Statistical Properties of the Correlation Method for Mapping QTL in Double Haploid Population Hong Li Department of Biological Engineering Chongqing technology and Business University Chongqing ,China lhctbu163.com) Abstract: This paper analyzed the statistical properties of the correlation method used for mapping quantitative trait loci (QTL) in double haploid (DH) population. This method has considered the effect of chiasma interference in the condition of double crossing over. It makes the mapping of QTL more precise and reliable. It discussed the test of two linkage QTLs, and the estimation of the epistasis deviation between QTLs using the correlation coefficient R when there are several QTLs affect the quantitative trait of interest and not linkage with each other. Keywords: marker loci; DH population; correlation coefficient; chiasma interference; epistasis 1. Introduction The methods of mapping quantitative trait loci have been studied by many scholars. The statistical methods include mean and variance analysis, maximum likelihood estimation, moments estimation, correlation analysis and likelihood ratio test, etc. 12. In the research of correlation method, Hu et al. (1995) put forward the correlation method of mapping QTL using single marker locus in RIL and DH populations, and further extended the situation of flanking markers 3. Although the correlation method has some advantages, it didnt consider the effect of chiasma interference to the mapping of QTL. In order to perfect the correlation method of QTL mapping using flanking markers, the author further puts forward the correlation method of mapping QTL using flanking markers in the condition of chiasma interference, and analyzes the mathematical property of the correlation coefficient through numerical simulation. 2. Material and method A Linkage model and genetic materials According 3, the flanking markers loci are A-a, B-b; QTL is Q-q. If the three loci are linkage, there are three linkage sequences. As the II type and III type linkage sequences are similar, we only consider the expected genotypic frequencies of the I type and II type linkage sequences in DH population. According to Haldane mapping function, there are in the I type linkage sequence, and in the II type linkage sequence. Here, C is the coefficient of coincidence. ra and rb are the recombination values between QTL and marker A and marker B, respectively. B Linkage examination of flanking markers and one QTL According to the method of 2, we give an assignment value to flanking marker as follow: AABB=2, AAbb=0, aaBB=0, aabb=2. The value of QTL is the observed value of quantitative trait. When there is only one QTL affecting the quantitative trait, and no environmental factors effect, we may obtain the simple correlation coefficient between flanking markers loci and QTL from the assignment pair-wise data. assignment pair-wise data. The genetic expected value of the correlation coefficient R is: I type linkage sequence (1) II type linkage sequence (2) Here, nxxyy denotes the observed number of genotype xxyy, and N denotes the observed babaabrCrrrr2+=babbabarCrrrr2+=( )()()() ()()NrrrrrrCrnnrCrrrREbababaabaaBBAAbbbaba21111+=( )()NnnrREaabbAABBb+=21978-1-4244-4713-8/10/$25.00 2010 IEEE number of all genotypes. According statistical knowledge, the sample variance is (3) We may examine the linkage of flanking markers and QTL using the test statistic t. (4) C The test of linkage QTLs When there are two linkage QTLs, we may obtain the correlation coefficient between flanking markers and the two linkage QTLs. If we assume that r1 is the recombination value between marker A and QTL1, and r2 is the recombination value between marker B and QTL2, and rab is the recombination value between marker A and B, then we may obtain the following expression. From the formula, we may know that when , has its maximum value and when , . For a given size of genetic effects, the power of tests for QTL detection is affected by many factors, such as the total number of individuals genotyped in population, the recombination rate between the marker locus and the putative QTL, the missing marker data, and analysis approach, etc. 456789. D The detection of epistasis deviation In the above, we only consider one QTL, but when there are many interaction QTLs, this method cant obtain the precision location of QTLs, but we may use the following method to examine the epistasis deviation between different QTLs. When QTLs didnt linkage with each other, we may obtain the correlation coefficient between flanking markers and QTLs. Here, k denotes the number of QTLs, m is the number of the flanking markers, ra and rb are the recombination values between QTL and marker A and marker B, respectively. is the additiveadditive additive epistasis deviation of different QTLs, and VY is the phenotypic variance of the quantitative trait of interest. E The sample amount required for linkage examination(DH system number) From (4), it may obtain: 3. Result and Conclusion A The characteristics of the correlation coefficient When one QTL links up with K flanking markers in one chromosome, the linkage sequence of QTL and (k-1) flanking markers is II type linkage sequence (or III type linkage sequence), only one flanking markers link up with QTL in I type linkage sequence. When no QTL locate