Response to mass selection when an identified major gene is segregating

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Response to mass selection when an identified major

gene is segregating

Ricardo Pong-Wong, John A. Woolliams

To cite this version:

Original

article

Response

to

mass

selection

when

an

identified

major

_gene

is

_segregating

Ricardo

_Pong-Wong*

John A. Woolliams

Roslin Institute

_(Edinburgh),

Roslin,

Midlothian EH25

9PS,

UK

(Received

28 October

1997; accepted

2 June

₁₉₉₈₎

Abstract - A deterministic model to

predict

response with mass selection when a

major

locus is

_segregating

is

_presented.

The model uses a selection index framework in which the

_weight

of the different _components included in the index are

_adjusted

to describe the different methods of selection

_using

_genotype information as selection

criteria. The _response over

_multiple

_generations to several methods of selection

_using

either the whole genotype effect

_(genotypic

_methods)

or

_only

the Mendelian

_sampling

deviation of the

_major

locus

_(Mendelian

_methods)

was

_compared

with selection

using

only performance

record

_{(phenotypic method).}

Relevant differences in response between

using

and

_ignoring

information on the

major

_gene were observed

_only

when the favourable allele was at a low

frequency.

When the

_major

locus had a

_completely

additive

effect,

all the

genotypic

or Mendelian methods had a

_higher

cumulated

genetic

gain

in the first 3-4

_generations

of selection but this

_advantage

was lost thereafter.

In the

_long

_term, without

_exception,

all methods

_using

_genotype information of an

additive

_major

gene had lower cumulated

gain

than

_phenotypic

selection over a wide

range of parameters. The reason for the

_{long-term loss,}

was a reduction in the

_intensity

of selection

_applied

to the

_{polygenic background arising}

from

_increasing

the differences

in the selective

_advantage

between _genotype groups. The same trend was observed

when the favourable allele of the major locus was

_completely

recessive or

_dominant,

with the

_exception

of the cases of a

large

recessive locus

(over

one

phenotypic

standard

deviation)

where the extra

_{early gain}

from

using

genotype information was maintained

in the

long

term. This was

_{explained by}

the

_inefficiency

of the

phenotypic

selection to fix the favourable allele due to the

_{linkage disequilibrium built-up}

between the

major

locus and the

_polygenic

effects. Differences in the

_inbreeding

rate were also observed between these methods: the

_genotypic

methods had the

_{highest inbreeding}

rate while the Mendelian had the lowest. The difference in the

_inbreeding

rate was

mainly

observed in the first

_generations

of selection and increased with lower

_starting

frequency

of the

_major

locus.

_@

_{Inra/Elsevier,}

Paris

major gene

/

indice

/

gain

/

inbreeding

/

loss

*

Correspondence

and reprints

Résumé - Réponse à la sélection massale en _présence d’un gène majeur iden-tifié. On

présente

un modèle déterministe de

prédiction

de la

_réponse

à la sélection massale

quand

un

gène majeur

est en

ségrégation.

Le modèle utilise le cadre de la théorie des index avec modifications des

pondérations

concernant les différentes composantes de

l’index,

en fonction des différentes méthodes de sélection. Les

réponses,

sur

_{plusieurs générations,}

à

_plusieurs

méthodes de sélection utilisant soit

l’ensemble des effets

_{génotypiques}

_(méthodes

_{génotypiques)}

ou seulement la déviation

d’échantillonnage

mendélien au locus

_majeur

(méthodes mendéliennes)

ont été

com-parées

à la sélection utilisant

_uniquement

les

_performances

_{(méthode phénotypique).}

Des différences

_{appréciables}

selon la

_prise

en _compte ou non de l’information sur le

_{gène majeur}

ont été observées

_{uniquement quand}

l’allèle favorable était à une basse

fréquence. Quand

le locus

_majeur

avait un effet

complètement additif,

toutes les méthodes

génotypiques

ou mendéliennes ont

_engendré

un

_{progrès génétique}

cu-mulé

plus

élevé durant les 3-4

premières générations

mais cet _avantage a été ensuite

perdu.

Sur le

_long

_terme, pour une

_large

_gamme de

_paramètres

et sans

_exception,

toutes les méthodes utilisant l’information

_génotypique

pour un

_{gène majeur}

addi-tif ont

_engendré

un

_gain

cumulé inférieur à la sélection

_{phénotypique.}

La raison de cette _perte à

_long

terme a été une réduction de l’intensité de sélection

_appliquée

à

l’arrière-plan

polygénique,

liée à

_{l’augmentation}

des différences

_d’avantage

sélectif entre groupes

génotypiques.

La même tendance a été observée

quand

l’allèle

fa-vorable au locus

_majeur

a été

_{complètement}

récessif ou

dominant,

à

l’exception

des cas d’un

_gène

à effet

_important

(plus

d’un

écart-type

phénotypique)

et avec récessivité

où le

gain

lié à l’utilisation de l’information

génotypique

se maintient

_{plus longtemps.}

Ceci a _pu être

expliqué

par l’inefficacité de la sélection

phénotypique

à fixer l’allèle

favorable à cause du

déséquilibre

de liaison induit entre le locus

_majeur

et les ef-fets

_{polygéniques.}

Les méthodes

_{génotypiques}

ont créé

_plus

de

_{consanguinité}

_que les méthodes mendéliennes. Ceci a été observé surtout dans les

_{premières générations}

et

a été d’autant

_{plus important}

_que la

_fréquence

initiale de l’allèle favorable au locus

majeur

était faible.

