CN108197435B - Marker locus genotype error-containing multi-character multi-interval positioning method - Google Patents

Marker locus genotype error-containing multi-character multi-interval positioning method Download PDF

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CN108197435B
CN108197435B CN201810083627.6A CN201810083627A CN108197435B CN 108197435 B CN108197435 B CN 108197435B CN 201810083627 A CN201810083627 A CN 201810083627A CN 108197435 B CN108197435 B CN 108197435B
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佟良
周影
马春华
孙晓霞
邹大伟
付丽
耿艳秋
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Suihua University
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Abstract

The invention provides a marker locus genotype-based multi-character multi-interval positioning method with errors, which comprises the following steps: obtaining a molecular genetic map; determining the recombination rate gamma between the marks at both sides of the ith mark intervaliAnd the recombination rate gamma between the QTL and the upper marker locus in the ith marker intervali1(ii) a According to the recombination rate gamma between the marks at both sides of the ith mark intervaliAnd the recombination rate gamma between the QTL and the upper marker locus in the ith marker intervali1Generating marker genotypes and QTL genotypes; obtaining a marker genotype containing errors according to the configured error rate; configuring a parameter true value; obtaining a phenotype character observation value through the model; configuring initial values of parameters; iterating the parameter expression deduced by the EM algorithm until convergence; and repeating the loop for N times to obtain the average value and the mean square error of the parameter estimation as target values. The invention can well solve the problem that the marker gene information contains errors, and can estimate the error rate of the marker gene information.

