Breeding values for identified quantitative trait loci under selection

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Breeding values for identified quantitative trait loci

under selection

Jack C.M. Dekkers

To cite this version:

Original

article

Breeding

values

for identified

quantitative

trait loci under selection

Jack C.M. Dekkers

Department

of Animal

_Science,

225C Kildee

_Hall,

Iowa State

_University,

Ames, IA,

50011, USA

(Received

26 March

_{1999; accepted}

6

_July

₁₉₉₉₎

Abstract - The use of an identified

_quantitative

trait locus

_(QTL)

in selection

requires

the

_integration

of

_breeding

values

_(BV)

for the known

_QTL

with estimates of

_polygenic

BV. For a

QTL

with two

_alleles,

BV for the

QTL

are

traditionally

based on the allele substitution

effect,

a = a +

_{d(q - p),}

where a and d are additive and dominance

effects,

and p

and

q are gene

frequencies

in the current

_generation.

It is shown here that to maximize

_{single generation}

_response, BV for a

_QTL

with dominance must be derived based on _gene

_frequencies

_among selected mates rather than

_frequencies

in the current

_(unselected)

generation.

Because selection affects gene

frequencies

that in turn affect

_optimal

BV for the

_QTL,

_gene substitution effects must

be derived

_{numerically. Response}

from selection on

_optimized

versus standard BV

for the

QTL

was evaluated for a _range of _parameters. Benefits of

optimal

selection

were _greatest for intermediate _gene

_frequency

and increased with a

_magnitude

of

additive and dominance effects up to 9 %. Extra response was

_negligible

for gene

frequencies

less than 0.05 or _greater than 0.85. In

conclusion, strategies

for

marker-assisted selection that aim to maximize short-term response must account for the

effects of dominance and

_changes

_{in gene}

_frequency

at the

_QTL

on

_performance

of

future progeny.

©

Inra/Elsevier,

Paris

marker-assisted selection

_/

dominance

_/

_breeding values

_/

_quantitative trait loci

Résumé - Valeurs génétiques pour des loci quantitatifs identifiés en situation de

sélection. L’utilisation d’un locus

quantitatif

(QTL)

identifié en sélection nécessite

l’intégration

des valeurs

_génétiques

_(BV)

_pour le

QTL

connu avec les estimées

des BV

polygéniques.

Pour un

_QTL

avec deux

allèles,

les BV à un

_QTL

sont

traditionnellement basées sur l’effet de substitution

allélique,

a = a ₊

_{d(q - p),}

où a et d sont les effets additifs et de dominance et où p et q sont les

_fréquences

géniques

à la

_{génération présente.}

On montre ici _{que pour} maximiser la

_réponse

à

une seule

_génération

de

_sélection,

les BV _pour un

QTL

avec dominance doivent être

calculées à

_partir

des

_{fréquences géniques parmi}

les

_conjoints

sélectionnés

_plutôt

que des

_fréquences

dans la

_{génération présente}

non sélectionnée. Parce _que la sélection

affecte les

_{fréquences géniques qui}

à leur tour affectent les BV

_optimales

_pour le

QTL,

les effets de substitution de

_gènes

doivent être calculés

_{numériquement.}

La

réponse

à la sélection sur la valeur

_{génétique optimisée}

ou

_classique

_pour le

_QTL

a

été évaluée pour une série de

_paramètres.

Les bénéfices de la sélection

_optimale

ont été

plus importants

pour les

fréquences

de

gène

intermédiaires et ont

_{augmenté jusqu’à}

9 % avec

_{l’importance}

des effets additifs et de dominance. La

_{réponse supplémentaire}

a été

_négligeable

_pour les

fréquences géniques

inférieures à 0,05 ou

_supérieures

à

0,85. En

_conclusion,

les

_stratégies

de sélection assistée par marqueurs

qui

maximisent

la

_réponse

à court terme doivent tenir _compte des effets de dominance et des

changements

de

_{fréquence génique}

au

_QTL

sur la

_performance

de la descendance future.

_©

_{Inra/Elsevier,}

Paris

sélection assistée par marqueurs

/

dominance

_/

valeur génétique

/

locus

quanti-tatif

1. INTRODUCTION

Permanent

_{genetic improvement}

for

_quantitative

traits is created

_by

selection

on the additive effects of _genes that affect the trait of interest. Additive

effects are termed

_breeding

values and form the basis for

_{genetic improvement}

programs in livestock and

plants.

An individual’s

breeding

value is defined as

the

_{expected performance}

of progeny under random

mating

(4!.

