Prediction of annual genetic gain and improvement lag between populations

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Prediction of annual genetic gain and improvement lag

between populations

Jm Elsen

To cite this version:

Original

article

Prediction of

annual

_{genetic gain}

and

_{improvement lag}

between

_populations

JM Elsen

INRA, Station dAm6lioration Genetique des _Animaux, BP 27, 31326 Castanet Todosan, France

(Received

4 December 1991; accepted 18 November

₁₉₉₂₎

Summary - An _approach for _computing the _{expected genetic gain} and the _improvement

lag between _{subpopulations,} based on matrix

algebra,

is _proposed. This is a _{generalization}

of the classical Rendel and Robertson

₍₁₉₅₀₎

formula, whose main feature is a _comparison

of successive _generation mean values. A _{simple example} is _given.

selection response / genetic gain / gene flow

R.ésumé - Prédiction du progrès génétique annuel et du décalage génétique entre sous-populations. Une _approche du calcul de _{l’espérance} du _{progrès génétique} et du

décalage

entre _{sous-populations,} basée sur _{l’algèbre matricielle,} est _proposée. Il s’agit d’une _{généralisation} de la _{formule classique} de Rendel et Robertson

_(1950),

dont la

carac-téristique

principale est de comparer les valeurs moyennes des générations successives. Un

exemple

simple

est donné.

réponse à la sélection _{/ gain génétique /} flux de _gènes

INTRODUCTION

The formula of Rendel and Robertson

₍₁₉₅₀₎

for

_estimating

the annual

_genetic

_gain

is well suited to closed

_{homogeneous populations.}

It may be used

directly

when there

is

_only

one _type of

_breeding

animal _per sex. In other _cases, such as _progeny test

designs

where known and tested males are both

_reproducing,

the formula has to

be

_adapted

_(Lindh6,

_1968).

Bichard

₍₁₉₇₁₎

showed how _{to process} a hierarchical

population

and how to estimate the _improvement

_lag

between

_{subpopulations.}

These methods are based on

_comparisons

between the mean additive

_genetic

values of successive

_generations.

More

_recently,

iterative methods

_(Hill,

_1974;

Elsen and

order to take account of the year

by

year

change

of

genetic

values.

They

are well fitted for the

_description

of hierarchical

_populations.

In the _{present paper,} we _propose a new method for

_estimating

the

_genetic

progress and

improvement

lag

between

_{subpopulations,}

which can be

_applied

to

these

_{heterogeneous}

populations.

Like the method of Rendel and Robertson

_(1950),

our

procedure

is based on a

_comparison

of successive

_generations

and _is,

thus,

of a

simpler

formulation than iterative methods.

METHODS

Description

of the

_population

We consider

_only

stable

_populations

where the selection

_policy

_(selection

pressure,

organisation

of

_matings)

and structure are constant.

The

_population

is subdivided _{into groups} of

_breeding

animals. Let

_X

_i

be the mean

_genetic

value of the ith _group. The model

_gives

the value of the vector X of

X

i

given

the mean values at the

_{previous generation, Yi.}

The _groups are defined in the

following

_way. Two individuals

belong

to the same group, i:

- if

they

are of the same sex; - if the

probabilities

that their

_respective

_parents of the same sex

_belong

to the same

_{group j}

(Pij)

are

equal;

- if

they

have

_{equal probabilities}

of

_being

_parents of individuals

_belonging

to the same _group of the next

_generation.

Generations are defined here in a relative _way: let us consider the

_population

at a

_given

time. All individuals

belonging

to group i at this moment constitute a

generation

of this group.

By

definition their

parents

which

belong

to

_{group j}

are from the

_previous

_(parental)

_generation

of

_{group j}

relative to i. With this

definition,

this

_{parental generation}

of

_{group j}

does not

_comprise

the same individuals if

considering

their

_offspring

from group i or from group i’. Its mean

genetic

value will be noted

₅

_j

_;

for group i.

Computations

of the mean

_genetic

values _Xi and

_Y

_ji

are made

_{by considering}

the

_individuals,

at

_{birth, prior}

to _any selection.

The

_principle

of the method is to write

relationships

between groups of one

generation

and groups of the

previous

one. _{For group i,} we have:

where M and 7 are male and female

_breeding

animals

_{respectively,}

and where

_0!!

is the deviation between the mean value of

_{group j}

at

_birth,

_Y!i,

and the value of those individuals from this _group which will

_actually

be _parents of _{group i.}

On the other

_hand,

due to the

_genetic

_progress

_(OG

_per

_year),

we have:

Solution

Pooling

relations

_[1]

and

_!2!,

we _get

Or,

in matrix

_notation,

where A is the matrix _{of aij and} H the vector of

_E

_{a!A! —}

_L

_ji

_AG),

which we

j

shall write H = A -

LAG,

A and L

being

the vectors of

L

a

ij

a

ji

and

E

a

ij

l

ji

,

j j

respectively.

Case of a

_single

_population

In this case matrix A is stochastic.

Indeed,

the events &dquo;sire

_{(or dam)}

of an individual of _group

_{i belongs}

to _group

_j&dquo;,

defined over the different

_{groups j}

of the

population,

form a

_complete

_system of _events, and

The

_{largest eigenvalue}

of A is 1. Let V

_(transpose

_V

_T

₎

be the

_eigenvector

corresponding

to 1. This vector V may

easily

be found

by

substitution since

V

T

A

=

V

T

(note

that _to

simplify

this substitution one of the elements of Y may be fixed to

_1).

..

