Balancing selection response and inbreeding by including predicted stabilised genetic contributions in selection decisions

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Balancing selection response and inbreeding by

including predicted stabilised genetic contributions in

selection decisions

Jr Brisbane, Jp Gibson

To cite this version:

Original

article

Balancing

selection

_response

and

inbreeding by including

predicted

stabilised

_genetic

contributions

in

selection decisions

JR Brisbane

JP

Gibson

University of Guelph,

Centre

for

Genetic

Improvement of Livestock,

Department of

Animal and

_{Poultry Science,}

_Guelph,

ON N1G 2W1, Canada

(Received

30

September 1994; accepted

30

_August

₁₉₉₅₎

Summary - A selection _strategy is

_investigated

which should

_improve

_upon

_methodology

previously

introduced for

_{reducing inbreeding by including genetic relationships}

in

selec-tion decisions. The new _strategy includes

_predictions

of stabilised

_genetic

contributions

of _parents to descendants in selection decisions. An additive infinitesimal genetic model

is assumed with discrete

_generations

of selection and random

_mating

of selected parents.

Stochastic simulation is used to compare rates of

_inbreeding

and

_{genetic gain}

from the

strategy

using relationships

with those from the strategy

using predicted genetic

contri-butions. The latter strategy

gives slightly higher genetic gain

at a

_given

level of cumulate

inbreeding,

but the

_advantage

is

_small,

and the calculations are more

_complex

and difficult

to

_apply

in

_practice,

and therefore the

_previous

_strategy

_using

_{relationships}

is more useful for

_{practical application.}

inbreeding

/

selection

_/

_{genetic gain}

*

Present address: Canadian Centre for Swine

_Improvement,

2200

_{Walkley Road, Ottawa,}

ON K1G

_4G8,

Canada.

Résumé - Un compromis entre la _réponse à la sélection et la _{consanguinité} obtenu

en considérant les contributions génétiques à l’équilibre des parents sélectionnés.

Une

_stratégie

de sélection qui devrait améliorer la

_{méthodologie précédemment suggérée}

pour réduire la

consanguinité

par l’utilisation des relations

génétiques

dans les décisions

de sélection est étudiée. Cette nouvelle

_stratégie

utilise la

_prédiction

de la contribution

génétique

à

l’équilibre

des parents à leurs descendants dans les décisions de sélection. On a

supposé

un modèle

_{polygénique additif}

avec des

générations

discrètes de sélection de même

que la panmixie entre les parents sélectionnés. Une modélisation

_stochastique

a été utilisée

supérieur

pour un niveau donné de

consanguinité. Cependant

cet avantage est

_faible

et les calculs sont

_{plus complexes}

et

_{plus difficiles}

à

_appliquer

en _pratique. Par

_conséquent,

la

stratégie

utilisant les relations

_génétiques

s’avère

_plus

utile.

consanguinité

/

sélection

_/

progrès génétique

INTRODUCTION

In most

_breeding

schemes a balance between

_{genetic gain}

and

inbreeding

is

sought.

Increased

_genetic

_gain

in the short term is

_usually

associated with increased

inbreeding

which leads to decreased

_{genetic gain}

in the

_long

_term, due to declines

in fitness and

_genetic

variance. Evaluation

_using

the records of all relatives

_(eg,

best linear unbiased

_prediction

_using

an animal

model),

increased female

_reproductive

rates

_(eg,

use of

multiple

ovulation and

embryo

transfer or in vitro

_embryo

production),

and selection of animals at a _{younger age}

_using

pedigree

rather than progeny

information,

lead to increased

inbreeding.

Various studies

(eg,

Toro and

Perez-Enciso,

1990;

Verrier et

_al,

_1993;

_Grundy

et

_al,

_1994;

_Wray

and

_Goddard,

1994)

have

_investigated

selection methods for

reducing

inbreeding

while

_maintaining

high

rates of

_{genetic gain.}

Brisbane and Gibson

₍₁₉₉₅₎

showed that a selection

strategy

(using

adjusted

estimated

breeding

value and denoted

_ADJEBV)

that includes

_{genetic relationships}

in selection decisions

_gives

_greater

_{genetic gain}

at a

given

level of cumulated

_inbreeding

than selection on a

_family

index with reduced

weight

on sib

information,

selection on an index

_omitting

some sib

information,

or

selection on an index with a restriction on the number of full-sibs selected. The

objective

of this

_study

is to

_investigate

the extent to which the ADJEBV method

can be

improved by including

a

_prediction

of the stabilised

_genetic

contributions of selected animals in the

_prediction

of their effect on

inbreeding.

