A reduced animal model with elimination of quantitative trait loci equations for marker-assisted selection

Le modele animal propose par 1 Hoeschele pour la selection assistee par marqueurs est modifie ici en modele animal reduit (MAR). Cette approche MAR permet d'evaluer simultanement les effets fixes, les valeurs genetiques additives des individus parents et les effets additifs de locus lies aux locus marqueurs pour les seuls individus marques ou qui fournissent des liens de parente entre des descendants marques. La matrice de covariance residuelle correspondante est donnee, ainsi que les formules permettant de remonter aux individus non parents. Un exemple numerique est egalement traite.


INTRODUCTION
A procedure for marker-assisted selection (MAS) using best linear unbiased prediction (BLUP; Henderson, 1973Henderson, , 1975Henderson, , 1984 was first proposed by Fernando and Grossman (1989), showing how marker information can be utilized in an animal model (AM) for simultaneous evaluation of fixed effects, additive genetic effects due to quantitative trait loci (QTL) unlinked to the marker loci (ML) and additive effects due to marked QTL (MQTL). Later, certain authors presented various types of procedures to incorporate marker information in BLUP, taking into consideration multiple markers, using a reduced animal model (RAM), or combining the MQTL effects and the effects of alleles at the remaining QTL into the total additive genetic merits. Goddard (1992) extended Fernando and Grossman's (1989) model for flanking markers and discussed the use of RAM. Cantet and Smith (1991) derived a RAM version of the AM model of Fernando and Grossman (1989). Also, an AM method to reduce the number of equations per animal to one was presented by van Arendonk et al (1994), combining information on MQTL and QTL unlinked to ML into one numerator relationship matrix. A RAM approach to the AM of van Arendonk et al (1994) is also available (Saito and Iwaisaki, 1997). Hoeschele (1993) worked with an AM of the total additive genetic merits and the additive effects for MQTL alleles, and indicated that if some of the animals to be evaluated do not have marker data and do not provide relationship ties among genotyped descendants with known marker data, the MQTL equations for such animals can be eliminated, showing that the inverse of a covariance matrix among the total additive genetic merits and the needed additive effects of the MQTL alleles can be obtained directly. When applied to genetic evaluations in which only a small fraction of the animals are genotyped for ML and the remaining fraction do not provide marker data, Hoeschele's (1993) procedure can have the large advantage of reducing the number of equations to be solved.
In this paper, a RAM version of the model of Hoeschele (1993) is described.
The current approach does not require the MQTL equations for parent animals that were not marker genotyped and that do not provide relationship ties among marker genotyped descendants.
THEORY For simplicity, one MQTL is assumed in the derivations.