Description

Marker locus genotype error-containing multi-character multi-interval positioning method
Technical Field
The invention belongs to the technical field of molecular biology, and particularly relates to a marker locus genotype-based error-containing multi-character multi-interval positioning method.
Background
Human genetic diseases and crop phenotypic traits are almost related to genetic genes, and for a long time, most studies neglect the situation that the marker genotype contains errors for the convenience of the study. However, due to the precision of the apparatus, there is a possibility that the information of the marker gene has an error. In order to prevent genetic diseases and better utilize beneficial genes in germplasm resources, a novel QTL positioning method is provided. The analysis method can well solve the problem of multi-character gene positioning of the marker genotype with errors.
Disclosure of Invention
The invention can well solve the problem that the marker gene information contains errors, and can estimate the error rate of the marker gene information. Therefore, the method can be used for estimating the number, the position and the effect of the QTL influencing the phenotypic character more accurately under the condition of known phenotypic values and marker gene information containing errors.
The invention can be realized by adopting the following method:
a multi-character multi-interval positioning method based on marker locus genotype containing errors comprises the following steps:
step one, obtaining a molecular genetic map;
step two, determining the recombination rate gamma between the marks at the two sides of the ith mark intervaliAnd the recombination rate gamma between the QTL and the upper marker locus in the ith marker intervali1；
Step three, according to the recombination rate gamma between the marks at both sides of the ith mark intervaliAnd the recombination rate gamma between the QTL and the upper marker locus in the ith marker intervali1Generating marker genotypes and QTL genotypes;
step four, obtaining a marker genotype containing errors according to the configured error rate;
step five, configuring a true value of a parameter;
sixthly, obtaining a phenotype character observation value through the model;
step seven, configuring initial values of the parameters;
step eight, iterating the parameter expression deduced by the EM algorithm until convergence;
and step nine, repeating the loop for N times to obtain the average value and the mean square error of the parameter estimation as target values.
Further, the observed value of the phenotypic character is obtained through a model, specifically through a statistical model
Figure BDA0001561745610000021
Obtaining an observed value of the phenotypic trait, wherein,
Figure BDA0001561745610000022
is a phenotypic character matrix, q is q intervals closely linked,
Figure BDA0001561745610000023
and
Figure BDA0001561745610000024
respectively represent ith QTL genotype indicative vectors of n individuals, then,
ξji＝1，ηji-1/2 when
Figure BDA0001561745610000025
ξji＝0，ηji1/2, when
Figure BDA0001561745610000026
ξji＝-1，ηji-1/2 when
Figure BDA0001561745610000027
ai＝(ai1,ai2...ait) And di＝(di1,di2...dit) Respectively representing additive effect and dominant effect vectors of the ith QTL on the t phenotypic characters; e ═ Eji}n×tIs a residual matrix, here ejiIs the random error of the ith phenotypic characteristic of the jth individual with a mean of 0, cov (e)ji,ejl)＝σil＝ρσiσl,i,l＝1,...,t。
Further, the additive effect and dominant effect vector of the ith QTL on the t phenotypic characters form a QTL effect matrix of
Figure BDA0001561745610000028
In summary, the present invention can be implemented by the following method:
a multi-character multi-interval positioning method based on marker locus genotype containing errors comprises the following steps: step one, obtaining a molecular genetic map; step two, determining the recombination rate gamma between the marks at the two sides of the ith mark intervaliAnd the recombination rate gamma between the QTL and the upper marker locus in the ith marker intervali1(ii) a Step three, according to the recombination rate gamma between the marks at both sides of the ith mark intervaliAnd the recombination rate gamma between the QTL and the upper marker locus in the ith marker intervali1Generating marker genotypes and QTL genotypes; step four, obtaining a marker genotype containing errors according to the configured error rate; step five, configuring a true value of a parameter; sixthly, obtaining a phenotype character observation value through the model; step seven, configuring initial values of the parameters; step eight, iterating the parameter expression deduced by the EM algorithm until convergence; and step nine, repeating the loop for N times to obtain the average value and the mean square error of the parameter estimation as target values.
The beneficial effects are that:
1. the genotype of the marker locus is considered to contain errors, and the positioning is more accurate.
2. The precision degree of an instrument for obtaining the marking information is not required to be the best, and the cost is saved.
3. A parametric estimate of the marker locus genotype error rate is given.
Drawings
FIG. 1 is a diagram of a random simulation structure of a multi-character multi-interval positioning method based on errors of marker locus genotypes, provided by the invention;
FIG. 2 is a diagram illustrating QTL effect estimates and mean square error values for different heritability;
FIG. 3 is a schematic diagram of recombination rate and covariance matrix estimation for different heritability;
FIG. 4 shows QTL mapping results for gallstone formation.
Detailed Description
The present invention provides an embodiment of a multi-trait multi-interval localization method based on errors in marker locus genotypes, and in order to make the technical solutions in the embodiments of the present invention better understood and make the above objects, features and advantages of the present invention more obvious and understandable, the technical solutions of the present invention are further described in detail below with reference to the accompanying drawings:
the invention firstly provides a multi-character multi-interval positioning method based on marker locus genotype containing errors, as shown in figure 1, comprising the following steps:
s101, acquiring a molecular genetic map;
the data in this example is from Wittenburg et al 2003, a data set containing 305F 2 children. Phenotypic data selection for gallstone (gallstone) weight and High-density lipoprotein (High-density lipoprotein), markers D4Mit31 and D4Mit126 are located at chromosome 4, 51.3cM, 71cM, respectively, markers D10Mit66 and D10Mit34 are located at chromosome 10, 49cM, 62cM, respectively, and 4 marker loci constitute two marker intervals.
S102, step two, determining recombination rate gamma between marks at two sides of the ith mark intervaliAnd the recombination rate gamma between the QTL and the upper marker locus in the ith marker intervali1；
