Another look at multiplicative models in quantitative genetics

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Another look at multiplicative models in quantitative

genetics

Christine Dillmann, Jean-Louis Foulley

To cite this version:

Original

article

Another look

at

_{multiplicative}

models

in

_quantitative

_genetics

Christine Dillmann

Jean-Louis

_Foulley

a

Station de

_{génétique végétale, INA-PG,}

ferme de _Moulon, 91190

_{Gif-sur-Yvette,}

France

Station

de

_{génétique quantitative}

et

_appliquée,

Institut national

de la recherche

_agronomique,

78352

_{Jouy-en-Josas cedex,}

France

(Received

29

_April

1998;

accepted

22

_September

₁₉₉₈₎

Abstract - This paper reviews basic

theory

and features of the

multiplicative

model

of gene action. A formal

decomposition

of the mean and of the

_genotypic

variance is

_presented.

Connections between the statistical _parameters of this model and those of the factorial

_{decomposition}

into

_additive,

dominance and

_epistatic

effects are also

emphasized.

General formulae for the

genotypic

covariance among inbred relatives

are

_given

in the case of

_{linkage equilibrium.}

It is shown that

_neglecting

the

_epistatic

components of variation makes the

_{multiplicative}

model a

_{pseudo-additive}

_one, since this

_{approximation}

does not break the _strong

_dependency

between mean and

variance effects. Similarities and differences between the classical

_polygenic

’additive-dominance’ and the

_{multiplicative}

gene action

approaches

are outlined and discussed.

Numerical

_examples

for the biallelic case are

_produced

to illustrate that comparison.

©

Inra/Elsevier,

Paris

multiplicative gene action

/ covariance

among relatives

/ inbreeding

*

Correspondence

and _reprints E-mail: dillmann@moulon.inra.fr

Résumé - Un autre _regard sur le modèle _{multiplicatif} en _{génétique quantitative.}

Cet article

présente

la théorie et les

principales caractéristiques

du modèle

multipli-catif d’action des

_gènes.

Une

_{décomposition}

formelle de la moyenne et de la

vari-ance

_génotypique

_permet d’établir les relations entre les

_{paramètres statistiques}

de ce

modèle et ceux issus de la

_{décomposition}

factorielle de l’effet des

_gènes

en effets

addi-tifs,

de dominance et

_{d’épistasie.}

Une formule

_générale

de la covariance entre

appar-entés dans une

_{population consanguine}

en

_équilibre

de liaison est

_proposée.

On montre que les composantes

épistatiques

de la variabilité

génétique

peuvent être

_{négligées ;}

le

modèle

_{multiplicatif}

devient alors un modèle

pseudo-additif, l’approximation

ne supp-rimant pas la forte liaison entre moyenne et variance. Les similitudes et les différences

entre le modèle

_polygénique

« additif-dominance

» classique

et le modèle

_{multiplicatif}

d’action des

_gènes

sont discutées et illustrées par des

exemples

dans le cas

_{biallélique.}

©

Inra/Elsevier,

Paris

1. INTRODUCTION

Most models for

quantitative

characters in

_evolutionary

genetics proceed

from a few

_concepts

_{developed by}

Fisher

[15]

in

_applying

Mendel’s laws to

_complex

characters:

_genetic

variation is due to a _very

_large

number of

independent

loci whose effects are _very small and of about the same

_magnitude

at each locus. The use of a statistical linear

decomposition

of the

_genotypic

value

into mean effects of _genes and allelic interaction effects within and between loci

_justifies

the use of a multifactorial model.

_Furthermore,

the

_assumption

of an infinite number of loci without

_epistasis

leads to normal distribution

theory

with

_properties

that allow

_prediction

of

_changes

in moments of traits for

populations subjected

to different

_evolutionary

forces such as drift or selection

!2!.

However,

phenotypes

may be controlled

by

other mechanisms of gene action.

The

_optimum

model

_[45],

the

_{multiplicative}

model

_[24]

and the

synergistic

model

_[27]

have been

_proposed

as alternatives to the additive or

additive-dominance models.

One of the basic features of the

multiplicative

model is that it creates allelic interactions between loci and introduces a

dependency

between mean

and variance of a trait. Indirect evidence for

_{multiplicative}

_{gene action}

_(MGA)

has been

provided by studying

the distributions of

_breeding

values for

_complex

traits. Such distributions are

expected

to be skewed under MGA

_!35!,

or to become Gaussian after

_logarithmic

transformation. Such evidence has been found for

_height

_!11!

and

growth

[10]

in mice and on fruit

_weight

in tomatos

[14,

35!.

More

_{generally, production}

traits are

_{multiplicative}

!21!.

Grain

_yield

in maize is the

_product

of the number of seeds and mean

_weight

of seeds. In

the same _way,

_prolificacy

of domestic mammals is the

_product

of ovulation

rate and

_embryo

survival. There is also some

_experimental

evidence for the

applicability

of such a model to the

biosynthesis

chain of

anthocyanin

in

flowers

_!38!.