_©

_{Inra/Elsevier,}

Paris

gène majeur

/

index

_/

progrès génétique

/

consanguinité

1. INTRODUCTION

Although

selection in farm animals has been

_successfully

carried out

assum-ing

the infinitesimal

_model,

the

_discovery

of

_single

_genes

_{having large}

effects on

quantitative

traits and advances in DNA

technology

has increased the interest of

_using

_genotype

information to

_improve

_response to selection.

_{Additionally,}

statistical methods to obtain estimates of the effects of such _genes are also

becoming

more reliable

_(e.g.

_[8,

_11,

_12,

_16]).

The benefits of

_combining

both the

_genotype

and

performance

information

has

mostly

been assessed in terms of the short- and the medium-term

_genetic

response relative to traditional

_phenotypic

selection. The

_general

conclusions

are that the use of the

_genotype

information from a

_major

_gene or a marker linked to the _gene

_{significantly}

increases the short-term

_genetic

_response

[2,

13, 17, 18,

20].

However,

Gibson

_[6]

also

_reported

that methods

_using

genotype

information may have a detrimental effect in the

long-term

cumulated

gain.

Therefore,

further studies are still

_required

to understand the factors

affecting

the short- and

long-term

response to selection when a

_major

locus is

segregating.

poly-genic

effect and a

_single

locus with a

_major

effect)

is defined. Recursive equa-tions for

_predicting

the

_change

in the

_{genetic level,}

the

_polygenic

variance and the _gene

_frequency

of the

_major

locus over

_{multiple generations}

of selection are

_presented.

The

_linkage

_{disequilibrium}

between the

_major

locus and the

polygenic

effects

built-up

with selection is also calculated. A selection index framework to combine both

_genotype

and

_performance

information is used to describe different

_{opportunities}

for selection.

_Using

this

framework,

several methods of selection are

_compared

across a wide _range of

_parameters.

The

comparison

was made in terms of short- and

_long-term

_response, the level of

inbreeding

accumulated after several

_generations

of selection and the

proba-bility

of

_losing

the favourable allele

_during

the selection _process.

_Comparison

of risks associated with _gene assisted selection

_(GAS)

such as

inbreeding

have received little information to date.

2. METHODS

2.1. Deterministic

_genetic

model 2.1.1. Notation

A

_quantitative

trait is assumed to be

_genetically

affected

_by

a

polygenic

effect and the

_major

effect of a

_single

diallelic locus

(A

and

_B).

Before selection in the base

_population,

the

_frequency

of the favourable allele

_(A)

is _p, and the three

_possible

_genotypes

_(AA,

_AB,

_BB),

are assumed to have

_{Hardy-Weinberg}

equilibrium frequencies,

and be in

_{linkage equilibrium}

with the

_polygenic

effect.

Following

the same notation as Falconer

!4!,

the

single

_gene has an additive effect

_(a),

defined as half the difference between the effects of both

_homozygote

genotypes

(i.e.

a =

(G

A

p -

)/2),

BB

G

and a dominance effect

_(d)

defined as the deviation of the effects of the

_heterozygote

_genotype

from the _average value of both

_homozygote

_genotype

effects

_(i.e.

d =

-

AB

G

(G

AA

₊

G

BB

)/2).

The

additive

_genetic

variance

_{explained by}

the

_single

locus is

_{o, q 2 (}

₀

_,1

₌

_{2p(1 —}

_p)a

₂

_),

where a is the _{average gene} substitution

_equal

to: a +

_{d(1 -}

_2p)

_[4].

It is also assumed that all individuals have known

_genotype

and the effects of the

_major

locus is also known without error.

Individuals within a

_genotype

class can be

_{distinguished by considering}

the

_genotypes

of their

_parents.

The

_genotype

effect of an individual

_is,

_then,

decomposed

into two different

_components:

_i)

the average effect of its

parents’

genotypes

(MG);

and

_ii)

the

_remaining

_(MS)

defined here as the Mendelian

sampling

term of the

_major

locus

_(i.e.

G = MS ₊

MG).

The

_component

MG

represents

the

_family

mean effect due to the

_single

locus,

and MS the deviation of the individual from the average

family

effect. When the effect of the

single

locus is

_completely

additive

_(i.e.

d =

0),

three

groups with different MS value

can be

_{distinguished}

in each of the three

_genotype

classes. The

_possible

MS

values for these _groups are:

_+a,

_+a/2

or 0 for

_{homozygotes AA;}

_+a/2,

0 or

- a/2

for

_{heterozygotes AB;}

and

_0,

_-a/2

or -a for

homozygotes

BB.

_Knowing

the

_genotype

and the MG term of an individual determines the value of its MS

term.

Hence,

the total

_population

is classified into nine different groups defined

Mendelian

_sampling

terms

_k(k

=

1,2,3)

distinguished

with _a

completely

addi-tive locus. The mean

_polygenic

effects for each _group

_jk,

is _pjk with

vari-ance

Jfl,,!,

and their

_frequencies

in the whole

_population

are

_ubj

_k

_,

where

£ £ u

Jjk

= 1. In the base

population

all the

groups have the same

expecta-j k

tion and variance for the

_polygenic

_{effects, equal}

to zero and

_Va,

_{respectively.}

The environmental variance

_J

_{d ,}

is

_equal

across

_generations

and groups. The

initial

_polygenic

_heritability

_h),

in the base

_population

is

_{Va/(Va +}

_Jd

_{) .}

The total

_genetic

effects GV

_(single

locus and

_polygenic

_effects)

of individuals within each

_group

_k

is

_normally

distributed with the

_following

_expectation

and variance:

and the

_phenotypic

values

_(y)

have the same

_expectation

as

_equation

_(1),

but with an additional variance due to environment

_(0,!2).