in the chromosome, the simple correlation coefficient is little. It cant arrive the significant level. When there are several QTLs in one chromosome, the correlation coefficient of QTL and flanking markers may arrive the significant level in several flanking markers. It shows that QTL locate in these flanking markers. In the I type sequence, the recombination fraction between flanking markers may be calculated from the observed number: . When , the correlation coefficient is denoted as Rmiddle(I). Substitute rab , ra , rb, into the expression of E(R) in the I type sequence, we obtain the following expression of Rmiddle(I). When , the correlation coefficient has the same value in the I type linkage and the II type linkage. The correlation coefficients that arrive the significant level must larger than . From the statistical knowledge , we may obtain . If rFQ denotes the distance of QTL and the middle point of flanking markers. When , we may obtain (ra=0, or rb=0, =1). It means that QTL is located ()abrR=1()abrR=1()()2122=NRSR( )()()212=NRRRt( )21122+RtRNabFQrr21=CCrrrabba2211=()122,2,2,2+=NNNtNtRNnnraaBBAAbbab+=( )() ()()()()()()()ababababababababmiddleCrCCrrCrCrCrCrCCCrCCCIR2112211221112141211313222+=0,=babarrr()()()21212121222112+=YddababVirrrrrrRE022R0abr21abr22R( )()()()()2121212111111 =+=YdddkmbambambambakVirrCrrrrCrrREkmmmmmmmmkdddi 212,NR on the boundary of flanking markers. When 12FQabrr, rb0 (or ra0), it may obtain . In this condition, (1, 1-2rb1, 1), . When the chiasma interference is negative (C1), . When there is no chiasma interference (C=1, =1), . B Parameter transformation and homologous examination One could transfer the simple correlation coefficient R into normal variable z using the formula: . For a sample with system number n, . In the other word, has the property of 2 distribution with d.f.=1. When there are multiple samples, we may calculate the simple correlation coefficient Ri. It may obtain the transfer variable zi. If is the average mean of zi , the following equation is true: . It may be write as . The 2 value with d.f.=m has been divided into two items. The first item has d.f.=(m-1), it may be used in the homologous test. The second item has d.f.=1, it may be used to examine the significance level of the combined data when the samples are homologous. This method has considered the effect of chiasma interference in the condition of double crossing over. It makes the mapping of QTL more precise and reliable 1011. The method of 3 didnt consider the condition of chiasma interference. It will invalidate the mapping of QTL when the chiasma interference is serious. C Numerical simulation and comparison of two linkage sequences In order to understand the relationship between Rmiddle(I) and rab , C and , we use Monte Carlo method to simulate the change of Rmiddle(I) at different rab , C and . From Table 2, we may know that, when C is infinite, the Rmiddle(I) decreases as rab increasing. But when rab is infinite, the larger C is, the smaller Rmiddle(I) will be. TABLE I. THE RELATIONSHIP BETWEEN Rmiddle(I) AND rab , C AND . rab C rb = ra Rmiddle(I) 0.02 1.5 0.010 0.995 0.9865 0.02 1.0 0.010 1.000 0.9897 0.02 0.5 0.010 1.007 0.9924 0.05 1.5 0.026 0.987 0.9659 0.05 1.0 0.026 1.000 0.9735 0.05 0.5 0.025 1.013 0.9808 0.10 1.5 0.054 0.979 0.9274 0.10 1.0 0.053 1.000 0.9446 0.10 0.5 0.051 1.033 0.9592 0.15 1.5 0.086 0.954 0.8548 0.15 1.0 0.082 1.000 0.9074 0.15 0.5 0.078 1.043 0.9363 0.20 1.5 0.123 0.930 0.7375 0.20 1.0 0.113 1.000 0.8660 0.20 0.5 0.106 1.059 0.9109 0.25 1.5 0.167 0.900 0.6297 0.25 1.0 0.146 1.000 0.8165 0.25 0.5 0.134 1.077 0.8827 0.30 1.5 0.228 0.853 0.5827 0.30 1.0 0.184 1.000 0.7560 0.30 0.5 0.163 1.098 0.8517 0.35 1.5 0.35 1.0 0.226 1.000 0.6794 0.35 0.5 0.194 1.120 0.8169 ()abrR=1()abrR1() ()1 21babRrr12131, 0nNz()23 zn _z21212+=mmCCrrrabba2211=() ()abbrrR=121()abrR=1 00.20.40.60.811.200.10.20.30.4The recombination fraction between markers,rabThe correlation coefficient,Rmiddle(I) Figure1.The value of correlation coefficient Rmiddle(I) at different recombination fraction of markers, rab, and C. We have also analyzed the change of Rrb=0 with rab using numerical simulation. From Table 2, we know that Rrb=0 decreases as rab increases. It means that there is a negative correlation between Rrb=0 and rab. TABLE II. THE VALUES OF Rrb=0 AT DIFFERENT rab rab 0.01 0.03 0.05 0.07 0.10 0.12 0.15 Rrb=0 0.995 0.985 0.975 0.964 0.949 0.938 0.922 rab 0.17 0.20 0.23 0.25 0.27 0.30 Rrb=0 0.911 0.894 0.877 0.866 0.854 0.837 The research of the statistical methods for QTL mapping has obtained many advances in recent years. In the region of correlation method for QTL mapping, there are some unsolved problems. But in the future, researchers will put forward some better statistical methods for QTL mapping, and applied these achievements in the genetic breeding of crops and livestock. 1 Y. Q. Zhou, Q. X. Yang, G. N. Zhang, Biology genetic marker and its application, 1st ed, Chemical Industry Publication, Beijing: 2008, pp. 246-248. 2 Z. Hu, X. Zhang, C. Xie, G. R. McDaniel, D. L. 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Li, “ A correlation method for mapping quantitative trait loci using flanking markers in three-points backcross with cross-interference”, Journal of Biomathematics, vol. 16(4), pp. 473-479, 2001. C=1.5 C=1.0 C=0.5