Case of a

composite

population

The

_general

_equation

is still of type

(4!,

but here we have:

where

_h

_{X, hh!A}

and _hH are vectors and matrices

_specific

to

_{subpopulations}

h and h’. In

_particular,

an annual

_{genetic gain}

AG

h

,

specific

to

_population

h,

is found for each

_h

_H.

As a

whole,

the matrix A is still

stochastic,

but each of its

hh

A

elements is not

necessarily

of this _type.

Thus,

we have:

where U is the column

_eigenvector

_(made

of

_1’s)

and VT the row

_eigenvector

corresponding

to the

_eigenvalue

1. The T matrix is such that RT = TR = 0.

The

_lag

between 2

_{groups k}

and k’ is

_given

_by

_g

_T

_X,

where

_g

_T

is a row vector

with all its elements zero, except the elements

_{corresponding}

to the groups

k(g

k

=

1)

and

_{k’(g!! _ -1).}

The

_lag

between 2

_{subpopulations,}

which could be defined

_by

the difference in mean values of

productive

animals

(milking

cows, slauthtered

lambs...),

will most often be

_given

_by

the

_lag

between 2 groups

belonging respectively

to these

2

_{subpopulations}

and defined on an

equivalent

basis.

Nevertheless,

one can

_imagine

that in some instances the level of a

subpopulation

_may be characterized

by

a

weighted

sum

L

9k

X

k

.

k

Thus,

as RX is a vector all of whose elements are

_equal,

g

T

RX

= 0 and the

lag

E is:

EXAMPLE Model

Let a

_population

_comprise

a nucleus and a base. In the

nucleus,

as in the

base,

the selection is on maternal

performance only.

Good females

(selection

_pressures

O

F1F

,

insemination. The AI sires are sons of elite dams

_(selection

pressure

ð-F1A1)

and AI

sires.

Among

the sires used in the

base,

a fraction

d

2

was born in the

_{nucleus, given}

the diffusion of

_{genetic gain.}

These males are chosen from among those born from artificial _{insemination,} with a selection pressure of

_O

_F

_,

_AZ

on maternal value.

The mean values of the

_breeding

animals will be denoted:

. _{for the nucleus:}

9 for the base:

Noting

that

_tlF1F1

=

tl

F1B1

and

tl

F2F2

=

tl

F2B2’

equation

[4]

is

The annual

_{genetic gain}

becomes .

Noting

that the

_eigenvector

V of the matrix A is

_(1V

_T

_,

_0),

we find that

E is

_easily

deduced. Numerical

_application

We consider the

_simple

situation where all the

_dam-progeny

_generation

_lengths

(LF1Fl1LF1Al1LF1Bl1LF1A2,LF2F2,LF2B2)

are _{5 years} and

_sire-progeny

_generation

lengths

(LA1F1’ LB1F1’ LA1A1’ LA1B1’ LB1B1’ LA2F2’ LB2F2’ LA1A2’ LA2B2’ LB2B2)

are 3 years.

It is also assumed that the females are not recorded in the base

_(O

_FZF2

=

O

FZ

s

2

=

0),

and

that,

in the

nucleus,

the

dam-daughter

are the best

50%,

the dam-AI sire are the best 10% and the dam-natural

_{mating sire,}

the next 20%. Given a common _accuracy h = 0.5 for the

dam,

the selection

differentials,

in standard

where i is the selection

_intensity

function,

assuming

the trait

_normally

distributed. With these

_assumptions,

we find

The

_lag

E = w!H is then

CONCLUSION

The main

_difficulty

of the method is the definition of _groups. A

_particular

_population

may be

analysed

in different ways. The smaller the number of groups, the more

easily

the

_eigenvector

V and the inverse matrix M-1 will be

_found,

but the more difficult will be the correct

_writing

of matrix A and vectors 0 and L. There are 2 extreme cases: the first is one in which

_only

2 _groups are

considered,

in

keeping

with the classical demonstration of the formula of Rendel and Robertson

_(1950);

the second is one in which individuals of the same _group have the same _age, similar

to the model of Hill

₍₁₉₇₄₎

and Elsen and

_Mocquot

_(1974).

Finally,

it should be

_emphasized

that the

_preceding

results are

_only

_asymptotic

and need constant selection pressure and

population

structure in the

_long

run.

ACKNOWLEDGMENT

The critical comments of an _{anonymous reviewer} are _{gratefully acknowledged.}

REFERENCES

Bichard M

₍₁₉₇₁₎

Dissemination of

genetic improvement

through

a livestock

industry.

Anim Prod

_13,

401-411 1

Ducrocq

V,

(auaas

RL

₍₁₉₈₈₎

Prediction of

_genetic

_response to truncation selection across

_generations.

J

_Dairy

Sci

_71,

2543-2553

Elsen

_JM,

_Mocquot

JC

₍₁₉₇₄₎

Recherches _pour une Rationalisation

_Technique

des Schemas de Selection des Bovins et des Ovins. Bull Tec

_D6p

G6n6t Anim No

_17,

INRA,

Paris

Elsen JM

₍₁₉₈₀₎

Diffusion du

_{progrès g6n6tique}

dans les

_populations

avec

gene-rations

_{imbriqu6es: quelques}

_propri6t6s

d’un modèle de

_pr6vision.

Ann Genet Sel Anim

_12,

49-80

Hill WG

₍₁₉₇₄₎

Prediction and evaluation

_{of response}

to selection with

_overlapping

generations.

Anim Prod

_{18, 117-140}

Lindh6 B

₍₁₉₆₈₎

Model simulation of AI

_breeding

within a dual purpose breed of

cattle. Acta

_Agric

Scand

_18,

33-41

Rendel

_JM,

Robertson A

₍₁₉₅₀₎

Estimation of

_genetic

gain

in milk

_{yield by}

selection