Combining

the different

_subgroups

with the same

_genotype

_j,

the mean

polygenic

effects of the combined groups and their variance are:

where the first term of the variance arises from the

_polygenic

variance within each MS group and the second term from the differences between the mean

effect of each MS group. The same

_polygenic

_parameters

for the overall

population

(i.e.

p and

Jf

l )

can also be calculated

_using

formulae

(3)

and

(4),

but the summation is over the

_parameters

of the three combined

_genotype

_groups.

The

_polygenic

variance of the whole

_population

₍

_U2

₎

_can,

_then,

be divided into

two

_components

_according

to their sources: the within

_genotype

variance

(a

;

w)

and the between

genotype

variance

_{(U2 ab ),}

_{being equal}

to the first and second

components

of formula

_(4),

_{respectively.}

Before

_selection,

the between

_genotype

polygenic

variance is zero since all groups have the same mean

polygenic

effects.

2.1.2. Selection index

The total

_genetic

effects

_affecting

an individual can be divided into four

components:

the MS and MG effects due to the individual’s

_genotype

at the

single locus,

the mean

polygenic

effects of the

_genotype

_group the individual

belongs

to and its deviation from _{the group} mean.

Assuming

that all individuals

are

_known,

a

_general

selection index used to calculate their estimated

_breeding

values for truncation selection is of the form:

and its

_expectation

and variance within each group are:

where the

_components

of the index are the estimators of the four

_genetic

components.

BS and BG are the

_breeding

values due to the

_components

of the

_major

locus

_(MS

and

_MG),

BU is an estimator of the mean

_polygenic

effect of each

_genotype

_{group, ttj,} and BE is the

_{remaining polygenic}

effect confounded with the environmental deviation

_(i.e.

BE =

y -

_{G!k -}

BU!,)

and its

_expectation

is zero.

Assuming

random

mating,

the

_breeding

value due to a biallelic

_single

locus

_accounting

for its dominance deviation is estimated

_using

the _average gene substitution

(a).

Thus the

respective

breeding

value of individuals with

genotype

AA,

AB and BB are

2(1 - p)a, (1 — 2p)a

and _-pa

[4].

Then,

the

breeding

values of the candidates to selection and their

parents

required

to calculate BS and BG are estimated

_using

the new value of _a, recalculated

with the _{current gene}

_frequency

in the group of candidates to selection. The estimation of BU is dealt with in

_{Appendix A,}

but in the

_{suggested prediction}

model,

it is assumed to be estimated with

_negligible

error

(i.e. BU

is the

true

_p

_j

_).

In

_practice,

the estimation of the mean

polygenic

effect within each

genotype

group with small error would

_only

be

_possible

with

_large

_population

sizes.

The use of the selection index

_given

in

_equation

(5)

as a selection criteria

allows the

_flexibility

to

_change

the relative

_weight

_given

to each of the

_genetic

components.

For

_instance,

_increasing

the relative

_weight

given

to BS and BG would increase the _average selective

_advantage

of individuals with the most favourable

_genotype,

_yielding

a faster

_change

in the

_frequency

of the favourable

allele. The

_optimization

of the selection index under different

_assumptions

and its

_relationship

with some methods of selection described

_previously

in the

literature is

_explained

in

_Appendix

B. 2.1.3. Selection response

At each

_generation

_(assumed

to be

_discrete),

the

proportion

of selected

parents

of sex

x (x = m, f )

is _7rx. Since truncation selection is

applied,

a

threshold

_point

T can be found

numerically

fulfilling

the condition that the

proportion

of individuals with index score

_greater

that T over the nine groups

is 7rx. Thus the contribution of each group to the selected

parents

is _7rjk,x, such that

_L

_!r!!!! = _7rx.

Knowing

7 r jk ,

x and

!!!,!,

other

_polygenic

_parameters

jk,x

The difference in selective

_advantage

due to the

_single

_gene effect affects the

intensity

of selection

_(i

_jk

_,

_x

₎

applied

to the

_polygenic

effect in each _group

_jk.

It is

_expected

that individuals with the

_{poorest genotype}

would,

on _average, have a

_greater

_polygenic

effect if

_they

are to be selected over candidates with a more favourable

_genotype.

Similarly,

since the

_intensity

of selection varies

between _groups, the reduction in

_polygenic

variance due to the Bulmer effect

[1]

is also

expected

to be different.

Linkage disequilibrium

between the

_major

locus

genotype

effect and the

polygenic

effect

_is,

_then,

created in the selected

parents,

where

_S

_A

_p

<

_S

_AB

_,

_X

<

_-5’BB!;

and ! !A,! > !a,AB,x ! aa,BB,x

_(with

overdominance the selected

_{heterozygotes}

may be ranked

differently).

Assuming

that selected

_parents

are

_randomly

mated and there is

equal

family

size for each

_{mating pair,}

the

_genetic

_parameters

in the

_offspring

_generation

(denoted

with

_*

₎

are

_expected

to be:

and

where

₍

₇

_r

_jk

_,

_m

₇

_r

_jk

_,j)

is

_proportional

to the

_probability

of a sire from _group

_jk,

m

the

_probability

of a

_{mating pair}

from _groups

_jk,

m and

_{jk, f having}

an

_offspring

j

*

k

*

,

given

Mendelian inheritance.

The

_polygenic

variance within each

_{offspring’s}

group has three different

sources:

i)

the variance within each

_mating

_group;

ii)

the variance due to differences in the

_expected

mean

_polygenic

effect between

_mating

_pairs;

and

iii)

the

_polygenic

Mendelian

_sampling

variance. The reduction in variance due

to selection

_[1]

_affecting

the variance within

_{mating pairs}

was accounted for in formula

_(10).

