A reduced animal model approach to predicting total additive genetic merit for marker-assisted selection

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A reduced animal model approach to predicting total

additive genetic merit for marker-assisted selection

S Saito, H Iwaisaki

To cite this version:

Original

article

A reduced animal model

_approach

to

_predicting

total

additive

_genetic

merit

for marker-assisted selection

S Saito

H Iwaisaki

1

Graduate School

of

Science and

Technology;

2

Department

of

Animal

_{Science, Faculty of Agriculture,}

Niigata University, Niigata 950-21, Japan

(Received

22

_{February 1996; accepted}

15 October

₁₉₉₆₎

Summary -

Using

a _system of recurrence

_equations,

best linear unbiased

prediction

applied

to a reduced animal model

(RAM)

is

presented

for marker-assisted selection. This

approach

is a RAM version of the method with the animal model to reduce the number of

equations

per animal to one. The current RAM

_approach

allows simultaneous evaluation of fixed effects and total additive

genetic

merit which is

_expressed

as the sum of the

additive

_genetic

effects due to

_quantitative

trait loci

_(QTL)

unlinked to the marker locus

(ML)

and the additive effects due to the

_QTL

linked to the ML. The total additive

_genetic

merits for animals with no _progeny are

_{predicted by}

the formulae derived for

_backsolving.

A numerical

_example

is

_given

to illustrate the current RAM

_approach.

marker-assisted selection

_/

reduced animal

_{model / best}

linear unbiased prediction

/

total additive genetic merit

_/

combined numerator _relationship matrix

Résumé - Utilisation d’un modèle animal réduit pour prédire la valeur _génétique

globale dans la sélection assistée par marqueur. Sur la base d’un

_{système d’équations}

de

_récurrence,

la méthode du meilleur

_prédicteur

linéaire sans biais

_appliquée

à un modèle animal réduit

_(MAR)

est

_présentée

_pour la sélection assistée par marqueur. Cette méthode

est une version MAR de celle du modèle animal pour réduire à un le nombre

_{d’équations}

par animal. Cette méthode MAR permet d’estimer simultanément les

effets fixés

et la

valeur

_{génétique globale, qui}

est la somme des

effets génétiques additifs

des locus de

caractère

_quantitatif

_(QTL)

non liés au locus marqueur et des

_{effets additifs}

des

QTL

liés au locus _marqueur. La valeur

génétique globale

des animaux sans descendance est

prédite

par un

système d’équations

reconstitué à

partir

du

système

principal.

Un

exemple

n2imëriqué

est donné pour illustrer la méthode MAR

présentée

ici.

sélection assistée par marqueur

/

modèle animal réduit

/

meilleur prédicteur linéaire

sans biais

_/

valeur génétique additive totale

_/

matrice de _parenté combinée

*

INTRODUCTION

Marker-assisted selection

_(MAS)

is

_expected

to contribute to

_genetic

_progress

_by

increasing

accuracy of

selection,

by

reducing generation

interval and

by

increasing

selection differential

_(eg,

Soller, 1978;

Soller and

Beckmann, 1983;

Smith and

Simpson,

1986;

Kashi et

_{al, 1990;}

Meuwissen and van

_Arendonk,

1992),

especially

for

lowly

heritable traits

_(Ruane

and

_Colleau,

_1995).

Fernando and Grossman

₍₁₉₈₉₎

_{presented methodology}

for the

_application

of

best linear unbiased

prediction

(BLUP;

Henderson, 1973,

1975,

1984)

to MAS in animal

_breeding.

_Using

an animal model

(AM)

with additive effects for alleles at a marked

quantitative

trait locus

(MQTL)

linked to a marker locus

(ML)

and additive effects for alleles at the

_{remaining quantitative}

trait loci

_(QTL)

which are

not linked to the

_ML,

_they

showed the

_approach

to simultaneous evaluation of fixed

effects,

effects of

_MQTL

alleles,

and effects of alleles at the

remaining

QTL.

The number of

_{equations required}

in the AM

_approach

is

_f

+

_q(2m

+

₁₎

where

_f,

q and

m are the number of fixed

effects,

the number of animals in the

pedigree

file and the number of

MQTLs,

respectively.