Wherein, suppose F2Each marker interval of the population has at most one QTL, the ith marker interval of the jth individual
Figure BDA0001561745610000041
When known, the QTL genotype within the marker interval
Figure BDA0001561745610000042
Conditional probability
Figure BDA0001561745610000043
See table 1.
TABLE 1 conditional probability of QTL genotypes given the known genotype of the markers
Figure BDA0001561745610000044
ri＝γi1/γi
S103, configuring initial values of the parameters;
and step four, iterating the parameter expression deduced by the EM algorithm until convergence, and obtaining a parameter estimation value as a final result.
Preferably, the statistical model is
Figure BDA0001561745610000045
Wherein the content of the first and second substances,
Figure BDA0001561745610000046
is a phenotypic character matrix, q is q intervals closely linked,
Figure BDA0001561745610000047
and
Figure BDA0001561745610000048
respectively represent ith QTL genotype indicative vectors of n individuals, then,
ξji＝1，ηji-1/2 when
Figure BDA0001561745610000049
ξji＝0，ηji1/2, when
Figure BDA0001561745610000051
ξji＝-1，ηji-1/2 when
Figure BDA0001561745610000052
ai＝(ai1,ai2...ait) And di＝(di1,di2...dit) Are respectively provided withExpressing additive effect and dominant effect vectors of the ith QTL on the t phenotypic characters; e ═ Eji}n×tIs a residual matrix, here ejiIs the random error of the ith phenotypic characteristic of the jth individual with a mean of 0, cov (e)ji,ejl)＝σil＝ρσiσl,i,l＝1,...,t。
Preferably, the QTL effect matrix formed by the additive effect and dominant effect vectors of the ith QTL on the t phenotypic characters is
Figure BDA0001561745610000053
Here, based on the EM algorithm, detailed inference is given for the parameter Ω ═ C, Σ, γ, θ in the present embodiment. The method comprises the following specific steps:
with respect to the inference of the likelihood function, the full likelihood function with respect to the parameter vector Ω can be expressed in the form
Figure BDA0001561745610000054
Having a complete log-likelihood function of
Figure BDA0001561745610000055
Where Y isj＝(Yj1,...,Yjt)，Xj＝(Xj1,...,Xj(q+1))，
Figure BDA0001561745610000056
Wherein Y isji(j 1., n, i 1., t) denotes the ith phenotypic trait value of the jth individual, Xji(j 1, 1., n, i 1.,. q +1) and
Figure BDA0001561745610000061
respectively representing the marker genotype of the ith marker of the jth individual and the marker genotype containing errors,
Figure BDA0001561745610000062
indicates the QTL genotype within the i-th marker interval of the j-th individual.
Further with respect to the calculation of the Q function,
Figure BDA0001561745610000063
further calculations regarding the a posteriori probability are made such that,
Figure BDA0001561745610000064
further make
Figure BDA0001561745610000065
Here, the
Figure BDA0001561745610000066
Represents Xj,Yj,Ω(s)Under the condition of
Figure BDA0001561745610000067
The conditional probability of the kth value of (a). Order to
Figure BDA0001561745610000068
Expressing the genotype error rate, and further
Figure BDA0001561745610000069
Representing the joint error rate of the jth individual. In each iteration, when true genotype XjGiven a value compared to genotype
Figure BDA0001561745610000071
Number k of erroneous gene codes in q +1 marker locijIs calculable.
Deriving an iterative expression of the QTL effect matrix C by derivation:
Figure BDA0001561745610000072
r in this case(s)And M(s)Is expressed as
Figure BDA0001561745610000073
Figure BDA0001561745610000074
Where # denotes the hadamard product of the two vectors. The iterative formula of Σ is:
Figure BDA0001561745610000075
Figure BDA0001561745610000076
is an n x 3qD ═ D (D)1,D2,D3,...,Dcq) When considering additive and dominant effects, c is 2.
When c is 2
Figure BDA0001561745610000081
QTL genotype and indicative function of marker interval in which the genotype is located
Figure BDA0001561745610000082
Having the following expression
Figure BDA0001561745610000083
Where j is 1,., n, i is 1,., q. The explicit expression of the recombination rate γ can be obtained by the above indicative function and Q function
Figure BDA0001561745610000084
Error rate theta explicit expression can be derived through Q function
Figure BDA0001561745610000085
The results of the tests presented in FIG. 4 show the presence of QTLs affecting gallstone weight and High Density Lipoprotein (HDL) at chromosome 10, 56cM, and chromosome 4, 62 cM. QTL in the 56cM position has the function of inhibiting the formation of gallstone and promoting High Density Lipoprotein (HDL). QTL in 62cM position has the action of promoting the formation of gallstone and inhibiting high-density lipoprotein (HDL).
The invention relates to a marker locus genotype-based multi-character multi-interval positioning method with errors, which comprises the following steps of: step one, obtaining a molecular genetic map; step two, determining the recombination rate gamma between the marks at the two sides of the ith mark intervaliAnd the recombination rate gamma between the QTL and the upper marker locus in the ith marker intervali1(ii) a Step three, according to the recombination rate gamma between the marks at both sides of the ith mark intervaliAnd the recombination rate gamma between the QTL and the upper marker locus in the ith marker intervali1Generating marker genotypes and QTL genotypes; step four, obtaining a marker genotype containing errors according to the configured error rate; step five, configuring a true value of a parameter; sixthly, obtaining a phenotype character observation value through the model; step seven, configuring initial values of the parameters; step eight, iterating the parameter expression deduced by the EM algorithm until convergence; and step nine, repeating the loop for N times to obtain the average value and the mean square error of the parameter estimation as target values.
The invention can well solve the problem that the marker gene information contains errors, and can estimate the error rate of the marker gene information. The Quantitative Trait Locus (QTL) positioning method based on the marker genotype error enables people to estimate the number, the position and the effect of the QTL (quantitative trait locus) influencing the phenotypic trait more accurately under the condition of the known phenotypic value and the marker gene information containing errors.
The above is a specific example of the multi-trait multi-interval localization method based on the error-containing marker locus provided by the present invention, and the example is only used to help understand the method of the present invention and its core idea, and the content of the present specification should not be construed as a limitation to the present invention. All modifications which can be derived or suggested by a person skilled in the art from the disclosure of the invention are considered to be within the scope of the invention.