Selection can be

made on estimates of the collective additive effects of _genes on the trait without

knowledge

of the _genes involved. Such estimated

_breeding

values

_(EBV)

can

be derived based on

_phenotypic

records of the individual and its relatives. To

date,

most programs for

_improvement

of additive

genetic

merit in livestock and

_plants

have relied on selection based on EBV derived from

_phenotypic

records.

_{Increasingly,}

_however,

information is

_becoming

available on the effects of individual _genes that affect

_{quantitative traits,}

so-called

_quantitative

trait loci

_(QTL).

Information on

QTL

can be combined with EBV derived from

phenotypic

records to

_improve

rates of

_{genetic improvement.}

Use of information from identified

_QTL

_{(major genes)}

in selection for

quantitative

traits was first described

_by

Neimann-Sorensen and Robertson

[13].

They

developed

procedures

to

_weight

information from an identified

QTL

with

_phenotypic

information

_using

selection index

_procedures

_(8!,

based

on the amount of

_genetic

variance

_{explained by}

the

_QTL.

Smith

_[15]

and Smith and Webb

_[16]

extended these

_procedures

and

_compared

the rates of _response from one

_generation

of selection on this index to the _response from selection on

_phenotypic

information alone. Lande and

_Thompson

_[10]

derived selection criteria

_combining

information from

_genetic

markers linked

to

_QTL

with

_{phenotypic information,}

_using

the selection index

_theory.

Marker information was combined into a marker _score, which was

_equal

to the sum

of the _average effects associated with markers.

_Average

effects were defined as allele substitution effects and derived as

_partial

coefficients of

_regression

of

phenotype

on number of marker alleles

_(10!.

Soller

_[17]

considered the discrete

nature of effects at an identified

QTL

in

_predicting

_response to selection. Selection was on an index of the

_breeding

value for the

_QTL,

which was assumed

to be known without _error, and an EBV for

_polygenic

effects.

_Pong-Wong

and

Woolliams

_[14]

showed that the discrete index used

_by

Soller

_[17]

is

_equivalent

to the indexes of Smith

_[15]

and Lande and

_Thompson

_[10]

when

_QTL

effects

The indexes described

by

the above authors were

_designed

to maximize the average

genetic

level of progeny when mated to a random _group of unselected

parents.

In

_particular,

the effects of identified

_QTL

or markers used in these

indexes were derived based on their _average effects in an unselected

_population.

In

_{practical breeding}

_programs,

_however,

selection takes

place

in both sexes,

and selected

_parents

are mated to a selected rather than an unselected _group

of mates. This may

change

the average effect of alleles for genes that express dominance.

The

_impact

of selection of mates on the

_breeding

value for identified

_QTL

was

_recognized

_by

Larzul et al.

_!11!,

who

_developed

a deterministic model for

selection on a combination of an identified

_QTL

and

_polygenes

in a

_breeding

program with

overlapping generations. Breeding

values and their estimates were

obtained in an iterative manner within the context of the defined selection program. Larzul et al.

(11!,

however,

did not consider the nature of

optimal

breeding

values for identified

QTL,

nor did

they investigate

the

impact

of

the use of

optimal

versus standard

breeding

values for the identified

QTL

on

selection _response.

The

_objectives

of this _{paper were,}

_therefore,

to derive

_breeding

values for identified

_QTL

that maximize the _response to

_{single generation}

selection and to evaluate the

_advantage

of selection based on

_optimum

breeding

values

over selection based on conventional

_breeding

values for

_single

_genes. A

_single

identified

_QTL

with known effects is considered for

simplicity,

but

_implications

for selection on marked

_QTL

or when

_QTL

effects are not known without error

are discussed. The

_objectives

of this _paper are

important

relative to the use of

information of individual _{genes in}

_{genetic improvement}

programs.

2. METHODS 2.1. Notation

Consider

_generation

0 of an unselected

population

of infinite size with

discrete

_generations

and in

_{gametic phase equilibrium}

_!1!.

The

_population

is recorded for a

_quantitative

trait that is affected

by

an identified

_QTL

and

unlinked

_polygenes.

All individuals are

genotyped

for the

QTL prior

to their

age of selection. The

QTL

has two

alleles,

B and

b,

with

frequencies

po and qo.

Following

Falconer and

_MacKay

_!4!,

_genotypic

values for the

_QTL

are _a, d and

- a for individuals with

_genotypes

G

i

_equal

to

BB,

Bb and

_{bb, respectively.}

Parameters and notation for the identified

_QTL

are summarized in table L Effects and

_frequencies

of alleles at the

_QTL

are assumed to be known without

error.