_Furthermore,

the

_{multiplicative}

model turns out to be the common

choice in resource allocation

_models,

when

_considering

a trade-off between life

history

traits such as seed and

_{pollen production,}

or survival rate

_(biomass)

and

reproduction

[12, 13, 41]).

Theory

of the

_{multiplicative}

model was worked out

_by

Cockerham

_[7]

who

proposed

a

_partition

of the

_genotypic

variance and

expressed

the amount of non-additive variation due to

_{multiplicative}

effects of genes. As he

pointed

out,

non-additive variation was rather small for realistic values of the total

genotypic

coefficient of variation

_(less

than 40

_%).

It was

_suggested

that the

multiplicative

model was

_formally

an additive one.

_However,

this model has been used as an

explanation

for heterosis

_!21,

_37!.

The authors

_emphasized

the

_analogy

for the

decomposition

of the mean between a

multiplicative

model at the trait level

(one

trait

_being

the

_product

of several

_traits)

and a

_{multiplicative}

model at the _gene level

_{(multiplicative}

_gene

_action).

_Using

_generation

_means, a test for the existence of

_{multiplicative}

effects was

_proposed

[33]

and some evidence

for such

_{multiplicative}

effects was found in fava beans.

_{Finally, considering}

a

trait

_{governed by}

_{genes with}

_{multiplicative}

effects and

_{undergoing stabilizing}

selection,

Gimelfarb

_[19]

showed that MGA enhances some ’hidden’

_variability

which can be

_expressed

as additive variation when selection is relaxed. This

(4!.

Another consequence of

epistasis

is the increase of the additive variation in

finite

_{[9, 20]}

or subdivided

_[44]

_populations.

In this context, it seems worthwhile

to reconsider some

_implications

of the

_{multiplicative}

model.

The purpose of this paper is threefold:

i)

to

_present

a formal

_{decomposition}

of the mean and of the

_genotypic

variance under

_MGA,

_ii)

to make connections between the statistical

_parameters

of this model and those of the classical

de-composition

of the

_genotypic

value into its

_additive,

dominance and

_epistatic

components,

and

_iii)

to derive exact and

_approximate

formulae for the

covari-ance _among inbred relatives under MGA.

2. DECOMPOSITION OF THE GENOTYPIC VALUE

2.1. Classical

_theory

The

_{decomposition}

of the

_genotypic

value of an individual was derived

by

Fisher

_[15]

based on the ’factorial’ method of

_{experimentation}

and later

generalized

by

Kempthorne

(25!.

It

_proceeds

from the factorial

_{decomposition}

of

_genotypic

values in a

_{panmictic population}

of infinite size.

Consider a character determined

by

S autosomal loci. Let L be the set of all

possible

genotypes

at the S

loci,

and

G

z

be the random variable

_designating

the

genotypic

value of an individual chosen at random in the

_population

with z E L.

The realized value gz can be

partitioned

into different effects and interactions

within and between loci:

I...

where i and k refer to the

paternal

allelic forms at loci s and

t,

respectively,

and j

and l to the maternal allelic forms at loci s and

_t,

_{respectively;}

_{tL is} the

general

mean; ais is the average

(or additive)

effect of allele i at locus s;

(

3

ij

s

is the first order interaction

_{(or dominance)}

effect between alleles i

_{and j}

_at

locus

s;

(ŒŒ

k

k

,

is the first order interaction

(or

additive

by

additive

_effect)

between the additive effects of allele i at locus s and of allele k at locus t.

In a

_{large panmictic population,}

_supposing

that all the loci are in

_linkage

where

_A

_,

₂

_a#

and

_{(TAA 2}

_represent

the

_additive,

dominance and additive

_by

additive

_components

of

_variance,

_{respectively,}

_{and pi,} is the

_frequency

of allele

i at locus s in the

population.

Other

_components

of the

_genetic

_variance,

such as the additive

_by

dominance

(

QAD

)

and the dominance

_by

dominance

(012 D

D

)

epistatic

variances _may be derived in the same _way. If the loci are in

linkage disequilibrium

in the

_population,

extra covariance terms _among effects at those loci must be

_added,

and the

_expression

of variance

_components

becomes somewhat

_{complex, especially}

under selection and assortative

_mating

_!2,

_{28, 40!.}

2.2. Partition of the mean and variance under MGA

Let

_As

be the effect of alleles at locus s for a

randomly

chosen individual

having

z as

_genotype,

and _a,,!s be the realized value of this random variable

given

z =

_(ij)

at locus s.

_{Then, by}

definition of

MGA,

the

_genotypic

value is

One can _express the mean _p and the variance

(

7b

of

G

as functions of the mean _as

= ¿ Pij s aij

and variance

as

= ! pij

s (a

ijs

-

as)2

of the

A

s

s.

