as sixteen-parent MAGIC populations), the genetic components of detected QTLs (containing only additive effects) should be treated as fixed effects, and the additive and environmental interactions, additive-additive effects, and their interactions with the environment of detected QTLs should all be treated as random effects, thus developing a new method for QTL mapping in homozygous MAGIC populations based on a mixed linear model. Therefore, this study can construct an effect coefficient matrix for homozygous MAGIC populations generated in different ways, such as recombinant inbred lines and doubled haploid systems, within a mixed linear model framework, using the Markov chain algorithm combined with orthogonal decomposition, targeting the specific transition probability matrix of MAGIC populations. Based on the corresponding coefficient matrix, one-dimensional and two-dimensional QTL scanning is performed across the entire genome, and the genetic parameters of the detected QTLs are estimated. By analyzing simulated data at different heritability and sample size levels, the performance of the new method in terms of statistical power, false positive rate, and effect estimation accuracy is evaluated. Based on the proposed model and algorithm, corresponding software is developed for QTL mapping analysis in homozygous MAGIC populations. Detailed methods include:

[0048] (1) Orthogonal decomposition of genetic effects

[0049] Unit-point model: Assuming a QTL in a four-parent homozygous MAGIC population has four different genotypes (AA, BB, CC, DD) from the four parents, the effect size of each genotype can be represented by the following unit-point model.

[0050]

[0051] Where g i (i = 1, 2, 3, 4) represents the homozygous genotype value of the i-th parent, μ is the population mean, specifically the mean of the effect values ​​of these four homozygous genotypes, and a i (i = 1, 2, 3, 4) represents the additive effect value of the homozygous genotype of the i-th parent. These are the coefficients corresponding to the additive effects α1, α2, and α3 after orthogonal decomposition. According to the Cockerham genetic model, the genetic variance can be divided into variance components of each genetic effect, which must satisfy the constraint ∑ i p i a i =0, because theoretically, in a four-parent homozygous MAGIC population, E(p) = 0. i Since ) = 0.25, the constraint condition can be simplified to ∑ i a i =0. Under this constraint, the four homozygous genotypes still have three degrees of freedom remaining (see Table 1), therefore, they can be determined by... The three effects obtained after orthogonal decomposition are as follows and the corresponding coefficient vector of the three orthogonal decompositions. For example, This represents the coefficient vector corresponding to effect α1, where This represents the coefficient corresponding to effect α1 under the AA genotype, and is also denoted by matrix. This is the additive orthogonal decomposition matrix for a homozygous MAGIC population with four parents. It's worth noting that in a single-site model with only additive effects, orthogonal decomposition does not affect the unbiasedness of the estimation of the additive effects (a1, a2, a3, a4) for the four homozygous genotypes. However, the coefficients corresponding to the additive effects after orthogonal decomposition can conveniently be used to obtain the coefficients corresponding to the additive-additive effects after orthogonal decomposition. To facilitate extension to a two-locus (additive-additive) mapping model and to achieve the decomposition of additive and additive-additive variance components, orthogonal decomposition was still performed on the additive effects in the single-site model.

[0052] Table 1. Orthogonal decomposition of additive effects in the homozygous MAGIC population of the four parents.

[0053]

[0054] Note: AA, BB, CC, and DD represent different genotypes from the four parents, respectively. This is the transpose of the additive effect coefficient vector after the three orthogonal decompositions.

[0055] Two-locus model: Assuming a homozygous MAGIC population with four parents, two QTLs have four different genotypes from each of the four parents: AA, BB, CC, DD and A′A′, B′B′, C′C′, D′D′. The genotype value for each pair of genotype combinations can be represented by the following two-locus model.

