A method to test algorithms for incorporating genetic marker data in BLUP.

Abstract

Proc. Assoc. Advmt. Anim. Breed. Genet. Voll2 A METHOD TO TEST ALGORITHMS FOR INCORPORATING GENETIC MARKER DATA IN BLUP B.E. Clarke'' and B.P. Kinghorn' `Department of Animal Science, University of New England, Armidale, NSW 235 1 *CSlRO Division of Animal Production, Armidale, NSW 2350 SUMMARY A method has been developed for simulating the gametic relationship matrix at a quantitativepait locus (QTL) for a pedigreed population with full or partial marker genotype information. This has provided a p0werfi.d basis for the comparison of different methods of genetic evaluation of individuals at the QTL. The method can be extended to cover multiple linked genetic markers. This work can be used to help ensure a rational basis for marker assisted selection breeding programs. Keywords: Genetic marker, best linear unbiased prediction, quantitative trait loci INTRODUCTION Genetic markers provide information about the transmission of quantitative trait loci (QTL) alleles from parents to offspring. The use of this genetic information in the prediction of breeding values for domestic livestock has meant that additional additive gametic effects for individual QTL can be added to the usual mixed model equations to obtain best linear unbiased predictions (BLUE) of additive genetic effects. These extra gametic effects are included in a gametic relationship matrix (GEM) which contains probabilities of identity by descent of QTL alleles between gametes where paternally and maternally inherited alleles are considered separately. Fernando and Grossman (1989) described rules to build the inverse of the GRM directly taking into account recombination rates between genetic markers and QTL. This involved assigning paternal and maternal origin to marker alleles. A recursive method to build the same GRM was developed by van Arendonk et al. (1994). However, use of either of these methods can result in loss of information when parental origin of marker alleles is unknown. Wang et al. (1995) and Bink and van Arendonk (1994) described how to build the GRM and its inverse when parental origin may not be known, with the method of Wang et al. including an inbreeding coefficient for a QTL conditional on observed marker genotypes. These methods differ in computational ease and the assumptions required to build the GRM and its inverse, and cover both knowledge of parental origin of marker alleles and no knowledge of parental origin. The GRh4 contains probabilities that genetic marker alleles are identical by descent. This is an estimation of the relationship between individuals for QTL alleles. The true gametic relationship matrix would contain indicators (either 0 or 1) whether QTL alleles are identical by descent. However, without knowledge of QTL alleles carried by an individual, the GRM built using marker information is the best estimate of the true gametic relationships. 213 Proc. Assoc. Advmt. Anim. Breed Genet. Vol12 A method was developed to build the true gametic relationship matrix using simulated genotype information which is based on known genetic marker information. This method may be used to compare different methods of building the GRM when parental origin is either known or unknown. The effect of differing GRMs on estimation of breeding values is also illustrated. QTL SIMULATED GRM Given pedigree and genetic marker information the true GRM was simulated. QTL alleles linked to known marker alleles were simulated for individual animals and a GRM was built based on the QTL information. This could be identified as the true GRM as it was not only built using QTL instead of marker alleles, but consisted of only O's and l's for any one population, these being the known incidences of QTL alleles being identical by descent. This was possible through tracing the inheritance of simulated QTL alleles throughout the whole population. A single autosomal marker locus was assumed. The marker locus and the QTL were both biallelic (alleles 1 and 2). Marker and QTL alleles linked with a recombination rate of 0.1 were simulated for each animal. In the case that simulated marker genotypes did not match known marker genotypes, the markers and the QTL alleles were resimulated. A QTL genotype and associated phenotype were simulated for each animal within each replicate population. This information could then be used for breeding value estimation. Genotype values were simulated as a, d and -a for genotypes A,A,, AlAz and A,A,. True breeding values were calculated as 2crr, ai + a2 and 2a2, where: a is the average effect of a gene substitution (a = al - a2) and al = q (a + d (q - p)), a2 = -p (a + d (q - p)), where p is the frequency of A, . Phenotypes were simulated as a function of QTL genotype and environment, where z is a normally distributed random number N(0, 1): phenotype = genotype + fi z. Using a fixed pedigree structure and known marker alleles (Table 1) many replicate populations were simulated with the same marker genotypes but varying linked QTL alleles. These GRMs were averaged over replicates and then averaged over total number of replicates to provide a simulated GRM (sGRM). Replicate populations continued to be generated until sGRM elements all changed by less than 10' with the addition of the last replicate. GRM COMPARISON Using the same pedigree and marker information as in Table 1, gametic relationship matrices were built using the methods of Fernando and Grossman (1989), van Arendonk et al. (1994), Wang et al. (1995) and the above mentioned simulated QTL method, labelled Method 1,2,3 and sGRM respectively. A comparison of these different methods required a qualitative assessment of individual elements in the relationship matrix and then an evaluation of the quantitative importance of resulting differences in estimation of breeding values. Breeding values were estimated according to the mixed model equations given by Fernando and Grossman (1989). 