THE EFFECT OF LINKAGE ON THE MEAN VALUE OF INBREDS DERIVED FROM A RANDOM MATING POPULATION

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T H E EFFECT O F LINKAGE ON THE MEAN VALUE OF INBREDS DERIVED FROM A RANDOM MATING POPULATION1

I. M. R. VAN AARDE

Statistical Laboratory, Iowa State Uniuersity, Ames Iowa 50010

Manuscript received February 1, 1973 Revised copy received June 3, 1974

ABSTRACT

An expression is derived which accounts for the effect of linkage on the mean value of diploid inbreds. The original population is taken to be in Hardy- Weinberg equilibrium. It is shown that linkage will accelerate inbreeding depression. The precise nature of the acceleration is worked out for some special cases.

H E purpose of this paper is to exhibit an expression for the mean genotypic Tvalue of some trait in a population that has been derived (without selection) from a Hardy-Weinberg population of diploid ancestors where there is arbitrary linkage epistasis. It is assumed that there are no position effects.

In the case of no linkage, KEMPTHORNE (1957) has shown that the mean of the inbred population deviates from the mean of the ancestors by a quantity which he expresses in terms of Wright’s F as

FD,

+

FZD,

+

F3D,

+

. .

.

+

F”D,

.

Here n is the number of loci, and Dk involves only the “dominance x dominance

x

. .

.

x dominance” deviations from k loci, as defined for the ancestral population. The coefficient of Dk arises from the product rule for the probability of k independent events as F X F X F

x

. .

.

x F from

k

factors.

With linkage, however, the probability that the genes of a n inbred individual are identical by descent at both of two tightly linked loci is more nearly F than

F X F . Hence, their joint contribution to inbreeding depression will be similar to

D,

rather than to D,. Similar reasoning applies to a block of k tightly linked loci. The aim is to show precisely what the relationships are. This is accomplished by deriving a result which amounts to replacing the k-th power of the one-locus inbreeding coefficient in KEMPTHORNE’S expression by the k-locus generalized coefficient to account for linkage. As such, the result is almost self-evident. Therefore, KEMPTHORNE’S development and notation is used as far as possible. Journal Paper No. J-7443 of the Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No.

1661). The research reported here was supported partially by the National Science Foundation, Grant No. 19218 and the National Institutes of Health, Grant No. 13827.

Present address: Faculty of Agriculture, University of Stellenbosch, Republic of South Africa.

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The Mean Of A n Inbred Population

Let random individuals be drawn from the ancestral population and let matings be made so as to derive (without selection of any sort) a n inbred individual, X . A given mating procedure specifies an infinite population of inbreds from which X is a random individual. W e shall now derive a general expression for the mean of such a population of inbreds.

The derivation is facilitated by working with the generalized inbreeding coefficient of X with respect to a given set of loci: This is the probability that at every locus in the set, the genes that

X

receives from its parents ( X , and X I ) are identical by descent. For the inbreeding coefficients of

X

with respect to the sets of loci

we make the notations respectively.

{ i } , { i d , {iYLA}, *

.

* 7

Fa7 F ~ J , Fa3k7

. .

,

Consider the two genes of X at locus i . We make the notations

A% = (the gene that X receives from X , ) , and

A%$ = (the gene that X receives from XI).

XI

W e extend the notation by writing

(A:;, A:;)

=

(A;:, A",) - 1

when the genes are copies of different ancestral genes (indexed by .z: and z:), and

(A:;, A i $ ) E (At%, A,",)

when the genes are identical by descent, both being copies of a single ancestral gene (indexed by z a ) . The ancestral genes are random; that is to say, each one may be identical in state to any one of an arbitrary number of allelomorphic forms,

A ; , A ; , A ; ,

.

,

which occur at locus i with arbitrary frequencies,

P:,

P:, P;,

.

7

b1,,

.:I,

kz,,

z",, (2307 2:) 7

. . .

,

( z ? , z:),

respectively. Hence we obtain n pairs of random variables,

such that their joint distribution is characterized by the following four properties: i) Members of different pairs are independent because the ancestral popu-

lation is in Hardy-Weinberg equilibrium.

ii) Members of the same pair are (for the same reason) independent when they index different ancestral genes.

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L I N K A G E A N D I N B R E E D I N G 1247

iv) Members of the triplet (zk, 2:; 2%) always have the same marginal distri-

bution, given by the probabilities P

( 2

= j ) = p : for all ( i , j ) .

The genes of

X

at locus

i

can now be characterized by the equations

A i .

= ( 1 - S i ) Aii

+

Si A $ , and

2," a1

Aii = ( 1 - Si) Aii

+

Si Aii,

$1 Zo

where we introduce n random Kmnecker deltas,

SI, s2, s 3 ,

.

,

S", such that

P ( S i = l ) = F i , p ( S i

P ( S i = l , S i = l , s k = 1 )

= F . .

z 3 k

,

1, Si = 1 ) = F . . 2 3 7

and so on.

We now express the geno,typic value of X in terms of the formal praduct

;;

( 1

+

a i i ) ( 1

+

a i i )

4 01

i=l

which has precisely the same meaning as the corresponding expression used by

KEMPTHORNE

(1957) ; we have merely adjoined indices to identify the genomes

that X receives from X , and XI, respectively. Precisely as in

KEMPTHORNE'S

development, the expression must be expanded into a sum of 22" terms. The term unity represents the mean of the random bred population of ancestors, p R . The remaining 22n- 1 terms represent deviations. For instance,

represents the "additive x additive

x

dominance" deviation associated with gene Ai* at locus

i

,

gene Ai at lo~cus

i ,

and genes Akk and Akk at locus I t . ,

$1

4

5 0 "1

which is (ai ai

dk)

i j k in K E M P T H O R N E ' S notation.