[0056] g ij =μ+a i +a′ j +aa ij

[0057] Similarly, where g ij (i, j = 1, 2, 3, 4) represent the genotype combinations of 4 homozygous individuals at each of the two loci, resulting in a total of 16 effect values. μ is the population mean, specifically the mean of these 16 genotype combination effect values. i (i = 1, 2, 3, 4), a j ′(j=1,2,3,4),aa ij (i, j = 1, 2, 3, 4) represent the additive effect of the first QTL, the additive effect of the second QTL, and the sum of the secondary and tertiary effects of the two QTLs, respectively. Similarly, the constraint condition ∑ must be satisfied. i p i a i =∑ j p j a j ′=∑ i，j p i p j aa ij =∑ i p i aa ij =∑ j p j aa ij =0, because in the four-parent homozygous MAGIC population, E(p) i )=E(p j Since ) = 0.25, the constraint condition can be simplified to ∑ i a i =∑ j a j ′=∑ i，j aa ij =∑ i aa ij =∑j aa ij =0. Using the additive orthogonal decomposition coefficients obtained from the unit point model, the additive coefficients of the first and second QTLs in the two-site model are encoded in three columns each according to Table 1 (see Table 2), namely ω 10 ω 11 ω 12 ω 13 ω 14 ω 15 Furthermore, because the coefficients after orthogonal decomposition have a recursive property, we can use ⊙(Hadamard product) to obtain the pairwise orthogonal 9 columns, adding the intrinsic coefficients as follows: ω1=ω 10 ⊙ω 13 , ω2=ω 10 ⊙ω 14 ,ω3=ω 10 ⊙ω 15 ,ω4=ω 11 ⊙ω 13 ,ω5=ω 11 ⊙ω 14 ,ω6=ω 11 ⊙ω 15 ,ω7=ω 12 ⊙ω 13 ,ω8=ω 12 ⊙ω 14 ,ω9=ω 12 ⊙ω 15 And the corresponding additive central effects αα1, αα2, αα3, αα4, αα5, αα6, αα7, αα8, αα9 after orthogonal decomposition. Similarly, let the matrix be... This is the orthogonal coefficient matrix for the four-parent MAGIC population with added indirect effects. The orthogonal coefficient matrices for the eight-parent and sixteen-parent MAGIC populations will not be elaborated upon here, as the large number of parents would significantly increase the difficulty and complexity of orthogonal decomposition according to the aforementioned model and assumptions. Therefore, the Hadamard matrix can be used to recursively generate the corresponding orthogonal coefficient matrices. wait.

[0058] Table 2 Orthogonal decomposition of additive symptom effect in homozygous MAGIC populations of four parents

[0059] genotype Genotype value Genotype frequency <![CDATA[ω 10 ]]> <![CDATA[ω 11 ]]> <![CDATA[ω 12 ]]> <![CDATA[ω 13 ]]> <![CDATA[ω 14 ]]> <![CDATA[ω 15 ]]> AAA′A′ <![CDATA[μ+a1+a′1+aa 11 ]]> 0.0625 1 1 1 1 1 1 AAB′B′ <![CDATA[μ+a1+a′2+aa 12 ]]> 0.0625 1 1 1 1 -1 -1 AAC′C′ <![CDATA[μ+a1+a′3+aa 13 ]]> 0.0625 1 1 1 -1 1 -1 AAD′D′ <![CDATA[μ+a1+a′4+aa 14 ]]> 0.0625 1 1 1 -1 -1 1 BBA′A′ <![CDATA[μ+a2+a′1+aa 21 ]]> 0.0625 1 -1 -1 1 1 1 BBB′B′ <![CDATA[μ+a2+a′2+aa 22 ]]> 0.0625 1 -1 -1 1 -1 -1 BBC'C' <![CDATA[μ+a2+a′3+aa 23 ]]> 0.0625 1 -1 -1 -1 1 -1 BBD′D′ <![CDATA[μ+a2+a′4+aa 24 ]]> 0.0625 1 -1 -1 -1 -1 1 CCA′A′ <![CDATA[μ+a3+a′1+aa 31 ]]> 0.0625 -1 1 -1 1 1 1 CCB′B′ <![CDATA[μ+a3+a′2+aa 32 ]]> 0.0625 -1 1 -1 1 -1 -1 CCC′C′ <![CDATA[μ+a3+a′3+aa 33 ]]> 0.0625 -1 1 -1 -1 1 -1 CCD′D′ <![CDATA[μ+a3+a′4+aa 34 ]]> 0.0625 -1 1 -1 -1 -1 1 DDA′A′ <![CDATA[μ+a4+a′1+aa 41 ]]> 0.0625 -1 -1 1 1 1 1 DDB′B′ <![CDATA[μ+a4+a′2+aa 42 ]]> 0.0625 -1 -1 1 1 -1 -1 DDC′C′ <![CDATA[μ+a4+a′3+aa 43 ]]> 0.0625 -1 -1 1 -1 1 -1 DDD′D′ <![CDATA[μ+a4+a′4+aa 44 ]]> 0.0625 -1 -1 1 -1 -1 1

[0060] Note: AAA′A′ to DDD′D′ represent 16 genotype combinations at two loci, ω 10 ω 11 ω 12 and ω 13 ω 14 ω 15The three additive effect coefficients of the first and second QTLs in the two-site model.

