Comparison of multiple-trait genetic evaluation accuracy using region marker relationships with traditional best linear unbiased prediction ( BLUP )

Accuracy of multiple-trait genetic evaluation based on allelic relationships with traditional best linear unbiased prediction (BLUP) was compared through computer simulation. Firstly, a base population (N = 100) was simulated; the population was half male/half female. After reaching linkage disequilibrium between marker and QTL, phenotypic and genotypic records were simulated for the last generation. For each animal in the base population, three chromosomes (each one Morgan in length) was created; on each chromosome, 200 markers and 50 QTLs were randomly located. Mutation rate in each generation was 2.5 × 10. Total allelic relationships consider variety of genetic relationships among relatives and employ information of non-relatives as well. In evaluation using total allelic relationships, three separate allelic matrixes were required. To form such matrixes, firstly, marker effects were estimated for each trait using mixed model of BLUP. Then, based on these effects, particular markers of each trait as well as common markers among two traits were determined and used to form allelic relationship matrixes. Correlation between actual and estimated breeding values in the last generation was evaluated based on 10 iterations. Statistical comparison was performed using t-student test. A significant difference was observed between evaluation accuracy of studied two methods.


INTRODUCTION
In traditional selection, genetic evaluation is conducted using phenotypic and pedigree information and the relationship matrix is formed based on pedigree information and shows the common portion of genes in two animals (Christensen et al., 1996).In multiple-trait analysis, two or more traits of animals are evaluated simultaneously.For this purpose, genetic and phenotypic correlations are *Corresponding author.E-mail: Rostami_ankasi@yahoo.com.Tel: +989113951564.Fax: +981912226605.
Abbreviations: BLUP, best linear unbiased prediction; TBV, True breeding values; SNPs, single nucleotide polymorphisms; TAR, total allelic relationship; MAS, marker assisted selection; QTL, quantitative loci.used (Mrode, 2005).Henderson and Quaas (1976) used best linear unbiased prediction (BLUP) for multiple-trait evaluation.MAS techniques are applied as a tool for selection in plant and animal association.These techniques use molecular markers to determine QTL genetic pattern (Dekkers and Hospital, 2002).Molecular markers information, which is in relation with QTLs can promote genetic improvement (Soller and Beckman, 1983).One of the problems of marker assisted selection is that it explains only a limited part of total genetic variance (Hayes, 2007).
Practically, single markers are not able to determine the great portions of genotype variance, and MEBV is not accurate enough to do so.For solving this problem, the polygenetic effects (genetic effects, which was not determined by markers) in model are to be considered or multiple markers shall be used.Some markers may be used to determine similar QTL, when using multiple markers for prediction of breeding value (Vander Voort, 2004).
Single nucleotide polymorphisms (SNPs) are the most frequent form of diverse forms of DNA on a genome.Due to their relative less mutation and as their genotype can be determined more easily, therefore they are recommended more than other genetic markers (Kolbehdari et al., 2007).Nejati Javaremi et al. (1997) recommended Total Allelic Relationship (TAR) as an appropriate alternative for additive relationship found from pedigree.They show that use of allelic relationship matrix for the evaluation using BLUP increases the accuracy of evaluation significantly.There are differences between total allele relationship and pedigree method to determine relationship matrix: 1.In pedigree method, the members of base generation are considered non-family members; therefore their inbreeding is regarded as zero.But, in total allelic relationships, members of base generation are to some extent homozygous for specific loci; therefore allelic relationships of animals of the base generation are not zero.2. In pedigree method, the progeny of a specific mating are assigned similar relationship and no diversity is considered.Where-as, total allelic relationships uses the diversity arising out of Mendelian sampling (Nejati Javaremi et al., 1997).
Although many researches have been conducted using simulation for genetic evaluation by single markers, no research has been carried out on multiple-trait genetic evaluation using multiple markers and allelic relationship matrix.In the present research, we used marker genotypes with high content of markers to determine allelic relationships and pedigree relationship matrix was substituted with allelic relationship matrix in mixed model equations of BLUP.The objective of this research is to study the effects of allelic relationship on the accuracy of multiple-trait genetic evaluation using multiple markers and compare the accuracy of evaluation with that of traditional BLUP.

Structure of population
Firstly, we simulated a base population with the statistic size of 100 members, so that half of these animals were male and others were females.This structure was used in a fixed manner for 110 generations.No selection was done for the convenience of conducting research, and mating in each generation was performed randomly.Consequently, each animal has averagely two progenies in the subsequent generation and the statistic size and the effective population size were approximately the same.In several generations, random mating was performed for creating linkage disequilibrium between marker and QTL and the algorithm of Meuwissen and Luo was used for inbreeding calculation.
The population size of last generation was increased by 10 times and phenotypic and genotypic records of animals of the last generation were simulated.

