Covariance between relatives for a marked quantitative trait locus

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Covariance between relatives for a marked quantitative

trait locus

T Wang, Rl Fernando, S van der Beek, M Grossman, Jam van Arendonk

To cite this version:

Original

article

Covariance between

relatives

for

a

marked

quantitative

trait

locus*

T Wang

1

RL Fernando

S

van

der

Beek

M

Grossman

JAM

van

Arendonk

1

University

of Illinois, Department of

Animal

_{Sciences, 1207,}

W

_{Gregory Drive, Urbana,}

IL

_{61801, USA;}

2

Wageningen

Agricultural University, Department of

Animal

Breeding,

PO Box

338,

6700 AH

_Wageningen,

The Netherlands

(Received

6

_{April 1994; accepted}

lst

_February

₁₉₉₅₎

Summary - Best linear unbiased

_prediction

_(BLUP)

can be

_applied

to marker-assisted selection. This

_{application requires computation}

of the inverse of the conditional covariance matrix

_(G

_v

₎

of additive effects for the

_quantitative

trait locus

_(QTL)

linked to the marker locus

_(ML),

_given

marker genotypes. This paper presents

theory

and

_algorithms

to construct

G

v

and to obtain its inverse

_efficiently.

These

_algorithms

are

suf&ciently

_general

to accommodate situations

₍₁₎

where

paternal

or maternal

_origin

of marker alleles cannot

be determined and

₍₂₎

where the marker genotypes of some individuals in the

_pedigree

are unknown.

genetic marker

_/

marker-assisted selection

_/

best linear unbiased prediction

/

co-variance between relatives

_/

gametic relationship

*

Supported

in _part

_by

the Illinois

_{Experiment Station,}

Hatch

_Projects

35-0345

_(RLF)

and 35-0367

_(MG).

**

Correspondence

and reprints.

Résumé - Covariance entre apparentés pour un locus de caractère quantitatif marqué. La meilleure

prédiction

linéaire sans biais

(BL UP)

s’applique

à la sélection assistée par marqueur. Cela demande d’inverser la matrice

(G

v

)

des covariances entre

apparentés

des

_{effets génétiques additifs}

du locus

_quantitatif

lié au locus _marqueur,

covariances conditionnelles aux

génotypes

du _marqueur. Cet article

présente

la théorie et les

algorithmes

pour établir

G

v

et pour obtenir son inverse d’une manière

ef!îcace.

Ces

algorithmes

sont assez

_généraux

_pour

_prendre

en _compte des situations

_i)

où

_l’origine

paternelle

ou maternelle des allèles marqueurs ne _{peut pas} être

déterminée,

ii)

où le

génotype

marqueur de certains individus dans le

pedigree

n’est pas connu.

marqueur génétique

/

sélection assistée par marqueur

/

meilleure _prédiction linéaire

INTRODUCTION

Theory

for covariance between relatives

_provides

the basis for use of data from rel-atives in

_genetic

evaluation. At

_present,

_genetic

evaluations in animal

_populations

are

_primarily

obtained

_by

best linear unbiased

prediction

(BLUP;

Henderson

1973)

using

trait

_phenotypes

_(T-BLUP).

Due to advances in molecular

_{biology, genetic}

markers are

becoming increasingly

available for use in

_genetic

evaluation. Several

approaches

for use in

_genetic

evaluation

_using

marker

_genotypes

and trait

pheno-types

have been discussed

_(Geldermann,

_1975;

_Soller,

_1978;

Soller and

Beckmann,

1982;

Smith and

_Simpson,

_1986;

Kashi et

al,

1990).

In

_addition,

Fernando and Grossman

₍₁₉₈₉₎

described how BLUP can be used for

_genetic

evaluation

using

marker

_genotypes

and trait

_phenotypes

_(TM-BLUP).

Some

_strategies

have been

proposed

to make TM-BLUP

_{computationally}

efficient

_(Cantet

and

Smith, 1991;

Hoeschele,

1993;

van Arendonk et

_al,

_1994).

TM-BLUP has also been extended to accommodate

_multiple

markers

_(Goddard,

_1992;

van Arendonk et

_{al 1994).}

TM-BLUP

_requires

_computation

of the inverse of the conditional covariance matrix

_(G

_v

₎

of additive effects for the

_quantitative

trait locus linked to the marker

locus, given

marker

_genotypes.

To

_compute

this

_inverse,

Fernando and Grossman

(1989)

provided

an

algorithm

that

required

information on the

parental

(paternal

or

maternal)

origin

of marker

_alleles,

in addition to information on marker

_genotypes.

The

_parental

_origin

of marker alleles in an

individual, however,

is not

_always

known. For

example,

if 2

_parents

and their

offspring

each has

_genotype

A

l

A

2

at the same marker

_locus,

marker allele

A

1

in the

_offspring

could have descended from either of the

parents,

thus the

parental origin

of

A

1

in the

_offspring

is unknown.

The

_objective

of this paper is to

present

theory

and

_algorithms

to

_compute

the conditional covariance matrix and its inverse when

_{parental origin}

of the marker alleles may not be known.

Theory

and

_algorithms

are

developed

for

pedigrees

where the marker

_genotype

of each individual is known

_(complete

marker

_data).

Application

of this

_theory

is

_given

for

_pedigrees

where the marker

_genotype

of some

individuals is unknown

_(incomplete

marker

_data).

Wang

et al

₍₁₉₉₁₎

_presented,

without

_proof,

a recursive

_equation

to construct

G

v

and an efficient

algorithm

to

compute

its inverse. This recursive

_equation

has been used

_by

van Arenonk et al

₍₁₉₉₄₎

and Hoeschele

_(1993).