[0061] (2) Construction of the parental origin probability matrix

[0062] Unlike biparental populations, the parental origin of each molecular marker and the QTL between markers in the MAGIC population needs to be inferred. Therefore, the probability that each marker and the QTL between markers originate from each parent can be obtained by using the Markov chain algorithm based on the recombination information between markers, and then the corresponding parental origin probability matrix can be constructed.

[0063] Taking the MAGIC population of four-parent recombinant inbred lines as an example, assuming the population size is n, the number of molecular markers on a linkage group is m, and the genetic distance between markers is known, the markers are arranged in the following order: M1, ..., M m In this population, each molecular marker of any individual should have a definite parental origin, let x k Representing an individual in M k The parental origin of the gene, x k The values ​​1, 2, 3, and 4 represent the individual's position in M. k The gene originates from the first, second, third, and fourth parents, i.e., the x-gene is defined as... k For this individual in M k The "parental type" at the location; z k The representative only observed this individual and four parents in M k The genotype at location M can be used to infer the individual's M k The parental origin of the gene, denoted by z. k For this individual in M k The "phenotype" at a location. For example, M k The genotypes of the individuals from the first to the fourth parent were AA, aa, aa, and aa, respectively, while the individual exhibited M... k The genotype at this location is AA, therefore the individual M can be directly determined through observation alone. k The gene originates from the first parent. In this case, x k =z k =1, like M in this individual k Such a notation is referred to as the "full observation" notation in this paper; however, if M k The parental genotypes are AA, AA, aa, aa in sequence. This individual is in M... k The genotype at this location is AA. Observation alone cannot determine whether this individual's locus originated from the first or second parent, i.e., z k={1, 2}, such a marker is called a "partial observation" marker in this article. Therefore, for the "partial observation" marker of this individual, using only the information at this marker position is not sufficient to infer the accurate parental origin. It is necessary to further combine the marker information on both sides and use the Markov chain algorithm to calculate the probability that this locus comes from different parents. For x k Taking the conditional probabilities for different values of x and conditioned on the "phenotypes" of all markers on this linkage group, in this article, we use P(x k |z1,..., z m ) to represent it. If the two adjacent markers of the "partial observation" M k marker, that is, M k-1 and M k+1 are both "complete observation" markers at this individual, then without considering interference, this conditional probability completely depends on the "phenotype" of the M k marker, the "parental types" of M k-1 and M k+1 ("the "phenotype" of a "complete observation" marker is the "parental type"), and the recombination rates between M k and M k-1 and between M k and M k+1 , and is independent of other markers on the linkage group. Then we can get the following equation

[0064] P(x k |z1,..., x k-1 , z k , x k+1 ,.., z m ) = P(x k |x k-1 , z k , x k+1 )

[0065] If one or both of the adjacent markers on both sides are not "complete observation" markers, then continue to extend to one side or both sides of the adjacent markers until reaching a "complete observation" marker or the last marker at the end of this linkage group. Let M i and M l (i < k < l) be the two nearest "complete observation" markers on both sides. If there are no "complete observation" markers on both sides or on one side, then M i = M1 or M l = M m Or M i = M1, M l = M m . Calculate the probability P(x k |z i [[ID=�0]],..., z l ), and by Bayes' formula, we can get

[0066]

[0067] Where P(x) k ) is x k Let q be the prior probability. k ={P(x k )} (4×1) The corresponding row vector is:

[0068] q k T =[P(x k =1), P(x) k =2), P(x k =3), P(x k =4)]

[0069] Similarly, there is the following definition:

[0070]

[0071] It is worth noting that, without considering interference, let x k For a given value, the following conclusion can be drawn:

[0072] P(z i , ..., z l |x k )=P(z i , ..., z k |x k , z k+1 , ..., z l )P(z k+1 , ..., z l |x k )

[0073] =P(z) i , ..., z k |x k )P(z k+1 , ..., z l |x k )

[0074] Therefore, we need to calculate {P(x)} k |z i , ..., z l )} (4×1) That is, p k It can be expressed by the following equation:

[0075]

[0076] Where ⊙ represents the Hadamard product of vectors.