Under

ij

&dquo;

ij

’

linkage equilibrium,

the

A

s

s

are

_independent

so that

Under the same

_assumption,

-E(G!) = n!(!)’

and the

_expression

for the

8

variance is

In

_equation

_(5a),

_QG

is a

_product

of _sums, but _may be

_{alternatively expressed}

as a sum of

products

of means and variances because the

product

of mean

effects over the S loci cancels out due to statistical

_{independence.}

Denote

_by

A the set of the S loci and r the set of all

_possible

subsets of

_A,

the null set

excepted,

and then

where U stands for

_{any element}

of >,. For

_example,

with two

_loci,

2.3.

_{Relationships}

with

_parameters

of the factorial method

This section deals with the different

_components

of

_genotypic

values under MGA

_resulting

from the

_application

of the factorial method. Mathematical de-tails and derivations are

_given

in

_Appendix

A.

_They

follow

_{straightforwardly}

from the

_general

_approach

_{of Kempthorne}

_(26!.

Note that a formal

decomposi-tion of this model limited to the mean deviation effects has

_already

been

_given

Let

_/-Lijs

=

E(Gz !

z ==

_ij

_s

₎

be the conditional

_expectation

of the

_genotypic

value

_G

_z

_given

the

_(ordered)

_genotype

_{z being ij}

at locus s. The additive effect of allele i at locus s is defined as _ais =

E

j

(!tt!) —/!.

Using equation

(4)

fora

and

factoring

I1

at, this effect can be

_expressed

as _ai, =

C !

p

j

s

aij

s

-

as) (

I1

d

t .

tis

j

to !s

!

The first term can be

interpreted

as an additive effect _among the

_a

_ijs

values at locus s. Denote this effect

_by

Then,

Thus,

under

_MGA,

the additive effect of an allele at locus s is the

product

of the additive effect of the allele among the effects of

genotypes

at locus s times the

_product

of mean

_genetic

effects at the other loci.

Similarly,

the dominance effect

₍

₃

_ij

between alleles

_{i and j}

_at

locus s is the

product

of

_fl

_at and the dominance effect

_(3;j,

_among the _a;j, at locus _s, i.e.

t54s c

’

and

Using equations

(6a)

and

_(6b),

the additive

_by

additive effect

_(aa)is!t

pertaining

to allele i at locus s and allele k at locus t is:

Thus,

in the

multiplicative model,

the

_genetic

_components

_{(a, (3)}

at a locus level

_depend

_upon the mean

_genotypic

values at other loci. More

_precisely,

_any interaction effect can be

expressed

as the

product

of the additive and dominance effects among the

genotypic

effects at each locus times the

product

of mean

genotypic

effects at different loci. For

_instance,

the additive

_by

dominance

_(a

₍

_3),

dominance

_by

dominance

₍

_/

_3/3)

and additive

_by

additive

_by

additive

_(aaa)

Using

formulae

_{(6b), (7b), (8)}

and

_(9abc),

one can derive the

_expression

for the different variance

_components

_(see

_Appendix

_B).

The additive

ge-netic variance

_(or2A)

is the sum, over all

_loci,

of the

_product

of the additive

genetic

variance

_(oa

₅

₎

_among the

_ai!

_values

at each locus s times the

product

of the

_squared

mean effects at the other loci:

where

S ?

!2 2 B

Note that

_equation

_(lOa)

can

_{alternatively}

be written as

QA

=

J-l2

2&dquo; ,

as

for

_{1i! #}

0. This shows

_that,

under

_MGA,

variance

_components

are related to

squared

coefficients of variation at each locus.

Similarly,

the dominance variance

_(a#)

can be

_expressed

as the _sum, over

loci,

of the dominance variance

₍

_Qd

_{_}

_‘

₎

_among

_{the /3;j s}

_elements

times the

_product

of the

squared

mean effects at the other loci:

&dquo;

where

The additive

_by

additive

_epistatic

variance reduces to:

while the additive

_by

dominance

_{(a fi!),}

the dominance

_by

dominance

₍

_QDD

₎

and the additive

by

additive

by

additive

_{(a fi ! !)}

_genetic

variances are:

Hence,

each variance

component

can be

_{easily expressed}

as the combination of a

_genetic

variance at one locus

(or

_product

of variances at different

_loci)

times

_squared

mean effects at the

_remaining

loci. The total

_genetic

variance

such

_partitions.

The

_highest

order variance

_corresponds

to the

_{(S - l)th}

order

interaction

Table I illustrates the

_partition

of the

_genetic

variance as

_expressed

ana-lytically

in formulae

_(10ab),

_(llab),

_(12abc)

for a trait controlled

_by

MGA.

Clearly,

the additive and dominance

_components

of variance

_depend

not

_only

on additive and dominance

_genetic

effects at each locus but also on _average

genotypic

values at the other loci.

2.4. Covariances between

_arbitrary

relatives

Extensions of those formulae to covariances between relatives can be

_easily

derived. De

_Jong

and Van

_Noordwijk

_[12]

_gave the

_expressions

for covariances between non-inbred relatives and between

_life-history

traits for some models of resource allocation. Those results are now extended to the case of

inbreeding.