In the

_present

_paper, we _prove that the recursive

equation

holds when marker data are

complete,

but does not hold

_generally

when marker data are

_incomplete.

Chevalet et al

₍₁₉₈₄₎

have described a method to

_compute

_G

_v

_given

marker

phenotypes.

This method does not

_{require knowing}

the

parental origin

of marker alleles and can accommodate

_missing

marker

_phenotypes.

The

method,

_however,

is not

_{computationally}

feasible for the

_{large pedigrees typically}

encountered in animal

breeding. Computation

of the conditional covariance matrix and its inverse become feasible

_{by conditioning}

on marker

genotypes

instead of marker

phenotypes.

NOTATION AND ASSUMPTIONS

Consider a

_single

_polymorphic

marker locus

(ML)

_closely

linked to a

_quantitative

trait locus

_((aTL),

which will be referred to as the marked

QTL

(MQTL).

Assume

denote 2 alleles at the

_ML,

and let

_QI

and

_Q;

denote

MQTL

alleles linked to

_M/

and

_Ml

as shown below

If the 2 marker alleles for individual i are

_known,

then

_they

will be

_arbitrarily

labelled as

Mi

and

M

2

.

For

example,

_suppose individual i has marker alleles

A

3

and

_A

_l

_,

then

A

3

can be labelled as

Mi

and

A

1

as

M?,

or

_A

₁

can be labelled as

M!

and

A

3

as

M?.

If the 2 marker alleles for individual

_i,

_however,

are

unknown,

Mi

can be _any of the marker alleles

_segregating

in the

_population,

and

_M2

can also be any of the marker alleles. For

example,

suppose there are 3 marker alleles

(A

l

,

A

2

,

and

₃

₎

_A

_segregating

in the

_population,

then

_M

₂

₁

can be

_,

_A

_l

A

2

,

or

_A

₃

_,

and

M

2

can also be

_A

₁

_,

_A

₂

_,

or

_A

₃

_.

Further,

let

_vl

and

_v?

be the additive effects of

_Q!

_and

_Q2,

and let

_w

=

Var( vI) =

Var(v!)

be their

_variance,

for i =

1, ... , n.

Observed marker

_genotypes

are denoted

_by

Gobs.

COVARIANCE OF

_MQTL

EFFECTS GIVEN COMPLETE MARKER DATA

The conditional covariances of additive effects of

_MQTL

alleles will be derived

separately

for alleles between individuals and for alleles within an individual.

Covariance between individuals

Suppose

s and d are

_parents

of

i,

and j

is not a direct descendant of

_{i (fig 1).}

The conditional covariance of the additive effects of

_MQTL

alleles

_Qk

_i

and

Q

in individuals i and

_{j, given}

the observed marker

_genotypes

_(Gobs),

is

where k

i

and

_k

_j

can be 1 or

2,

and

Pr(Q7

i

==

Q)&dquo; )

Gobs)

is the conditional

probability

that

_Q7

_i

is identical

_by

descent to

Q/

_{given Gobs}

_(eg,

Fernando and

_Grossman,

1989).

Because individuals s and d are

_parents

of

_i,

Q7

i

can be identical

_by

descent to

Q

in 1 of 4 _ways:

1.

_Q7

_i

descended from

_Q!

and

_Q;

was identical

by

descent to

Q)! ,

denoted

by

(Q7i {= Q;, Q; == Q!j)

2.

_Q7

_i

descended from

_Q!

and

_Q;

was identical

by

descent to

Q!j,

denoted

_by

(Q7i {= Q;, Q; == Q!j)

3.

_Q7

_i

descended from

_Qà

and

_Qà

was identical

_by

descent to

Q!j,

denoted

_by

Fig

1. Chromosome

_{fragments containing}

the ML and the

MQTL

for individuals s,

d,

i

and

_j.

4.

_Q!i

descended from

_Q!

and

_Q§

was identical

by

descent to

Q!j,

denoted

_by

!!..<-

r)2 r)2 &mdash;

r)!-’’)

Therefore,

the

_probability

in

_[1]

can be written as

Because

_{individual j}

is not a direct descendant of individual

_i,

and marker

genotypes

of s and d are

known,

the conditional

sampling

of

Q7

i

from s or d is

independent

of alleles

_{in j}

_being

identical

_by

descent to alleles in s or d

(fig 1),

given Gobs. Thus,

the

_probability

in

_[1]

can be

computed

recursively

as

Equation

[3]

was first

_given

_by

_Wang

et al

_(1991).

It will be shown later that

_[3]

does not hold

_generally

when marker data are

_incomplete.

Generalizing

in

_(3!,

Pr(Q7

i

«

Q;

P

IG

obs

)

is the conditional

_probability

that allele

Q/°

in

_offspring

i descended from allele

Qp

in

_parent

_p = _{s or} d for

k

i

,

k

P

= 1 _or 2.

This conditional

_probability

will be referred to as the

_probability

of descent for a

for

_k

_i

= 1 or 2 and p = s or

_d,

where p = r when

_k

_P

= 1 and

p = 1 - r when

_k

_P

=

_2,

and where _r is the recombination _rate between the ML and

MQTL.

Further,

_{Pr(Miki !}

M;

P

¡G

obs

)

is the conditional

probability

that marker allele

_Mk’

in

_offspring

i descended from marker allele

M!’

in

_parent

_p,

_given

the

_pedigree

and marker

_genotypes.