Polygenic

effects for the

_quantitative

trait conform to the infinitesimal

genetic

model

_[4].

After

_accounting

for effects at the identified

_QTL,

the

phenotypic

standard deviation of the trait is _{op and}

_heritability

is

h

2

.

_Breeding

values for

_polygenic

effects are estimated with _accuracy r&dquo;, for males and _{r f}

for

females,

resulting

in a standard deviation of estimated

breeding

values

for

_polygenic

effects

_equal

to am =

r&dquo;,h!P

for males and cry =

r

f

h

Qr

for females. With

_polygenic

_breeding

values estimated based on own

_performance,

r

m =

r = h. The results derived

here, however,

apply

to estimates of the

information from

relatives,

or based on the best linear unbiased

_prediction

methods,

with a model that includes a

_QTL

_genotype

as a fixed effect

(e.g.

[9]).

Consider the selection of a fraction

_Q

₉

of males and

Q

d

of females to

_produce

the next

_generation

_{(generation 1).}

_Mating

of selected

_parents

is at random.

Selection is

_by

truncation on an EBV that combines the

_breeding

value for the identified

QTL

with an estimate of the

polygenic

breeding

value:

where

_A2!k

is the total EBV for animal k of

_{sex j}

_(male

or

female)

and

_QTL

genotype

i (BB,

Bb or

bb),

_gi! is the

_breeding

value for the

_QTL

for individuals

with

_QTL

_genotype

i of sex

_j,

as a deviation from the

_QTL

_breeding

value

for individuals with

_genotype

Bb

_(g

_Bb

_,

_j

=

0),

and

Û

ijk

is an estimate of the

polygenic breeding

value for animal

_{ijk. Following}

Falconer and

_MacKay

_!4!,

breeding

values for the

_QTL

for individuals with

_genotypes

_BB,

Bb and bb are

equal

to

_+2q

_o

_a

_o

_,

_(q

_o

_{- p}

_o

_)a

_o

_,

and

_-2p

_o

_a

_o

_,

where ao is defined as the _average allele substitution effect and is

_equal

to ao = _{a +}

(q

o

-

p

o

)d.

When selection

is within a

_{generation, QTL}

_breeding

values can for

_simplicity

be deviated

from the

_breeding

value of the

_heterozygote

without

_changing

the

_ranking

of

individuals

_{by subtracting}

_(q

_o

_-

_p

_o

_)a

_o

_.

This results in

_adjusted

_QTL

_breeding

values _gij

_equal

to

_+a

_o

_,

0 and -ao

(see

table

7).

2.2.

_{Optimal QTL breeding}

values

Under random

_mating

to selected

_mates,

the EBV of an individual that is

expected

to maximize _response from the current to the next

_generation

can

be derived as two times the

_expected

mean of _progeny conditional on the

information available. Consider an individual

_ijk

with

_QTL

_{genotype G}

_i

and

polygenic

EBV

_equal

to

_Û

_ijk

_,

which is mated at random to a _group of mates

with

_QTL

gene

frequencies

pm and qm and average

polygenic

EBV

equal

to

Let

pi and qi denote the

frequency

of

gametes

carrying

the B and b allele:

p

i

equals 1,

1/2

and 0 for

G

i

equal

to

BB,

Bb and

bb,

and q

i

=

_{1 - p}

_i

_.

Under

random

_mating

to selected

_mates,

B and b

_gametes

are combined at random

to B and b

gametes

with

_frequencies

_pm and _q&dquo;,,.

_Again

taken as a deviation

genotype

G

i

can be derived as:

Using

the fact that pi+ qi= 1 and

p

m

+ q

m

=

1,

the latter

equation

can be

simplified

to:

Resulting

QTL

breeding

values are

equal

to +a.&dquo;,,, 0 and -a.&dquo;,, for individuals

with

_genotypes

_BB,

Bb and bb. Note that this result is consistent with the

quantitative genetic

theory

[4, 17],

except

that the gene substitution effect a

m is based on _gene

frequencies

_among selected mates rather than

_frequencies

among all selection candidates.