_Similarly

the variance

arising

from the

_polygenic

Mendelian

sampling

is also

expected

to be reduced with the accumulation

_{of inbreeding}

in the selected

_parents.

_However,

this effect is not accounted for with the

_present

deterministic model.

Since the distribution of

parental

genotypes

will differ among the

genotypic

classes in the

_{offspring generation,}

a

_proportion

of the

_{disequilibrium}

created

during

selection of the

_parents

is retained. In the

_offspring

_generation,

the

mean

polygenic

effect within each

_genotype

_group

(/’*AA, /

_-lÀ

_B’

_/

-lj’m)

and its

variance

_(a;*A

_A’

_a;*A

_B’

₀

_a,B

_B

₎

are no

_longer

_expected

to be the same. In the

overall

_population,

this

_{disequilibrium}

results in the _appearance of a

_negative

covariance between the

major

locus and the

_polygenic

effects,

and the

_polygenic

variance between

_genotypes

₍

_Q

_ab)

is no

longer

zero. The measurement of these variance

_components

is described in

_Appendix

A

_(note,

_{however, they}

are not

required

for the

prediction

of the response to

_selection).

Since the

_offspring

become the candidates for selection in the next

_round,

the

_parameters

calculated for the

_offspring

_generation

_can,

_then,

be used

recursively

to estimate

_parameters

of

_{subsequent generations.}

In each round of

selection,

new

_{linkage disequilibrium}

between the

_major

locus

_genotype

and the

polygenic

effect is created and maintained until the favourable allele is fixed. The differences in the selective

_advantage

_responsible

for this

_{disequilibrium}

will _vary due to

_changes

in the

_parameters

of the next

_generation

such as the

group

frequencies

_ub

_jk

_,

the

polygenic

variance and the

linkage disequilibrium

carried over from the

_previous

round of selection.

2.1.4.

_Comparison

with stochastic simulations

In order to test the accuracy of the

predictions

obtained

_using

the

_present

deterministic

_approach

over

_multiple

_generations,

_they

were

_compared

with

re-sults from stochastic simulation

_using

a thousand

_replicates.

In the simulated

population,

the base group was assumed to be

_composed

of 360 unrelated indi-viduals

₍₁₈₀

males and 180

_females).

The initial

_polygenic

and environmental variances were considered to be 0.2 and

0.75,

respectively.

The

_segregating

ma-jor

locus was

_completely

additive

(a

=

0.443,

d =

0)

and the

starting frequency

of the favourable allele was 0.15

_(i.e.

_J

₎

₌

_0.05).

At each

_generation

all indi-viduals were scored with the relevant index and 30 males and 60 females with

the

_highest

estimated

_breeding

values were selected to be the

_parents

of the next

_generation

_(i.e.

_proportion

selected 7rm =

1/6,

!rf =

1/3).

Each male _was

mated

_{hierarchically}

to two females

_randomly

chosen from the selected _group

to

_produce

six

_offspring

_per female

_(three

males,

three

_females).

The same

se-lection _process was then

_applied

to the

_offspring

to

_produce

the

_subsequent

(i.e.

the

_{polygenic breeding}

value of an

_offspring

was simulated to be the mean

polygenic

breeding

value of its

_parents

_plus

a Mendelian deviation drawn from a normal distribution with mean zero and variance

_{(Va/2)[1- (F}

_S

+

_Fd)/2!,

where Va is the

_polygenic

variance in the base

population

and

_F

_S

and

_F

_d

the

inbreeding

coefficient of the

_{offspring’s}

sire and

dam,

respectively).

Figure

1 shows the evolution of the total and the

_polygenic

means as well as the

change

in the _gene

frequency

of the

_major

locus obtained with both

the deterministic and the stochastic

_approaches

under two different methods of selection. In the

_{early generations}

the results from the deterministic

_approach

have

_good

agreement

with the stochastic

_results,

but a small overestimation

of the

_polygenic

_response was observed later. After 19

_generations

of selection

the overestimation of the

_polygenic

response for both methods of selection was

8

_%,

_representing

0.32

_phenotypic

standard deviation. Most of the

_discrepancy

between these two

_prediction

_approaches

is

_{explained by}

the fact that the loss in

_polygenic

variance due to

_inbreeding

is not taken into account with the deterministic

_approach.

The cumulated overestimation of the deterministic

approach

after 19

generations

was reduced to

_only

2

%

_(i.e.

0.09

_op)

when the

inbreeding

level observed in the stochastic simulation was used to

_adjust

the

polygenic

variance in the deterministic formulae

_(results

not

_shown).

For the

population

size

and the gene

frequency

assumed in the stochastic

simulations,

the error associated with the estimation of BU

(as

a

_predictor

of

p

j

)

was very small and affected the

agreement

between

_predictions

obtained with the stochastic and the deterministic model very little.

2.2.

_Comparison

between different methods of selection

Two methods of selection

_(genotypic

and Mendelian

_selection)

_using

geno-type

information in the selection _process when a known

_major

locus is

segregat-ing

were

_compared

with the traditional

_phenotypic

selection. Three different variants

_(I,

II and

_III)

for both the

_genotypic

and the Mendelian methods were considered in the

_comparison.

The methods of selection were based _upon

varying

the

_{weight given}

to the different

components

included in the selec-tion index

_given

in

_equation

₍₅₎

_(see

table

_1).

In the

_genotypic

methods both

components

of the

_major

locus are included in the index while the Mendelian methods

weight

the

_major

locus

_only

by

its BS

_component

_(i.e.

/3

BG

= 0).

In this

_study

the

_genotypic

and Mendelian methods of selection are referred as the gene assisted selection

(GAS)

methods.