Therefore,

the

_application

of the AM

approach

may be limited to smaller data sets.

Accordingly,

Cantet and Smith

(1991)

derived

a reduced animal model

(RAM)

version of Fernando and Grossman’s

approach, by

which the total number of

equations

to be solved could be

_considerably

reduced. The total additive

genetic merit, ie,

the sum of the value for

polygenic

effects and

gametic

effects can be

predicted directly

by

an AM

_procedure

(van

Arendonk et

_al,

1994).

The number of

equations

required

in the

_procedure

is

_{f +q}

_since

the number of

_equations

per animal is reduced to one

_{by combining}

information on the

MQTL

and the

_remaining

_QTL

into one numerator

_relationship

matrix.

BLUP methods for MAS

require computation

of the inverse of the conditional

covariance matrix of additive effects for the

_MQTL

alleles. Fernando and Grossman

(1989)

derived an

algorithm

to

_compute

the

_inverse,

which

_requires

not

_only

information on marker

_genotypes

but also information on the

_parental

_origin

of marker alleles.

_Wang

et al

₍₁₉₉₅₎

extended Fernando and Grossman’s work to

situations where

_paternal

or maternal

origin

of marker alleles can not be determined

and where some marker

_genotypes

are uninformative.

In this _paper, a RAM

_approach

to the

_prediction

of total additive

_genetic

merit is

presented.

The number of

equations

in the

_system

for this RAM

_approach

becomes of the order

_f

_{+ ql} where ql is the number of

parental

animals.

Also,

a small numerical

example

is

_given

to illustrate the current

_approach.

THEORY

In the

derivations,

one

_MQTL

and one observation _per animal are assumed for

simplicity.

The conditional covariance matrix between additive effects of the

_MQTL

alleles, given

the marker

_information,

is based on the recursive

equation

which was

AMs for MAS

An AM discussed

_by

Fernando and Grossman

₍₁₉₈₉₎

is written as

where _{y is} the n x 1 vector of

_{observations, 8}

is the

_f

x 1 unknown vector of fixed

effects,

u is

_the

_{q x 1 random} vector with the additive

genetic

effects due to

_QTL

not linked to the

_ML,

v is the

_2q

x 1 random vector with the additive effects of the

MQTL

alleles,

e is the n x 1 random vector of residual

_effects,

and

_X,

Z and P are n x

f, n

x _q and q x

_2q

known incidence

matrices, respectively.

The

expectation

and

_dispersion

matrices for the random effects are assumed to be

where

_A,!

is the numerator

_relationship

matrix for the

_QTL

not linked to the

_ML,

A

v

is the

_{gametic relationship}

matrix for the

MQTL,

I is an

_identity

_matrix,

and

a

u

2,a2

and

or

2

are the variance

components

for the additive

_genetic

effects due to

QTL

unlinked to the

_ML,

for the additive effects of the

_MQTL

alleles and for the residual

effects,

respectively.

The mixed model

equations

for

equation

[1]

are

where Œu =

a£ la£

and _Œv=

af

l

l

afl.

On the other

hand,

the total additive

genetic

merit is

_expressed

as the sum of the additive

_genetic

effects due to

_QTL

not linked to the ML and the additive effects of the

_{MQTL alleles,}

or a = u + Pv.

_Then,

as discussed

_by

van Arendonk et al

(1994),

equation

[1]

can be written as

With the model

_!3!,

the variance-covariance structure for the total additive

_genetic

merit a is

_given

_by

where

A

a

is the combined numerator

_relationship

_matrix,

and

_{a 2(= o!}

+

_2a!)

is the variance

_component

of the total additive

_genetic

merit.

_Assumptions

on the

expectation

and

_dispersion

_parameters

for the random effects in the model

_[3]

are

As described

_by

van Arendonk et al

_(1994),

the mixed model

_equations

are

The

_proposed

RAM

_approach

for MAS

The vectors y, u and v in

_equation

[1] can

be

partitioned

as y =

[yp’

Yo! ! ,

u =

[up’

uo’ !’

and v =

( v

P

’

’ ]’,

V

0

respectively,

where the

subscripts

_p and o

refer to animals with progeny and without progeny,

respectively.