[0077] Furthermore, construct the following from marker M k To mark M k+1 (also the marker M) k+1 To mark M k The transition probability matrix of )

[0078]

[0079] Where r k This represents the recombination rate between the two markers (the cumulative recombination rate of the recombinant inbred line population). Then, using this transition probability matrix, we can obtain:

[0080]

[0081] Where c is a column vector with all elements equal to 1, and assume z k+1 ={1, 2, 3}, then in the above formula I k+1 With z k+1 Correspondingly, z k+1 ={1, 2, 3} indicates that the individual is in M k+1 The gene at the marker can only come from the first three parents, so the corresponding I k+1 Only the first, second, and third elements on the diagonal are 1, and all other elements are 0. That is, in z... k+1 When the value is {1, 2, 3}, the individual is marked M. k Transfer to M k+1 The conditional probability at x k+1 In the case of 4, the probability of moving to the fourth parent is 0. More simply, we define H... k+1 (r k )=H(r k )I k+1 ,therefore

[0082]

[0083] The same

[0084]

[0085] With z k Corresponding matrix I k also with I in k+1 With z k+1 They have a similar correspondence.

[0086] At this point, the individual is in M k The probability that the marker at a given location originates from a different parent has been obtained using the Markov chain algorithm, i.e., p. k Combined with Mk p where there are individuals k ,remember For the i-th individual in M k The probability p of the parental origin at the location k Therefore, M can be obtained in the four-parent homozygous MAGIC population. k Parental origin probability matrix of all individuals Combined with the additive effect orthogonal coefficient matrix mentioned in the previous section M can be obtained k The coefficient matrices of the additive effects α1, α2, and α3 after orthogonal decomposition Similarly, the coefficient matrices for eight-parent and sixteen-parent lines can be derived from... and Furthermore, the above-mentioned Markov chain-based method for inferring parental origin probability is also applicable to markers with missing genotypes, or to QTLs to be tested between two markers.

[0087] (3) Establishment of statistical genetic models

[0088] Suppose a homozygous MAGIC population of size n with four parents (eight or sixteen parents are also acceptable, but we will use four parents as an example here). Genetic experiments are conducted under p different environments, and it is assumed that the variation of the target trait is influenced by s QTLs (abbreviated as Q1, Q2, ..., Q5). s The model is controlled by t pairs of QTLs interacting. The coefficient matrix is ​​obtained by combining the orthogonal decomposition of QTL genetic effects and the Markov chain algorithm, and the following hybrid linear model is constructed:

[0089]

[0090] Where y ij Let μ be the phenotypic value of the i-th individual in the j-th environment, and μ be the population mean; k For Q k The additive effect vector (α) after orthogonal decomposition 1k α 2k α 3k ) T (Fixed effect), x ki For the i-th individual located in Q k The coefficient vector of the additive effect at the location x hi For the i-th individual located in Q h The coefficient vector of the additive effect at point aa kh For Q k With Q h The summation of the central effect vector (αα) after orthogonal decomposition 1kh ,...,αα 9kh )T (Fixed effect); e j Represents the main effect (random effect) of the j-th environment; ae kj For Q k The additive effect vector of the interaction between the environment and the environment (αe) 1kj αe 2kj αe 3kj ) T (Random effects); aae khj For Q k With Q h The interaction vector between the centrality and the environment (αe) 1khj ,ααe 2khj , ..., ααe 9khj ) T (Random effects); ε ij For residuals, The Kronecker product of vectors, for example

[0091] Its matrix expression is as follows:

[0092]

[0093] Where y is an n×1 phenotypic value vector; μ is the population mean vector; and b is the effect vector. A =[a1 T a2 T , ..., a s T ] T b AA =[aa1 T aa2 T , ..., aa t T ] T , and The corresponding coefficient matrices are X A X AA U E , and (The coefficient matrix of fixed effects is as follows) For example, among which The remaining coefficients can be obtained similarly; for the coefficient matrix of random effects, we have... For example, U E (This is the coefficient matrix corresponding to the main environmental effects); Let I(U) be an n×1 residual effect vector; r+1Let be an n×n identity matrix. Specifically, when constructing the genetic model corresponding to a sixteen-parent homozygous MAGIC population, the additive symptom effect aa needs to be included. v As a random effect, and changing its corresponding matrix form X AA b AA for