The selection indices for the variants I and II are the result from the

op-timization

_using

classical index

_theory

to maximize immediate

_{genetic gain,}

from either

accounting

for or

_ignoring

the

linkage

disequilibrium

between the

major

locus and the

_polygenic

effects

_(see

_Appendix

_B).

The

_genotypic

I and II are also

_equivalent

to the maximum _accuracy and direct selection methods described

by

Gibson

_[6].

For

_{genotypic III,}

the relative

_weight

_given

to the

components

BG is the same as it would be with

_phenotypic

selection. For the

case of Mendelian

_III,

the

_{weight given}

to the BS is

_updated

in each

_generation

to maximize _{response in} the current round of selection. This was

_required

since the

_optimum

_weight

in Mendelian methods

_depends

on the _gene

_frequency,

favourable allele

_(see

_Appendix

_B).

It is

important

to note that variant I of both methods

_(i.e.

when

_accounting

for the

_{disequilibrium)}

are the

_only

cases

where the two

polygenic

components

(BU

and

_BE)

are

_assigned

a different

weight

in the selection index. Thus the

precision

in which the mean

_polygenic

effect is estimated affects variant I. In the other variants where both

_polygenic

components

have the same

weight,

the distinction between both

polygenic

sources becomes irrelevant.

2.3. Criteria of

_comparison

The effects of each alternative of selection on the short- and

_long-term

cumulated

_genetic

_response were

_{compared using}

the deterministic model

previously

described. This

_comparison

was carried out over a _{range of different}

heritabilities and the size and

_degree

of dominance of the

_major

gene effects. Further criteria of

_comparison

were the

_inbreeding

coefficient cumulated over

several

_generations

of selection and the

_probability

of

_losing

the favourable allele when its

_starting

_frequency

was low. These

_comparisons

were made

_using

stochastic simulation as the

_present

deterministic model does not account for them.

Most of the

_comparisons

were carried out with a common set of

_parameters.

In this set the

_polygenic

and the environmental variance were 0.20 and

0.75,

respectively

(i.e.

polygenic heritability

_{hP =}

0.21).

The

_major

locus had a

completely

additive effect

_(a

=

0.443,

d =

0)

and the

starting frequency

of the

favourable allele was 0.15

_(i.e.

_J

₎

₌

0.05;

total

_heritability

h

2

=

_0.25).

The

proportions

of males and females selected were 0.16 and

_0.33,

_{respectively.}

Changes

in initial

_{!9 were}

made

_{by altering}

the gene

frequency

and its effect. The

_polygenic

heritability

was modified

by

altering

the environmental

variance while

_keeping

constant the

_polygenic

variance

_(Va

=

0.2),

thus the

3. RESULTS

3.1.

_Response

to selection

_(using

the deterministic

_model)

3.1.1. Short- and

_long-term

cumulated response with an additive locus

The

_predicted

cumulated _responses to selection over the

_generations

when the

_major

locus is

_completely

additive are shown in table II. When the

starting

frequency

of the favourable allele was

_0.15,

all the GAS methods achieved

greater

cumulated

_genetic

_response than the traditional

_phenotypic

selection

during

the

_{early generations}

of selection. The

_superiority

of these methods over

the traditional

_phenotypic

selection

_peaked

after 2-3

_generations

of

_selection,

ranging

from 10

%

of extra

_gain

for the Mendelian methods to 30

%

obtained with the

_genotypic

schemes.

However,

the extra cumulated response of these methods over the

_phenotypic

selection

_gradually

diminished and

_disappeared

after 6-7

_generations.

After the favourable allele had been fixed with all the methods of selection

_(see

results of

_generation

_20),

the GAS methods

_yielded

a lower cumulated

_genetic

_response than the

_phenotypic

selection. In the

_longer

term,

their loss in the cumulated

_gain

relative to the

_phenotypic

selection was

of

_{comparable magnitude}

to the maximum benefit

_(extra

cumulated

_gain)

_they

had in

_early

_generations.

Since the

_{genetic gain}

_per

_generation

after fixation is

_expected

to be the same for all the methods

(since

_genetic

_response is

only

due to

_polygenic

_gain),

the difference in the cumulated _response between these methods becomes

permanent.

A similar trend was found when the

starting

frequency

of the favourable allele was 0.85 but at lower timescale and differences. The extra

_gain

achieved

_{using genotypic}

selection was

_only

12

%

for the first

_generation,

and

_disappeared

after 2-3

_generations.

The short-term benefit

_using

Mendelian methods was

_{only marginal}

or at worst null

_(results

not

_shown).

Figure 2

shows the

_genetic

response achieved in

generations

1 and 30 with a

range of

polygenic

heritabilities

(where

Va was held

constant)

and effects of the

major

locus with

_starting

_frequency

of 0.15.

_(Since

the trends were similar in

most of the GAS methods not all of them are

shown.)

The _{extra response}

achieved

_by

the cases of

_genotypic

selection in the first round of selection

was

_greater

with lower

_{polygenic heritability}

and a

larger

effect of the

single

locus, confirming

the results

_previously

_reported

_by

Lande and

_Thompson

_!13!.

However,

as in table

_II,

_greater

_gain

in the short term tended to be associated with a

larger

_permanent

loss in the

longer

term. For the case of Mendelian

methods,

the

_advantage

over

_phenotypic

selection in

_{early generations}

was observed

only

with low

_polygenic

heritability.

When the

_{starting frequency}

was

_0.85,

the effects of all selection methods in the cumulated _response were

only marginal

in both the

_early

and later

_generations

_(results

not

_shown).