Structure of genome
For each animal of the base population, a genome comprising of three chromosomes, which was one Morgan in length, has been created.On each chromosome, 200 marker loci and 50 QTLs were located randomly.It was assumed that both markers and QTLs were made of two alleles with similar allele frequencies.
Mutation rate of each generation for two-allele markers and QTLs was 2.5 x 10 -5 .Mutations changed allele situation from 1 to 2 and 2 to 1.

Linkage disequilibrium between marker and QTL
The limited size of the effective population (N = 100) created linkage disequilibrium between marker and QTL.To calculate r 2 , the following formula was used:

Genotypic information
Genotypic information of the animals of the last generation was simulated.For this purpose, the effects of QTL for each trait were sampled from the standard deviation of the genotypic value of the loci of that trait (a_sd).The standard deviation of the genotypic value of the second trait (a_sd2) were determined using regression coefficient and in form of a function of the standard deviation of the genotypic value of the first trait (a_sd1), in order to create the genetic correlation among traits.Moreover, the effects of QTL of the traits were sampled using normal distribution.

Phenotypes simulation
As there was no selection, it was not needed to simulate phenotypic records for all generations and phenotypic records were simulated only for the animals of the last generation.Firstly, the true breeding values (TBV) of animals were simulated for each trait.True breeding values were determined in form of sum of allelic locating effects of different loci of each animal.These breeding values were used to determine phenotypes.For each animal, two phenotypic records (P1 and P2) were simulated.Contrary to univariate evaluation, in which phenotypic value is determined by adding a sampled error from normal distribution to true breeding value, the multiple-trait evaluation uses vector of phenotypic values, vector of true breeding values, vector of random numbers and Cholesky decomposition of environmental variance.P = TBV + e + µ e = Tz R e = ) var( ) , 0 ( ~2 ~e N e σ P = phenotypic vector including the records of number of egg and weight of egg for each animal, measuring 2 x 1; TBV = vector of true breeding values of the traits of each animal, measuring 2 x 1; T = Cholesky decomposition of the matrix of environmental (co)variance of the traits, measuring 2 x 2; Z = vector of random numbers including normal numbers, measuring 2 x 1.
Each positive semi-definite symmetric matrix of R can be stated in form of TT'.T matrix is triangular.The matrix of T can be calculated using the following formula.
The i th diagonal element in the matrix of T is as follows: And the none-diagonal element related to the i th row and k th column, which is below the diameter of the matrix of T is:

Genetic evaluation using traditional BLUP
The two trait model is as follows:

Var
G is an additive genetic (co) variance matrix for animal effect, whose elements are as follows: 1. g11 = additive genetic variance for the direct effects of trait 1 2. g12 = g21 = additive genetic covariance between two traits 3. g22 = additive genetic variance for the direct effects of trait 2.

Mahboubeh et al. 7783
A is the relationship matrix among animals and R is the matrix of (co)variance of remaining effects, whose elements has been shown using rij.In this model, mean is the only fixed effect and error is random effect.It is also supposed that the intraction effect among genotype and environment is zero.Mixed model equations are as follows: To solve mixed equation, the method of iteration has been used, and for this purpose Gauss-Seidel iteration method (GS) is used.

Allelic relationship matrix
To determine allelic relationship between two animals in a locus, each allele of the first animal has been compared with alleles of second animal.Subsequently, the extent of similarity of alleles in a locus has been determined using the following formula (Nejati Javaremi et al., 1997): Here, Iij shows the similarity of i th allele of an animal with j th allele of the second animal.If two alleles are similar and the same, IiJ, the result will be one, and if they are not the same, the result will be zero.Total allelic similarity for two animals with locus L, is calculated using the following formula (Nejati Javaremi et al., 1997):

Genetic evaluation using allelic relationship matrix
One of the characteristics of TAR is that a separate allelic relationship matrix is formed for each trait.Contrary to pedigree method, which uses only one matrix for multiple-trait evaluation, TAR needs three matrixes for bi-variate genetic evaluation.Matrix Here, 2 G is total genetic variance and n is the total number of markers.The statistic model used to estimate marker effects is as follows.In the following model, mean and markers were used as fixed and random effects, respectively.= total mean.Xij = genotype of j th marker of i th animal (0, 1, and 2 are the signs of homozygosis for the first allele, hetrozygosis and homozygosis for the second allele, respectively).aj = the random effect of j th marker.ei = remaining random effect.
The univariate evaluation was conducted once for number of egg and another time for weight of egg in order to determine the marker effects for both traits.The marker, whose effect was two percent (2%) more than genetic standard deviation of the trait in question for both traits, was considered common marker.If the effect of marker is two percent (2%) more than genetic standard deviation for only one trait, this marker has been considered the specific marker of that trait.Otherwise, the marker in question was deemed as neutral marker.So, the specific and common markers of any trait were determined, and the matrixes of allelic relationships were formed.In forming the matrix of A , the common markers of both traits were used.