Claims (3)

1. A multi-character multi-interval positioning method based on marker locus genotype containing errors is characterized by comprising the following steps:
step one, obtaining a molecular genetic map;
step two, determining the recombination rate gamma between the marks at the two sides of the ith mark intervaliAnd the recombination rate gamma between the QTL and the upper marker locus in the ith marker intervali1；
Step three, according to the recombination rate gamma between the marks at both sides of the ith mark intervaliAnd the recombination rate gamma between the QTL and the upper marker locus in the ith marker intervali1Generating marker genotypes and QTL genotypes;
step four, obtaining a marker genotype containing errors according to the configured error rate; based on the EM algorithm, the parameters Ω ═ C, Σ, γ, θ are estimated in detail as follows:
with respect to the inference of the likelihood function, the full likelihood function with respect to the parameter vector Ω can be expressed in the form:
Figure FDA0003465017610000011
the complete log-likelihood function is:
Figure FDA0003465017610000012
Yj＝(Yj1,...,Yjt)，Xj＝(Xj1,...,Xj(q+1))，
Figure FDA0003465017610000013
wherein Y isji(j 1., n, i 1., t) denotes the ith phenotypic trait value of the jth individual, Xji(j 1, 1., n, i 1.,. q +1) and
Figure FDA0003465017610000014
respectively representing the marker genotype of the ith marker of the jth individual and the marker genotype containing errors,
Figure FDA0003465017610000015
(ii) represents the QTL genotype within the ith marker interval of the jth individual;
further calculations regarding the Q function are:
Figure FDA0003465017610000016
Figure FDA0003465017610000021
further calculations regarding the posterior probability are:
Figure FDA0003465017610000022
further make
Figure FDA0003465017610000023
Wherein the content of the first and second substances,
Figure FDA0003465017610000024
represents Xj,Yj,Ω(s)Under the condition of
Figure FDA0003465017610000025
Conditional probability of the kth value of (1), order
Figure FDA0003465017610000026
Expressing the genotype error rate, and further
Figure FDA0003465017610000027
Representing the joint error rate of the jth individual; in each iteration, when true genotype XjGiven a value compared to genotype
Figure FDA0003465017610000028
Number k of erroneous gene codes in q +1 marker locijIs calculable;
deriving an iterative expression of the QTL effect matrix C by derivation:
Figure FDA0003465017610000029
wherein R is(s)And M(s)The expression of (a) is:
Figure FDA00034650176100000210
Figure FDA0003465017610000031
where # denotes the Hadamard product of two vectors; the iterative formula of Σ is:
Figure FDA0003465017610000032
wherein the content of the first and second substances,
Figure FDA0003465017610000033
is an n x 3qD ═ D (D)1,D2,D3,...,Dcq) When considering additive effect and dominant effect, c is 2, and when c is 2
Figure FDA0003465017610000034
QTL genotype and indicative function of marker interval in which the genotype is located
Figure FDA0003465017610000035
Having the following expression:
Figure FDA0003465017610000036
where j 1., n, i 1., Q, an explicit expression of the recombination rate γ can be obtained from the above indicative function and the Q function
Figure FDA0003465017610000037
An error rate thetajdominant expression can be derived by the Q function:
Figure FDA0003465017610000038
step five, configuring a true value of a parameter;
sixthly, obtaining a phenotype character observation value through the model;
step seven, configuring initial values of the parameters;
step eight, iterating the parameter expression deduced by the EM algorithm until convergence;
and step nine, repeating the loop for N times to obtain the average value and the mean square error of the parameter estimation as target values.
2. The method of claim 1, wherein the model-based multiple trait multiple interval localization with errors in genotype at marker loci is used to obtain phenotypic trait observations, in particular by statistical modeling
Figure FDA0003465017610000041
Obtaining an observed value of the phenotypic trait, wherein,
Figure FDA0003465017610000042
is a phenotypic character matrix, q is q intervals closely linked,
Figure FDA0003465017610000043
and
Figure FDA0003465017610000044
respectively represent ith QTL genotype indicative vectors of n individuals, then,
ξji＝1，ηji-1/2 when
Figure FDA0003465017610000045
ξji＝0，ηji1/2, when
Figure FDA0003465017610000046
ξji＝-1，ηji-1/2 when
Figure FDA0003465017610000047
ai＝(ai1,ai2...ait) And di＝(di1,di2...dit) Respectively representing additive effect and dominant effect vectors of the ith QTL on the t phenotypic characters; e ═ Eji}n×tIs a residual matrix, ejiIs the random error of the ith phenotypic characteristic of the jth individual with a mean of 0, cov (e)ji,ejl)＝σil＝ρσiσl,i,l＝1,...,t。
3. The method of claim 2, wherein the additive effect of the ith QTL on the t phenotypic traits and the dominant effect vector form a QTL effect matrix comprising QTL effect matrices of
Figure FDA0003465017610000048
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Inventor before: Geng Yanqiu

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