THEORY

The selection

_objective

is assumed to be M =

G

n

-

D !

_F

_n

_,

where

G

n

and

F!

are

the

_genetic

mean and mean

_inbreeding

coefficient after n

_generations

of

_selection,

and D is the value of a unit of

inbreeding

relative to a unit of

_{genetic gain.}

If the

genetic

contribution of an ancestor is the

_proportion

of genes

originating

from that

ancestor,

then with discrete

_{generations, equal family}

sizes

_prior

to

_selection,

and N!rc sires and

_{Nf dams}

in each

generation,

each sire has a contribution of

_1/(2Nm)

and each dam a contribution of

11(2N

P

to _progeny

_prior

to selection.

Contribu-tions of individual sires and dams to the _gene

_pool

in

_{subsequent generations}

vary

depending

on the

_genetic

merit of the sire or dam and on Mendelian

_sampling

and environmental contributions to the estimated

_breeding

values

_(EBV)

of

de-scendants, although

the average contribution remains at

_1/(2Nm)

across sires and

1/(2N

fJ

across dams. The

_genetic

contribution of an ancestor reaches a stable value

after a sufficient number of descendant

_generations.

Wray

and

Thompson

(1990)

is

_equal

to one

_quarter

of the sum of _squares of stabilised

_genetic

contributions of

any

generation

of ancestors to descendants.

The mean

_relationship

_among animals of any

generation

is a

_weighted

sum of

squares of contributions from all ancestors to

_parents

of that

_generation

_(Wray

and

Thompson,

1990;

Brisbane and

_Gibson,

_1995).

If it is assumed that contributions of all ancestors to animals in

_generation

t have reached their stabilised

_values,

then the mean

relationship,

_an,

_among animals in

_generation

_n, where n >

_t,

is

equal

to the mean

relationship

_among animals in

_generation

t

_plus

the

_weighted

sum of _squares of contributions to

_parents

of

_{generation n}

from ancestors in

generations

between t and n. Under this

_assumption,

_an is

_equal

to at

plus

a

term

_independent

of the selection decision in

_generation

_t,

and in

_generation

t

it seems reasonable to use at as a

_predictor

of the effect of the selection decision

of

F

n

.

The

_assumption

that contributions of ancestors to animals in

_generation

t

have reached their stabilised values is not

_true,

but

_changes

in the contributions

of ancestors of

_generation

t in

_subsequent

_generations

will be influenced to a

_large

degree by

Mendelian

_sampling

and environmental

_effects,

which are random events.

There is a

_positive

linear

_regression

of stabilised contribution on the

breeding

value of an ancestor

_(Wray

and

_Thompson,

_1990).

Given a consistent selection

strategy

followed in each

_generation,

this

_regression

should enable some

_prediction

of

_changes

in contributions of ancestors of

_generation

t in

_subsequent

_generations.

The selection

_strategy

_proposed

here is to use the sum of _squares of

_predicted

stabilised contributions of ancestors of

_generation

t as a

_predictor

of the effect of the selection decision of

_F!.

Breeding

values are not

_known,

but are

_estimated,

and an individual’s EBV

should be of some use in

_predicting

its stabilised

_genetic

contributions. The

usefulness of the EBV will

_depend

on its _accuracy. When evaluation is based on an

index of the records of collateral

_relatives,

and no

pedigree

information is

_used,

the

covariance between stabilised

genetic contribution,

Voo,i and EBV for animal

i,

at

a

_given

true

_breeding

_value,

_A

_i

_,

is _zero, since

_prediction

errors are not inherited. In this

_{situation, using}

conditional

_covariance,

and

_neglecting

the effect of selection

on the variance of EBV and the

_genetic

_{variance among}

_parents,

we have.