Then Cantet and Smith

₍₁₉₉₁₎

discussed the RAM version of the model of Fernando and Grossman

(1989).

In the AM

given

as

_equation

(3!,

the vector a is

partitioned

as a

= [ap’

a

o

’] ,

where

_ap

and ao

represent

the total additive

_genetic

merit for the

_parents

and for the

_non-parents,

_{respectively.}

With the similar idea used in _y,

_X,

Z and _e, the

RAM of

_equation

_[3]

can be written as

For the

_RAM,

it is

_necessary

that ao is

expressed

as a linear function of

_ap.

Then we utilize a

_system

of recurrence

_equations,

as follows

where K is a matrix

_relating

ao to

_ap

and is defined

_by

and (p

is the vector of the residual effects. The vectors

_ap

and ao are

_expressed

as

_ap

=

up

+

_Ppvp

and ao = _{Uo +}

o

,

o

v

P

respectively.

Moreover,

_Uo and _vo can

be

_{represented by}

_{linear functions of up and vp,}

_respectively

_(Cantet

and

_Smith,

1991).

The additive

genetic

effects due to

_QTL

not linked to the ML of an animal can be described as the sum of the _average of those of its

_parents

and a Mendelian

sampling effect,

or

where the matrix T has zero elements

_except

for 0.5 in the column

pertaining

to a known

_parent,

and m is a vector of the Mendelian

_sampling

effects.

The

_relationship

between Vo and vp is written as

where B is a matrix

_relating

the additive

MQTL

effects of animals to those of

parents,

and E is a vector of the

_segregation

residuals. If the situations where the

parental

origin

of marker alleles is not determined are

considered,

as discussed

by

and d stand for the sire and the dam of animal

_i,

_{respectively,}

in scalar notation

equation

[10]

is

_rewritten,

as follows

where v!

(1

= 1 or 2 and x =

i,

s or

d)

are the

_{corresponding}

elements of vo and

vp.

The coefficients

b!k

(k

=

1,2,3

3 or

4)

_are the conditional

probabilities

that

Q!,

is a _copy of

_QP

(m

= 1 or 2 and _p = _{s or}

d),

_given

the marker

information,

where

Q!

stands for the

_MQTL

allele linked to the allele

_M!

at the

_QTL

_(Wang

et

_al,

1995).

Also,

ei

and

e?

are the segregation residual effects.

Consequently,

in

_equation

_[8]

the vector

_{corresponding}

to animal i of K can be

computed

as

where

_A

_a

_{p, A}

_u

_p

and

_A

_v

_p

are

_appropriate

submatrices of

A

a

,

A

u

and

A

v

,

respec-tively,

t

i

is the vector

_{corresponding}

to animal i of

_T,

qiis the matrix

corresponding

to animal i of

B,

1 is the vector

₍

1

_{1 )!}

and 0 stands for the direct

_product

op-erator.

Using

equation

[7]

in

_equation

_[6]

_gives

and further

_equation

_!11!

can be

_arranged

as

For this model

_(12!,

the

_assumptions

for

_expectations

and

dispersion

parameters

of

ap

and 0 are

_{given by}

where the matrix R is

_expressed

as

If we denote 4P +

Io0!

_by

R

o

,

then the inverse matrix of

R

o

can be

obtained,

as follows

with s. =

Ro, i-lro, 21,

where

_r

_oj

_is

the subvector

_{corresponding}

to animal i of

_R

_o

_,

which contains elements for animals 1 to i - 1.

Thus,

the mixed model

equations

for

_equation

_[12]

are written as

Backsolving

for animals with no progeny

The total additive

_genetic

effects of animals with no _progeny can be

_predicted

from the

_following

_equations

The inverse of 0 can be obtained

_according

to

_equation

_!14!.

EXAMPLE

We use a small

example

data set

_including

six

_animals,

four animals

_having

_progeny and two animals with no _progeny, as

_given

in table I.

and the matrix W in

_equation

_[12]

is

The inverse matrix of R is

_given

as

and

the vector of

_right-hand

side is

Consequently,

the vector of solutions for

equation

[15]

is

_given

as

and

also,

the vector of back-solutions in

_equation

_[16]

is

While the orders of the mixed model

equations

in the AMs of Fernando and

Grossman

₍₁₉₈₉₎

and van Arendonk et al

₍₁₉₉₄₎

and in the RAM of Cantet and Smith

₍₁₉₉₁₎

are

_20,

8 and

14, respectively,

that in the current RAM

approach

is

6,

because animals 5 and 6 are

_non-parents.