[0094] (4) QTL mapping and genetic parameter estimation

[0095] Based on the corresponding coefficient matrix, one-dimensional and two-dimensional scans of QTLs were performed across the entire genome, and stepwise regression was used for model selection to eliminate false positive QTLs. To estimate the genetic parameters of the detected QTLs, the least second-order unbiased estimation (MINQUE) method was used to estimate the variance components of each random effect in the mixed linear full model. Based on the variance component estimates, the adjusted unbiased prediction method (AUP) was used to predict the random effect values; the generalized least squares (GLS) method was used to estimate the fixed effects in the model. The obtained estimates were used as prior values ​​in the Markov chain Monte Carlo process. Based on the Gibbs sampling technique, the distribution samples of each parameter were extracted, and the parameters were estimated and significance tests were performed to make corresponding statistical inferences. This computational framework of one-dimensional and two-dimensional scanning combined with parameter estimation has been successfully applied to the QTL localization analysis of complex traits in biparental breeding populations (Yang et al., Bioinformatics, 2007, 23(12), 1527-1536).

[0096] (5) Other

[0097] The innovation of this invention compared with the prior art is as follows:

[0098] a) The addition variance and the addition of central variance were decomposed using orthogonal decomposition, which improved the localization power of QTL and reduced the false positive rate.

[0099] b) Compared with existing MAGIC population QTL mapping and genetic parameter estimation models that do not include epistasis and gene-environment interaction effects, the model proposed in this method includes epistasis and environmental interaction effects, which is more consistent with the genetic characteristics of complex traits and can more effectively resolve the genetic structure of complex traits.

[0100] c) The model proposed in this invention is a hybrid linear model, which has great flexibility and can be easily extended to more parents and MAGIC populations of any generation.

[0101] The invention will be further explained below with specific examples.

[0102] Example 1

[0103] (1) Software

[0104] Based on the existing software QTLNetwork2.0 (http: / / ibi.zju.edu.cn / software / ), a CPU-parallelized QTLNetwork-MP was developed using C++ to locate additive, additive-additivity, and interaction QTLs between these two populations and the environment in homozygous MAGIC populations (RIL and DH populations) with four, eight, and sixteen parents, and to estimate the corresponding genetic parameters.

[0105] (2) Materials

[0106] The performance of the new QTL mapping and parameter estimation method for the new MAGIC population was tested through simulation (using the software QTLNetwork-MP). The R package mpMap2 was used to simulate a four-parental recombinant inbred line MAGIC population with a population size of 500. The linkage group number was set to 3, the genetic distance of each linkage group was 200 cM, and it contained 201 uniformly distributed markers (i.e., the distance between adjacent markers was 1 cM). The specific simulation strategy is as follows: Figure 1 As shown, a total of 9 QTLs were simulated: 1-20, 1-100, 1-180, 2-10, 2-100, 2-170, 3-10, 3-90, and 3-170. These included 4 additive QTLs and 3 pairs of additive-similarity QTLs (1-20 represents the QTL at 20 cM on the first linkage group). Three heritability levels were set: 0.4, 0.6, and 0.8. The corresponding residual variances were calculated using the heritability formula. And according to the normal distribution The simulation yields individual residuals, which in turn lead to individual phenotypic values. It's important to note that the simulated markers are encoded as 1, 2, 3, and 4 (meaning the QTL originates from the first, second, third, and fourth parents), rather than traditional molecular markers like SNPs. Therefore, after generating a complete phenotype using markers encoded by parental origin, for each marker, we need to convert the parental-encoded marker into SNP form AA or aa. This allows the offspring to also obtain their own SNP markers, more closely resembling the situation in real-world data analysis where most markers or QTLs do not have directly known parental origins. This further verifies the accuracy of the Markov algorithm—the accuracy of inferring parental origin from molecular markers (such as SNPs).