Letting

ps and q9 be the

frequencies

of B and b among selected males and p

d and qdthe

frequencies

among selected

females,

optimal

breeding

values for the

_QTL

become

_equal

to _+a,, 0 and -as for sires and

_equal

to +ad, 0 and

- a

d for

dams,

with:

2.3. Numerical

_procedures

for derivation of

_{optimal QTL}

breeding

values

The

_problem

with the

_{implementation}

of the

_procedures

described above for selection on the identified

_QTL

is that

_{optimal breeding}

values for the

_QTL

in index

₍₄₎

_depend

on the _gene

_frequency

of the

_QTL

_among

_mates,

_which

in turn

_depends

on the selection that takes

place

_among mates

_{and, therefore,}

on the index used for selection. This means that

_optimum

_breeding

values

cannot be derived

_{analytically,}

but that iterative

_procedures

are

_required.

These

procedures,

which are derived

below,

involve the

_prediction

of _gene

_frequencies

among selected sires and dams for

given

values of as and ad, followed

by

updating

a9and ad in an iterative manner based on the new

frequencies

_among

2.3.1. Deterministic model for selection on

_{given QTL breeding}

values

For each _{sex, truncation} selection on index

Â

ijk

=

2(q

i

-

1/2)a

m

₊

u2!!

involves selection across three Normal distributions that

_correspond

to

individuals with

_QTL

_genotypes

_BB,

Bb and

bb,

as illustrated in

_figure

1. Distributions have means

equal

to _+as, 0 and -as for sires and

equal

to

+ a d

, 0 and -ad for dams. The standard deviation of the three distributions is

equal

to am for males and Q

_for

females. The

_frequency

of each distribution

is determined

by

the

_frequency

of

_QTL

_genotypes

_among selection

_candidates,

which under random

_mating

is

_equal

to

_P6

_,

_2p

_o

_q

_o

and

_qo

in

_generation

0.

For a

_given

set of

_frequencies,

means

(based

on

a! )

and standard deviations

of the three

_{distributions,}

a

_unique

truncation

_point

_Cj exists across the three

distributions for

_{sex j}

_that

results in the correct selected fraction

_(Q

_s

for males and

_Q

_d

for

_females).

Let

_f

_ij

and _xij be the selected fraction and standardized truncation

_point,

_{respectively,}

for the distribution of EBV for individuals with

QTL

genotype

i of sex

_j.

The

_unique

truncation

_point

on the EBV

_scale,

_cj,

relates to the standardized truncation

_points

_xzj based on:

where _ilij is

_equal

to

_+a

_j

_,

0 and _-aj for

_genotypes

_BB,

Bb and bb.

Also,

the

following relationships

must exist between the standardized truncation

_points

x2! :

In

_addition,

the

_f

_ij

fractions selected from distribution

_ij,

which are

_equal

to 1 -

_1>(

_Xij

_),

where 4) is the cumulative distribution function for a standard normal

_{distribution,}

must

_satisfy

a constraint on the overall fraction selection:

Equations

(7)-(9)

uniquely

define the truncation

_point

_c!. Even for

_given

distribution

_parameters,

an

analytical

solution does not exist but _Cj must be

solved

_iteratively.

Iteration can be based on a Newton method

_algorithm,

as

developed

by

Ducrocq

and

_Quaas

_!3!,

or on a bisection method as

_suggested

_by

Gibson

_[6]

and

_given

in

_Appendix

I. Once the

_unique

truncation

_point

_cj has been

_obtained,

the

_{QTL frequency}

_among selection candidates

_(p

_s

and

_p

_d

₎

can

be derived from

With random

_mating

of selected

_parents,

the _average

_genetic

value of progeny can be derived based on

where _U, is the _average

_polygenic

value of progeny. This value ul can be

QTL

genotypes

and sexes as:

2.3.2. Iterative

_procedure

for

_deriving

_{optimal QTL}

_breeding

values

Iterative

_procedures

for

_finding

the

_unique

truncation

_points

for

_given

allele substitution effects must be

_incorporated

within an iterative

_procedure

for

_finding

the

_{optimal QTL}

substitution effects as and ad. The

following

procedure

can be used:

3)

Find the

_unique

truncation

_points

_c! and _cd and fractions

_selected,

_f

_ij

_,

based on the

_procedures

described in section 2.3.1.

4)

Compute

the

_frequency

of

_QTL

alleles among selected

parents

p, and pd, based on

_equation

(10).

5)

Using

the new solutions for _p, and _{, dp}

_compute

new values for as and ad

as: as = _{a +}

(q

d

-

Pd

)d

and

ad

= a +

_(q

_s

_-

_p

_s

_)d.

A

_{multiplicative}

relaxation factor _may be

_{required here,}

_reducing

_changes

in as and ad from one iteration

to

_another,

to allow convergence.