The differences in the short- and

_long-term

cumulated _response observed with these methods of selection were related to the

_{weight given}

in the selection index to the

_major

locus relative to the

_polygenic

effects. The extra

_gain

in the

early

generations

obtained with the

genotypic

and the Mendelian methods was

achieved

_through

a faster increase in the

frequency

of the favourable

allele,

but

those methods with lower rate of

polygenic gain

in the

_previous

_generations

had less cumulated

_genetic

_response. Over all the methods of selection a faster increase in the

_frequency

of the favourable allele in a

_generation

was

_always

related with a lower

_gain

in the

_polygenic

effects. The maximum

_gain

in

the

_polygenic

effects for a

_single

round of selection was obtained when the

3.1.2. Effect of

_accounting

for the

_linkage

_{disequilibrium}

The

_{linkage disequilibrium}

is taken into account in variant I of the

_genotypic

and the Mendelian methods

_by

assigning

the

optimum

weight

to BU. The selection method

_genotypic

I

_performed

better than

_genotypic

II over the whole selection _process

_confirming

the results

_previously

_reported

in the literature

_!6!.

Nevertheless,

this benefit

represented only

a

marginal

increase in response to

selection. For the second round of

_selection,

the cumulated

_gain

obtained with

genotypic

I was less than 2

_{% greater}

than the

_genetic

_response observed with

genotypic

II. In the

_long

term the loss in the cumulated

_genetic

response of

genotypic

I was 10

%

smaller than that observed with

_genotypic

II.

_Assigning

the correct

_weight

to BU in the Mendelian method did not

_yield

_any

_benefit,

in terms of extra

_gain.

In this case the

_{re-optimization}

of the selection index

considering

the

_frequency

of each _group rather than

_using

the estimate obtained from classical index

theory

(i.e.

Mendelian

_III),

was more

_important

to ensure

maximum

_genetic

_progress.

3.1.3. Effect of the

_degree

of dominance

Comparisons

of different GAS methods when the effect of the favourable allele

_(A)

is

_{completely additive,}

dominant or recessive are shown in

figure

3 for p = 0.15 and _a = 0.443

(when

p = 0.85 the trend _was the _same but _{at a} smaller

scale).

The most beneficial situation of

_using

GAS

_methods,

in terms of

greater

short-term _response, was when the favourable allele was recessive and at low

frequency. Moreover,

their

_{long-term genetic gain}

was even

_greater

for the cases when the effect of the recessive

_major

locus was

larger

than one

phenotypic

standard deviation

_(figure

_4),

_contrasting

with the case of an additive locus

where a loss in the

_long-term

cumulated

_{gain always appeared}

unavoidable

(see

figure

!).

The trend with the favourable allele

_being

dominant was similar to those observed in the case of a

_completely

additive locus but at a lower scale

(results

not

_shown).

3.2. Level of

_inbreeding

The

_inbreeding

accumulated after ten

_generations

of selection for two cases

with different

_starting

_gene

_frequencies

is shown in table III. The

_highest

level of

_inbreeding

was obtained when selection was carried out

_{using genotypic}

methods;

while the lowest was achieved with the Mendelian methods. The

inbreeding

rate varied over

_generations,

with the

highest

rate observed in

early

generations

before the favourable allele was fixed. The

_greatest

differences in the level of

_inbreeding

were obtained when the

_frequency

of the favourable allele was low. Where the

_starting

frequency

of the favourable allele was

_0.15,

the

inbreeding

level of

_genotypic

I and Mendelian II was 2.8

_{% greater}

and 5.7

%

smaller, respectively,

than the

_inbreeding

level accumulated with

phenotypic

selection. Where the

_{starting frequency}

was

_0.05,

the

_inbreeding

_{coefficients,}

3.3.

_Probability

of

_losing

the favourable allele

Table IV shows the

_probability

of

_losing

the favourable allele when its

starting

frequency

was 0.05. For the variants I BU was estimated in three

different _ways:

_i)

the mean

_phenotypic

value

_adjusted

_by

the

_major

locus

effects;

ii)

by regressing

the

_adjusted

phenotype

on the favourable

alleles;

and

iii)

the same as

(ii)

but

_restricting

the difference to between -cY and 0

_(see

Appendix

A for more information on the

_regression

_approach).

Because the average selective

advantage

of a

_genotype

_group

_depends

on the

_genotype

effects and their mean

_polygenic

_effects,

the restriction

_imposed

in

_(iii)

it was ensured that the

_genotype

group with the most favourable

_genotype

will

_always

have the

_greatest

selective

_advantage.

As

_expected

those methods which

_assign

a

_greater

_weight

to the

major

genotype

had lower

probability

of

_actually

_losing

a rare favourable allele.

However,

the method of

_estimating

B U when

_{using genotypic}

I and Mendelian I methods of selection had a

_great

impact

on the

probability

of

_losing

the favourable allele. Unless the differences in the

_polygenic

mean between

_genotype

groups were

_restricted,

the

probability of losing

the favourable allele was

_large,

especially

when the

_single

locus had a small effect. The error associated with

the estimation of BU when

_only

few individuals

_belong

to a

_given

_genotype

group can be

_quite

_{large overcoming}

the

_greater

selective

_expected

from the individuals with the better

_genotype.

4. DISCUSSION

range of situations to

give

a

_picture

of the

_potential

benefit of

using

information of a

_major

locus

during

_{selection, including}

the

_impact

on

_inbreeding

and the

probability

of

_losing

the favourable allele.

None of the indices

_{studied, genotypic}

or

_Mendelian,

were able to resolve the conflict between the short-term and the

_long-term

benefits of

_GAS,

with the

_exception

of rare recessive alleles of

_large

effect. This

_finding

is similar in

outcome to Gibson

_[6],

Ruane and Colleau

_(17!,

Fournet et al.

_[5]

and Larzul et al.