where r is the accuracy of evaluation and

afi

is the additive

_genetic

variance. It follows that the

_{regression, b,}

of stabilised

_genetic

contribution on EBV is

equal

to

the

_regression

of stabilised

_genetic

contribution on true

_{breeding value,}

since

where

_b

_v

_,

_A

is the

_regression

of stabilised

_genetic

contribution on true

_breeding

value. If v,,!,i,. is the

predicted

stabilised

_genetic

contribution of animal

i,

based on the

on the true

_{breeding value,}

then

Using

the

_EBV,

it is

_possible

to account for a

_proportion

r

2

of

the variance

of stabilised

genetic

contributions which is associated with

breeding

value. In the

case where

_genetic

evaluation includes

pedigree

information, prediction

errors are

inherited to

_{some extent,}

and

_cov(foo!,

₎

_i

_A

_iI

_EBV

> 0. Therefore

_cov(!oo,,,

_EBV

_i

₎

>

r

2

cov(v

CX)

,

i

A

i

)

’

This means that b >

_b

_v

_,

_A

and

i

,*)

VC

X),

V(

>

_r

₂

_V(vCX),

_i

_,**).

The

_regression

of stabilised

genetic

contribution on ancestral

breeding

value is

the same for both sexes of

_descendants,

but is different for each sex of ancestors. If

b!y

denotes the

_regression

of stabilised contribution to descendants of sex _y from ancestors of sex x on the

breeding

value of those

ancestors,

where x = _m

(males)

and _y = m or

_f,

then

6! =

(Nf/ Nm) . b

_f

_y

(Wray

and

Thompson,

1990).

b

mx

will be referred to as

_b

_m

and

_y

_f

_b

as

_b

_f

_.

In the ADJEBV

_strategy

of Brisbane and

Gibson

_(1995),

where contributions of all ancestors are assumed to have reached their stabilised

_values,

v is a column vector with elements 1 to Nm

_equal

to

_1/(2Nm)

and elements Nm+ to

_{Nm+Nf equal}

to

_{I/ (2N}

_J

_).

The

_population

selection criterion

to be maximised is

’

where as,

ddand asdare the mean

relationships

_among selected

sires,

_among selected

dams,

and between selected sires and

_dams,

EBV

s

.

and

_EBV

_d

_,

are the mean EBV

of selected sires and

_dams,

and k is an

arbitrary

constant. The selection

strategy

attempts

to maximise this function in each

_generation.

We now

_replace

v with a vector _{v 00,*’}

_Here,

and later in this _paper, the

_subscript

*

is used to denote a

prediction

based on the EBV. Element i of v 00,* is !

and

where

b

m

and

_b

_f

are the

_regressions

of stabilised

_genetic

contribution on EBV for sires and dams. The

_population

selection criterion to be maximised when

_selecting

parents

in

_generation

t is now

where

A

tt

is the

_relationship

matrix _among the selected

parents

and k is an

arbitrary

constant.

_v)

_A

_tt

_v

_00,’

differs from

v’A

tt

v

in that

_{relationships involving}

parents

of

_higher

than average

EBV,

and therefore

_higher

than _average

_predicted

stabilised

genetic contribution,

are

_given

more

weight.

It can be shown that

v’ 00, Attv,,,,.

is a

weighted

sum of squares of

predicted

stabilised contributions

METHODS

An additive infinitesimal

_{genetic model,}

discrete

generations

of

_selection,

and random

_mating

in a hierarchical

design

are assumed. Stochastic simulation is

used with

_methodology

as

_given

_by

Brisbane and Gibson

(1995).

The units of

genetic

merit are base

population

_genetic

standard

deviations,

(TAD =

1,

and in each

_generation

there are Nrra

_sires,

_Nf

_dams,

and

n

w/

2

_progeny of each sex _per

dam. The selection method based on

_predicted

stabilised

_genetic

contributions is

denoted

_ADJEBV(R).

Both ADJEBV and

_ADJEBV(R)

are

_simulated,

and the

balance of

_inbreeding

and

_{genetic gain}

achieved

_by

each after 6

_generations

of

selection is

_compared.

Parameters of Nm =

Nf =

5,

_wn =

12,

and

h

2

= 0.5 are used.

A small

_population

size and

_simple

structure are used to minimise the substantial

computation

involved in the simulation of

_ADJEBV(R),

but results may

apply

more

broadly,

since the behaviour of ADJEBV was consistent across a wide _range

of

_population

sizes and

_parameters

_(Brisbane

and

Gibson,

1995).