We consider here the

variability

of a neutral trait

_governed

by independent

loci in a

_{large, possibly inbred, population.}

Under those

_hypotheses,

the loci are

_expected

to be in

_{linkage equilibrium}

and the

A

s

s

are

_independent,

so that

s

E

(

G*

z)

=

n

E

(

A

S

).

Now,

E

(A

S

)

=

_Fs

_a

=

a

9

+f

z

do

s

,

where

f

z

is the

probability

8=1

of

_identity

_by

descent between two

_homologous

alleles of an individual

(denoted

here as

z)

drawn at random in the

_population,

and

_do,

_{= L Pi}

_s

_f3

_i

_is

is the

i

&dquo;

average dominance effect in the

homozygous

population. Therefore,

the first

moment of the distribution of

_G

_z

is

Under the same

_assumptions

and

_using

the same notation as in

_equation

(B.1),

the

_genotypic

covariance between two individuals z and

z’

is defined as:

Hence,

the

_problem

reduces to

_calculating

the covariance between relatives at

one locus.

Following

the basic results obtained

_by

Fisher

_(15!,

_Wright

_[45]

and Malecot

(30!,

the

_general

_expression

for covariances between relatives was first derived

by

Cockerham

_[5]

and

_Kempthorne

_{[25, 26]}

under the

_assumptions

of random

mating

and

_{linkage equilibrium.}

The case of linked loci was

_investigated

later

_by

Cockerham

_[6]

and Schnell

_(36!.

The case of inbred relatives was

independently

covariance between two individuals

_{(z, z’)}

from an inbred

population

under MGA can be written as:

where the

A

i

s

are the

_{probabilities}

of

_identity

_modes;

_wzz, is Malecot’s coefficient of

_kinship

between individuals

_{(z, z’);}

_a

_fl

=

2!p!a!!

and

a

§

= &dquo; i &dquo;

2

i

j

j

,

are

the classical additive and dominance

components

of the

zj

ij

2

genetic variance, respectively,

in a

large panmictic

population

under

linkage

equilibrium;

_Qdos

=

2 ¿;

P

is f3t

is

and d 2

_are the variance and

squared

_mean

.°

i

’ &dquo; ’

of the dominance

_{effects, respectively,}

in the

_{homozygous population,}

and

(J’ a

dos

=4

¿; Pi

s

0;7, f3ti

s

is the covariance between additive and dominance effects i

in the same

_population.

Formula

_(15a)

_encompasses three new moment

_parameters

defined in the

homozygous population resulting

from the condition of full

_identity

between

homologous

genes. This formula also involves six functions of the

elementary

identity

coefficients.

_Using

for instance

_identity

measures introduced

_by

Zhao-Bang Zeng

and Cockerham

_[46],

i.e. -yl =

!1

₊

4 I(A

2

+

A

3

+

A

4

+

_A

₅

_),

6

1

=

A

9

₊

A

12

,

6

2

=

!1

and

b

3

=

!1

₊

A

6

,

it can be

_{alternatively}

written in a more condensed form as:

The same

_reasoning

_applies

to the

_genotypic

variance and leads to

The

_expressions

for variances and covariances between inbred relatives in

(15abc)

are

_products

of sums and _may be

_decomposed

as in

_(lOa-13)

into

order,

5S(S - 1)

epistatic

components

of the second

_order,

_{5S(S - 1)(S - 2)}

epistatic

components

of the third order and so on. Each variance

_component

is the sum of variances at one or more loci times

_squared

mean effects at the

remaining

loci. Each covariance component is the sum of covariances at each locus times

products

of mean effects at the

_remaining

loci.

With our

assumptions

(independent

loci,

infinite size

population),

those

expressions

are much

_simpler

than the

_{corresponding expressions}

in the full factorial

_{decomposition}

_{[16, 43].}

In

_fact,

the

_only

coefficients of

_identity

that

are needed in

(15bc)

are

obviously

the ones

_{corresponding}

to identities between four alleles at a

_single

locus.

Under

_{linkage disequilibrium,}

additional covariances

_{(between loci)}

occur in the

_expression

of total

_{genetic variance,}

the effects of which on the vari-ances and covariances are

essentially

unknown. For

_instance,

the total

_genetic

variance

_comprises

five

_components

for two

_polymorphic

loci in the absence of dominance

_effects,

i.e. an additive variance

(a fi),

an additive

by

additive

epistatic

variance

_{(a fi! ) ,}

covariances between additive effects

₍

_QA

_,A),

between additive and

_epistatic

effects

_{(a fi !! ) ,}

and between

_epistatic

effects

_(!AA,AAO

In the

_general

_case,

_higher

order terms are also involved. Note that in this

case,

the

_{corresponding expression}

for the

_population

mean

(14a)

also involves

_high

order

_{identity by}

descent coefficients.

2.5.

_{Approximations}

In this

_section,

we will show that

_neglecting

the

_epistatic

variance compo-nents leads to much

_simpler

_expressions

for the covariance between relatives.