This conditional

_probability

will be referred to as the

_probability

of descent for a marker allele

(PDM).

There are 8 PDMs for each

individual,

and their

_computations

are

_explained

in

Appendix

A. Note that the PDMs and

PD(as

associated with the unknown

_parent(s)

are undefined.

Equation

[4]

explicitly

shows the

_relationship

between

_PDQs

and PDMs in scalar notation. For

_convenience,

it is rewritten in matrix notation as

where

Covariance within an individual

The conditional covariance between additive effects

_vi

_{t and v?}

_of

MQTL

alleles

_(!

_z and

_Q?

_in

individual i with

_parents

s and

_{d, given}

_Gobs,

can be written from

[1]

as

where

_fi

=

Pr(Ql

> Q/ )Gobs)

is the conditional

probability

that 2

homologous

alleles at the

_MQTL

in individual i are identical

_by

_{descent, given Gobs.}

_{Thus, f}

_i

is the conditional

_inbreeding

coefficient of individual i for the

_{MQTL, given}

_Gobs.

This is different from

_{Wright’s inbreeding coefficient,}

which is the conditional

_probability

that 2

_homologous

alleles at _any locus in individual i are identical

_{by descent, given}

only

the

_pedigree.

The

pair

of 2

_homologous

alleles at the

_MQTL,

_Ql

and

_Q?,

in individual i descended from 1 of the

_following

parental pairs:

(Qs,

Qd),

(Q9,

Qd),

8

Q’)

or

s

Q§) .

Let

_T,!skd

denote the event that the

_pair

of alleles in i descended from the

Because

_{(QI = Q2!Tks!d> Gobs)}

_implies

_{(QSS - Qdd !Gobs),}

_[10]

becomes

The

_Pr(T

_k

_g

_kdI

_G

_obs

₎

can be

_expressed

in terms of

_PD(as

_(see

_Appendix

_G!

as

For

_example,

where

_B

_i

_{(l, k)}

are elements

_{of B}

_i

in

_(5!.

If 1 of the denominators in

_!12!

is zero, then the entire

_{corresponding}

term is set to zero.

Tabular method to construct covariance matrix

_G

_v

The conditional covariance matrix

_(G

_v

₎

between additive effects of

_MQTL

alleles can be

_written,

from

[1]

and

!9!,

as

where A is the matrix of conditional

_{probabilities}

that

the

2

_homologous

alleles

at

_MQTL

are identical

by descent, given Gobs.

The matrix A includes a row and column for each of the 2

_MQTL

alleles in each individual. Thus the order of A is

_2n,

where n is the number of individuals in the

_pedigree.

This matrix is the conditional

gametic

relationship

matrix

_(Smith

and

_Allaire,

_1985),

_{given Gobs.}

It follows that each

_diagonal

element of this matrix is

_unity.

The tabular method to construct A is

_explained

below.

Following

Henderson

_(1976),

individuals are ordered such that

_parents

_precede

their _progeny, and individuals 1

_through

b are considered to be unrelated and non-inbred.

_Thus,

the _upper left submatrix of A is an

_identity

matrix of order

_2b,

which is

_{expanded sequentially by}

the 2 rows and 2 columns

_{corresponding}

to individual

i,

for i =

b + 1, ... , n,

as follows:

Let 81

=

2(i &mdash; 1)

+ 1 and

6f

=

2(i &mdash; 1)

+ 2 be the row indices of A

corresponding

to the 2

_MQTL

alleles

_Ql

_and

_Q2

_of

individual i. From

_!3!,

the elements of the 2

rows 6/

and

_{6i ,}

_{corresponding}

to the 2

_MQTL

alleles of individual i with

_parents

s

for j

=

61 -

1,

where

_B

_i

_(L,

_k)

were defined in

_!6!.

Element

_À

_p

₆₁

=

fi,

where

f

i

is

_given

in

_!11!.

Elements of columns

_6!

_and

_8;

_are obtained

_by

_symmetry.

If 1

parent

is

_unknown,

terms

_involving

the unknown

parent

are

_dropped

from

!14!.

For

_convenience,

the tabular

_algorithm

described above can be written in matrix notation. Let

_A

_i

_-

_i

be the upper left submatrix of A

expanded

up to i - 1. For individual

_i,

with

_parents

s and

_{d, A}

i

_

1

is

_expanded

to

Ai

as

where

and

In

_{(17!, q’}

is a 2 x

_2(i-1)

matrix with at most 8 non-zero

_elements,

which are from

B

i

and are located in columns

6s, 8;,

6d

_and

6d.

The above tabular

_algorithm

to construct A is similar to that used to construct the numerator

_relationship

matrix

_(Emik

and

Terrill, 1949; Henderson,

1976).

Further,

A

_plays

the same role in

_prediction

of

_MQTL

effects as the numerator

relationship matrix, A,

does in

_prediction

of

_breeding

values.

ALGORITHM TO INVERT COVARIANCE MATRIX OF

_MQTL

ALLELE EFFECTS

Theory

Tier and S61kner

_(personal

communication,

1994)

and van Arendonk et al

₍₁₉₉₄₎

used

_partitioned

matrix

_theory

to

_develop

rules to invert the numerator

_relationship

matrix

_efficiently

for

populations

with unusual

_{relationships.}

A similar

approach

is used here to invert A

_efficiently.

From

_[13],

_Gj!

=

_{A -1 /a!.}

In

_general,

the inverse of

_A

_i

_,

_partitioned

as in

!15!,

can be obtained as

where

_Di

=

Ci - q§Aj- i qi

is 2 x 2 matrix

(Searle, 1982).