ADJEBV

Following

Brisbane and Gibson

_(1995),

in each

_generation

Nm sires and

_{Nf dams}

are

initially

selected

by

truncation on EBV based on a

_family

selection index of

the individual record and the records of the 11 full sibs. The selected group is then

modified as follows.

_Adjusted

EBV are calculated for selected and unselected males as

where

_a

_s

_,

_i

_.

and

.

!

,

sd

Q

are the mean

_{relationships}

of male i with selected sires and

with selected

_dams,

and

_EBV,,

_i

_.

is the EBV of male i.

_Adjusted

EBV are calculated

analogously

for

_females,

and mean

_{relationships}

are calculated in such a _way that

the

relationship

of the animal with itself carries the same

_weight

for a selected

animal as for an unselected animal. The unselected male with the

highest adjusted

EBV

replaces

the selected male with the lowest

_adjusted

EBV. If the

_population

selection criterion

_given

_by

_[3]

is increased the switch is

accepted

and

_adjusted

EBV for animals recalculated to account for the

_change

in the _{selected group.}

_Switching

and

_updating

of

_adjusted

EBV

_{continues, alternating}

between the sexes until the

population

selection criterion cannot be increased

_(see

Brisbane and

_{Gibson, 1995,}

for further

_details).

Maximising

the mean

_adjusted

EBV of selected females and of

selected males maximises the

population

selection

_criterion,

but the _process does

not

_guarantee

_finding

the selected group which

gives

this result.

ADJEBV(R)

In each

generation

Nm sires and

_Nf

dams are selected

initially by

truncation on

EBV based on a

family

selection index of the individual record and the records of

where ql =

Nm/(Nm — 1)

if male i is

currently selected,

or 1 if male i is

_currently

unselected,

and q2 =

(Nm - 1)/Nm.

q

1 and q2 are

multipliers

which are

required

to obtain fair

_comparison

of selected and unselected

_{animals, accounting}

for

the

relationship

of the animal with itself as for ADJEBV

_(Brisbane

and

Gibson,

1995).

w,,,, i

. is the

predicted

stabilised

genetic

contribution of the ith male selection

candidate,

calculated from the deviation of its EBV from the mean of those of

the selected sires

_{using equation}

_(4!.

If the ith male selection candidate is

_currently

selected then _w!,;,* will appear in the vector _{v,,,.. as,i!} is the

_relationship

between the ith male selection candidate and the

_{jth currently}

selected

_sire,

and _a!d.tj is the

relationship

between the ith male selection candidate and the

_jth

_currently

selected dam.

_Adjusted

EBV of females are obtained

_{by analogy.}

The _process of

_switching

and

_updating

of

_adjusted

EBV then continues as described for

_ADJEBV,

using

the

population

selection criterion

_{given by}

_(6!.

Initially

b

m

and b

are

unknown,

because

they depend

on the selection

_strategy

of which

_they

themselves are to be

_part.

_Initially,

therefore 2 500

_replicates

are

run with

b

m

=

b

=

0,

equivalent

to the ADJEBV

method,

since voo,. = _v.

b

m

and

_b

_are calculated

_{retrospectively using}

the EBV of base sires and

_dams,

and their

genetic

contributions to progeny in

generation

7. In this

_example,

the

expected

regression

of

_{asymptotic genetic}

contribution is the same in each _sex, and so the _average of

_b.&dquo;,

and

_b

_f

_,

_b,

is taken. The simulations are then

_repeated

with

3 000

_{replicates using}

the average estimated value of

b,

and a new estimate of b is

obtained from the

_resulting

_generation

7

_regressions.

This

_cycle

is continued until the average value of b calculated is close to that used in the selection method. The

process is

repeated

for various values of k in

equation

!8!,

in order to determine the

performance

of the

_strategy

in terms of the rate of

_{genetic gain}

achieved _{at any} level of

_{inbreeding, compared}

to that of ADJEBV.

RESULTS AND DISCUSSION

Table I shows

_examples

of the realised values of the

_regression

of

_genetic

contribu-tion to

_generation

7 on EBV for base sires and dams in each

_cycle

of simulation for various values of k. The

_regressions

increase as k

decreases,

as

expected

since more

emphasis

is

_put

on EBV

_during

selection. The

_regressions

move toward _convergence

after 3 or 4

_cycles.