In a

panmictic population

and under

MGA,

the ratio of the

_{non-epistatic}

variance

_components

to the total

_genotypic

variance

_depends

on the number of loci and on the total

_genotypic

coefficient of variation

_(CV

=

a

G/

»

[7].

S

_{j2 !2}

₁

From

(5a), C1

1 +

CV!)

=

fl

C1

+ 2

_a2)

In the

_symmetrical

case From

(5a),

1 +

CV2

?=i !

1

a_,

+ ! .

In the

symmetrical

case

B

7

8=1 B

_as

_as

/

with allelic effects and

_{frequencies being}

the same at each

_locus,

it reduces / B /

_!2

_!

S

to

C1

+

CV

21

=

C1

+

a2

2 +

a2 )

S !7!.

Using equations

(

lOa

)

and

₍

_lla

₎

and to

1

+

/

=

1

a

ä!

a&dquo;/

[7].

Using equations

(lOa)

and

_(l1a)

and / / B / i/s

rearranging,

we obtain

( a! + 2 ab

)

=

I

1 + C

V

2

)

2 -1 I . Whatever the

rearranging,

we obtain

_aG

=

S CV 2

. Whatever the (7! CV

number of

_loci,

this _{ratio is very} close to 1 for values of the total

genotypic

coefficient of variation lower than

40 %.

Hence

_{a$ = a fi}

+

_a

_b

and the total

Similar

_{approximations}

to that

_given

in

_equation

_(16a)

_apply

in the case of

inbreeding.

The covariance between inbred relatives reduces to

and the

_genotypic

variance _may be

approximated by

Note that the

_{approximations}

_(16abc)

are tantamount to

_assuming

that the

genotypic

value

_G

_z

can be written

(apart

from a

_constant)

as

Those

_{approximations}

will be checked

_numerically

in the next section.

Formally,

as outlined

_by

Cockerham

(7!,

this

_{approximation}

makes the

mul-tiplicative

model an additive one without

_epistasis

and the two could not a

priori

be

_{distinguished}

from data.

However,

it does not break down the

depen-dency

between mean and

_variance,

which is one of the main characteristics of the _presence of

_epistasis:

the

_genotypic

variance is a sum of

_products

of means and variances at different loci. In other

words,

the

_genetic

variance at each locus is

_{weighted by}

mean effects at the other loci. As

_inbreeding

affects both mean and variance effects at each locus

_(equations

14a and

_l6bc),

it should be

possible

to

_distinguish

between the two models

by comparing

different levels of

inbreeding

for the same

_population.

3. NUMERICAL RESULTS: THE BIALLELIC CASE

Numerical results

_presented

here

_rely

_upon a biallelic

_symmetrical

model.

S loci in

_{linkage equilibrium}

are

considered,

with allelic

frequencies being

the same at each locus in the base

_{panmictic population. Genotypic}

effects of the three

possible

genotypes

at one locus were set to

_M+a,

M+d

and

_M-a,

where M is the

_mid-parent

value and a and d the additive and dominance

deviation,

respectively.

Parameter values for

M,

a and d were assumed to be the same at

each locus. We defined 6 =

d/a

_as the _constant

degree

of dominance.

3.1. Base

_population

and the mean effect of locus s is as = M ₊

(p - q)a

₊

2pqd

where

p is the

frequency

of the favourable allele and q = 1 -

p. Under MGA and with allelic effects and allelic

_{frequencies being}

the same at each

_locus,

the total

_genetic

variance is

_{given by}

_equation

_(5a)

so that

Equating

these two formulae for

_{or allows}

us to _express the additive deviation

(a)

as a function of the mean

(!t),

the total

_genotypic

coefficient of variation

(CV),

the allelic

_frequency

_(p)

and the

_degree

of dominance

_(6):

Similarly,

M is

_given

_by

The total

_genotypic

coefficient of variation of the base

_population

is assumed to be

_equal

to

_CU

_o

=

0.2,

and the mean of the base

_population

is

_equal

to

p, = 1. We also took 6 = 1

corresponding

_to

complete

dominance _at each

locus,

or 6 = 3

_{corresponding}

to overdominance.

_Using

the

_{approximations}

in

_equation

_(16abc),

the total

_genetic

variance of the base

_population

is

Var(G

o

)

N

6’cr!as

and

the initial

_squared

coefficient of variation is

2

Cl0

X5

S !2 ] .

! _{&dquo;o ! !} !’ 0

as

2 3.2. Inbred

_population

We consider now an inbred

_population

derived from the base

_population

by changing

the

_reproductive

behaviour of the individuals and

_forcing

inbred

matings

during

t generations.

In this case, the five variance

_components

in the biallelic model were

given by

Chevalet and Gillois

!3]

and Mather and Jinks

[32].

They

can

easily

be

_expressed

as functions of

as,

the contribution of one locus to the

_genetic

variance. Let us define

h

z

,

the

_heritability

_{(narrow sense)}

as

so that

(J!

=

h

2

(J;

and

aj!

=

dÕ

=

(1 -

h

2

)(

J

;.