From

!18!,

the contribution

of individual i to

_Ai

is

_{given by}

the second term on the

_right-hand

side of this

Because of the sparse structure of qi as shown in

(17!, qiA

i

_

l

qi

can be written as

B

ics

,

d

B§,

where

_d

_,

_s

_C

is the 4 x 4 conditional

_{gametic relationship}

matrix for

parents

of

_i,

s and

d,

the elements of which are in

₁

_,

_A

_{_}

_Z

and

_B

_i

is the matrix of

PDQs

defined in

_(6!.

Thus

If

_{fi, f}

_s

and

_f

_d

are

_nulle,

then

where

₁

₂

is a 2 x 2

_identity

matrix.

The submatrix

_qiDilq!

in

_[18]

is a _{square matrix} of order

2(i &mdash; 1)

that contains

only

16 non-zero

_elements,

which are

_given

_by

l

Bi.

i

Dz

B

The submatrix

Di

l

qi

is a matrix of order 2 x

_{2(i -}

₁₎

that contains

_only

8 non-zero

elements,

which are

given

by

DilB!.

Thus,

there is a total of 36 non-zero elements

_contributing

to

_Ail

i from individual i. For

_convenience,

these 36 non-zero elements are collected into a 6 x 6 matrix:

Because

_W

_i

contains all contributions to

_l

_i

_A

from

individual

_i,

we refer to it as the ’contribution matrix’. The

_position

of contribution element

_W

_i

_(l,

_k)

is

_given

_by

element

_Il

_i

_{(1, k),}

so we define the

_{corresponding}

_’position

matrix’ for

_W

_i

as

where 6b

=

2(a-1)+b

for a = _s,

d,

or i and b = 1 _or 2. If both

_parents

of individual

i are

_known,

then all elements in

_Ii

_i

are defined. If at least 1

_parent

is

_unknown,

then elements in

_II

_i

associated with the unknown

_parent(s)

are not defined.

Because qi has at most 8 non-zero

_elements,

and the

positions

of these elements are

_simple

functions of s and

_d,

[18]

leads to an efficient

_algorithm

to invert

_A,

where the number of arithmetic

_operations

for

_inverting

is

_proportional

to

_2n,

the size of A. It is

_noteworthy

that any

symmetric positive

definite matrix can be inverted

using

!18!.

Unless _qi is _sparse and the

_positions

of the non-zero elements can be determined

_easily,

this

approach

will not be efficient. Note that

_[19]

_requires

_C

_s

_,

_d

_,

which is from

_A

_i

_{_}

₁

_.

Thus for an inbred

_pedigree,

_C

_s

_,

_d

needs first to be

_computed,

similar to the situation where

_inbreeding

coefficients need first to be

computed

when Henderson’s

_{rapid algorithm}

_(Henderson,

₁₉₇₆₎

is used to invert a numerator

Algorithm

1. Set

A-

1

equal

to the null matrix. 2. For individual

_i,

i =

_{1, ... ,}

_n:

(a)

if both

_parents

are

_unknown,

then add Is to

_A

_6i

_l

_bi

and

_A

_a

_i

₁₆₂

(b)

if at least 1

_parent

is

_known,

then:

i)

compute B

i

according

to

_[5]

ii)

compute

D

i

according

to

_[19]

for

inbreeding

or

_[20]

for

non-inbreeding

iii)

compute W

i

according

to

_[21]

iv)

for each ’defined’ element in

_II

_i

_,

add element

_W

_i

_{(l, k)}

to

A-’

at the

_{position given by}

_{Hi (I, k)}

NUMERICAL EXAMPLE WITH COMPLETE MARKER DATA Consider the

pedigree

of 5 individuals in table I. These 5 individuals are numbered

sequentially

so that

_parents

_precede

their

_offspring,

and are assumed to be from a

population

with marker allele

frequencies

of

p(A

d

=

0.7,

p(A

2

)

=

0.1,

and

p(A

3

)

= 0.2. For

_convenience,

_we assumed that

J

fl =

1.0 and _r = 0.1. For this

example,

genotype

A

Z

A

2

is

_assigned

to individual

_2,

so that marker data are

complete.

Computing

PDMs

The PDMs are undefined for individuals 1 and

2,

because their

_parents

are unknown. Individual 3 has

_parents

1 and 2.

_Thus,

as shown in

_{Appendix A,}

the 8 PDMs for individual 3 can be

computed

as

for

_k

₃

_{, kp, p}

= 1 or

_2,

where

_,

_l

_G

_G

₂

_,

and

_G

₃

_represent

marker

_genotypes

of individuals

1, 2,

and 3. The

_right-hand

side of

_[23]

can be

_computed

from Mendelian

For individual

_4,

the

_paternal

_parent

is unknown.

_Thus,

PDMs for individual 4 can be

_computed

as

for

k

4

, k

2

= 1 _or 2 where

G

u

=

AiA!

is the ordered marker

genotype

for the unknown

_paternal

_parent.

The _upper limit of the summation is the number of marker alleles

segregating

in the

_population.

The

_resulting

PDMs are

The first 2 columns in

_S

₄

are undefined because the

_paternal

_parent

is unknown. For individual

_5,

both

_parents

are known.

_{Thus, computation}

of PDMs for individual 5 is similar to that for individual

_3,

and the

_resulting

PDMs are

Constructing

A

Individuals 1 and 2 are unrelated and non-inbred

_{(table I),}

thus the _upper left submatrix of the conditional

_gametic

_relationship

matrix A is an

_identity

matrix of

order 4. This submatrix will be

_expanded

_by

the tabular method for individuals

3,

4,

and

_5,

as shown below.