_[14]

who used different

_approaches

to the

_{problem. Moreover,}

the

_present

results also show that the situations when the use of

_genotype

information is

expected

to

_yield

a

_greater

benefit in the short

_term,

tend to be associated

to a

larger negative

effect in the

long-term

cumulated _response. The loss in

long-term

genetic gain

increased with smaller

_{heritability, larger}

effect of the

The conflicts arise from the

higher

short-term response from a faster increase

in the favoured allele and a reduced

_long-term

_gain

in

_polygenic

effects

[5,

_14,

16,

17].

The

_higher

_average selective

advantage

of individuals with the most

favourable

_genotype

results in a lower selection _{pressure among} them. Further

increases in the selective

_advantage

of an allele results in a

_greater

_proportion

of individuals with this allele

_being

selected,

but also further decreases in the selection _pressure

_applied

to the

_polygenic

effects. The selective

_advantage

is increased

_by

_increasing

the

_weight

in the index or an increase in the _average effect. In the overall

population,

this

_greater

_proportion

of individuals selected from the _groups with lower selection _pressure would

_yield

an unavoidable loss in the

_intensity

of selection

applied

to the

_polygenic

effects

_(see

_figure

Bl in

Appendix

B).

This

explains

both:

_i)

that the maximum

_{polygenic gain}

_per

generation

for _any of these methods was

_predicted

to be when the

_major

locus

was fixed and the

population

was no

longer

subdivided in different _groups; and

ii)

the observation that the

_greater

the initial selective

_advantage

the

_greater

the

_long-term

loss.

_Although

methods

_assigning

_greater

_weight

to the

_major

locus have faster fixation time of the favourable

allele, they

are also

_expected

to have

_greater

_polygenic

_gain

_during

the

_period

after

_they

fix the favourable allele but before it is achieved

by

the

_phenotypic

selection method. In most of the cases studied

_here,

this latter

_{period only partially compensated}

for the

initial loss of

_{polygenic gain.}

Exceptions

to this

_{general picture}

were observed and were limited to cases

with a favourable recessive allele at low

_starting

_frequency

and with

large

effect

(

N

a >

_This

refines the results of Fournet et al.

_[5]

and Larzul et al.

_[14]

for

mass selection who studied recessive alleles of

_large

effect. The use of

_genotype

information with recessive

_single

genes of smaller effect still

presented

the

negative

effect in the

_long-term

cumulated

_gain

relative to

_{phenotypic selection,}

but the

_magnitude

was

_{substantially}

smaller than when the favourable allele was additive or dominant. The benefits of GAS are due to the

_inefficiency

of the

phenotypic

selection in

_fixing

a recessive locus. In the first round of selection the

_heterozygote

individuals

_(AB)

have,

on _average, the same chance of

_being

selected as those individuals

homozygous

(BB)

to the less favourable

allele,

reducing

the rate of

_change

in the

_frequency

of the favourable allele. However in

_{subsequent generations linkage}

_{disequilibrium}

is

_built-up,

and the average selective

_advantage

of the

_heterozygote

individuals is less than the

_homozygote

BB since their mean

_polygenic

effects are

_expected

to be smaller

_(some

ABs will have AA

_parents).

_Considering

that

_major

genes with

greater

effects are

expected

to

_{yield larger linkage disequilibrium,}

the

_phenotypic

selection is also

expected

to be less efficient

_fixing

such loci. Thus the beneficial effect of

_using

genotype

information remains in the

_long

term

_only

for recessive locus with

large

effects. Fournet et al.

_[5]

and Larzul et al.

_[14]

also studied the case with

progeny

testing

but in this case the AB individuals are distinct from the BB

when

_genotypes

are not recorded.

A

_{justification}

for

_increasing

the difference in the selective

_advantage

between

genotype

groups would be to reduce the

_probability

of

_losing

a favourable

allele,

yet

whilst the benefits _{may appear}

_obvious,

the _poor results observed for the selection methods

_genotypic

I and Mendelian I are

_alarming.

The chance of

losing

the rare favourable allele with these methods _{were, in} some _cases,

_greater

associated with errors in the estimation of the mean

polygenic

effects of each

genotype

group due to the small size of the

genotype

group with the rare

allele. For the

_population

size assumed in this

_study,

the mean

_polygenic

effect

between the different _groups was estimated with such error that the

_greater

selective

_advantage

of individuals with the most favoured

_genotype

was not

always

secured.

These results raise some doubts about the

_{practical desirability}

of

_using

a

selection index which accounts for the difference in the mean

_polygenic

effects.

The error in

_estimating

the mean

polygenic

effects of each

_genotype

_group did not affect the

_performance

of other methods since all

_polygenic

_components

had the same

weight

in the index. Since the extra _response to selection

using

genotypic

I

_(where

mean

_polygenic

effects are assumed

_{known, precisely,}

and are accounted

_for)

relative to

_genotypic

II

_(where

_they

are

ignored)

it _may be that unless the

_population

is _very

_large

_genotypic

II is more robust.

The different selection indices resulted in differences in the

inbreeding

coef-ficient at the time of fixation and these differences _may affect the assessment of the

_long-term

results from the deterministic model. The lowest cumulated

in-breeding

was observed for the Mendelian methods while the

_genotypic

methods showed the

_highest.

This difference in cumulated

_inbreeding

was more

accen-tuated when the

_starting

_frequency

was low.

Grundy

et al.

_[7]

showed that selection on the Mendelian

_sampling

term gave reductions in the

inbreeding

rate of up to 24

%.

Hill et al.

_[10]

also showed the value of

_selecting

for

_family

deviation in

_reducing

the

_inbreeding

rate.