_Similarly,

_{one can} define b =

Therefore,

from

_equation

_(l6abc),

the variance of allelic effects at one locus

in an inbred

_population

is

and the covariance of allelic effects at one locus between inbred relatives is

The total

_genetic

variance of the

_population

also

_depends

on

At _any time

_t,

the

_{expected genetic}

variance is

_equal

to

while under

_AGA,

_Var(G

_t

₎

reduces to

_!3!:

Note that the terms between brackets in the

_right

hand sides of

equation

(18ab)

depend only

on allelic

frequencies

and on the

_degree

of dominance.

It would therefore be

_possible

to _compare the two models of _{gene action}

_by

expressing

the

genetic

variances at time t in units of the total

genetic

variance

in the base

_population.

It is also clear from

_equation

_(l8ab)

that in the biallelic

case, the

only

difference between AGA and MGA is the coefficient

m

2

(

S

-

1

)

weighting

the

_inbreeding

variance under MGA.

Figure

1 illustrates the evolution of the

_expected

total

_genetic

variance with the mean

_inbreeding

coefficient of the

_populations.

Under AGA with

_complete

dominance

_{(figure lc),}

the

_genetic

variance increases with

_f

_z

_.

The more the

frequencies

of the favourable alleles

_depart

from

_1/2,

the

_higher

are the values

of the total

_genetic

variance. Under AGA with overdominance

_{(figure 1 d),}

the

genetic

variance decreases with

_f

_z

for intermediate

_frequencies

of the favourable allele

_(p

= 0.4 _or

0.6).

Under MGA with

_{complete dominance,}

_two

_major

differences occur as

_compared

to the AGA case.

_First,

for low

_frequencies

of

the favourable

allele,

the

_genetic

variance decreases with

inbreeding

( figure 1 a),

as seen in the case of AGA with overdominance.

Second,

the

_genetic

variance

increases with the

_frequency

of the favourable alleles. This

_phenomenon

is

enhanced

by

an increase in the number of loci

_governing

the trait

(figure lb).

However,

such results

rely

upon the

approximations

made in

_equation

_(l6bc),

total

_genetic

variance at time

_t,

i.e.

/ BS

with

af

+

a

2

a7ae

corresponding

exactly

to the definition of the total

( z

!

! !

F,

genetic

variance of the

_population.

Figure

2a and b shows the residual

epistatic

component

of the total

_genetic

variance as a function of the mean

inbreeding

coefficient,

f

z

,

and for the same

genetic

models as in

figure

1 and

b, respectively.

It can be seen that the

magnitude

of the

_epistatic

variance

components

depends

on allelic

_frequencies.

For _very low

frequencies

of the favourable

alleles,

the

_epistatic

variance never

exceeds 10

%

of the additive and dominance

components.

It

_drops

to less than 1

%

of the additive and dominance

components

in other cases and _may

4. DISCUSSION

This

_study

was

_primarily

concerned with the statistical

_properties

of the

multiplicative

model of _{gene action.} This model can be seen as a

_good

candidate to

_explain

some features of traits observed in

_{physiological}

or biochemical

studies,

as well as in classical

quantitative genetics experiments

(see

Introduction).

It was

found,

as

predicted by

Cockerham

!7!,

that the

epistatic

_components

of variance can be

neglected

under MGA when the total

_genotypic

coefficient of

variation is not too

_large.

It is then

_possible

to describe a trait

_{by invoking}

_only

additive and dominance effects at each locus. Tractable formulae for variance

components

are obtained

_{by neglecting}

the terms

_{corresponding}

to

_products

of variances at each locus. Those

_{approximations}

were shown to be valid

mean and variance under MGA. Allelic effects and variances at each locus are

weighted by

the effect of other loci. This

_phenomenon

_may have two main consequences.

First,

means and variances at each locus are not affected in the same _way

2

Q2

by

inbreeding.

Let us define

CV£

_A

2 = E !!a_,

GV1

Do

= L

_°2°

°

,

and so on. It -2 A ₂

s

_G!

s

_Q

_{’ FS}

turns out from

_equation

_(16b)

that the

_squared

total

_genotypic

coefficient of

variation of an inbred

_population

can be

_expressed

as

Var

(

Gz

)

_/1+

_{fz)GVA 2 +}

_(1-

_z

2 +

_z

+

_z

+

_fz(1-

_z

2 ^&dquo;’’ _l

_{.f )}

_{A+(’-f-)CV6+f-CV!D,,+f,CV6o+f!,(’-f!,)CV!2}

0 z

This formula

_clearly

indicates that the coefficients

_affecting

the CVs in the

right

hand side of the above

_equation

are the same as those

_affecting

variance

components

under AGA. This means that under

_inbreeding,

the formal

similar-ity

betwen MGA and AGA is not at the variance

_level,

but rather at the level of the total

_genotypic

coefficient of variation. Such results have been observed

in alfalfa

_by

Gallais

_!17!,

who

_pointed

out that the

_genotypic

coefficient of

vari-ation

_provided

a better scale than the

_genetic

variance to linearize the effect of

_inbreeding

_depression

on the

_genetic

variation. Note that the

_analytical

ex-pression

obtained here relies on some

_strong

_assumptions

_(linkage

_equilibrium

and infinite

_population)

which are discussed below.