The matrix

_B

₃

of

_PD(as

for individual 3 with

_parents

1 and 2 is

_computed

_using

S

3

according

to

_[5]:

Now,

from

_[14],

elements

_A5

_,

_j

and

_,

_,

_j

_A6

_{for j}

=

l, ... , 4,

which

correspond

to

individual

3,

are

_computed

as linear functions of elements in the first 4 rows, which

correspond

to the

_parents

1 and 2:

Diagonal

elements

_A

₅

_,

₅

and

₆

_,

₆

_A

for individual 3 are

_unity.

Off-diagonal

element

_.!6,5,

which is defined as the conditional

_inbreeding

coefficient in

_!10!,

is null because the

of elements

_A

₅

_,

_j

_{for j =}

_{1, ... ,}

5 and

_!6,!

_{for j}

₌

_{l, ... ,}

6 are

The

_{corresponding}

column elements are obtained

by

_symmetry.

The

_PD(as

for individual 4 are

_{computed using}

!5!:

For individual

_4,

numerical values of elements

_A

₇

_,

_j

_{for j}

₌

1, ... ,

7 and

_)..8,

_j

for

j = 1,...,8 are

The

PD(as

for individual 5 are

_computed

_using

!5!:

To

_compute

_f

₅

defined in

_!10!,

we need

-

3

Pr(Q3

Q!4IGobs)

and

Pr(T

kak4

IG

obs

)

for

k

3

, k

4

= 1 or 2.

_{Probabilities,}

Pr(Q3

3

-

Q!4IGobs),

have

_already

been

_computed

as

Probabilities,

Pr(r!![Go6s);

can be obtained

_according

to

_[12]

as

Similarly,

Pr(Tl2!Gobs)

=

41/100,

Pr(T2l!Gobs)

=

41/100,

and

Pr(T

22I

G

obs

)

For individual

_5,

numerical values of elements

_>’9,

_j

_{for j =}

_{1, ... ,}

9 and

_A

_l

_o,j

for

j = 1,...,10 are

The conditional

_{gametic relationship}

matrix

_(A)

is

Inverting

A

Set

A-

1

to the null matrix. For each of the 5

_individuals,

the contribution matrix

W

i

and

_{corresponding}

_position

matrix

_H

_i

are

computed

as described below. The inverse of A is obtained

_by

_adding

elements

_W

_i

_(l,

_k)

to

A-

1

at

_positions

indicated

by

elements

_{IIi(l, k).}

For the first 2

_individuals,

the

_parents

are unknown.

_Thus,

add Is to

_{AIL A2!,}

A3!

and

_A4!.

For individual

_3,

_PD(as

_(B

₃

₎

can be obtained as shown earlier. Because individual 3 is not

_{inbred, D}

₃

=

I

2

-

B3B!,

from

(20!.

Matrix

W

3

is _in

table II and

_II

₃

is in table III.

Similarly,

for individual

_4,

matrices

_W

₄

and

₁₁

₄

are in tables II and III. Note that 1

_parent

of individual 4 is unknown. Those elements in

_W

₄

and

₁₁

₄

associated with the unknown

_parent

are undefined.

From the

_{previous section,}

individual 5 is inbred

_{( f}

₅

=

0.045).

Thus,

[19]

is used

to obtain

_D

₅

=

C5 - B5C3,4B!,

where

C

5

and

C

3

,

4

_were

_computed

_in the

_previous

section:

The

A-

1

matrix is

COVARIANCE OF

_MQTL

EFFECTS GIVEN INCOMPLETE MARKER DATA

Algorithms

to construct and invert the conditional

_{gametic relationship}

matrix

_(A),

given complete

marker

_data,

are based on the recursive

_equation

!3).

In

_deriving

[3]

from

_!2!,

it was

_{assumed, given complete}

marker

_data,

that events

Q7; {::::

_Q§

and

Qs -

_Q

_i

_kj

for example,

are

_{independent. They}

_may not

_always

be

_independent,

however,

when marker

_genotypes

of the

_parents

are unknown.

_Thus,

_although

[2]

holds for

_complete

and

incomplete

marker

data,

[3]

may not hold for

_incomplete

marker data.

Therefore, algorithms developed

for

_complete

marker data cannot be

_{directly applied,}

in

_general,

to

_pedigrees

with

_incomplete

marker data. In this

section,

we first demonstrate that

_[3]

may not hold when marker

_genotypes

of

parents

are unknown. A

_strategy

to accommodate

_pedigrees

with

_incomplete

marker data is then

_presented.

The

_pedigree

in table I is used to demonstrate that

_[3]

_{may not} hold when marker

_genotypes

of the

_parents

are unknown. In this

_pedigree,

marker

_genotype

of individual

2,

the maternal

parent

of individuals 3 and

4,

is unkown.

_Thus,

as shown

_below,

_{Pr(Q4 -}

_Q2)

cannot be

_{computed using}

_!3!.

From

_!2!,

.

The last 2 terms in

_[26]

are null because the

_QTL

alleles in the unknown

_parent

of individual 4 cannot be identical

_by

descent to

_QTL

alleles in individual 3. In

deriving

[3]

from

_!2!,

it was

_{assumed, given Gobs,}

that

Q4 ! Q’

and

Qz = Q2,

for

example,

are

_independent,

ie

Because the marker

_genotype

for the maternal

_parent

of individual 4 is

_unknown,

Given the

parents’

genotypes,

the

_genotypes

of

_offspring

are

_independent.