214 Proc. Assoc. Advmt. Anim. Breed Genet. Voll2 Table 1. Set pedigree and marker information Animal 1 2 3 4 5 6 Sire 1 1 3 3 A six animal, three generation pedigree (Table 1) was used, with QTL alleles simulated according to the following Dam Marker Genotype parameter values: QTL allele frequency 11 in the base population p = 0.5, a = 1, d = 12 0, Va = 2pq[a + d (q - p)]2, Ve = 0.5. 2 12 Polygenic effects were assumed to be 2 11 100,000 replicate equal to zero. 21 4 populations were simulated according to 21 5 the pedigree and marker information in Table 1, therefore, giving a unique set of and associated phenotypes for each replicate population. This allowed estimated (EBVs) to be predicted using the different gametic relationship matrices at each QTL genotypes breeding values replicate. Individual elements of the GRMs built using the 4 different methods were compared using a sum of squared differences (Equation 1) between equivalent elements (i> in two GRMs (A and B). The EBVs resulting from solving the mixed model equations which included the inverse of the GRMs were compared according to correlation coefficients of the estimated (X) versus the true (X) breeding values. The higher the correlation coefficient represented the method that gave breeding values which were closest to the truth. difference = 'z(ai - bi)2 i=l. RESULTS The methods of Fernando and Grossman (1989) and van Arendonk et al. (1994) gave the same GRM (Table 2). This assumed that the first marker allele given for each animal represented the marker inherited from the sire of the individual and that the second marker allele was from the dam. This assumption was valid for all animals other than animal number 6, whose parental origin of marker alleles is uncertain. The GRM built using the method of Wang et al. (1995) was almost identical to the sGRM. These two only differed Corn the first two methods in variances and covariances relating to animal number 6. The standard errors on the elements of the simulated GRM were extremely small, giving confidence in their values. Table 2. Sum of squares of differences between elements in the four different GRMs Method 1 Method 2 Method 3 Method 2 0.000000 Method 3 2.590268 2.590268 sGRM 2.592401 2.592401 0.000014 Over 100,000 replicate populations the following means and standard deviations of correlations were produced (Table 3). Due to occasional lack of variance in the true breeding values, it was not possible to calculate correlations between the EBVs and true breeding values for some replicates. From the 100,000 replicates 215 Proc. Assoc. Advmt. Anim. Breed Genet. Vol I2 77,963 provided information to calculate correlations. These results (Table 3) indicate significant differences in me?in correlations between all methods. Table 3. Means and standard deviations of correlations between estimated and true breeding values from 77,963 replicates of different methods of building the gametic relationship matrix MesIr 0.6917 0.6889 0.699% 0.702 1 Standard deviation 0.2127 0.2133 0.2124 0.2100 DISCUSSION Although the relationship matrices built by Fernando and Grossman (1989) and van Arendonk et al. (1994) were identical, their methods to build inverses gave different results. The inverse of the GRM built using the rules of Fernando and Grossman did not give the correct inverse of the GRM for this particular pedigree. This explains the differences in correlations (Table 3). Fernando and Grossman's method of building the GRM and its inverse is far less computationally demanding than that of van Arendonk et al. However, the latter method may be more tractable for multiple QTL. Method 1 Method 2 Method 3 sGRM The method of Wang et al. (1995) appears to be most closely related to the correct gametic relationship matrix. This would suggest that this method should be used in preference to the other published methods. This is especially the case with the given pedigree which contained animals whose parental origin of marker alleles was unclear. Jn establishing the importance of such differences between elements within the GRM to estimating breeding values the results indicate that the consequences of error in using the less accurate methods may be relatively unimportant. Although this study was carried out using only a single marker locus linked to a quantitative trait locus, it may be used to compare new methods which involve any number of genetic markers linked to the loci of important traits. The small pedigree was engineered with an appropriate loop to distinguish between GRM building methods. The sGRM is built using a gene drop method which aborts non-conforming pedigrees. This is a disadvantage of the method that it is unable to be used on larger pedigrees. However, the aim of the method is to investigate the properties of unusual GRMs which are not only a feature of larger pedigrees. ACKNOWLEDGMENTS This work was funded by a joint CSIRO/UNE Postgraduate Research Scholarship. REFERENCES Bink, M.C.A.M. and van Arendonk, J.A.M. (1994) Proc. Prod 21~233. Fernando, RL. and Grossman, M. (1989) Genet. Sel. Evol. van Arendonk, J.A.M., Tier, B. and Kinghom, B.P. (1994) Wang, T., Fernando, R.L., van der Beek, S., Grossman, Genet. Sel. Evol. 27~251. 5th World Cong. Genet. Appl. Livest. 21:467. Genetics 137:3 19. M. and van Arendonk, J.A.M. (1995) 216