6)

Repeat

steps

2

_through

5 until as and ad converge to stable solutions. Once

_optimal

solutions have been

_obtained,

the

_{expected genetic}

level among progeny can be determined based on

_equations

(11)

and

(12).

Note that the

starting

values for this iterative

_procedure,

which are set in

_step

_1,

_provide

results for classic selection with a known

QTL.

2.4.

_{Optimal QTL}

_breeding

values with

_{gametic phase}

disequilibrium

In section

_2.3,

the

_{parental generation}

was assumed to be in

_gametic

_phase

equilibrium.

When

_{gametic phase}

_{disequilibrium}

is

_present

in the

parental

population

as a result of

_{prior selection,}

_average

_polygenic

values will differ

by

QTL

genotype;

with truncation

selection,

individuals with the favorable

QTL

genotype

tend to have lower

_polygenic

values. This

_{disequilibrium}

must be

_incorporated

in selection decisions. Let

₇

_!

_ij

be the _average

_{polygenic breeding}

value for

_{QTL genotype}

i for sex

_j.

Under random

_mating

of selected

_parents,

average

polygenic

values will be

equal

for male and female progeny and

equidistant

between the three _progeny

_genotypes,

and hence let

_UBB

_,

_j

₌

usb,! -!B6,j -u66,j

= 6.

Assuming

6 can be estimated with sufficient

accuracy,

gametic

phase disequilibrium

between the

_QTL

and

_polygenes

can be accounted

for in the selection index as follows

(e.g. !14!)

where

_Û

_ijk

is now the individual’s

_polygenic

EBV as a deviation from the

average

polygenic breeding

value for individuals of

QTL

genotype

i and sex

j.

Based on

_{this, optimal QTL}

allele substitution effects can be derived as

Note that because 6 is

_negative,

a

_{gametic phase disequilibrium}

will reduce

the _average allele substitution effect associated with the

_QTL.

3. RESULTS

3.1. One

_generation

_response

Methods for the

_optimization

of

_single

_generation

_response were

_applied

and the _responses were

_compared

to selection on an index in which

breeding

values for the

QTL

were derived based on

_frequency

in the

_{parental generation}

(a

= a +

_{(1 -}

_2p

_o

_)d).

These two

_strategies

will be referred to as

_optimal

and standard

_{gene-assisted}

selection

_(GAS),

respectively.

Figure

2 compares

the _response to one

_generation

of

optimal

GAS to _response to standard

GAS,

as a function of

_frequency

of the favorable allele at the

QTL.

The results are shown for

_varying

levels of additive and dominance effects at the

polygenic

effects

_(or),

which is what determines the selection response for the

QTL

for

_polygenes,

rather than relative to the

_genetic

or

phenotypic

standard

deviation.

_Therefore,

the results in

_{figure 2}

hold for

_{specified magnitudes}

a and

d in terms of standard deviations of EBV but

regardless

of

_heritability

and

phenotypic

standard deviations. Relative

QTL

effects in

_figure

_{2 can,}

_however,

be converted to values relative to the

_genetic

standard deviation

_{by multiplying}

a and d

by

the _accuracy of EBV and to values relative to the

phenotypic

standard deviation

_{by multiplying}

a and d

_by

_accuracy and the _square root of

heritability.

For

_example,

with

_polygenic

EBV based on own

_phenotype

alone for a trait with

heritability equal

to

_0.25,

and a

phenotypic

standard deviation

equal

to one, one standard deviation of EBV converts to 0.5

_genetic

standard deviations

_(accuracy

=

0.5)

and to 0.25

phenotypic

standard deviations

(square

root of

_heritability

=

0.5).

_Hence,

_a

QTL

with _a = 1!

_represents

_a

gene with

only

moderate effects for a trait with low

_{heritability.}

For

_figure

_2,

the standard deviation and accuracy of EBV is assumed

equal

for males and females.

Over a

_single

_generation,

benefits of

_optimal

GAS over standard GAS were

the

_greatest

for

_{QTL frequencies}

between 0.3 and 0.5 and increased with the

magnitude

of additive and dominance effects at the

_major

_gene

_{(figure 2).}

Extra _response was

_negligible

for _gene

_frequencies

less than 0.05 and

greater

than 0.85. Extra _response was

_greater

than 8

%

for

QTL

with

large

effects

Figure 3

shows the effect of selection

_intensity

on extra _response from

_optimal

selection for a

_QTL

with

_complete

dominance and a = 1Q. The benefit of

optimal

selection increased with the

_intensity

of selection. Selection of 5

%

among males and 40

%

among females had similar results as selection of 20

%

for both males and females.