Therefore,

Pr(Qi « Q’, Q’

=

!w3I!’robs)

can be

_computed

_{by conditioning}

on the

genotype

of individual 2

_(parent

of individuals 3 and

₄₎

as

The

_{probabilities required}

in the above

_computation

are

From the above

table,

Pr(Q4 ! Q)]G

obs

)

and

_Pr(Q!

=

Q3IG

obs

)

can also be

The values of

_{Pr(<! 4=}

_Q2!Gobs)

=

1/24,

Pr(Q2

=

Q3IGobs)

=

1/2,

and

Pr

(Q’ 4 - <- Q2, Q2

=

Q3!Gobs) =

3/400

illustrate that

Pr(! 4= Q

,

Ql

-

l

Q2

i

G

ob,,) =

A

Pr(‘‘!4 ! Q

(

w2 =

‘

2l

Pr

obs)

G

Q

2

3

ob.,)

Because

_[3]

_may not hold when marker

_genotypes

of

parents

s and d are

unknown,

the tabular

_algorithm

for

_complete

marker data cannot be

_{applied directly}

to construct

_A,

_{given incomplete}

marker data. The tabular

_algorithm

can be

used,

however,

to construct A

_{given incomplete}

marker

_data,

as described below..

Let S2 be the set of all

_possible

marker

_genotype

_{configurations}

for individuals with unknown

_genotypes,

and let

Gobs

be the observed marker

_genotypes

for individuals with known

_genotypes.

The conditional

_{gametic relationship}

matrix

given

incomplete

marker

data,

_A

_IGobs’

can then be

computed

as

where

_A

_lw

_,G

_Ob8

is the conditional

_gametic

relationship

matrix

_given

marker

geno-types

w for individuals with unknown

genotypes

and

Gobs

for individuals with known

genotypes,

and

_Pr(w

_I

_G

_obs

₎

is the conditional

_probability

of individuals with unknown

_genotypes

_having

marker

_genotypes

w,

given

marker

genotypes

Gobs

for individuals with known

_genotypes.

The matrix

_A

_lw

_,

_GOb8

can be constructed

using

the tabular method

_given

_complete

marker

_data,

and the

_probability

_Pr!Go!)

can be

computed

as

where

_{Pr(w, G}

_obs

₎

can be

computed efficiently

(Elston

and

Stewart, 1971;

Bonney,

1984).

The

conditional

_{gametic relationship}

matrix

_(A)

for the

_pedigree

in table

I,

computed

using

!27!,

is

unknown

_genotypes.

_Further,

an efficient

algorithm

to invert

_AI

_Gobs

has not been found.

_Therefore,

2

_approximate

methods to

_compute

_A¡

_Gobs

and its inverse are

presented:

1)

We have

_already

shown that

_[3]

may not hold for

incomplete

marker data

because,

given

Gobs,

Q7i !

Q!

and

_{Q9 =}

_Q

_ki

in

_[2],

for

_example,

_may not be

_independent.

If we

_ignore

this

_dependency,

then

[15]

and

(18!,

which are based on

(3!,

can be used to

_approximate

A and its inverse. This

_{approximation}

will

_require

PDMs for individuals with

_incomplete

marker data. For individual

_i,

with unknown marker

genotypes

for

_parents

s and

d,

PDMs can be

_computed

as

where each summation is over all

_possible

_genotypes

at the ML. If

_G

_s

_,

_G

_d

_,

or

G

i

is not

_missing,

then the

_{corresponding}

summation should be

dropped

from

!28!.

The

_computation

of

_{Pr(G,,Gd,GilG!b,)}

can be _very

_{time-consuming}

when a

_large

number of individuals have unknown marker

_genotypes.

An

_{approximation}

for

_Pr(G

_s

_,

_G

_d

_,

_{Gi ] Gobs)}

can be

_obtained,

however,

by conditioning only

on marker

information of ’close’ relatives of

_i,

s and

_{d, where,}

for

_example,

a set of ’close’ relatives for an individual could be its

parents,

sibs and

_offspring.

The conditional

gametic relationship

matrix

_(A),

for the

_pedigree

in table

_{I, using}

this

approxima-tion is

The consequence of this

approximation

is that the summation in

[27]

has been

brought

into inside of A and

_performed

on

_B

_i

_(or

_S

_i

_,

see

[5]).

2)

Let wmax be the

genotype

configuration

in S2 with the

_{largest probability.}

Given w

max and

Gobs,

[15]

and

[18]

can be used to

_approximate

A and its inverse. Sheehan et al

₍₁₉₉₃₎

_proposed

a

_sampling

_scheme

to

compute

the

_probability

of

genotype

configurations.

For

the

_pedigree

in table

_{I, given}

_Gobs,

_G

_z

=

A

i

A

2

has

the

_largest

.conditional

_probability

_(2/3)

among all

possible

genotypes

for

G

2

, ie

w

max =

(Gz

=

AiA

2

).

Thus,

[15]

_can be used _{to construct} A with

G

Z

=

_A

_i

_A

₂

_.

_The

The consequence of this

approximation

is that the

_resulting

A is conditional on wmax ·

A measure of how well an

_{approximation}

_compares to the exact method is the correlations

coefficient,

r’exact,a.pprox;between upper

off-diagonal

elements of

_AI

_Gob

_{s ,}

computed

exactly

by

!27!,

and

_{corresponding}

elements

_{computed by}

_approximate

methods. For the

_pedigree

in table

I,

Texact,approxl = 0.9877 for

approximation

1 and _{?’exact,approx2} = 0.8735 for

_{approximation}

2.