The solutions obtained

by

the current

approach

are the same as the

_{corresponding}

ones calculated

according

to AMs of

Fernando and Grossman

₍₁₉₈₉₎

and van Arendonk et al

(1994).

DISCUSSION

For marker-assisted selection

_{using BLUP,}

the AM

_approach

was

presented

first

by

Fernando and Grossman

_(1989),

and its RAM version was described

_by

Cantet and Smith

_(1991).

These AM and RAM

_approaches

permit

best linear unbiased

estimation of fixed effects and simultaneous BLUP of the additive

_genetic

effects

due to

_QTL

unlinked to the ML and the additive effects due to the

_MQTL.

On the other

hand,

van Arendonk et al

₍₁₉₉₄₎

discussed an AM method to reduce the number of

_equations

per animal to one

_{by combining}

information on

MQTL

and

_QTL

unlinked to the ML into one numerator

_relationship

matrix. Their method allows the

_prediction

of

_only

the total additive

_genetic

merit in addition to the estimation of fixed effects.

_Accordingly,

however,

the size of mixed model

equations

required

in their method can be smaller than those for the

approaches

by

Fernando and

Grossman

₍₁₉₈₉₎

and Cantet and Smith

_(1991).

The current

_approach

is a RAM version of the method

presented by

van

Arendonk et al

_(1994),

and is

given using

a

_system

of recurrence

_equations.

In this

RAM

_approach,

the conditional covariance matrix for the

_MQTL

can be

computed

by

the method described

_{by Wang}

et al

₍₁₉₉₅₎

which does not

_require

_assigning

the

_origin

of the marker alleles and accounts for inbred

_parents.

With the current

approach,

there is a reduction

_expected

in the size of mixed model

equations

since

for the random effects

_only

the

_equations

for

_parental

animals are

required

and

the number of

_equations

per

parental

animal is

only

one.

_However,

one feature of the current method is that the matrix R defined in

_equation

_[13],

_essentially

R

o

= 0 +

IoQe,

is not

_diagonal,

and needs to be inverted before introduction into

equation

(15!.

The

_computing

_algorithm

shown in this _paper could be one of the

strategies

for the

_practical

calculation. Another feature of our

approach

is that

reduction of

_computing

time. Further

_comparisons

between the current RAM and

other

approaches,

for relative

_{computational properties,}

are needed.

Hoeschele

₍₁₉₉₃₎

derived an AM

_{approach considering}

_equations

for total addi-tive

_genetic

merits and additive effects due to the

_MQTL,

where

_MQTL

_equations

for animals not

typed

and certain other animals are absorbed. The

method,

for realistic

_situations,

would also lead to a

_large

drop

in the number of

_equations

re-quired.

A RAM consideration of Hoeschele’s

_approach

has been

_given

_by

Saito and Iwaisaki

_(1996).

The BLUP methods for

_{MAS, including}

the current RAM

_{approach, require}

the

_knowledge

of the recombination rate

_(r)

between the ML and the

_MQTL

and

the additive

_genetic

variance

_explained

_by

_MQTL

₍

_Q

_v).

Since true values of these

parameters

are

_usually

unknown in

_practice,

it is _necessary that

_they

are estimated.

As

_discussed,

_eg,

_by

Weller and Fernando

_(1991),

van Arendonk et al

₍₁₉₉₃₎

and

Grignola

et al

_(1994),

with the

_assumption

of effects of

_MQTL

alleles

_normally

distributed,

these

_parameters

can be estimated

_by

the likelihood-based methods such as restricted maximum likelihood

_(Patterson

and

_{Thompson, 1971).}

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Cantet

RJC,

Smith C

₍₁₉₉₁₎

Reduced animal model for marker assisted selection

_using

best linear unbiased

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221-233

Fernando

_RL,

Grossman

_M(1989)

Marker assisted selection

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best linear unbiased

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467-477

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_{I, Meyer}

K

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