_Weighting

the

_major

locus

_{by only}

its Mendelian

_sampling

term reduces the extent of between

_family

selection and so _may be

expected

to reduce the

_inbreeding

rate

_during

fixation. The in-creased

_inbreeding

rate observed with the

_genotypic

methods is consistent with the results from Ruane and Colleau

_[17]

who

_reported

a similar trend when

genotype

information of a marker linked to a

_major

_{gene is} used

during

the

selection _process. Greater

_inbreeding

rates will affect the

_long-term

_response

through

the

_greater

rate of loss of

_polygenic

variance. Thus

_{predictions using}

the deterministic model are

_expected

to overestimate the

_long-term

_response of the

_genotypic

methods relative to the

_phenotypic

selection and underestimate those for Mendelian methods.

Finally,

the

_general

framework

_proposed

here for

_studying

response to mass selection when a

_major

locus is

_segregating

_may

_provide

a valuable tool to assess the value of _gene assisted selection schemes in more

specific

situations. This

_study

considered

_only

some

_specific

cases of how the

_major

locus and

the

_polygenic

_components

_may be

_weighed

in a selection _process.

However,

the selection index

approach

used in this

_study

would allow

_testing

of _any

_possible

combination of

_weights

_given

to both

_genetic

sources. The

_antagonism

between

the short- and the

_long-term

_genetic

response first

reported by

Gibson

[6]

and confirmed

_by

further studies

_including

this one, appears to

suggest

that there is

no

_general

formulae for

_{maximizing genetic}

_response across the whole selection

process when a

_major

locus is

_segregating.

Dekkers and Van Arendonk

[3]

showed that selection _{response at} a

_given

timescale _may,

_however,

be

_optimized

by

modifying

the

_{weight given}

to the

_major

locus across the different rounds

of selections. Their method for

_finding

the

_{optimum weight}

to maximize

_gain

The model described is also

_sufficiently

flexible for extension to

_multiple

alleles and

_overlapping

_generations.

The latter case was considered

_by

Larzul et al.

_[14]

but their model does not account for the Bulmer effect and the

weight

given

to the

_major

locus relative to the

_polygenic

effect is less obvious

to

_change.

However the results of this

_study

_suggest

that the

_{implementation}

of GAS

_requires

clear definition of the

_{objectives resolving}

the conflicts of

_early

response with

long-term

response,

inbreeding

and allele loss.

ACKNOWLEDGEMENT

The authors

acknowledge

financial support from the Milk

_Marketing

Board of

England

and Wales and The

Ministry

of

Agriculture,

Fisheries and Food. The authors thank Naomi

_Wray

for

_providing

the _programme for stochastic simulation.

REFERENCES

!1!

Bulmer

M.G.,

The effect of selection on

genetic variability,

Am. Nat. 105

(1971)

201-211.

[2]

De

_{Koning G.J.,}

Weller

J.1., Efficiency

of direct selection on

_quantitative

trait

loci for a two-trait

_{breeding objective,}

Theor.

Appl.

Genet. 88

₍₁₉₉₄₎

669-677.

[3]

Dekkers

_{J.C.M., Van}

Arendonk

_{J.A.M., Optimum}

selection on identified quan-titative trait

_loci,

Proc. 6th World

_Cong.

Genet.

_Appl.

Livest. Prod. 26

₍₁₉₉₈₎

361-364.

[4]

Falconer

_D.S.,

Introduction to

_{Quantitative Genetics,}

3th

_{ed., Logman}

Scien-tific and

Technical, Essex,

1989.

[5]

Fournet

_F.,

Elsen

_J.M.,

Barbieri

_M.E.,

Manfredi

_E.,

Effects of

_including

ma-jor

gene information in mass selection: a stochastic simulation in a small

population,

Genet. Sel. Evol. 29

₍₁₉₉₇₎

35-56.

[6]

Gibson

J.P.,

Short-term

_gain

at _{the expense} of

_long-term

_response with

selec-tion of identified

_loci,

Proc. 5th World

Cong.

Genet.

_Appl.

Livest. Prod. 21

₍₁₉₉₄₎

201-204.

[7]

Grundy B.,

Luo

_Z.W.,

Villanueva

_B.,

Woolliams

_J.A.,

The use of Mendelian

indices to reduce the rate of

_inbreeding

in selection programmes, J. Anim. Breed.

Genet. 115

₍₁₉₉₇₎

39-51.

[8]

Guo

S.W., Thompson E.A.,

A Monte Carlo method for combined

_segregation

and

_{linkage analysis,}

Am. J. Hum. Genet. 51

₍₁₉₉₂₎

1111-1126.

[9]

Hazel

L.N.,

The

genetic

basis of

constructing

selection

indexes,

Genetics 28

(1943)

476-490.

[10]

Hill

_W.G.,

Caballero

_{A., Dempfle}

L., Prediction of _{response to} selection within

_families,

Genet. Sel. Evol. 28

₍₁₉₉₆₎

379-383.

[11]

Janss

_L.L.G.,

Van Arendonk

_{J.A.M., Thompson R., Application}

of Gibbs

sampling

for inference in a mixed

_major

_{gene-polygenic}

inheritance model in animal

populations,

Theor.

Appl.

Genet. 91

₍₁₉₉₅₎

1137-1147.

[12]

Kennedy B.W., lauinton M., Van

Arendonk

_{J.A.M., Estimation}

of effects of

single

genes on

_quantitative

_traits, J. Anim. Sci. 70

₍₁₉₉₂₎

2000-2012.

[13]

Lande

_{R., Thompson R., Efficiency}

of marker-assisted selection in the

im-provement of

_{quantitative traits,}

Genetics 124

₍₁₉₉₀₎

743-756.

[14]

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.