To further examine these

_{approximations}

_{rexa!t,apProxi and rexa!c,appTOx2} were

computed

for a

_pedigree

of 99 individuals with 3

_generations.

The first

_generation

consisted of 3

_grandsires,

each mated with 12

_granddams.

The second

_generation

consisted of 2 sires and 10 dams from each

_grandsire

for a total of 6 sires and 30 dams. Each sire was

_randomly

mated with 4

_{dams, avoiding}

full-sib and halfsib

matings.

The third

_generation

consisted of 2

_grandsons

and 2

_{granddaughters}

from each sire for a total of 12

grandsons

and 12

_{granddaughters.}

Marker

_genotypes

were assumed

_missing

for the 30 maternal

_granddams.

Thus covariances were

_only

computed

for the

_remaining

69 individuals in the

_pedigree.

Marker

_genotypes

for these 69 individuals were

_{generated randomly. Granddaughters}

and dams without progeny were

_assigned

_missing

marker

genotypes

with

_probability

0.6.

Exact and

_approximate

covariances were

_computed

for 20

_{randomly generated}

marker

_genotype

_{configurations.}

_{The average for rexact,approxi} was 0.8923 and for

!’exact,approx2 was 0.8939. The effect of these

_{approximations}

on marker-assisted

genetic

evaluation needs to be studied.

DISCUSSION

Theory

and

_algorithms

are

_presented

here to construct the conditional covariance matrix between relatives for a marked

_quantitative

trait locus

_(G

_v

=

Au v 2)

and

construct

_A!Gobs

for

_incomplete

marker data may not be efficient for

large pedigree.

Therefore,

we

presented

2 alternative

_strategies

to

_approximate

_A!Gobs

and its inverse. Simulation results indicate that the 2

_{approximations}

are similar because

they

have similar correlations with the exact method

₍

_?’exact

_,

_a

_pp

_rox

_{i =}

_0.8923,

rexact,approx2 !

0.8939).

Approximation

(1)

is

preferred, however,

because it may be difficult to search for wmax when a

_large

number of individuals have unknown marker

genotypes.

We also

_presented

an

_algorithm

to

_compute

the conditional

_inbreeding

coefficient

( fi)

for a

QTL

_{given Gobs,}

which is different from

_Wright’s

_inbreeding

coefficient. This conditional

_inbreeding

coefficient is the

_probability

that the 2

_homologous

alleles at the

_MQTL

in an individual are identical

_by

descent

_given

the

_pedigree

and marker

information,

whereas

_{Wright’s inbreeding}

coefficient is the conditional

probability

that the 2

_homologous

alleles at _any locus in an individual are identical

by

descent

_given

_only

the

_pedigree.

A numerical

_example

is used to show that

equation

!3!,

which is the basis of tabular method to construct

_G

_v

_,

does not hold

generally

when marker data are

_incomplete.

In most

_{practical situations,}

marker information will not be available on distant ancestors.

_Thus,

TM-BLUP cannot be

computed.

One of the 2

_{approximations}

presented

in this paper,

however,

can be

_employed

to

_compute

_A

_IG

_o

_bs’

Thus available marker information can be used to obtain

_improved

_genetic

evaluations

by approximate

TM-BLUP.

_Further,

in

_general,

information on distant ancestors has little

_impact

on

_genetic

evaluations.

If the ML and

_MQTL

are in

_{linkage disequilibrium,}

marker data

_provide

information on the first moment of

_MQTL

effects. In this

_{situation, regression}

techniques

can be used for

_genetic

evaluation

_using

marker and trait information

(Lande

and

_Thompson,

_1990;

_Zhang

and

Smith,

1992).

If the ML and

_MQTL

are in

_linkage

_equilibrium,

marker data do not

_provide

information on the first moment of the

_MQTL

effects. Even with

equilibrium, however,

marker data do

_provide

information on covariances of

_MQTL

effects. In this

_situation,

TM-BLUP can be used for

_genetic

evaluation

_{by fitting MQTL}

effects as random effects within animal

(Fernando

and

_Grossman,

_1989;

Cantet and

_Smith,

_1991;

_Goddard,

_{1992; Hoeschele,}

1993).

Genetic evaluation

by

TM-BLUP

_requires

_knowledge

of

_genetic

_parameters,

such as r and

o, v 2.

This is also true for

_T-BLUP,

which

_requires

_knowledge

of

genetic

variances and covariances. In

_practice,

true values of

_genetic

_parameters

are unknown and estimates are used in their

_places.

Both restricted maximum likelihood and maximum likelihood

approaches

can be used to estimate

_parameters

required

for TM-BLUP

_(Weller

and

_Fernando,

_1991).

Ideally,

marker-assisted selection will be based on

multiple

marker loci. When the

_linkage

phase

between

_flanking

marker loci is known in addition to the

_parental

origin

of marker

_alleles,

the method

_presented

_by

Goddard

₍₁₉₉₂₎

for

multiple

markers can be used for TM-BLUP. Further research is needed for TM-BLUP

_using

multiple

markers when both the

_linkage

_phase

between

_flanking

marker loci and the

ACKNOWLEDGMENT

The authors would like to thank an _anonymous reviewer for

_pointing

out an error in the

manuscript.

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₍₁₉₇₈₎

The use of loci associated with _quantitative traits in

_dairy

cattle

improvement.

Anim Prod 27, 133-139

Soller

_M,

Beckmann JS

₍₁₉₈₂₎

Restriction

_{fragment length polymorphisms}

and

_genetic

improvement.