Second,

if a favourable allele is fixed

_by

selection in a

_{given population,}

fixation will increase the mean effect of the locus and decrease its variance. These two effects _may cancel out in

_equation

_(5a)

and result in no

_change

for the total

_genetic

variance of the trait. Under

_AGA,

the same

_phenomenon

would lead to a

_systematic

decrease of the total

_genetic

variance.

Despite

such

_{important qualitative differences,}

the two models can

hardly

be

distinguished.

In an outbred

_population,

the absence of a

_significant

amount of

epistatic

variation may be

_interpreted

in two different _ways: as

_originating

from

polygenic

additive-dominance

_genetic

determinism or from

_{multiplicative}

_gene

action.

_Similarly,

it is difficult to

_distinguish

between AGA with overdominance and MGA without overdominance in the _presence of

inbreeding.

Note that

multiplicative

gene action can also be viewed as a

_parsimonious

_explanation

for heterosis:

_complete

dominance under MGA can

_explain

some

_patterns

of

change

of the

_inbreeding

_genetic

variance as does overdominance under AGA. It is nevertheless

_possible

to test

_{multiplicative}

_{gene action}

_by

_comparing

different levels of

_inbreeding

for the same

_population.

_Melchinger

et al.

_[33]

defined a

_{multiplicative}

factor and

_proposed

a test based on the

_comparisons

of the means of different inbred

_generations.

Our results

_suggest

a

_possible

test at the variance level restricted to

_{populations exhibiting}

low

frequencies

of the favourable alleles. In this _case, whatever the

_degree

of

_dominance,

the

_genetic

variance is

_expected

to increase with

_inbreeding

under

_AGA,

and to decrease under MGA.

The numerical results

_presented

here in the biallelic case were obtained

_by

assuming equal

allelic effects and

_frequencies

for each locus. The main reason for that was to

_simplify

the

_{complex algebra generated by}

MGA.

_However,

this

_assumption

should not alter the

_general

trend of the results. We checked

epistatic

components

rarely

exceed 10

%

of the total

_genetic

_variance,

as

_long

as the total

genotypic

coefficient of variation does not exceed 40

%

(results

not

shown).

Up

to now, the exact distribution of allelic effects over loci

_governing

a trait is not

_known,

even

though

results

concerning

the distribution of

QTLs,

which can be detected in a

_population,

seem to indicate a

_L-shaped

distribution

[29,

31,

34].

Relying

upon

QTL

detection

results,

unequal

gene effects may

concern a maximum of 20

%

of loci that _govern a

_given

trait.

Equal

allelic

effect is an

_{implicit assumption}

in the

polygenic

additive-dominance model

[2, 15!.

In our

opinion,

the

_strongest

assumption

here is the

equal frequencies

hypothesis.

Even without random

genetic drift,

and with the same selection

pressure

acting

on each

_locus,

the allelic

frequencies

_may not be

_expected

to be the same because of mutation.

Most of the results

_presented

here are also

heavily dependent

on the

hypothesis

of statistical

independence

between loci. This

_hypothesis

restricts the

_analysis

to the case of

independent

loci and

large populations. However,

such situations _{may exist in} artificial inbred

populations

created

_by

breeders. In

plant breeding,

for

_{example, populations}

of 300 to 500

_reproducing

individuals

are common, with

linkage disequilibrium

restricted to loci situated on the same chromosome

_(Dillmann

and

Charcosset,

pers.

comm.).

In

general,

random

genetic

drift in finite

_populations,

as well as

_linkage

between

loci,

_generates

multilocus identities

by

descent

_[16,

39,

42,

43].

In that _case, the

validity

of our

_{approximations}

remains to be checked.

But, equation

(16b)

stresses the

_importance

of mean effects at each locus in

_evolutionary

processes, when

epistasis

is

involved,

and

_provides

a

good

basis to

study

the evolution of

genetic

variation under

inbreeding

for MGA traits.

As for the effect of

inbreeding,

we were

_only

concerned with

_expectations

of

the

_parameters.

Those

_parameters

also have a variance which _may be calculated

[46].

As

_experimental

studies

_always

involve a finite number of

populations,

and often a

_unique

_one, the variation around

expected

values _may be

important.

In

_particular,

_genetic

drift and selection

generate

not

_only

variation between

populations

in mean

_performance,

but also in within

population

variance which contributes

_indirectly

to the variation in selection _response

_[1,

_23!.

It makes the

_intensity

of selection fluctuate and therefore

_changes

the

_population

means

at the next

_generation.

Due to the interaction between mean and

_variance,

those fluctuations may even be enhanced

by multiplicative

_{gene action.} We are

presently studying

the combined effects of selection and random

genetic drift,

including

also the case of linked loci.

ACKNOWLEDGEMENTS

We thank P.