After the second

_cycle

of

_simulation,

_subsequent

_changes

in the values of the

_regressions

are much smaller than the standard errors. This means

initialise the random number

_generator

for _every

_cycle

of the simulation. Thus each

cycle

began

with the same

_replicated

base

populations,

and the _sequence of

regres-sions

_converged

to a value

_specific

to those

_populations.

The standard errors of the

regressions

reflect the

sampling

variance associated with each estimate across base

populations.

There was a correlation of around -0.24 between realised values of

b

m

and

b

f

across

_replicates,

which contributed to a small reduction in the standard error of the mean

_{regressions given}

in table I.

Figure

1 shows

_{genetic gain}

_plotted

_against

cumulate

inbreeding

at

_generation

7 for ADJEBV and

_ADJEBV(R).

The k values used are

_given

with the

figure,

and for

both

_lines,

_points

at

_greater

cumulate

_inbreeding

values are

_always

obtained

_using

smaller values of k.

_ADJEBV(R)

_gives

up to 0.03 units more

_{genetic gain}

than

ADJEBV at a

_given

rate of

_inbreeding.

This

_advantage

is

_small,

but

_{statistically}

significant,

since the standard errors of the mean

_gains

are 0.012 to 0.013 with 3 000

_replicates

used. A

_{large heritability}

_(0.5)

was used so that the EBV and the

_predicted

_genetic

contributions would be more

_accurate,

and the

ADJEBV(R)

strategy

would be

compared

in a favourable situation. It was shown earlier

_that,

neglecting

some effects of selection on

_variances,

a

proportion

r

2

of

the variance of

genetic

contributions associated with

_breeding

value is associated with EBV. When the

_heritability

is

_lower,

the variance of the

_predicted

_genetic

contributions is

_lower,

and

_ADJEBV(R)

becomes more similar to ADJEBV. As

_heritability

approaches

zero,

predicted

contributions

approach

1/(2Nm)

for all sires and

1/(2N

/

)

These results

_clearly

indicate that failure to include

_prediction

of

_genetic

con-tributions in method ADJEBV causes trivial loss of

performance

compared

to

ADJEBV(R),

which is fortuitous

_given

the

_difficulty

in

_obtaining

the

where H is the

_{proportionate}

reduction in the variance of the EBV after

_selection,

given

by

H =

i(i — x)

where i and x _are the selection

_intensity

and truncation

point

on the standardised normal distribution

assuming

an infinite

_population

size.

_Also,

the

_regression

coefficient in

_equation

_[2]

is unaffected

_by

selection. From these

_results,

it follows that

_using

the

_EBV,

we account for a

_proportion

Q = r

2

(1 - H)/(1 - Hr

2

)

of the variance of stabilised

genetic

contributions which is associated with

_breeding

value. As selection

_{intensity increases,}

_Q

decreases from

a value

r

2

towards

zero.

_However,

the

_regression

of stabilised

_genetic

contribution on true

_breeding

value increases with selection

_intensity

_(Wray

and

_Thompson,

1990)

and so the total variance of stabilised

_genetic

contributions associated with

breeding

value increases. The overall effect of selection

_intensity

on the

_advantage

of

_ADJEBV(R)

over ADJEBV is therefore not clear.

Further simulation work is

_required

to determine the effects of selection

inten-sity

and finite

_population

size on the

_advantage

of

ADJEBV(R)

over ADJEBV.

With

_{overlapping generations,}

where some animals breed

_longer

and contribute more _progeny than

others,

there will be more variation in the stabilised

_genetic

contributions,

and

_greater

_potential

for

_ADJEBV(R)

to

_outperform

ADJEBV.

REFERENCES

Brisbane JR

₍₁₉₉₄₎

Control and

_prediction

of

_inbreeding

in

_{genetic improvement}

schemes for livestock. PhD

_{Thesis, University}

of

_{Guelph, Guelph,}

ON

Brisbane JR, Gibson JP

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Balancing

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₍₁₉₉₄₎

A note on

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Anim Prod

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_Optimisation

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Verrier

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_Long-term

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Theor

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Wray NR,

Goddard ME

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Prediction of rates of

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