Calculation of breed direct and maternal genetic fractions and breed specific direct and maternal heterozygosity for crossbreeding data

Teaching, research, and herd breeding applications may require calculation of breed additive contributions for direct and maternal Brazilian Journal of Genetics Calculation of breed direct and maternal ... http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0100-845519... 2 of 14 5/2/2008 1:54 PM genetic effects and fractions of heterozygosity associated with breed specific direct and maternal heterosis effects. These coefficients can be obtained from the first NB rows of a pseudo numerator relationship matrix where the first NB rows represent fractional contributions by breed to each animal or group representing a specific breed cross. The table begins with an NB x NB identity matrix representing pure breeds. Initial animals or representative crosses must be purebreds or two-breed crosses. Parents of initial purebreds are represented by the corresponding column and initial two-breed cross progeny by the two corresponding columns of the identity matrix. After that, usual rules are used to calculate the NB column entries corresponding to breeds for each animal. The NB entries are fractions of genes expected to be contributed by each of the pure breeds and correspond to the breed additive direct fractions. Entries in the column corresponding to the dam represent breed additive maternal fractions. Breed specific direct heterozygosity coefficients are entries of an NB x NB matrix formed by the outer product of the two NB by 1 columns associated with sire and dam of the animal. One minus sum of the diagonals represents total direct heterozygosity. Similarly, the NB x NB matrix formed by the outer product of columns associated with sire of dam and dam of dam contains breed specific maternal heterozygosity coefficients. These steps can be programmed to create covariates to merge with data. If X represents these coefficients for all unique breed crosses, then the reduced row echelon form function of MATLAB or SAS can be used on X to determine estimable functions of additive breed direct and maternal effects and breed specific direct and maternal heterosis effects. INTRODUCTION Several methods are available to model breed additive and interaction effects for records of crossbred animals. With designed experiments, the coefficients are the same for groups by generation. In the more general cases of composites or unstructured breeding plans, calculation of the coefficients is time consuming at best. The purpose of this note is to outline computational procedures to simplify calculation of fractions of inheritance from ancestral breeds as well as fractions of breed specific heterozygosity for an animal and its dam. An additional section will outline a simple way to determine what functions of breed and heterozygosity effects are estimable from statistical solutions. MATERIAL AND METHODS Breed effects can be modelled as breed combinations with linear contrasts used to separate breed effects and breed interactions (e.g., Dickerson, 1969; Wyatt and Franke, 1986). Breed effects can also be modelled as covariates with fractional contributions (e.g., Robison et al., Brazilian Journal of Genetics Calculation of breed direct and maternal ... http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0100-845519... 3 of 14 5/2/2008 1:54 PM 1981). In that case, Westells rules for genetic groups can be used equivalently with breeds corresponding to the proxy parents (Westell et al., 1984, 1988). Breeding values from using Westells rules incorporate breed solutions weighted automatically by their fractional contributions to the animal plus a genetic deviation from that function. With Westell groups effects due to heterozygosity would need to be modelled separately, probably as covariates. The equivalent model with breed effects as covariates would result in solutions for regression coefficients and for random genetic deviations. Predicted breeding values would then be constructed from linear functions of the estimated regression coefficients for breeds with weighting by fraction of inheritance by breed plus the predicted random genetic deviation for the animal. The additional effects of breed crosses are usually modelled as interaction effects which are equivalent to expected heterozygosity. Heterozygosity effects can be expressed as general (the breeds in the cross all interact equally) or specific (each breed cross has a different effect). Heterozygosity effects can be modelled as covariates in either case (or in intermediate cases where, for example, continental by continental crosses have the same effect, continental by English breeds may have a different effect, and English by indicus breeds may have still a different effect). In this note calculation of general and breed specific heterozygosity coefficients will be described with special cases requiring obvious modifications. The methods for calculating breed fractions and heterozygosity fractions will be described with a simple example which will illustrate the concepts. Sketches of programming steps to do the calculations will also be given. Pedigree files will be assumed available tracing back to the first crosses between breeds or even further. At some point the parents will be assumed unknown but the breeds of the parents will be known. Animals and parents will have unique identification with parent identification numbers smaller than their progeny (necessary only for use of standard methods of calculating inverse of numerator relationship matrix). The example will be for three animals with records: Animal 60 with sire 40 and dam 50 Animal 70 with sire 60 and dam 50 Animal 80 with sire 60 and dam 70 Sire 40 is breed A. Dam 50 is a cross of breeds B and C. (The procedure requires originating parents be a purebred or a two-breed cross). The method will involve calculation of a pseudo tabular relationship matrix (Van Vleck, 1993). To facilitate that calculation, the original identification numbers need to be recoded to begin with 1. In fact, the first breed must be recoded to 1 followed by other breeds and then animals in identification order. Brazilian Journal of Genetics Calculation of breed direct and maternal ... http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0100-845519... 4 of 14 5/2/2008 1:54 PM Recoding identification numbers The MTDFNRM program in the MTDFREML package of programs for estimating variance components (Boldman et al., 1995) can be used for the recoding. Assume the pedigree file is as follows: