The effect of linkage on the additive by additive covariance between relatives 1

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The effect of linkage on the additive by additive

covariance between relatives 1

Liviu R. Totir, Rohan L. Fernando

To cite this version:

Original

article

The

_{effect of linkage on}

the additive

by

additive covariance

between relatives

1

Liviu

R.

Totir,

Rohan

L. Fernando

Department

of Animal

_Science,

Iowa State

_University,

Kildee

_Hall,

Ames,

IA

_50011,

USA

(Received

22

_September

_{1997; accepted}

17

_August

₁₉₉₈₎

Abstract - The additive x additive

_relationship

coefficient needs to be calculated in

order to _compute

_genetic

covariance between relatives. For linked

_loci,

the compu-tation of this coefficient is not as

_simple

as for unlinked loci. Recursive formulae

are

_given

to _compute the additive x additive

_relationship

coefficient for an

_arbitrary

pedigree.

Based on the recursive

_formulae,

numerical values of the desired coefficient for selfed or outbred individuals are examined. The method

presented provides

the means to _compute the additive x additive

_relationship

coefficient for any situation

assuming linkage.

The effect of

_linkage

on the covariance was examined for several

pairs

of relatives. In the absence of

_{inbreeding, linkage}

has no effect on the

parent-offspring

covariance. All of the other

_{relationships}

examined were affected

_{by linkage.}

As recombination rate increased from 0.1 to

_0.5,

in

_descending

order of _percentage

change

in the

_covariance,

the

_{relationships}

ranked as follows: first

_cousins,

double first

cousins,

grandparent-grandoffspring,

half

_{sibs, aunt-nephew,}

full

_sibs,

parent-offspring.

With

_inbreeding,

the

_{parent-offspring}

covariance is also affected

_{by linkage.}

©

Inra/Elsevier,

Paris

additive x additive _relationship coefficient

_/

covariance between relatives

_/

identity by descent

1

Journal Paper

No. J-17555 of the Iowa

_Agriculture

and Home Economics

Experi-ment

Station, Ames,

Iowa.

_Project

No. 1307, and

_{supported by}

Hatch Act and State of Iowa funds

*

Correspondence

and

_reprints

E-mail: rohan@iastate.edu

Résumé - Effet du _linkage sur les covariances entre _apparentés pour les in-teractions de type additif x additif. En cas

_{d’épistasie,}

le calcul de la

covari-ance

génétique

entre

_apparentés

nécessite le calcul du coefficient de

_parenté

pour les termes d’interaction additif x additif.

Quand

les loci sont

_liés,

le calcul de ce coefficient n’est pas aussi

simple

que dans le cas de loci

indépendants.

Des

dans le cas d’un

_{pedigree quelconque. À partir}

des formules

_récursives,

des valeurs

numériques correspondant

au cas d’individus issus d’autofécondation et de par-ents sexués sont examinées. La méthode

_présentée

fournit le moyen de calculer

le coefficient de

parenté

additif x additif pour toute situation

_impliquant

le

link-age. L’effet du

linkage

sur la covariance a été examiné _pour

plusieurs

_paires

d’apparentés.

En l’absence de

consanguinité,

le

linkage

n’a pas d’effet sur la

covariance

_{parent-descendant.}

Toutes les autres

_parentés

examinées ont été af-fectées par le

linkage. Quand

le taux de recombinaison _augmente de 0,1 à 0,5,

les

_{parentés présentées}

suivant l’ordre décroissant de sensibilité des covariances sont les suivantes : cousins

_germains,

cousins issus de

germains,

grands-parents

petits-fils, demi-frères, oncle-neveux, pleins-frères, parent-descendants.

En cas de

consanguinité,

la covariance

_{parent-descendant}

est aussi affectée par le

linkage.

©

Inra/Elsevier,

Paris

coefHcient de _parenté additif x additif

_/

covariance entre _apparentés

_/

identité par descendance mendélienne

1. INTRODUCTION

Genetic covariance between relatives can be

_expressed

as a linear combi-nation of

_genetic

variance

_components.

In order to

_compute

the covariance

between

_relatives,

coefficients associated with the variance

_components

need to

be calculated from

_{pedigree relationships.}

Additive and dominance

_relationship

coefficients can be

computed

through

several methods for

_arbitrary

_pedigrees

[4-6,

8,

10!.

The additive x additive

relationship

coefficient between unlinked loci can be obtained as the _square of the additive

_relationship

coefficient

_(7!.

When loci are

linked,

the additive x additive

_relationship

coefficient cannot

be

_{computed simply}

as the _square of the additive

_relationship

coefficient. Now this coefficient _may

_depend

on the recombination rate and it has been derived for several common

relationships

[2,

_3,

12).

A

_general

_approach

for

_computing

the additive x additive

_relationship

coefficient for collateral relatives has been

developed by

Schnell

_[9].

For

_{general pedigrees,}

this

_approach

becomes _very

complicated.

More

_recently,

_Thompson

_!11!

has described a recursive

_approach

for

_computing

two-locus

_{identity probabilities}

that can be

applied

to _any

pedigree.

In this _paper we _present

_{independently}

derived recursive formulae that are different from those of

_Thompson

for

_computing

the additive x additive

relationship

coefficient for an

_{arbitrary pedigree.}

These formulae can be used to

examine the effect of

_linkage

on the additive x additive

_relationship

coefficient for _any

_pair

of relatives. Based on the results obtained in this _paper, the situations when the effect of

_linkage

on the additive x additive covariance between relatives can be

_ignored

are examined. Some

_examples

are

_given

here

and a C++

implementation

of the recursive method with some numerical

examples

is available on the Web at

_{http://www.public.iastate.edu/}

_N

_rohan

by

following

the link Software.

2. THEORY

at each locus. The additive x additive

_genotypic

value of an individual I with

alleles k and

k’

at the first locus and alleles 1 and I’ at the second locus can be

written as

where 6 is the additive x additive effect.

_Similarly,

the additive x additive

genotypic

value for an individual J with alleles

n, n’, p

and

p’

is

The additive x additive contribution to the covariance between I

and

J can

be written as a sum of 16 covariances.

Each of the 16 covariances can be written as the

_product

between one-fourth

of the additive x additive variance

_component

_(V

_AA

₎

and a

_probability

that

pairs

of alleles are identical

_by

descent

(IBD).

For

example

C

OV

(b

kl

, 6,,,)

in

equation

(3)

is

where

_{Pr(k -}

n, I -

p)

is the

_probability

that the allele k of individual I is IBD with allele n of individual J and allele of I is IBD with allele p of J. The

additive x additive

_relationship

coefficient

_(!I,!)

is one-fourth of the sum of

the 16 IBD

_{probabilities}

_{corresponding}

to the 16 covariances in

_equation

_(3).

Each of these

_{probabilities}

can be obtained

recursively

as

_explained

below.

3. RECURSIVE COMPUTATION OF IBD PROBABILITIES

The

principle

underlying

the recursive method for

_computing

IBD

_{probabilities}

is first described for a

_single

locus. Then we show how to

_compute

_recursively

IBD

_{probabilities}

for two loci. 3.1.

_Single-locus

_computations

The basic

_{principle underlying}

the recursive method is that the maternal

(paternal)

allele at a

_given

locus in an individual is a _copy of either the maternal or

_paternal

allele at the same locus of its mother

(father).

To

illustrate,

consider an individual I with

_parents

S and D. The maternal and

_paternal

alleles of

I,

for

_example,

at locus A are denoted

_by

_AI

and

A{.

Based on the

_principle

mentioned

_above,

the

_probability

that the maternal allele of I is IBD to the

where,

for

example,

_A

_I

_f

_-

A

B

is the condition that

_A

₁

is a _copy of

!4!.

If J

is not a descendent of

I, equation

(5)

can be

_simplified

to

However, equation

(6)

is not true when J is a descendent of

I,

because

now the IBD

_{relationships}

between

A!

and

_AD

and between

A!

and

AD

depend

on whether

AI

is a _copy of

_AD

or of

!4!,.

In order to take

_advantage

of

_equation

_(6),

it is _{necessary to} determine whether I or J is _younger, and

always

recurse on the _younger allele.

_Using

this

_procedure

the recursion can

be

performed

until both alleles in an IBD

relationship

are from founders. In

founders,

the IBD

_probability

between two different alleles is defined to be null and is

_unity

for an allele with itself. Several authors have used recursion to

compute

IBD

_{probabilities}

between alleles at a

single

locus

[6,

_8,

10!.

3.2. Two-locus

_computations

The

_principle

used here

_is,

as for the

_single-locus

_case, that the maternal

(paternal)

allele of an individual can be traced back to its mother’s

_(father’s)

maternal or

_paternal

allele. Consider

_computing

the additive x additive

rela-tionship

coefficient

₍

₁

₎

_I

_,

_J

₎

between I and

_J,

where I is younger than J. The

parents

of I are denoted

by

S and D.

_Using

the same notation as in the

single-locus case for alleles at locus

B,

the

_probability

in

_equation

₍₄₎

can be written as

where we have assumed that k and l are the maternal alleles of

I,

and n and p are maternal alleles of J. For notational

_simplicity

the

_probability

in

equation

(7)

will be denoted

by

Pr((Al , BI ) - (Am, BJ)]

’

Now,

!1,

can be

written as

Note that the

_pairs

of alleles from I can be classified into two

_types:

those that can be

_thought

of as

being

either a recombinant

_gamete

from I or those

probability

the

_pair

of alleles from I is of the non-recombinant

_type.

This

_pair

is

a _copy of either one of the two non-recombinant or one of the two recombinant

gametes

of D.

_{Thus, using recursion,}

this

_probability

can be written as

where r is the recombination rate between A and B. The

_pairs

of alleles from I in the first

_eight

_{probabilities}

are of the non-recombinant

_type,

and can be

computed

as shown in

_equation

_(8).

The

_pairs

of alleles from I in the last

_eight

probabilities

are of the recombinant

_type.

For

_example,

in the ninth

_probability

the

pair

of alleles from I is

_(Am,

_{BI ).}

In this

_pair

_(Am)

is either the maternal

or the

_paternal

allele of

_D,

and

(BI )

is either the maternal or the

paternal

allele of S.

_{Thus, using recursion,}

the ninth

_probability

can be written as

This

_probability

is not a function of the recombination rate between A and B because

_{(A1 )}

and

_(Bf)

are inherited

_{independently}

from D and S.

In the two IBD

_{probabilities computed above,}

the

pair

of alleles that were traced back were from the same individual.

_However,

when recursion is continued it will be necessary to trace back alleles that

_belong

to two different individuals. For

_example,

if S and D are _younger than

_{J, computing}

the first

_probability

in

_equation

₍₉₎

will

_{require tracing}

back alleles from S and

D to alleles of their

_parent.

General rules to

_compute

IBD

_{probabilities}

that accommodate all cases encountered in recursion are described below.

Consider

_computing

_{Pr[(Ax, B!) ==}

_{(Aw , Bz)],}

where alleles in the first

pair

are from individuals X and

Y,

alleles in the second

_pair

are from individuals

W and

_Z,

and

_superscripts

_{!, y, w,} z = m or

_{f .}

Without loss of

_generality,

we assume that X _{is younger} than W and Y _{is younger} than Z. All cases

encountered in recursion can be classified into two

_types:

where

_(A

_X

_,

_BY)

is of

the non-recombinant

_type

_(type-1);

or where

_(!4!-,B!)

is of the recombinant

3.2.1. Recursion for

type-1

cases

Type-1

cases are encountered when X = Y and x =

y.

Now,

if the condition

is

_true,

then

_Pr[(A

_x

_,

_{By) (Aw ,}

_BZ)!

_{= 1;}

if the condition c is not

_true,

but all four alleles are from founders

_then,

Pr!(AX,

By

(Aw,

B’)]

=

_0,

because

different alleles in founders are assumed to be not IBD.

Suppose

condition c is not

_true,

none of the four alleles is from a

founder,

and alleles at one of the two loci are the same. For

_example,

if X =

W,Y !

Z,

x = w = m and z =

f, then

Pr!(AX, BY ) _

(A!,, Bz)!

can be

recursively

computed

as

where P is the mother of X.

_Here,

_AX

and

_!4!

are the same

_{allele, and,}

therefore,

in the desired

_probability

we have

only

three different alleles. As a

result,

only

Hi

is not traced back to its

_parental

alleles. Note that here and in

all

_type-1

cases both alleles

_A

_X

and

BY

are traced back to the same

_parent;

as a

_result,

recombination rate enters into the formula for recursion.

Suppose

condition c is not

_true,

none of the four alleles is from a

founder,

and alleles at neither of the two loci are the same. For

_example,

if

X #

_W,

Y #

Z x = _{m, w} = m and z =

f, then

Pr!(AX, BY ) - (Am,

B

i

)]

can be

recursively computed

as

where P is the mother of X. This is the same situation

_given

by

equation

(8).

3.2.2.Recursion for

_type-2

cases

Type-2

cases are encountered when X = Y and

x 7!

y or when

X #

Y. Even

is

_true,

_fr[(!4!,

_{BY) -}

_(!4!,

_{B § )}

= 1. If condition c is not true and all four al-leles are from founders

_then,

fr[(7l!-,

BY) -

(!4!,

Bz)]

= 0.

Suppose

_now that

X = Y = Z = W but

z #

y and

w #

z. For

_example,

if x = _{m, y =}

f ,

w =

f

and z = _m, then

where

_{(AX, Bm)}

is of the non-recombinant

type.

Recursion can then be done as described for

_type-1

cases.

Suppose

that condition c is not true and alleles at

_only

one of the two

loci are from founders.

_Then,

if the alleles from the founders are not the

same,

P7-[(!,.S!) =

(Aw , Bz )]

=

0;

on the other

_hand,

if the alleles from

the founders are the _same, recursion will be

applied

to the other locus. For

example,

if

_A

_X

and

_Aw

are the same founder

allele,

Y !4 Z, x = w = m, y = m

and z =

f, then

Pr!(AX, BY) - (.4!,.Bj!)]

_can be

recursively computed

as

where R is the mother of Y.

_Here,

_A

_X

and

_!4!

are the same

_allele,

and it is not traced back to

_parental

alleles because X = W is _a founder. As a

_result,

only

By

is traced back to its

_parental

alleles. Note that here and in all

_type-2

cases the alleles

_Ax

and

_BY

are traced back to different

_parents;

as a

_result,

recombination rate does not enter into the formula for recursion.

Now _suppose condition c is not

_true,

none of the four alleles is from

_founders,

but alleles at one of the two loci are the same. For

example if,

X =

W, Y !

Z,

x = w = m, y = m and z =

_f,

then alleles at locus A are the same and

fr[(!4!,-B!-) = (Aw , Bz )]

can be written

_recursively

as

where P is the mother of X and R is the mother of Y.

_Again,

_!4!-

and

_!4!,

are

the same

_allele,

and as a result in the desired

_probability

we have

_only

three

different alleles.

_Thus,

the

_only

allele that is not traced back is

_Bfzl

Finally,

suppose condition c is not

_true,

none of the four alleles is from

X:A

W,

Y #

Z,

x =

m, y = m, w = _m and _{z =}

f,

then

Pr!(AX, BY ) _

(!4!,

B

i

)]

can be

_recursively

_computed

as

where P is the mother of X and R is the mother of Y.

_Now,

in the desired

probability

we have four different

alleles,

and

_only

_AX

and

_By

are traced back.

4. NUMERICAL EXAMPLES

The recursive formulae are used here to examine the effect of

_linkage

on the additive x additive

_relationship

coefficient. Cockerham

_[2]

stated that the covariance between two

_relatives,

where one is an ancestor of the

_other,

is not

affected

by

linkage.

Schnell

_[9]

as well as

Chang

[1]

showed that the

_previous

statement is not

_always

true. It can be shown that the covariance between

a

_parent

and its non-inbred

_offspring

is not affected

_{by linkage.}

_However,

the covariance between a

_parent

and its inbred

offspring,

as well as between

grandparent

and

_{grandoffspring,}

will be affected

_by

_linkage.

Consider first the covariance between

parent

(W)

and a non-inbred

_offspring

(X).

The additive x additive

_relationship

coefficient

(ox,w)

can be

_computed

using

two-locus

computations. However,

of the 16

_{probabilities,}

only

four are non-zero because the

_parents

of X are assumed to be unrelated. For

example,

if W is the mother of

_X,

two-locus

_computation

reduces to

where A and B are the two loci. Note that the four

_{probabilities}

in

_equation

₍₁₆₎

are of

_type

1 and as a result we can write

because the recombination rate cancels out in

_equation

_(17).

As a result the

Assume now that X is

inbred,

its

_parents

_being

full sibs. Assume also that the

_parents

of W are unrelated. In this case all 16

_{probabilities}

in section 3.2

will have non-zero

_values,

and

_!X,w

is

_given

_by

Note that in this case the recombination rate will affect the covariance

between

_parent

and

_offspring.

Consider now

_computing

the additive x additive

_relationship

coefficient

!G,W

between

_grandparent

_(W)

and

_{grandoffspring}

_(G).

Let W be the

ma-ternal

_grandparent

of

_{G, X}

the

_daughter

of W and the mother of

G,

and Y

the father of G.

_Again,

_O

_G

_,

_W

can be written

_using

two-locus

_computation.

As

in the

_{parent-offspring}

_case, there are

_only

four

_{probabilities}

that are non-zero

because Y is considered to be unrelated to W.

_Applying

_equation

₍₁₁₎

to the four

_{probabilities}

in

_equation

₍₁₉₎

_gives

As a result

_{!G yv}

can be written as

which is a function of the recombination rate r.

The recursive method was used to

_compute

numerical values of the additive

x additive

relationship

coefficient for different relatives and different

recombi-nation rates

_{(table 1).}

As

_expected,

when

_linkage

is absent

_(r

=

0.5)

the additive x additive coefficient is

_equal

to the _square of the additive coefficient. In the absence of

_linkage,

the

_genetic

covariance will be identical for certain

pairs

of relatives. For

example,

the covariance between

_{grandparent-grandoffspring,}

half sibs and

_aunt-nephew,

is

_equal

to 0.25

_V

_A

+ 0.0625

_V

_AA

_.

_However,

if loci

are

linked,

the

_genetic

covariance for these

pairs

of relatives will not be the same

(table 1).

The numerical values of the additive x additive

_relationship

coefficient

increase as the

_linkage

becomes

_tighter

_(r

becomes

_smaller).

As a

_result,

when

we assume that

linkage

is

absent,

the additive x additive variance

_component

will be overestimated.

Numerical values for the additive x additive

_relationship

coefficient for full sib and for

_{parent-offspring}

_{relationships,}

after several

_generations

of

_selfing,

are

_given

in tables II and III. In this

design,

individuals in

_generations

i are the

offspring

of selfed individuals from

generation

i - 1. The numerical values in

table II are for the

_relationship

between the

offspring

of a

_single

selfed individual

from

generation

n. The numerical values in table III are for the

_relationship

between a

_parent

in

generation n

and its

offspring

in

generation

n + 1. Note

that after

_{t generations,}

if

_linkage

is

_absent,

the additive x additive

relationship

coefficient for full sibs has the same value as the additive x additive

relationship

coefficient for

_{parent-offspring.}

When

_linkage

is

_present

the two values are

different. The additive x additive

_relationship

coefficient of a founder with _any

individual obtained

_{through selfing}

will be

_always

one. The numerical value of

5. DISCUSSION

This paper describes a recursive method to

_compute

the additive x additive

relationship

coefficient for

_{arbitrary pedigrees}

in the presence of

linkage.

The

additive x additive

_relationship

coefficient can be described as one-fourth the sum of 16 two-locus IBD

_{probabilities}

that can be

recursively

traced back to

known values. We have

_given

five recursive

_equations

to

_compute

these IBD

probabilities,

where the

_origin

of the _younger

_pair

of alleles is traced back to

the

previous

generation.

Thompson

[11]

gave six recursive

equations

to address the same

_problem.

However her

_approach

differs from ours. Some of these differences are

briefly

described below

_using

our notation.

_{Thompson’s approach}

is based on recursive

equations

for two-locus IBD

probabilities involving only

the alleles of

parent

P in its

_offspring

X or

X’,

where P is not

_Y,

W or Z nor an ancestor of any

of them. Further her recursive

_equations

are linear combinations of one- and

two-locus IBD

_{probabilities}

while our

_equations

are linear combination of

only

two-locus IBD

_{probabilities}

and do not involve one-locus IBD

_{probabilities.}

While all the recursive

_{equations given}

in the

present

paper are based on

tracing

alleles back to the

_{previous generation,}

not all of

_Thompson’s

_[11]

Thompson

!11!,

which in our notation becomes

where alleles

_AX

and

Ay,

are from

parent

P. This

_equation

is obtained

by

observing

that alleles

_AX

and

_AX,

will be the same with

probability

one

half;

if the two alleles are the same, then the two-locus IBD

probability

on the left hand side of

_equation

(25)

becomes the one-locus

probability

Pr(BY

=

_B

_z

_);

if the two alleles are not the _same, the two-locus IBD

_probability

is

_P

_r[(Ap,

_By)

(A

P

,

_Bz)!.

In

_contrast,

we trace back the alleles

_A

_X

and

_BY

to the

_{previous generation.}

Suppose

x = x’ = _m

and

y = m, then the two-locus IBD

probability

on the

where P is the mother of X and

_X’,

and R is the mother of Y. This is

_clearly

different from

_equation

_(25).

_Although

these two

_approaches

use different recursive

_equations

the final results for the IBD

probabilities

are identical. This demonstrates that there is more than one

approach

to

compute

IBD

probabilities by

recursion.

Based on the recursive method described in this paper, numerical values

of the desired coefficient for selfed or outbred individuals are

_given.

_Using

the

_computer

program available at

_{http://www.public.iastate.edu/}

_N

_rohan,

the effect of

_linkage

on the additive

_by

additive covariance can be examined for

any

type

of

_{relationship.}

This would be useful to examine the

_potential

bias in

covariance estimates when

_linkage

is

_{ignored. Figure}

_{1 gives}

the rate of

_change

in the additive

_by

additive covariance for several

_{relationships.}

_{Relationships}

with flatter curves are less biased

_{by linkage.}

Other

_applications

are in

linkage

_analysis

and the identification of

_pairwise

relationships

based on data at linked loci

_(11!.

REFERENCES

[1]

Chang H.L.,

Studies on estimation of

genetic

variances under non-additive

gene

action,

Ph.D.

thesis, University

of Illinois at

_{Urbana-Champaign,}

1988.

[2]

Cockerham

C.C.,

Effects of

linkage

on the covariance between

relatives,

Ge-netics

_{41 (1956) 138-141.}

[3]

Cockerham

C.C.,

Additive

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additive variance with

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Emik

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M.,

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A.

_(Ed.),

The Genetic Structure of

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1974.

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D.L.,

Genotypic

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[7]

Kempthorne O.,

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S.,

Maki-Tanila

A., Genotypic

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dominance and

_inbreeding,

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65-91.

(11!

Thompson E.A.,

Two-locus and three-locus gene

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