Backsolving in combined-merit models for marker-assisted best linear unbiased prediction of total additive genetic merit

Restitution des solutions pour la valeur genetique additive totale en cas de prediction BLUP utilisant des marqueurs. On decrit la procedure de restitution des solution completes pour la valeur genetique totale a partir des solutions d'un modele animal reduit. On peut obtenir egalement des solutions completes pour les effects genetiques additifs lies a un QTL marque et les effects lies aux autres genes. Ces solutions sont identiques a cells du modele animal de Fernando et Grossman.


INTRODUCTION
In recent years, a large number of genetic polymorphisms, for example, restricted fragment length polymorphisms (eg, Botstein et al, 1980), variable numbers of tandem repeats (eg, Jeffreys et al, 1985;Nakamura et al, 1987) and random amplified polymorphic DNA (eg, Williams et al, 1990), are being detected by molecular techniques. If these are linked to quantitative trait loci (QTLs) affecting quantitative economic traits and are useful as the genetic markers, then markerassisted prediction of breeding values may be conducted as discussed by Fernando and Grossman (1989). These authors first presented an animal model (AM) procedure to incorporate marker information in a best linear unbiased prediction (Henderson, 1973(Henderson, , 1975(Henderson, , 1984. Following the work of these authors, various models and procedures for the marker-assisted best linear unbiased prediction have been described further (eg, Cantet and Smith, 1991;Goddard, 1992;Hoeschele, 1993;van Arendonk et al, 1994;Togashi et al, 1996;Iwaisaki, 1996, 1997b). Van Arendonk et al (1994) presented a combined-merit model, or the AM model combining the additive effects due to marked (aTLs (MQTLs) and the effects of alleles at the remaining (aTLs into the total additive genetic merit. A reduced animal model (RAM) version of the combined-merit model is also available (Saito and Iwaisaki, 1997b). With these models, the number of systems of equations to be solved is relatively reduced; however, the best linear unbiased predictors (BLUP) of the additive effects of the MQTL alleles and those of the remaining (aTLs are not given directly, even if one wishes to know the values for certain animals.
The objective of this paper is to describe the procedures for computing the backsolving of the MQTL-and the remaining (aTL-effects in the cases of the combined-merit AM and RAM.

THEORY
Backsolving in the combined-merit AM Assuming a MQTL and one observation per animal for simplicity, the AM discussed by Fernando and Grossman (1989) is written as In contrast, the combined-merit AM of van Arendonk et al (1994) is expressed as with a = u + (I 9 &reg; 1')v, where y is the n x 1 vector of observations, (3 is the f x 1 vector of fixed effects, u is the q x 1 random vector of additive genetic effects due to alleles at the QTLs not linked to the marker locus, v is the 2q x 1 random vector of additive effects of the MQTL alleles, a is the q x 1 random vector of the total additive genetic merits or breeding values, e is the n x 1 vector of random residuals, X and Z are n x f and n x q known incidence matrices, respectively, Iq is an identity matrix whose dimension is q, 1 is the column vector ( I 1 )', and 0 stands for the direct product operator. For model (2!, the expectation and dispersion matrices for the random effects are assumed to be with G = A u afl + (Iq &reg; 1')A&dquo;(Iq 01)a! and R = I n af , where A u is the numerator relationship matrix for the (aTLs not linked to the marker locus, A v is the gametic relationship matrix for the MQTL, In is an identity matrix whose dimension is n, and Q!, ol2and Q e are the variance components for the additive effects due to alleles at the (aTLs unlinked to the marker locus, for the additive effects of the MQTL alleles and for the residuals, respectively.
The BLUP of the total additive genetic merits, hence, are obtained by solving the following mixed model equations (MME)  (Saito and Iwaisaki, 1997b) is written as where y, X and (3 are the same as in equations [1] and !2!, ap is the appropriate subvector of a and the subscript p refers to animals with progeny, e is the n x 1 residual effects, and W is the incidence matrix. With model [6], the assumptions for expectation and dispersion parameters of the random effects are where Gp is the appropriate submatrix of G, and R r is further expressed as equation [13] of Saito and Iwaisaki (1997b).
The BLUP of the total additive genetic merits for parent animals are then obtained by solving the following MME In the case of the RAM, the BLUP of additive genetic effects due to (aTLs unlinked to the marker locus and additive effects due to MQTL as obtained by solving the MME for the full model, or equations !1!, are given by the two steps for backsolving for up and Vp and then for i Z and v o , where the subscript o refers to animals without progeny. That is, considering Cov(!uP' vp'l' , ap')[Var(ap)]-l and Cov([up' vP'!', A')!Var(0)!-1, the BLUP of up and vp are first computed as where 0 = y -X(3° -Wap, A u p, Ay and R o are the appropriate submatrices of A u , A v and R r , respectively, K is a matrix relating a o to a P , T has zero elements except for 0.5 in the column pertaining to a known parent of animal i, and B is a matrix relating the additive MQTL effects of the animals to those of the parents and contains zero elements except for at most four non-zero elements in each row, which are the conditional probabilities for the MQTL (Wang et al, 1995). For details, see Saito and Iwaisaki (1997b).
Then, with Up and V; provided, the BLUP of Uo and v o are further obtained as where m and e represent the vectors of the Mendelian sampling effects and the segregation residuals predicted, respectively, which are given as where (x u = o r 2/0,2, a, = U2/or2, S = y o -X ol 3° -Tu P -(I. <8 1')BQ, D is the diagonal matrix whose diagonal elements equal 0.5 -0.25(F, + F d ) with the inbreeding coefficients of the sire and the dam, F, and F d , and G, is the blockdiagonal matrix (Saito and Iwaisaki, 1997a), in which each block is calculated as where A V( i ) and B!i! are appropriate submatrices of A v and B, respectively, which correspond to the parents of animal i, and f i is the inbreeding coefficient for the MQTL (Wang et al, 1995).

DISCUSSION
The systems of equations in the combined-merit model approach may be compact, relative to that for the AM of Fernando and Grossman (1989), even if the number of MQTLs is high. Compared with the combined-merit AM, the RAM version, applied to species where the fraction of non-parents is high, would lead to a further reduction of the size of the system of equations, although the sparseness in the coefficient matrix of the MME would be adversely affected.
With these models, the inverse covariance matrix of the total additive genetic merits for individual animals or for parent animals in the pedigree file is needed, and moreover the RAM version requires R r to be inverted before it can be introduced into equations !7!. For these calculations, certain computing algorithms are available, as discussed by van Arendonk et al (1994) and Saito and Iwaisaki (1997b). Rapid development in computing power may make applications of this type of approach attractive, especially when a large number of markers are considered.
The most relevant information in selecting animals would be the predictors of the total additive genetic merits, which are given directly by the combined-merit model approach. When the models are applied, and one further wishes to compute BLUP of additive genetic effects due to (aTLs not linked to the marker locus and/or additive effects due to the MQTL for all or a part of animals, this can be done by using the procedures for backsolving, as just demonstrated in this paper. The backsolutions derived are equivalent to the solutions for the Fernando and Grossman AM. However, the backsolving obviously requires additional computations. Hence, examination of the most efficient numerical techniques would definitely be needed. As an approach, the use of certain transformation techniques might be useful. For the situation where one absolutely needs the solutions in the full model, further research would also be necessary to determine the relative efficiencies of the combined-merit models for computing as compared to the model of Fernando and Grossman (1989) for both cases, single or multiple markers.