Figure

4 shows

the

_relationship

between

optimal

allele substitution effects at

the

_QTL

and _gene

_frequency

for a

QTL

with

_complete

dominance and with 5

%

selected among males and 40

%

among females. The standard allele substitution effect

_changes

with gene

frequency

in a linear _manner, based on a =

a+(q-p)d.

Optimal

allele substitution effects

_changed

in a nonlinear _manner,

_depending

on

_QTL

_frequency

_among mates.

_Optimal

allele substitution effects were lower

than the standard substitution effects. For

_{females, optimal}

substitution effects

were _up to 75

%

lower than standard substitution effects.

_Optimal

allele substitution effects were more

greatly

affected for females than males because

selection

_intensity

was

_greater

for

_{males, and, therefore,}

_QTL

_frequency

differed

more

_drastically

from

_QTL

_frequency

_among all candidates for selected males

effect would occur

_(results

not

_shown);

_{optimal breeding}

values are

_greater

than standardized

_breeding

values under selection because

_breeding

values

(a

+

_{(q - p)d)}

increase with _p for

_negative

d. This increase in

_{QTL breeding}

values will increase the

_emphasis

on

_QTL

relative to

_polygenes.

3.2.

_{Multiple generation}

response

Responses

to

_optimal

and standard GAS were also

compared

over

multiple

generations, starting

from a

population

in

gametic phase

equilibrium.

QTL

allele substitution effects were

updated

each

_generation

for both

optimal

and

standard GAS to account for the

changes

in _gene

_frequency

and

_{gametic phase}

disequilibrium

between the

_QTL

and

_polygenes.

_Polygenic

means

_by

_genotype

class were assumed known without error.

_Polygenic

variance was assumed to remain constant.

Figure 5

shows the extra cumulative benefit of selection on

_optimal

over

until gene

frequencies

were between 0.3 and 0.5 and then decreased. This trend

is consistent with the

_relationship

between

_single

_generation

response and gene

frequency

observed in

_{figures 2}

and 3. Extra cumulative responses after ten

generations

were

relatively

small

(less

than 2

%

for the chosen

examples).

4. DISCUSSION

The

_objective

of this paper was to derive

_breeding

values for a

_single

locus

that,

when used in combination with EBV for

_polygenic

effects,

maximize

_single

generation

response to selection based on

_{expected performance}

of _progeny.

Single

locus

_breeding

values thus derived were

equivalent

to

_breeding

values derived on the standard

quantitative genetic theory

[4]

but with the average

effect of allele

_{substitution,}

a derived from _gene

frequency

_among

_mates,

rather

the difference between

_optimal

and standard

_breeding

values for an individual locus therefore

_depends

on the

degree

of

dominance, d,

and the effect of selection

on the _gene

_frequency

_among selected mates. The latter

_depends

on selection

emphasis

that is

placed

on the individual locus and its effect and

_frequency.

With

_phenotypic

selection and when the trait is affected

_by

a

_large

number of

genes of minor

effect,

the effect of selection on _gene

_frequency

will be

_small,

and, hence,

the difference between

_optimal

and standard

_breeding

values for a

single

locus will be minimal. With direct selection on

_QTL

of sizeable

_effect,

selection _can,

_however,

have a substantial

_impact

on _gene

_{frequencies, and,}

therefore, optimal

QTL

breeding

values can differ

significantly

from standard

breeding

values for a locus with dominance. This is illustrated in

_figure

_4.

The

importance

of derivation of

optimum

breeding

values for a

_single

locus lies in the current advances in molecular

_genetics,

which lead to the

_uncovering

of loci that affect

_quantitative

_traits,

either

_by

direct identification or

_indirectly

through

linked

genetic

markers. Use of this information in

_{genetic improvement}

involves

_combining

information on identified

_QTL

with EBV for the collective

effects of other _genes that affect the trait

_{(polygenic effects).}

The results from this _paper show

_that,

if the

QTL

exhibits

dominance,

substantial additional

genetic

progress can be made over a

_single

_generation

if

_breeding

values for the

QTL

take into account the effect of selection on _gene

frequencies

_{among mates.}

Although

benefits were small for

_QTL

with moderate additive and dominance

effects,

improvements

of up to 9

%

in

single generation

response were observed

for

_QTL

with

_larger

additive and dominance effects

_(see

_figure

_2).