XI "0 4 $1

The i-th factor in (1) must be

( 1

+

( 1

+

+) when Si = 0, and

(1 + a t ; ) ( 1 + a t ; ) when S i = 1

.

Therefore, we express the factor as

[ I + (1-Si) aii

+

si

a5i1 [ I + (l-Si) a;;

+

si

a 3

.

a0

Note that when we expand the product, ambiguous terms involving both and

zi or both zi, and zivanish, because (l--Si)Si and S i ( l - S i ) vanish. Mo're generally, we replace ( 1 ) with the formal product

l? [1+ (1-Si) aii

+

6 % ai;] [1+ (1-Si) aii

+

si

a y

.

( 2 )

ZO 21

i=l

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represents pR. The remaining terms involve a’s or mixed products of a’s which represent deviatioas.

The random variables z:; z:,

2,

for

i

= 1,2,3,

. . .

,n, are statistically independent of the Kronecker deltas, because (with no selection) the state of an ancestral gene has no bearing on contingencies which may cause copies of that gene to be transmitted to descendents. Hence, to obtain the mean of the inbred population, pI, we expand (2), identify the representations of p R and the deviations, and take the expectation in two stages as follows: First we take the expectation only with respect to

{zk;z4,zi

1 i =

1,2,3,.

.

,n}

.

Properties (i), (ii), (iii) and (iv) show that all terms involving one or more of the variables

will vanish, leaving only p R and terms involving quantities like

{zk,

I

i =

1,2,3,

. . .

,n}

di

= E [(di)-i.il ^ f

,

dij = E [ ( d i d ’ ) zizizjzjl dijk = E [ (&didk) 2 i z i z j z j z ~ z k ]

,

{ S i

1

i = 1 , 2 , 3 , . . . , n } .

(Si), = S i f o r all i

,

and so on. The second stage consists of taking the expectation with respect to Here we simply have to note that

and

Then we obtain the general result

E ( S i S j S k .

.

.) = F i j k . . . for all

i,

j , k,

.

. .

pr = p R

+

&

F i

di

i- ,Z F i j dij i- Z Fijk dijk i-

.

2. 1 . 4 i<j<k

The coefficients Fi take the same value, Wright’s F , for all loci. With no linkage, so that the result reduces to the form given by KEMPTHORNE (1957) who makes the notations

F i j p . . . = F X

F

X F X

.

,

D i = ( z d i ) , i

D , = (

i < j

dij),

D 3 = ( i z < k d i i k ) , - . .

The Eflect Of Linkage In General

Linkage tends to accelerate the effects which progressive inbreeding may have on the population mean, because

F”

>

F2, Fn,k 2 F3, FtIki

>

F‘,

.

. .

I n particular, if loci 1 and 2 are very tightly linked (almost aliases)

F,,,

N F,, and F,, f o r

i =

1,2,3,.

. .

, n ,

F,,,, Fi, ’ F ,

F,,, and F z C 3 for i,j = 1,2,3,.

.

,n,

(i#i),

and so on. Consequently,

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L I N K A G E A N D I N B R E E D I N G

will approximate the contribution of a single locus to PI,

will approximate the Contributions of pairs of loci to pi,

will approximate the contributions of triplets of loci to pi, and so on.

If loci 1,2,3,

. .

.

,k,

are a tightly linked block of loci, the quantity

(dl,

+

d,,

+

d,,,) for

i

= 1,2,3,

.

. .

,n,

(dl,j

+

d,ij

+

dlz,j) for i,j = 1,2,3,

. . .

,n,

_(i#j),

(di

+

dz

+

diz

+

&

+

di,

+

dz,

+

dim

+

.

+

di,,

.

. . k) will approximate the contribution of a single locus to PI.

1249

The Effect Of Linkage In Special Cases

The generalized inbreeding coefficients, F,,

F,,,

F , , k ,

.

,

were introduced by SCHNELL (1961). In any given instance their values will depend on the mating procedure and the linkage relationships. For instance, if X is derived by one generation of selfing, SCHNELL obtains

F %

=(vi?>,

F,, =

(%),

(1 +A:, )

,

and

F 2 , k =

( % I 3

(1

+At]

+

y k +

7

wherein each X is a linkage parameter defined as

1 - 2 (the corresponding recombination rate).

Thus 0

<

A,,

<

1, where A,, vanishes if loci

i

and

i

are unlinked and A,, approaches unity if loci

i

and j are tightly linked. Hence, for one generation of selfing,

p I = p R + ( % ) X d , + ( 1 / 2 ) ' Z ( 1 + A 2 ) d , , +

. . .

.

k %<I 23

Similarly, for one generation of parent-off spring mating, and for one generation of half-sib mating,

The coefficients of d, decrease geometrically over the three mating procedures, but those of d,, decrease less rapidly because (1

+

A,,) 2 (1+ A:,).

We note again that when h,, approaches unity, the terms in d,, d,, and d,,,

reduce to F (dl+d2+d12) which corresponds to the contribution of a single locus.

PI P R

+

(%)*

+

(1/2)"

(1

+

Xz

) (1

+

A,,)

4,

+

. .

$1

p.1 p E

+

(

1 / 2 ) 3

Z d,

+

( i / ) 6

Z (1

+

A* ) (1

+

A1j)' d,j

+

. .

% % < I % I

L I T E R A T U R E C I T E D

KEMPTHORNE, O., 1957 York.

SCHNELL, F. W., 1961

Mi: 937-957.

A n Introduction to Genetic Statistics. John Wiley and Sons, Inc., New

Same general formulations of linkage effects in inbreeding. Genetics