In: 2nd World

_Congress

Genet

_Appl

Livest

Prod, Madrid,

Editorial

_Garsi,

Madrid, Spain,

vol 6, 396-404

van Arendonk

JAM,

Tier

_{B, Kinghorn}

BP

₍₁₉₉₄₎

Use of

multiple genetic

markers in

prediction

of

_breeding

values. Genetics

_137,

319-329

Weller

JI,

Fernando RL

₍₁₉₉₁₎

Strategies

for the

improvement

of animal

_{production using}

marker-assisted selection. In: Gene

_{Mapping: Strategies, Techniques}

and

_Applications

(LB

Schook,

HA

_Lewin,

DG

_McLaren,

_eds),

Marcel

_Dekker,

New

_{York, USA,}

305-328

Zhang W,

Smith C

₍₁₉₉₂₎

_Computer

simulation of marker-assisted selection

utilizing

linkage disequilibrium.

Theor

_Appl

Genet 83, 813-820

APPENDIX A

Theory

for

_computation

of PDMs

Let

_G

_s

=

M; M;,

G

d

=

MIM2

and

G

i

=

Mi M2

be the marker

_genotypes

of 2

parents

s and d and their

_offspring

i. Given

_C

_s

_,

G

d

and

_G

_i

_,

the

_probability

that

Mik

i

descended from

Mp!

does not

_depend

on other information in the

pedigree.

Thus,

_{Pr(Mf° «}

MpP!Go6s)

=

Pr(Miki !

M;

P

ICs, C

d

, Ci),

which _can be obtained

as

The numerator and denominator of

_[All

are

easily computed

from Mendelian

principles.

For

_example,

if 2

_parents

and their

_offspring

each has marker

_genotype

A

l

A

2

,

ie

_G

_s

=

Ms Ms

=

A

A

2

,

I

G

d

=

MlM2 -

2

A

A

l

and

G

i

=

Mi M

2

=

A

l

A

2

,

then

Thus,

Pr(M1 ! M

;

IC

s

,

G

d

,

G

i

)

=

1/2.

Other

examples

are listed below.

Eight

PDMs for each individual i are collected into matrix

_S

_i

_,

which is defined in

_!7!.

APPENDIX B

Theory

for

_computation

_{of PDQs}

Because

Qf°

and

_{M ki}

are on the same chromosome of individual

_i,

each must have descended from the same

_parent.

_Thus,

Pr M,&dquo; 4--

Mp&dquo;54,, Qk° ! QPP !Gobs)

is null.

Now,

There are 2

_{probabilities}

on the

_right-hand

side in

!B2!.

The first

_probability

is a PDM for individual i

_(see

_[All

for its

_{computation).}

The second

_probability

can be

_expressed

in terms of PDMs and of the recombination rate r between the ML and the

MQTL

as

_explained

below.

Given

_Mf°

_«

M:!,

the

_probability

that

_Qf°

descended from

QP

P

does not

depend

on other information in the

pedigree.

Thus,

If

_k

_§

₌

_kp,

then recombination has not taken

place,

so that

If

_{k’ p 54}

_kp,

then recombination has taken

place,

so that

The

_PD(as,

_{Pr( Q7}

_i

_{=

Q

k,

_!Gobs),

for

_ki,

_k

_P

=

_{1, 2,}

_can be obtained

by using

the

above in

_!B2!:

where p = s or d.

In _summary,

for

k

i

= 1 or 2 and p = s or

_d,

_{where p =} r when

_kp

= 1 and

p = 1- r when

kp

= 2. Note that

_PDC!s

are now

expressed

in terms of PDMs and r.

APPENDIX C

Theory

for

_computation

_{ofPr(TkskdIGobs)}

The event that the

_pair

of alleles

_(<!,<3!)

in individual i descended from

_parental

pair

(Qs

s

,

Qd

d

)

(fig 1),

is denoted

_by

_kskd

_T

for

_k

_s

_k

_d

= 1 _or 2. This _event can occur in 1 of _{2 ways:}

1.

_Q¡

descended from

_Q!s

and

_Q/

from

_Qd

_d

_,

denoted

_{(Q! 4= Q!8, Q}

₇

_.;=

Q!d)

2.

_Ql

descended from

_Qk

_d

and

_Q2

from

_Qk

_s

denoted

_{(Q$ «}

_{0!,Q! !=}

_Qs

_s

₎

Given the

_pedigree

and marker

genotypes,

the

_probability

of

_T

_kskd’

which is denoted

_by

_{Pr(T!!!!Go!),}

can be written as

Consider the first

_probability

on the

right-hand

side in

[Cl],

which can be

expressed

as

where

_{Pr(Q2 !}

Q!!G!)

is a

PDQ

and

Pr(Q/ « Q!Q? !

Qa

d

, G

obs

)

can be

expressed

in terms of

PDQs

for individual

_i,

as

explained

below.

Observe that event

_{Q/ «}

s is

implied by

Q} {= Qss;

therefore,

Further,

Thus,

[C3]

can be rewritten in terms of

PD(as

as

After

_substituting

_[C4]

in

_!C2!,

the first

_probability

on the

right-hand

side in

[Cl]

can be written in terms of

PD(as.

The same

_approach

is

_applied

to the second

probability

in

_!C1!.

_Then,

_obs

₎

_G

_kskJ

_Pr(T

can be

_expressed

in terms of

_PD(as

as

If 1 of the denominators in

_[C5]

is zero

(indicating

the event in 1 of the terms in