Brabant,

G. de

Jong,

M.

_Rose,

J.W. James and two _anonymous

reviewers for their

_helpful

comments on the

_manuscript;

and C. Chevalet and A.

Gallais for valuable discussions and

suggestions

on the section about

inbreeding.

We are

grateful

to R.L. Fernando for his

_English

revision of the

_manuscript.

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APPENDIX A:

_{Decomposition}

of the

_genotypic

value under MGA

according

to the factorial method Additive effects

From

_equation

_(3),

and

_{assuming linkage equilibrium}

In the factorial

method,

the additive effect of allele i at locus s is defined as

where

_symbols

are the same as in the text.

_Using

the

_expression

₍₃₎

_{for p}

and

factoring

fl

at, one obtains the

_expression

(6b)

for the additive effect of allele

tics

i at locus s under MGA. Dominance effects

The dominance effect

_(3ij

between alleles i

_{and j}

_at

locus s is defined

classically

as

The

_expression

_(7b)

is obtained

_{by using}

formulae

_(A.1)

for _Itij, and

_(6b)

for ais

and ajs’ and

again factoring

fl

at.

tops s

Epistatic

effects

As

_pointed

out in the

_text,

the factorial

_{decomposition applied}

under MGA

generates

epistatic

effects. The additive

_by

additive effect

_(aa)t!

_pertaining

to allele i at locus s and allele k at locus t is defined as:

where /!.i+.sk+t is the

_expectation

of

_genotypic

values for individuals

_having

received

_gene

i at locus s and _gene k at locus t from one of their

_parents

(e.g. sire),

the genes transmitted

by

the other

_parent

being

any gene drawn at random in the

_population.

Under

_{linkage equilibrium}

Using

the

_expression

for

_at

_given

in

_equation

_(6a)

and

_putting

it into

_equations

(A.4

and

_A.5)

_gives

APPENDIX B: Partition of the

_genotypic

variance under MGA Consider the same

_{decomposition}

as in

_equation

₍₁₎

with realized values

replaced

by

random variables

_pertaining

to the same

_genetic

effects defined for a random

_by

chosen individual in the

_population

where the

_symbols

_{i and j}

_coding

for the alleles are

_{replaced by}

the

_integers

1 and 2

_designating

the genes transmitted

by

the sire and

dam, respectively

(those

figures being

omitted in dominance effects for the sake of

_simplicity).

The same

_assumptions

are made as before

(i.e.

infinite

_{population size, linkage}

equilibrium

and

_panmixia),

_resulting

in

_orthogonal

_{decomposition}

with

inde-pendent

random variables.

Additive

_genetic

variance

₍

_QA

₎

By

definition

and,

because the

paternal

and maternal

components

are

_playing

the same s / B

roles,

Var(a

ls

)

=

Var(a

2

,) =

! pis a2s

and

_a

_fi

= 2

!

C ! pis a2s J .

The 7,

&dquo;

8

=1 !

&dquo;/

additive

_genetic

variance

_(lOa)

is then obtained

_by

_using

the

_expression

of ai

s in

equation

(6b),

and

by setting

for the additive variance among

_at

values at locus s. Dominance

_genetic

variance

₍

_QD

₎

Similarly,

QD

= ¿ Var(,8

s

).

Knowing

that

_,8

_i

_j,

=

,8ij

s

( I1

_lit

and

_letting,

s

&dquo;

&dquo;B!!t 7

s _s#t

as before

one obtains the

_expression

(lla)

for the dominance

_genetic

variance.

As

_{previously, paternal}

and maternal contributions are

equivalent,

and the four

elementary

variances are

_equal.

Thus

(omitting

_subscripts

for

_parental

contri-butions)

a fi !

= 4 £ £ Var((aa)!t) ,

with

_Var((aa)st)

_{=!!pispkt(aa)2skt.}

s t>.s I k

&dquo;

Now, using equation

(8)

for

_(a!)i,!!t

and

_equation

_(B.3)

for the

_relationship

between

_a:s

and

_{d!_, ,}

we have

Finally,

Other variance

_components

Additive

_by

dominance

_{genetic variance,}

dominance

_by

dominance

_genetic

variance,

as well as variances

_pertaining

to

_higher

order interactions can be derived in the same _way. For

_instance,

the

_expression

for additive

by

additive

epistatic

variance

₍

_QAAA

₎

can be obtained

_along

the same lines as in

_equation

(B.6).

In that case, there are

_eight

variance terms for

_{(aaa) s}

_tu

_paternal

and maternal contributions which are

_equal.

As ¿p

d

aT,)

2

=

Q

a,g/2,

this

i

’

factor 8 cancels out with

_(1/2)

₃

_owing

to the introduction of the

_product

!as!atQau.

As there are 3!

_{possible permutations}

of

_s,

t and u in

_(aaa)St!,

which are

_equivalent,

the final

_expression

for

QAAA

is obtained

_{by summing}

up

elementary

contributions over different

_s,

t and u loci and

_{by dividing by}

3 !