Simulation of genetic parameters
Parameters applied for the traits of the number of egg and weight of egg was shown in Table 3 (Ghazikhani Shad et al., 2007).
Genetic and environmental covariance of these two traits was -3.8 and 0.708, respectively, and their genetic correlation was equal to -0.2654139.
Upon formation of G and R matrixes, their converse was formed to be used in evaluation with BLUP mixed equations.The Matrixes of G -1 and R -1 was as follows:

R
The genetic correlations simulated using TP -BLUP and TAR -BLUP were shown for different iterations in Figure 2.
The genetic correlations simulated using both methods were compared through t-student statistic test for different iterations.The difference between these two strategies was not significant in the level of one percent (1%) (t0.01, 2, 9).

Evaluation accuracy
The correlation between true and estimated breeding values of the last generation was calculated.Correlations were provided based on ten iterations.Moreover, the genetic mean of the base population tends to zero in either iteration of simulation.The accuracy of evaluation of the number of egg has been shown in the Figure 3, using multiple-trait methods of TAR -BLUP and TP -BLUP.
The numbers of ten different iterations of these two strategies were compared using t-student statistic test and the difference of the accuracy of evaluation by TP -BLUP and TAR -BLUP was significant in the level of five percent (5%) (t0.05,2,9).The values of multiple-trait evaluation accuracy for the trait of number of egg using different methods have been shown in Table 4.
The evaluation accuracy of the trait of weight of egg measured by multiple trait TAR -BLUP and TP -BLUP methods has been shown in Figure 4.
The comparison of the values of ten different iteration of these two strategies were conducted using t-student statistic test and the difference of genetic evaluation accuracy of the weight of egg was significant in the level of one percent (t0.01, 2, 9).The values of multiple-trait evaluation accuracy for the trait of weight of egg using different methods have been shown in Table 5.

RESULTS
In this study, the simulation program was applied dynamically in order to be used for various numbers of animals, chromosomes, markers, QTLs, generations, different recombination rate and mutation.The structure of base population has been shown in Table 1.
After 110 random mating generations, the amount of linkage disequilibrium was 0/19 between marker and QTL.The changes of linkage disequilibrium along generations have been shown in Figure 1.The number of simulated specific and polutropic genes was shown in Table 2.

DISCUSSION
In the present study, the multiple-trait genetic evaluation accuracy based on allelic relationships was compared with traditional BLUP through computer simulation.Nejati Javaremi et al. (1997)     X ± Sd = 0.51111 ± 0.0608581 the additive genetic relationships among families and it also uses non-family information of animals.Therefore, using total allelic relationships, which has exacter information in comparison with pedigree method, increases the accuracy of evaluation.
Moreover, in multiple-trait evaluations with traditional BLUP, a similar relationship matrix is used for all traits.Whereas, animals have different allele relationships for various traits, therefore, separate matrixes are formed, which is another advantage of applying allelic relationship.Markers were assigned effects for each trait which are specific, and common markers were determined to form allelic relationship matrixes.So using allelic relation-

For
In this model, yi = vector of the observations for i th trait Bi = vector of fixed effects for i th trait ai = vector of random effects of animal for i th trait Xi and Zi are matrix of the coefficients relating the records of i th trait to fixed and animal random effects, respectively.It is supposed that:

1
In this model: yi = record of i th animal.
of trait one and common markers of both traits were used.In forming the matrix of A 1 22 − , markers of trait 2 and common markers of both traits were applied.In forming matrix of A 1 12 −

Figure 4 .
Figure 4. Multiple-trait evaluation accuracy of the weight of egg using TAR -BLUP and TP -BLUP.

Table 1 .
Structure of the Simulated Base Population.

Table 2 .
Number of simulated specific and polutropic genes.

Table 3 .
Parameters used for the trait in question.
using allelic relationship matrix increased the accuracy of evaluation.The total allelic relationships use relationship information more precisely, since this method considers

Table 4 .
Values of evaluation accuracy of the trait of the number of egg by TAR -BLUP and TP -BLUP methods.

Table 5 .
Values of evaluation accuracy of the trait of the weight of egg by TAR -BLUP and TP -BLUP methods.