Greatest benefits for the use of

optimal

over standard

QTL breeding

values were obtained

for _gene

_frequencies

in the

_parental

_generation

between 0.3 and

0.5,

depending

on the

magnitude

of the

QTL

effects. For a

QTL

with

_positive

_dominance,

genetic

variance contributed

_by

the

_{QTL and, therefore,}

the

_opportunity

to

change

gene

frequency

is

greatest

for this _range of _gene

_frequencies

_(4!.

The use of

_{optimal QTL}

_breeding

values over successive

_generations

resulted in

_greater

cumulative _response than the use of standard

QTL breeding

values

(figure 5),

although

the benefit of

_optimal

over standard

_breeding

values

decreased over

_generations.

It is

_important

to note that the

optimal

QTL

breeding

values derived here maximize

_{single generation}

_responses but _may

not maximize cumulative _response over

_{multiple generations.}

This has been illustrated

_by

several authors

_{(e.g. [7, 14!)}

for additive _genes, for which standard

QTL

breeding

values maximize

_{single generation}

_response, and

_by

others

_(e.g.

[11])

for

_QTL

with dominance. The reason for the

_{suboptimality}

of

QTL

selection

_strategies

that maximize

_{single generation}

_response over

_multiple

generation

is that selection

_changes

not

_only

the

_population

mean but also

population

parameters

(frequency

and,

thereby,

variance at the

_{QTL) !2!.}

_Single

generation

selection

_thereby

affects

_{opportunities}

for _{response in}

_subsequent

generations.

Manfredi et al.

_[12]

and Dekkers and Van Arendonk

_[2]

_developed

methods to

_optimize

_QTL

selection over

_{multiple generations.}

The additional benefit of

_multiple

_{generation optimization}

over

_single

_{generation optimization}

will be

_investigated

in

_subsequent

work.

In the

_present

_{study, QTL}

_genotypes

could be observed

_directly,

and the effect of the

_QTL

was assumed known without error. In _{many cases,}

_QTL

on

changes

in

frequency

at the

QTL and, therefore,

the difference between

optimal

and standard

breeding

values. With

_uncertainty

about estimates of

QTL effects,

the effect of selection on

QTL frequencies

may be difficult to

predict.

This will increase the errors of

_prediction

of

optimal breeding

values.

It must also be noted that derivation of

optimal QTL breeding

values

_requires

estimates of additive

_(a)

and dominance

_(d)

effects at the

_QTL,

as well as an estimate of the

frequency

of the

QTL.

These estimates may be difficult to obtain in outbred

_populations

based on linked markers. For

example,

the

best linear unbiased

_prediction

method

developed

by

Fernando and Grossman

[5]

and extended

by

others for the

_{incorporation}

of marker information in

breeding

value estimation estimates the _average effect of the

QTL,

rather than

separate

additive and dominance effects. For non-additive

QTL,

the

_resulting

QTL breeding

value estimates will

_depend

on the

_{QTL frequency}

_among mates

of animals that contributed information to estimate the

_QTL

effect. With selection on the

_QTL,

the

_QTL

_frequency

_{among mates} of these animals may not be the same as the

QTL

frequency

_among individuals to which animals

that are selected based on the

QTL

will be mated. The same holds for the

multiple regression

methods

_{suggested by}

Lande and

_Thompson

_!10!,

in which marker effects are estimated as linear coefficients of

_regression

of

phenotypes

on number of marker alleles.

Implementation

of

optimal QTL breeding

values

in

_strategies

for marker-assisted selection in outbred

populations, therefore,

requires

further

_{investigation.}

ACKNOWLEDGMENTS

Financial

_support

from the Iowa Pork Producers Association

through

the National Pork Board for

aspects

of this work is

_{greatly appreciated.}

This is Journal

Paper

No. J-18323 of the Iowa

_Agriculture

and Home Economics

Experiment Station, Ames, Iowa,

USA

_(Project

No.

₃₄₅₆₎

and

_{supported by}

the Hatch Act and State of Iowa funds.

REFERENCES

[1]

Bulmer

M.G.,

The Mathematical

Theory

of

_{Quantitative Genetics,}

Clarendon

Press,

Oxford,

1980.

[2]

Dekkers

_{J.C.M., Van}

Arendonk

J.A.M., Optimizing

selection for

_quantitative

traits with information on an identified locus in outbred

populations,

Genet. Res.

(1998)

257-275.

[3]

Ducrocq V., Quaas R.L.,

Prediction of

_genetic

response to truncation selection

across

_generations,

J.

Dairy

Sci. 71

₍₁₉₈₈₎

2543-2553.

[4]

Falconer

D.S., MacKay T.F.C.,

Introduction to

_{Quantitative Genetics,}

Long-man,

Harlow,

UK, 1996.