[0107] (3) Results

[0108] We compiled the performance of the new mapping method on simulated data under three different genetic structure assumptions. As shown in Table 3-8, based on different heritability assumptions, in most cases, both additive and additive-augmented QTLs achieved good statistical power (greater than 0.8). The simulation results were as expected: the statistical power of QTLs increased with the increase of the proportion of genetic variance and the increase of overall heritability, while the false positive rate (FDR) decreased accordingly. Among them, the FDR of additive-augmented QTL mapping, which involves a pair of QTLs, was much higher than that of single additive QTL mapping. In terms of genetic parameter estimation, the estimated values ​​of additive and additive-augmented effects were good in line with the simulation preset values. However, since additive-augmented effects require the combination of the parental origin probability matrix of two loci (the error of Markov chain inference is multiplied) and the estimated effect number is four times that of additive effects (in four parents), the accuracy of the additive effect estimation is much better than that of the additive-augmented effect. Furthermore, it can be seen that when gene interactions account for a certain genetic proportion in the genetic structure of complex traits, their statistical power is also quite considerable. If the detected additive sympathetic QTLs can be included in the model, not only can the genetic explanation rate be improved, but it will also be more in line with the biological laws of inheritance, leading to a more accurate and in-depth understanding of genes.

[0109] Table 3. Simulation preset values ​​for four additive QTL effects

[0110] QTL 1-20 1-100 1-180 2-10 a1 1.00 1.41 1.73 1.41 a2 -2.00 -2.83 -3.46 -2.83 a3 -3.00 -4.24 -5.2 -4.24 a4 4.00 5.66 6.93 5.66

[0111] Table 4. Simulation results of four additive QTLs at different heritability rates (0.4 and 0.6).

[0112]

[0113] Table 5 Simulation results of four additive QTLs at a heritability of 0.8

[0114]

[0115] Table 6. Simulation preset values ​​for three pairs of additive contingent QTL effects.

[0116]

[0117]

[0118] Table 7. Simulation results of three pairs of additive sympathetic QTLs at different heritability rates (0.4 and 0.6).

[0119]

[0120]

[0121] Table 8. Simulation results of three pairs of additive sympathetic QTLs at a heritability of 0.8.

[0122]

[0123]

[0124] (4) Other

[0125] Finally, it is particularly important to note that the examples given above are merely specific embodiments of the present invention. Obviously, the present invention is not limited to the above embodiments, and many variations are possible. All variations that can be directly deduced or conceived by those skilled in the art from the content disclosed in this invention are considered to be within the scope of protection of this invention.

Claims

1. A QTL mapping method for homozygous multi-parent high-generation interbreeding populations, characterized in that, Includes the following steps: (1) Orthogonal decomposition of genetic effects: Based on the Cockerham genetic model, the total genetic variance is decomposed into the variance components of each genetic effect. Then, combined with the constraints satisfied by each effect, pairwise orthogonal coefficient vectors are constructed to obtain the orthogonal coefficient matrix of the homozygous multi-parent high-generation intercross population. The aforementioned genetic effects are additive and additive in situ genetic effects; (2) Construction of the parental origin probability matrix: Using the recombination information between markers, the probability that each marker and the QTL between markers originate from each parent is obtained through the Markov chain algorithm. A parent origin probability matrix is ​​constructed, and the effect coefficient matrix is ​​constructed by combining the corresponding orthogonal coefficient matrix in step (1). (3) Establishment of statistical genetic models: Size is A homozygous multiparent high-generation interbreeding population, in Genetic experiments were conducted under different environments, and the variation of the target trait was affected. A QTL control, abbreviated as , which contains For the interactions between QTLs, the coefficient matrix is ​​obtained by combining the orthogonal decomposition of QTL genetic effects and the Markov chain algorithm, and the following mixed linear model is constructed: ； in For the i-th individual in the i-th... Phenotypic values ​​in an environment, It is the group mean; for The additive effect vector after orthogonal decomposition is taken as the fixed effect. For the first Individuals located at The coefficient vector of the additive effect at the location, For the first Individuals located at The coefficient vector of the additive effect at the location, for and The summation of the intermediate effects vector after orthogonal decomposition is used as a fixed effect. Indicates the first The main effects of each environment are treated as random effects. for With the environment The additive effect vector between the environment and the environment is considered as a random effect. for and The addition of superordinate and environmental factors The interaction effect vector is taken as a random effect. For residuals, The Kronecker product represents the vector; Obtain the corresponding coefficient matrix; (4) QTL mapping and estimation of its genetic parameters: Based on the corresponding coefficient matrix in step (3), one-dimensional and two-dimensional scans of QTLs are performed across the entire genome to obtain QTLs and their corresponding genetic parameters.