A RAM for MAS
If each animal in the relevant population has only one observation, the AM of Fernando and Grossman (1989) can be arranged, and a RAM can be obtained as described by Cantet and Smith (1991).
The AM is written as where Y ( nx1 ) is the vector of observations, 0 (f x 1) is the vector of fixed effects, &dquo;(9x1) is the random vector of the additive genetic effects due to QTL not linked to the ML, V ( 2 q xi ) is the random vector of the additive effects of the MQTL alleles, e( nx1 ) is the random vector of residual effects, and Xi!,x f), Zinxq) and Piq x29 ) are the known incidence matrices, respectively. The subscripts represent the sizes of the vectors and the matrices. The expectation and dispersion matrices for the random effects are usually assumed as where A! is the numerator relationship matrix for the QTL unlinked to the ML, A v is the gametic relationship matrix for the MQTL, I is an identity matrix, and !u, w and Q e are the variance components for the polygenic effects due to QTL unlinked to the ML, the additive effects of MQTL alleles and the residual effects, respectively. The mixed-model equations (MMEs) are given as where a u = U e / U u and a v = Qe !w ! Then the vectors y, u and v in equation [1] can be partitioned as respectively, where the subscripts p and o refer to animals with progeny and without progeny, respectively. Also, Uo and v o are further expressed as follows and where T is a matrix relating U p to up and has zeros except for 0.5 in the column pertaining to a known parent, m is a vector of the Mendelian sampling effects, B is a matrix relating v o to v P and contains at most four non-zero elements in each row if the parental origin of marker alleles cannot be determined (Hoeschele, 1993;van Arendonk et al, 1994;Wang et al, 1995), and e is a vector of segregation residuals. where the matrices A u p and A,, are appropriate submatrices of A u and A v respectively, and where Ip is an identity matrix, D is a diagonal matrix with diagonal elements, di = 1 -0.25(6 ss (i) + a dd (i)), where 6 5s (i) and 6 dd ( i ) are the diagonal elements of A u p corresponding to the sire and the dam of animal i, and G, is a block diagonal matrix (Saito and Iwaisaki, 1996) in which each block is calculated as where f i is the conditional inbreeding coefficient of animal i for the MQTL, given the marker information, according to Wang et al (1995).
Consequently, the MMEs for equation [6] are given by The information on recombination rate between the ML and the MQTL and the variance components required in the BLUP approaches for MAS could be obtained, for instance, by the restricted maximum likelihood or the maximum likelihood procedures (eg, Weller and Fernando, 1991;van Arendonk et al, 1993;Grignola et al, 1994).
The RAM containing the total additive genetic effects and only the MQTL effects needed Consider the following transformation matrix where I(qp ) and I( 2 qp ) are identity matrices, Pp is the qp x 2qp incidence matrix, and ap is the qp x 1 subvector of a, the vector of the overall genetic values.
Then equation [6] can be written as and using equation [10] in equation [11] gives where L = -Z tPP + W. Also, the inverse covariance matrix of the total additive genetic effects and the additive effects of MQTL alleles is given by (Hoeschele, 1993), or Therefore, the expectation and dispersion parameters of ap, vp and 4! in equation (12! are given as Then the MMEs for equation [12] are Now, for animals which have unknown marker data and have only one progeny with known marker data in the relevant population, the equations for the additive effects of MQTL alleles for BLUP may not be needed and may be eliminated (Hoeschele, 1993). If some of the additive effects of the MQTL alleles with a nonparent animal, its sire or its dam can be eliminated, then formula [7] is no longer true.
Since the vectors of the total additive genetic effects and the needed additive effects of MQTL alleles can be represented as gp = (ap vP!!' for parent animals and as go = [a' 0 v*']' &dquo; for non-parent animals, where v* and v* are subvectors of vp and v,, respectively, a system of recurrence equations for animal i can be utilized (Hoeschele, 1993), or where bi and t:( i ) are vectors of corresponding partial regression coefficients and residual effects, respectively. Then the vector e( j ) in equation [15] can be partitioned into the residuals of a o ( i ) and v*k 0( i) (k = 1 or 2), or m * and e *k , which are uncorrelated, and the dispersion parameters for these vectors are given by the diagonal matrix D * and the block diagonal matrix G!, respectively. For animal i, the diagonal of D * and the block of G! can be calculated by and with the definition being provided later in the section Computing dispersion parameters of residual effects, where the matrix B! relates V!(i) to vp*( i) and has at most four non-zero elements of the conditional probabilities, as in tables I or II of Hoeschele (1993). Therefore the dispersion parameters of the residual effects in equation [7] must be replaced by Consequently, the MMEs are given by where G*-' is the inverse covariance matrix of the total additive genetic effects and the needed additive effects of MQTL alleles for the parent animals, which is computed according to the methodology as described by Hoeschele (1993), and L * = -ZtPp + W * . The matrices with asterisks represent the appropriate submatrices of the corresponding matrices in equation !14!.

Backsolving for non-parent animals
The additive genetic effects and needed additive effects of MQTL alleles for nonparent animals can be computed by where L* is the appropriate submatrix of L * , and the vectors m * and 8 are given by with equations [17] and !18!.

