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

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A reduced animal model with elimination of quantitative

trait loci equations for marker-assisted selection

S Saito, H Iwaisaki

To cite this version:

Original

article

A reduced animal model with elimination

of

_quantitative

trait

loci

_equations

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

26 March

1996; accepted

12

_July

₁₉₉₆₎

Summary - A reduced animal model

_(RAM)

version of the method with the animal model

_{proposed by}

Hoeschele for marker-assisted selection is

_presented.

The current RAM

approach

allows simultaneous evaluation of fixed

effects,

the total additive

genetic

merits

for _parent animals and the additive effects due to

_quantitative

trait loci linked to the marker locus

_only

for animals which have the marker data or

_{provide relationship}

ties among descendant animals with known marker data. An

appropriate

covariance matrix

of the residual effects is

_given,

and formulae for

_backsolving

for non-parent animals are

presented.

A numerical

_example

is also

_given.

marker-assisted selection

_/

best linear unbiased prediction

/

reduced animal model

_/

total additive genetic effect

_/

additive effect of marked QTL alleles

R.ésumé - Un modèle animal réduit avec élimination d’équations relatives aux locus

de caractères _quantitatifs pour la sélection assistée par marqueurs. Le modèle animal

proposé

par I Hoeschele pour la sélection assistée par marqueurs est

_modifié

ici en modèle animal réduit

_(MAR).

Cette

approche

MAR permet d’évaluer simultanément les

effets

fixés,

les valeurs

_génétiques

additives des individus parents et les

_{effets additifs}

de locus liés aux locus marqueurs pour les seuls individus

marqués

ou

qui

fournissent

des liens de

parenté

entre des descendants

_marqués.

La matrice de covariance résiduelle

_{correspondante}

est

_donnée,

ainsi _que les

_formules

_permettant de remonter avx individus non _parents. Un

é!émple numérique

est

_également

traité.

sélection assistée par marqueurs

/

meilleure _prédiction linéaire sans biais

/

modèle

animal réduit

_/

valeur _génétique additive totale

_/

effet additif de locus _quantitatif

marqué ,

*

INTRODUCTION

A

_procedure

for marker-assisted selection

_(MAS)

_using

best linear unbiased pre-diction

_(BLUP;

_{Henderson, 1973, 1975,}

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).

where

₍

_nx1

_{) is}

_Y

the vector of

observations,

₁₎

_is

₀

_x

_(f

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 au =

U

e /

U

u

and _av =

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 vo are further

_expressed

as follows

and

where T is a matrix

_relating

Up 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

voto _vP 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. Thus

_equation

_[1]

can be written as a RAM

_by

using

the

_appropriate

matrices

_Zt

and W.

With this

_RAM,

the

_assumptions

for

_expectation

and

_dispersion

parameters

of

up,

vp

and 4)

are

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

_i

₎

₆

₍

_dd

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

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

ad-ditive

_genetic

effects and the additive effects of

_MQTL

alleles is

_given

_by

(H’)-’[Var(u’ v!)’]-lH-1

(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

non-parent

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

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

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₎

can be

applied

to

_computing

the matrices D* and

_GE.

*

For the matrix

_D

_*

_,

if an animal i has one or both of its

_parents

with the known marker data in the marker

file,

D* in

_equation

_[19]

_equals

_Dor2

_in

_equation

_[7].

For an animal i with the both

parents

in the

_{pedigree file,}

if one or both of the

parents

are not retained in the marker

_file,

the

_regression

coefficients

_bi

in

_equation

!17!

are

computed by

_equation

[16]

with

_equations

[26]

and

[27]

or with

_equations

!28!

and

_[29]

in Hoeschele

_(1993),

_{respectively,}

and then

_d(

_i)

can be

_{computed by}

equation

[17]

with

_Var(a

_o

₍

_i

_{)) =}

_0,2,

where

₎

_j

_gp(

and

_a

_o

₍

_i

₎

are

_equivalent

to _gi,par and

a

i in Hoeschele

(1993),

respectively. Also,

if one of the

parents (eg, s)

is retained in the marker

_file,

and the additive effect of the

_MQTL

allele

_(eg,

_v?)

derived from another

parent

(eg, d)

which is not retained in the marker file may be

eliminated,

then

_d!i)

_equals

that

_given

as

_equation

[25]

in 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

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).

The

_{gametic relationship}

matrix

For

_equation

_!19),

the matrix R* is

_given

as

where for the

non-parent

animals 5 and 6

Therefore,

each effect for the

parent

animals can be obtained

_{by solving}

the

following

MME

_(see

next

_page)

_given

as

_equation

[20]

with the matrices mentioned

above.

_{Moreover, using equations}

_[21]

and

_[22]

with

m

*

and

8

given

as

_equation

!23!,

each effect of the

_non-parent

animals is

_given.

The

m

*

and

e

*

for this

_example

are

The solutions for fixed effects are 111.7257 and

_97.9823,

and those for the random

effects are listed in table

_III,

_together

with the estimates from the AM

_approach

of 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

informa-tion 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 attrac-tive 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.

REFERENCES

Cantet

RJC,

Smith C

₍₁₉₉₁₎

Reduced animal model for marker assisted selection

_using

best linear unbiased

prediction.

Genet Sel

Evol 23,

221-233

Fernando

_RL,

Grossman M

₍₁₉₈₉₎

Marker assisted selection

_using

best linear unbiased

prediction.

Genet Sel

Evol 21,

467-477

Goddard ME

₍₁₉₉₂₎

A mixed model for

_analyses

of data on

multiple genetic

markers.

Theor

A

PP

I

Genet

_83,

878-886

Grignola FE,

Hoeschele

_{I, Meyer}

K

₍₁₉₉₄₎

_Empirical

best linear unbiased

_prediction

to map

QTL.

In:

Proceedings of

the 5th World

Congress

on Genetics

Applied

to Livestock

Production, University

of

Guelph, Guelph, Canada,

21, 245-248

Henderson CR

₍₁₉₇₃₎

Sire evaluation and

_genetic

trend. In: Animal

Breeding

and Genetics

Symposium

in Hov,or

_of

Dr

_Jay

L

Lush,

American

_Society

of Animal Science and American

_Dairy

Science

_{Association, Champaign, IL, USA,}

10-41

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