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

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Backsolving in combined-merit models for

marker-assisted best linear unbiased prediction of total

additive genetic merit

S Saito, H Iwaisaki

To cite this version:

Note

Backsolving

in combined-merit

models

for

marker-assisted

best

linear unbiased

prediction

of

total

additive

_genetic

merit

S Saito

H

Iwaisaki

2

1

Graduate

School

_of

Science and

_Technology;

2

Department

of

Animal

_{Science, Faculty of Agriculture,}

Niigata University, Niigata

950-21,

Japan

(Received

6

_May

1997;

accepted

12

_August

₁₉₉₇₎

Summary - The

procedures

for

backsolving

are described for combined-merit models for marker-assisted best linear unbiased

prediction,

or for the animal and the reduced animal models which contain fixed effects and random effects of total additive

_genetic

merits and

residuals.

_Using

the best linear unbiased

_predictors

_(BLUP)

of the total additive

_genetic

merits and the

_residuals,

with the _present

_procedures,

the BLUP of additive

_genetic

effects due to

_quantitative

trait loci

_(QTLs)

unlinked to the marker locus and additive effects due to the marked

_QTL

are also obtained. These backsolutions are identical to the solutions

in the Fernando and Grossman animal model.

best linear unbiased prediction / marker-assisted selection

_/

combined-merit model

_/

backsolving / additive effect of marked QTL alleles

Résumé - Restitution des solutions pour la valeur génétique additive totale en cas

de prédiction BLUP utilisant des marqueurs. On décrit la

_procédure

de restitution

des solutions

_complètes

pour la valeur

génétique

totale à partir des solutions d’un modèle animal réduit. On peut obtenir

_également

des solutions

_complètes

_pour les

effets génétiques

additifs

liés à un

_{QTL marqué}

et les

_effets

liés aux autres

gènes.

Ces solutions sont

identiques

à celles du modèle animal de Fernando et Grossman.

meilleure prédiction linéaire non biaisée

_/

sélection assistée par marqueur

/

restitu-tion des solurestitu-tions

_/

QTL marqués

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

marker-assisted

_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, 1975,

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;}

Saito and

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

_{® 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

with G =

A

u

afl

₊

(Iq

® 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)

Then,

in the case of the

AM,

denoting

Cov([u’

v’]’,

a

/

)[Var(a)]-

l

by H’,

the BLUP of additive

_genetic

effects due to

_(dTLs

unlinked to the marker locus and additive effects due to the

_MQTL

are further

_given

_by

Backsolving

in the combined-merit RAM The RAM

_(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

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

ao to aP, 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 vo 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 (xu =

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

block-diagonal

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

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.

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