[5]

Fernando R.L., Grossman M., Marker-assisted selection

using

best linear

un-biased

prediction,

Genet. Sel. Evol. 21

₍₁₉₈₉₎

467.

[6]

Gibson

J.P.,

An Introduction to the

_Design

and Economics of Animal

Breeding

Strategies,

Course notes, Centre for Genetic

_Improvement

of

Livestock, University

of

Guelph, Canada,

1995, p. 149.

[8]

Hazel

_L.N.,

The _genetic basis of

_constructing

selection

_indexes,

Genetics 28

(1943)

476-490.

[9]

Kennedy B.W., Quinton M.,

Van Arendonk

_J.A.M.,

Estimation of effects of

single

genes on

_quantitative

_traits, J. Anim. Sci. 70

₍₁₉₉₂₎

2000-2012.

[10]

Lande

_{R., Thompson R., Efficiency}

of marker-assisted selection in the

im-provement of

_{quantitative traits,}

Genetics 124

₍₁₉₉₀₎

743-756.

!11!

Larzul

_{C., Manfredi E., Elsen J.M.,}

Potential

gain

from

_{including major}

_gene information in

_breeding

value

_estimation,

Genet. Sel. Evol. 29

₍₁₉₉₇₎

161-184.

[12]

Manfredi

E.,

Barbieri

M.,

Fournet

_F.,

Elsen

J.M.,

A

_dynamic

deterministic model to evaluate

breeding strategies

under mixed

inheritance,

Genet. Sel. Evol. 30

(1998)

127-148.

[13]

Neimann-Sorensen

A.,

Robertson

A.,

The association between blood groups and several

production

characteristics in three Danish cattle

breeds,

Acta

Agric.

Scand. (1961)

161-196.

[14]

Pong-Wong R.,

Woolliams

J.A., Response

to mass selection when an

identi-fied

major

gene is

segregating,

Genet. Sel. Evol. 30

(1998)

313-337.

[15]

Smith

_{C., Improvement}

of metric traits

_{through specific genetic loci,}

Anim. Prod. 9

₍₁₉₆₇₎

349-358.

[16]

Smith

_C.,

Webb

_A.J.,

Effects of

_major

_genes on animal

_{breeding strategies,}

Z. Tierzucht.

_Zuchtgsbiol.

98

₍₁₉₈₁₎

161-169.

[17]

Soller

_M.,

The use of loci associated with

_quantitative

effects in

_dairy

cattle

improvement,

Anim. Prod. 27

₍₁₉₇₈₎

133-139.

APPENDIX I: Bisection method to determine

_unique

truncation

point

to select across

_multiple

Normal distributions

Selection of a fraction

_Q

_by

truncation across three distributions with

frequencies

pi, mean _pi

_(i

=

1, 2, 3)

and standard deviation _0’!. Let _c be the

unique

truncation

_point

on the

_original

scale and xi and

_f

_i

the standardized

truncation

_point

and fraction selected for distribution i. Based on the definition

of a standardized truncation

_point,

xi =

(c - p

i

)lo

i

and

f

i

= 1 —

1

>(

Xi

),

where 1> is the cumulative Normal distribution function.

Then,

truncation

_point

c must be chosen such that

_{p, f, +p2/2 +}

_P3

_h

=

_Q.

The

_following

iterative

_procedure

can be used to find truncation

_point

c

(based

on

_(6!).

1)

For all

i,

find the standardized truncation

_point

xi

corresponding

to 1 -

_1>(

_Xi

₎

=

_{Q using}

the inverse Normal distribution function.

2)

Convert standardized truncation

_{points x}

_i

to the

_original

scale based on

c

i = _{xzaz +}

!,. Choose the lowest ci as lower bound for

C

(c

d

and the

_highest

c

i as the _upper bound for c

(c

U

)

(c

must lie between cL and

c

u

).

3)

Compute

the

_midpoint

between cL and cu c,i,l =

(c

L

₊

c

U

).

4)

Compute

standardized truncation

_{points corresponding}

to cM for each

distribution: xi =

(c

M

-

J

-

li

)/a

i

and the

corresponding

_proportions

selected:

fi

= 1 -

4

$

(

X

z ) .

5)

Compute

the total

_proportion

selected as:

Q,1,1 = p

l

f

l

+ pzf2 +

_P3

_f

₃

_.

6)

If

_{]OM &mdash;<3! <}

_convergence

_criterion,

the

unique

truncation

_point

has been

found: c = _cM

7