Computing dispersion parameters of residual effects
First, as stated by Hoeschele (1993), for each marker used in the population, the needed additive effects of MQTL alleles are determined, and the list of the genotyped animals is created; this is referred to as the marker file. The following rules presented by Hoeschele (1993) Hoeschele (1993).
For the matrix GE, if both parents of animal i are retained in the marker file, it can be calculated by equation (18!. With no inbreeding, then Var(v!(i)) = I!2x2)w 2 and Var(vP!i)) = 1(4x4)!, where the subscripts represent the size of the identity matrices. With inbreeding, however, the matrix G* equals G,ov2 in equation [7], and Var(v 1 v 2 1 v2 v! v f ) must be computed from a list of the additive effects of MQTL alleles for all animals in the pedigree file as described by Wang et al (1995). If both parents of animal i are known, and only one of them (eg, d) is retained in the marker file, we have as given by Hoeschele (1993), where t = o!/<7!, and /! is the conditional inbreeding coefficient of sire for the MQTL, given the observed marker information, as described by Wang et al (1995). If the MQTL effects of both parents are eliminated, we have If one parent of an animal i is unknown, rows and columns pertaining to this parent in Var(v; (i) ) are deleted.

EXAMPLE
Six animals (animals 1-6) are considered for an illustration. Marker information on the animals is given in figure 1. Animal 1 has no record, and animals 2-6 have only one record each. The vector of records for animals 2-6 is [80 120 90 110 115].
Since parent animals 2 and 3 have no marker data, the additive effects of MQTL alleles for these animals can be eliminated. In addition, non-parent animal 5 is also eliminated, since the additive effect of an MQTL allele linked to M 4 was derived from its parent animals 3. Therefore, the needed additive effects of MQTL alleles are vi and v4 (k = 1 or 2) for parent animals and vl and Vk for non-parent animals, where v! represent the additive effects of MQTL alleles for animal i.
In this paper, r = 0.05 is the recombination rate assumed between the ML and the MQTL. The assumed values for variance components are Q u = 0.9, w = 0.05, or2= 1 and U2 = 1.5. The inverse covariance matrix of the total additive genetic effects and the needed MQTL effects for parent animals are given in table I, which are computed as described by Hoeschele (1993 Fernando and Grossman (1989). The number of equations in the AMs of Fernando and Grossman (1989) and Hoeschele (1993) and in the RAM of Cantet and Smith (1991) are 20, 15 and 14, respectively, while that in the current RAM approach is ten. The solutions obtained by the current approach are equal to the corresponding ones in those methods.

DISCUSSION
The application of BLUP to MAS is expected to be useful for accelerating genetic progress through increasing accuracy of selection, reducing generation interval and increasing selection differential (eg, Soller, 1978;Soller and Beckmann, 1983;Smith and Simpson, 1986;Kashi et al, 1990;Meuwissen and van Arendonk, 1992;Ruane and Colleau, 1995). The RAM approach presented permits the best linear unbiased estimation (BLUE) of fixed effects and simultaneous BLUP of the total additive genetic merits for parent animals and the additive effects of the QTL alleles linked to the marker alleles only for animals that have known marker data or provide relationship ties among at least two descendants with known marker data. The current model is the RAM version of the AM derived by Hoeschele (1993) and is also equivalent to the AM of Fernando and Grossman (1989), which allows BLUE of fixed effects and simultaneous BLUP of the additive genetic effects due to QTL unlinked to the ML and the additive effects due to MQTL alleles.
Genetic evaluation of animals in the current population often requires information on their ancestors in the analysis. On the other hand, marker information will only be available on current animals, probably a limited number of elite animals even in the near future. Therefore, a substantial fraction of animals considered in the analysis may not be marker genotyped. Hoeschele's (1993) approach is attractive for these situations, since for the MQTL effects it requires the equations only for genotyped animals and ancestor animals connecting between any two animals with marker information provided. Thus, if many of the animals in the population do not have the marker data, the equations for the additive effects of MQTL alleles for such animals may not be needed with the current RAM approach. Moreover, since for the random effects, only the equations for parent animals are required, the number of equations may be considerably reduced in the current approach.