2. The QTL mapping method for homozygous multi-parent high-generation interbreeding populations according to claim 1, characterized in that, When one QTL in a high-generation intercross population of homozygous polyzygotic parents has multiple different genotypes from the four parents sequentially, the effect value of each genotype is represented by the following unit point model: ； in For the first Genotype values ​​of homozygous parents For the first The additive effect values ​​of homozygous genotypes of each parent. , , These are the additive effects after orthogonal decomposition. , , The corresponding coefficients; satisfying the constraints. ; When two QTLs in a high-generation intercross population of homozygous polyzygotic parents have four different genotypes from each of the four parents, the genotype value for each pair of genotype combinations is represented by the following two-locus model: ； in The values ​​represent the combined values ​​of four homozygous genotypes at each of the two loci, for a total of 16 effect values. , , These are the effect values ​​of the additive effect of the first QTL, the additive effect of the second QTL, and the sum of the sum of the sum of the sums ... Satisfying the constraints .

3. The QTL mapping method for homozygous multi-parent high-generation interbreeding populations according to claim 2, characterized in that, When the high-generation interbreeding population of homozygous multiparents is an eight-parent homozygous multiparent high-generation interbreeding population or a sixteen-parent homozygous multiparent high-generation interbreeding population, the corresponding orthogonal coefficient matrix is ​​recursively generated using the Hadamard matrix.

4. The QTL mapping method for homozygous multi-parent high-generation interbreeding populations according to claim 1, characterized in that, In step (2), the markers also include markers with missing genotypes, and the QTLs between markers also include QTLs between the two markers to be tested.

5. The QTL mapping method for homozygous multi-parent high-generation intercross populations according to claim 1, characterized in that, In step (3), the corresponding coefficient matrix expression is as follows: ； Where 𝐲 is The phenotypic value vector; 𝛍 is the population mean vector; effect vector , , , and The corresponding coefficient matrices are respectively , , , and ; for The residual effect vector; for The identity matrix.

6. The QTL mapping method for homozygous multi-parent high-generation intercross populations according to claim 5, characterized in that, When constructing a statistical genetic model for a high-generation intercross population of sixteen homozygous parents, the effect of situation will be added. As a random effect, the corresponding coefficient matrix expression Replace with , .

7. The QTL mapping method for homozygous multi-parent high-generation interbreeding populations according to claim 1, characterized in that, In step (4), in order to estimate the genetic parameters of the detected QTL, the least second-order unbiased estimation method is used to estimate the variance components of each random effect in the mixed linear full model, and the adjusted unbiased prediction method is used to predict the random effect value based on the variance component estimate; the generalized least squares method is used to estimate the fixed effects in the model.

8. The QTL mapping method for homozygous multi-parent high-generation interbreeding populations according to claim 7, characterized in that, The obtained variance component estimates are used as prior values ​​in the Markov chain Monte Carlo process. Based on the Gibbs sampling technique, distribution samples of each parameter are drawn, and the parameters are estimated and significance tests are performed to make corresponding statistical inferences.

9. The QTL mapping method for homozygous multi-parent high-generation interbreeding populations according to claim 1, characterized in that, Step (4) also includes using stepwise regression to select the model and eliminate false positive QTLs.

10. A system for implementing the QTL mapping method for homozygous multi-parent high-generation interbreeding populations as described in claim 1, characterized in that, The statistical genetic model is as follows: ； in For the first The individual in the first Phenotypic values ​​in an environment, It is the group mean; for The additive effect vector after orthogonal decomposition is taken as the fixed effect. For the first Individuals located at The coefficient vector of the additive effect at the location For the first Individuals located at The coefficient vector of the additive effect at the location, for and The summation of the intermediate effects vector after orthogonal decomposition is used as a fixed effect. Indicates the first The main effects of each environment are treated as random effects. for With the environment The additive effect vector between the environment and the environment is considered as a random effect. for and The addition of superordinate and environmental factors The interaction effect vector is taken as a random effect. For residuals, The Kronecker product represents the vector; Based on the statistical genetic model, the corresponding coefficient matrix is ​​obtained.

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